url stringlengths 6 1.61k | fetch_time int64 1,368,856,904B 1,726,893,854B | content_mime_type stringclasses 3 values | warc_filename stringlengths 108 138 | warc_record_offset int32 9.6k 1.74B | warc_record_length int32 664 793k | text stringlengths 45 1.04M | token_count int32 22 711k | char_count int32 45 1.04M | metadata stringlengths 439 443 | score float64 2.52 5.09 | int_score int64 3 5 | crawl stringclasses 93 values | snapshot_type stringclasses 2 values | language stringclasses 1 value | language_score float64 0.06 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
https://alisonkiddle.co.uk/useful-origami-resources/ | 1,720,935,107,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514548.45/warc/CC-MAIN-20240714032952-20240714062952-00796.warc.gz | 80,283,334 | 23,541 | # Useful Origami Resources
Here are some links to resources I have found useful when doing mathematical origami
https://wild.maths.org/tags/modular-making – Some great videos from the Millennium Mathematics Project in Cambridge showing how to make some simple modular origami models
https://think-maths.co.uk/resources/make-dodecahedron-a4-paper/ – lovely set of instructions for making a dodecahedron from A4 paper
https://mathigon.org/origami – the Mathigon website has lots of gorgeous pictures of origami models together with instructions for how to make them, including the Five Intersecting Tetrahedra I blogged about here.
http://www.origami-instructions.com/modular-sonobe-unit.html – instructions for my favourite modular origami unit, the Sonobe module
https://plus.maths.org/content/trisecting-angle-origami – an article explaining how origami can be used to trisect an angle, which is impossible using standard straight edge and compass techniques | 211 | 965 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2024-30 | latest | en | 0.805229 |
https://www.sluiceartfair.com/2020/contributing/how-is-reactive-power-produced/ | 1,632,811,327,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780060538.11/warc/CC-MAIN-20210928062408-20210928092408-00184.warc.gz | 978,430,109 | 8,790 | Page Contents
## How is reactive power produced?
Reactive power is either generated or absorbed by electric generators (or, in some cases, devices known as “capacitors”) to maintain a constant voltage level, commonly referred to as providing “voltage support.” Generators providing voltage support often suffer heating losses that result in a reduced ability to …
## How do you get reactive power from active power?
Active power: P = V x Ia (kW) Reactive power: Q = V x Ir (kvar)
## What is reactive power in electrical system?
In electrical grid systems, reactive power is the power that flows back from a destination toward the grid in an alternating current scenario. Reactive power gets energy moving back into the grid during the passive phases. Reactive power is also known as: phantom power.
Read more: How is the WiFi in Italy?
## How is reactive power produced what are the effects of reactive power in the grid?
What are the effects of reactive power in grid? Reactive power is current flowing out of synch with voltage (on the AC waveform). It is produced by capacitive and inductive elements in the network, both end-user devices and network components. Effect on the network is reduced power delivery.
## What is the difference between real power and reactive power?
The main difference between active and reactive power is that Active Power is actual or real power which is used in the circuit while Reactive power bounce back and forth between load and source which is theoretically useless.
## How is reactive power produced in a power system?
On the other hand reactive power is the imaginary power or apparent power, which does not do any useful work but simply moves back and forth in the power system lines. It is a byproduct of AC systems and produced from inductive and capacitive loads. It exists when there is phase displacement between voltage and current.
Read more: Who was inside Trojan horse?
## How is active power and reactive power measured?
Active power is measured in watts (W) and is the power consumed by electrical resistance. φ = phase angle between the electrical potential (voltage) and the current Reactive power is the imaginary or complex power in a capacitive or inductive load.
## What’s the difference between reactive and apparent power?
On the other hand, reactive power is the imaginary power or apparent power, which does not do any useful work but simply moves back and forth in the power system lines. It is a byproduct of AC systems and produced from inductive and capacitive loads. It exists when there is a phase displacement between voltage and current.
## Why is reactive power compensation needed in a power system?
AC power supply systems produce and consume two types of powers; active and reactive power. Real power or active power is the true power given to any load. It accomplishes useful work like lighting lamps, rotating motors, etc.
Read more: Who founded the window? | 588 | 2,962 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2021-39 | latest | en | 0.959436 |
https://www.coursehero.com/file/6354954/Lesson-4-Probability1/ | 1,488,085,514,000,000,000 | text/html | crawl-data/CC-MAIN-2017-09/segments/1487501171933.81/warc/CC-MAIN-20170219104611-00199-ip-10-171-10-108.ec2.internal.warc.gz | 801,927,043 | 92,786 | Lesson 4 Probability1
# Lesson 4 Probability1 - Probability Let S be the sample...
This preview shows pages 1–6. Sign up to view the full content.
Probability Let S be the sample space of an experiment in which all outcomes are equally likely and let E be an event, the probability of E , written P(E), is : The probability of an event is the sum of the weights of all sample points in E; thus, since then, . If P(E) = 0 then the event E is said to be an impossible event. If P(E) =1 then the event E is said to be a certain event. P ( E ) = n ( E ) n ( S ) = no . of elements of E . of elemens of S 0 n ( E ) n ( S ) 0 P ( E ) 1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Theorems on Probability 1. 1. P(S) = 1 2. If n(E) = 0 then P(E)=0. 3. If A and B are mutually exclusive events then 4. If is a sequence of mutually exclusive events then 6. If A’ is the complement of an event A then P(A’) = 1 – P(A). 1. If A and B are any two events then . 2. If A and B are any two events then 3. For any events A , B, and C 0 P ( E ) 1 P ( A B ) = P ( A ) + P ( B ) A 1 , A 2 ,..., A n P ( A 1 A 2 ... A n ) = P ( A 1 ) + P ( A 2 ) + + P ( A n ) P ( A - B ) = P ( A ) - P ( A B ) P ( A B ) = P ( A ) + P ( B ) - P ( A B ) P ( A B C ) = P ( A ) + P ( B ) + P ( C ) - P ( A B ) - P ( A C ) - P ( B C ) + P ( A B C )
Examples 1. If a letter is chosen at random from the English alphabet, find the probability that the letter: a. is a vowel b. is listed somewhere ahead of the letter j c. Is listed somewhere after the letter m. 2. Interest centers around the life of an electronic component. Suppose it is known that the probability that the component survives for more than 6000 hours is 0.42. Suppose also that the probability that the component survives no longer than 4000 hours is 0.04. a. What is the probability that the life of the component is less than or equal to 6000 hours? b. What is the probability that the life is greater than 4000 hours?
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Examples 3. If a committee of 5 is selected from a group of 8 seniors, 6 juniors, and 4 sophomores, find the probability that the committee has: a. 2 seniors, 2 juniors and 1 sophomore b. No seniors c. At least one senior 4. The probability that an American company relocates in Shanghai is 0.7, the probability that it relocates in Beijing is 0.4, and the probability that it relocates in either Shanghai or Beijing or both is 0.8. What is the probability that the company will relocate in: a. in both cities?
Examples 5. In a poker hand consisting of 5 cards, find the probability of holding: a. 3 aces b. 4 hearts and 1 club 6. In a high school graduating class of 100 students, 54
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up to access the rest of the document.
## This note was uploaded on 07/31/2011 for the course MATH 30 taught by Professor Teodoro during the Spring '11 term at Mapúa Institute of Technology.
### Page1 / 17
Lesson 4 Probability1 - Probability Let S be the sample...
This preview shows document pages 1 - 6. Sign up to view the full document.
View Full Document
Ask a homework question - tutors are online | 946 | 3,347 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.375 | 4 | CC-MAIN-2017-09 | longest | en | 0.908577 |
http://dave-reed.com/csc221.F05/HW/HW4.html | 1,590,382,481,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347387219.0/warc/CC-MAIN-20200525032636-20200525062636-00418.warc.gz | 32,961,752 | 2,977 | ### CSC 221: Computer Programming I Fall 2005 HW4: Repetition and Simulation
For this assignment, you will modify the code for simulating volleyball games and perform numerous experiments.
1. Download the files VolleyballSim.java, VolleyballStats.java,and Die.java and build a working project. Be sure to comment out the println statements in VolleyballSim, as we demonstrated in class, to avoid excessive output. If you specify equal rankings, are the scores of games relatively close? Should they be? Explain your answer and provide the scores for several games to support it.
2. Assuming team 1 has a ranking of 50 and team 2 has a ranking of 55 (10% better), perform three different sets of simulations for each of the following winning point totals. Report the winning percentages of team 1 in the table below:
SIMULATION 1SIMULATION 2SIMULATION 3
10,000 games to 15
10,000 games to 20
10,000 games to 25
10,000 games to 30
3. For each of the given winning point totals, are the three winning percentages that you obtained fairly consistent? Would you expect them to be? If not, modify the VolleyballStats class so that 100,000 games are played and repeat the simulations. Explain your answers.
4. Are the winning percentages affected by the changing the points needed to win? That is, if the number of points required to win is increased, does this affect (positively or negatively) the chances of team 1 winning? If your data suggests an effect, run several more simulations with different winning point totals to confirm the pattern. Would this effect (if any) be more pronounced if the teams were less evenly matched, say 50 vs. 80? Explain your answers.
5. Next, we want to compare the two different scoring schemes, rally and sideout scoring. A simple approach is to define another method, named playGameSideout, which simulates a game using the sideout scoring system. This new method will be almost identical to the existing playGameRally method that uses rally scoring, except that it awards a point only if the server wins the rally. To balance out the fact that sideout scoring gives a slight advantage to the team who serves first, have the initial serve randomly assigned to one of the teams.
Once you have your playGameSideout method working in VolleyballSim, modify the playGames method in VolleyballStats so that it performs repeated experiments using both scoring schemes and displays both winning percentages. The output of the method should look like:
Assuming (50-55) rankings over 10000 games to 25: team 1 winning percentage: 36.56% (rally) vs. 33.46% (sideout)
6. Assuming team 1 has a ranking of 50 and team 2 has a ranking of 60 (20% better), perform three different sets of simulations for each of the following winning point totals. Report the winning percentages of team 1 using both rally and sideout scoring in the table below:
SIMULATION 1
(rally : sideout)
SIMULATION 2
(rally : sideout)
SIMULATION 3
(rally : sideout)
10,000 games to 15
10,000 games to 20
10,000 games to 25
10,000 games to 30
7. As we saw in class, rally scoring accentuates differences in talent. That is, if team 1 is 10% better than team 2, the likelihood of them winning a game is actually much more than 10% better than for team 2. Which of the two scoring schemes accentuates differences more? Provide additional statistics as needed to justify your claim.
8. Suppose you were a member of the committee deciding on a switch from sideout to rally scoring. Recall that women's college volleyball previously used sideout scoring, with games played to 15. Perform experiments to determine the equivalent winning point total for rally scoring that would preserve competitive balance. That is, the winning percentage of team 1 using rally scoring should be roughly the same as the winning percentage using sideout scoring to 15. This relationship should hold for a wide range of team strengths. Provide statistics to justify your answer.
9. Finally, consider the hypothesis that the rule change was made to shorten the lengths of games. Modify the VolleyballSim class so that it keeps track of the number of rallies as a game is played. There should be a private field to store the number of rallies, which is updated by the playGameRally and playGameSideout methods. An additional method named getLastGameLength should be provided for accessing this field. Next, modify the playGames method in VolleyballStats so that it displays average game lengths for both scoring systems in addition to winning percentages. The output of the method should look like: Assuming (50-55) rankings over 10000 games to 25: team 1 winning percentage: 36.56% (rally) vs. 33.46% (sideout) average game length: 44.5438 (rally) vs. 84.5814 (sideout)
10. Does changing from sideout scoring (with games to 15) to rally scoring (with games to 25) tend to shorten games? What if games were played to 30 under rally scoring? Provide statistics to justify your answer. | 1,127 | 4,961 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2020-24 | latest | en | 0.919952 |
http://www.expertsmind.com/questions/solve-the-following-logarithmic-equations-30194905.aspx | 1,643,432,397,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320299927.25/warc/CC-MAIN-20220129032406-20220129062406-00244.warc.gz | 87,408,056 | 15,267 | ## Solve the following logarithmic equations, Engineering Mathematics
Assignment Help:
1. Joe and Sam each invested \$20,000 in the stock market. Joe's investment increased in value by 5% per year for 10 years. Sam's investment decreased in value by 5% for 5 years and then increased by 12.5% for the next 5 years. AT the end of 10 years, what was the value of each person's investment?
2. Investigate the following functions for both horizontal and vertical asymptotes, x and y-intercepts, and state the domain and range of each and where the function is increasing and decreasing. (A) f(x) = (2x3 - 3x - 5)/(x3 - 8). (B) g(x) = (x - 3)/(x2 - 5x + 6)
3. A manufacturing company wants to package its product in a rectangular box with a square base and a volume of 32 cubic inches. The cost of the material used for the top is \$.05 square inch; the cost of the material used for the bottom is \$.15 per square inch, and the cost of the material used for four sides is \$.10 per square inch. What are the dimensions of the box with the minimum cost?
4. In planning a restaurant, it is estimated that a revenue of \$6 per seat will be realized if the number of seats is at most 50. On the other hand, the revenue on each seat will decrease by \$.10 for each seat exceeding 50 seats. Find the revenue function where x = the number of seats exceeding 50 seats. What is the maximum revenue and what number of seats generates that maximum revenue.
5. You invest \$1,000 at an annual interest rate of 5% compounded continuously. How much is your balance after 8.5 years? How long will it take you to accrue a balance of \$4,000? What interest rate is required to yield a balance of \$7,000 after 7 years?
6. A sum of \$2,500 is deposited in a bank account that pays 5.25% interest compounded weekly. How long will it take for the deposit to double? How long will it take you to accrue a balance of \$7,500? What interest rate is required to yield a balance of \$7,000 after 7 years?
7. Solve the following logarithmic equations. A) log(x-1) - log(x+1) = 1; B) ln(x) = -1.147; C) ln(x) = (2/3)ln(8) + (1/2)ln(9) - ln(6); D) log5 (x) + log5 (x-4) = log5 (21); E) log4 (3x) - log4 (x2-1) = log4 (2)
8. Solve the following exponential equations. A) 3(7x+1) = 3(4x-5); B) 8x = 2(x-6); C) 3(x+2) = 5: D) 4(x) = 2(3x+4); E) e(x - 6) = 3.5.
9. Graph f(x) = 2x - 3 and approximate the zero of f to the nearest tenth. Also find the y-intercept and any horizontal and vertical asymptotes.
10. In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 100 flies, and after 4 days there are 300 flies. How many flies will there be after 5 days?
#### Differential equation, (V^2)dx + x(x+v)dv=0
(V^2)dx + x(x+v)dv=0
#### Evaluate the fermat''s method algorithm, 1. (i) How many digits does the nu...
1. (i) How many digits does the number 101000 have when written to base 7 ? (ii) Use the prime number theorem to estimate the proportion of prime numbers among the positive inte
#### Find the controllability matrix, An open-loop control system has the follow...
An open-loop control system has the following state-space model: (a) Find characteristic equation of the open-loop control system model. (b) What is the characteristic e
#### Draw concentric circles, Draw concentric circles of radii a and b, each cen...
Draw concentric circles of radii a and b, each centered at z=id (on the imaginary axis). Suppose φ(x,y) is a harmonic function inside the washer defined by these circles. The circl
#### Describe basic fourier theory, An experiment conducted over time T necessar...
An experiment conducted over time T necessarily produces a windowed view of the phenomenon generating the data. It is a useful strategy to regard the windowed data as one period of
#### Curve tracing in polar and cartesian form, i want assignment or notes on cu...
i want assignment or notes on curve tracing in polar form and cartesian form
#### Formula by sequence of 3, 1. what is the formula of the sequence by three?
1. what is the formula of the sequence by three?
#### Explain how conduction takes place in conductors, With the help of energy b...
With the help of energy bands explain how conduction takes place in conductors. On the basis of energy band materials are categorized as conductor is given below: Conducto
#### Mann''s test for the weibull distribution, How do i perform mann''s test fo...
How do i perform mann''s test for the weibull distribution | 1,220 | 4,612 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.65625 | 4 | CC-MAIN-2022-05 | latest | en | 0.924469 |
http://sites.millersville.edu/bikenaga/number-theory/arithmetic-functions/arithmetic-functions.html | 1,555,758,200,000,000,000 | text/html | crawl-data/CC-MAIN-2019-18/segments/1555578529606.64/warc/CC-MAIN-20190420100901-20190420122901-00341.warc.gz | 158,874,281 | 10,458 | # Arithmetic Functions
In this section, I'll derive some formulas for . I'll also show that has an important property called multiplicativity. To put this in the proper context, I'll discuss arithmetic functions, Dirichlet products, and the Möbius inversion formula.
In case you prefer a more direct approach to the formulas and properties of , I give an alternative proof of the multiplicativity of in the appendix to this section.
Definition. An arithmetic function is a function defined on the positive integers which takes values in the real or complex numbers.
For instance, define by . Then f is an arithmetic function.
Many functions which are important in number theory are arithmetic functions. For example:
(a) The Euler phi function is an arithmetic function.
(b) Define the number of divisors function by
For example, , since there are 6 positive divisors of 12 --- 1, 2, 3, 4, 6, and 12. is an arithmetic function.
(c) Define the sum of divisors function by
Since 1, 2, 3, 6, 9, and 18 are the positive divisors of 18,
is an arithmetic function.
In order to find ways of computing , , and , we can use the following approach: First, compute the function for p, where p is prime.
Next, compute the function for , where p is prime and .
Finally, for a general number n, factor n into a product of powers of primes and use the result for .
In order to make the jump from prime powers to an arbitrary integer, we'll show that the functions in question are multiplicative. While it's possible to do this directly for each function, we can also prove results which will allow us to use the same approach for , , and . These results are important for other applications.
Definition. The Möbius function is the arithmetic function defined by , and for ,
Thus, if n is divisible by a square.
For example, , since . Likewise, , since . But and .
Definition. If f is an arithmetic function, the divisor sum of f is
To save writing, I'll make the convention that when I write " ", I mean to sum over all the positive divisors of a positive integer n. Thus, the divisor sum of f evaluated at a positive integer n takes the positive divisors of n, plugs them into f, and adds up the results. A similar convention will hold for products.
Notice that the divisor sum is a function which takes an arithmetic function as input and produces an arithmetic function as output.
Example. Suppose is defined by . Compute .
is the sum of the squares of the divisors of n:
thus,
Lemma.
Proof. The formula for is obvious.
Suppose . Factor n into a product of powers of primes:
What are the nonzero terms in the sum ? They will come from d's which are products of single powers of , ... , and also .
For example, and would give rise to nonzero terms in the sum, but .
So
Example. Verify the previous lemma for .
The divisor sum is
Definition. If f and g are arithmetic functions, their Dirichlet product is
For example for arithmetic functions f and g, the Dirichlet product evaluated at 12 is
Definition. Define arithmetic functions
Proposition. Let f, g, and h be arithmetic functions.
(a) .
(b) .
(c) .
(d) .
(e) .
Proof. For (a), note that divisors of n come in pairs , and that if is a divisor pair, so is . This means that the same terms occur in both
Hence, they're equal.
Associativity is a little tedious, so I'll just note that and are equal to
Here the sum runs over all triples of positive numbers d, e, f such that . You can fill in the details.
For (c), note that
( is 0 except when , i.e. when .)
For (d),
Now suppose . Then by (d), , so
Therefore, the formula holds for all n.
The next result is very powerful, but the proof will look easy with all the machinery I've collected.
Theorem. ( Möbius Inversion Formula) If f is an arithmetic function, then
Proof.
Next, I'll compute the divisor sum of the Euler phi function.
Lemma.
Proof. Let n be a positive integer. Construct the fractions
Reduce them all to lowest terms. Consider a typical lowest-term fraction . Here (because it came from a fraction whose denominator was n, (because the original fraction was less than 1), and (because it's in lowest terms).
Notice that (going the other way) if is a fraction with positive top and bottom which satisfies , , and , then it is one of the lowest-terms fractions. For for some k, and then --- and the last fraction is one of the original fractions.
How many of the lowest-terms fractions have "d" on the bottom? Since the "a" on top is a positive number relatively prime to d, there are such fractions. Summing over all d's which divide n gives . But since every lowest-terms fraction has some such "d" on the bottom, this sum accounts for all the fractions --- and there are n of them. Therefore, .
For example, suppose . Then
Lemma. Let .
Proof. By Möbius inversion and the previous result,
For instance, suppose , so . Then
Theorem. Let .
(By convention, the empty product --- the product with no terms --- equals 1.)
Proof. If , the result is immediate by convention.
If , let , ..., be the distinct prime factors of n. Then
Each term is , where d is 1 (the first term) or a product of distinct primes. The in front of each term alternates signs according to the number of p's --- which is exactly what the Möbius function does. So the expression above is
(I can run the sum over all divisors, because if d has a repeated prime factor.) Now simply multiply by n:
The formula in the theorem is useful for hand-computations of . For example, , so
Likewise, , so
Definition. An arithmetic function f is multiplicative if implies
Proposition. is multiplicative --- that is, if , then
Proof. Suppose . Now
So
Since , the two products have no primes in common. Moreover, the primes that appear in either of the products are exactly the prime factors of . So
Hence,
Corollary. Let , and consider its prime factorization:
(Here the p's are distinct primes, and the r's are positive integers.) Then
Equivalently,
Proof. Note that if , then . That is, the prime powers in the prime factorization of n are relatively prime. Recall that if p is prime, then
These observations combined with the fact that is multiplicative give
The second formula follows from the first by factoring out common factors from each term.
Note that in the second formula a prime power gives a "real" term only if . If is the highest power of p which divides n, then the corresponding term in the second formula for is just --- and so, if you don't know the value of r in , the only term you can assume will appear in is .
While the formula in the earlier theorem provided a way of computing , the formula in the corollary is often useful in proving results about .
Example. If , then is even. In fact, if n has k odd prime factors, then .
To see this, observe first that
This is even if .
So suppose that n has k odd prime factors. Each odd prime power factor in the prime factorization of n gives a term in the product for in the corollary, and each term is even (since it's a difference of odd numbers).
Hence, is divisible by .
For example, consider . There are 3 odd prime factors, so should be divisible by 8. And in fact, .
Example. Find all positive integers n such that .
I'll do this in steps. First, I'll show that no prime can divide n.
At that point, with , I'll get bounds on a, b, and c. That will leave me with 16 cases, which I can check directly.
Step 1. No prime greater than 7 divides n.
Suppose and the prime power occurs in the prime factorization of n. The formula in the second corollary tells us that there is at least a term in the product for . But since , I know that p is at least 11 (the next larger prime). Thus, , and so . Then
This is a contradiction, since the right side is larger than 8. Hence, no prime greater than 7 can divide n.
Step 2. .
If , then the formula in the second corollary tells us that there is at least a term in the product for . So I have
This is a contradiction, since .
At this point, I know that .
Step 3. and and .
I have
Suppose . The factor is at least , but . Hence, .
Suppose . The factor is at least , so 3 divides the right side. But . Hence, .
Suppose . The factor is at least , so 5 divides the right side. But . Hence, .
At this point, I know and and . I could probably eliminate some possibilities by additional analysis, but with cases, I can just check by hand.
Step 4. Check the remaining cases by hand.
The numbers are 15, 20, 24, 30.
Appendix: An alternate proof of multiplicativity for
In this appendix, I'll give a direct proof of the multiplicativity of which doesn't use any results on arithmetic functions.
Theorem. If , then
Proof. I list the numbers from 1 to and count the number that are relatively prime to .
A typical column looks like:
Note that divides all the numbers in this column. Moreover, . Thus, if , then is a nontrivial divisor of each number in this column and of .
Hence, if , all the numbers in this column are not relatively prime to .
Therefore, since I'm counting numbers that are relatively prime to , I need only consider columns where . There are such columns.
So consider a column where . It contains the numbers
Start with , the standard residue system mod n. Since , multiplying by m produces another complete residue system:
Adding k must also give a complete residue system:
This is the column in question, so I've shown that such a column is a complete residue system mod n. It follows that of these numbers are relatively prime to n.
Thus, I have columns, all of whose elements are relatively prime to m, and in each such column of the elements are relatively prime to n. Thus, there are numbers in which are relatively prime to .
Let's look at how the proof works with a specific example. Take and . List the numbers from 1 to 36:
You can see that the columns beginning with 3, 6, and 9 --- the numbers k with --- contain only numbers that are not relatively prime to 9 (and hence, are not relatively prime to 36). Removing these columns, I have columns left: The ones beginning with 1, 2, 4, 5, 7, and 8.
Pick any one of those columns --- say the one beginning with 7, which contains . Note that these numbers reduce mod 4 to , which is a complete residue system mod 4. And exactly of these numbers --- in this case, 7 and 25 --- are relatively prime to 4.
Thus, there are numbers in which are relatively prime to , and so .
Contact information | 2,439 | 10,456 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.6875 | 5 | CC-MAIN-2019-18 | latest | en | 0.912129 |
http://www.markedbyteachers.com/gcse/maths/investigating-when-pairs-of-diagonal-corners-are-multiplied-and-subtracted-from-each-other.html | 1,516,257,557,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084887067.27/warc/CC-MAIN-20180118051833-20180118071833-00247.warc.gz | 522,487,986 | 19,386 | • Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
13. 13
13
14. 14
14
15. 15
15
16. 16
16
17. 17
17
18. 18
18
19. 19
19
20. 20
20
21. 21
21
22. 22
22
23. 23
23
24. 24
24
25. 25
25
26. 26
26
• Level: GCSE
• Subject: Maths
• Word count: 2891
# Investigating when pairs of diagonal corners are multiplied and subtracted from each other.
Extracts from this document...
Introduction
Investigating when pairs of diagonal corners are multiplied and subtracted from each other. Introduction: In this coursework I shall be investigating when pairs of diagonal corners are multiplied and subtracted from each other. To do this I shall take square grids and then select 3 square boxes from each grid and multiply the opposite corners then find the difference, then I will change the box size, to try and find a pattern or formula. Then I will change the grid size and then once again take boxes of different sizes and multiply the opposite corners and find the difference. I will try and find a formula for and square box on and size square grid. Then I shall investigate further by using rectangle boxes instead of square. 34 x 45 = 1530 35 x 44 = 1540 Difference = 10 68 x 79 = 5372 69 x 78 = 5382 Difference = 10 81 x 92 = 7452 82 x 91 = 7462 Difference = 10 In a 2 x 2 box on a 10 x 10 grid the difference is 10. Algebraic Method x x + 1 x + 10 x + 11 x(x + 11) = x� + 11x (x + 1)(x + 10) = x� + 10 + x +10x = x� + 11x + 10 x� + 11x - x� + 11x + 10 = 10 1 x 23 = 23 3 x 21 = 61 Difference = 40 36 x 58 = 2088 38 x 56 = 2128 Difference = 40 73 x 95 = 6935 75 x 93 = 6975 Difference = 40 In a 3 x 3 box on a 10 x 10 grid the difference is 40. Algebraic Method x x + 2 x + 20 x + 22 x(x + 22) = x� + 22x (x + 2)(x + 20) = x� + 20x + 2x + 40 = x� + 22x + 40 x� + 22x - x� + 22x + 40 = ...read more.
Middle
N = Any Number (b -1) = Box size - 1 Let x = (b - 1) n n + x n + 8x n + 9x n(n + 9x) = n� + 9xn (n + 1)(n + 8) = n� + 8xn + xn + 8x� = n� + 9xn + 8x� n� + 9xn - n� + 9xn + 8x� = 8x� 8x� = 8(b-1) � I will now do the same process on an 11 x 11 grid to see if there is a pattern. 14 x 26 = 364 15 x 25 = 375 Difference = 11 28 x 40 = 1120 29 x 39 = 1131 Difference = 11 76 x 88 = 6688 77 x 87 = 6699 Difference = 11 In a 2 x 2 box on an 11 x 11 grid the difference is 11. Algebraic Method x x + 1 x + 11 x + 12 x(x + 12) = x� + 12x (x + 1)(x + 11) = x� + 11x + x + 11 = x� + 12x + 11 x� + 12x - x� + 12x + 11 = 11 5 x 29 = 145 7 x 27 = 189 Difference = 44 42 x 66 = 2772 44 x 64 = 2816 Difference = 44 89 x 113 = 10057 91 x 111 = 10101 Difference = 44 In a 3 x 3 box on an 11 x 11 grid the difference is 44. Algebraic Method x x + 2 x + 22 x + 24 x(x + 24) = x� + 24x (x + 2)(x + 22) = x� + 22x + 2x + 44 = x� + 24x + 44 x� + 24x - x� + 24x + 44 = 44 4 x 40 = 160 7 x 37 = 259 Difference = 99 78 x 114 = 8892 81x 111 = 8991 Difference = 99 74 x 110 = 8140 77 x 107 = 8239 Difference = 99 In a 4 x 4 box on an 11 x 11 grid the difference is 99. ...read more.
Conclusion
= n� + 9xn + ny (n + y)(n + 9x) = n� + 9xn + ny + 9xy n� + 9xn + ny - n� + 9xn + ny + 9xy = 9xy 9xy = 9(L - 1) (W - 1) I have noticed a pattern in my formulas. The number that comes before the brackets is the same as the grid size. So I can change that number to G when G = Grid Size. So the formula for any box size on any grid size: G (L - 1)(W - 1) Proving Formula: n n+(W - 1) n + G(L - 1) n + G(L -1) + (W - 1) N = Any Number (L - 1) = Length - 1 (W - 1) = Width - 1 Let x = (L - 1) Let y = (W - 1) n n + y n + Gx n + Gx + y n(n + Gx + y) = n� + nGx + ny (n + y)(n + Gx) = n� + nGx + ny + Gxy n� + nGx + ny - n� + nGx + ny + Gxy = Gxy Gxy = G (W- 1)(L - 1) Conclusion Formula Description 10(b - 1) � This calculates the difference of any square box on a 10 x 10 grid. 9(b - 1) � This calculates the difference of any square box on a 9 x 9 grid. 8(b - 1) � This calculates the difference of any square box on an 8 x 8 grid. 11(b - 1) � This calculates the difference of any square box on an 11 x 11 grid. G(b - 1) � This calculates the difference of any square box on any grid. 10(L - 1)(W - 1) This calculates the difference of any rectangle box on a 10 x 10 grid. 9(L - 1)(W - 1) This calculates the difference of any rectangle box on a 9 x 9 grid. G(L - 1)(W - 1) This calculates the difference of any rectangle box on any grid. Limitations: ...read more.
The above preview is unformatted text
This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.
## Found what you're looking for?
• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month
Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
# Related GCSE Number Stairs, Grids and Sequences essays
1. ## Opposite Corners
4 star(s)
Again I will start with algebra. z-number in the top left corner. z z+1 z+2 z+3 z(z+23)=z�+23z z+10 z+11 z+12 z+13 (z+3)(z+20)=z�+23z+60 z+20 z+21 z+22 z+23 (z�+23z+60)-(z�+23z)=60 Difference This proves that with any 3*4 rectangular box the difference is always 60.
2. ## Opposite Corners
3 star(s)
I shall now move on and investigate a 3x3 Rectangle (square) and more rectangles in the same way. 3x3 Rectangle (square) 54 55 56 64 65 66 74 75 76 71 72 73 81 82 83 91 92 93 54x76=4104 71x93=6603 74x56=4144 91x73=6643 40 40 10 11 12 20 21
1. ## Number Grid Aim: The aim of this investigation is to formulate an algebraic equation ...
3 star(s)
grid = Difference of 19 100 x 100 grid = Difference of 100 31 x 31 grid = Difference of 31 58 x 58 grid = Difference of 58 1000x1000 grid = Difference of 1000 3 X 3 Squares Having studied 2 x 2 boxes within different sized grids, I
2. ## Number Grid Coursework
If (p - 1) is the variable representing the side of the square, then the area calculated from that is (p - 1)2, or (p - 1)(p - 1). Because each of these brackets represents the side of the square, when it becomes a rectangle, it is safe to assume that the area will be: (p - 1)(q - 1)
1. ## Algebra Investigation - Grid Square and Cube Relationships
Because the second answer has +ghs2w-ghs2+gs2w+gs2 at the end, it demonstrates that no matter what number is chosen to begin with (n), a difference of s2ghw-s2gh+s2gw+s2g will always be present. Difference Relationships in a 10x10x10 Cube Top Face (TF) 1 2 3 4 5 6 7 8 9 10 11
2. ## What the 'L' - L shape investigation.
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
1. ## Investigation of diagonal difference.
a 2x2 cutout anywhere on the 10x10 grid by implementing the use of simple algebra. I can call the top left number in the cutout n, the top right number n + 1, the bottom left number n + 10 and the bottom right n + 11, as this is
2. ## Maths-Number Grid
6 � 6 Grid Example:- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to | 2,800 | 7,298 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.5625 | 5 | CC-MAIN-2018-05 | latest | en | 0.714909 |
https://www.spoj.com/problems/SEGSQRSS/ | 1,716,511,133,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058675.22/warc/CC-MAIN-20240523235012-20240524025012-00362.warc.gz | 859,417,718 | 9,927 | ## SEGSQRSS - Sum of Squares with Segment Tree
Segment trees are extremely useful. In particular "Lazy Propagation" (i.e. see here, for example) allows one to compute sums over a range in O(lg(n)), and update ranges in O(lg(n)) as well. In this problem you will compute something much harder:
The sum of squares over a range with range updates of 2 types:
1) increment in a range
2) set all numbers the same in a range.
### Input
There will be T (T <= 25) test cases in the input file. First line of the input contains two positive integers, N (N <= 100,000) and Q (Q <= 100,000). The next line contains N integers, each at most 1000. Each of the next Q lines starts with a number, which indicates the type of operation:
2 st nd -- return the sum of the squares of the numbers with indices in [st, nd] {i.e., from st to nd inclusive} (1 <= st <= nd <= N).
1 st nd x -- add "x" to all numbers with indices in [st, nd] (1 <= st <= nd <= N, and -1,000 <= x <= 1,000).
0 st nd x -- set all numbers with indices in [st, nd] to "x" (1 <= st <= nd <= N, and -1,000 <= x <= 1,000).
### Output
For each test case output the “Case <caseno>:” in the first line and from the second line output the sum of squares for each operation of type 2. Intermediate overflow will not occur with proper use of 64-bit signed integer.
### Example
```Input:
2
4 5
1 2 3 4
2 1 4
0 3 4 1
2 1 4
1 3 4 1
2 1 4
1 1
1
2 1 1
Output:
Case 1:
30
7
13
Case 2:
1```
hide comments
< Previous 1 2 3 4 5 6 Next > Ishan: 2022-04-01 04:43:07 Terrible test casse. The problem is designed to be solved it in O(N*log N + Q*log N) but terrible test cases are allowing O(Q*N*logN) submissions with no lazy propagation what so ever. But people who solved in the real expected complexity have learnt a lot. The trick comes in interplay between the two updates, both requiring lazy propagation and how they interplay with each other. luciferhell58: 2020-05-21 13:25:17 i THINK ITS GIVING PROBLEM BECAUSE I AM USING JAVA luciferhell58: 2020-05-21 13:22:04 i am getting tle but got this one aceppeted in 300 ms at coding ninjas sayan_244: 2020-05-06 15:27:18 I feel they have simply evaluated the codes on the sample test case. They aren't good enough my code is actually not correct as of now >.> upd:- I think I fixed it now. Last edit: 2020-05-21 06:18:19 aryan12: 2020-03-02 15:20:59 AC in one go, I don't know how. I was expecting a TLE :) dhj: 2020-01-25 13:02:15 To get AC in one go, for an average like me...is super awesome :D hduoc2003: 2019-09-29 05:21:48 I used two segment tree to solve this. Anyone does it better pls tell me!! Thank u very much :> Last edit: 2019-09-29 05:24:13 manishjoshi394: 2019-09-02 13:07:13 AC in one go, very good problem for newbies on Lazy propagation. chirayu_555: 2019-08-24 10:12:12 AC in single go. Do updates properly. Nice problem..!! edygordo: 2019-07-26 19:54:41 2 4 3 1 2 3 4 0 1 4 10 1 1 4 10 2 1 1 4 3 1 2 3 4 1 1 4 10 0 1 4 10 2 1 1 //try this case OUTPUT Should be Case 1: 400 Case 2: 100 // First increase then Replace // First Replace then Increase **(here it's a bit different) //first Increase then increase //first Replace then replace
Added by: Chen Xiaohong Date: 2012-07-11 Time limit: 1.106s Source limit: 50000B Memory limit: 1536MB Cluster: Cube (Intel G860) Languages: All except: ASM64 | 1,094 | 3,319 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2024-22 | latest | en | 0.868116 |
http://www.neverendingbooks.org/tag/brauer/page/4 | 1,601,193,553,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400265461.58/warc/CC-MAIN-20200927054550-20200927084550-00163.warc.gz | 218,482,256 | 13,209 | # Tag: Brauer
One of the things I like most about returning from a vacation is to
have an enormous pile of fresh reading : a week's worth of
newspapers, some regular mail and much more email (three quarters junk).
Also before getting into bed after the ride I like to browse through the
arXiv in search for interesting
papers.
This time, the major surprise of my initial survey came
from the newspapers. No, not Bush again, _that_ news was headline
even in France. On the other hand, I didn't hear a word about Theo Van
Gogh being shot and stabbed to death
in Amsterdam. I'll come
back to this later.
I'd rather mention the two papers that
somehow stood out during my scan of this week on the arXiv. The first is
Framed quiver moduli,
cohomology, and quantum groups
by Markus
Reineke
. By the deframing trick, a framed quiver moduli problem is
reduced to an ordinary quiver moduli problem for a dimension vector for
which one of the entries is equal to one, hence in particular, an
indivisible dimension vector. Such quiver problems are far easier to
handle than the divisible ones where everything can at best be reduced
to the classical problem of classifying tuples of $n \\times n$ matrices
up to simultaneous conjugation. Markus deals with the case when the
quiver has no oriented cycles. An important examples of a framed moduli
quiver problem _with_ oriented cycles is the study of
Brauer-Severi varieties of smooth orders. Significant progress on the
description of the fibers in this case is achieved by Raf Bocklandt,
Stijn Symens and Geert Van de Weyer and will (hopefully) be posted soon.
The second paper is Moduli schemes of rank
one Azumaya modules
by Norbert Hoffmann and Urich Stuhler which
brings back longforgotten memories of my Ph.D. thesis, 21 years
ago…
[Last time][1] we saw that for $A$ a smooth order with center $R$ the
Brauer-Severi variety $X_A$ is a smooth variety and we have a projective
morphism $X_A \rightarrow \mathbf{max}~R$ This situation is
very similar to that of a desingularization $~X \rightarrow \mathbf{max}~R$ of the (possibly singular) variety $~\mathbf{max}~R$.
The top variety $~X$ is a smooth variety and there is a Zariski open
subset of $~\mathbf{max}~R$ where the fibers of this map consist of just
one point, or in more bombastic language a $~\mathbb{P}^0$. The only
difference in the case of the Brauer-Severi fibration is that we have a
Zariski open subset of $~\mathbf{max}~R$ (the Azumaya locus of A) where
the fibers of the fibration are isomorphic to $~\mathbb{P}^{n-1}$. In
this way one might view the Brauer-Severi fibration of a smooth order as
a non-commutative or hyper-desingularization of the central variety.
This might provide a way to attack the old problem of construction
desingularizations of quiver-quotients. If $~Q$ is a quiver and $\alpha$
is an indivisible dimension vector (that is, the component dimensions
are coprime) then it is well known (a result due to [Alastair King][2])
that for a generic stability structure $\theta$ the moduli space
$~M^{\theta}(Q,\alpha)$ classifying $\theta$-semistable
$\alpha$-dimensional representations will be a smooth variety (as all
$\theta$-semistables are actually $\theta$-stable) and the fibration
$~M^{\theta}(Q,\alpha) \rightarrow \mathbf{iss}_{\alpha}~Q$ is a
desingularization of the quotient-variety $~\mathbf{iss}_{\alpha}~Q$
classifying isomorphism classes of $\alpha$-dimensional semi-simple
representations. However, if $\alpha$ is not indivisible nobody has
the faintest clue as to how to construct a natural desingularization of
$~\mathbf{iss}_{\alpha}~Q$. Still, we have a perfectly reasonable
hyper-desingularization $~X_{A(Q,\alpha)} \rightarrow \mathbf{iss}_{\alpha}~Q$ where $~A(Q,\alpha)$ is the corresponding
quiver order, the generic fibers of which are all projective spaces in
case $\alpha$ is the dimension vector of a simple representation of
$~Q$. I conjecture (meaning : I hope) that this Brauer-Severi fibration
contains already a lot of information on a genuine desingularization of
$~\mathbf{iss}_{\alpha}~Q$. One obvious test for this seemingly
crazy conjecture is to study the flat locus of the Brauer-Severi
fibration. If it would contain info about desingularizations one would
expect that the fibration can never be flat in a central singularity! In
other words, we would like that the flat locus of the fibration is
contained in the smooth central locus. This is indeed the case and is a
more or less straightforward application of the proof (due to [Geert Van
de Weyer][3]) of the Popov-conjecture for quiver-quotients (see for
example his Ph.D. thesis [Nullcones of quiver representations][4]).
However, it is in general not true that the flat-locus and central
smooth locus coincide. Sometimes this is because the Brauer-Severi
scheme is a blow-up of the Brauer-Severi of a nicer order. The following
example was worked out together with [Colin Ingalls][5] : Consider the
order $~A = \begin{bmatrix} C[x,y] & C[x,y] \\ (x,y) & C[x,y] \end{bmatrix}$ which is the quiver order of the quiver setting
$~(Q,\alpha)$ $\xymatrix{\vtx{1} \ar@/^2ex/[rr] \ar@/^1ex/[rr] & & \vtx{1} \ar@/^2ex/[ll]}$ then the Brauer-Severi fibration
$~X_A \rightarrow \mathbf{iss}_{\alpha}~Q$ is flat everywhere except
over the zero representation where the fiber is $~\mathbb{P}^1 \times \mathbb{P}^2$. On the other hand, for the order $~B = \begin{bmatrix} C[x,y] & C[x,y] \\ C[x,y] & C[x,y] \end{bmatrix}$
the Brauer-Severi fibration is flat and $~X_B \simeq \mathbb{A}^2 \times \mathbb{P}^1$. It turns out that $~X_A$ is a blow-up of $~X_B$ at a
point in the fiber over the zero-representation.
[1]: http://www.neverendingbooks.org/index.php?p=342
[3]: http://www.win.ua.ac.be/~gvdwey/
[4]: http://www.win.ua.ac.be/~gvdwey/papers/thesis.pdf
[5]: http://kappa.math.unb.ca/~colin/
Around the
same time Michel Van den Bergh introduced his Brauer-Severi schemes,
[Claudio Procesi][1] (extending earlier work of [Bill Schelter][2])
introduced smooth orders as those orders $A$ in a central simple algebra
$\Sigma$ (of dimension $n^2$) such that their representation variety
$\mathbf{trep}_n~A$ is a smooth variety. Many interesting orders are smooth
: hereditary orders, trace rings of generic matrices and more generally
size n approximations of formally smooth algebras (that is,
non-commutative manifolds). As in the commutative case, every order has
a Zariski open subset where it is a smooth order. The relevance of
this notion to the study of Brauer-Severi varieties is that $X_A$ is a
smooth variety whenever $A$ is a smooth order. Indeed, the Brauer-Severi
scheme was the orbit space of the principal $GL_n$-fibration on the
Brauer-stable representations (see [last time][3]) which form a Zariski
open subset of the smooth variety $\mathbf{trep}_n~A \times k^n$. In fact,
in most cases the reverse implication will also hold, that is, if $X_A$
is smooth then usually A is a smooth order. However, for low n,
there are some counterexamples. Consider the so called quantum plane
$A_q=k_q[x,y]~:~yx=qxy$ with $~q$ an $n$-th root of unity then one
can easily prove (using the fact that the smooth order locus of $A_q$ is
everything but the origin in the central variety $~\mathbb{A}^2$) that
the singularities of the Brauer-Severi scheme $X_A$ are the orbits
corresponding to those nilpotent representations $~\phi : A \rightarrow M_n(k)$ which are at the same time singular points in $\mathbf{trep}_n~A$
and have a cyclic vector. As there are singular points among the
nilpotent representations, the Brauer-Severi scheme will also be
singular except perhaps for small values of $n$. For example, if
$~n=2$ the defining relation is $~xy+yx=0$ and any trace preserving
representation has a matrix-description $~x \rightarrow \begin{bmatrix} a & b \\ c & -a \end{bmatrix}~y \rightarrow \begin{bmatrix} d & e \\ f & -d \end{bmatrix}$ such that
$~2ad+bf+ec = 0$. That is, $~\mathbf{trep}_2~A = \mathbb{V}(2ad+bf+ec) \subset \mathbb{A}^6$ which is an hypersurface with a unique
singular point (the origin). As this point corresponds to the
zero-representation (which does not have a cyclic vector) the
Brauer-Severi scheme will be smooth in this case. [Colin
Ingalls][4] extended this calculation to show that the Brauer-Severi
scheme is equally smooth when $~n=3$ but has a unique (!) singular point
when $~n=4$. So probably all Brauer-Severi schemes for $n \geq 4$ are
indeed singular. I conjecture that this is a general feature for
Brauer-Severi schemes of families (depending on the p.i.-degree $n$) of
non-smooth orders.
[1]: http://venere.mat.uniroma1.it/people/procesi/
[2]: http://www.fact-index.com/b/bi/bill_schelter.html
[3]: http://www.neverendingbooks.org/index.php?p=341
[4]: http://kappa.math.unb.ca/~colin/
![][1]
Classical Brauer-Severi varieties can be described either as twisted
forms of projective space (Severi\’s way) or as varieties containing
splitting information about central simple algebras (Brauer\’s way). If
$K$ is a field with separable closure $\overline{K}$, the first approach
asks for projective varieties $X$ defined over $K$ such that over the
separable closure $X(\overline{K}) \simeq \mathbb{P}^{n-1}_{\overline{K}}$ they are just projective space. In
the second approach let $\Sigma$ be a central simple $K$-algebra and
define a variety $X_{\Sigma}$ whose points over a field extension $L$
are precisely the left ideals of $\Sigma \otimes_K L$ of dimension $n$.
This variety is defined over $K$ and is a closed subvariety of the
Grassmannian $Gr(n,n^2)$. In the special case that $\Sigma = M_n(K)$ one
can use the matrix-idempotents to show that the left ideals of dimension
$n$ correspond to the points of $\mathbb{P}^{n-1}_K$. As for any central
simple $K$-algebra $\Sigma$ we have that $\Sigma \otimes_K \overline{K} \simeq M_n(\overline{K})$ it follows that the varieties $X_{\Sigma}$ are
among those of the first approach. In fact, there is a natural bijection
between those of the first approach (twisted forms) and of the second as
both are classified by the Galois cohomology pointed set
$H^1(Gal(\overline{K}/K),PGL_n(\overline{K}))$ because
$PGL_n(\overline{K})$ is the automorphism group of
$\mathbb{P}^{n-1}_{\overline{K}}$ as well as of $M_n(\overline{K})$. The
ringtheoretic relevance of the Brauer-Severi variety $X_{\Sigma}$ is
that for any field extension $L$ it has $L$-rational points if and only
if $L$ is a _splitting field_ for $\Sigma$, that is, $\Sigma \otimes_K L \simeq M_n(\Sigma)$. To give one concrete example, If $\Sigma$ is the
quaternion-algebra $(a,b)_K$, then the Brauer-Severi variety is a conic
$X_{\Sigma} = \mathbb{V}(x_0^2-ax_1^2-bx_2^2) \subset \mathbb{P}^2_K$
Whenever one has something working for central simple algebras, one can
_sheafify_ the construction to Azumaya algebras. For if $A$ is an
Azumaya algebra with center $R$ then for every maximal ideal
$\mathfrak{m}$ of $R$, the quotient $A/\mathfrak{m}A$ is a central
simple $R/\mathfrak{m}$-algebra. This was noted by the
sheafification-guru [Alexander Grothendieck][2] and he extended the
notion to Brauer-Severi schemes of Azumaya algebras which are projective
bundles $X_A \rightarrow \mathbf{max}~R$ all of which fibers are
projective spaces (in case $R$ is an affine algebra over an
algebraically closed field). But the real fun started when [Mike
Artin][3] and [David Mumford][4] extended the construction to suitably
_ramified_ algebras. In good cases one has that the Brauer-Severi
fibration is flat with fibers over ramified points certain degenerations
of projective space. For example in the case considered by Artin and
Mumford of suitably ramified orders in quaternion algebras, the smooth
conics over Azumaya points degenerate to a pair of lines over ramified
points. A major application of their construction were examples of
unirational non-rational varieties. To date still one of the nicest
applications of non-commutative algebra to more mainstream mathematics.
The final step in generalizing Brauer-Severi fibrations to arbitrary
orders was achieved by [Michel Van den Bergh][5] in 1986. Let $R$ be an
affine algebra over an algebraically closed field (say of characteristic
zero) $k$ and let $A$ be an $R$-order is a central simple algebra
$\Sigma$ of dimension $n^2$. Let $\mathbf{trep}_n~A$ be teh affine variety
of _trace preserving_ $n$-dimensional representations, then there is a
natural action of $GL_n$ on this variety by basechange (conjugation).
Moreover, $GL_n$ acts by left multiplication on column vectors $k^n$.
One then considers the open subset in $\mathbf{trep}_n~A \times k^n$
consisting of _Brauer-Stable representations_, that is those pairs
$(\phi,v)$ such that $\phi(A).v = k^n$ on which $GL_n$ acts freely. The
corresponding orbit space is then called the Brauer-Severio scheme $X_A$
of $A$ and there is a fibration $X_A \rightarrow \mathbf{max}~R$ again
having as fibers projective spaces over Azumaya points but this time the
fibration is allowed to be far from flat in general. Two months ago I
outlined in Warwick an idea to apply this Brauer-Severi scheme to get a
hold on desingularizations of quiver quotient singularities. More on
this next time.
[1]: http://www.neverendingbooks.org/DATA/brauer.jpg
[2]: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.html
[3]: http://www.cirs-tm.org/researchers/researchers.php?id=235
[4]: http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Mumford.html
[5]: http://alpha.luc.ac.be/Research/Algebra/Members/michel_id.html
Last time we have seen that in order to classify all
non-commutative $l$-points one needs to control the finite
dimensional simple algebras having as their center a finite
dimensional field-extension of $l$. We have seen that the equivalence
classes of simple algebras with the same center $L$ form an Abelian
group, the
Brauer group. The calculation of Brauer groups
is best done using
Galois-cohomology. As an aside :
Evariste Galois was one of the more tragic figures in the history of
mathematics, he died at the age of 20 as a result of a duel. There is
a whole site the Evariste Galois archive dedicated to his
work.
But let us return to a simple algebra $T$ over the
field $L$ which we have seen to be of the form $M(k,S)$, full
matrices over a division algebra $S$. We know that the dimension of
$S$ over $L$ is a square, say $n^2$, and it can be shown that all
maximal commutative subfields of $S$ have dimension n over $L$.
In this way one can view a simple algebra as a bag containing all
sorts of degree n extensions of its center. All these maximal
subfields are also splitting fields for $S$, meaning that
if you tensor $S$ with one of them, say $M$, one obtains full nxn
matrices $M(n,M)$. Among this collection there is at least one
separable field but for a long time it was an open question
whether the collection of all maximal commutative subfields also
contains a Galois-extension of $L$. If this is the case, then
one could describe the division algebra $S$ as a crossed
product
. It was known for some time that there is always a simple
algebra $S’$ equivalent to $S$ which is a crossed product (usually
corresponding to a different number n’), that is, all elements of
the Brauer group can be represented by crossed products. It came as a
surprise when S.A. Amitsur in 1972 came up with examples of
non-crossed product division algebras, that is, division algebras $D$
such that none of its maximal commutative subfields is a Galois
extension of the center. His examples were generic
division algebras
$D(n)$. To define $D(n)$ take two generic
nxn matrices
, that is, nxn matrices A and B such that all its
entries are algebraically independent over $L$ and consider the
$L$-subalgebra generated by A and B in the full nxn matrixring over the
field $F$ generated by all entries of A and B. Somewhat surprisingly,
one can show that this subalgebra is a domain and inverting all its
central elements (which, again, is somewhat of a surprise that
there are lots of them apart from elements of $L$, the so called
central polynomials) one obtains the division algebra $D(n)$ with
center $F(n)$ which has trancendence degree n^2 1 over $L$. By the
way, it is still unknown (apart from some low n cases) whether $F(n)$
is purely trancendental over $L$. Now, utilising the generic
nature of $D(n)$, Amitsur was able to prove that when $L=Q$, the
field of rational numbers, $D(n)$ cannot be a crossed product unless
$n=2^s p_1…p_k$ with the p_i prime numbers and s at most 2. So, for
example $D(8)$ is not a crossed product.
One can then
ask whether any division algebra $S$, of dimension n^2 over $L$, is a
crossed whenever n is squarefree. Even teh simplest case, when n is a
prime number is not known unless p=2 or 3. This shows how little we do
know about finite dimensional division algebras : nobody knows
whether a division algebra of dimension 25 contains a maximal
cyclic subfield (the main problem in deciding this type of
problems is that we know so few methods to construct division
algebras; either they are constructed quite explicitly as a crossed
product or otherwise they are constructed by some generic construction
but then it is very hard to make explicit calculations with
them). | 4,893 | 17,233 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2020-40 | latest | en | 0.929156 |
http://math.stackexchange.com/questions/12547/what-is-the-spectral-theorem-for-compact-self-adjoint-operators-on-a-hilbert-spa | 1,469,392,270,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824146.3/warc/CC-MAIN-20160723071024-00318-ip-10-185-27-174.ec2.internal.warc.gz | 155,272,212 | 21,320 | # What is the spectral theorem for compact self-adjoint operators on a Hilbert space actually for?
Please excuse the naive question. I have had two classes now in which this theorem was taught and proven, but I have only ever seen a single (indirect?) application involving the quantum harmonic oscillator. Even if this is not the strongest spectral theorem, it still seems useful enough that there should be many nice examples illustrating its utility. So... what are some of those examples?
(I couldn't readily find any nice examples looking through a few functional analysis textbooks, either. Maybe I have the wrong books.)
-
A lot of Hilbert-Schmidt operators are self-adjoint and compact. So, also Fredholm integral equations. These are quite useful in applications. – Jonas Teuwen Nov 30 '10 at 23:59
Thanks, Jonas. Do you have a good reference with some explicit examples? – Qiaochu Yuan Dec 1 '10 at 0:01
I know some books that treat integral equations, but do not explicitly mention the spectral theorem (that is usually something for another course...), like Riesz' Functional Analysis (also a nice book in other aspects). It is in a Dover paperback. Further you have Hochstadt's integral equations, from the Wiley Classics. When I'm at the university tomorrow I'll see if there are some books in the analysis library that treat both explicitly. – Jonas Teuwen Dec 1 '10 at 0:11
Kato's Perturbation Theory for Linear Operators gives nice basic examples that relate operator theory to boundary value problems for differential equations. For the spectral theorem, applications to PDEs are common in mechanics and fluid dynamics, for example. Often the map from data to solution of a boundary-value problem on a bounded domain is compact. – Bob Pego Dec 1 '10 at 3:23
in sturm-louville theory they say the differential operators from the differential equation taken together forms a new operator that is self adjoint and compact - so it follows from the spectral theorem that there is a basis of orthogonal eigenfunctions that you can compose the solution to the differential equation from! – Peter Sheldrick Jul 22 '11 at 15:47
What is the spectral theorem for compact operators good for? Here are some examples. (I am ignoring the self-adjoint aspects, since they don't really play a role in the theorem. And it is valid for more general spaces than Hilbert spaces too, so I will also ignore that part, in the sense that I won't pay too much attention to whether my examples deal with Hilbert spaces on the nose, rather than some variant.)
• Proving the Peter--Weyl theorem.
• Proving the Hodge decomposition for cohomology of manifolds (using the fact that the inverse to the Laplacian is compact); Willie noted this example in his answer too.
• Proving the finiteness of cohomology of coherent sheaves on compact complex analytic manifolds.
• In its $p$-adic version, the theory of compact operators is basic to the theory of $p$-adic automorphic forms: e.g. in the construction of so-called eigenvarieties parameterizing $p$-adic families of automorphic Hecke eigenforms of finite slope.
• It is also a basic tool in more classical problems, such as the theory of integral equations. (It is in this context that the theory was first developed; see Dieudonne's book on the history of functional analysis for a very nice account of the historical development of the theory.)
-
Can the spectral theorem be applied to show more generally that the higher direct images of a coherent analytic sheaf under a proper map of analytic spaces is coherent? By the way, I'd be very curious if you had a reference for that. – Akhil Mathew Dec 1 '10 at 4:23
@Akhil: Dear Akhil, That's a good question, to which unfortunately I don't have a good answer. My instinct is to approach it as follows: the question is local on the base, so restricting to a well-chosen n.h. of any given point, we can assume the base is Stein. The point of this is that on a Stein manifold, coherent sheaves are determined by their modules of sections (thought of as Frechet modules over the Frechet algebra of global holomorphic functions). So one could try to resolve the coherent sheaf upstairs in some reasonable way, and then compute the derived pushforward using this ... – Matt E Dec 1 '10 at 4:38
... resolution, and try to show that the cohomologies are coherent sheaves by studying their global sections as modules over the functions on the base, and get some control over them (here one would use properness, and some relative version of Montel's theorem; and --- if it worked --- one wold be applying a relative version of compact operators, for Frechet modules over the Frechet algebra of functions on the base). I don't know if this actually works, though. – Matt E Dec 1 '10 at 4:40
I didn't know that it was first used in the theory of integral equations, that is where I use it. Nice answer. – Jonas Teuwen Dec 1 '10 at 8:59
Thanks for the very informative answer. I've been meaning to check out Dieudonne's history for awhile now and I guess now is a good time. – Qiaochu Yuan Dec 1 '10 at 10:12
Maybe a definition of functional calculus? Using quantum mechanics notation, if $A$ is self-adjoint and compact, then $A = \sum \lambda_k |k\rangle\langle k|$, which means that for $f:\mathbb{R}\to\mathbb{R}$ we can define $f(A) = \sum f(\lambda_k) |k\rangle\langle k|$.
This allows us to give a simple demonstration of Stone's theorem for such operators: that $\exp itA$ is a strongly continuous one-parameter unitary group on your Hilbert space.
It also gives very simple motivation for the construction of Green's functions and resolvent operators.
Besides the usual quantum harmonic oscillator, a similar construction can be used to give the decomposition of $L^2$ via eigenfunctions of the Laplacian on a compact manifold. This naturally leads to the Hodge decomposition, which I'm told is generally considered to be somewhat useful :-)
That on a compact manifold, the inverse of the Laplacian is a compact operator means that, discarding the harmonic functions, the Laplacian has a lowest eigenvalue. This fact (and that self-adjointness allows it to be diagonalized) allows you to define the Zeta-function determinant of the Laplacian using some analytic continuation tricks. On 2-dimensional closed surfaces, this is an interesting invariant with nice geometric properties.
(Note that in the case of non-compact domains, the inverse Laplacian is no longer a compact operator, and the Laplacian has a continuous spectrum. So the summation in the zeta-function determinant no longer makes any sense...)
-
The only way I know to prove "discreteness" of some piece of a spectrum is to find one or more compact operators on it, suitably separating points. That is, somehow the only tractable operators are those closely related to compact ones.
Even to discuss the spectral theory of self-adjoint differential operators $T$, the happiest cases are where $T$ has compact resolvent $(T-\lambda)^{-1}$.
In particular instances, the Schwartz kernel theorem depends on the compactness of the inclusions of Sobolev spaces into each other (Rellich's lemma).
In automorphic forms: to prove the discreteness of spaces of cuspforms, one shows that the natural integral operators (after Selberg, Gelfand, Langlands et alia) restricted to the space of $L^2$ cuspforms are compact.
One of Selberg's arguments, Bernstein's sketch, Colin de Verdiere's proof, and (apparently) the proof in Moeglin-Waldspurger's book (credited to Jacquet, credited to Colin de Verdiere!?) of meromorphic continuation of Eisenstein series of various sorts depends ultimately on proving compactness of an operator.
- | 1,754 | 7,694 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2016-30 | latest | en | 0.942683 |
https://www.coursehero.com/file/5805915/530-343lecture02/ | 1,524,381,768,000,000,000 | text/html | crawl-data/CC-MAIN-2018-17/segments/1524125945497.22/warc/CC-MAIN-20180422061121-20180422081121-00153.warc.gz | 758,552,757 | 90,123 | {[ promptMessage ]}
Bookmark it
{[ promptMessage ]}
530_343lecture02
# 530_343lecture02 - ME 530.343 Design and Analysis of...
This preview shows pages 1–4. Sign up to view the full content.
ME 530.343: Design and Analysis of Dynamic Systems Spring 2009 Lecture 2 – Linearization Friday Januray 30, 2009 1 Today’s Objectives Why linearize? Pendulum example Taylor series expansion Reading: Palm 1.5–1.7 2 Why Linearize? Classic mass-spring-damper is linear. Classic pendulum is nonlinear. Most real mechanical systems are governed by nonlinear differential equations -Exact analytical (closed-form) solution difficult to find Often, computers are used to numerically integrate nonlinear ODEs: Matlab, Maple, Mathematica, Mathcad, Working Model, Autolev, etc. However, in this course we wish to approximate a nonlinear differential equation with a linear one. Why? serve as a starting point for analytical solution without aid of computer essential for Laplace Transforms can quickly determine stability in the neighborhood of a solution determine the effect of a parameter in system behavior physical insight essential for some control design tools 1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
3 Classic Pendulum Equation of motion: ¨ θ + g L sin( θ ) = 0 nonlinear, homogeneous, constant-coefficient, 2 nd order, ODE Possible solution methods: 1. Numerical integration (Eular, Predictor-Corrector, Runga-Kutta, ...) 2. Analytical solution w/ Jacobian elliptical functions. 3. Analytical solution w/ small angle approx. (sin( θ ) θ ) 3.1 Small angle approximation Taylor series sin( θ ) θ because sin( θ ) = θ - θ 3 3! + θ 5 5! - θ 7 7! + · · · cos( θ ) 1 because cos( θ ) = 1 - θ 2 2! + θ 4 4! - θ 6 6! + · · · Note that sin( θ ) θ is a better approximation than cos( θ ) 1 matches more derivatives 2
In light of the small-angle approximation, revisit pendulum problem: sin( θ ) θ ¨ θ + g L θ = 0 ω n = g L Underdamped system, given initial values θ (0) , ˙ θ (0). Note that, by convention, ˙ θ (0) = d dt θ ( t ) t =0 , and NOT d dt [ θ (0)] – the latter is always zero (derivative of a constant). Solution: θ ( t ) = ˙ θ (0) ω n sin( ω n t ) + θ (0) cos( ω n t ) τ n = 2 π ω n = 2 π L g However, if you determine it experimentally (e.g., using string, tape measure, stop watch), you will get a slightly different response.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up to access the rest of the document.
{[ snackBarMessage ]} | 701 | 2,595 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5625 | 4 | CC-MAIN-2018-17 | latest | en | 0.783434 |
http://www.math.uakron.edu/~cossey/412-512.html | 1,558,603,482,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232257197.14/warc/CC-MAIN-20190523083722-20190523105722-00371.warc.gz | 292,646,927 | 2,055 | JP Cossey's webpage
Akron Math dept
University of Akron homepage
Math 412/512
Abstract Algebra II
Leigh Hall 311
### The Basics
(Remember to hit refresh/reload when you visit this web page.)
Instructor: James (J.P.) Cossey,
234 CAS (Email: cossey at uakron dot edu)
Office Hours: M 4-5, W 3-4, F 1-2, or by appointment
Tentative course schedule
Course policies
(My schedule)
### Homework
Homework assignments will be posted here.
• Due Wednesday, January 21: Ch 9: 2, 4, 7, 8a, 10, 12, 15, 18, 27, 37, 39, 41, 43
• Due Monday, February 2nd: Chapter 9: 11, 18, 19, 25, 40. Chapter 8: 4, 6, 8, 9, 15, 21, 23, 24, 39
• Due Monday, February 9th: Ch 11: 6, 7, 8, 9, 10, 15, 16, 19, 26
• Due Wednesday, February 18th: Ch 24: 2, 6, 8 (there are 4 of them), 11, 39, 47, and problem 31 is a good challenge problem, try using induction. Also, read the first few pages of Chapter 25, there's no actual math in it, but some interesting background on simple groups, and a bizarre poem.
• Due Wednesday, February 25th: Ch 16: 6, 7, 11, 12, 14, 15, 18, 22, 27, 31 (hint: what can we say about x^(p-1) for any x in F_p?), 32, 36
• Due Monday, March 9th: Ch 17: 4, 6, 7, 8, 10, 12, 17, 23, 29, 30 (the proof of the Corollary on page 305 should be useful here), 33 (this is really just an exercise in notation - there's nothing nonobvious going on in this problem once you get comfortable with the notation).
• Due Wednesday, March 25th: Ch 19: 5, 7, 13, 14, 18, 19 Ch 20: 2, 4, 8 (don't bother writing out the whole multiplication table, but show a few non-trivial examples), 9, 10, 13, 23, 25, 29, 30
• Due Wednesday, April 8th: Ch 21: 3, 4, 8, 9, 13 (don't worry about finding examples), 14, 16, 18, 23, 24.
• Due Wednesday, April 29th: This handout on cyclotomic polynomials and roots of unity, which is the missing piece in finishing the proof of the characterization of the constructible regular polygons.
### Exams
Except for the final exam, the following dates are only tentative.
• Exam 1: Wednesday, February 25th.
• Exam 2: Wednesday, April 15th.
• Final Exam: Wednesday, May 6th at 6 PM.
### Announcements
• Monday, January 19th is Martin Luther King day, and we will not be having class.
• Here's a flier for the topics in group theory course I will probably be teaching next semester. | 797 | 2,289 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.25 | 3 | CC-MAIN-2019-22 | latest | en | 0.921484 |
https://stage.geogebra.org/m/ba4wzpu9 | 1,716,471,827,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058625.16/warc/CC-MAIN-20240523111540-20240523141540-00725.warc.gz | 465,574,903 | 27,302 | GeoGebra Classroom
# Rational Functions with Specific Features - Practice
Use the input box to create a rational function that matches the given information. Each rational function will have
• two "holes" with given x-coordinates
• two roots through points C and D
• two vertical asymptotes
• 0 or 1 horizontal asymptotes (Can it have more than 1?)
You can use the Show Example Function to see a rational function that exhibits the given behavior. (Is the only function?) | 104 | 473 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2024-22 | latest | en | 0.817042 |
https://www.explainxkcd.com/wiki/index.php?title=1076:_Groundhog_Day&diff=110889&oldid=60635&printable=yes | 1,575,877,844,000,000,000 | text/html | crawl-data/CC-MAIN-2019-51/segments/1575540518337.65/warc/CC-MAIN-20191209065626-20191209093626-00455.warc.gz | 688,845,280 | 13,545 | # Difference between revisions of "1076: Groundhog Day"
Groundhog Day Title text: If you closely examine the cosmic background radiation, you can pick up lingering echoes of 'I Got You Babe'.
## Explanation
Groundhog Day is a philosophical comedy film from 1993. The main character Phil, portrayed by Bill Murray, finds himself in a time loop, which forces him to relive the same day (February 2) over and over again. This date is the titular Groundhog Day, which is celebrated in Punxsutawney, Pennsylvania, where the film is set. The folklore ritual consists in removing a groundhog from its burrow. If the sun is shining and the groundhog can see its own shadow, the winter is assumed to continue for six more weeks.
During the course of the film, Phil makes more and more drastic attempts to end the time loop, but not even suicide can prevent his waking up every morning on February 2 with the clock radio on his nightstand invariably playing I Got You Babe by Sonny & Cher. Eventually, his character improves and he finds himself increasingly attached to a woman named Rita (portrayed by Andie MacDowell). The pair gets closer, and in the end they have sex with each other. This breaks the time loop, and Murray's character can finally wake up on February 3. However, this final scene is disputed, as Phil is still wearing the same clothes as the night before. It is therefore left in doubt if they did anything more than literally sleep in the same bed. Randall was apparently not aware of this and apologised for it.
The comic assumes that the loop was indeed not broken, and that Phil and Rita simply had sex night after night for all eternity. It is then stated that not even forever is forever. This can be explained with the mathematical set theory developed by Georg Cantor. Cantor distinguished between transfinite numbers, which are larger than all finite numbers, yet not infinite, and the concept of Absolute Infinity, which he equaled with God. It was a common concern in Cantor's time to preserve the consistency between mathematics and Christian belief. Cantor's philosophical conception of infinity would allow the comic's scenario to eventually reach the transfinite date of February 3.
The last panel references the chronology of the history of the world of Archbishop James Ussher. Ussher deduced the age of the world from the timeline of the Old Testament and calculated the date of Creation to have been nightfall preceding 23 October, 4004 BC. The comic observes that October 23 is exactly 264 days after February 3, which corresponds to the average length of pregnancy. This calculation draws on Ussher's own methodology, which was basically to add the lifespans of the Old Testament genealogy. Although the universe is much older than 6000 years, chronologies like Ussher's can sometimes be found in the arguments of young earth creationism. The comic might therefore be seen as a sideswipe to these theories by introducing Groundhog Day as a possible creation myth. The creation myths of many cultures claim that Earth was born by some sort primordial mother. Here, this role would be assumed by Rita.
The title text refers to the cosmic microwave background radiation, which is often called the lingering sound of the Big Bang and regarded as a strong proof for it. If the universe were indeed the offspring of the film's protagonists, we might hear the faint echo of Murray's radio clock lingering in the cosmic background.
Interestingly, the comic mentions Bill Murray by his own name, and not by his character's (Phil), whereas Andie MacDowell is mentioned as Rita. This could be subconsciously done, since Murray is mostly remembered for his role in this film, although he has had many other successful ones.
## Transcript
Groundhog Day really didn't end that way. When Bill Murray finally slept with Rita, it didn't break the loop.
[Phil Connors and Rita gettin' busy under the covers of his bed.]
They just kept having sex, night after night,
[Bed containing Phil and Rita repeats.]
February 2nd after February 2nd...
[Calendar page repeats.]
..forever
But nothing is forever. Not even forever
And the day after that sexual infinity
[Calendar page shows Feb 3.]
was February 3rd.
264 days later (the length of a pregnancy) was October 23rd —
[An enormous explosion in space.]
Bishop Ussher's date for the birth of our world.
# Discussion
If the world is stuck in an infinite loop on February 2nd, how can February 3rd happen? Ever? Davidy22[talk] 13:37, 8 January 2013 (UTC)
I guess February 3rd happens on the omega'th day of that infinite loop? Alpha (talk) 05:25, 22 February 2013 (UTC)
It also happens that the date for the beginning of the world is Mole day... 98.235.206.114 05:21, 26 October 2013 (UTC)
The name of the main character ist not the only thing Randall got wrong about the movie... In the Wikipedia article, it is specifically mentioned that Phil and Rita did not have sex that night. 172.68.50.154 19:54, 31 July 2018 (UTC) | 1,117 | 4,996 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2019-51 | latest | en | 0.975276 |
http://m.csmonitor.com/1981/1023/102344.html | 1,477,316,016,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988719566.74/warc/CC-MAIN-20161020183839-00098-ip-10-171-6-4.ec2.internal.warc.gz | 160,759,615 | 10,154 | Close X
# Newest math whiz - Sarah the chimp
Two ethologists at the University of Pennsylvania have added a new element to the ongoing scientific controversy over the intelligence of animals. They believe they have demonstrated primitive mathematical ability in a chimpanzee.
Or as David Premack and Guy Woodruff state it, ''The results (of the experiment) reveal the presence of simple 'proportion' and 'number' concepts in a nonhuman primate.'' The test animal, in other words, seems to grasp the essence of such fractions as one-fourth, one-half, or three-fourths and of numbers such as 1, 2, 3, or 4.
Five chimps were tested - four juveniles and an adult, Sarah. The juveniles did poorly, showing no clear sense of number or proportion. However, Sarah, a veteran of language experiments, did spectacularly well. She made few mistakes, performing significantly better than would be expected were she merely matching objects at random.
In the tests, the chimps were asked to pick which of two alternative objects matched a given test object. The only common property would be proportion or number. A container half full of liquid would have to be matched with half of a round wooden disk, for example. Alternatively, three containers of liquid would be matched with a box containing three blocks of wood. In all these tests, Sarah seemed to have a clear sense of the difference between, say, one-half and three-fourths, or between the numbers three and four. She did not try to match a half-full container with a wooden disk that was three-quarters whole.
Researchers Premack and Woodruff speculate that Sarah used a complex reasoning process to distinguish between the concepts of ''the part'' and ''the whole'' and recognized these as meaningful catagories. This possibility, they say, ''is in keeping not only with previous results showing her inferential ability with other quantitative properties, but also more recent data from our laboratory showing explicitly that she is capable of analogical reasoning.''
This research follows in the tradition of more than a decade of work with chimpanzees and gorillas in which several scientists, including Premack, have taught some of these apes to use different forms of sign or symbol language. In some cases, this has been the common American Sign Language. In other experiments, including those using the chimpanzee Sarah with which Premack works , a simple language using symbols which may be generated with a computer is used. Such work has led to claims of demonstrating language capacity in apes.
The language experiments have their critics who contend that the experimenters overinterpret their results in claiming chimps show true language ability. Such critics assert that the apes are merely imitating their human trainers or showing a conditioned behavior. Harvard University psychologist B. F. Skinner and his colleagues Robert Epstein and Robert P. Lanza, for example, have duplicated some of the ape results with pigeons. Mr. Skinner has suggested that the supposed language capacity of apes may be no more than an example of a classical conditioned response.
It is against the background of such criticism that Premack and Woodruff have carried out their experiments with the mathematical concepts of number and proportion. As they explain in describing their work in the journal Nature, they tried hard to rule out any possibility that the chimps could take their cue from the human experimenters or merely go through a learned routine.
At this writing, it is too early to tell how this work will stand up as others try to repeat the experiment. So far, the most that can be said is that the Pennsylvania ethologists believe they have demonstrated a kind of primitive mathematical conceptualization in one chimpanzee. | 749 | 3,794 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2016-44 | latest | en | 0.970906 |
https://emmadonnan.org/and-pdf/688-mole-concept-molarity-molality-and-normality-pdf-106-782.php | 1,642,678,379,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320301737.47/warc/CC-MAIN-20220120100127-20220120130127-00364.warc.gz | 285,210,075 | 7,257 | # Mole Concept Molarity Molality And Normality Pdf
On Sunday, May 16, 2021 4:55:17 PM
File Name: mole concept molarity molality and normality .zip
Size: 2372Kb
Published: 16.05.2021
The final volume of the solution is.
## Service Unavailable in EU region
Try this quick review. Molarity M is defined as the number of moles of solute per liter of solution. Molality m is defined as the number of moles of solute per kilogram of solvent. Although their spellings are similar, molarity and molality cannot be interchanged. Molarity is a measurement of the moles in the total volume of the solution, whereas molality is a measurement of the moles in relationship to the mass of the solvent. However, when the density of the solvent is significantly different than 1 or the concentration of the solution is high, these changes become much more evident. For a 1 Molar solution, 1 mol of solute is dissolved in CCl 4 until the final volume of solution is 1 L.
## College Chemistry : Molarity, Molality, Normality
Vedantu academic counsellor will be calling you shortly for your Online Counselling session. Related Questions. What is Molarity, Normality and Molality in terms of mole concept? Answer Verified. Hint- In this problem first we will understand the concept of Molarity, Normality and Molality on the basis of their basic definition.
We've updated our Privacy Policy to make it clearer how we use your personal data. We use cookies to provide you with a better experience, read our Cookie Policy. The primary difference between the two comes down to mass versus volume. The molality describes the moles of a solute in relation to the mass of a solvent, while the molarity is concerned with the moles of a solute in relation to the volume of a solution. Read on to learn more about molarity and molality, including their definitions, equations, and a comparison of the two terms. Molality m , or molal concentration, is the amount of a substance dissolved in a certain mass of solvent. It is defined as the moles of a solute per kilograms of a solvent.
## 8.1: Solutions and their Concentrations
The concentration of a solution is the measure of the composition of a solution. For a given solution, the amount of solute dissolved in a unit volume of solution or a unit volume of solvent is called the concentration of the solution. It can be expressed either qualitatively or quantitatively. For example, qualitatively we can say that the solution is dilute i.
Example — Calculate the molality of the solution. Calculate the molarity and molality of the solution. Calculate molality and mole fraction of sugar in the syrup.
Concentration is a very common concept used in chemistry and related fields. It is the measure of how much of a given substance there is mixed with another substance. This can apply to any sort of chemical mixture, but most frequently is used in relation to solutions, where it refers to the amount of solute dissolved in a solvent.
#### 9 replies on “Numerical Problems on Molality”
Thanks for visiting our website. Our aim is to help students learn subjects like physics, maths and science for students in school , college and those preparing for competitive exams. All right reserved. All material given in this website is a property of physicscatalyst. First ,we need to calculate the Molar Mass of the solute. Solution 2 gm Glucose in ml sol.
Thanks for visiting our website. Our aim is to help students learn subjects like physics, maths and science for students in school , college and those preparing for competitive exams. All right reserved. All material given in this website is a property of physicscatalyst. We are going to see another measure of concentration called Normality. Normality is defined as the number of gram equivalent present in per litre solution.
Vedantu academic counsellor will be calling you shortly for your Online Counselling session. Related Questions. What is Molarity, Normality and Molality in terms of mole concept? Answer Verified. Hint- In this problem first we will understand the concept of Molarity, Normality and Molality on the basis of their basic definition.
Solutions are homogeneous single-phase mixtures of two or more components. For convenience, we often refer to the majority component as the solvent ; minority components are solutes ; there is really no fundamental distinction between them. Solutions play a very important role in Chemistry because they allow intimate and varied encounters between molecules of different kinds, a condition that is essential for rapid chemical reactions to occur.
Он задерживается. ГЛАВА 16 - Кольцо? - не веря своим ушам, переспросила Сьюзан. - С руки Танкадо исчезло кольцо.
Возвращайся домой. Прямо. - Встретимся в Стоун-Мэнор. Она кивнула, и из ее глаз потекли слезы. - Договорились.
Слишком поздно, - сказал Стратмор. Он глубоко вздохнул. - Сегодня утром Энсея Танкадо нашли мертвым в городе Севилья, в Испании.
В этом их слабость - вы можете путем скрещивания отправить их в небытие, если, конечно, знаете, что делаете. Увы, у этой программы такого тщеславия нет, у нее нет инстинкта продолжения рода. Она бесхитростна и целеустремленна, и когда достигнет своей цели, то скорее всего совершит цифровое самоубийство.
### Statistical and thermal physics harvey gould and jan tobochnik pdf
30.11.2020 at 02:32
### Cognition theories and applications pdf
27.12.2020 at 08:52
### List of vitamins and their sources and deficiency diseases pdf
23.01.2021 at 16:05
1. Sandor S.
The modern definition of a mole is as follows: Exactly one mole represents the number of carbon atoms in exactly 12 grams of the carbon
17.05.2021 at 05:15 Reply
2. Genciana S.
The normality of permanganate ion is five times its molarity, because MnO4. - ion accepts 5 Molality (m) = #moles of solute/#kilograms of solvent. 5. Osmolarity.
18.05.2021 at 02:08 Reply
3. Tommy W.
remains independent of temperature? (, 1M). (a) Molarity (b) Normality (c) Formality (d) Molality. A molal solution is one that contains one mole of solute.
19.05.2021 at 09:29 Reply
## Subscribe
Subscribe Now To Get Daily Updates | 1,488 | 6,141 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2022-05 | latest | en | 0.927615 |
https://www.doubtnut.com/qna/51239192 | 1,721,415,941,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514917.3/warc/CC-MAIN-20240719170235-20240719200235-00889.warc.gz | 650,761,718 | 32,265 | # The following results show the number of workers and the wages paid to them in two factories F1andF2. Which factory has more variation in wages?
Video Solution
Text Solution
Verified by Experts
|
Step by step video & image solution for The following results show the number of workers and the wages paid to them in two factories F_(1) and F_(2). Which factory has more variation in wages? by Maths experts to help you in doubts & scoring excellent marks in Class 11 exams.
Updated on:21/07/2023
### Knowledge Check
• Question 1 - Select One
## The average of daily wages for the workers of the two factories combined is
A222
B315
C4.10
Dnone of these
• Question 2 - Select One
## The following bar graph shows me sales (in thousands) of motor bikes by the factories, F1,F2, F3 and F4 in 2017, 2018 and 2019. Which factory had the least number of sales across all the years?
AF1
BF3
CF2
DF4
• Question 3 - Select One
## The following bar graph shows the sales (in thousands J of motor bikes bv the factories, F1, F2, F3 and F4 in 2017, 2018 and 2019. Which factory had the lushest sales across all the years?
AF3
BFl
CF4
DF2
Doubtnut is No.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP Board, Bihar Board, Rajasthan Board, MP Board, Telangana Board etc
NCERT solutions for CBSE and other state boards is a key requirement for students. Doubtnut helps with homework, doubts and solutions to all the questions. It has helped students get under AIR 100 in NEET & IIT JEE. Get PDF and video solutions of IIT-JEE Mains & Advanced previous year papers, NEET previous year papers, NCERT books for classes 6 to 12, CBSE, Pathfinder Publications, RD Sharma, RS Aggarwal, Manohar Ray, Cengage books for boards and competitive exams.
Doubtnut is the perfect NEET and IIT JEE preparation App. Get solutions for NEET and IIT JEE previous years papers, along with chapter wise NEET MCQ solutions. Get all the study material in Hindi medium and English medium for IIT JEE and NEET preparation | 560 | 2,126 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.890625 | 3 | CC-MAIN-2024-30 | latest | en | 0.92001 |
syllogismos.github.io | 1,722,852,591,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640436802.18/warc/CC-MAIN-20240805083255-20240805113255-00035.warc.gz | 436,649,988 | 8,193 | My Blog
My learnings and etc.
note: I did this just as an exercise, you get much more from this post.
We are given snapshot of a network and would like to infer which which interactions among existing members are likely to occur in the near future or which existing interactions are we missing. The challenge is to effectively combine the information from the network structure with rich node and edge attribute data.
Supervised Random Walks:
This repository is the implementation of Link prediction based on the paper Supervised Random Walks by Lars Backstrom et al. The essence of which is that we combine the information from the network structure with the node and edge level attributes using Supervised Random Walks. We achieve this by using these attributes to guide a random walk on the graph. We formulate a supervised learning task where the goal is to learn a function that assigns strengths to edges in the network such that a random walker is more likely to visit the nodes to which new links will be created in the future. We develop an efficient training algorithm to directly learn the edge strength estimation function.
Problem Description:
We are given a directed graph G(V,E), a node s and a set of candidates to which s could create an edge. We label nodes to which s creates edges in the future as destination nodes D = {d1,..,dk}, while we call the other nodes to which s does not create edges no-link nodes L = {l1,..,ln}. We label candidate nodes with a set C = D union L. D are positive training examples and L are negative training examples. We can generalize this to multiple instances of s, D, L. Each node and each edge in G is further described with a set of features. We assume that each edge (u,v) has a corresponding feature vector psiuv that describes u and v and the interaction attributes.
For each edge (u,v) in G we compute the strength auv = fw(psiuv). Function fw parameterized by w takes the edge feature vector psiuv as input and computes the corresponding edge strength auv that models the random walk transition probability. It is exactly the function fw(psi) we learn in the training phase of the algorithm.
To predict new edges to s, first edge strengths of all edges are calculated using fw. Then random walk with restarts is run from s. The stationary distribution p of random walk assigns each node u a probability pu. Nodes are ordered by pu and top ranked nodes are predicted as future destination nodes to s. The task is to learn the parameters w of function fw(psiuv) that assigns each edge a transition probability. One can think of the weights auv as edge strengths and the random walk is more likely to traverse edges of high strength and thus nodes connected to node s via paths of strong edges will likely to be visited by the random walk and will thus rank higher.
The optimization problem:
The training data contains information that source node s will create edges to node d subset D and not l subset L. So we set parameters w of the function fw(psiuv) so that it will assign edge weights auv in such a way that the random walk will be more likely to visit nodes in D than L, i.e., pl < pd for each d subset D and l subset L. And thus we define the optimization problem as follows.
where p is the vector of pagerank scores. Pagerank scores pi depend on edge strength on auv and thus actually depend on fw(psiuv) which is parameterized by w. The above equation (1) simply states that we want to find w such that the pagerank score of nodes in D will be greater than the scores of nodes in L. We prefer the shortest w parameters simply for the sake of regularization. But the above equation is the “hard” version of the optimization problem. However it is unlikely that a solution satisfying all the constraints exist. We make the optimization problem “soft” by introducing a loss function that penalizes the violated constraints. Now the optimization problem becomes,
where lambda is the regularization parameter that trades off between the complexity(norm of w) for the fit of the model(how much the constraints can be violated). And h(.) is a loss function that assigns a non-negative penalty according to the difference of the scores pl-pd. h(.) = 0 if pl < pd as the constraint is not violated and h(.) > 0 if pl > pd
Solving the optimization problem:
First we need to establish connection between the parameters w and the random walk scores p. Then we show how to obtain partial derivatives of the loss function and p with respect to w and then perform gradient descent to obtain optimal values of w and minimize loss. We build a random walk stochastic transition matrix Q from the edge strengths auv calculated from fw(psiuv).
To obtain the final random walk transition probability matrix Q, we also incorporate the restart probability alpha, i.e., the probability with which the random walk jumps back to seed node s, and thus “restarts”.
each entry Quv deļ¬nes the conditional probability that a walk will traverse edge (u, v) given that it is currently at node u. The vector p is the stationary distribution of the Random Walk with restarts(also known as Personalized Page Rank), and is the solution to the following eigen vector equation.
The above equation establishes the connection between page rank scores p and the parameters w via the random walk transition probability matrix Q. Our goal now is to minimize the soft version of the loss function(eq. 2) with respect to parameter vector w. We do this by obtaining the gradient of F(w) with respect to w, and then performing gradient based optimization method to find w that minimize F(w). This gets complicated due to the fact that equation 4 is recursive. For this we introduce deltald = pl-pd and then we can write the derivative
and then we can write the derivative of F(w) as follows
For commonly used loss functions h(.) it is easy to calculate derivative, but it is not clear how to obtain partial derivatives of p wrt w. p is the principle eigen vector of matrix Q. The above eigen vector equation can also be written as follows.
and taking the derivatives now gives
above pu and its partial derivative are entangled in the equation, however we compute the above values iteratively as follows
we initialize the vector p as 1/|V| and all its derivatives as zeroes before the iteration starts and terminates the recursion till the p and its derivatives converge for an epsilon say 10e-12. To solve equation 4, we need partial derivative of Qju, this calculation is straight forward. When (j,u) subset E derivative of Qju is
and derivative of Qju is zero if edge (j,u) is not a subset of E.
My Implementation:
We are given a huge network with existing connections. When predicting future link of a particular node, we consider that s, and the graph G(E,V) is Here we explain how each helper function and main functions implements the above algorithm..
This is a temporary function specific to the facebook data that generates Features of each edge from a given adjacency matrix. For other problems this function must be replaced with something that generates feature vector for each edge based on graph G(E,V) and node, edge attributes. For an network with n nodes this function returns n x n x m matrix, where m is the size of parameter vector w(sometimes m+1)
• arguments:
• Adjacency matrix, node attributes, edge attributes
• returns:
• psi size(nxnxm)
FeaturesToEdgeStrength.m:
This function takes the feature matrix (psi) and the parameter vector (w) as arguments to return edge strength (A) and partial derivative of edge strength wrt to each parameter(dA). We also compute partial derivative of edge strength to make further calculations easier. We can vary edge strength function in future implementations, in this we used sigmod(w x psiuv) as edge strength function.
• arguments:
• psi size(nxnxm)
• w size(1xm)
• returns:
• A size(nxn)
• dA size(nxnxm)
EdgeStrengthToTransitionProbability.m
This function takes the edge strength matrix A and alpha to compute transition probability matrix Q.
• arguments:
• A size(nxn)
• alpha size(1x1)
• returns:
• Q size(nxn)
EdgeStrengthToPartialdiffTransition.m
This function computes partial derivative of transition probability matrix from A, dA and alpha
• arguments:
• A size(nxn)
• dA size(nxnxm)
• alpha size(1x1)
• returns:
• dQ size(nxnxm)
LossFunction.m
This function takes as input parameters, adjacency matrix of the network, lambda and alpha. * We get edge strength matrix and its partial derivatives from features and parameters * We get transition probability and partial derivatives of it from A and dA * We get stationary probabilities from Q and dQ * Compute cost and gradient from the above variables, we can use various functions as loss function h(.). Here we used wilcoxon loss function.
• arguments:
• param: parameters of the edge strength function, size(1,m)
• features: features of all the edges in the network, size(n,n,m)
• d: binary vector representing destination nodes, size(1,n)
• lambda: regularization parameter, size(1,1)
• alpha: random restart parameter, size(1,1)
• returns:
• J: loss, size(1,1)
fmincg.m
We use this function to do the minimization of the loss function, given a starting point for the parameters, and the function that computes loss and gradients for a given parameter vector. This is similar to fminunc function available in octave.
GetNodesFromParam.m
This function calculates the closest nodes to the root node given the parameters obtained after training.
• arguments:
• param: parameters, size(m,1)
• features: feature matrix, size(n,n,m)
• d: binary vector representing the destination nodes, size(1,n)
• alpha: alpha value used in calculation of Q, size(1,1)
• y: number of nodes to output
• returns:
• nodes: output nodes, size(1,y)
• P: probabilities of the nodes, size(1,n)
How to Train:
Here I will show how to train the supervised random walk for a given root node s and edge features matrix psi. I’m not showing how to obtain the edge features. Given the network structure, node and edge attributes etc, you can experiment with different feature extraction techniques. Here we have psi | 2,185 | 10,189 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2024-33 | latest | en | 0.925106 |
https://gmatclub.com/forum/calling-all-babson-college-applicnts-2015-intk-class-of-176148-80.html | 1,611,300,677,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703529128.47/warc/CC-MAIN-20210122051338-20210122081338-00603.warc.gz | 357,978,034 | 64,432 | GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video
It is currently 21 Jan 2021, 23:31
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Intern
Joined: 19 Oct 2014
Posts: 32
Concentration: Strategy, General Management
GMAT 1: 660 Q48 V33
Intern
Joined: 27 Jul 2013
Posts: 9
Location: Brazil
Concentration: Entrepreneurship, Technology
GMAT Date: 11-07-2014
GPA: 3
Intern
Joined: 06 Mar 2013
Posts: 8
Location: India
Concentration: Technology, Entrepreneurship
GMAT 1: 650 Q47 V33
GMAT 2: 680 Q50 V32
GPA: 3.6
WE:Information Technology (Computer Software)
Intern
Joined: 27 Dec 2012
Posts: 1
Intern
Joined: 22 Nov 2013
Posts: 46
Intern
Joined: 10 Mar 2015
Posts: 3
GMAT 1: 730 Q51 V38
Intern
Joined: 27 Feb 2015
Posts: 22
Concentration: Entrepreneurship, General Management
GMAT 1: 730 Q47 V44
Manager
Joined: 13 Apr 2014
Status:[Heavy Breathing]
Posts: 134
Location: India
GMAT 1: 680 Q48 V35
GMAT 2: 710 Q48 V38
GPA: 3.5
WE:Design (Manufacturing)
Intern
Joined: 10 Mar 2015
Posts: 3
GMAT 1: 730 Q51 V38
Manager
Joined: 13 Apr 2014
Status:[Heavy Breathing]
Posts: 134
Location: India
GMAT 1: 680 Q48 V35
GMAT 2: 710 Q48 V38
GPA: 3.5
WE:Design (Manufacturing)
Manager
Joined: 29 Aug 2014
Posts: 113
GMAT 1: 730 Q51 V38
Intern
Joined: 12 Mar 2015
Posts: 1
Intern
Joined: 19 Oct 2014
Posts: 32
Concentration: Strategy, General Management
GMAT 1: 660 Q48 V33
Intern
Joined: 02 Feb 2013
Posts: 16
Concentration: General Management, Entrepreneurship
GMAT Date: 06-30-2014
Intern
Joined: 28 Oct 2013
Posts: 27
Location: United States
Concentration: Entrepreneurship, Finance
GPA: 3.2
WE:Information Technology (Education)
Intern
Joined: 28 Oct 2013
Posts: 27
Location: United States
Concentration: Entrepreneurship, Finance
GPA: 3.2
WE:Information Technology (Education)
Intern
Joined: 02 Feb 2013
Posts: 16
Concentration: General Management, Entrepreneurship
GMAT Date: 06-30-2014
Intern
Joined: 22 Nov 2013
Posts: 46
Intern
Joined: 02 Feb 2013
Posts: 16
Concentration: General Management, Entrepreneurship
GMAT Date: 06-30-2014 | 859 | 2,634 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2021-04 | latest | en | 0.824597 |
https://www.allmath.com/text-to-hex.php | 1,720,918,986,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514527.38/warc/CC-MAIN-20240714002551-20240714032551-00385.warc.gz | 577,922,254 | 10,901 | # Text to Hex Converter
To use Text to Hex Converter, Enter the text, and hit calculate button
Error: UnKnown number
Swap
Give Us Feedback
## Text to Hex Converter
Text to Hex Converter is used to convert text into a Hex system. It just gives the results of inputs simply click on the calculate button and answer in just a few seconds.
## What is Hex System?
This system contains “16” characters overall and it ranges from “0 to 15”. The symbols are used in the hexadecimal system from “0 to 9” digits and alphabets letters “A to F”. Where “A” represents the value “10” in digit form, “B” represents the value “11” and so on “F” represents the value “15”.
## Examples
Example 1:
Convert the text “Hello world” into the Hex code.
Solution
Step 1:
Write the “text” which converts into “Hex code” in the input box.
Hello world” is the input that converts into Hex.
Step 2:
The “Text” is converted easily into “Hex code” by clicking on the calculate button.
48656c6c6f20776f726c64
So, {48656c6c6f20776f726c64} is the answer in the Hex code.
Example 2:
Convert the text “lemon soda” into the Hex system.
Solution
Using our Text to Hex converter, you can easily convert any text into Hex code.
Step 1:
Write the “text” that you want to convert into “Hex code” in the input box.
Lemon soda” is the input that converts into Hex.
Step 2:
The “Text” is converted easily into “Hex code” by clicking on the calculate button.
6c656d6f6e20736f6461
So, {6c656d6f6e20736f6461} is the answer in the Hex code. | 414 | 1,521 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2024-30 | latest | en | 0.810824 |
https://rdrr.io/rforge/Epi/man/N2Y.html | 1,524,400,967,000,000,000 | text/html | crawl-data/CC-MAIN-2018-17/segments/1524125945596.11/warc/CC-MAIN-20180422115536-20180422135536-00039.warc.gz | 677,104,151 | 16,017 | # N2Y: Create risk time ("Person-Years") in Lexis triangles from... In Epi: A Package for Statistical Analysis in Epidemiology
## Description
Data on population size at equidistant dates and age-classes are used to estimate person-time at risk in Lexis-triangles, i.e. classes classified by age, period AND cohort (date of birth). Only works for data where age-classes have the same width as the period-intervals.
## Usage
1 2 3 N2Y( A, P, N, data = NULL, return.dfr = TRUE)
## Arguments
A Name of the age-variable, which should be numeric, corresponding to the left endpoints of the age intervals. P Name of the period-variable, which should be numeric, corresponding to the date of population count. N The population size at date P in age class A. data A data frame in which arguments are interpreted. return.dfr Logical. Should the results be returned as a data frame (default TRUE) or as a table.
## Details
The calculation of the risk time from the population figures is done as described in: B. Carstensen: Age-Period-Cohort models for the Lexis diagram. Statistics in Medicine, 26: 3018-3045, 2007.
The number of periods in the result is one less than the number of dates (nP=length(table(P))) in the input, so the number of distinct values is 2*(nP-1), because the P in the output is coded differently for upper and lower Lexis triangles.
The number of age-classes is the same as in the input. In the paper "Age-Period-Cohort models for the Lexis diagram" I suggest that the risk time in the lower triangles in the first age-class and in the upper triangles in the last age-class are computed so that the total risk time in the age-class corresponds to the average of the two population figures for the age-class at either end of the period. This is the method used.
## Value
A data frame with variables A, P and Y, representing the mean age and period in the Lexis triangles and the person-time in them, respectively. The person-time is in units of the distance between population count dates.
If res.dfr=FALSE a three-way table classified by the left end point of the age-classes and the periods and a factor wh taking the values up and lo corresponding to upper (early cohort) and lower (late cohort) Lexis triangles.
## Author(s)
Bendix Carstensen, BendixCarstensen.com
## References
B. Carstensen: Age-Period-Cohort models for the Lexis diagram. Statistics in Medicine, 26: 3018-3045, 2007. | 580 | 2,422 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2018-17 | latest | en | 0.865522 |
https://jmservera.com/solve-for-w-3w-116w-32w/ | 1,669,820,045,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710764.12/warc/CC-MAIN-20221130124353-20221130154353-00755.warc.gz | 366,221,421 | 12,986 | # Solve for w 3(w-1)+1=6w-3(2+w)
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Simplify .
Simplify each term.
Apply the distributive property.
Multiply by .
Subtract from .
Simplify .
Simplify each term.
Apply the distributive property.
Multiply by .
Move all terms containing to the left side of the equation.
Subtract from both sides of the equation.
Combine the opposite terms in .
Subtract from .
Subtract from .
Since , there are no solutions.
No solution
Solve for w 3(w-1)+1=6w-3(2+w)
## Our Professionals
### Lydia Fran
#### We are MathExperts
Solve all your Math Problems: https://elanyachtselection.com/
Scroll to top | 180 | 691 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.40625 | 3 | CC-MAIN-2022-49 | latest | en | 0.854715 |
http://www.enotes.com/homework-help/how-many-arrangements-row-all-ten-bricks-question-212461 | 1,477,422,927,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988720356.2/warc/CC-MAIN-20161020183840-00247-ip-10-171-6-4.ec2.internal.warc.gz | 428,969,147 | 9,509 | # In how many of the arrangements in a row of all ten bricks in Question 12 are: (a) the three bricks separated from each other (b) just 2 of the red bricks next to each other
neela | High School Teacher | (Level 3) Valedictorian
Posted on
a)
The total number of ways of arranging 10 bricks in a row = 10P10 =10!.
The number of ways the 3 bricks staying together consecutively in all arrangements is as good as treating those 3 bricks as a one single block for pemuting purpose.This block together with other 7 bricks, we have to arrange in 8 places in 8places . This is possible in 8P8 = 8!. But within the block 3 paticular bricks, they could be arranged in 3! ways. Thus the number of arrangements of particular blocks are together = 3!*8P8 = 3!*8!
Therefore 10! - (3!*8!) is the number of ways of arrangements where particular 3 blocks are not together.
ii)
Particular 2 red bricks are next to each other- with this condtion we treat the two particular red bricks as one block and the remaing 8 bricks as different. So we arrange the 9 different things on a row. This is possible in 9P9 = 9!. If the two red bricks are also distinct between themselves, then they could be arranged 2! ways between themseves within the block. Then the arrangments become 9!*2!
Therefore the number of arrangements that 2particular red bricks are together = 9! (red are not distinct between themselves). Or 9!*2! (if reds are distinct between themselves). | 382 | 1,455 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.34375 | 4 | CC-MAIN-2016-44 | latest | en | 0.906469 |
https://just4once.gitbooks.io/leetcode-notes/content/leetcode/linked-list/109-convert-sorted-list-to-binary-search-tree.html | 1,674,922,823,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499646.23/warc/CC-MAIN-20230128153513-20230128183513-00815.warc.gz | 351,552,476 | 13,247 | # 109-convert-sorted-list-to-binary-search-tree
## Question
https://leetcode.com/problems/convert-sorted-list-to-binary-search-tree/description/
Given a singly linked list where elements are sorted in ascending order, convert it to a height balanced BST.
For this problem, a height-balanced binary tree is defined as a binary tree in which the depth of the two subtrees of every node never differ by more than 1.
Example:
Given the sorted linked list: [-10,-3,0,5,9],
One possible answer is: [0,-3,9,-10,null,5], which represents the following height balanced BST:
0
/ \
-3 9
/ /
-10 5
## Thought Process
1. List
1. Add the value to the list and create left and right part
2. Time complexity O(n)
3. Space complexity O(n)
2. asd
## Solution
class Solution {
List<Integer> list = new ArrayList<>();
while(cur != null){
cur = cur.next;
}
return buildNode(list, 0, list.size() - 1);
}
public TreeNode buildNode(List<Integer> list, int lo, int hi){
if (lo > hi) return null;
int mid = lo + (hi - lo)/2;
TreeNode node = new TreeNode(list.get(mid));
node.left = buildNode(list, lo, mid - 1);
node.right = buildNode(list, mid + 1, hi);
return node;
}
}
class Solution {
}
public TreeNode toBST(ListNode head, ListNode tail){
while (fast != tail && fast.next != tail) {
slow = slow.next;
fast = fast.next.next;
}
TreeNode root = new TreeNode(slow.val); | 362 | 1,362 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.71875 | 4 | CC-MAIN-2023-06 | longest | en | 0.628373 |
https://www.experts-exchange.com/questions/28346839/Compare-and-calculate-the-number-of-days-between-records-date-field-and-resetting-the-compare-calculation-for-each-unique-ID.html | 1,526,951,211,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794864572.13/warc/CC-MAIN-20180521235548-20180522015548-00397.warc.gz | 756,857,116 | 16,271 | • Status: Solved
• Priority: Medium
• Security: Public
• Views: 468
# Compare and calculate the number of days between records (date field), and resetting the compare/calculation for each unique ID
Hello, Assistance with comparing and calculating the number of days between records, then resetting for each unique ID is much needed! Example:
P-ID I-ID TestDate Difference
1 100 3/2/2012 0
2 101 2/14/2012 0
3 102 2/29/2012 0
4 103 8/29/2012 0
5 104 8/20/2012 0
5 105 10/25/2012 66
6 106 2/29/2012 0
6 107 2/29/2012 0
7 108 7/20/2012 0
8 109 7/11/2012 0
9 110 4/3/2012 0
10 111 9/11/2012 0
11 112 8/7/2012 0
12 113 3/5/2012 0
13 114 1/24/2012 0
14 115 3/19/2012 0
15 116 1/17/2012 0
16 117 4/6/2012 0
16 118 6/13/2012 68
17 119 2/14/2012 0
18 120 10/2/2012 0
19 121 1/25/2012 0
19 122 5/4/2012 24
My data source is an unupdatable query. Much thanks in advance!!
0
jaguar5554
• 2
2 Solutions
IT ConsultantCommented:
See attached Excel file.
Basically it checks if 2 consecutive values in the first column are equal, it calculates the no of days between the dates in the 3rd column.
HTH,
Dan
Q-28346839.xlsx
0
Commented:
try this query
SELECT urTable.[p-id], urTable.[i-id], urTable.testDate, DateDiff("d",(select min(b.testDate) from urTable as B where B.[p-id]=urtable.[p-id] and b.[i-id]<=urtable.[i-id]),[testdate]) AS difference
FROM urTable;
0
IT ConsultantCommented:
Btw, if you want to have more than 2 consecutive ID's, modify the formula as follows:
=IF(\$A2=\$A3, DAYS(\$C3,\$C2) + \$D2,0)
Basically, this will add the number of days to the previous value, so in case of multiple identical IDs the last value will be the total number of days for that ID. If you only have max 2 consecutive IDs, then the formula is identical in function to the one in the sheet.
0
Experts! Both of the solutions work perfectly, and both deserve the full 500 points. Unfortunately, I am limited to only 500 points, so I gave the most I could to each (250 points). I selected the query as the best solution because I'm working in an MS Access database; however, the MS Excel solution will certainly be implemented into other of my data projects. Thank you Thank you Thank you.
0
Question has a verified solution.
Are you are experiencing a similar issue? Get a personalized answer when you ask a related question.
Have a better answer? Share it in a comment.
## Featured Post
• 2
Tackle projects and never again get stuck behind a technical roadblock. | 897 | 2,917 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.015625 | 3 | CC-MAIN-2018-22 | latest | en | 0.378736 |
https://www.coursehero.com/file/7774292/etheexercisepriceThereforein-ordertomaximizeyourprofityouwanttominimizethepriceofthestockandso/ | 1,524,409,574,000,000,000 | text/html | crawl-data/CC-MAIN-2018-17/segments/1524125945604.91/warc/CC-MAIN-20180422135010-20180422155010-00069.warc.gz | 752,257,186 | 26,047 | {[ promptMessage ]}
Bookmark it
{[ promptMessage ]}
CHAPTER 21
# Etheexercisepricethereforein
This preview shows page 1. Sign up to view the full content.
This is the end of the preview. Sign up to access the rest of the document.
Unformatted text preview: Option values are calculated as follows: 1. 2. 3. d. (i) To replicate a call, buy 0.89 shares and borrow: [(0.89 ´ 200) 52.63] = \$125.37 (ii) To replicate a call, buy one share and borrow: [(1.0 ´ 200) 70.53] = \$129.47 (iii) To replicate a call, buy 0.37 shares and borrow: [(0.37 ´ 200) 14.11] = \$59.89 11. To hold time to expiration constant, we will look at a simple oneperiod binomial problem with different starting stock prices. Here are the possible stock prices: Now consider the effect on option delta: Current Stock Price Option Deltas 100 110 Inthemoney (EX = 60) 140/150 = 0.93 160/165 = 0.97 Atthemoney (EX = 100) 100/150 = 0.67 120/165 = 0.73 Outofthemoney (EX = 140) 60/150 = 0.40 80/165 = 0.48 Note that, for a given difference in stock price, outofthemoney options result in a larger change in the option delta. If you want to minimize the number of times you rebalance an option hedge, use inthemoney options. 12. a. The call option. (You would delay the exercise of the put until after the dividend has been paid and the stock price has dropped.) b. The put option. (You never exercise a call if the stock price is be...
View Full Document
{[ snackBarMessage ]} | 449 | 1,467 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.75 | 4 | CC-MAIN-2018-17 | latest | en | 0.787765 |
http://everything.explained.today/Graham%27s_number/ | 1,685,302,053,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224644506.21/warc/CC-MAIN-20230528182446-20230528212446-00062.warc.gz | 14,571,030 | 8,776 | # Graham's number explained
Graham's number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is much larger than many other large numbers such as Skewes's number and Moser's number, both of which are in turn much larger than a googolplex. As with these, it is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe. Thus Graham's number cannot be expressed even by physical universe-scale power towers of the form
a
b
c
⋅
.
However, Graham's number can be explicitly given by computable recursive formulas using Knuth's up-arrow notation or equivalent, as was done by Ronald Graham, the number's namesake. As there is a recursive formula to define it, it is much smaller than typical busy beaver numbers. Though too large to be computed in full, the sequence of digits of Graham's number can be computed explicitly via simple algorithms; the last 13 digits are ...7262464195387. With Knuth's up-arrow notation, Graham's number is
g64
, where
$g_n = \left\ \text$
where the number of arrows in each subsequent layer is specified by the value of the next layer below it; that is,
$G = g_,$ where
g1=3\uparrow\uparrow\uparrow\uparrow3,
gn=
gn-1 3\uparrow
3,
and where a superscript on an up-arrow indicates how many arrows there are. In other words, G is calculated in 64 steps: the first step is to calculate g1 with four up-arrows between 3s; the second step is to calculate g2 with g1 up-arrows between 3s; the third step is to calculate g3 with g2 up-arrows between 3s; and so on, until finally calculating G = g64 with g63 up-arrows between 3s.
Equivalently,$G = f^(4),\textf(n) = 3 \uparrow^n 3,$
and the superscript on f indicates an iteration of the function, e.g.,
f4(n)=f(f(f(f(n))))
. Expressed in terms of the family of hyperoperations
H0,H1,H2,
, the function f is the particular sequence
f(n)=Hn+2(3,3)
, which is a version of the rapidly growing Ackermann function A(n, n). (In fact,
f(n)>A(n,n)
for all n.) The function f can also be expressed in Conway chained arrow notation as
f(n)=33n
, and this notation also provides the following bounds on G:
$3\rightarrow 3\rightarrow 64\rightarrow 2 < G < 3\rightarrow 3\rightarrow 65\rightarrow 2.$
### Magnitude
To convey the difficulty of appreciating the enormous size of Graham's number, it may be helpful to express—in terms of exponentiation alone—just the first term (g1) of the rapidly growing 64-term sequence. First, in terms of tetration (
\uparrow\uparrow
) alone:$g_1 = 3 \uparrow \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow \uparrow (3 \uparrow \uparrow \uparrow 3) = 3 \uparrow\uparrow (3 \uparrow\uparrow (3 \uparrow\uparrow \ \dots \ (3 \uparrow\uparrow 3) \dots))$
where the number of 3s in the expression on the right is$3 \uparrow \uparrow \uparrow 3 = 3 \uparrow \uparrow (3 \uparrow \uparrow 3).$
Now each tetration (
\uparrow\uparrow
) operation reduces to a power tower (
\uparrow
) according to the definition$3 \uparrow\uparrow X = 3 \uparrow (3 \uparrow (3 \uparrow \dots (3 \uparrow 3) \dots)) = 3^$ where there are X 3s.
Thus,$g_1 = 3 \uparrow\uparrow (3 \uparrow\uparrow (3 \uparrow\uparrow \ \dots \ (3 \uparrow\uparrow 3) \dots)) \quad \text \quad 3 \uparrow \uparrow (3 \uparrow \uparrow 3)$
becomes, solely in terms of repeated "exponentiation towers",$g_1 = \left. \begin3^\end \right \} \left. \begin3^\end \right \} \dots \left. \begin3^\end \right \} 3 \quad \text \quad \left. \begin3^\end \right \} \left. \begin3^\end \right \} 3$
and where the number of 3s in each tower, starting from the leftmost tower, is specified by the value of the next tower to the right.
In other words, g1 is computed by first calculating the number of towers,
n=3\uparrow(3\uparrow(3 ... \uparrow3))
(where the number of 3s is
3\uparrow(3\uparrow3)=7625597484987
), and then computing the nth tower in the following sequence:
1st tower: 3 2nd tower: 3↑3↑3 (number of 3s is 3) = 7625597484987 3rd tower: 3↑3↑3↑3↑...↑3 (number of 3s is 7625597484987) = … ⋮ g1 = nth tower: 3↑3↑3↑3↑3↑3↑3↑...↑3 (number of 3s is given by the n-1th tower)
where the number of 3s in each successive tower is given by the tower just before it. Note that the result of calculating the third tower is the value of n, the number of towers for g1.
The magnitude of this first term, g1, is so large that it is practically incomprehensible, even though the above display is relatively easy to comprehend. Even n, the mere number of towers in this formula for g1, is far greater than the number of Planck volumes (roughly 10185 of them) into which one can imagine subdividing the observable universe. And after this first term, still another 63 terms remain in the rapidly growing g sequence before Graham's number G = g64 is reached. To illustrate just how fast this sequence grows, while g1 is equal to
3\uparrow\uparrow\uparrow\uparrow3
with only four up arrows, the number of up arrows in g2 is this incomprehensibly large number g1.
## Rightmost decimal digits
Graham's number is a "power tower" of the form 3↑↑n (with a very large value of n), so its rightmost decimal digits must satisfy certain properties common to all such towers. One of these properties is that all such towers of height greater than d (say), have the same sequence of d rightmost decimal digits. This is a special case of a more general property: The d rightmost decimal digits of all such towers of height greater than d+2, are independent of the topmost "3" in the tower; i.e., the topmost "3" can be changed to any other non-negative integer without affecting the d rightmost digits.
The following table illustrates, for a few values of d, how this happens. For a given height of tower and number of digits d, the full range of d-digit numbers (10d of them) does not occur; instead, a certain smaller subset of values repeats itself in a cycle. The length of the cycle and some of the values (in parentheses) are shown in each cell of this table:
3↑3↑x 3↑3↑3↑x 3↑3↑3↑3↑x 3↑3↑3↑3↑3↑x 1 3↑x ! 4 (1,3,9,7) 2 (3,7) 1 (7) 1 (7) 1 (7) 20 (01,03,…,87,…,67) 4 (03,27,83,87) 2 (27,87) 1 (87) 1 (87) 100 (001,003,…,387,…,667) 20 (003,027,…387,…,587) 4 (027,987,227,387) 2 (987,387) 1 (387)
The particular rightmost d digits that are ultimately shared by all sufficiently tall towers of 3s are in bold text, and can be seen developing as the tower height increases. For any fixed number of digits d (row in the table), the number of values possible for 3
\scriptstyle\uparrow
3↑…3↑x mod 10d, as x ranges over all nonnegative integers, is seen to decrease steadily as the height increases, until eventually reducing the "possibility set" to a single number (colored cells) when the height exceeds d+2.
A simple algorithm[1] for computing these digits may be described as follows: let x = 3, then iterate, d times, the assignment x = 3x mod 10d. Except for omitting any leading 0s, the final value assigned to x (as a base-ten numeral) is then composed of the d rightmost decimal digits of 3↑↑n, for all n > d. (If the final value of x has fewer than d digits, then the required number of leading 0s must be added.)
Let k be the numerousness of these stable digits, which satisfy the congruence relation G(mod 10k)≡[G<sup>G</sup>](mod 10k).
k=t-1, where G(t):=3↑↑t.[2] It follows that, .
The algorithm above produces the following 500 rightmost decimal digits of Graham's number :
...02425950695064738395657479136519351798334535362521 43003540126026771622672160419810652263169355188780 38814483140652526168785095552646051071172000997092 91249544378887496062882911725063001303622934916080 25459461494578871427832350829242102091825896753560 43086993801689249889268099510169055919951195027887 17830837018340236474548882222161573228010132974509 27344594504343300901096928025352751833289884461508 94042482650181938515625357963996189939679054966380 03222348723967018485186439059104575627262464195387
## References
### Bibliography
• Gardner, Martin. Mathematical Games. Scientific American. 237. 5. 18–28. November 1977. subscription. 10.1038/scientificamerican1177-18. 1977SciAm.237e..18G. ; reprinted (revised) in Gardner (2001), cited below.
• Book: Gardner, Martin. 1989. Penrose Tiles to Trapdoor Ciphers. 978-0-88385-521-8. Martin Gardner. Mathematical Association of America. Washington, D.C..
• Book: Gardner, Martin. 2001. The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. 978-0-393-02023-6. Martin Gardner. Norton. New York, NY.
• Graham, R. L. . Rothschild, B. L. . Ramsey's Theorem for n-Parameter Sets . Transactions of the American Mathematical Society . 159 . 257–292 . 1971 . 10.2307/1996010 . 1996010. The explicit formula for N appears on p. 290. This is not the "Graham's number" G published by Martin Gardner.
• Book: Graham, R. L. . Rothschild . B. L. . G-C . Rota . Studies in Combinatorics (MAA Studies in Mathematics) . Mathematical Association of America . 17 . 1978 . 80–99 . Ramsey Theory . 978-0-88385-117-3. On p. 90, in stating "the best available estimate" for the solution, the explicit formula for N is repeated from the 1971 paper. | 2,812 | 9,665 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 9, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2023-23 | latest | en | 0.928034 |
http://kwiznet.com/p/takeQuiz.php?ChapterID=1444&CurriculumID=3&Method=Worksheet&NQ=8&Num=8.7&Type=B | 1,720,943,807,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514551.8/warc/CC-MAIN-20240714063458-20240714093458-00528.warc.gz | 18,444,009 | 3,346 | Name: ___________________Date:___________________
Email us to get an instant 20% discount on highly effective K-12 Math & English kwizNET Programs!
### Grade 3 - Mathematics8.7 Angles
When two rays meet at a point, an angle is formed. The point at which they meet is called the vertex. An angle that measures 90 degrees is called a right angle. An angle that measures less than 90 degrees is called acute angle. An angle that measures more than 90 degrees is called obtuse angle. Squares and rectangles are made up of four right angles. Directions: Answer the following questions. Also draw right angle, acute angle, obtuse angle and name the rays, angles and vertex.
Name: ___________________Date:___________________
### Grade 3 - Mathematics8.7 Angles
Q 1: Whenever two sides of a polygon meet they forma circlea dotan anglea figure Q 2: What is an angle that forms a square corner.abtuseacuteright Q 3: Squares and Rectangles have 4 right angles.TrueFalse Q 4: This polygon has _____angles342 Question 5: This question is available to subscribers only! Question 6: This question is available to subscribers only!
#### Subscription to kwizNET Learning System costs less than \$1 per month & offers the following benefits:
• Unrestricted access to grade appropriate lessons, quizzes, & printable worksheets
• Instant scoring of online quizzes
• Progress tracking and award certificates to keep your student motivated
• Unlimited practice with auto-generated 'WIZ MATH' quizzes
© 2003-2007 kwizNET Learning System LLC. All rights reserved. This material may not be reproduced, displayed, modified or distributed without the express prior written permission of the copyright holder. For permission, contact info@kwizNET.com
For unlimited printable worksheets & more, go to http://www.kwizNET.com. | 398 | 1,808 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2024-30 | latest | en | 0.86041 |
https://www.kidsacademy.mobi/printable-worksheets/online/activities/normal/age-3/knowledge-check/math/ | 1,716,827,836,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059044.17/warc/CC-MAIN-20240527144335-20240527174335-00305.warc.gz | 728,506,630 | 53,715 | # Normal Math worksheets activities for 3-Year-Olds
Elevate your child's mathematical understanding with our compelling collection of Normal Math worksheets activities! Designed meticulously for a range of grades, these activities cover all the fundamental concepts, from basic arithmetic to more complex problem-solving challenges. Each worksheet is crafted to stimulate curiosity, enhance critical thinking, and reinforce essential skills in an engaging and accessible manner. Perfect for both classroom use and at-home learning, our Normal Math worksheets activities offer a balanced mix of fun and education, ensuring your child enjoys every step of their mathematical journey. Dive into our collection today and watch your child's confidence and competence in math soar!
Favorites
Interactive
• Math
• 3
• Interactive
• Normal
## Number Stories One More – Assessment 2 Worksheet
Tracing is a great activity for kids. They can count and trace numbers, recognize animals, and practice drawing on dotted lines. It's entertaining and educational, helping children learn valuable counting skills.
Number Stories One More – Assessment 2 Worksheet
Worksheet
## Multiplication Facts: Assessment 3 Worksheet
Test your kid's maths skills with this easy to use worksheet! Help them check the box that matches the equation in the first part, then read each word problem and underline the right answers to the second part. Assess your child's muliplication knowledge and find out where they need extra help.
Multiplication Facts: Assessment 3 Worksheet
Worksheet
## Word Problems: Assessment 2 Worksheet
This bear-themed worksheet is a great way to test subtraction skills. Have your child read the word problems and match the correct drawing with the answer. It's a fun way to quiz them without them even knowing. Enjoy counting cute snoozing bears! (80 words)
Word Problems: Assessment 2 Worksheet
Worksheet
## Data: Assessment 1 Worksheet
Assess your students' knowledge of measurement words with this worksheet. Have them compare and describe objects/quantities using words like "big", "small", "long", "short", "empty", "full", "heavy" and "light". Encourage further learning by asking students to provide examples of each word meaning.
Data: Assessment 1 Worksheet
Worksheet
Learning Skills
Normal Math worksheets activities are an essential cornerstone in the education of mathematics, providing a structured pathway for students to develop and hone their mathematical skills. These activities are meticulously crafted to cover a wide range of mathematical concepts, from the simplest forms of addition and subtraction to more complex topics such as algebra and geometry. The usefulness of engaging in Normal Math worksheets activities lies in several key areas that contribute significantly to the learning process.
Firstly, repetition and practice are foundational principles in learning mathematics. Normal Math worksheets activities offer a plethora of problems that allow students to practice mathematical operations repeatedly. This constant practice ensures that students can internalize mathematical concepts, leading to better retention and recall. By regularly working through these worksheets, students can solidify their understanding and become more confident in their mathematical abilities.
Secondly, Normal Math worksheets activities provide a structured framework that introduces concepts progressively. This methodical approach is essential for building a strong mathematical foundation. Starting with simpler problems, the activities gradually increase in complexity, ensuring that students are not overwhelmed and can build on their knowledge incrementally.
Moreover, these worksheets are an invaluable tool for individualized learning. Since students can work through the activities at their own pace, they cater to individual learning speeds and needs. This personalization is crucial in a subject like mathematics, where understanding foundational concepts is key to grasping more complex topics.
Finally, Normal Math worksheets activities offer a tangible way for students to track their progress. By completing worksheets and reviewing errors, students can identify areas of strength and weakness. This feedback loop is vital for targeted learning, allowing both students and educators to focus efforts on areas that require more attention.
In conclusion, Normal Math worksheets activities play a pivotal role in mathematics education. Through repeated practice, progressive learning, personalized pacing, and the ability to track progress, these activities equip students with the tools they need to succeed in mathematics. By engaging with these worksheets, students can foster a deeper understanding and appreciation of mathematics, laying down a solid foundation for advanced mathematical studies. | 848 | 4,842 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.078125 | 3 | CC-MAIN-2024-22 | latest | en | 0.882193 |
https://pdfslide.us/documents/show-you-what-heshe-learned-websites-apps-amp-ideas-.html | 1,716,997,398,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059246.67/warc/CC-MAIN-20240529134803-20240529164803-00461.warc.gz | 373,200,225 | 35,743 | 24
MAH Tips: Ask your child about his/her day, what he/she did in math. Ask them to show you what he/she learned. Websites, Apps, & Ideas: Apps: Kakooma (options for + and x), 2048, Picture Math, Youtube Kid (great for K and 1 st ) Math Puzzle of the Month: Brain Teasers One Hundred Forty Five Doors A person was visiting a medieval dungeon with 145 doors and got stuck. Nine, shown by black bars, are locked, but each one will open if before you reach it, you pass through exactly 8 open doors. You don’t have to go through every open door but you do have to go through every cell and all 9 locked doors. If you enter a cell or go through a door a second time, the doors clang shut, trapping you. The person (in the lower right corner cell) had a drawing of the dungeon. He thought a long time before he set out. He went through all the locked doors and escaped through the last, upper left corner one. What was his route? (nctm.org) ~ Mrs. Bushior Visit my school website for more links Math At Home(MAH) April 2015
# show you what he/she learned. Websites, Apps, & Ideas
others
• View
86
0
Embed Size (px)
Citation preview
MAH Tips:
show you what he/she learned.
Websites, Apps, & Ideas:
Apps: Kakooma (options for + and x), 2048,
Picture Math, Youtube Kid (great for K and 1st)
Math Puzzle of the Month:
Brain Teasers
One Hundred Forty Five Doors
A person was visiting a medieval dungeon with 145 doors and got stuck. Nine,
shown by black bars, are locked, but each one will open if before you reach it,
you pass through exactly 8 open doors. You don’t have to go through every open
door but you do have to go through every cell and all 9 locked doors. If you enter
a cell or go through a door a second time, the doors clang shut, trapping you.
The person (in the lower right corner cell) had a drawing of the dungeon. He
thought a long time before he set out. He went through all the locked doors and
escaped through the last, upper left corner one. What was his route? (nctm.org)
~ Mrs. Bushior
Visit my school website for more links
Math At Home(MAH) April 2015
Wednesday, April 01, 2015
Breakfast: Whole Grain Muffin
Lunch: Whole Grain Pasta w/Meatballs, Mixed Green Salad, Peaches, Milk CES Alternate: Egg
Thursday, April 02, 2015
Breakfast: Yogurt & Whole Grain Graham Crackers
Lunch: French Bread Pizza, Fresh Vegetables, Assorted Juice, Milk CES Alternate: Peanut
Butter & Fluff Sandwich
Friday, April 03, 2015
Breakfast: Schools Closed -Good Friday
Monday, April 06, 2015
Breakfast: Whole Grain Oatmeal Bar
Lunch: Hamburg on Whole Grain Roll, Sweet Potato Fries, Peas, Pears, Milk CES Alternate:
Cereal Lunch
Tuesday, April 07, 2015
Breakfast: Whole Grain Bagel & Cream Cheese
Lunch: French Toast Sticks, Sausage, Applesauce, Assorted Juice, Milk CES Alternate: Tuna
Wednesday, April 08, 2015
Breakfast: Whole Grain Muffin
Lunch: Whole Grain Pasta with Meat Sauce, Mixed Green Salad, Peaches, Milk CES Alternate:
Thursday, April 09, 2015
Breakfast: Yogurt & Whole Grain Graham Crackers
Lunch: Chicken Nuggets, Butter Egg Noodles, Spinach, Assorted Juice, Milk CES Alternate:
Peanut Butter & Fluff Sandwich
Friday, April 10, 2015
Breakfast: Whole Grain Cinnamon Bun
Lunch: Whole Grain Pizza, Fresh Carrots, Milk CES Alternate: Bagel Lunch
Monday, April 13, 2015
Breakfast: Schools Closed - Spring Recess
Tuesday, April 14, 2015
Breakfast: Schools Closed - Spring Recess
Wednesday, April 15, 2015
Breakfast: Schools Closed - Spring Recess
Thursday, April 16, 2015
Breakfast: Schools Closed - Spring Recess
Friday, April 17, 2015
Breakfast: Schools Closed - Spring Recess
Monday, April 20, 2015
Breakfast: Whole Grain Oatmeal Bar
Lunch: Hotdog on Whole Grain Roll, Baked Beans, Cucumbers, Applesauce, Milk CES
Alternate: Cereal Lunch
Tuesday, April 21, 2015
Breakfast: Whole Grain Bagel & Cream Cheese
Lunch: Chicken Patty on Whole Grain Roll, Lettuce, Tomato, Sweet Potato Fries, Assorted
Juice, Milk CES Alternate: Tuna Salad Sandwich
Wednesday, April 22, 2015
Breakfast: Whole Grain Muffin
Lunch: Whole Grain Pasta with Meatballs, Mixed Green Salad, Mixed Fruit, Milk CES
Thursday, April 23, 2015
Breakfast: Yogurt & Whole Grain Graham Crackers
Lunch: Chicken Fajita Wrap, Rice, Carrots, Assorted Juice, Milk CES Lunch: Earth Day-Bag
Lunch: Ham Sandwich, Turkey Sandwich or PB&J, Goldfish Grahams, Fresh Fruit, Fresh
Carrots, Milk
Friday, April 24, 2015
Breakfast: Whole Grain Cinnamon Bun
Lunch: Quesadilla Pizza, California Blend Vegetables, Peaches, Milk CES Alternate: Bagel
Lunch
Monday, April 27, 2015
Breakfast: Whole Grain Oatmeal Bar
Lunch: Mozzarella Sticks, Sauce, Broccoli, Applesauce, Milk CES Alternate: Cereal Lunch
Tuesday, April 28, 2015
Breakfast: Whole Grain Bagel & Cream Cheese
Lunch: Taco Wrap, Rice, Corn, Assorted Juice, Milk CES Alternate: Tuna Salad Sandwich
Wednesday, April 29, 2015
Breakfast: Whole Grain Muffin
Lunch: Whole Grain Pasta with Meat Sauce, Mixed Green Salad, Peaches, Milk CES Alternate:
Thursday, April 30, 2015
Breakfast: Yogurt & Whole Grain Graham Crackers
Lunch: Macaroni & Cheese, Green Beans, Bread & Butter, Assorted Juice, Milk CES Alternate:
Peanut Butter & Fluff Sandwich
Press Release from Willington Education Association
Charles Kramer Scholarship for Willington Residents
In May 2015, scholarships (ranging between \$500.00 and \$1000.00
depending on how many are awarded) will be available to graduating high school seniors who:
• Are current residents of Willington
• Graduated from Hall Memorial School • Plan to pursue a career in teaching
• Submit a completed application by April 30, 2015
The Charles Kramer Scholarship fund was created in honor of Charles Kramer, a former teacher at Hall Memorial School in Willington. Applications are available on the Willington Public
School website and the guidance offices of E.O. Smith High School, and at Windham Technical School. Applications must be submitted to the Charles Kramer Scholarship Committee, Hall School, 1 River Rd., Willington, CT 06279, by April 30, 2015.
Charles Kramer Scholarship Application for Prospective Teachers
Name in full: ______________________________________________Age:___________
Telephone:________________________ Email: ______________________________ Father's Occupation;______________________________________
Mother's Occupation:_________________________________
Other dependent members of family: _________________________________________
I plan to attend the following college in September:________________________
Explain why you are applying for this scholarship:______________________________
______________________________________________________________________________
High School Average:___________________
High School Activities: _________________________________________________
_____________________________________________________________________
Please submit a short essay on an attached sheet that explains why you want to be a teacher.
If there was a particular teacher who positively affected your life, include that information in the essay.
Application is due by April 30, 2015 Send to The Charles Kramer Scholarship Committee, c/o Hall Memorial School,
111 River Rd. , Willington, Ct. 06279
CENTER SCHOOL – SUCCESS SEPTO Group Number: 990030219
www.yankeecandlefundraising.com
IMPORTANT DATES: Start Date: Monday, March 2, 2015 End Date: Tuesday, March 31, 2015 Delivery: Week of April 27, 2015 MONEY COLLECTION: Please collect all money while taking orders. Make all checks payable to: SUCCESS SEPTO
IMPORTANT NOTES: • All orders and money must be returned NO LATER than March 31, 2015. • Before you turn in your packet, be sure that the order and amount of payment received match. • Please retain both the PINK copy of the order form and the BROCHURE for reference at product delivery. • Return the YELLOW and WHITE copies with your order packet. Please print clearly and bear down hard. • Be sure that the SELLER AND CENTER SCHOOL SUCCESS SEPTO is printed neatly at the top of
your order form. • PLEASE, NO DOOR-TO-DOOR SALES - just to relatives, friends, neighbors and parents' co-workers.
Thank you for your support! Cathy Britschock Susan Rogers Fundraising Coordinator/SEPTO President The Yankee Candle Co. Regional Consultant 860-429-8333 home 860-989-8879 cell
Have (out of town) friends & family shop online
@ www.yankeecandlefundraising.com
ONLINE SHOPPING
Click on the 'Seller Iogin' button and fill out the 'Signup Po Ne M Seller' form; use the group number above to register. Follow instructions to send shopping emails to friends & family. Happy Shopping!
990030219
What if next school year there was no Willington PTA?
What does the PTA do for our schools each year???
∙ Raises funds that are dispersed for various events to benefit the
students of both CES and Hall School. ∙ Gives 100% of raised funds to the schools. ∙ Operate a \$10,000 annual budget. ∙ Provides each Center school student with a grade specific t-shirt for use
throughout the year. ∙ Gives every student funds towards field trips. ∙ Enrichment money for special presentations such as the Chinese acrobats
and the Internet Safety Presentations. ∙ The Center School Yearbook, which captures a school memory for every
student, is created by parent and teacher volunteers. ∙ Raises funds to purchase school equipment such as lighting for our
concerts and performances at Hall school and recess equipment for our children.
∙ Plans and holds events for the teachers and staff including the holiday cookie buffet and teacher appreciation week.
∙ Coordinates and plans fun and memorable social events for the entire school community such as dances, bowling, ice cream socials, the fun run and 5K etc.
∙ Funds the stores at both schools. ∙ Solicits donations from town business to assist with social events. ∙ Collects and processes Box Tops for Education to raise money for the
PTA budget.
This is all achieved every year by volunteers.
Parents just like you who are:
∙ Willing to give a few hours of their time. ∙ Willing to use their creativity and talents. ∙ Willing to ensure our children's school years are an amazing experience. ∙ Willing to share their opinions and ideas for the benefit of our
children.
Are YOU willing to give your time to help our schools strive for the excellence our children deserve?
Membership Meetings are once a month for 1 hour. Sadly this year at most meetings there were less then 8 members in attendance. With more volunteers, the PTA can achieve so much more for our children! MANY HANDS MAKE LIGHT WORK. This year the work has been heavy with very few hands. IF THERE ARE NOT ENOUGH PARENTS WILLING TO VOLUNTEER, THE WILLINGTON PTA WILL BE DISSOLVED BEFORE THE NEXT SCHOOL YEAR. There will be no extra funding, no Daughter Dance, no Sons Bowling, no field trip money, no CES yearbook, and no enrichment money for special events and equipment. The Willington PTA Nominating Committee is seeking those interested in being nominated for the following positions:
President: Preside at all meetings of the Willington PTA, serve as an ex-officio member of all committees except the nominating committee and coordinate the work of the officers and committees of the Willington PTA in order that purposes may be promoted.
Vice President: Act as an aide to the President and perform such other duties as may be provided for by these bylaws prescribed by the parliamentary authority or directed by the Executive Board.
Treasurer(s): Co-Treasurer will have designated roles (A or B). Role A: Have custody and maintain a full account of fund and keep a full and accurate account of receipts and disbursements. Role B: Create and maintain the budget using a template and create and present monthly treasurers report.
Secretary: Record minutes of all meetings of the PTA, keep file of all minutes and agendas and have a current copy of bylaws available at all PTA sponsored events. Membership Chair: Coordinate annual Ice Cream Social and Membership Drive, solicit new members and maintain membership listings and committee volunteers and any newly added throughout the year.
The Willington PTA will be electing officers for the 2015-2017 2 year term on Tuesday May 19th at 6:30 pm during the Monthly Membership meeting
at the Willington Public Library.
If you would like our schools to continue to benefit from the PTA’s work and have an interest in filling one of the above positions,
and direct you to our nominating committee.
President - Rachel Pierce - [email protected] V. President – Sue Heacox – [email protected]
Treasurer – Melissa McKinnon – [email protected] Secretary – Hanna Prytko – [email protected] Membership– MaryBeth Luchon – [email protected]
Come to Track Nine DinerFriday, April 10 2015
5:00 p.m. to 8:00 p.m.
Proceeds to offset costs of the
8th Grade Dance for Willington Students
Come enjoy delicious entrees and dessert!
Track Nine Diner12 Tolland Turnpike
Willington, CT
Bring family and friends for great food and a fun night out and help us to providethe 8th Grade Graduating Class of 2015 a class dance to remember!
Sponsored by the 8th Grade Dance Parent CommitteeQuestions: Contact Samantha Hills, Committee Chairwoman at [email protected]
YOU DON’THAVE TO
COOK!
Willington Youth, Family & Social Services
“Where Community Grows” 40 Old Farms Rd
Willington, CT 06279 Tel: 860-487-3118 Fax: 860-487-3125 [email protected]
Dear Willington Families, Willington Youth, Family & Social Services is pleased to offer a professionally staffed, fun & safe camp for your children (grades K-6).during their April vacations from school. With new, longer camp hours; you can be sure that your children are having a great time while you are busy at work! Your child(ren) will enjoy cooking, hiking, arts & crafts, playground time, games, movies, sports, music, and more!
Camp Details Dates: Monday April 13th through Friday April 17th
Times: Camp Hours 9:00 am – 4:00 pm Location: Willington Hill Fire House
Fees: 5 Day Camp Fee \$150.00
(Full or partial scholarships available for eligible households
Campers: Grades K -6 students. Lunch program not available, but snacks & drinks provided. Dress to play! Registration form & payment must be returned at school main
offices/ WYFSS no later than 3:00 PM Monday April 6, 2015.
Visit WYFSS on the web at: www.willingtonct.org to view our other programs or to download scholarship information. We encourage you to email us with any question at: [email protected]. We’ll see you at camp!
Willington Youth, Family & Social Services
2015 April Vacation
Camper Registration
______________________________________________________________________________ Camper Vital Information Camper’s Name: _____________________________________ DOB: ________________ Gender: ___ Grade: ___ Street Address: ___________________________________________ Willington, 06279 Primary Phone: (___) _______________ Secondary Phone: (___) ________________ Parent/Guardian Full Name(s):_____________________________________________________ _____________________________________________________________________________ Please register my child for (check all that apply) 5 Day April Camp \$150.00 Payment Enclosed \$___________
Scholarships available for qualified families…Please call 860-487-3118
_________________________________________________________________________________
Emergency/Medical information Please contact in case I cannot be reached in the case of sickness or emergency (other than parent(s) or guardian(s) the child resides with) 1’ 1”St Alt Contact Name: ____________________________________ Relationship: __________ Primary Phone: (___) _______________ Secondary Phone: (___) ________________
2’nd Alt Contact Name: __________________________________ _ Relationship: ___________ Primary Phone: (___) _______________ Secondary Phone: (___) ________________
Physician: ___________________________ Phone: (___) _______________ Dentist: _____________________________ Phone: (___) _______________ Does your child have any allergies, medical conditions, or behavioral challenges that we should be aware of in order to keep him/her safe, comfortable and happy while at camp? ______________________________________________________________________________
___________________________________________________________________________________________
Release, Waiver and Assumption of Liability
and Consent For Medical Treatment
I, the undersigned, by registering my child to attend WYFSS April Camp Program understand the nature and risks associated with the participation in those activities. I hereby grant my child permission to participate. I am aware that participation is at one’s own risk. I acknowledge that the activity, equipment and facilities may pose a risk of personal injury. I am also aware that each participant is responsible for his or her own safety. I hereby waive and release myself, my heirs, executors or administrators of any and all claims and damage we ever had or now have, against the Town of Willington, its successors and assigns, employees, agents and representative for any and all kinds of injury, including but not limited to personal injury and/or property damage suffered by my child, myself, family members or friends while participating in this program. Consent for Medical Treatment of Minors, as the parent or legal guardian of the above named player, I hereby give consent for emergency medical care prescribed by a fully licensed Doctor of Medicine or Doctor of Dentistry. This care may be given under whatever conditions are necessary to preserve the life, limb or well being of my dependent.
______________________________@_________._____
I certify that the information contained on this form is accurate and complete. I furthermore accept the terms and conditions of the above waiver.
____________ __________________________________ Date Signature of Parent or Guardian
Registration and payment must be returned to HMS/CES main offices or WYFSS office no later than 3:00 PM Monday April 7, 2015
Please make checks payable to: “WYFSS April Camp”
THANK YOU!
REFUND POLICY: If April Vacation Camp does not run due to low enrollment; your full payment will be returned to you. Single-day refunds will not be issued for absence; except in cases where a medical reason exists, and a note from a physician is provided.
Recyclables Craft Event for Kids
Saturday, April 25, 2015 Willington Public Library – Craft Room
10:30 to 12:00
In celebration of Earth Day, the 7th grade members of Girl Scout Troop 65292 are hosting a recycling-themed craft event for children.
Dear Willington Residents, In celebration of Earth Day, the 7th grade members of Girl Scout Troop 65292 are hosting a recycling-themed craft event for children on April 25, 2015 at the Willington Public Library. To prepare for this event, we are seeking donations of clean, recycled household items. The following items are needed:
• Toilet paper/ paper towel tubes • Egg cartons • Empty, washed, used K-cups
These items can be dropped off in donation boxes from 3/14/15 – 4/11/15 at the following locations:
• Willington Town Office Building • Hall Memorial School (in the gym lobby) • Willington Public Library • Transfer Station (swap shed)
Our Cadet Troop is organizing this community event to help us earn our Silver Award. We really appreciate your help in our efforts! If you have any questions, you can contact Michelle Tharp at 860-429-1491. Thank you, Brenna Leech, Grace Tharp, Rachel Cook and Emma Kennedy
WOW 50 YEARS The Joe Green Memorial Easter Egg Hunt
Open to Willington residents, newborn to 8th grade
Starts at 12:00 noon sharp
Come by 10:30 to get your kids in on the free bike raffle
And to see the bunny come in by a fire truck.
Where: Willington Hill Fire Dept;(24 Old Farms Rd.)
Time: Egg hunt starts at 12:00 noon
Registration: for the bike raffle starts at 10:30
Please print and hang up at home QUESTIONS 860-429-2993 OR 860-418-9672
Project Based curriculum that teaches
confidence, encourages self-
expression and enriches social
development.
Activities to inspire physical and
mental growth, teamwork and
imagination
Healthy snacks, physical activity and
daily homework help
Fully licensed by the State of
Connecticut.
Bussing provided from Center
Elementary to the YMCA Child Care
Program
Choose from 2, 3, or 5 day option
Care Provided 2:30-6:00pm
Care available for school vacation
days and snow days
information, 860-872-7329
Financial Assistance Available
Held at Hall Memorial
School
Serving Center Elementary
and Hall Memorial School
INDIAN VALLEY FAMILY YMCA
11 Pinney Street, Ellington Ct. 06029
Ph. 860-871-0008 Fax. 860-871-2550
www.ghymca.org
LEARN.
GROW.
THRIVE. YMCA After School Care
2015 Vacation Bible School Monday, July 27
th – Friday, July 31st
9 AM – 12 Noon
at the Federated Church of Willington
Rte 32., South Willington
Closing Program –Friday at 11:00 a.m
**Pizza Lunch – Friday at 12 NOON
STORIES! CRAFTS! SONGS! SNACKS! GAMES! MISSION! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Cost: \$15/child if received by June 30th (Maximum per family: \$35) \$17/child after June 30th (Maximum per family: \$40) {financial assistance is available if needed} TO REGISTER – Please use one form per child and mail check paid to: The Federated Church of Willington, 132 River Rd., Willington, CT 06279. Please put “VBS Registration” in the memo line. Questions? Call 860-429-9911 or email us at [email protected]
Name _____________________________ Home Phone #_______________________
Date of Birth ____/_____/_____ Allergies__________________________________________
Grade – circle one – Nursery (3yr or 4 yr) or entering K 1 2 3 4 5 6 7 8 9 10 11 12
Parent Names _______________________________________________________________
Emergency Phone #’s (1) _________________________ (2) _________________________ | 5,151 | 22,135 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.984375 | 4 | CC-MAIN-2024-22 | latest | en | 0.944724 |
https://nrich.maths.org/public/leg.php?code=6&cl=2&cldcmpid=2791 | 1,569,041,238,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514574265.76/warc/CC-MAIN-20190921043014-20190921065014-00017.warc.gz | 588,553,621 | 9,076 | # Search by Topic
#### Resources tagged with Place value similar to Diagonal Sums:
Filter by: Content type:
Age range:
Challenge level:
### There are 75 results
Broad Topics > Numbers and the Number System > Place value
### Diagonal Sums
##### Age 7 to 11 Challenge Level:
In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?
### Being Resourceful - Primary Number
##### Age 5 to 11 Challenge Level:
Number problems at primary level that require careful consideration.
##### Age 5 to 11 Challenge Level:
Try out this number trick. What happens with different starting numbers? What do you notice?
### Oddly
##### Age 7 to 11 Challenge Level:
Find the sum of all three-digit numbers each of whose digits is odd.
### Napier's Bones
##### Age 7 to 11 Challenge Level:
The Scot, John Napier, invented these strips about 400 years ago to help calculate multiplication and division. Can you work out how to use Napier's bones to find the answer to these multiplications?
### Spell by Numbers
##### Age 7 to 11 Challenge Level:
Can you substitute numbers for the letters in these sums?
### Being Resilient - Primary Number
##### Age 5 to 11 Challenge Level:
Number problems at primary level that may require resilience.
### Coded Hundred Square
##### Age 7 to 11 Challenge Level:
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
### Being Collaborative - Primary Number
##### Age 5 to 11 Challenge Level:
Number problems at primary level to work on with others.
### Six Is the Sum
##### Age 7 to 11 Challenge Level:
What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?
### Which Scripts?
##### Age 7 to 11 Challenge Level:
There are six numbers written in five different scripts. Can you sort out which is which?
### ABC
##### Age 7 to 11 Challenge Level:
In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?
### Being Curious - Primary Number
##### Age 5 to 11 Challenge Level:
Number problems for inquiring primary learners.
### The Thousands Game
##### Age 7 to 11 Challenge Level:
Each child in Class 3 took four numbers out of the bag. Who had made the highest even number?
### Song Book
##### Age 7 to 11 Challenge Level:
A school song book contains 700 songs. The numbers of the songs are displayed by combining special small single-digit cards. What is the minimum number of small cards that is needed?
### Round the Three Dice
##### Age 7 to 11 Challenge Level:
What happens when you round these three-digit numbers to the nearest 100?
### One Million to Seven
##### Age 7 to 11 Challenge Level:
Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?
### Even Up
##### Age 11 to 14 Challenge Level:
Consider all of the five digit numbers which we can form using only the digits 2, 4, 6 and 8. If these numbers are arranged in ascending order, what is the 512th number?
### Trebling
##### Age 7 to 11 Challenge Level:
Can you replace the letters with numbers? Is there only one solution in each case?
### Alien Counting
##### Age 7 to 11 Challenge Level:
Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.
### Subtraction Surprise
##### Age 7 to 14 Challenge Level:
Try out some calculations. Are you surprised by the results?
### All the Digits
##### Age 7 to 11 Challenge Level:
This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?
### Number Detective
##### Age 5 to 11 Challenge Level:
Follow the clues to find the mystery number.
### Nice or Nasty for Two
##### Age 7 to 14 Challenge Level:
Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your opponent.
### Round the Dice Decimals 2
##### Age 7 to 11 Challenge Level:
What happens when you round these numbers to the nearest whole number?
### Arrange the Digits
##### Age 11 to 14 Challenge Level:
Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500?
### Multiply Multiples 3
##### Age 7 to 11 Challenge Level:
Have a go at balancing this equation. Can you find different ways of doing it?
### Round the Dice Decimals 1
##### Age 7 to 11 Challenge Level:
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
### Multiply Multiples 2
##### Age 7 to 11 Challenge Level:
Can you work out some different ways to balance this equation?
### Multiply Multiples 1
##### Age 7 to 11 Challenge Level:
Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?
### Digit Sum
##### Age 11 to 14 Challenge Level:
What is the sum of all the digits in all the integers from one to one million?
### Dicey Operations in Line for Two
##### Age 7 to 11 Challenge Level:
Dicey Operations for an adult and child. Can you get close to 1000 than your partner?
### Which Is Quicker?
##### Age 7 to 11 Challenge Level:
Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?
### Becky's Number Plumber
##### Age 7 to 11 Challenge Level:
Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?
### Cayley
##### Age 11 to 14 Challenge Level:
The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?
### Just Repeat
##### Age 11 to 14 Challenge Level:
Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?
### What Do You Need?
##### Age 7 to 11 Challenge Level:
Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?
### Football Sum
##### Age 11 to 14 Challenge Level:
Find the values of the nine letters in the sum: FOOT + BALL = GAME
### Eleven
##### Age 11 to 14 Challenge Level:
Replace each letter with a digit to make this addition correct.
### Six Times Five
##### Age 11 to 14 Challenge Level:
How many six digit numbers are there which DO NOT contain a 5?
### Two and Two
##### Age 11 to 14 Challenge Level:
How many solutions can you find to this sum? Each of the different letters stands for a different number.
### Tis Unique
##### Age 11 to 14 Challenge Level:
This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.
### Method in Multiplying Madness?
##### Age 7 to 14 Challenge Level:
Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?
### Reach 100
##### Age 7 to 14 Challenge Level:
Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.
### X Marks the Spot
##### Age 11 to 14 Challenge Level:
When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .
### Skeleton
##### Age 11 to 14 Challenge Level:
Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.
##### Age 11 to 14 Challenge Level:
Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E
##### Age 5 to 11 Challenge Level:
Who said that adding couldn't be fun?
### Double Digit
##### Age 11 to 14 Challenge Level:
Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. . . .
### Three Times Seven
##### Age 11 to 14 Challenge Level:
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why? | 2,041 | 8,593 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.921875 | 4 | CC-MAIN-2019-39 | latest | en | 0.84585 |
https://www.coursehero.com/file/9010070/ECE15A-Homework1-Winter2014-1/ | 1,519,617,667,000,000,000 | text/html | crawl-data/CC-MAIN-2018-09/segments/1518891817999.51/warc/CC-MAIN-20180226025358-20180226045358-00241.warc.gz | 849,436,339 | 183,470 | {[ promptMessage ]}
Bookmark it
{[ promptMessage ]}
ECE15A-Homework1-Winter2014 (1)
# ECE15A-Homework1-Winter2014 (1) - ECE15AHomework#1...
This preview shows pages 1–3. Sign up to view the full content.
ECE 15A Homework #1 Winter 2014 1. Do the following conversion problems: (5p) (a) Convert decimal 19.2225 to binary. Assume that the integral and fractional parts of the number can use 8 bits each. Integral: 19/2 = 9+1 9/2 = 4+1 4/2 = 2+0 2/2 = 1+0 1/2 = 0+1 10011 00010011 Fractional: .2225*2 =0+.445 .445*2 =0+.89 .89*2 =1+.78 .78*2 =1+.56 .56*2 =1+.12 .12*2 =0+.24 .24*2 =0+.48 .48*2 =0+.98 .00111000 Answer: 00010011.00111000 (5p) (b) Calculate the binary equivalent of 1/17 up to 8 places. Then convert from binary to decimal. How close is the result to 1/17? Fraction to Binary: 1/17*2 =0+2/17 2/17*2 =0+4/17 4/17*2 =0+8/17 8/17*2 =0+16/17 16/17*2 =1+15/17 15/17*2 =1+13/17 13/17*2 =1+9/17 9/17*2 =1+1/17 .00001111 Binary to Decimal: 0(1/2)+0(1/4)+0(1/8)+0(1/16)+1(1/32)+1(1/64)+1(1/128)+1(1/256) = 0.05859375 Comparison:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
1/17 = 0.0588235294117647 repeating .00001111 = 0.05859375 % Error Calculation: |0.0588235294117647 0.05859375| / |0.0588235294117647| approx 0.390625 % error or 0.00022977941 absolute difference 2. A computer has a word length of 8 bits, including sign. Obtain 1’s and 2’s complement of the following binary numbers: (2p) (a) 10101010 1’s complement 01010101 (flip) = 1+4+16+64 = 85 2’s complement 01010101 + 1 (flip+1) = 01010110 = 2+4+16+64 = 86 (2p) (b) 01010011 01010011 = 1+2+16+64 = 83 = (2p) (c) 00000001 00000001 = 1 = 1 (2p) (d) 10000000 01111111 (flip)
This is the end of the preview. Sign up to access the rest of the document.
{[ snackBarMessage ]}
### Page1 / 7
ECE15A-Homework1-Winter2014 (1) - ECE15AHomework#1...
This preview shows document pages 1 - 3. Sign up to view the full document.
View Full Document
Ask a homework question - tutors are online | 820 | 2,021 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.21875 | 4 | CC-MAIN-2018-09 | latest | en | 0.665852 |
http://math.stackexchange.com/questions/tagged/complex-numbers+fourier-series | 1,412,241,182,000,000,000 | text/html | crawl-data/CC-MAIN-2014-41/segments/1412037663739.33/warc/CC-MAIN-20140930004103-00131-ip-10-234-18-248.ec2.internal.warc.gz | 196,174,445 | 15,046 | Tagged Questions
82 views
Fourier series without Fourier analysis techniques
It is known that one can sometimes derive certain Fourier series without alluding to the methods of Fourier analysis. It is often done using complex analysis. There is a way of deriving the formula ...
57 views
7 views
How can we derive $\frac{1}{2j}\mathbf{F}^{-1}[Y_b(f-f_c)-Y_b^*(-f-f_c)]=\frac{\sqrt{2}}{2j}\mathbf{F}^{-1}[Y(f)u(f)-Y(f)u(-f)]$?
When I was reading digital communication theory, I couldn't derive following equation $$\frac{1}{2j}\mathbf{F}^{-1}[Y_b(f-f_c)-Y_b^*(-f-f_c)]=\frac{\sqrt{2}}{2j}\mathbf{F}^{-1}[Y(f)u(f)-Y(f)u(-f)]$$ ...
19 views
Find Complex Fourier coefficients for $f(x) = \sum^5_{m=1} (\frac{-1}{2})^m \cos(2^mx)$
This is a question from a book that I'm trying to solve and I don't know how. Find Complex Fourier coefficients for $f(x) = \sum^5_{m=1} (\frac{-1}{2})^m \cos(2^mx)$ Can you please give me some ...
16 views
Hyprecomplex datapoint
I was working on 2D dataset 512$\times$512. To be more precise,its the datapoints are collected in dimensions t1 and t2. All the data points are complex in the form a+b$i$. To get the frequency ...
40 views
Complex Conjugate (FourierSeries)
Let $f$ be a piecewise continuous complex function on the interval $[\pi,\pi]$ and $$f(x) \sim \sum_{n=-\infty}^{\infty}c_{n}e^{inx} \tag{*}$$ be its complex Fourier ...
116 views
Is the matrix Wn from the DFT a Hermitian operator?
A homework question asks me whether or not the matrix $W_N$ from the matrix representation of the Direct Fourier Transform is a Hermitian operator. From what I understand an Hermitian operator does ...
56 views
Confused between multiple representations of Fourier Series' formula
I have never used the formula for Fourier Series with the representation that the instructor of the above video is using that involves $k$ and $\omega$. Instead, I use $n$ and $\pi$. Now, suppose ...
211 views
What is the odd Fourier extension of $\sin x \cos(2x)$?
odd half range extension of $$f(x) = \sin x \cos(2x)\text{ with limits 0 to \pi}$$
297 views
Fourier Analysis
I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function? Can someone provide me simple information ...
55 views
Proving that two representations of a Fourier series are the same
I have to show that $$\sum_{n=0}^\infty A_n\cos\left({xn\frac{2\pi}{T}-\theta_n}\right) \equiv \sum_{n=-\infty}^\infty c_n \mathrm{e}^{\left({ixn\frac{2\pi}{T}}\right)}$$ I have tried two ...
322 views
Real part of an integral with complex argument
This is a paper about Fourier cosine series approximation to option pricing problem. The coefficient $A_k$ is defined as $$A_k=\frac{2}{b-a}\int_a^bf(x)\cos\left(k\pi\frac{x-a}{b-a}\right)dx$$ Then ...
Consider an $n$-sided convex polygon $P$ that contains the origin in the complex plane. Let the $j$-th vertex be denoted $z_j = r_j e^{i\theta_j}$ ($0 \leq \theta_j < 2 \pi$) for $j= 1 \dots n$. ...
I'm having trouble wrapping my head around the following issue. My book solves a problem without using complex exponential solution like $C_1 e^{it}$ and using either $A \cos(t) + B \sin(t)$ or \$A ... | 952 | 3,238 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0} | 3.59375 | 4 | CC-MAIN-2014-41 | latest | en | 0.850933 |
https://crypto.stackexchange.com/questions/39406/noise-of-ciphertexts-in-lwe-rlwe-based-fhe | 1,653,562,156,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662604794.68/warc/CC-MAIN-20220526100301-20220526130301-00021.warc.gz | 239,622,620 | 66,889 | # Noise of ciphertexts in LWE/RLWE based FHE
Often times $[\langle \textbf{c}, \textbf{s} \rangle]_q$ is referred to as the noise associated to the ciphertext $\textbf{c}$, and that decryption is correct when the norm of the noise is $< q/2$. But shouldn't $\langle \textbf{c}, \textbf{s} \rangle$ be the noise instead of $[\langle \textbf{c}, \textbf{s} \rangle]_q$, since $[\langle \textbf{c}, \textbf{s} \rangle]_q$ means the reduction mod $q$ already happened and so the norm of $[\langle \textbf{c}, \textbf{s} \rangle]_q$ is always $< q/2$?
• Have you read it on BGV paper? Aug 17, 2016 at 19:25
• Yes, I'm currently working through it and this is one the confusions that came up for me.
– sycs
Aug 17, 2016 at 21:39
• $\langle c ,s\rangle$ cannot be defined as the noise, since we have $\langle c,s\rangle= 2e + m + uq$ for some integer $u$. The reduction mod $q$ removes the term $uq$ so that we have access to the noise term, which is $e$, or $2e$, or $2e + m$ depending on the definition. In BGV paper, what they really want is $|2e + m| < q/2$. But it is true that the way they defined everything does not seem good, because the reduction maps to $(-q/2, ..., q/2]$, so what they defined as the noise will always be smaller than $q / 2$ (or exactly equal to $q/2$, which is very very unlikely). Apr 9, 2021 at 8:01
This is because of the decryption equation: If two ciphertexts differ ($\mathbb{Z}$-component-wise..., not homomorphically) in a multiple of $q$, they both decrypt to the same thing because of the modular reduction. Therefore, upon reception of a 'big' ciphertext, it makes sense to take modular reduction of this ciphertext and then look at its noise. Conversely, to cheat silly attackers one could add 'false noise' in this fashion (severely augmenting ciphertext size, however)
• If I am right, is there a good explanation for why $[\langle \textbf{c}, \textbf{s} \rangle ]_q$ is referred to as the noise and not $\langle \textbf{c}, \textbf{s} \rangle$?
• They write $[\langle\vec{c},\vec{s}\rangle]_q$ in order to explicitly distinguish it from the similar-looking-but-distinct $[\langle\vec{c},\vec{s}\rangle]_p$ for $p < q$. In particular, modulus switching was "the new thing" in BGV, and their key idea revolves around these two types of terms (so they really want to keep them separate). Aug 21, 2016 at 6:28
• Maybe I'm misinterpreting what $[\langle \textbf{c}, \textbf{s} \rangle]_q$ means. Is my assumption that $[\langle \textbf{c}, \textbf{s} \rangle]_q$ means the reduction mod $q$ already went through so then the norm of $[\langle \textbf{c}, \textbf{s} \rangle]_q$ is always $< q/2$ correct? | 798 | 2,640 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2022-21 | longest | en | 0.867592 |
https://mcqsadda.online/mcqs-on-units-and-measurements/ | 1,679,822,348,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296945440.67/warc/CC-MAIN-20230326075911-20230326105911-00213.warc.gz | 436,910,420 | 31,325 | # Mcqs on units and measurements
5/5 - (1 vote)
Test your understanding on the topic of units and measurements by Mcqs on units and measurements help in understanding the basics of this topic can help you solve a variety of problems. In this post, we’ll be exploring multiple-choice questions related to units and measurements and providing the answers to help you test your knowledge. So let’s get started!
Physics. Measurements/units and measuring devices and scales
Units and measurements are the cornerstones of scientific study. They are the fundamental building blocks of our world, and understanding them is essential for making sense of the data we have. That’s why we’ve compiled this post – to help you get a better grasp on the basics of units and measurements. We’ll be covering topics such as the International System of Units (SI units) and their sub-divisions, the differences between the US customary system and the metric system, and the concept of dimensional analysis. We’ll also be exploring the different ways in which these units and measurements are used in the world of science. we’ve included a series of MCQs based on the topics we’ll be discussing. With these, you’ll be able to test your knowledge on the key concepts and make sure you’re up-to-date on the latest information about units and measurements. So let’s get started!
Q1. The SI unit of power is
(a) Hertz
(b) Volts
(c) Watt
(d) Neutron
(c) Watt
Q2. Frequency is measured in
(a) hertz
(b) metre/second
(d) watt
(a) hertz
Q3. The SI unit of electric power is :
(a) Ampere
(b) Volt
(c) Coulomb
(d) Watt
(d) Watt
Q4. The SI unit of the force is
(b) Fermi
(c) Newton
(d) Rutherford
(c) Newton
Q5. The SI unit of work is
(a) Joule
(b) Neutron
(c) Watt
(d) Dyne
(a) Joule
Q6. Light-year is the SI unit of
(a) Energy
(b) Intensity
(c) Age
(d) Distance
(d) Distance
Q7. The SI unit of electric current is :
(a) Ampere
(b) Volt
(c) Coulomb
(d) Watt
(a) Ampere
Q8. Angstrom is a SI unit of
(a) velocity
(b) energy
(c) frequency
(d) wavelength
(d) wavelength
Q9. What is the unit of pressure?
(a) Newton / sq. metre
(b) Newton-metre
(c) Newton
(d) Newton/metre
(a) Newton / sq. metre
Q10. The smallest unit of length is
(a) Micron
(b) Fermimetre
(c) Angstrom
(d) Nanometre
(b) Fermimetre
Q11. Dobson’ Unit is used for the measurement of
(a) Thickness of Earth
(b) Thickness of Ozone layer
(c) Thickness of Diamond
(d) Measurement of Noise
(b) Thickness of Ozone layer
Q12. The velocity of wind is measured by :
(a) Barometer
(b) Anemometer
(c) Hydrometer
(d) Wind vane
(b) Anemometer
Q13. Which of the following device is used to measure electric current
(a) Voltmeter
(b) Ammeter
(c) Voltameter
(d) Potentiometer
(b) Ammeter
Q14. Which one of the instruments is used to measure atmospheric pressure?
(a) Hydrometer
(b) Barometer
(c) Manometer
(d) Hygrometer
(b) Barometer
Q15. Richter scale is used for measuring
(a) Amplitude of seismic waves
(b) Intensity of light
(c) Velocity of sound
(d) Intensity of sound
(a) Amplitude of seismic waves
Q16. Fathometer is used to measure :
(a) Earthquake
(b) Depth of sea
(c) Rain
(d) Sound intensity
(b) Depth of sea
Q17. The device used for detecting lie is known as
(a) Gyroscope
(b) Pyrometer
(c) Polygraph
(d) Kymograph
(c) Polygraph
Q18. The density of milk is measured by?
(a) Hygrometer
(b) Hydrometer
(c) Barometer
(d) Lactometer
(d) Lactometer
Q19. A medical device used for listening to the sounds of the heart
(a) Viscometer
(b) Hydrometer
(c) Stethoscope
(d) Lactometer
(c) Stethoscope
Q20. A detection system that uses radio waves to determine the range, angle or velocity of objects
(a) SONAR
(c) Wind vane
(d) Lactometer
Q21. It gathers and measures the amount of liquid precipitation over a set period.
(c) Wind vane
(d) Rain Gauge
(d) Rain Gauge
Q22. It is a transducer that converts sound waves into electrical signals.
(b) Microphone
(c) Microtome
(d) Nephoscope
(b) Microphone
Q23. Which of the following instruments helps to detect or locate submerged submarines and icebergs.
a) Audiometer
(b) Galvanometer
(c) Sextant
(d) SONAR
(d) SONAR
Mcqs on units and measurements class 11
Subscribe
Notify of | 1,169 | 4,293 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.734375 | 4 | CC-MAIN-2023-14 | longest | en | 0.825548 |
https://issuu.com/vinku/docs/chemistry_class_xii_chapter_1/16 | 1,508,618,306,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187824894.98/warc/CC-MAIN-20171021190701-20171021210701-00293.warc.gz | 710,273,361 | 17,905 | those of the first layer as shown in Figs. 1.18 and 1.19. This pattern of layers is often written as ABCABC ........... This structure is called cubic close packed (ccp) or face-centred cubic (fcc) structure. Metals such as copper and silver crystallise in this structure. Both these types of close packing are highly efficient and 74% space in the crystal is filled. In either of them, each sphere is in contact with twelve spheres. Thus, the coordination number is 12 in either of these two structures. 1.6.1 Formula of a Compound and Number of Voids Filled
Earlier in the section, we have learnt that when particles are closepacked resulting in either ccp or hcp structure, two types of voids are generated. While the number of octahedral voids present in a lattice is equal to the number of close packed particles, the number of tetrahedral voids generated is twice this number. In ionic solids, the bigger ions (usually anions) form the close packed structure and the smaller ions (usually cations) occupy the voids. If the latter ion is small enough then tetrahedral voids are occupied, if bigger, then octahedral voids. Not all octahedral or tetrahedral voids are occupied. In a given compound, the fraction of octahedral or tetrahedral voids that are occupied, depends upon the chemical formula of the compound, as can be seen from the following examples.
d e
h s
T i l R b E u C p N re Š e b o t t
Example 1.1 A compound is formed by two elements X and Y. Atoms of the element Y (as anions) make ccp and those of the element X (as cations) occupy all the octahedral voids. What is the formula of the compound?
Solution The ccp lattice is formed by the element Y. The number of octahedral voids generated would be equal to the number of atoms of Y present in it. Since all the octahedral voids are occupied by the atoms of X, their number would also be equal to that of the element Y. Thus, the atoms of elements X and Y are present in equal numbers or 1:1 ratio. Therefore, the formula of the compound is XY.
Example 1.2 Atoms of element B form hcp lattice and those of the element A occupy 2/3rd of tetrahedral voids. What is the formula of the compound formed by the elements A and B?
Solution The number of tetrahedral voids formed is equal to twice the number of
o n
atoms of element B and only 2/3rd of these are occupied by the atoms of element A. Hence the ratio of the number of atoms of A and B is 2 Ă— (2/3):1 or 4:3 and the formula of the compound is A4B3.
Locating Tetrahedral and Octahedral Voids We know that close packed structures have both tetrahedral and octahedral voids. Let us take ccp (or fcc) structure and locate these voids in it. (a) Locating Tetrahedral Voids Let us consider a unit cell of ccp or fcc lattice [Fig. 1(a)]. The unit cell is divided into eight small cubes.
Chemistry
16
Chemistry Class XII Chapter 1
Chemistry Class XII Chapter 1 - The Solid State | 735 | 2,921 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.109375 | 3 | CC-MAIN-2017-43 | latest | en | 0.938649 |
http://willbraffitt.org/math/math13.html | 1,550,356,936,000,000,000 | text/html | crawl-data/CC-MAIN-2019-09/segments/1550247481122.31/warc/CC-MAIN-20190216210606-20190216232606-00518.warc.gz | 298,208,822 | 1,381 | # HUES 6th Grade Math Choice #13 02-May-2000
Hollis Upper Elementary School, Hollis, New Hampshire
1. 1995-96 Annual 7th grade math contest
1.9. There are 25% more new windows in my house than old ones.
What is the ratio of the number of new windows to the number of old ones?
1.19. I played video games at the arcade from 10:53 AM to 5:05 PM the same day.
When was I halfway done at the arcade that day?
1.20. What is the largest prime factor of 77,000,000?
1.28. If the radius plus the circumference of a certain circle is 6? + 3,
how long is the diameter of the circle?
1.38. Of 200 bears, 150 sing and 60 dance. Of those who dance, 40 sing.
How many bears neither sing nor dance?
1.39. What is 1% of 1%?
1.40. What is the hundreds' digit of 1 + 2 + 3 + ... + 998 + 999 + 1000?
Source: The Math League
http://www.mathleague.com
Don Braffitt | 275 | 863 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.046875 | 3 | CC-MAIN-2019-09 | latest | en | 0.864385 |
https://it.mathworks.com/matlabcentral/cody/problems/73-replace-nans-with-the-number-that-appears-to-its-left-in-the-row/solutions/502310 | 1,576,077,339,000,000,000 | text/html | crawl-data/CC-MAIN-2019-51/segments/1575540531917.10/warc/CC-MAIN-20191211131640-20191211155640-00122.warc.gz | 403,922,930 | 15,622 | Cody
Problem 73. Replace NaNs with the number that appears to its left in the row.
Solution 502310
Submitted on 17 Sep 2014 by Pasumarthi Viswanath
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
Test Suite
Test Status Code Input and Output
1 Pass
%% x = [NaN 1 2 NaN 17 3 -4 NaN]; y_correct = [0 1 2 2 17 3 -4 -4]; assert(isequal(replace_nans(x),y_correct))
2 Pass
%% x = [NaN 1 2 NaN NaN 17 3 -4 NaN]; y_correct = [ 0 1 2 2 2 17 3 -4 -4]; assert(isequal(replace_nans(x),y_correct))
3 Pass
%% x = [NaN NaN NaN NaN]; y_correct = [ 0 0 0 0]; assert(isequal(replace_nans(x),y_correct))
4 Pass
%% x = [1:10 NaN]; y_correct = [ 1:10 10]; assert(isequal(replace_nans(x),y_correct)) | 268 | 752 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2019-51 | latest | en | 0.545082 |
http://list.seqfan.eu/pipermail/seqfan/2010-October/006112.html | 1,642,998,839,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320304471.99/warc/CC-MAIN-20220124023407-20220124053407-00065.warc.gz | 36,424,683 | 2,373 | # [seqfan] Distinct poker hands
David Scambler dscambler at bmm.com
Tue Oct 5 04:46:28 CEST 2010
```In poker (usually) the suite does not matter except insofar as there are cards with matching suites. For a normal deck of 13 ranks and 4 suites there are C(52, 5) 5-card hands, but when suite symmetries are removed there are only 134459 distinct hands.
Keeping 4 suites and 5-card hands, varying the number n of ranks per suite generates the following number of distinct hands:
S4(n) = 0, 0, 6, 57, 272,901,2376,5362, 10808,19998,34602,56727,88968,134459,196924, 280728,390928,533324,714510,941925,1223904,...
Varying the number of suites and 5-card hands,
1 suite: S1(n) = A000389 = C(n,5)
2 suites: S2(n) = A053132(n+2), n>=3 otherwise zero = C(2n-4,5)
3 suites: S3(n) = 0,0,2,27,152,551,1536,3598,7448,14058,24702,40997,64944,98969,145964, 209328,293008,401540,540090,714495,931304,...
5 suites: S5(n) = 0, 1,12,78,328,1027,2628,5824,11600,21285,36604,59730, 93336,140647,205492,292356,406432,...
Adding a joker to the normal deck
Sj1(n) = 0,1,15,99,405,1231,3072,6671,13070,23661,40237,65043, 100827,150891,219142,310143,...
Adding two indistinguishable jokers to the normal deck:
Sj2(n) = 0,2,21,119,453,1326,3238,6937,13470,24234,41027,66099, 102203,152646,221340,312853,...
Cheers
dave
``` | 514 | 1,309 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2022-05 | latest | en | 0.710628 |
https://championcapitalgroup.com/darbhanga-lok-qwkh/a57840-fraunhofer-diffraction-from-single-and-double-slits-pdf | 1,660,265,617,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882571536.89/warc/CC-MAIN-20220811224716-20220812014716-00445.warc.gz | 170,921,019 | 9,460 | Figure \(\PageIndex{2}\) shows a single-slit diffraction pattern. Shape and intensity of a Fraunhofer diffraction remains constant. A computer scan of a double-slit interference pattern (slit width 0.08 mm and slit separation 0.50 mm) is shown at left. Fraunhofer Diffraction from a Single Slit16-1. -Compare the diffraction patterns of a single-slit and a double slit. simpliÞes to with! Fraunhofer diffraction by a single slit . Fraunhofer Diffraction Equipment Green laser (563.5 nm) on 2-axis translation stage, 1m optical bench, 1 slide holder, slide with four single slits, slide with 3 diffraction gratings, slide with four double slits, slide with multiple slits, pinhole, slide containing an etched Fourier transformed function, screen, tape measure. Young’s double slit diffraction x O-Gs ϑ amplitude intensity Od 2O d O d. 21 Young’s double slit diffraction x s 22 Young’s double slit diffraction x s 23 Single slit diffraction x y s amplitude intensity 24 Single-slit diffraction SINC FUNCTION. Single slit: Double slit: Three slits: Five slits: Diffraction and interference : Interference only: Under the Fraunhofer conditions, the light curve of a multiple slit arrangement will be the interference pattern multiplied by the single slit diffraction envelope. Diffraction problem basics (reminder) 2. Consider a slit of width w, as shown in the diagram on the right. In optics, Fraunhofer diffraction (named after Joseph von Fraunhofer), or far-field diffraction, is a form of wave diffraction that occurs when field waves are passed through an aperture or slit causing only the size of an observed aperture image to change due to the far-field location of observation and the increasingly planar nature of outgoing diffracted waves passing through the aperture. Preview . 16-1. If the width of the slits is small enough (less than the wavelength of the light), the slits diffract the light into cylindrical waves. This assumes that all the slits are identical. Figure 38.4b is a photograph of a single-slit Fraunhofer Active Figure 38.4 (a) Fraunhofer diffraction pattern of a single slit. Consider a plane wave front incidents on the slit of width ‘d’. Fraunhofer diffraction at single slit. In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.. Fraunhofer diffraction, diffraction by single slit, double-slit diffraction 2 = − @ 0 A ( − )----- (1) Where r is the optical path length from the interval ds to the point P. the amplitude 0 is divided by r because the spherical waves decrease in irradiance with distance, in accordance with the inverse square law, that is, Diffraction from a double slit. Use of a convex lens for observation of Fraunhofer diffraction pattern Photo of diffraction with Helium Neon laser: Index Diffraction concepts Fraunhofer diffraction . The intensity at point P 1 on the screen is obtained by applying the Fraunhofer diffraction theory at single slit and interference of diffracted waves from the two slits. Diffraction pattern produced by a single slit 4. Assume that the slit is very long in the direction perpendicular to the page so that we can neglect diffraction effects in the perpendicular direction. The light passing through the slit will converge by converging lens on screen which is at a distance ‘D’ from the slit. In fraunhofer diffraction, source and the screen are far away from each other. Educator Price. 2) Calculate the slit width, which produces the single-slit diffraction pattern, and observe how the slit width affects the diffraction pattern. Qty: Add to Cart. (b) Photograph of a single-slit Fraunhofer diffraction pattern. U.S. The positions of all maxima and minima in the Fraunhofer diffraction pattern from a single slit can be found from the following simple arguments. (# i, # o) u s (x, y ) r e ik r d S Fresnel-Kirchhof f diffraction integral Fraunhofer dif fraction in 1D ! The equation was named in honour of Joseph von Fraunhofer although he was not actually involved in the development of the theory. In the double-slit diffraction experiment, the two slits are illuminated by a single light beam. This is exactly the answer we saw last lecture, Laser light is much more coherent than light from conventional sources. No particular interesting phenomena are observed. Fraunhofer Diffraction from a Single Slit • Consider the geometry shown below. Fraunhofer Diffraction by Double Slit. Diffraction pattern remains in a fixed position. Diffraction from a single slit. Physics with animations and video film clips. 2.8 Fraunhofer diffraction at a double slit Two rectangular slits parallel to one another and perpendicular to the plane of the paper, width of each slit is say and width of the opaque portion is , figure 2.3. One example of a diffraction pattern on the screen is shown in Figure \(\PageIndex{1}\). Diffraction refers to various phenomena that occur when a wave encounters an obstacle or opening. To obtain Fraunhofer diffraction, the single-double plane diffraction grafting is used. According to rectilinear propagation of light, it is expected that, the central bright spot at ‘o’ and there is dark on either side of ‘o’. Diffraction from Single and Double Slits Alamdar Hussain, Rabiya Salman and Muhammad Sabieh Anwar Syed Babar Ali School of Science and Engineering, LUMS Version 1; March 31, 2014 Diffraction is one of the remarkable consequences of the wave nature of light. We can develop an expression for the far field of a continuous array of point sources of uniform amplitude and of the same phase. Fraunhofer diffraction pattern for a slit 2b In this case, the problem is a single Fourier transform (in x), rather than two of them (in x and y): 00111 exp jk Ex xx Aperturexdx z The aperture function is simple: 1 1 1 0otherwise bx b Aperture x How very satisfying! Double Slit Diffraction. The purple line with peaks of the same height are from the interference of the waves from two slits; the blue line with one big hump in the middle is the diffraction of waves from within one slit; and the thick red line is the product of the two, which is the pattern observed on the screen. Diffraction through a Single Slit. What's Included. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Bulletin of Physics Projects, 1, 2016, 24-28 26 The variations of dark fringe width were studied as we move away from the central bright maxima in order to validate the experimental findings with the well established theory of Fraunhofer diffraction due to a single slit. "Fraunhofer diffraction", a coherent source of parallel light is required and the laser provides such a source. In this experi- ment you will study diffraction patterns for single slit and double slit arrangements. = k sin "Note: U s(!) Fig. In Fraunhofer diffraction, diffraction obstacle gives rise to wavefronts which are also plane. Principles Diffraction experiments provide evidence of the wave character of light. Coherence, which is the extent (in space or time) to which the beam of light is in phase with itself, is necessary for the observation of interference. The solid line with multiple peaks of various heights is the intensity observed on the screen. Here we begin to see what amazing things it does in more than one dimension. Diffraction phenomena always occur when the free propagation of light is changed by obstacles such as iris diaphragms or slits. In a single slit diffraction, light spreads out in a line perpendicular to the slit. Lecture aims to explain: 1. Calculation of the diffraction integral for a long slit 3. Light rays are collected by the converging lens L and interference patterns are seen in the screen FT. Let a plane wave front be incident on the surface XY. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. We have seen it already in one-dimensional situations such as interferometers and reflection from thin films. Light passing through a single slit forms a diffraction pattern somewhat different from those formed by double slits or diffraction gratings, which we discussed in the chapter on interference. Single and Double Slit Comparison. Interference is a crucial part of the physics of diffraction. Continuous broadside array of point sources of length a. Physclips provides multimedia education in introductory physics (mechanics) at different levels. Young's experiment with finite slits: Physclips - Light. A photograph of the actual laser pattern is shown above. 1) Observe Fraunhofer diffraction and interference from a single slit, double-slit and multiple-slit (a diffraction grating). 3: Fraunhofer diffraction patterns for (a) single slit, (b) double slit and (c) triangular slits. \$940. 25 Phasors –single-slit Fresnel diffraction O NO SLIT SLIT CORNU SPIRAL. 53 Experiment 7 Diffraction at a single and double slit Apparatus: Optical bench, He – Ne Laser, screen with slits, photocell, micro-ammeter. Fraunhofer Diffraction Lecture Notes for Modern Optics based on ... Diffraction from a single slit We simulate the gggeometrical arrangement for the Frounhofer diffraction byyp placinggp a point source at the focal point of a lens and a screen at the focal point of a lens after the amerture. The width of the slit is W. The Fraunhofer diffraction pattern is shown in the image together with a plot of the intensity ... Single-slit diffraction of Electric Field using Huygens ' Principle. In optics, the Fraunhofer diffraction equation is utilized to show the diffraction of waves when the diffraction design is seen at a significant distance from the diffracting object (in the far-field locale), and furthermore when it is seen at the focal plane of an imaging lens. (Drawing not to scale.) -Investigating diffraction at a double slit and measuring the wavelength of the laser. It is a product of the interference pattern of waves from separate slits and the diffraction of waves from within one slit. The pattern consists of a central bright fringe flanked by much weaker maxima alternating with dark fringes. Loading... Unsubscribe from Pankaj Physics Gulati? i!!" Fraunhofer diffraction ¥Fraunhofer diffraction as Fourier transform ¥Convolution theor em: solving difÞcult diffraction pr oblems (double slit of Þnite slit width, diffraction grating) lecture 7 Fourier Methods Fourier Methods u p = ! Phasor sum to obtain intensity as a function of angle. DIFFRACTION AT A SINGLE SLIT || FRAUNHOFER DIFFRACTION AT A SINGLE SLIT WITH NOTES || हिंदी में || Pankaj Physics Gulati. Aperture. The complete solution for examining interference and diffraction patterns from laser light passing through various single-slits and multiple-slits. – Fraunhofer diffraction of slits and circular apertures – Resolution of optical systems • Diffraction of a laser beam. A plane wave is incident from the bottom and all points oscillate in phase inside the slit. Purpose of the experiment: To measure the intensity distribution due to diffraction due to single and double slits and to measure the slit width (d) and slit separation (a). And minima in the double-slit diffraction experiment, the two slits are illuminated a... Slit arrangements k sin `` Note: U s (! the answer we saw last lecture, Fraunhofer at! Wave character of light is required and the laser provides such a source physics.. Wave front incidents on the screen various single-slits and multiple-slits mechanics ) at different levels provide evidence the! Converge by converging lens on screen which is at a double slit patterns of a single-slit Fraunhofer Figure! Consider a plane wave front incidents on the slit will converge by converging lens screen... The interference pattern of waves from within one slit diffraction grating ) propagating wave laser light is much coherent! Secondary source of the wave character of light changed by obstacles such as interferometers and reflection from films... Line perpendicular to the slit will converge by converging lens on screen which is at a single slit as in... We begin to see what amazing things it does in more than one dimension diffraction '', a fraunhofer diffraction from single and double slits pdf of. By a single slit, ( b ) double slit and measuring the wavelength of the pattern! And multiple-slit ( a ) Fraunhofer diffraction '', a coherent source of parallel light is and... ( b ) double slit single-slit Fraunhofer diffraction pattern wave front incidents on the screen are far away each! Education in introductory physics ( mechanics ) at different levels from separate slits and the.... Solution for examining interference and diffraction patterns from laser light passing through various single-slits and multiple-slits observation Fraunhofer! As a function of angle things it does in more than one dimension c triangular! A line perpendicular to the slit the single-slit diffraction pattern gives rise to wavefronts which are plane... Fraunhofer diffraction pattern from a single slit diffraction, diffraction obstacle gives rise to wavefronts which are plane..., double-slit fraunhofer diffraction from single and double slits pdf multiple-slit ( a ) Fraunhofer diffraction of a single-slit and a double.. Width, which produces the single-slit diffraction pattern of width ‘ d ’ the... Slit || Fraunhofer diffraction from a single slit a function of angle pattern consists a... • diffraction of slits and the screen are far away from each other much weaker maxima with! (! slit slit CORNU SPIRAL the complete solution for examining interference and diffraction patterns laser. Systems • diffraction of slits and the laser provides such a source produces the single-slit diffraction pattern 16-1, produces... Can develop an expression for the far field of a double-slit interference of. On screen which is at a distance ‘ d ’ screen is shown at left and slit separation 0.50 )! Of diffraction, which produces the single-slit diffraction pattern of a convex lens for observation of Fraunhofer remains... Examining interference and diffraction patterns from laser light passing through the slit multimedia... Observation of Fraunhofer diffraction at a double slit and double slit arrangements various phenomena that occur when the free of... When the free propagation of light is required and the diffraction patterns from laser light is by. Observe Fraunhofer diffraction, diffraction obstacle gives rise to wavefronts which are plane... Diffraction experiment, the two slits are illuminated by a single slit and ( c ) triangular.... – Resolution of optical systems • diffraction of waves from within one slit by a single •. Also plane Fraunhofer although he was not actually involved in the Fraunhofer diffraction pattern of waves from separate and., and Observe how the slit Joseph von Fraunhofer although he was not actually involved in the diffraction... B ) photograph of the actual laser pattern is shown at left the of... The geometry shown below propagation of light 38.4b is a crucial part the! = k sin `` Note: U s (! laser provides such a source slit NOTES. Study diffraction patterns for single slit diffraction, diffraction obstacle gives rise to which! All fraunhofer diffraction from single and double slits pdf oscillate in phase inside the slit within one slit is changed obstacles. At different levels character of light is much more coherent than light from conventional.... ) Calculate the slit width 0.08 mm and slit separation 0.50 mm ) is shown above fraunhofer diffraction from single and double slits pdf theory as and... Line perpendicular to the slit of width w, as shown in Figure \ ( \PageIndex { 2 \... Experiment, the two slits are illuminated by a single slit, b... Diffraction remains constant various single-slits and multiple-slits each other light from conventional sources is changed by obstacles such as diaphragms... Observation of Fraunhofer diffraction pattern on the right slit will converge by converging lens on screen is. Spreads out in a line perpendicular to the slit will converge by converging on! Of waves from separate slits and the laser double-slit diffraction experiment, the two slits illuminated. ’ from the bottom and all points oscillate in phase inside the slit from separate slits and the of... Calculation of the wave character of light is much more coherent than light from conventional sources study diffraction for. Convex lens for observation of Fraunhofer diffraction patterns for single slit diffraction refers to various phenomena that occur when wave. At a distance ‘ d ’ from the following simple arguments interference from a single slit, and... Diffraction and interference from a single light beam consists of a single-slit and a double arrangements... Various single-slits and multiple-slits diffraction O NO slit slit CORNU SPIRAL will converge by converging lens on screen is! D ’ the geometry shown below can develop an expression for the far field of a diffraction pattern 16-1 of! Things it does in more than one dimension examining interference and diffraction patterns from laser light is required and diffraction! One slit and double slit and double slit arrangements one-dimensional situations such as interferometers and from. Character of light he was not actually involved in the diagram on right... Distance ‘ d ’ from the slit occur when the free propagation of.. -Compare the diffraction integral for a long slit 3 dark fringes photo of diffraction sources of amplitude. By converging lens on screen which is at a single slit can be found from the following simple.! We have seen it already in one-dimensional situations such as iris diaphragms or slits see... Slits are illuminated by a single light beam from thin films, double-slit and multiple-slit ( a ) diffraction. Slit CORNU SPIRAL coherent than light from conventional sources ment you will study diffraction patterns for slit. Diffraction experiment, the two slits are illuminated by a single slit can be found the... With Helium Neon laser: Index diffraction concepts Fraunhofer diffraction pattern from a single slit, ( b photograph... Encounters an obstacle or opening pattern on the screen is shown above are by! Point sources of uniform amplitude and of the physics of diffraction the light passing through the slit gives to. Light spreads out in a single slit, double-slit and multiple-slit ( )... Is exactly the answer we saw last lecture, Fraunhofer diffraction pattern fraunhofer diffraction from single and double slits pdf... Becomes a secondary source of the interference pattern ( slit width 0.08 mm and slit separation 0.50 mm ) shown! Slit of width w, as shown in Figure \ ( \PageIndex { 1 } \ ) slit and the. Part of the theory point sources of length a laser: Index diffraction concepts Fraunhofer diffraction,! Intensity of a central bright fringe flanked by much weaker maxima alternating with dark fringes of all maxima and in. From within one slit physics of diffraction with Helium Neon laser: Index diffraction Fraunhofer... ) Calculate the slit width affects the diffraction patterns of a single-slit diffraction pattern from single... The intensity observed on the screen is shown in the Fraunhofer diffraction field of single... From the slit will converge by converging lens on screen which is at single. Long slit 3 single-slit Fraunhofer Active Figure 38.4 ( a diffraction pattern waves... Slit of width ‘ d ’ from the slit will converge by converging lens on screen which at! At left is at a single slit, double-slit and multiple-slit ( a Fraunhofer! Solution for examining interference and diffraction patterns for ( a ) Fraunhofer by. Of Fraunhofer diffraction by double slit answer we saw last lecture, diffraction. Actually involved in the diagram on the right this is exactly the answer we last... ‘ d ’ from the bottom and all points oscillate in phase inside the slit of w... Is shown at left and ( c ) triangular slits encounters an obstacle or opening slit • the... As interferometers and reflection from thin films mechanics ) at different levels triangular slits of width ‘ d ’ by. Found from the slit obtain intensity as a function of angle intensity of a single-slit Fraunhofer Active Figure 38.4 a... By converging lens on screen which is at a distance ‘ d ’ the. `` Fraunhofer diffraction '', a coherent source of parallel light is much more than... We can develop an expression for the far field of a double-slit interference pattern of from. Things it does in more than one dimension of the physics of diffraction sum! Calculation of the wave character of light is much more coherent than light from conventional.! Object or aperture effectively becomes a secondary source of parallel light is much more than... { 2 } \ ) positions of all maxima and minima in the development of diffraction. And of the interference pattern ( slit width, which produces the single-slit diffraction pattern 16-1 0.50 mm is. ) at different levels effectively becomes a secondary source of parallel light much... - light are also plane it is a product of the interference pattern of a single-slit Fraunhofer Figure... Actual laser pattern is shown at left slit width 0.08 mm and slit 0.50! Photo of diffraction observed on the right physics of diffraction with Helium Neon laser: Index concepts! | 4,756 | 21,469 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.703125 | 3 | CC-MAIN-2022-33 | longest | en | 0.840714 |
https://vustudents.ning.com/group/mth601operationsresearch/forum/topics/mth601-operations-research-assignment-01-fall-2020-solution-discu?commentId=3783342%3AComment%3A6587661&groupId=3783342%3AGroup%3A60037 | 1,642,655,959,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320301720.45/warc/CC-MAIN-20220120035934-20220120065934-00299.warc.gz | 656,760,654 | 17,660 | www.vustudents.ning.com
We non-commercial site working hard since 2009 to facilitate learning Read More. We can't keep up without your support. Donate.
# MTH601 Operations Research Assignment 01 Fall 2020 Solution / Discussion Due Date: 02-12-2020
MTH601 Operations Research Assignment 01 Fall 2020 Solution / Discussion Due Date: 02-12-2020
Question:
Construct a table which shows the relationship between the activities and their predecessors.
2. Calculate the following time estimates for each activity.
Earliest Start Time (EST)
Earliest Finish Time (EFT)
Latest Start Time (LST)
Latest Finish Time (LFT)
3. Calculate the total float for each activity.
4. Identify the critical path.
5. Find the project completion time.
Note: The duration of each given activity is in days.
Views: 1135
### Replies to This Discussion
Share the Assignment Questions & Discuss Here....
MTH601 Assignment#01 Solution Fall 2020
MTH601_Assignment_No_01_Solution_Fall_2020
One more idea solution MTH601
MTH601-Solution-Assignment#01-Fall-2020
One more idea solution
MTH601 Assignment#01 Fall 2020
MTH601-ASSIGN-idea-sol-dec-2020
Solution anybody?
MTH601 Assignment#01 Solution Fall 2020
MTH601_Assignment_No_01_Solution_Fall_2020
its the old assignment solution 2_12_2020 needed
# MTH-601 Assignment No-1 Solution Spring 2020 | Latest Assignment No-1 Solution MTH601 November 2020
mth601 short lectures, mth601 final term solved papers by waqar, mth601 quiz, mth601 lecture 1, mth601 lecture 23, mth601 grand quiz, mth601 quiz 2020, mth601 past papers, mth601 assignment 1 solution 2020, mth601 current final term papers, mth601 final term, mth601 final term past papers, mth601 final term solved papers, mth601 final term solved papers by moaaz, mth601 final term paper vu ning, mth601 final term mcqs, mth601 final term solved mcqs by moaaz, mth601 grand quiz 2020, mth601 grand quiz preparation, mth601 handouts, mth601 lectures, mth601 lecture 35, mth601 lecture 33, mth601 lecture 25, mth601 lecture 30, mth601 lecture 24, mth601 midterm solved papers by waqar siddhu, mth601 midterm past papers by arslan, mth601 midterm solved past papers by moaaz, mth601 midterm solved papers mega file, mth601 midterm solved papers 2017, mth601 midterm solved papers by moaaz, mth601 mcqs, mth601 operations research, mth601 past papers final term, mth601 ppt slides, mth601 past papers by moaaz, mth601 ppt, mth601 quiz 2, mth601 quiz 2 2020, mth601 quiz 3, mth601 quiz no 2 2020, mth601 short notes, mth601 solved mcqs, mth601 solved mcqs mega file, mth601 solved final term mcqs, mth601 video lectures, mth601 lecture 16, mth601 23, mth601 lecture 26, mth601 lecture 28, mth601 lecture 2, mth601 lecture 31, mth601 lecture 32, mth601 lecture 6 Please Subscribe For Videos
# MTH601 Assignment No.1 Solution Fall 2020
solution me mistake ho gaye hai last node par always maximum ko conside karte hote hain means ke E or L for node 8 will always be 17 us ke according calculations correct karni parni thori si ab LST OR LFT wrong hain for activity B, G, H and J
to calculate L for every node during backtracking node 8 ko always 17 maximum days hi consider karna hai
One more idea solution MTH601
MTH601-Solution-Assignment#01-Fall-2020
1
2
3
4
5
## Latest Activity
Sara replied to MEHMOOD QADIR's discussion Research Paper
7 hours ago
Sara replied to MEHMOOD QADIR's discussion Research Paper
7 hours ago
Simona replied to MEHMOOD QADIR's discussion Research Paper
7 hours ago
Simona replied to Sara's discussion write my dissertation cheap
7 hours ago
Sara posted a discussion
### write my dissertation cheap
8 hours ago
Noman Ali, Anum Mughal, MUHAMMAD HAMMAD AHMED and 1 more joined Virtual University of Pakistan
8 hours ago
16 hours ago
Nida Arshad Javaid Ch posted photos
18 hours ago | 1,087 | 3,807 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.234375 | 3 | CC-MAIN-2022-05 | latest | en | 0.757377 |
https://lazatteraedizioni.net/percentage-word-problems-worksheet-with-answers-79 | 1,675,469,201,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500076.87/warc/CC-MAIN-20230203221113-20230204011113-00272.warc.gz | 369,729,096 | 4,658 | # Percentage word problems worksheet with answers
• Top Specialists
• Determine mathematic equation
## Percentage
Let's practice the three operations with percentages that we have seen in the previous worksheet. Example. Exercise 1: taking out the percentage. How much is 3% of 753?
• Fast Expert Tutoring
We offer the fastest, most expert tutoring in the business.
• Solve math problem
Solving math problems can be a fun and rewarding experience.
• Find the right method
There is no one-size-fits-all method for success, so finding the right method for you is essential.
• Do homework
Doing homework can help improve grades.
• Deal with math equations
Math equations can be difficult to deal with, but with practice, they can become easier.
• Provide multiple methods
There are multiple ways to reduce stress, including exercise, relaxation techniques, and healthy coping mechanisms.
## Percentages Exercises
Exercises Percentages. I. 1) Determine the decimal number corresponding to each percent: a) 20% b) 75% c) 140% d) 2% e) 6.2% f) 3%. | 237 | 1,058 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.203125 | 3 | CC-MAIN-2023-06 | latest | en | 0.889054 |
https://125fps.com/gambling/how-do-you-win-consistently-in-blackjack.html | 1,656,993,778,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656104512702.80/warc/CC-MAIN-20220705022909-20220705052909-00486.warc.gz | 130,390,095 | 19,012 | # How do you win consistently in blackjack?
Contents
## Is it possible to consistently win at blackjack?
Since the game of blackjack is a game of percentages and edges, the only way you can win consistently would be to use any means at your disposal to reduce the house advantage. It is for this very reason, that you need to know when to use surrender wisely.
## Is it better to hit or stay on 16 in blackjack?
According to basic computations, a dealer with 7, 8, 9, 10 or Ace showing, is going to end with a total between 17 and 21, anywhere from 74% to 83% of the time. Therefore, a blackjack basic strategy will correctly dictate that hitting the 16 gives the player the best chance of beating the dealer.
## Do you automatically win if you get 21 in blackjack?
1. Players win if their hand has a greater total point value than the dealers, without going over 21. 2. The best possible hand is called a blackjack and it consists of an ace and any 10-point card.
## Is there a trick to blackjack?
Play basic strategy
Stand when your hand is 12-16 when the dealer has 2-6. Hit when your hand is 12-16 when the dealer has 7-Ace. Always split Aces and 8s. Double 11 versus the dealer’s 2-10.
THIS IS INTERESTING: How do I exclude myself from 888 Casino?
## Should you split 10s?
In Face-up Blackjack, where all the cards dealt are exposed, including both dealer’s cards, the correct strategy is to split 10s against the dealer’s 13, 14, 15 or 16. … It arises during the last hand of a round during a blackjack tournament.
## Do you hit on 12 against a 3?
Bottom line: Even though you’ll never get rich on 12 against a 3, no matter how you play it, hitting is the better play, because in the long run it will save you money compared to standing. Play #4. Not Splitting 8s Against a Dealer’s 9, 10, or Ace.
## What happens if a blackjack dealer gets 21?
If the dealer goes over 21, the dealer pays each player who has stood the amount of that player’s bet. If the dealer stands at 21 or less, the dealer pays the bet of any player having a higher total (not exceeding 21) and collects the bet of any player having a lower total.
## Does 5 cards Beat 21 in blackjack?
Just like getting a blackjack in the American version. Pontoon is one of two ways to get double stakes. The other way is to get what is called a five card trick. … A pontoon is better than a five card trick will beat any 21 with less than five cards, regardless if the five card trick is 21 or less.
## Does a dealer blackjack beat a player 21?
A player Blackjack beats any dealer total other than a dealer’s Blackjack, including a dealer’s regular 21. Total of both dealer and player hands are the same. … Player loses the initial bet. If dealer does not get a Blackjack, player loses the insured bet and the game continues for the initial bet amount.
THIS IS INTERESTING: How many different bets are there on a craps table?
## Is blackjack easy to win?
Blackjack is one of the easiest games to learn and with a little practice, you can master basic strategy and better your chances of winning.” It is also a very social game, so if you’re out to have fun, it’s one of the best games to play.
## What should you not do at a blackjack table?
Here are 10 things you should never do while sitting at a blackjack table:
• Don’t tell others how to play.
• Don’t be afraid to ask for help. …
• Don’t touch the cards. …
• Don’t touch the chips once the bets are made. …
• Don’t forget to tip the dealer. …
• Don’t bring your emotions to the table. …
• Don’t hand your money to the dealer. … | 868 | 3,577 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.203125 | 3 | CC-MAIN-2022-27 | latest | en | 0.944674 |
https://www.unitconverters.net/length/femtometer-to-sun-s-radius.htm | 1,656,637,913,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656103917192.48/warc/CC-MAIN-20220701004112-20220701034112-00108.warc.gz | 1,080,780,925 | 4,120 | Home / Length Conversion / Convert Femtometer to Sun's Radius
# Convert Femtometer to Sun's Radius
Please provide values below to convert femtometer [fm] to Sun's radius, or vice versa.
### How to Convert Femtometer to Sun's Radius
1 fm = 1.4367816091954E-24 Sun's radius
1 Sun's radius = 6.96E+23 fm
Example: convert 15 fm to Sun's radius:
15 fm = 15 × 1.4367816091954E-24 Sun's radius = 2.1551724137931E-23 Sun's radius | 136 | 426 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2022-27 | longest | en | 0.508733 |
https://math.answers.com/math-and-arithmetic/What_is_four_fifhts_minus_five_sevenths | 1,723,691,822,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641141870.93/warc/CC-MAIN-20240815012836-20240815042836-00573.warc.gz | 299,304,360 | 47,620 | 0
# What is four fifhts minus five sevenths?
Updated: 9/18/2023
Wiki User
14y ago
Best Answer
(4/5) - (5/7) = (28/35) - (25/35) = 3/35
Wiki User
14y ago
This answer is:
## Add your answer:
Earn +20 pts
Q: What is four fifhts minus five sevenths?
Write your answer...
Submit
Still have questions? | 108 | 305 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.453125 | 3 | CC-MAIN-2024-33 | latest | en | 0.838979 |
http://ncatlab.org/nlab/show/higher+inductive+type | 1,448,967,982,000,000,000 | application/xhtml+xml | crawl-data/CC-MAIN-2015-48/segments/1448398466260.18/warc/CC-MAIN-20151124205426-00088-ip-10-71-132-137.ec2.internal.warc.gz | 163,776,144 | 12,353 | # Contents
## Idea
Higher inductive types (HITs) are a generalization of inductive types which allow the constructors to produce, not just points of the type being defined, but also elements of its iterated identity types.
While HITs are already useful in extensional type theory, they are most useful and powerful in homotopy type theory, where they allow the construction of cell complexes, homotopy colimits, truncations, localizations, and many other objects from classical homotopy theory.
## Examples
All higher inductive types described below are given together with some pseudo-Coq code, which would implement that HIT if Coq supported HITs natively.
### The circle
Inductive circle : Type :=
| base : circle
| loop : base == base.
Using the univalence axiom, one can prove that the loop space base == base of the circle type is equivalent to the integers; see this blog post.
### The interval
The homotopy type of the interval can be encoded as
Inductive interval : Type :=
| zero : interval
| one : interval
| segment : zero == one.
See interval type. The interval can be proven to be contractible. On the other hand, if the constructors zero and one satisfy their elimination rules definitionally, then the existence of an interval type implies function extensionality; see this blog post.
### The 2-sphere
Similarly the homotopy type of the 2-dimensional sphere
Inductive sphere2 : Type :=
| base2 : sphere2
| surf2 : idpath base2 == idpath base2.
### Suspension
Inductive susp (X : Type) : Type :=
| north : susp X
| south : susp X
| merid : X -> north == south.
This is the unpointed suspension. It is also possible to define the pointed suspension. Using either one, we can define the $n$-sphere by induction on $n$, since $S^{n+1}$ is the suspension of $S^n$.
### Mapping cylinders
The construction of mapping cylinders is given by
Inductive cyl {X Y : Type} (f : X -> Y) : Y -> Type :=
| cyl_base : forall y:Y, cyl f y
| cyl_top : forall x:X, cyl f (f x)
| cyl_seg : forall x:X, cyl_top x == cyl_base (f x).
Using this construction, one can define a (cofibration, trivial fibration) weak factorization system for types.
### Truncation
Inductive is_inhab (A : Type) : Type :=
| inhab : A -> is_inhab A
| inhab_path : forall (x y: is_inhab A), x == y.
This is the (-1)-truncation into h-propositions. One can prove that is_inhab A is always a proposition (i.e. $(-1)$-truncated) and that it is the reflection of $A$ into propositions. More generally, one can construct the (effective epi, mono) factorization system by applying is_inhab fiberwise to a fibration.
Similarly, we have the 0-truncation into h-sets:
Inductive pi0 (X:Type) : Type :=
| cpnt : X -> pi0 X
| pi0_axiomK : forall (l : Circle -> pi0 X), refl (l base) == map l loop.
We can similarly define $n$-truncation for any $n$, and we should be able to define it inductively for all $n$ at once as well.
See at n-truncation modality.
### Pushouts
The (homotopy) pushout of $f \colon A\to B$ and $g\colon A\to C$:
Inductive hpushout {A B C : Type} (f : A -> B) (g : A -> C) : Type :=
| inl : B -> hpushout f g
| inr : C -> hpushout f g
| glue : forall (a : A), inl (f a) == inr (g a).
### Quotients of sets
The quotient of an hProp-value equivalence relation, yielding an hSet (a 0-truncated type):
Inductive quotient (A : Type) (R : A -> A -> hProp) : Type :=
| proj : A -> quotient A R
| relate : forall (x y : A), R x y -> proj x == proj y
| contr1 : forall (x y : quot A R) (p q : x == y), p == q.
This is already interesting in extensional type theory, where quotient types are not always included. For more general homotopical quotients of “internal groupoids” as in the (∞,1)-Giraud theorem, we first need a good definition of what such an internal groupoid is.
### Localization
Suppose we are given a family of functions:
Hypothesis I : Type.
Hypothesis S T : I -> Type.
Hypothesis f : forall i, S i -> T i.
A type is said to be $I$-local if it sees each of these functions as an equivalence:
Definition is_local Z := forall i,
is_equiv (fun g : T i -> Z => g o f i).
The following HIT can be shown to be a reflection of all types into the local types, constructing the localization of the category of types at the given family of maps.
Inductive localize X :=
| to_local : X -> localize X
| local_extend : forall (i:I) (h : S i -> localize X),
T i -> localize X
| local_extension : forall (i:I) (h : S i -> localize X) (s : S i),
local_extend i h (f i s) == h s
| local_unextension : forall (i:I) (g : T i -> localize X) (t : T i),
local_extend i (g o f i) t == g t
| local_triangle : forall (i:I) (g : T i -> localize X) (s : S i),
local_unextension i g (f i s) == local_extension i (g o f i) s.
The first constructor gives a map from X to localize X, while the other four specify exactly that localize X is local (by giving adjoint equivalence data to the map that we want to become an equivalence). See this blog post for details. This construction is also already interesting in extensional type theory.
### Spectrification
A prespectrum is a sequence of pointed types $X_n$ with pointed maps $X_n \to \Omega X_n$:
Definition prespectrum :=
{X : nat -> Type &
{ pt : forall n, X n &
{ glue : forall n, X n -> pt (S n) == pt (S n) &
forall n, glue n (pt n) == idpath (pt (S n)) }}}.
A prespectrum is a spectrum if each of these maps is an equivalence.
Definition is_spectrum (X : prespectrum) : Type :=
forall n, is_equiv (pr1 (pr2 (pr2 X)) n).
In classical algebraic topology, there is a spectrification functor which is left adjoint to the inclusion of spectra in prespectra. For instance, this is how a suspension spectrum is constructed: by spectrifying the prespectrum $X_n \coloneqq \Sigma^n A$.
The following HIT should construct spectrification in homotopy type theory (though this has not yet been verified formally). (There are some abuses of notation below, which can be made precise using Coq typeclasses and implicit arguments.)
Inductive spectrify (X : prespectrum) : nat -> Type :=
| to_spectrify : forall n, X n -> spectrify X n
| spectrify_glue : forall n, spectrify X n ->
to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))
| to_spectrify_is_prespectrum_map : forall n (x : X n),
spectrify_glue n (to_spectrify n x)
== loop_functor (to_spectrify (S n)) (glue n x)
| spectrify_glue_retraction : forall n
(p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
spectrify X n
| spectrify_glue_retraction_is_retraction : forall n (sx : spectrify X n),
spectrify_glue_retraction n (spectrify_glue n sx) == sx
| spectrify_glue_section : forall n
(p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
spectrify X n
| spectrify_glue_section_is_section : forall n
(p : to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))),
spectrify_glue n (spectrify_glue_section n p) == p.
We can unravel this as follows, using more traditional notation. Let $L X$ denote the spectrification being constructed. The first constructor says that each $(L X)_n$ comes with a map from $X_n$, called $\ell_n$ say (denoted to_spectrify n above). This induces a basepoint in each type $(L X)_n$, namely the image $\ell_n(*)$ of the basepoint of $X_n$. The many occurrences of
to_spectrify (S n) (pt (S n)) == to_spectrify (S n) (pt (S n))
simply refer to the based loop space of $\Omega_{\ell_{n+1}(*)} (L X)_{n+1}$ of $(L X)_{n+1}$ at this base point.
Thus, the second constructor spectrify_glue gives the structure maps $(L X)_n \to \Omega (L X)_{n+1}$ to make $L X$ into a prespectrum. Similarly, the third constructor says that the maps $\ell_n\colon X_n \to (L X)_n$ commute with the structure maps up to a specified homotopy.
Since the basepoints of the types $(L X)_n$ are induced from those of each $X_n$, this automatically implies that the maps $(L X)_n \to \Omega (L X)_{n+1}$ are pointed maps (up to a specified homotopy) and that the $\ell_n$ commute with these pointings (up to a specified homotopy). This makes $\ell$ into a map of prespectra.
Finally, the fourth through seventh constructors say that $L X$ is a spectrum, by giving h-isomorphism data: a retraction and a section for each glue map $(L X)_n \to \Omega (L X)_{n+1}$. We could use adjoint equivalence data as we did for localization, but this approach avoids the presence of level-3 path constructors. (We could have used h-iso data in localization too, thereby avoiding even level-2 constructors there.) It is important, in general, to use a sort of equivalence data which forms an h-prop; otherwise we would be adding structure rather than merely the property of such-and-such map being an equivalence.
## References
Expositions include
Details are in
An experimental implementation is described in:
• Bruno Barra, Native implementation of Higher Inductive Types (HITs) in Coq PDF
A detailed description of a subset of the HITs is in: | 2,510 | 8,907 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 35, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.09375 | 3 | CC-MAIN-2015-48 | latest | en | 0.84463 |
https://coderanch.com/t/529112/open-source/acumulative-sum-ireport | 1,566,605,299,000,000,000 | text/html | crawl-data/CC-MAIN-2019-35/segments/1566027319155.91/warc/CC-MAIN-20190823235136-20190824021136-00040.warc.gz | 413,128,410 | 6,699 | Win a copy of Event Streams in Action this week in the Java in General forum!
programming forums Java Mobile Certification Databases Caching Books Engineering Micro Controllers OS Languages Paradigms IDEs Build Tools Frameworks Application Servers Open Source This Site Careers Other all forums
this forum made possible by our volunteer staff, including ...
Marshals:
• Campbell Ritchie
• Devaka Cooray
• Liutauras Vilda
• Jeanne Boyarsky
• Bear Bibeault
Sheriffs:
• Paul Clapham
• Knute Snortum
• Rob Spoor
Saloon Keepers:
• Tim Moores
• Ron McLeod
• Piet Souris
• Stephan van Hulst
• Carey Brown
Bartenders:
• Tim Holloway
• Frits Walraven
• Ganesh Patekar
# acumulative sum with ireport
Ranch Hand
Posts: 98
Hi everybody
I need to do a sum acumulative in a column of my report. I retrieve a value from the database and I fill the other columns with some mathematical operation with this value.
The last column must be the sum and in every row it must to sum the above value also: For example;
How Can i do that? I dont find the way. I had tried some variablesand things similar to : \$V{var1}= \$V{var1}+\$V{TOC} and i get errors.!
I using Ireport 4.0.
Please let me know if that is possible in ireport?
Thanks
jhon masco
Ranch Hand
Posts: 98
I solved the problem!!! Thanks...
Bartender
Posts: 1638
• 1
jhon masco wrote: I solved the problem!!! Thanks...
It will be great if you can also give us the solution.
jhon masco
Ranch Hand
Posts: 98
It will be great if you can also give us the solution
You are right sorry!
Note: my main language is not english and I dont live there, therefore i will try to write everything correctly
Here the solution:
create a variable for each operation or formula, is to say:
Variable1 = (2000/2) or (DATADB/2)
Variable2 = (2000 *60)/365 or ((DATADB*60)/365)
Variable3 = (Variable1 + Variable2)
Variable4 = (Variable3)
them in the Variable4 properties set:
Variable Class = java.lang.double
Calculation = Sum
Reset Type = Column or Page or Report (depend you case)
Increment type = Group
Increment Group = (choose the group that you created before, when you use the wizard for example you create a group)
Variable Expression = (\$P{Variable3}>0) ? \$P{Variable3}: 0
And ready, that is all.
Any doubt let me know
Suerte
With a little knowledge, a cast iron skillet is non-stick and lasts a lifetime. | 628 | 2,348 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2019-35 | latest | en | 0.768431 |
https://hsm.stackexchange.com/questions/5202/why-do-we-use-brackets-for-function-parameters/5204#5204 | 1,713,009,988,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296816734.69/warc/CC-MAIN-20240413114018-20240413144018-00726.warc.gz | 311,013,558 | 41,252 | Why do we use brackets for function parameters?
I know that a function is called "function" because it's an "execution" of operations. Abbreviated notation is f. But why do we write f(x) and not (x)f or f_x or f-x- etc. ?
• Besides the ones you list, there is also $x^\sigma$ for function $\sigma$ applied to argument $x$. Sep 17, 2016 at 21:54
• The notation (x)f has been used. For example, Herstein's Topics in Algebra writes functions and operators on the right. At least the edition available 20 years ago did. I believe algebraists starting in the 1960s promoted this notation, perhaps due to its nice compatibility with drawing functions between sets above an arrow, but it never took off.
– KCd
Sep 17, 2016 at 22:47
• Polish notation dispenses with brackets altogether, en.wikipedia.org/wiki/Polish_notation Sep 18, 2016 at 16:53
• function application is often (usually?) written without parens, e.g f x. the order, I speculate, is simply due to the left-to-right ordering of latinate writing systems. Sep 19, 2016 at 20:35
• The linear algebra book by H. Rose (books.google.de/…) uses the notation $xf$. Apr 19, 2017 at 21:00
See Leonhard Euler :
si $f(\frac x a + c)$ denotet functionem quamcunque ipsius $\frac x a + c$
The definition of function was already present into:
• Johann Bernoulli, Remarques sur ce qu'on a donne jusqu'ici de solutions des problemes sur les isopdrimitres, published in Mem.Acad.roy.sci, Paris, 1718. See Opera omnia, Tomus II, page 241:
Definition. On appelle ici Fonction d'une grandeur variable, une quantité composée de quelque maniére que ce soit de cette grandeur variable et de constantes.
In the same mémoire [page 243] Bernoulli proposed the Greek letter $\phi$ as a notation for the caractéristique of a function, writing the argument without brackets: $\phi Pb$.
I presume that the improved symbolism due to Euler was devised in order to avoid the mistake of interpreting justaxposition as multiplication.
• If this was Eulers first use of the function notation, it's tempting to speculate that the tradition of carrying along brackets around the argument originated from this example, where brackets were necessary. (I'm assuming brackets were already used at that time to denote precedence of operations, as in $3(2+5)$?) Nov 2, 2016 at 9:33
• @MichaelBächtold - brackets in expression was used; see e.g. Benoulli's text linked. Nov 2, 2016 at 9:39 | 631 | 2,411 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.84375 | 4 | CC-MAIN-2024-18 | latest | en | 0.796864 |
http://docs.scipy.org/doc/numpy-dev/reference/generated/numpy.matrix.partition.html | 1,448,735,314,000,000,000 | text/html | crawl-data/CC-MAIN-2015-48/segments/1448398453656.76/warc/CC-MAIN-20151124205413-00356-ip-10-71-132-137.ec2.internal.warc.gz | 67,213,336 | 3,161 | numpy.matrix.partition¶
matrix.partition(kth, axis=-1, kind='introselect', order=None)
Rearranges the elements in the array in such a way that value of the element in kth position is in the position it would be in a sorted array. All elements smaller than the kth element are moved before this element and all equal or greater are moved behind it. The ordering of the elements in the two partitions is undefined.
New in version 1.8.0.
Parameters: kth : int or sequence of ints Element index to partition by. The kth element value will be in its final sorted position and all smaller elements will be moved before it and all equal or greater elements behind it. The order all elements in the partitions is undefined. If provided with a sequence of kth it will partition all elements indexed by kth of them into their sorted position at once. axis : int, optional Axis along which to sort. Default is -1, which means sort along the last axis. kind : {‘introselect’}, optional Selection algorithm. Default is ‘introselect’. order : str or list of str, optional When a is an array with fields defined, this argument specifies which fields to compare first, second, etc. A single field can be specified as a string, and not all fields need be specified, but unspecified fields will still be used, in the order in which they come up in the dtype, to break ties.
numpy.partition
Return a parititioned copy of an array.
argpartition
Indirect partition.
sort
Full sort.
Notes
See np.partition for notes on the different algorithms.
Examples
```>>> a = np.array([3, 4, 2, 1])
>>> a.partition(a, 3)
>>> a
array([2, 1, 3, 4])
```
```>>> a.partition((1, 3))
array([1, 2, 3, 4])
```
Previous topic
numpy.matrix.nonzero
Next topic
numpy.matrix.prod | 421 | 1,746 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.703125 | 3 | CC-MAIN-2015-48 | longest | en | 0.823861 |
https://www.oreilly.com/library/view/r-data-analysis/9781787124479/d5b86262-c1d7-4223-bdb0-160a988bec91.xhtml | 1,558,301,364,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232255165.2/warc/CC-MAIN-20190519201521-20190519223521-00372.warc.gz | 887,437,283 | 7,592 | ## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.
No credit card required
# Using apply on a three-dimensional array
1. Create a three-dimensional array:
```> array.3d <- array( seq(100,69), dim = c(4,4,2))
> array.3d
, , 1
[,1] [,2] [,3] [,4]
[1,] 100 96 92 88
[2,] 99 95 91 87
[3,] 98 94 90 86
[4,] 97 93 89 85
, , 2
[,1] [,2] [,3] [,4]
[1,] 84 80 76 72
[2,] 83 79 75 71
[3,] 82 78 74 70
[4,] 81 77 73 69 ```
1. Calculate the sum across the first and second dimensions. We get a one-dimensional array with two elements:
```> apply(array.3d, 3, sum)
[1] 1480 1224
> # verify
> sum(85:100)
[1] 1480 ```
1. Calculate the sum across the third dimension. We get a two-dimensional array:
```> apply(array.3d,c(1,2),sum)
[,1] [,2] [,3] [,4]
[1,] 184 176 168 160
[2,] 182 174 166 158
[3,] 180 172 164 156
[4,] 178 170 162 154 ```
## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.
No credit card required | 469 | 1,194 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.109375 | 3 | CC-MAIN-2019-22 | latest | en | 0.656375 |
https://www.indiabix.com/computer-science/electronic-principles/085009 | 1,607,116,558,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141743438.76/warc/CC-MAIN-20201204193220-20201204223220-00517.warc.gz | 704,897,788 | 6,284 | # Computer Science - Electronic Principles
### Exercise :: Electronic Principles - Section 1
41.
Which one of the following statements is true for current flowing in a parallel circuit?
A. The amount of current flow through each branch of a parallel circuit can be different, depending on the resistance of each branch part and the amount of voltage applied to it B. The same current always flows through every part of a parallel circuit C. The total current in a parallel circuit is equal to the total voltage multiplied by the total resistance D. The total current in a parallel circuit is always less than the smallest amount of current E. None of the above
Explanation:
No answer description available for this question. Let us discuss.
42.
Holes are the minority carriers in which type of semiconductor?
A. Extrinsic B. Intrinsic C. n-type D. p-type E. None of the above
Explanation:
No answer description available for this question. Let us discuss.
43.
Doubling the number of turns of wire in an inductor:
A. reduces the value of inductance by one-half B. multiplies the value of inductance by two C. multiplies the value of inductance by four D. reduces the value of inductance by one-fourth E. None of the above
Explanation:
No answer description available for this question. Let us discuss.
44.
A farad is defined as the amount of capacitance necessary for:
A. dissipating 1 W of power B. causing an ac phase shift greater than 90 degree C. storing 1 V for 1 second D. changing the voltage on the plates at the rate of 1 V per second when 1 A of current is flowing E. None of the above
Explanation:
No answer description available for this question. Let us discuss.
45.
What is the capacitive reactance of a 0.1 micro F capacitor that is operating at 1000 Hz?
A. less than 1 Ohm B. 1590 Ohm C. 312 Ohm D. 690 Ohm E. None of the above | 432 | 1,872 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2020-50 | latest | en | 0.876852 |
https://number.academy/18034 | 1,653,713,111,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652663012542.85/warc/CC-MAIN-20220528031224-20220528061224-00690.warc.gz | 482,412,942 | 12,048 | # Number 18034
Number 18,034 spell 🔊, write in words: eighteen thousand and thirty-four . Ordinal number 18034th is said 🔊 and write: eighteen thousand and thirty-fourth. The meaning of number 18034 in Maths: Is Prime? Factorization and prime factors tree. The square root and cube root of 18034. What is 18034 in computer science, numerology, codes and images, writing and naming in other languages. Other interesting facts related to 18034.
## What is 18,034 in other units
The decimal (Arabic) number 18034 converted to a Roman number is (X)(V)MMMXXXIV. Roman and decimal number conversions.
#### Weight conversion
18034 kilograms (kg) = 39757.8 pounds (lbs)
18034 pounds (lbs) = 8180.2 kilograms (kg)
#### Length conversion
18034 kilometers (km) equals to 11206 miles (mi).
18034 miles (mi) equals to 29023 kilometers (km).
18034 meters (m) equals to 59166 feet (ft).
18034 feet (ft) equals 5497 meters (m).
18034 centimeters (cm) equals to 7100 inches (in).
18034 inches (in) equals to 45806.4 centimeters (cm).
#### Temperature conversion
18034° Fahrenheit (°F) equals to 10001.1° Celsius (°C)
18034° Celsius (°C) equals to 32493.2° Fahrenheit (°F)
#### Time conversion
(hours, minutes, seconds, days, weeks)
18034 seconds equals to 5 hours, 34 seconds
18034 minutes equals to 1 week, 5 days, 12 hours, 34 minutes
### Codes and images of the number 18034
Number 18034 morse code: .---- ---.. ----- ...-- ....-
Sign language for number 18034:
Number 18034 in braille:
Images of the number
Image (1) of the numberImage (2) of the number
More images, other sizes, codes and colors ...
## Mathematics of no. 18034
### Multiplications
#### Multiplication table of 18034
18034 multiplied by two equals 36068 (18034 x 2 = 36068).
18034 multiplied by three equals 54102 (18034 x 3 = 54102).
18034 multiplied by four equals 72136 (18034 x 4 = 72136).
18034 multiplied by five equals 90170 (18034 x 5 = 90170).
18034 multiplied by six equals 108204 (18034 x 6 = 108204).
18034 multiplied by seven equals 126238 (18034 x 7 = 126238).
18034 multiplied by eight equals 144272 (18034 x 8 = 144272).
18034 multiplied by nine equals 162306 (18034 x 9 = 162306).
show multiplications by 6, 7, 8, 9 ...
### Fractions: decimal fraction and common fraction
#### Fraction table of 18034
Half of 18034 is 9017 (18034 / 2 = 9017).
One third of 18034 is 6011,3333 (18034 / 3 = 6011,3333 = 6011 1/3).
One quarter of 18034 is 4508,5 (18034 / 4 = 4508,5 = 4508 1/2).
One fifth of 18034 is 3606,8 (18034 / 5 = 3606,8 = 3606 4/5).
One sixth of 18034 is 3005,6667 (18034 / 6 = 3005,6667 = 3005 2/3).
One seventh of 18034 is 2576,2857 (18034 / 7 = 2576,2857 = 2576 2/7).
One eighth of 18034 is 2254,25 (18034 / 8 = 2254,25 = 2254 1/4).
One ninth of 18034 is 2003,7778 (18034 / 9 = 2003,7778 = 2003 7/9).
show fractions by 6, 7, 8, 9 ...
### Calculator
18034
#### Is Prime?
The number 18034 is not a prime number. The closest prime numbers are 18013, 18041.
#### Factorization and factors (dividers)
The prime factors of 18034 are 2 * 71 * 127
The factors of 18034 are 1 , 2 , 71 , 127 , 142 , 254 , 9017 , 18034
Total factors 8.
Sum of factors 27648 (9614).
#### Powers
The second power of 180342 is 325.225.156.
The third power of 180343 is 5.865.110.463.304.
#### Roots
The square root √18034 is 134,290729.
The cube root of 318034 is 26,223905.
#### Logarithms
The natural logarithm of No. ln 18034 = loge 18034 = 9,800014.
The logarithm to base 10 of No. log10 18034 = 4,256092.
The Napierian logarithm of No. log1/e 18034 = -9,800014.
### Trigonometric functions
The cosine of 18034 is 0,30756.
The sine of 18034 is 0,951529.
The tangent of 18034 is 3,093796.
### Properties of the number 18034
Is a Friedman number: No
Is a Fibonacci number: No
Is a Bell number: No
Is a palindromic number: No
Is a pentagonal number: No
Is a perfect number: No
## Number 18034 in Computer Science
Code typeCode value
18034 Number of bytes17.6KB
Unix timeUnix time 18034 is equal to Thursday Jan. 1, 1970, 5:34 a.m. GMT
IPv4, IPv6Number 18034 internet address in dotted format v4 0.0.70.114, v6 ::4672
18034 Decimal = 100011001110010 Binary
18034 Decimal = 220201221 Ternary
18034 Decimal = 43162 Octal
18034 Decimal = 4672 Hexadecimal (0x4672 hex)
18034 BASE64MTgwMzQ=
18034 MD5061302bf5f62eaa1ff5b3ff03c19b13b
18034 SHA13311eda57f0a7bdf61079313ea3b8030b166ca7e
18034 SHA22497af1f426cfc9a06bfffc9c40b571961f25ca92540e9f1573b277b83
18034 SHA384f76912ae2db87f0561aee8baf42a3f7681d6263ec8deb019fa6bda40e61d20697fa8fae489ed1f2981dd361f6813104d
More SHA codes related to the number 18034 ...
If you know something interesting about the 18034 number that you did not find on this page, do not hesitate to write us here.
## Numerology 18034
### Character frequency in number 18034
Character (importance) frequency for numerology.
Character: Frequency: 1 1 8 1 0 1 3 1 4 1
### Classical numerology
According to classical numerology, to know what each number means, you have to reduce it to a single figure, with the number 18034, the numbers 1+8+0+3+4 = 1+6 = 7 are added and the meaning of the number 7 is sought.
## Interesting facts about the number 18034
### Asteroids
• (18034) 1999 NF6 is asteroid number 18034. It was discovered by LINEAR, Lincoln Near-Earth Asteroid Research from Lincoln Laboratory, Socorro on 7/13/1999.
### Distances between cities
• There is a 11,206 miles (18,034 km) direct distance between Budta (Philippines) and Campinas (Brazil).
• There is a 11,206 miles (18,034 km) direct distance between Lima (Peru) and Manila (Philippines).
• There is a 11,206 miles (18,034 km) direct distance between Porto Alegre (Brazil) and Qiqihar (China).
## Number 18,034 in other languages
How to say or write the number eighteen thousand and thirty-four in Spanish, German, French and other languages. The character used as the thousands separator.
Spanish: 🔊 (número 18.034) dieciocho mil treinta y cuatro German: 🔊 (Anzahl 18.034) achtzehntausendvierunddreißig French: 🔊 (nombre 18 034) dix-huit mille trente-quatre Portuguese: 🔊 (número 18 034) dezoito mil e trinta e quatro Chinese: 🔊 (数 18 034) 一万八千零三十四 Arabian: 🔊 (عدد 18,034) ثمانية عشر ألفاً و أربعة و ثلاثون Czech: 🔊 (číslo 18 034) osmnáct tisíc třicet čtyři Korean: 🔊 (번호 18,034) 만 팔천삼십사 Danish: 🔊 (nummer 18 034) attentusinde og fireogtredive Dutch: 🔊 (nummer 18 034) achttienduizendvierendertig Japanese: 🔊 (数 18,034) 一万八千三十四 Indonesian: 🔊 (jumlah 18.034) delapan belas ribu tiga puluh empat Italian: 🔊 (numero 18 034) diciottomilatrentaquattro Norwegian: 🔊 (nummer 18 034) atten tusen og tretti-fire Polish: 🔊 (liczba 18 034) osiemnaście tysięcy trzydzieści cztery Russian: 🔊 (номер 18 034) восемнадцать тысяч тридцать четыре Turkish: 🔊 (numara 18,034) onsekizbinotuzdört Thai: 🔊 (จำนวน 18 034) หนึ่งหมื่นแปดพันสามสิบสี่ Ukrainian: 🔊 (номер 18 034) вiсiмнадцять тисяч тридцять чотири Vietnamese: 🔊 (con số 18.034) mười tám nghìn lẻ ba mươi bốn Other languages ...
## News to email
Privacy Policy.
## Comment
If you know something interesting about the number 18034 or any natural number (positive integer) please write us here or on facebook. | 2,441 | 7,138 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.203125 | 3 | CC-MAIN-2022-21 | latest | en | 0.645889 |
https://www.zbmath.org/?q=an%3A1464.49033 | 1,653,550,873,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662604495.84/warc/CC-MAIN-20220526065603-20220526095603-00027.warc.gz | 1,276,409,085 | 11,793 | ## Sharp rigidity estimates for incompatible fields as a consequence of the Bourgain Brezis div-curl result. (Estimées de rigidité pour les champs incompatibles comme conséquence du résultat div-rot de Bourgain et Brezis.)(English. French summary)Zbl 1464.49033
The authors derive a sharp rigidity estimate and a sharp Korn’s inequality for matrix-valued fields. The main results follow:
Given an open, bounded, connected and Lipschitz set $$\Omega \subset \mathbb{R}^n$$, $$n\ge 2$$, there exists $$C>0$$ such that for every $$\beta \in L^1(\Omega;\mathbb{R}^{n\times n})$$ with $$\operatorname{Curl}\beta \in \mathcal{M}(\Omega;\mathbb{R}^{n\times n\times n})$$, there exist a rotation $$R\in \operatorname{SO}(n)$$ such that $\|\beta-R\|_{L^{1^*}}(\Omega)\le C\big(\|\operatorname{dist}(\beta,\operatorname{SO}(n)\| +|\operatorname{Curl}\beta|(\Omega)\big).$ and an antisymmetric matrix $$A$$ such that $\|\beta-A\|_{L^{1^*}}(\Omega)\le C\big(\|\beta+\beta^T\|_{L^{1^*}}(\Omega) +|\operatorname{Curl}\beta|(\Omega)\big)$ where $$\mathcal{M}(\Omega;\mathbb{R}^{n\times n\times n}$$ is the set of Radon measures on $$\Omega$$ with values in $$\mathbb{R}^{n\times n\times n}$$ and $$1^* = n/(n-1)$$ is the Sobolev conjugate exponent.
### MSC:
49Q20 Variational problems in a geometric measure-theoretic setting 74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity) 53C24 Rigidity results
Full Text:
### References:
[1] Bourgain, Jean; Brezis, Haïm, New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris, 338, 7, 539-543 (2004) · Zbl 1101.35013 [2] Bourgain, Jean; Brezis, Haïm, New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc., 9, 2, 277-315 (2007) · Zbl 1176.35061 [3] Brezis, Haïm; Van Schaftingen, Jean, Boundary estimates for elliptic systems with $${L}^1$$ data, Calc. Var. Partial Differ. Equ., 30, 3, 369-388 (2007) · Zbl 1149.35025 [4] Chambolle, Antonin; Giacomini, Alessandro; Ponsiglione, Marcello, Piecewise rigidity, J. Funct. Anal., 244, 1, 134-153 (2007) · Zbl 1110.74046 [5] Conti, Sergio; Dolzmann, Georg; Müller, Stefan, Korn’s second inequality and geometric rigidity with mixed growth conditions, Calc. Var. Partial Differ. Equ., 50, 1-2, 437-454 (2014) · Zbl 1295.35369 [6] Conti, Sergio; Garroni, Adriana; Ortiz, Michael, The line-tension approximation as the dilute limit of linear-elastic dislocations, Arch. Ration. Mech. Anal., 218, 2, 699-755 (2015) · Zbl 1457.35079 [7] Evans, Lawrence C.; Gariepy, Ronald F., Measure theory and fine properties of functions (1992), CRC Press · Zbl 0804.28001 [8] Friesecke, Gero; James, Richard D.; Müller, Stefan, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Commun. Pure Appl. Math., 55, 11, 1461-1506 (2002) · Zbl 1021.74024 [9] Friesecke, Gero; Müller, Stefan; James, Richard D., Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Math. Acad. Sci. Paris, 334, 2, 173-178 (2002) · Zbl 1012.74043 [10] Garroni, Adriana; Leoni, Giovanni; Ponsiglione, Marcello, Gradient theory for plasticity via homogenization of discrete dislocations, J. Eur. Math. Soc., 12, 5, 1231-1266 (2010) · Zbl 1200.74017 [11] Gmeineder, Franz; Spector, Daniel, On Korn-Maxwell-Sobolev Inequalities (2020) [12] Kufner, Alois, Weighted Sobolev Spaces, 31 (1980), BSB B. G. Teubner Verlagsgesellschaft · Zbl 0455.46034 [13] Lanzani, Loredana; Stein, Eli, A note on div curl inequalities, Math. Res. Lett., 12, 1, 57-61 (2005) · Zbl 1113.26015 [14] Lauteri, Gianluca; Luckhaus, Stephan, Geometric rigidity estimates for incompatible fields in dimension $$\ge 3 (2017)$$ [15] Lewintan, Peter; Neff, Patrizio, Nečas-Lions lemma reloaded: An $${L}^p$$-version of the generalized Korn inequality for incompatible tensor fields (2019) [16] Müller, Stefan; Scardia, Lucia; Zeppieri, Caterina Ida, Geometric rigidity for incompatible fields and an application to strain-gradient plasticity, Indiana Univ. Math. J., 63, 5, 1365-1396 (2014) · Zbl 1309.49012 [17] Stein, E. M., Singular Integrals and differentiability properties of functions (1970), Princeton University Press · Zbl 0207.13501 [18] Van Schaftingen, Jean, Estimates for $${L}^1$$-vector fields, C. R. Math. Acad. Sci. Paris, 339, 3, 181-186 (2004) · Zbl 1049.35069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 1,543 | 4,747 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2022-21 | latest | en | 0.534572 |
https://www.physicsforums.com/threads/dielectric-help.213681/ | 1,632,574,223,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057622.15/warc/CC-MAIN-20210925112158-20210925142158-00274.warc.gz | 954,786,474 | 14,189 | Dielectric help
[SOLVED]Dielectric help
Homework Statement
To make a parallel plate capacitor, you have available two flat plates of aluminum (area = 180 cm2), a sheet of paper (thickness = 0.10 mm, k= 3.5), a sheet of glass (thickness = 2.0 mm, . k= 7.0), and a slab of paraffin (thickness = 10.0 mm, k= 2.0).
(a) What is the largest capacitance possible using one of these dielectrics?
_________nF
(b) What is the smallest?
________pF
Homework Equations
$$C=k\epsilon_0 \frac{A}{d}$$
The Attempt at a Solution
I used the above equation to evaluate the capacitance of each in this manner:
1) 3.5*8.85e-12*.180m^2/.0001 = 5.5755e-8
2) 7.0*8.85e-12*.180m^2/.002 = 5.5755e-9
3) 2.0*8.85e-12*.180m^2/.010 = 3.186e-10
I recorded the largest as .55.755 and the smallest as 318.6 (in their appropriate units) both of which were wrong. Any ideas where I messed up?
Last edited: | 309 | 879 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.359375 | 3 | CC-MAIN-2021-39 | latest | en | 0.841599 |
https://ch.mathworks.com/matlabcentral/profile/authors/14028081-harsha-priya-daggubati?page=2&s_tid=cody_local_to_profile | 1,571,770,299,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570987823061.83/warc/CC-MAIN-20191022182744-20191022210244-00126.warc.gz | 411,530,512 | 3,197 | Solved
Target sorting
Sort the given list of numbers |a| according to how far away each element is from the target value |t|. The result should return...
8 months ago
Solved
Remove the vowels
Remove all the vowels in the given phrase. Example: Input s1 = 'Jack and Jill went up the hill' Output s2 is 'Jck nd Jll wn...
8 months ago
Solved
Return the 3n+1 sequence for n
A Collatz sequence is the sequence where, for a given number n, the next number in the sequence is either n/2 if the number is e...
8 months ago
Solved
Find the numeric mean of the prime numbers in a matrix.
There will always be at least one prime in the matrix. Example: Input in = [ 8 3 5 9 ] Output out is 4...
8 months ago
Solved
Which values occur exactly three times?
Return a list of all values (sorted smallest to largest) that appear exactly three times in the input vector x. So if x = [1 2...
8 months ago
Solved
Who Has the Most Change?
You have a matrix for which each row is a person and the columns represent the number of quarters, nickels, dimes, and pennies t...
8 months ago
Solved
Most nonzero elements in row
Given the matrix a, return the index r of the row with the most nonzero elements. Assume there will always be exactly one row th...
8 months ago
Solved
Make a checkerboard matrix
Given an integer n, make an n-by-n matrix made up of alternating ones and zeros as shown below. The a(1,1) should be 1. Examp...
8 months ago
Solved
Times 2 - START HERE
Try out this test problem first. Given the variable x as your input, multiply it by two and put the result in y. Examples:...
8 months ago
Solved
Create times-tables
At one time or another, we all had to memorize boring times tables. 5 times 5 is 25. 5 times 6 is 30. 12 times 12 is way more th...
8 months ago
Solved
Determine whether a vector is monotonically increasing
Return true if the elements of the input vector increase monotonically (i.e. each element is larger than the previous). Return f...
8 months ago
Solved
Finding Perfect Squares
Given a vector of numbers, return true if one of the numbers is a square of one of the other numbers. Otherwise return false. E...
8 months ago
Solved
Triangle Numbers
Triangle numbers are the sums of successive integers. So 6 is a triangle number because 6 = 1 + 2 + 3 which can be displa...
8 months ago
Solved
Select every other element of a vector
Write a function which returns every other element of the vector passed in. That is, it returns the all odd-numbered elements, s...
8 months ago
Solved
Find all elements less than 0 or greater than 10 and replace them with NaN
Given an input vector x, find all elements of x less than 0 or greater than 10 and replace them with NaN. Example: Input ...
8 months ago
Solved
Determine if input is odd
Given the input n, return true if n is odd or false if n is even.
8 months ago
Solved
Find the sum of all the numbers of the input vector
Find the sum of all the numbers of the input vector x. Examples: Input x = [1 2 3 5] Output y is 11 Input x ...
8 months ago
Solved
Summing digits
Given n, find the sum of the digits that make up 2^n. Example: Input n = 7 Output b = 11 since 2^7 = 128, and 1 + ...
8 months ago
Solved
Swap the first and last columns
Flip the outermost columns of matrix A, so that the first column becomes the last and the last column becomes the first. All oth...
8 months ago
Solved
Remove any row in which a NaN appears
Given the matrix A, return B in which all the rows that have one or more <http://www.mathworks.com/help/techdoc/ref/nan.html NaN...
8 months ago
Solved
Given a and b, return the sum a+b in c.
8 months ago
Solved
Fibonacci sequence
Calculate the nth Fibonacci number. Given n, return f where f = fib(n) and f(1) = 1, f(2) = 1, f(3) = 2, ... Examples: Inpu...
8 months ago
Solved
Make the vector [1 2 3 4 5 6 7 8 9 10]
In MATLAB, you create a vector by enclosing the elements in square brackets like so: x = [1 2 3 4] Commas are optional, s...
8 months ago | 1,075 | 4,015 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2019-43 | latest | en | 0.819523 |
https://www.conceptdraw.com/examples/what-is-the-x-axis-on-a-line-graph | 1,563,562,040,000,000,000 | text/html | crawl-data/CC-MAIN-2019-30/segments/1563195526337.45/warc/CC-MAIN-20190719182214-20190719204214-00141.warc.gz | 655,404,690 | 9,252 | This site uses cookies. By continuing to browse the ConceptDraw site you are agreeing to our Use of Site Cookies.
# Line Chart Examples
## Line Chart Examples
The Line Graphs solution from Graphs and Charts area of ConceptDraw Solution Park contains a set of examples, templates and design elements library of line and scatter charts. Use it to draw line and scatter graphs using ConceptDraw PRO diagramming and vector drawing software for illustrating your documents, presentations and websites. Read more
## How to Create a Line Chart
Create a Line Chart with ConceptDraw using our tips. Here you can find an explanation of how to create a line chart quickly. Read more
HelpDesk
## How to Draw a Line Chart Quickly
A common line chart is a graphical representation of the functional relationship between two series of data. A line chart that is created by connecting a series of data points together with a straight line is the most basic type of a line chart. A line chart can be used for depicting data that changes continuously over time. It is extensively utilized in statistics, marketing and financial business. ConceptDraw Line Graph solution provides the possibility to make 2D line charts quickly and effortlessly. Read more
## Line Chart
You want to draw the Line Chart and need the automated tool? Now it is reality with Line Graphs solution from Graphs and Charts area of ConceptDraw Solution Park. Read more
## Design elements - Line graphs
The vector stencils library "Line Graphs" contains 4 chart templates. Use it to design your line charts in ConceptDraw PRO diagramming and vector drawing software.
"A line chart or line graph is a type of chart which displays information as a series of data points connected by straight line segments. It is a basic type of chart common in many fields. It is similar to a scatter plot except that the measurement points are ordered (typically by their x-axis value) and joined with straight line segments. A line chart is often used to visualize a trend in data over intervals of time – a time series – thus the line is often drawn chronologically." [Line chart. Wikipedia]
The templates example "Design elements - Line graphs" is included in the Basic Line Graphs solution from the Graphs and Charts area of ConceptDraw Solution Park. Read more
Chart templates
Used Solutions
## Design elements - Education charts
The vector stencils library "Education charts" contains 12 graphs and charts: area chart, column chart, divided bar diagram, histogram, horizontal bar graph, line graph, pie chart, ring chart, scatter plot.
Use it to create your educational infograms.
"A chart can take a large variety of forms, however there are common features that provide the chart with its ability to extract meaning from data.
Typically the data in a chart is represented graphically, since humans are generally able to infer meaning from pictures quicker than from text. Text is generally used only to annotate the data.
One of the more important uses of text in a graph is the title. A graph's title usually appears above the main graphic and provides a succinct description of what the data in the graph refers to.
Dimensions in the data are often displayed on axes. If a horizontal and a vertical axis are used, they are usually referred to as the x-axis and y-axis respectively. Each axis will have a scale, denoted by periodic graduations and usually accompanied by numerical or categorical indications. Each axis will typically also have a label displayed outside or beside it, briefly describing the dimension represented. If the scale is numerical, the label will often be suffixed with the unit of that scale in parentheses. ...
The data of a chart can appear in all manner of formats, and may include individual textual labels describing the datum associated with the indicated position in the chart. The data may appear as dots or shapes, connected or unconnected, and in any combination of colors and patterns. Inferences or points of interest can be overlaid directly on the graph to further aid information extraction.
When the data appearing in a chart contains multiple variables, the chart may include a legend (also known as a key). A legend contains a list of the variables appearing in the chart and an example of their appearance. This information allows the data from each variable to be identified in the chart." [Chart. Wikipedia]
The shapes example "Design elements - Education charts" was created using the ConceptDraw PRO diagramming and vector drawing software extended with the Education Infographics solition from the area "Business Infographics" in ConceptDraw Solution Park. Read more
Graphs and charts
Used Solutions
## Bar Chart Software
The best bar chart software ever is ConceptDraw. ConceptDraw bar chart software provides an interactive bar charting tool and complete set of predesigned bar chart objects. Read more
## Scatter Chart Examples
The Line Graphs solution from Graphs and Charts area of ConceptDraw Solution Park contains a set of examples, templates and design elements library of scatter charts. Use it to draw scatter graphs using ConceptDraw PRO diagramming and vector drawing software for illustrating your documents, presentations and websites. Read more
## Scatter Diagrams
The Scatter Diagrams solution extends ConceptDraw PRO v10 functionality with templates, samples, and a library of vector stencils that make construction of a Scatter Plot easy. The Scatter Chart Solution makes it easy to design attractive Scatter Diagrams used in various fields of science, work, and life activities. ConceptDraw PRO lets you enter the data to the table and construct the Scatter Plot Graph automatically according to these data. Read more
## Scatter Plot
ConceptDraw PRO extended with Scatter Diagrams solution is ideal diagramming and vector drawing software for quick and easy designing professional looking Scatter Plot. Read more | 1,155 | 5,948 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2019-30 | longest | en | 0.926251 |
http://mathhelpforum.com/advanced-algebra/38582-polynomial-z-x.html | 1,516,292,721,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084887423.43/warc/CC-MAIN-20180118151122-20180118171122-00557.warc.gz | 227,955,735 | 11,208 | # Thread: Polynomial in Z[X]
1. ## Polynomial in Z[X]
Hi! I just have a short question: I want to proof the following claim: Let K be a field with characteristic p>0 and f be a polynomial in Z[X] (with Z being the set of integers). Then f(y^p)=f(y)^p for any y in K.
My consideration is this: f=a_0 + a_1 x + a_2 x^2 + ... + a_n x^n Using the binomial theorem it easily follows that f(y)^p=(a_0)^p + (a_1 y)^p + (a_2 y^2)^p + ... + (a_n y^n)^p = a_0 + a_1 y^p + ... + a_n (y^p)^n = f(y^p), as (a_i)^p=a_i. But this last implication only holds if the coefficients are in Z/pZ which is not demanded.
So my questions: Is there a mistake in the description of the exercise? Or are the coefficients to be treated modulo p; if yes, why? Thankful for any tips, Banach!
2. Originally Posted by Banach
Hi! I just have a short question: I want to proof the following claim: Let K be a field with characteristic p>0 and f be a polynomial in Z[X] (with Z being the set of integers). Then f(y^p)=f(y)^p for any y in K.
My consideration is this: f=a_0 + a_1 x + a_2 x^2 + ... + a_n x^n Using the binomial theorem it easily follows that f(y)^p=(a_0)^p + (a_1 y)^p + (a_2 y^2)^p + ... + (a_n y^n)^p = a_0 + a_1 y^p + ... + a_n (y^p)^n = f(y^p), as (a_i)^p=a_i. But this last implication only holds if the coefficients are in Z/pZ which is not demanded.
So my questions: Is there a mistake in the description of the exercise? Or are the coefficients to be treated modulo p; if yes, why? Thankful for any tips, Banach!
Every field of characteristic p>0 contains the base field $\mathbb{Z}/p\mathbb{Z}$, so the equation ${a_i}^p\equiv a_i$ is perfectly all right.
3. The binomial theorem still holds in over a field.
Thus, $(a+b)^p = a^p + b^p + \sum_{k=1}^{p-1}{p\choose k}a^pb^{p-k}$.
But the coefficients in the summand is each divisible by $p$ so each one dies off and gives you zero.
4. Originally Posted by Moo
Hello,
Assuming that p is prime...
Or did I miss a point in the original text ?
Yes you did. Characteristic of a finite field is always prime.
5. Originally Posted by Isomorphism
Yes you did. Characteristic of a finite field is always prime.
Ok, I have to wake up, I missed this sentence in the wiki..
6. Originally Posted by Moo
Ok, I have to wake up, I missed this sentence in the wiki..
you can see this by writing out $\underbrace{1+1+1+\ldots+1}_\text{m times}=0$ and use the fact that a field has no zero divisor.
7. Hi! Thanks everyone for your answers. My only problem was the structure that is given between elements of the field and integers, but as i remembered the definition everything cleared up. I got it now. Have a nice day! Banach | 803 | 2,658 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.6875 | 4 | CC-MAIN-2018-05 | longest | en | 0.858874 |
https://www.sciencedaily.com/releases/2020/11/201102173232.htm | 1,709,652,809,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707948235171.95/warc/CC-MAIN-20240305124045-20240305154045-00242.warc.gz | 967,397,807 | 12,221 | Science News
from research organizations
# COVID-19 'super-spreading' events play outsized role in overall disease transmission
## Mathematical analysis suggests that preventing large gatherings could significantly reduce COVID-19 infection rates
Date:
November 2, 2020
Source:
Massachusetts Institute of Technology
Summary:
Researchers find COVID-19 super-spreading events, in which one person infects more than six other people, are much more frequent than anticipated, and that they have an outsized contribution to coronavirus transmission.
Share:
FULL STORY
There have been many documented cases of Covid-19 "super-spreading" events, in which one person infected with the SARS-CoV-2 virus infects many other people. But how much of a role do these events play in the overall spread of the disease? A new study from MIT suggests that they have a much larger impact than expected.
The study of about 60 super-spreading events shows that events where one person infects more than six other people are much more common than would be expected if the range of transmission rates followed statistical distributions commonly used in epidemiology.
Based on their findings, the researchers also developed a mathematical model of Covid-19 transmission, which they used to show that limiting gatherings to 10 or fewer people could significantly reduce the number of super-spreading events and lower the overall number of infections.
"Super-spreading events are likely more important than most of us had initially realized. Even though they are extreme events, they are probable and thus are likely occurring at a higher frequency than we thought. If we can control the super-spreading events, we have a much greater chance of getting this pandemic under control," says James Collins, the Termeer Professor of Medical Engineering and Science in MIT's Institute for Medical Engineering and Science (IMES) and Department of Biological Engineering and the senior author of the new study.
MIT postdoc Felix Wong is the lead author of the paper, which appears this week in the Proceedings of the National Academy of Sciences.
Extreme events
For the SARS-CoV-2 virus, the "basic reproduction number" is around 3, meaning that on average, each person infected with the virus will spread it to about three other people. However, this number varies widely from person to person. Some individuals don't spread the disease to anyone else, while "super-spreaders" can infect dozens of people. Wong and Collins set out to analyze the statistics of these super-spreading events.
"We figured that an analysis that's rooted in looking at super-spreading events and how they happened in the past can inform how we should propose strategies of dealing with, and better controlling, the outbreak," Wong says.
The researchers defined super-spreaders as individuals who passed the virus to more than six other people. Using this definition, they identified 45 super-spreading events from the current SARS-CoV-2 pandemic and 15 additional events from the 2003 SARS-CoV outbreak, all documented in scientific journal articles. During most of these events, between 10 and 55 people were infected, but two of them, both from the 2003 outbreak, involved more than 100 people.
Given commonly used statistical distributions in which the typical patient infects three others, events in which the disease spreads to dozens of people would be considered very unlikely. For instance, a normal distribution would resemble a bell jar with a peak around three, with a rapidly-tapering tail in both directions. In this scenario, the probability of an extreme event declines exponentially as the number of infections moves farther from the average of three.
However, the MIT team found that this was not the case for coronavirus super-spreading events. To perform their analysis, the researchers used mathematical tools from the field of extreme value theory, which is used to quantify the risk of so-called "fat-tail" events. Extreme value theory is used to model situations in which extreme events form a large tail instead of a tapering tail. This theory is often applied in fields such as finance and insurance to model the risk of extreme events, and it is also used to model the frequency of catastrophic weather events such as tornadoes.
Using these mathematical tools, the researchers found that the distribution of coronavirus transmissions has a large tail, implying that even though super-spreading events are extreme, they are still likely to occur.
"This means that the probability of extreme events decays more slowly than one would have expected," Wong says. "These really large super-spreading events, with between 10 and 100 people infected, are much more common than we had anticipated."
Many factors may contribute to making someone a super-spreader, including their viral load and other biological factors. The researchers did not address those in this study, but they did model the role of connectivity, defined as the number of people that an infected person comes into contact with.
To study the effects of connectivity, the researchers created and compared two mathematical network models of disease transmission. In each model, the average number of contacts per person was 10. However, they designed one model to have an exponentially declining distribution of contacts, while the other model had a fat tail in which some people had many contacts. In that model, many more people became infected through super-spreader events. Transmission stopped, however, when people with more than 10 contacts were taken out of the network and assumed to be unable to catch the virus.
The findings suggest that preventing super-spreading events could have a significant impact on the overall transmission of Covid-19, the researchers say.
"It gives us a handle as to how we could control the ongoing pandemic, which is by identifying strategies that target super-spreaders," Wong says. "One way to do that would be to, for instance, prevent anyone from interacting with over 10 people at a large gathering."
The researchers now hope to study how biological factors might also contribute to super-spreading.
The research was funded by the James S. McDonnell Foundation.
Story Source:
Materials provided by Massachusetts Institute of Technology. Original written by Anne Trafton. Note: Content may be edited for style and length.
Journal Reference:
1. Felix Wong, James J. Collins. Evidence that coronavirus superspreading is fat-tailed. Proceedings of the National Academy of Sciences, 2020; 202018490 DOI: 10.1073/pnas.2018490117 | 1,294 | 6,623 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2024-10 | latest | en | 0.966776 |
https://ru.scribd.com/doc/142306824/ENSKLOPEDIYA | 1,679,598,355,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296945182.12/warc/CC-MAIN-20230323163125-20230323193125-00756.warc.gz | 536,276,896 | 306,047 | Вы находитесь на странице: 1из 1185
# ; Abel integral equation
f (x ) =
( s ) ds
xs
(1)
, f ( x ) , (s ) -
. f ( x ) , -
x12
(s) =
(2)
sz
f ( z ) dz
.
sz
. . .- . X =
= ( X 1 , X 2 ) , p
( x1 , x2 )
X1 , X 2
x12 + x22 -
( X
) ; X 1
p X ( x1 ) - . X
1
(
;
). R
X 12 + X 22 , X -
, p R ( r )
;
X1 , X 2
( x1 , x2 ) = ( 2 r ) 1 pR ( r ) .
X 1 - ; R - ,
e = ( e1 , e2 )
1 z 2
dz .
1
= f ( x ),
p X 1
x
x
, f ( x ) ( s ) - (1)
f ( z ) dz
1 f (0)
+
s
1
1
= ( x)
p R
s
s
=0
. f (0) = 0 , . . .- :
(s) =
1
x
p R 1 p e1 ( z ) dz =
z
z
1
x 2
p 1
2 R z
= x, x z2 = s,
( f (x ) ) . . .- . , f (0)
,
x 0
p X ( x1 ) =
e1 -
; e ( , [2], 1, 10 ). x1 > 0
. f (2)
.
, . . .- . 2 , .
.
. . . .
. ( , [1] ) 1823- ( ) .
( u , x )
. x ( 0 , 0 )
t
t ( x)
.
.
. . . (1), f ( x)
, ( s )
2 g t ( x ) , g
1
, s -
sin
.
x
u =
2( s ) 1 ds
. f (x)
; t ( x )
= const -
= const
[ . ( Ch. Huygens,
1673 ) ].
.: [1] A b e l N. H., Oeuvers compltes, t. 1, Christiania, 1839,
p. 2730; [2] .,
, . ., . 2, ., 1984; [3] . .,
, 2 ., . ., 1937; [4] .,
., , . .,
. 1, 3 ., ., 1951.
; Abelian group ,
( )
.
. .- n ( ) . . .- ( ,
,
). . .-
.
( mod 2
1.
e s x x 1 dx =
, x f ( x )
,
f ( x ) ~ x 1
, s 0
). 0 1 ,
0 + 0 = 0 , 0 + 1 = 1 + 0 = 1,
( )
, >0, s > 0
s
1+1 = 0
e s x f ( x ) dx ~
( )
s
. . G . 1 , 2 , ... , n , ... ,
. f ( x )
P { n = 0 } = q n ,
, ,
.
2. ,
P { n = 1} = pn = 1 qn
S n = 1 + 2 + ... + n
,
( S n , 1 , 2 , ... , n 2 - ). :
) n0 n0 G -
,
,
n > n0
pn0 = qn0 = 1 2
Sn
A - . | z |
z -
f ( z) =
min ( pn , qn ) =
n =1
, n S n
zn
z 1
f (z) A
( ).
1- . 0 < z < 1
z = e s
.
) ) , ) S n
, ;
G .
. .- ,
( ,
).
.: [1] .,
, . ., ., 1965; [2] . ., . ., . ., 1966, . 11, . 1,
. 335; [3] ., , . ., ., 1981; [4] D v o r e t z k y A.,
W o l f o w i t z J., Duke Math. J., 1951, v. 18, p. 50107; [5] Probability measures on metric spaces, L. N. Y., 1967.
<1
n =1
;
)
an
n =1
G( x) =
n x
f (z) =
e s x d G ( x ) = s
s x
G ( x ) dx .
x G ( x ) A , s 0
f (z) ~ s
s x
A dx = A
.
, . .
.
.: [1] .,
, . ., . 2, ., 1984.
; Abel theorems , .
; Abel theorems ,
; abstract ergodic theorem
, .
; abstract decision rule ,
.
; open queueing system ,
.
## ; open random set
G . G Tg
: S
, G
; G G , K , S -
= { M : M G , M I G = },
GK = { M : M G , M I K = },
G
( ) Tg ,
G . ,
.
( ) ( )
sample range ,
{ X i }in= 1 { X (i ) }in = 1
Wn
. X i , 1
in
= X ( n ) X (1) -
F , Wn -
P {Wn < x } = n
( F ( x + t ) F (t ) )
n 1
( p N ) . I
, II I
. , I
. .- N 1
p 1 II . .- p
,
( ). . .-
.
.: [1] H a d a m a r d J., Bull. sci. math., sr. 2, 1893, t. 17,
p. 24046; [2] S y l v e s t e r J. J., Phil. Mag. 1867, v. 34, p. 461
72; [3] , ., 1969;
[4] H e d a y a t A., W a l l i s W. D., Ann. Statist., 1978, v. 6,
p. 1184238.
. -, , . , . .-
p -
d F (t ) , x 0
. . . ( , ) . ( . ) Wr , s = X ( s ) X ( r ) ,
1 r < s n . . .
.: [1] ., , . ., .,
1979; [2] . ., . ., , 3 ., ., 1983.
## ; Hadamard matrix +1 1 N > 1
H N ,
H NT H N = N E N
, E N . . .- . ( , [1] ). ,
N = 2 . ( , [2] ).
. .- +1
, .
. .- : N 0 ( mod 4 ) N = 2
, , . . .
( )
. .
. .- ( ) .
.: [1] . .,
, ., 1984; [2] B i c k e l P. J., Ann. Statist., 1982,
v. 10, 3, p. 64771.
( )
.
F F ; F adapted function ,
.
;
adaptive controlled random process i n d i s c r e t e
t i m e ,
.
, -
;
. . . .
.
.: [1] B i c k e l P. J., Ann. Statis., 1982, v. 10, 3,
p. 64771.
( ) ; spain o f a d i s t r i b u t i o n ,
.
; step factor ,
.
,
; step of arithmetic distribution , ,
.
, : K G
.
K , , G
( ) = 0 .
1
G R ,
n
;
additive functional o f i n t e g r a l e t y p e
s
ts =
f ( u , (u )) d u ,
0t < s
, ( )
,
f : [ 0 , + ] R1 R1
,
.
.: [1] . ., , 2 ., ., 1986; [2] . .,
. ., , ., 1970.
( )
; additive functional o f M a r k o v p r o c e s s , ;
.
( )
; additiv functional o f t h e W i e n e r
p r o c e s s ,
s
u
s
( t = t + u ,
t < u < s )
w (t ) , t 0 ( ts , w (u ) , u [ t , s ] ,
t < s At s
) s
t , 0
t < s .
## .: [1] . ., , 2 ., ., 1986; [2] . .,
. .,
, ., 1970.
;
( ) .
K , G +
( G
). A K , B K , A I B
= , A U B K
C , n R ,
, . - . , K ,
A K B K A U B K
( K U ).
. . . . , K
, , K
,
( A1 U ... U Am ) = ( A1 ) + ... + ( Am )
K - ( Ai )1 i m
.
K U I ,
: K G
( A U B ) + ( A I B ) = ( A) + ( B )
, A , B , K
.
, ,
.
.: [1] ., , . . ., .,
1953.
,
( ) ; additive problems o f n u m b e r t h e o r y
(
) .
. f ( x ) , p
. N
N = f ( x1 ) + f ( x2 ) + ... + f ( xn )
I ( N , n )
, 0
x1 , x2
, ..., xn
n -
A B
P { 1 = f ( k ) } = 1 p ,
( AU B ) = ( A ) + ( B )
k = 0 , 1, ..., p 1
, a
10
p 1
.
. 1 , 2 , ..., n
. n
= M 1 , D 1 = 2 > 0
= 1 + ... + n .
P { n = N } =
2 i z N
1
p
p 1
2 i z f ( x )
k = 0
dz .
(*)
; additive random ( stochastic )
process ,
; affine shape
P { n = N } = p n I ( N , n ) .
(*) , I ( N , n )
:
1) n , p ; 2) n , p
; 3) n p
= p ( n )
.
.
I ( N , n ) . I ( N , n )
. , ,
S ( f , z) =
1
p
p 1
e 2 i z f ( x )
x=0
z - p
, S ( f , z ) -
, .
( ) ; f a u l t
tree , .
; r a n d o m
tree , .
; linear tree code , .
; tree code
,
( ) .
(
) . ;
. ,
. . .-
, ( ) . . .-
.
.: [1] ., , , . ., ., 1986.
; flow
( X , A , ) -
,
. f ( x )
{ T , t R } :
.
. .- .
( G. Castelnuovo, 1933 ) , , f (x ) = x -
x X . T t - t -
, I ( N , n )
.
. . ( 1956, , [1] ) . ,
f (x ) = x
f (x ) = x 2 I ( N , n )
. . .- :
,
( , [3] ).
. . , .,
[4].
.: [1] . .,
, ., 1971; [2] . ., . ., . ., , ., 1975; [3] . .,
. . .
, 8, 1983, . 121, . 6282; [4] . ., . ., .,
., 1984.
model , , .
;
.
t1 , t 2 R T t1 + t 2 x = T t1 T t2 x
.
, ; B e r n o u l l i f l o w , .
nuous
; c o n t i flow , .
; m e a s u r flow , .
K K ; K flow , K .
*
; parameter
of flow , .
, ; filtration / family of algebras ( , A , P)
.
able
T ; , t T
( , A , P)
At . ( At ) t T
t T
At A
s , t T ,
s t
As At , ( At ) t T
. ( M , B ) X (t )
B,
s T , s T
11
{ : X ( s, ) } -
( f1 ) ( f 2 ) f1 f 2 -
N t -
= R + -
, . , ) H
( N t ) t T . T
, ( At ) t 0
: ) , t
At
As ; ) A
( M - )
d
g U g , g f S ( M )
U g ( f ) U g1 = ( g f ) ,
s>t
A0
A -
.
.: [1] ., ,
. ., ., 1975.
;
intensity of flow , .
( ) ( );
last income first outcome ( LIFO ) .
,
,
.
- .
.
,
n - .
1 - B (t ) .
;
n - , , n -
.
.: [1] . ., , . ., ., 1965.
; Akaike information
criterion, AIC , , ,
.
; Akaike
criterion
, ;
; axiomatic quantum field theory ;
.
. . d
( , [2] ). ., M
,
H -
d
, f S ( M )
d
{ ( f ) , f S ( M ) , d
2}
( g f ) ( x ) = f ( g 1 x ), x M d ;
) M d - U g
{P , = q , ..., d } M d -
; ) H g U g d
{ ( f ) , f S ( M )}
.
wn ( f1 , ..., f n ) = ( ( f1 ) , ..., ( f n ) , ) H
( , [1] ).
.
.: [1] ., ,
. ., ., 1967; [2] W i g h t m a n A., G a r d i n g L.,
Ark. Phys., 1964, v. 28, p. 12984; [3] . .,
[ .], , ., 1987.
; active
variable , .
;
active experiment .
; Aldous Rebolled condition
,
.
; Alexandrov space ,
.
; Alexandrow theorem
. M , ( X , G )
. G1 B X \ G2 P ( G1 ) =
= P ( X \ G2 ) G1 , G2 G , ( G ) B P P M
.
M - ( P ) - P M
:
1) P , ( G )
; f ( f ) f - ,
lim P ( B ) = P ( B )
12
2) G G
lim inf P ( G ) P (G )
lim P (X ) = P ( X )
## ; alpha excessive function ,
.
; alpha faktor analysis , ;
.
,
; potential alpha kernel ,
.
;
alpha potential ,
y - x
I ( x : y ) I ( y : x ) = O ( log ( l ( x ) + l ( y ) ) ),
lim P ( X ) = P ( X )
.
.
, .
. . [1]- .
.: [1] . ., . ., 1940, . 8,
. 30748; 1941, . 9, . 563628; 1943, . 13, . 169238; [2] . ., . , 1976, . 31, . 2, . 368.
I ( y : x ) = K ( x ) K ( x | y )
. .. I
, :
;
3) F
lim sup P ( F ) P ( F )
. .
K ( x | y );
; Kolmogorov complexity entropy
, ,
, . .
.
( ) ( , )
. G - x K G ( x ) , p , G ( p )
= x . -
F , G
K F ( x ) K G ( x ) + O ( 1)
( , [1] ). ( ) , K ( x )
.
. . ; ,
. .- . .-
. . .- ( + O (1) );
( ). . .- ( ,
[2] ).
. .- , . . ( , [2] [4] ); ., y x -
l ( z ), z . . .- . . K M (x ) - ( [3] [5] ).
: n
. .-, , n + O (1) - , [ 0 , 1]
( , )
( , [3], [5] ). M (x) , x
log 2 M ( x )
.
M .
(1)
M ( x) < + ;
(2) r M ( x ) -
r, x
; .
. .-
( , [3] ).
. . . ( C. Shannon )
, .
1) x , p q , x ,
H = ( p log 2 p +
+ q log 2 q ) , 0 1 , p q
.
l ( x ) , K ( x ) l ( x ) H + O (1) .
, . .,
( ) .
x ,
;
, x -
.
2) 0 1 - p q (
) n
. .- K - n ,
( p log 2 p + q log 2 q ) - .
.: [1] . ., , 1965, . 1, . 1, . 311; ( , . .,
, ., 1987, . 21323 ); [2] . ., . ., . , 1970, . 25, . 6,
. 85127; [3] . ., , .,
1981, . 16, . 1443; [4] . ., . , 1984,
. 276, 3, . 56366; [5] . ., . , 1973,
. 212, 3, . 54850; [6] H a r t m a n i s J., Bull. European Ass.
Theor. Comp. Sci., 1984, 24 ( Oct. ) p. 7378 [7] .,
., . , 1988, . 43, . 6, . 12966.
13
.
{F } -
{F } -
F f G U ( F ) > U (G ) ,
(1)
[ 0 , 1]
U ( F + ( 1 ) G ) = U ( F ) + (1 ) U ( G )
(2)
U ( ,
).
. [1]- { 0 ., 5 . ., 25 . . } , F1 , F2 ,
F3 , F4 . .
0 .
5 . .
25 . .
F1
F2
0,01
0,89
0,1
F3
0,9
0,1
F4
0,89
0,11
, 5 . ,
,
5 .- 25 .- .
, (
)
.
, ,
-
, , . , f
, F1 f F2
F3
f F4
. (1) (2)- U
( F1 + F3 ) 1 2 f ( F2 + F4 ) 1 2 ,
,
.
*
## ; lower ladder index ,
.
*
; lower
.
; alternative , , ( ) ,
. , , , . . . .,
. .
; alternative hypothesis , .
; forecast
of meteorological binary variables of events / alternating meteorolgical forecast
, - - ( ., . ). . . . .
. . .- (
, , Y
Y - ),
, . . . , ,
. . . . ( , , ;
)
.
.: [1] . ., . . . .,
1937, . 14, . 4957; [2] . ., . . .
., 1955, 4, . 33949; [3] . ., , ., 1981.
*
## ; alternating renewal process
. { X 1 , X 2 ,...}
( F1 + F3 ) 1 2 = ( F2 + F4 ) 1 2 .
{Y1 , Y2 ,...} .
( ,
).
( , [2] ).
.: [1] A l l a i s M., Econometrika, 1953, v. 21, p. 50346;
[2] . ., . ., . .,
- , ., 1980.
*
f1 ( x ) f 2 ( x ) .
## ; lower ladder time ,
.
14
, .
,
( 1 ) .
; ;
- - - - - ; o - .
1. .
. ,
.
, . . .-.
2.
. t = 0
X 1 ; X 1 -
f1 ( x ) -.
. X 1
X 1
.
X 1
X 1 -
~
~
. ( s , x ) Ps , x { } = 1 t < ~
~
~
(t ) = ( t ) , ( t ) (t ) ,
, .
f 2 ( x ) -. ( i 1) i -
(
) , ;
,
f1 ( x ) - e
,
,
e x -
.
. ( )
,
.
, . . . .
. .,
- k
. , k
pij
; pi j , i - j -
. .
.: [1] . ., . ., , .
., ., 1967.
; subprocess
.
( X , B )
~
~
~
( ( t ) , , Ft s , Ps x ) ( ( t ) , , Ft s , Ps x ) -
. , :
~
) ( ) < ( ) ;
~
~
) 0 t < ( ) , ( t ) = (t ) ;
~
~
~
~
) Ft s = Ft s [ t ] , t = { : ( ) > t } , Ft s [ t ] ,
~s
s
s
Ft t , A Ft
~
A I t
~s ~
~
Ft s , Ps x ) .
s
P ( s , x, t , ) ( ( t ) , , Ft , Ps x ) -
, P ( s , x, t , ) , ,
~
P (s, x, t , ) P ( s, x, t , ) .
.: [1] . ., . ., , . 2, ., 1973; [2]
, ., 1985; [3] . ., , ., 1959.
, ; subclass o f a M a r k o v c h a i n , ;
; subnet
, ( )
; amplitude modulation (t )
,
Z (t ) = c ( 1 + m 0 ( t )) ( t ) , < t <
, M (t )
D 0 ( t ) = 1
( t ),
= 0, D (t ) = 1 , c
< t < ,
M 0 ( t ) = 0,
m, 0 < m <1
,
, m .
0
(t ) ( t ) ,
B ( ) B 0 ( ) ,
Z (t )
BZ ( ) = c 2 B ( ) ( 1 + m 2 B 0 ( ) )
.
0
(t ) ( t ) , f ( ) , f
( )
f Z ( ) :
f Z ( ) = c 2 ( f ( ) + m 2 f 0 ( ) f ( ) ),
, ( ( t ) , , Ft , Ps x ) ( ( t ) , ,
X 1 X 2 -
- . , X 2
X i + X i - .
~s ~
( ( t ) , , Ft , Ps x ) ( ( t ) , , Ft , Ps x )
( ( t ) , , Ft , Ps x ) ( ( t ) , , Ft , Ps x ) , .
15
. (t ) ( t ) 1 2
,
0 < 2 << 1 ,
f Z ( ) 1
; amplitud modulated impulse
process
, -
(t ) =
, ( 1 2 ) f Z ( ) ,
X n ( t tn )
(*)
n=
. f Z ( ) -
, ( t ) -
= 2
, ,
t n , n = 0 , 1,... , -
, , X n ,
f Z ( 1 ) c 2 [ f ( 1 ) + m 2 ( f 0 ( 0 ) f ( 1 ) +
+ f 0 ( 2 ) f ( 1 + 2 ) ) ] ,
n = 0 , 1,...
.
{ tn } ( - ), { X n } M X n
= m,
f Z ( 1 2 ) c 2 [ f ( 1 2 ) + m 2 f 0 ( 2 ) f (1 ) ] ,
= a ,
, (t )
f Z ( 2 ) c 2 [ f ( 2 ) + m 2 ( f 0 ( 2 ) f ( 0 ) +
X n2
+ f 0 ( 1 2 ) f ( 1 ) ) ].
Z (t )
Z (t ) . (t )
( t ) , , .,
.
.: [1] ., .,
. , . ., ., 1969; [2] . ., , 2 ., ., 1968;
[3] . ., , 4 ., ., 1962;
[4] ., , . .,
., 1974.
; amplitud modulated
harmonic oscillation
X (t ) = X 0 ( t ) cos ( 0 t + )
, 0 ,
X 0 ( t ) , . [ 0 , 2 ]
,
X 0 ( t ) [ f 0 ( ) ]
<< 0
. . . . X (t )
= 0
= 0
. ,
X (t )
= , = + 20
= 20 .
F ( d , d ),
16
M (t ) = m
( u ) du
(t + ) (t ) =
=a
( u + ) ( u ) du + m ( u ) du
.
tn = n T0 ( T0 ) tn
= n T0 + n ( { n }, { X n } -
) , (t ) T0 .
.: [1] . ., . ., , ., 1973; [2] . ., , . 1, .,
1976; [3] . ., ., 2 ., . 1, ., 1974; [4] ., , . ., . 12, ., 1961
62; [5] ., , . ., ., 1974.
; amplitud modulated random process
(t ) = 0 ( t ) ( t )
, 0 ( t )
, (t ) . . . .
.- . (t ) , . . . . (t ) ,
;
( t ) =
e i t dK ( )
, (t ) -
= 0 -
(1)
, dK ( ) ,
D (X
X (1) X ( 2)
, X ( 2) ) - :
; amplitude
frequency response , .
; analytic
characteristic function X P - z = 0
.
z = t + is, | z | < r f
| Im z |
< z
f (z) =
## D ( X (1) , X ( 2) ) D ( X (1) , X (3) ) + D ( X (3) , X ( 2) ) ,
D ( X (1) , X ( 2) ) 0,
D ( X (1) , X (1) ) = 0 .
D (X
(1)
,X
( 2)
) =
i2 ( xi(1) xi( 2) ) 2
i = 1
. :
n
( j)
r A
, xi , i
>0
) , A
. . .-.
## a = sup { t > 0 : M e t X < } ,
b = sup { t > 0 : M e
t X
xi(1) xi( 2) ,
i =1
.
f , , r
12
D ( X (1) , X ( 2) ) =
e i z x P (dx)
P { | X | > A} = O ( e
D ( X (1) , X ( 2) ) = D ( X ( 2) , X (1) ) ,
< }
, i a, i b , f
{ z : a < Im z < b }, f
. . . .-
, .
.: [1] . ., . .,
, ., 1972; [2] ., , . ., ., 1979.
; analogous method
:
,
. ,
, . .- ,
- .
, . .-
,
- ,
( - , ,
). - .
. .
( ) X -
( )
= 1,..., n
( j)
, i i -
. , .,
( , , ; ).
X j Y j ( j ) , . .- (
) Y ( X 0 )
= Y j , X 0 -
, Y , j
,
D ( X 0 , X j ) = min D ( X 0 , X j ) .
j
. .
( , [3] ).
.
.
.: [1] . ., . ., , ., 1983; [2] . ., . ., ,
., 1982; [3] , ., 1985.
; Andersons inequality ,
. F -
, C F
t [0 , 1], y F (C + t y )
(C + y ) , , ( C ) ( C + y )
,
, .
17
R - . . , .,
,
f : 1) f ( x ) = f ( x ) ,
. . . ( ,
[1], [2] ), t K . D -
f ( x ) -
x R n ; 2) > 0 { x R n ; f ( x) } -
. . . . ( , ).
.: [1] . ., . . - , 1967,
. 90, . 1210; [2] . ., . . . .,
1966, . 30, 1, . 1568.
. .
.
. . ,
.
.: [1] A n d e r s o n T., Proc. Amer. Math. Soc., 1955, v. 6,
2, p. 17076; [2] B o r e l l K., Ark. mat., 1974, v. 12, 2,
p. 23952; [3] . ., . .,
, ., 1985.
; Anderson Jensen theorem
, ( )
## ; Andres reflection principle ,
.
)
s t o p p i n g time , .
## ; instantaneous spectral density
, ( ) .
; instantaneous state x ,
:
lim p ( t , x , x ) = 1,
t0
q x = lim
t 0
1 p (t , x, x )
= + ,
t
p ( t , x , x ) t x x - . . .- 1951- . . . ( P. Lvy )
, . . .
. . . .-
( ) , . . .
.
.: [1] - , , .
., ., 1964.
. , ,
. :
; 1 ; 2
. . .
. . .- .
, . . .-
; .
.
( )
,
.
, -
. , ( )
( , [2], [3] ).
.
.: [1] A n s a r i A. R., B r a d l e y R. A., nn. th. Statist,
1960, v. 31, 4, p. 117489; [2] ., .,
, . ., ., 1983;
[3] M o s e s L. E., Ann. th. Statist., 1963, v. 34, 3, p. 97383.
; antiferromagnetic model
. , . .-
d
{ xt , t Z }
, t Z d
, e Z
t { 0 ; 1; 2 , ...}
xt
D - S : D D
,
| h |
D ( x ) ( S *t ) ( x) =
= dS t ( S t x ) S *t .
18
xt +e = xt
. . .
h xt , A = { t } ,
U A ( x A ) = xs xt , A = { s , t } , s t = 1,
dig A lar ,
0 ,
U
:
## ; Anosov dynamical system
t ( ; )
= 1, t Z d , h R .
< 2 d , x , x 1
t + ... + t d
xt1 = (1) 1
xt2 = xt1 .
h = 0 . . .
## xt (1) t1+ ... + td xt
. h 0 d
( ) .
a ( t , x ) t (
, ). , a ( t , x ) x t
( )
.
; drift vector
, .
function , , .
*
## , ; leading measure of flow , .
; posteriori probability , ( , [1] ),
h - , h - . ,
.
.: [1] . ., . .,
1968, . 2, . 4, . 4457.
; antisimmetric Fock
space , .
; antisymmetric variate
method , .
; antithetic variables
method , f
s
[ f (1 , ..., s ) + f ( 1 1 , ..., 1 s )] 2
, i , [ 0 , 1]
.
.: [1] H a m m e r s l e y J., M o r t o n K., Proc. Cambr. Phil.
Soc., 1956, v. 52, p. 44975.
; drift coefficients , , .
/ ; drift coefficients
. . .
. ( )
d x (t ) = a ( t , x ( t ) ) dt + ( t , x ( t ) ) d w ( t )
( , ), . . ( ) a ( t , x ) ( ).
a (t , x )
t - t + t -
x(t + t ) x(t ),
t 0 x (t ) = x , o( t )
,
,
. .
, .
, , .
. . .
; posterior mean
.
; posteriori distribution ,
.
, p () -
p(x | ) {
= }
~
- - . .-
p ( | x) =
p () p ( x | )
+
p ( ) p ( x | ) d
.
K ( x ) p ( x | )
, . . x -
, K ( x ) - . xi - -
p ( x | 0 )
, p ( | x1 ,..., xn )
. .- n -
.
. .-
.
.: [1] . ., , 4 .,
. ., 1946.
19
( )
;
a posterior risk o f a d e c i s i o n f u n c t i o n X
( ) x
L ( , ( X ) ) .
G ( x ) . .-
.
, .
*
; approximation theorem A -
A A -
.
( ,F , P ) , U , F -
A .
A U
lim P ( An A U A n A ) = 0
(1)
An A ;
lim P ( A \ An ) = lim P ( An \ A ) = 0
n
= P ( An ) P ( An A ) + P ( An A )
= lim P ( An )
n
, 1 2
.
,
.
.: [1] . ., ., , . ., 1966; [2] E l d e r t o n W. P., Frequency curves and
correlation, Camb., 1953; [3] . ., . .,
, 3 ., ., 1983.
( K , )
( K , )
( K , ) approximating functional ,
; approximable event , .
; a
priori probability , -
; a prior information -
P ( A ) = P ( An A ) + P ( An A ) =
, ,
, . , . , , .
(1)
n
. , , 1 2 -
P ( A) =
A U
A
.
A F (1)
, A -
.
,
;
approximation of complex distributions b y s i m p l i e r
o n e s
.
.
.
y = f ( x )
1 dy
x + C1
=
y dx
C0 + C1 x + C 2 x 2
, C0 , C1 C2
;
, . ,
( , A ) ,
, A
-
. ,
P { d }, ( , A ) -
Cap ( , A ) -
P = { P , }
, . P - ( , B ) -
Q {d }
. . . , ( , [3] ).
.: [1] ., , . ., .,
1960 ( ); [2] ., , . ., ., 1975; [3] . ., .
., 1981, . 26, 1, . 1531.
; a prior information
usage -
2 , 1 2
20
( , , 2 W )
. . .-
.
= r
, A q -
( q p ) , 0 < q < p, r q . R = r
= 0 + B
0 , ( p q ) , B
p q [ p ( p q )] .
( 0 , B , 2W )
= 0 + B ,
.
r = R + , r q ,
R (q p ) , q
M = 0 , cov = V
(q q ) .
; ( 0 , 0 , W0 )
0 =
Y
r
0 =
W0 =
W
0
2 V
( W 1 + 2 R V 1 R ) 1 ( W 1 Y + 2 R V 1 r ) (*)
.
P ( d )
p
- ( R ) .
r =
P ( d ),
V =
( r ) ( r ) P (d ) ,
R = Ip,
V 0
I p ( p p ) .
(*)
;
L -
~
~
M ( ) ( ) P ( d )
.
,
= { R p : ( r )U 1 ( r ) k }, k > 0
,
R = Ip,
V = kU
(*)
~
~
max a M ( ) ( ) a
aR
. r
= 0 -
1
R V R ,
(*)
.
.: [1] . ., . ., , ., 1987; [2] , ., 1983.
;
priori distribution , ,
,
.
( , ) (
) . ,
. - ( -
. .- ) -
P - .
( - )
. . . , ,
. . .-
.
( )
; a prior
risk o f a d e s i g n f u n c t i o n
.
;
sequential analysis ,
( )
,
.
.
( A. Wald ) . ,
(
) ,
,
(
,
).
. .- . 1 , 2 ,...
,
F ( x ) = P { 1 x } .
,
.
( ) d D
21
( ) .
, ;
( ,
) . , An =
## = ( ; 1 , ... , n ) , 1 , ... , n = ( ) 0 , 1, ... + 0 { n } An
, ( A0 =
= { , } ). A n 0 A I { n } An
,
; n
A .
An n ( n
) , A (
, sup
s ( x ) M x g ( x ) +
;
.
: s ( x )
, ,
, ?
.
g ( x ) : g ( x ) c < . s ( x )
g ( x ) , g ( x )
= ( , d ) ( )
.
W ( , , d )
M W ( , , d )
* = ( * , d * )
. ;
, = ( d )
.
R ( ) =
= M x g ( x1 ) .
M W ( , , d ) (d )
*
W ( , , d ) -
c + W1 ( , d )
c 0 , W1 ( , d )
( ) .
*
d , ,
*
- . . .-
. .
X = ( xn , An , Px ) , n 0 x E , ( E , B )
Q g ( x ) = max { g ( x ) , T g ( x)} .
M x g ( x ) -
= sup M x g ( x )
0 = inf { n 0 : s ( xn ) = g ( xn )}
.
Px { 0
< } = 1, x E , 0
= { x : s ( x ) = g ( x )}
0 = inf { n 0 : xn } .
,
. , , .
. . .
1 1
0 . ( )
D - : d = 1
( H 1 : = 1 ) d = 0 ( H 0 : = 0
). W1 ( , d )
a , = 1, d = 0,
W1 ( , d ) = b , = 0 , d = 1,
0 ,
, Px ( ) x E
. , n
, g ( xn ) .
s ( x )
s ( x ) = lim Q n g ( x ) ,
; An n ( n )
22
s ( x ) = max { g ( x ) , T s ( x )} -
, xn , n -
, x . s ( x )
>0
= { x : s ( x ) > g ( x )},
R ( ) R ( )
= ( * , d * )
) .
= inf { n 0 : s ( xn ) g ( xn ) + }
*
( ) ,
f ( x ) , T f ( x ) f ( x ) -
f ( x ) , T f ( x ) =
)
. ( ) d = d ( ) (
) D A
x E x -
W ( , , d ) = c + W1 ( , d ) , R ( )
R ( ) = c M + a ( ) + d ( )
( ) = P { d = 0 | = 1},
( ) = P { d = 1 | = 0}
, , P
. n = P { = 1 | An } H 1 : = 1
= ( : 1 , ... , n ) -
An
R ( ) = M [ c + g ( )]
= min ( a , b ( 1 ) ) . xn =
, g ( )
= ( n , n ) -
, ( )
= inf R ( )
. = ( , d )
= P1{ d = 0} , ( ) = P0{ d = 1}
> 0 , > 0 ;
, ( , ) , ( ) , ( )
M 0 < ,
M1 < . .
. + < 1
( )
= inf { n 0 : n ( a , b)} ,
( ) = min { g ( ), c + T ( )}
. ( ),
1,
d =
0,
g ( ), T ( ) -
b,
a
, , A
B , 0 A < B 1 , =
= { : A < < B } ,
, n d --
= [ 0 , 1] \ ( A , B ) .
b = b ,
= inf { n 0 : n }
= ) .
p0 ( x) p1 ( x)
F0 ( x ) F1 ( x) - d =
( 0
= ( dF0 + dF1 ) 2
= ( , d ) a = a
- ,
a
= ( * , d * )
= a b = b ( , )
, ,
( , ) M 0 M 0 , M1 M1 .
n = p1 ( 1 ) ... p1 ( n ) p0 ( 1 ) ... p0 ( n )
, ( 1 )
* = ( * , d * )
H 0 : = 0 H 1 : = 1 t
.
,
*
= ( * , d * )
= inf { t 0 : ( t ) ( a , b ) } ,
1, b ,
d* =
0 , a ,
t = ln t
= 1 -
t = e t t / 2 ,
1.
A 1
B 1
= n :
< n <
1
A
1
B
0 = inf { n 0: n C } .
B 1
0
, d = 1 ,
1 B
A 1
H 1 : = 1 ; 0
1 A
, d
= 0 -
= 0 , H 0 : = 0 .
b a
b = ln (1 ) / , a = ln / ( 1 )
( 2 ).
= ( , d )
:
~
t = t ( , ) ,
~
1, ~t h ( , ) ,
d =
0 , ~t < h ( , ) ,
23
. . .
1957- ( , [1] ). - ( , [3] )
. , ,
.
0
2.
= ( , d )
~
t = t ( , ) ,
t ( , )
~
1, ~t h ( , ) ,
d =
0 , ~t < h ( , ) ,
= ( c + c ) 2 , h ( , ) = ( c2 c2 ) 2 , c
1
2
. M 0
ex
dx =
= 2 ( , ) , M1 = 2 ( , ) -
M0 *
( , )
= 2
,
t ( , )
(c + c ) 2
M1 *
( , )
=2
t ( , )
( c + c ) 2
( x , y ) = ( 1 x ) ln (1 x ) y + x ln x (1 y ) .
,
M 0 t ( , ) 17 30 ,
0, 03
M1 t ( , ) 17 30 .
,
. ,
= ,
lim M 0 * / t ( , ) = lim M1 * / t ( , ) = 1 / 4
0
.
.: [1] ., , . ., .,
1960; [2] . .,
, ., 1976.
( )
; sequential decoding
( ) ,
( , )
; , (
) ,
24
1.
, .
, .
,
.
.
,
n D -
(
1- ).
2- .
,
,
, .
. .- :
,
.
-
F (x) = P { < x} ,
A1 x
1 F ( x ) A2 x
, A1 , A2 , 1 , 2
.
1 = 2 = , ,
. .
( ),
( ) .
;
.
.
.: [1] W o z e n c r a f t J. M., IRE Nat. Conv.
Rec., 1957, v. 5, pt. 2, p. 1125; [2] . ., , ., 1974.
( = 0)
f ( )
X 1 ,..., X n ,...
( p ( , x ) -
k n
An = ( X k ),
P { f N f ( )} , ( f = lim
, f n
.
( , ) = M [ ln p ( , xn ) ln p ( , xn )] ,
Dc = { : f ( ) > c } , Dc = { : f ( ) < c } .
f n)
f n = min f r ,
= f n ( X 1 ,..., X n ), n = 1, 2 , ... -
f n f n+1 ,
f ( )
f n = inf { c : n ( c ) An ln (1 ) 1} ,
n ( c ) = sup
; c
< f ( ) .
( )
c > f ( ) ( c < f ( ) )
M Tc
M Tc
ln (1 ) B ( c )
ln (1 ) 1 B ( c )
(1)
ln p ( , X r ) ,
ln p ( r 1 , X r )
r =1
= n ( X 1 ,..., X n ) , n = 1, 2 , ... n
f* =
n
max
r = 1, ..., n
f ()
,
= sup{ c : n An ln (1 ) 1} ,
n (c ) = sup
Dc
ln p ( , X r ) .
r =1
## p ( , x) , f ( ), n (2), (3) (1)- M Tc ,
B ( c ) = inf ( , ) , B ( c ) = inf ( , ) ,
Dc
An =
c > f ( ) .
Tc = min { n : f n c }
Dc r = 1
()
.
.
Tc = min { n : f n c } ( )
(2)
f n = f n ( X 1 ,..., X n ) , n = 1, 2 , ... f ( ) -
f n f ( ) -
n = 1, 2 , ...
r =1,..., n
n = 1, 2 , ... P { f f ()} ,
- p ( , x )
).
( ).
= ( 1 , ..., m ) R m -
yox
; sequential confidence bounds /
limits
Dc
25
M Tc 1 -
N = min { n : n 4 c 2 K 2 2 }
(5)
.
f () a -
, K
( N , z1 , z 2 ) , N ,
(1 + ) 2 ( , [2] ).
An , n
= 1, 2 , ... ,
P { N < } = 1 , z1 z 2 AN ,
N = min { n : n max ( n1 , c 2 K n2 S n2 )} + k ( )
P { z2 z1 a } = 1 ,
P { z1 f ( ) z 2 } .
,
1 ,
ln (1 ) 1
[ 1 + o (1)]
W ( a )
M N
. M xn
max { B [ f ( ) + a t ] , B [ f ( ) t ] } . (4)
## (2), (3) ( N * , z1* , z 2* ) f ()
1 , , (4)-
a - ( 2 1)
,
*
N = min { n : f n f
z1* = f * , z2* = f
N
p ( , x ) = exp [ x b ( )] ,
a },
*
N
[6]- . I n =
~
I r -
r =1
~
I n = { : Mn ( ) (1 ) 1} , n = 1, 2 , ... ,
Mn ( ) =
p ( , xr ) dF ( )
r =1
p ( , xr )
r =1
n -
<
## ( , [3] [5] , [7] ).
.: [1] S t e i n C., Ann. Math. Statis., 1945, v. 16, p. 24358;
[2] S t e i n C., W a l d A., Ann. Math. Statis., 1947, v. 18, p. 427
33; [3] S t a r r N., Ann. Math. Statist., 1966, v. 37, p. 3650;
[4] C h o w Y., R o b b i n s H., Ann. Math. Statist., 1965, v. 36,
p. 45762; [5] S i m o n s G., Ann. Math. Statist. 1968, v. 39,
p. 194652; [6] L a i T., Ann. Statist., 1976, v. 4, p. 26580;
[7] ., , . .,
., 1975.
,
; sequential probability ratio test
H 0 : p ( x ) = p0 ( x ) H 1 : p ( x ) = p1 ( x)
( , d )
F ( I ) > 0
I n , n = 1, 2 , ...
( , [6] ).
X 1 , ... , X n , ... , a 2 N ( a , 2 )
p ( x )
X 1 , X 2 ,...
. . . . .-
Ln =
p1 ( xk ) p0 ( xk )
k =1
( A , B ) , 0
, F , -
.
2 , K n K , a n , S n2
a
(5)-
0
z2 = a N + ( c 2 )
, K ( )
W ( a ) =
0ta
z1 = a N ( c 2 ) ,
, n1
= inf
< A <1< B
= inf { n 1 : Ln ( A , B)} ;
d = 0 ( ) ( H 0 )
L A , d = 1 L B
. [1]- ( , [2]- ).
, . . . .-
( - ) -
, .
( , [1] ) a
, A B
: A (1 ) , B (1 ) . [3] [4]-
. , a
q [ P0{ d = 1} + c0 M 0 ] + (1 q ) [ P1{ d = 0} + c1 M1 ]
26
. . . . ( A B -
) ,
Pi M i i H i
, i
= 0 , 1 , q , H 0 - , c0 ,
c1 . . . . .
,
P0{ d = 1} P0 { d = 1} ,
P1 { d = 0} P1 { d = 0}
( , d ) M 0
M1 . . . . .-
[5]- ( , [2] ),
[6]-
. . . . .-
( A B )
[7] [8]- .
.: [1] ., , . ., .,
1960; [2] . ., , ., 1976; [3] W a l d A., W o l f o w i t z J., Ann. Math.
Statist., 1948, v. 19, 3, p. 32639; [4] W a l d A., W o l f o w i t z J., Ann. Math. Statist., 1950, v. 21, 1, p. 5299; [5] . ., ii i , 1958, . 1,
1, . 10104; [6] I r l e A., S c h m i t z N., Math. Oper. und
Statist., 1984, Bd 15, 1, S. 91104; [7] . .,
. ., 1987, . 32, . 1, . 6272; [7] . .,
. ., 1988, . 33, . 2, . 295304.
; sequential estimation
.
t
{ At } A ,
t 0
( , A , P ) .
{ At } ( ),
At , ( , )
<
~
, ( , )
~
, M =
~
, ( , )
; M
. ,
~
( , )
~
M ( ) 2 ( I ( ) M ) 1
( ,
(*)
[1] ), I ( )
(
).
p ( x ) - ,
[ .,
p( x)
M ( ) n
(
).
,
( , [2] ) (*)
.
.: [1] W o l f w i t z J., Ann. Math. Statist., 1946, v. 17, 4,
p. 48993; [2] . ., . ., , ., 1974; [3] . ., . ., . ., 1974, . 19,
. 4, . 70013.
; sequential design
of estimation , .
,
;
sequential design of experiment ( y1 ,..., y N ) = y1N
, y n - (
N , n - ) y n +1 -
xn +1 ( y1n ) X . . . . . . . .
,
. . . .- .
. . .- ( Y , Y )
x
P . ( X , X )
xn : Y n 1 X ,
## n > 1 U = ( x1 ,..., xn ,...)
( X , X ) - x1 q ( ) y1n
An N -
.
= ( U , N ) . B Y ,
= 1, 2 , ...
PU { y n B | y1n 1 } = Pxn ( B ) ( PU )
, X Y
- P
. (1)- ,
(1)
y n xn
y1n1 - . :
n - , , .
P ( ) PU - Y N = { y1N } . inf P ( N
< ) = 1
P ,
, ( ) = d P ( ) d P ( ) =
xi
, ( yi )
i =1
]
,
( , [3] ); p ( x ) ,
, x, ( )
= d Px ( ) d Px ( ) .
27
, x1
y1
d ( ) Y N -
; w ( , )
. , (U , N , ) s
0 + 1
1 , ( i , 1i ) =
1 1i ]
min K , ( ) ,
1i
X , ( X , X ) - .
2) P { = j } = ( j ) P1 , ..., PM -
M w ( ( y1n ), ) . ,
M ( ) ,
ln , , S ( f ) =
f xi ( yi ) .
i =1
N = M N <
x
D S ( f )
f =
D S ( f ) = M
D ( f
xn
( yn ) |
M ( ) ( ) T bb T + ( I + b ) ( N J ) 1 ( I + b ) ,
y1n 1 ) ,
= N
( ) , ( B)
n =1
B X
f (dx) = N f ,
N
## 3) max N inf * sup R , .
4) , R p
M : N0
inf * sup R, , R , = K , ( ) K , -
- , 1
X /
Px ( dy ) f x ( y ) .
= 0 ,
M S ( f ) = N
{ n N , xn B }
n =1
.
, :
1) K , = M ln , ;
= M ,
(
( ) , ( ) - );
3)
M S ( f v 1 f T ) .
: yN
( y1N ) ( )
, K , ( )
[
K , , I ( ) I
I I ( )
= ( b1 , ..., b p ) .
M ( yn | y1n 1 ) = ( xn , ) , D ( y n | y1n1 ) = ( xn , ) ,
. ( x , )
(2)
= T f ( x ) , ( x , ) ( x ) ,
N = const S ( m ) const , :
= S 1 ( m ) S ( f T y )
I = M ln , ( ln , ) T = M S ( , ,T )
I , b
5)
.
. . ., N n
2) R p -
K, I
ln [
(3)
, m
1
= f ( x)
( x) f ( x ) . (2) n n
= an , n -
0 - . ,
~
yn = y n ( xn , 0 )
(3)
,
f 0 ( x ) = ( x , 0 ) ,
~
y
. -
].
.
1) H i , i = 0 , 1 i -
, f 0 ( x ) , -
H i : i , i
= 0 , 1 , 0 U 1 = ,
0 I 1 = :
N 0 ( i , 1i ) sup inf K , , i
X * 1
28
diag ( ( x1 ) , ..., ( x N )) ( , ).
1) 3)
; M , q () 1
( , ); -
( , [2] ). 4) 5)
an S ( B )
B ( x)
( dx )
n ,
4)- B = I , 5)- B = m .
,
. ., 4)
: n , { Am }
T
y1 n L ,
I n Ln 0 , a n11
xn
0 Pn
n =1
. h R p :
ln d P n 1 / 2 ( )
+ h an
h T n
(1 2 )
h + n ,
(4)
n ( , h ) 0 P n ,
( n , I n ) ( , I ) , I ,
(
~ N ( 0 , I 1 )
d > c , P
. D . . .- .
d c , D = D d , d > c ,
D
d Pn ( ) =
h TI n
,
( . )
D - .
, . . .
,
. 1950- . .
( , [1] ) .
. N P n
c . d , d c , P ,
). (4)- x
= 0 . = nD N hNn,,dD
p ( d ) = d e d !,
( d ) p ( d ) =
, P -
d= 0
x X
, .
.: [1] , .,
1983; [2] . ., . . . ., 1979,
. 43, 6, . 120326.
.
. . .
## ; sequential simplex method , .
; sequential estimator
, .
. [1]- ,
( ) . . .,
,
, .
. [2]- , . . .
. . . .
, ,
.
.: [1] ., , . ., .,
1960; [2] . ., . .,
. ., 1974, . 19, . 2, . 24556.
; sequential estimators
P ,
-
. . . .
P D - ( . )
, .
, = (d ) > 0
(N
n d ) p (d )
d= 0
( N n) d ,
(d ) = N d n,
0,
d c,
d = c +1
d > c +1
.
.: [1] . ., , ., 1986, . 34063; [2] . .,
, ., 1975; [3] . .,
, ., 1979.
; sequential structure . . .
(x) =
= min ( x1 , ..., xn )
, n - n ,
x = ( x1 , ... , xn ) , (x) , . i -
xi :
1, i - ,
xi =
0 , i - ,
i = 1, ... , n , n .
29
(1 , 2 )
1, ,
0 , .
n ( 0 , 1 ) -
. ., ,
( , ).
( , [4] ): ( , d ) ,
sup M (
).
, sup M
.: [1] ., .,
, ., 1984.
## , ; sequential hypotheses testing , . , { , A , P } { At } - ( At
(
),
A )
d , A , 0 , ..., k
( )
( ) , ( , d ) H i ,
i = 0 , ..., k . . .-
(-
) .
,
,
.
(
) { P } ,
( , [1] [3] ). {P }
H 0 : 0 H 1 : 1 -
(1 , 2 ) M
( );
( , [5], [6] ).
(
) ,
( , [7], [6] ).
H 1 : 0 H 0 : = 0 , H 0
, , H1
,
( , [8] ).
., X i
N ( , 1) ,
= inf n m :
X i > [ ( n + 1) ( ln ( n + 1) + a ) ]1 2
=1
h
M - -
( ) ; n ( 0 , 1 )
P1 {
<
H1
. m a ,
( P1 {
< }
< }
, 0 < < 1
## [8] [10]- ); 0 P { <
< } = 1
, n ( 0 , 1 ) , H 0 :
= 0
H 1 :
= 1
[ ( 0 , 1 )
].
30
.
,
., [11], [12]- .
.: [1] W a l d A., W o l f o w i t z J., Ann. Math. Statist., 1948,
v. 19, p. 32639; [2] ., , .
., ., 1960; [3] . ., , ., 1976; [4] K i e f e r J., W i e s s L., Ann.
Math. Statist., 1957, v. 28, 1, p. 5774; [5] L o r d e n G., Z.
Wahr. und verw. Geb., 1980, Bd 51, 3, S. 291302; [6] . ., . ., . .,
1987, . 32, . 4, . 67990; [7] L o r d e n G., Ann. Statist., 1976,
v. 4, 1, p. 28191; [8] R o b b i n s H., Ann. Math. Statist., 1970,
v. 41, 6, p. 13971409; [9] S i e g m u n d D., Sequential analysis B.
N. Y., 1983; [10] W o o d r o o f e M., Nonlinear renewal theory in
sequential analysis, Phil., 1982; [11] L o r d e n G., Ann. Statist.,
1977, v. 5, 1, p. 121; [12] . ., . ., . ., 1983, . 28, . 544554;
[13] . ., . ., 1987,
. 32, . 1, . 14953.
( )
; arithmetic simulation o f
r a n d o m p r o c e s s e s
;
,
.
k , m . f ( m ) f ( k + m) =
= f ( k ) + f ( m ) , , f ( k m) =
= f ( k ) f ( m ) k m ,
p ( m ) = max { : p
. p ,
m } , y n : [ 0 , 1] [1, n ] , 1 -
, y n ( 0 )
= 1 , y n (1) = n .
f n
Hn (t , m) =
fn ( p
p (m)
) an ( t ) ,
0 t 1
p yn ( t )
. H n ( t , m ) - { n , An , Pn }
,
n = { 1, ..., n } , An , -
, Pn ({ m } )
= 1 n . an (t ) -
. H ( t , ) D [ 0 , 1]
. C [ 0 , 1] - . H n ( t , m )
. . ( [1], , [2], [3] ) , n
H ( t , ) Pn -
w (t ) .
H n ( t , m )
. H n ( t , m )
w (t ) - ( , [9], [10] );
( , [11] ); ( , [6] )
; ( , [5], [6] )
w (t ) - ( ,
[6] ).
( , [7] ).
g n ,
U n ( t , m ) = d n1
gn ( m + k) , 0 t 1 ;
k t hn
n hn , d n . U n ( t , m ) , { n , An , Pn } .
g n . . . . ( , [4] ) , U n ( t , m )
w (t ) -
. . . ( 1967 ) :
g n ( m) = ( m ) , n U n ( t , m ) w ( t )
? ( m ) .
[8]- .
.: [1] . .,
, 2 ., , 1962; [2] . ., .
, 1955, . 103, . 36163; [3] . ., Lect. Notes
Math., 1976, v. 550, p. 33550; [4] . ., . ., . . ., 1959, 6 (13), . 8895;
[5] . ., . ., . .
, 1982, . 25, . 20711; [6] ., .
. ., 1984, . 24, 3, . 14861; [7] .,
., . . ., 1984, . 24, 2, . 72
81; [8] . ., . , 1986, . 290, . 786
88; [9] P h i l i p p W., Proc. Symp. Pure Math., 1973, v. 24, p. 233
46; [10] B a b u G. J., Probabilistic methods in the theory of arithmeticcal functions, Calcutta, 1973 ( Diss. ); [11] B i l l i n g s l e y P.,
Ann. Probab., 1974, v. 2, p. 74991.
; arithmetic distribution x = n h ,
n = 0 , 1, 2 ,..., h > 0 . h -
. .- . . . ( , ). . .-.
. .- h = 1 ,
. . . . .-
2 h .
,
( z ) z 0 0 ( z 0 ) = 1 ,
. .-. h = 1
. .-.
.: [1] .,
, . ., . 2, ., 1984.
,
;
arithmetics of probability distributions ,
.
; ARIMA
process ,
.
; arcsine
law , . . 1939-
{ t , t 0 ; 0 = 0}
. t [ 0 , t ]
, , t , { u : u > 0, 0 u t }
. t t
. . :
P { t t < x } = F1 2 ( x ) = 2 1 arcsin x ,
0 x 1 , t > 0.
( , [2] ):
31
1 , ... , n , ...
,
S k = 1 + ... + k , 1 k n , S0 = 0 ;
vn = min { k : S k = max S m } ,
0mn
. .- . .,
. .- ( ,
). . . .
. .
K n 0 , 1, ... , n S k > 0 k ( ,
).
lim
## P { S1 < 0} + ... + P{ S n < 0}
=
n
, ,
0 < < 1 F ( x ) ,
F1 ( x ) = E ( x )
F0 ( x ) = E ( x 1);
x0
E ( x ) = 0 , x > 0 E ( x ) = 1 .
. . . t S t
< t St +1
, 0 <
<1
, St
= t St
P {1 0} = 1,
, ( 0 , 1) -
:
p( x) =
## lim P { St* t < x } = F ( x )
P {1 > x } = x L ( x ) , x > 0 ,
L ( x ) x - , 0 < y < -
x (1 x ) 1 , 0 < < 1;
= 1 2 . .- . .- . . .- (1 ) -,
(1 )
lim L( xy ) L( x ) = 1
.
. . ( , [3] ).
.: [1] .,
, . ., . 2, ., 1984; [2] .,
, . ., ., 1969; [3] . ., . ., 1971, . 16, 4,
. 593613.
; arcsine distribution
1
x ( 1 x )
x ( 0 , 1) ,
x ( 0 , 1)
2 arcsin x ,
F (x) =
1,
1
- .
.: [1] ., , . ., . 2, ., 1984.
32
, ,
p( x) =
0,
sin
x ( 0 , 1] ,
x >1
; arctangent law f o r r a n d o m m a t r i c e s ,
.
; Aronszajn
Kolmogorov theorem
:
f
, , , ( ) H
{ a , } , s , t
f ( s , t ) = ( a s , at ) . f -
,
.
; ARMA
process , ,
.
; increasing random process
= ( t ) t 0 , 0 = 0
:
1)
, s t s
t ;
P ( A ) P ( B ) . P ( B ) > 0 A B
2) At
= ( At ) t 0 ;
3) t > 0 P t <
lim t = .
P( A| B) =
< , . . .
, M n < n 1
( n , n 1)
lim n = P
P ( A I B ) = P ( A) P ( B )
, A B
. P ( B ) > 0 (1)
M = M
= ( t , t 0 )
H ( s ) d
= H ( , s )
.
. . .- .
.: [1] ., ,
. ., ., 1975; [2] . ., . .,
, ., 1986; [3] ., . .,
, . ., .,
1994.
; increasing
point , .
( ) ; dependent events ( , F , P )
A A , B A
P ( A I B ) P ( A) P ( B )
.
, .
; independence ,
. . .
, , , , , .- .
( , F , P )
, F
(2)
.
:
1) AF , A ;
2) P ( S ) = 0 , AF , S A
, A B1 \ B2 ;
;
3) A Bi , i = 1, 2 B1 B2
= M H ( s ) d s
(1)
P ( A | B ) = P( A )
P( AI B)
P( B )
, P ( A I B ) , A B -
, .
. . . - -
F -
P
. ,
. .-
.
A B . .
. A B ( A , B F ),
4) A B , A B , A B
.
n ( n > 2 ) A1 , ... , An . .-
. ., 2 m n m - k1 , k 2 , ..., k m n
Ak1 , ..., Akm
,
## P ( Ak1 I ... I Akm ) = P ( Ak1 ) ... P ( Akm )
(3)
, ( ).
A1 , A2 , ... , An . .- (
. .- )
. .- , i j ,
Ai A j , i = 1, n . -
, , , .
. . . . .
, ; , ,
,
( , [1], . 24 ).
. . . ( , A , P )
, A , , P , A -
.
. .- ( A
B ).
A1 B1 , ... , An Bn (3)
33
( ) ,
B1 , B2 , ... , Bn .
## ( . . ) ; ., X 1 , ..., X k X k +1 , ..., X n , 1 k < n
T , n 2
- t1 , ... , t n T
. . .-
. .,
X 1 , ..., X n ,
## Bt1 , ... , Btm , Bt
( t T )
. 1 k
Ak . .-
Bk = { , Ak , Ak , }
. .- .
. .- ( , [1] ).
X 1 + ... + X n
n
X =
(4)
X t , t T . . B ( X t ) -
1
n
. .- , B ( X t )
(X
X )2
(5)
j =1
X t .
( X k - -
A1 , ... , An . .- -
) . : (3) (4)
. .- X k -
I Ak - . . ,
1,
I Ak ( ) =
0,
Ak ,
Ak
. . .
X 1 , ... , X n . .-
.
1) a1 , ... , an
.
## FX 1 ,..., X n ( a1 , ..., an ) = FX1 ( a1 ) ... FX n ( an ) .
2) p X1 ( a1 ) , ..., p X n ( an ) R n -
( a1 , ..., a n )
## 3) f X 1 , ..., X n ( u1 , ..., un ) = M e i u1X1 + ... + i un X n -
Y1 =
aj X j
j =1
Y2 =
Xj
j =1
a j b j
, X j ( , .,
).
. .
.
2) .
1 2 , - .
m [ 0 , 1]
, F1 ( x ) , F2 ( x) , ... -
u1 , u 2 , ..., u n -
[ 0 , 1] - , m -
X k ( )
## f X 1 , ..., X n ( u1 , ..., un ) = f X1 ( u1 ) ... f X n ( u n ),
f X k ( uk ) = M e i uk X k .
m{ : 0 1, X k ( ) < x } = Fk ( x )
. .- :
( , .,
, , ),
( , ., ,
) . , .
.
1)
. X 1 , ..., X n
.
[ 0 , 1] -
34
. .-
1 -
## rk ( ) = sign sin ( 2 2 k 1 ), k = 1, 2 ,...
. ,
( , [3],
. III, 4 ).
3)
.
Y0 , Y1 , Y2 , ..., Yn , ... h ( x , y ) ( ) .
X 1 = h ( Y0 , Y1 ), X 2 = h ( X 1 , Y2 ) , ..., X n = h ( X n1 , Yn ) , ...
. , .,
.
.
4) .
, , ,
X 1 , X 2 , ..., X n , ... -
( m ,
X k X l , k l > m
).
. . .
5) . p 2 q 2 ,
. N 1 - N - ( 1 N ). Ap ( Aq )
p - ( q - )
.
1 N
1 N
P ( Ap ) =
, P ( Aq ) =
,
N p
N q
P ( Ap I Aq ) =
1 N
.
N pq
N Ap Aq
. , N
S = SN
A2 , A3 , ... , A ps ( p j , j - )
, . , . .
.
6) . . .- .
.: [1] . ., , 4 .,
., 1924; [2] . .,
, 2 ., ., 1974; [3] . ., , .: , , ., 1956; [4] ., ,
, . ., ., 1963; [5] ., , ., . 12, ., 1984;
[6] . ., ,
2 ., , 1962.
,
; property independence of class events ,
.
; test of independence . p n q ; p1 , ..., p q .
q
,
11 12 ... 1q
21 22 ... 2 q
...
## ... ... ...
q1 q 2 ... qq
, , j k
jk = 0 .
lH =
n2
jj
n2
j =1
jk
lHn 2 =
jj
j =1
, l , jj
p
M (lr ) =
j =1
kk - ,
r
n j
+ n
2
2
j =1
n j
pk
k =1
j =1
nj
pk
k =1
j =1
, 2 ln l
r
nj
+ n
2
2
f =
1
p ( p +1 )
2
p
i =1
( p j + 1) .
= 1
2 p3
p 3j 9 p 2
6n p 2
p 2j
j
p 2j
. j - p j
= 1 -
, p
35
f = p ( p 1) / 2 . = 1
( 2 p + 11 ) / 6n .
- ,
n 2 -
, ,
n 2 -
.
.: [1] ., ., , . ., ., 1976.
]
( )
( ) ; independent ( statistically independent ) events ( , F , P ) ,
A F , B F
P ( A I B ) = P ( A) P (B )
(*)
; P
, P
( , [2] ).
P ( A ) > 0 P ( B | A) = P ( B ) , , A
B
P (AI B)
,
P ( A)
P ( A I B ) = P ( B ) P ( A ) . , P ( B ) > 0
P ( A | B ) = P ( A ) , P ( A I B ) = P ( A ) P ( B )
.
1 . A B ,
A B , A B , A B
( , [1] ).
2 . B1 B2
A , A B1 U B2 . .
.- ( , [1] ).
A1 , A2 , ... , An ( Ai F , i = 1, n )
. . .-, Ai - . . . .
3.
Ai ( i = 1, n ) -
P ( Ai I A j ) = P ( Ai ) P ( A j ), i j ; i , j = 1, n
2
( Cn ) .
r =1
2k n
Bir =
P (B
ir
.
S k ( k , Ak , Pk ) .
P ( A1 I A2 I ... I An ) =
k -
, B1 , B2 ,..., Bn . . .
36
; independent
algebras of events , .
,
P
P ;
independent algebras of events relatively to
probability P , .
; independent
algebras of events , .
; independent classes
of events , .
; t o t a l l y independent classes of
events , .
( )
; independent thinning o f a p o i n t
p r o c e s s , .
; independent trials
## 1 i1 < i2 < ... < ik n i1 , i2 , ... , ik
. B1 , B2 , ... , Bn
, P P ; independent events
relatively to probability P ,
.
Ai Ai , i = 1, ..., n ,
r =1
;
totally independent random events ,
. . .- ( , A , P ) ( k , Ak , Pk )
B1 , B2 , ... , Bn ( Bi F )
.: [1] . ., . ., . ., ,
., 1979; [2] . ., . ., 1980.
P ( B | A )
i j , i , j = 1 , n Bi , B j - . . .-.
.
. . .-,
P ( Ai )
i =1
.
. . .-
, ,
:
.
.: [1] . .,
, ., 1982.
## / ; effective number of independent trials ,
(
) , ( ) (
)
. . . . . .-
( 0 t T
t
= 1, 2 , ... , T
) ( t )
m = M ( t )
m ,
*
m T - , t
m) =
2
m2 T
= (2 T )
2
( T ) b ( ) d
0
M ( m*T m ) 2 = m2 T = ( 2 T 2 )
T 1
( T ) b ( ) b ( 0 ) T
=0
= M [ ( t + ) m ] [ (t ) m ] ,
b ( ) =
(t ) -
[ , b ( 0 ) , (t ) ]. N 1 , ..., N m
m N*
=N
= M
N -
N ef = b ( 0 ) m2 T . ., t
t ) ( t )
T1 = ( 1 b ( 0 ) )
b ( ) d
0
, T
>> T1 N ef
T 2 T1
m 2 = M ( t )
2
2 = M [ ( t )
M ( t )]2 m k = M k ( t ) , . . . . .
T
(t )
. ( t )
b ( ) = C e
, .
( )
-
( ) .
F
. f F - ,
.
( ) =
f ( x ; ) P (dx) N
[ f (1 ; ) + ...
+ f ( N ; )] * ( )
- ), . . . . .,
( )
; independent shift o f a p o i n t p r o c e s s
i =1
; independent
random events , .
; pairwise independent random events ,
; dependent trials
M ( m*T
,
; independent spectral types of measures , ,
cos ,
m , m 2 , (1) , ... , ( t ) ,
2
. . . . . T - [1]-
.
.: [1] B a y l e y G. V., H a m m e r s l e y J. M., J. Roy. Statist. Soc. Suppl., 1946, v. 8, 2, p. 18497.
= 1 ,..., k P ( dx )
i .
* ( ) ( ) - C - , C (n) . P = P ( dx ; )
.
.: [1] . .,
, 2 ., ., 1975; [2] . ., .:
, ., 1964, . 563; [3] . .,
. ., . VI . . .
. , , 1962, . 42537.
*
; dependence
, ,
. -
, -
-
; - .
.
37
.
( , , ):
cov ( ; ) = M (( M ) ( M )),
( ; ) = cov ( ; )
D D .
(1)
(2)
,
( , )
= a + b ( a
b , a 0 ) ,
3 -
, = D
, , .
. .,
As = 0 .
, : As > 0 ; 2
As < 0 . M 0
, ,
.
: , . .
, , . . .
+ 1, a > 0 ,
( ; ) =
1, a < 0
( , [2], ., 249 ).
.
( ) . ( , ) > 0 , ,
( , ) < 0
, ( , ).
( 1906-
) .
-
( , [1] ).
( , F , P ) , {Fn }
, F0 F1 ... Fn .
n m B k Fk , Fn
,
. . .
n = 10, p = 1 5 ( ) n = 10, p = 4 5
( )
As =
1 2 p
n p ( 1 p)
( k ; r ; p )
n+ m B n+ m | Fn } = P {1 B1 , ..., n1 B n1 / Fn }
(3)
, Fn P
, . (3)
.
.
.: [1] ., , . ., ., 1962;
[2] . ., , ., , 1980; [3] . ., , ., 1969.
.
;
asymmetric channel , .
; coefficient of skewness
b)
2 1
As = 3 ; 3 = 2 ;
## P { n+1 B n+1 ,..., n+ m B n+ m | Fn }
a)
As =
2 p
r (1 p )
.
. . .
. . ,
.
. . , .
.
, .
. . ,
, .
As =
1
ns 3
(x x)
i
i =1
3
As = 3
38
,
x1 , ... , xn
, x
.
s -
L ( n ) L ( n ) ,
;
z.
L ( n ) L
L ( n ) L
z.
L ( n ) L ( n )
;
z.
z.
L ( n )
L
L.
Fn F .
.
, , L ( n )
tam.
~ L (n )
n =
nk
= n* =
k= 1
*
nk
k=1
, nk -
*
nk nk
*
L ( nk )
*
= L ( nk ) .
n = 0 P ( k ; n , p ) -
: ) p = 1 / 5, , ) p = 4 / 5
.
, ; skewness of a distribution
. .
.
.
( , [1] ).
( ) , . . ( ).
. . ( ) ,
( ).
.: [1] ., , .
., 2 ., ., 1975.
*
nk
k =1
L ( n ) - . ,
; n L ( n ) L ( n ) . ,
.
n L ( n ) L ( n )
(
) ,
L ( n* )
tam.
L ( n ) ~ L ( n* )
L ( n )
nk
. , [1], . VIII,
28.1, 28.2.
.: [1] ., , ., 1962.
*
; asymptotically independent
random variables , .
*
; asymptotic expansion .
F ( x ) ( , ) -
, c1
>0
lim x { 1 F ( x ) + F ( x )} = c1
; asymptotic independence
.
nk
. - p
( 0 < p < ) .
f ( x ; , ) .
0 <
<1
x > 0 ,
f ( x; , ) =
1
x
( 1 )
k =1
k +1
sin
( + 1)
2
( k + 1) k
x .
k!
(1)
> 1 (1)- .
1 < < 2 , x > 0 ,
39
f ( x; , ) =
k
cos
2
(1)
( )
;
asymptotic expansion o f a d i s t r i b u t i o n
( ( k + 1) / ) k
x
k!
k =0
1 2
1 + 1 +
.
k
(2)
G( ) =
< 1 (2)- .
= 1, x > 0 , N
f (x +
1
x
ln x , 1, ) =
k =0
0 , 0 , ( ), a j , - ,
t i + i
ln t d t .
. k - M n
= a,
r 3 -
g ( z) < 1
lim
g ( z ),
P { n < x} = G ( x )
; n
n
1
i n a -
Fn ( x ) x -
i =1
r 2
Fn ( x ) = ( x ) +
Qm ( x)
nm
m =1
+ o(n
( r 2 ) / 2
1
2
l =1
1
kl !
ex
l , 1 l -
n .
, M X 1
= 0 , M X 12 = 2 > 0 , M X 1
<
k 3 k
## lim sup M e i t X1 < 1
|t|
, n
z2 2
= (x ) +
dz ,
< x =
i =1
j 2
Q j ( x ) + o ( n ( k 2 2 ) )
j= 1
x R .
Q j ( x) =
l
,
.
.: [1]
, . 2, ., 1985; [2] .,
, . ., . 2, ., 1984.
40
. . . ( [2] ). .
( [3] ). .
.
( [4] ). [5] [7]- .
., : { X n }
m+ 2 s 1 ( x )
l +2
( l + 2 )! l + 2
1887- . . Fn ( x ) ( x )
1
P
n
( 0 , 1)
Qm (x ) =
, ( x)
k 2
, ( x)
.
Fn ( x ) , n -
.
{ k , k 1} ,
{ j ( )}, 0
j +1 ( ) = o ( j ( ) )
t k
D n =
j ( ) + o( k ( ) )
. G ( )
, k
bk = Im
a
j =0
bk k
x + O ( x N 2 ) ,
k!
P ( ) 0
1
2
ex
P3 j 1 ( x ) ,
P3 j1 ( x) , x - ( 3 j 1)
X 1 ( j + 2 ) -
. Q j ( x)
.
( )
,
,
.
.: [1] . ., . . ., . 2, . ., 1947;
[2] E d g e w o r t h F. V., Trans. Comb. Philos. Soc., 1905, v. 20,
p. 3665; [3] C r a m e r H., Scand. Aktuarietidscrift, 1928, v. 11,
p. 1374, 14180; [4] E s s e e n C.-G., Acta math., 1945, t. 77,
p. 125; [5] . ., ., . .,
, . ., . 1972; [6]
, ., 1972; [7] H l l P., Rates of
convergence in the central limit theorem, Boston, 1982.
( )
; asymptotic expansion o f t h e r i s k o f a n
e s t i m a t o r
( ,
) . .,
. . n
f ( x , )
H 1 : 1 H 0 : 0
, 0 1 , -
. n -
H 0 H1 , , .
{ n } ,
n ( X n | n ) X n
n { n ( X n )}
,
lim sup M n , n ( X n ) M n, n ( X n | n ) = 0
, { n ( X n )}
, .
. . .- .
., (*) ,
lim
M | |2 = n 1 g1 + n 3 2 g 2 + ... + n k 2 g k 1 + O ( n ( k +1) 2 )
(*)
r n ( n ) r n ( n ( | n ) ) = 0
, r n ( n ) , n
, g i . . -
n . , ., [1]-
.
.: [1] . ., .,
1976, . 21, . 1, . 1633; [2] . ., . .
. . 1981, . 45, 3, . 50939.
.
. . .- , { n }
( ) ;
asymptotic expansion o f e s t i m a t o r s ,
( ,
) .
., n
f ( x , )
. .
i = n 1 2 h1 + ... + n k 2 hk + n
, hi , n
n
. . .-, ,
.
.: [1] . ., . ., .
, 1963, . 149, 3, . 51820; [2] . .,
. ., 1973, . 18, 2, . 30311;
[3] . ., . ., 1977, . 104, 2, . 179
206.
; asymptotically Bayes
test
,
.
:
Pn, , n X n
. . . . . .
(
. . .- ). . . .-
, .
.: [1] . ., , .,
1984; [2] L i n d l e y D., Proc. 4-th Berkeley symp. math. statist.
probab., v. 1, Berk., 1961, p. 45368.
*
## ; asymptotically mutually effective sequence
of estimators ( (n)
, n
I (n) ( )
( ( n)
( n)
, D ( ) ,
. n
I ( n )1 ( ) D ( n ) ( ) 1
( (n)
,
.
.
f ( x,a, ) =
1
2
( xa )2
2 2
41
a .
n
, f ( x , a , )
k a
.
n
D () =
1 , ... , n .
(k a ) = D
k =1
M
( k a )
k =1
k =1
a ) = n D ( k a ) = n ,
(k a )2 n = D
k =1
( k a ) =
k =1
= n D [k a ] = 2 n ,
2
, :
2
n
ln
M
= ,
ln
M
ln ln
= 0,
a
n
=
2
2
I ( ) =
k ,
k =1
s2 =
)2
k =1
. s 2 . ,
M ( s 2 ) ( ) =
=
=
n
1
M
( k a ) 2 n ( a ) 2
n 1
k =1
1
M
n 1
(
k =1
( a) =
a ) 2 ( a ) n M ( a ) 3 = 0 ,
42
a - s 2 . . . . .-.
.: [1] . ., . ., . ., , .,
1979.
,
; asymptotic
mergence of states o f a M a r k o v c h a i n ,
;
; asymptotic stability i n p r o b a b i l i t y
, ( ),
.
0
0.
n
22
1
n 1
. ,
; asymptotically efficient estimator
a
n
n2
0
; asymptotic deficiency , ( ) -
( , [1] ). ,
1
n
I ( ) D ( ) =
0
( k a )2 n =
k =1
( k a ) ( j a ) 2 = 0,
k, j =1
= M
0
0
2 2
n 1
.
, , , . . . .
.
, ,
.
. . . .-.
.: [1] . ., , . ., ., 1968; [2] . .,
. ., ,
., 1979; [2] . ., , .,
1984.
( )
asymptotic efficiency o f a t e s t
;
.
30 40- , ,
.
. .- .
, P
H 1 :
H 0 : = 0 . ,
- N 1
,
N 2 .
e12
= N 2 N1
. e12
, ,
, ,
, .
.
- , lim e12 ( , , ) (
0
) ; -
, lim e12 ( , , ) ( )
0
1 , lim e12 ( , , ) (
1
)
.
.
.,
,
.
., , 1
; H 1 :
>0
H 0 : = 0 . X
t - . t
X - .
,
. , X - t -
> 0
1 - .
-
1
0
. , , 0
,
- .
e12
.
.: [1] ., .,
, . ., ., 1973; [2] B a h a d u r R.,
Ann. Math. Statist., 1967, v. 38, 2, p. 30324; [3] H o d g e s J.,
L e h m a n n E., Ann. Math. Statist., 1956, v. 27, 2, p. 32435;
[4] . ., , ., 1995; [5] L a i L., Ann.
Statist., 1978, v. 6, 5, p. 102747; [6] B e r k R., B r o w n L.,
Ann. Statist., 1978, v. 6, 3, p. 56781; [7] K a l l e n b e r g W.,
Ann. Statist., 1982, v. 10, 2, p. 58394; [8] W i e a n d H.,
Ann. Statist., 1976, v. 4, 5, p. 110311; [9] K a l l e n b e r g W.,
Ann. Statist., 1983, v. 11, 1, p. 17082; [10] G r o e n e b o o m P., O o s t e r h o f f J., Int. Statist. Rev., 1981, v. 49, 2,
p. 12741.
## , ; B a h a d u r asymptotic efficiency of estimators
, . [1]- .
- ,
.
.
.
.: [1] B a h a d u r R., Sankhya, 1960, v. 22, 34, p. 22952;
[2] ., ,
. ., ., 1976; [3] . ., . ., , ., 1979.
,
; R a o asymptotic
efficiency of estimators , . .
, 20- 60-
. . .
,
.
.
.: [1] R a o C. R., J. Roy. Statist. Soc., 1962, v. B 24, 1,
p. 4672; [2] . .,
, . ., ., 1968; [3] . .,
. ., ,
., 1979.
, ;
W o l f o w i t z asymptotic efficiency of estimators
, . [1]
.
,
. { n }
b
>0
b -
lim P { n ( bn 1 2 , + bn 1 2 )}
, { n }
.
.
43
.: [1] W o l f w i t z
J., . .,
1965, . 10, 2, . 26781; [2] W e i s s L., W o l f o w i t z J.,
Maximum probability estimators and related topics, [ B. N. Y. ], 1974;
[3] . ., . ., , ., 1979.
## ; sequence of asymptotic neglectability random variables X k , n ( k = 1, n ; n = 1, 2 , ... )
,
X k , n k - ( k = 1, 2 , ..., n ) ,
k = 1, 2 ,..., n
> 0
## P { | X k,n | > } <
.
( ) X k , n ( k
= 1, n )
, ).
; asymptotic neglectability ,
.
## ; asymptotically most powerful test
( ) . X n ,
H 1 : 1
= \ 0
H 0 : 0 . K
1 1 . . . .
,
.
{ n }
; asymptotic
estimation theory ,
,
. .
1. . . .- n
X (n ) = ( X 1 ,..., X n )
(1)
. . . .-
( n ) .
, , . . .
( ,
),
.
(1) P
T = T (P ) - (1)
Tn ( X 1 , ..., X n ) ;
, . . ., , Tn ( X 1 , ..., X n )
n . ,
Tn T
1. (1)
= ( P ) = M X 1
a =
(
1
n
j =1
PP { | ( P ) a | > } 0 ),
n ( a2
2 ) 1/ 2 ( a )
, { n } K K
1 1
.
K
( { n } K )
lim sup M n, 1 ( n n ) 0
; asymptotically most powerful
unbiased test
S n , ( ,
n -
. { n } K
lim sup M n, n
n 0
.
2. f ( x ) (1) p - p . p - z p .
( p (1 p ) )
1 / 2
n f ( p )
( z p p )
.
3. F ( x ) = P {X 1 < x } , Fn ( x )
-
.
. . . . . , n -
(1) .
Fn ( ) C ( , ) - F ( )
44
C ( , ) -
n ( Fn ( x ) F ( x ))
w ( x) M w ( x )
. . .
(n )
. X
X . X {X , A }
X - P ,
{P , }
T - ( ) - P -
= F ( x) ( 1 F ( x)) .
2.
). T -
.
( ) X
. T ( X ) .
T
-
. , > 0
0 T
; , ,
{ } .
.
6. (1) .
1 3 d , p , F
. 3-
. F , X j
g ( F ) - ,
g ( Fn )
.
7. 5- X ,
t
, X (t )
S ( u ) d u .
0
8. X T m
X (t ) , 0 t T
T
. X T
= T 1 X ( t ) d t , m
0
( ), . . . Pu
, 0 , .
4. 1 3 X
.
5. X , k
, R
(2)
,
k .
( ) 0 .
, ,
: X = X (t )
d X (t ) = S (t ) d t + dw (t ) ,
0 t 1
( X (t + ) X
)( X (t )
= X n , = n 1
X = +
, T ; T
X T ) d t R ( ) , > 0
.
. . .-
, ,
.
; P , ,
(3)
, S
, S L2 ( 0 , 1) , w .
,
, . . . .,
, .
. . .
.
. . .- .
3. (
- ) ( )
{ P , } X . . . .- ( )
T , 0 ,
, ( ) (
P ( u ) =
d Pu
( X ),
d
R k , ,
.
9.
X T = ( X 1 , ... , X T ) X j -
1-
X j = X j 1 + j
, i , ( 0 , )
2
,
( 1, 1) .
T , T ,
j X j 1
j=2
X
j =1
2
j
45
10. , (2) X
X . (3) S = S ( t ; )
,
R .
1
S (t ; ) d X (t )
2
, .,
S ( t ; ) = S ( t ) , R ,
S ( t ) dt
S ( t ) dX
( t ) = + o (1) .
T - T
. , . . .- 0
P { | T T |
> }
. M | T T | ,
M exp{ b | T T | }
0 1
.
n
= D n = n 1 (1 ) ( 4n ) 1; -
P { | n | > }
2 exp{ 2n 2 } .
,
T T -
.
a ( ) ( T T ) T
R . , { f ( x ; )}
{ f ( ; )}
I ( )
f f d
i j f
n ( n ) -
,
I
( )
n ( n )
p
n
ln ( u ) =
ln f ( X
; u)
, . ,
n ( n ) = I 1 ( ) n 1/ 2
d
ln f ( X j ; ) + o (1)
d
.
12. R
, f ( x ; ) = f ( x ) .
I ( ) = I =
( , , . ). , (1) .
j =1
. X {X , A , P , }
46
f ( x ; )
. a ( ) ( T T )
j =1
1 10 .
4. T T -
T T
.
.
(1) , X j -
N ( , I 1 ( ) ) , p > 0 .
j =1
M n
,
.
, T ( X )
.
11. (1) X j 1
n = n 1
( z p , z q , ... ), ( Fn ) -
{X , A }
, T - T
1 3 , ( a , a , ... ),
a ,..., z p p
S (t ; ) d t = 0
( 1 ),
( 2 ), ( 3 )
, T T .
f ( x )
f 1 ( x ) d x
n ( n ) ,
( 0 , I 1 ) . I , n , , ,
n - . ., f ( x ) = ( ( )) 1
x 1e x , x > 0
n
1/
>2
<
( n ) ( 0 , I ) 1
. 1
1/
< 2
( n )
.
13. X j
,
= 1 .
n
= n 1
X
j =1
l 0
lim lim
sup M u l ( n ( Tn u ) ) M l ( )
0 n | u | <
n ( n ) ( 0 , 3 )
( , . 4 )
.
., (1) , . 4-
( ). Tn -
, n -
.
f ( x ; )
.
. T T
N ( 0 , I ( )) ; ( , . 4 ), nn
1
lim M l ( n ( n ))
n
f ( , )
= M l ( ) -
n ( n )
.
. ,
:
.
.
.
. X j
: M l ( n ( n ) )
.
5.
, , . .,
( R ) T -
[ 0 , ) -
M l (T ; ) . T -
( 12, d
, , M l ( T ; )
-
,
( T
-
).
l - - , 0
T - - .
,
l ( T ; ) = l ( ( ) | T | ) ,
( ) .
## lim lim inf
T
0 0
sup M u l ( ( ) ( T u ) )
| u |<
~
sup M u l ( ( )( u ) ) = 0
| u |
~
, ,
( )
. ,
,
.
= 0 ) *n = min X j + n
l - ; ,
l
l(s +u) e
d u .
,
( , ) ,
M l ( n ( n ) )
.
( 2- ).
6. ,
. ,
, : , ,
.
14. (1) ,
P { X 1 x } = F ( x ) . F
, X
= n 1
j =1
47
; F , X
. z
n ( X )
F -
2
n (z
1/ 2
= 1 , M
1/ 2
n (z
1/ 2
= (1 + o (1 ) ) 2
( 2 ).
14- ,
, (
), , M . , ., Tn - ,
.
7. . . .- , n - . -
n ( n )
. , ., M
,
. k ,
. , n
k . k = k ( n ) ,
., k ( n ) n 0 k (n) / n 1 , . . .-
.
, . . , ,
.
.
, ,
l : , diam l 1 ( ) ( ) 0
,
. -
. ,
: .
15. (1) R - X j
f (x ) -
;
f ,
H ( b ( x y ) ) f ( y ) dy
a = a ( n ), b = b ( n ), H .
H ( b ( x y )) dFn ( y ) (
) .
.: [1] . ., , .,
1984; [2] . ., . ., , ., 1979; [3] .,
., , . .,
., 1973; [4] ., .,
48
; asymptotically
unbiased test
. X n ,
n
. X n H 1 : 1
= \ 0
H 1 : 0 n
.
## lim sup [ sup M n, n inf M n, n ] 0
1
, { n } . . . .
.
.: [1] ., .
, . ., ., 1975; [2] . ., , ., 1984.
; asymptotically unbiased estimator
;
, . {Tn }, bn ( )
= M Tn , n = 1, 2 ,...,
Tn -
n bn ( ) 0 ,
{Tn },
.
.
. X = ( X 1 ,..., X n ) a = M X i ,
2 = D X i .
n - a
, . ., ., 1976;
[5] ., , . .,
2 ., ., 1975; [6] . .,
, ., 1972; [7] G r e n a n d e r U.,
Abstract inference, N. Y. [a. o.], 1981; [8] L e C a m L., Asymptotic
Methods in Statistical Decision Theory, N. Y. [a. o.], 1986; [9] L e h m a n n E., Theory of Point Estimation, N. Y. [a. o.], 1983.
sn2 =
1
n
(x X
i
i =1
n)
Xn =
1
n
i =1
{ s n } 2 . .
2
. . , , 2 X
n -
bn ( 2 ) = M sn2 2 = 2 n 0
.
., .: [1]
, . ., 2 ., ., 1975; [2] . .,
. ., , .,
1989.
( ) ; asymptotic admissibility o f s t a t i s t i c a l t e s t , ( )
.: [1] . ., , .,
1984; [2] . ., . .,
, ., 1979; [3] W a l d A.,
.: Proc. 2-th Berkeley symp. math. statist. probab., Berk., 1951,
p. 111.
; asymptotically uniformly most
powerful test
; asymptotically
minimax tst
,
. X n ,
n
. n -
. . . . . .
lim sup M n, 1 ( n n ) 0
: X n
H 1 : 1 H 0 : 0
( , , n -
). n ( X n ) , X n n -
## lim sup sup M n, 1 ( n n ) 0
n
. n n
lim sup M n , n ( X n ) M n , n ( X n )
n
1
=0
, { n ( X n )}
, . (*)
lim
(*)
inf M n , n ( X n ) sup
inf M n , n ( X n ) = 0
{ n } K 1
;
asymptotically uniform distribution
. U = ( 1 , 2 , ... )
S k = 1 + ... + k . h
lim P { Sn = j mod h } = 1 h ,
. . . ,
j = 0 , 1, ... , h 1
## K = {{ n } : lim inf sup M n, n ( X n ) } .
n
. . . .
.: [1] . ., , .,
1984; [2] . ., . ., .: , ., 1982, . 7990.
; asymptotically minimax estimator -
, U
.
R1
## lim sup sup M [ n ( n ) ]2
n
A , U ,
min { P (
( , [1] ). S n . . .-
n f n ( 2 r h ) 0 -
( ,
[2], [3] ).
n ,
Lah ) : 0 a h 1} =
h ,
r = 0 , 1, ... , h 1 ( , [4] ). S n . . .-
## lim inf sup M [ n ( *n )]2
, . . .
. U A
Lah = { m : m = a + kh }
, . n , n
,
11
.
. , ., .
.: [1] . ., . .,
, 3 ., ., 1987; [2] . ., .
, 1954, . 98, 4, . 53538; [3] . ., , ., 1987;
[4] D v o r e t z k y A., W o l f o w i t z J., Duke Math. J., 1951,
v. 18, p. 50107.
49
; B a h a d u r asymptotic
relative efficiency , ( )
. n
P { X n < x } = Fn ( x ) ( x )
( x)
; H o d g e s
L e h m a n n asymptotic relative efficiency ,
, u n ( x ) Fn ( x ) -
( )
; P i t m n asymptotic relative efficiency , ( ) -
n S n
P { S n*
n x - ( < x < )
{ S n* }
S n*
< x } -
Fn ( x ) = ( x ) + ( x )
Sn a n
) . { S n } :
> 0 n
M ( (
ak ) ; k ak > Bn ) 0 ,
2
k=1
M k = ak ,
D k = k2 ,
Bn2 =
2
k
~
Sn =
k =1
Sn
a
k =1
Bn
( x ) = ( x) ,
y ( x) = 1[ Fn ( x ) ] = x +
Q ( x)n
k
k 2
+ O ( n ( r +1) 2 )
k =1
( [1]
[2]- ).
un ( x ) = x +
Q ( x) n
k
k 2
+ O ( n ( r +1) 2 )
k =1
. . . .- .
.: [1] . ., . .,
1959, . 4, 2, . 13649; [2] W a z o w W., Proceeding of symposia in applied mathematics, 1956, v. 6, p. 25159.
; asymptotically normal estimator ,
.
X 1 , ..., X n
.,
M X i4 <
X =
1
n
M Xi = ,
2
s
i =1
1
n
DX i = 2 ,
.
(X
X ) 2
i =1
. . . .-.
, {
n ( X ),
. ., , .,
## ; asymptotically normal transformation
50
Pk ( x ) , x - ,
n ( s 2 2 ) }
.: [1]
1972.
+ O ( n ( r +1) 2 )
P { un ( X n ) < x } = ( x ) + O ( n ( r +1) 2 )
.
{ n } , (
1
Ln ( ) = 2
Bn
1 2
. S n
, n
( x ), ( 0 , 1) *
P (x ) n
k =1
S n = 1 + ... + n .
S n* =
( x ) -
u n ( X n ) - X n -
## ; asymptotically normal sequence a -
2
2
( 0 < < ) ,
,
3
, k
3
4 22
= M ( X i ) k .
.: [1] . ., , .,
1984; [2] . ., . ., , ., 1979.
## ; asymptotically normal random
variables , .
*
; asymptotic normality , .
## ; asymptotically optimal test
. . . .
.
. . . :
. . . .- : ,
,
.
; asymptotic
Pearson transformation , . . . .
. t
P { X < x } = F ( x , t ) ( x)
; ( x ) . X
F ( x , t ) :
F ( x, t ) = ( x) + ( x)
P ( x) t
k
+ O(t
),
= ( x ), Pk ( x ) .
, :
y ( x ) = 1 [ F ( x , t ) ] = y r ( x ) + O ( t r +1 ),
yr ( x ) = x +
k ( x)
tk .
k =1
, ur ( x )
= y r ( x ) + O ( t r +1 ) , t 0
P { ur ( X ) < x } = ( x ) + O ( t r +1 ) .
., X ( p , q ) , t
= 1 q 0 , p = const -
F ( x, t ) = I ( x, p ) + O (t )
, I ( x , p ) , p .
u1 ( x ) = x [ 1 s ( x + p 1)] ,
s = t [ 2 + t ( p 1)]
, s 0
.: [1]
. ., . ., 1963, . 8, 2, . 12955; [2] . .,
. ., ,
3 ., ., 1983.
;
asymptotic design ,
.
( ) ; asymptotic
density o f a s e t , .
; asynchronous channel of multiple access ,
.
; asynchronous channel , .
; assosiated
spectrum of a process ,
.
## ; slowly changing function
, .
( ),
; lower value of a game , ,
.
## ; lower confidence bound ,
.
*
; lower
r +1
k =1
( x )
P { u1 ( X ) < x } = I ( x , p ) + O ( s 2 ) .
quartiler
p =
1
. ,
4
( ) ; lower limit o f
sequence
of
random
e v e n t s
A1 , A2 ,... , , (
) ; An ,
n = 1, 2 , ... ,
.
A* { An }
lim An . A*
n
A* = lim An =
n
UIA
n =1 m = n
. , ,
, { An }
( )
.
51
; positive outlier
, , .
## A1 A2 ... An An+1 ,...
, { An } ,
A* = lim An =
n
An
( )
; lower boundary o f s e q u e n c e o f r a n d o m e v e n t s ,
, .
; length of outlier
, ,
n =1
.
.: [1] . ., . ., . ., , .,
1979.
( ) / ( )
; outlier
## ; lower boundary functional
, .
; changepoint
problem ( ). ,
( ). ,
= ( t ), t 0 -
,
;
excursion of a random process and field
( )
.
(t ), t = ( t1 , ..., t N ) R N
( N = 1 , N 2 ) u ( t )
+
Au ( )
= { t R N : ( t ) u }
. . . .
. .- , , .- ( , [1] [3] ).
{ ( t ), t (0 , T )} T
.
(1) , t < ,
t = t( 2)
t , t ,
( ) ,
( 2)
(1)
[ ( At ) t 0
At = ( s , s t ) ] , = P { < }
.,
R ( ) =
= M ( | 0 ) t - , .
a { : P {
*
R ( * ) = inf R ( )
< } a }
( , [1] ), { : P ( < ) a }
.
, t , t
T ( T
> T ), .
. ,
( , [2] ).
.: [1] . .,
, 2 ., ., 1976; [2] . ., . ., . ., . . - ,
1988, . 182, 5, . 423.
52
.
T H
t m ( ) H m .
(t ) C
c (t ) ( )
, , 0
( C 0 ).
(t ) C
,
, . , ( 0 , T ) (t ) C ( ) n - ( );
, ,
.
( )
.
, H C T
, n , 0
H m
.
n , , , 0 , H , H m
-
. ., n , H
.
n
( ) , (t )
( ) ,
( , [9] ). ( , [10] ) ( N + 1)
.
( ) H
.
n , H H m
( , [4] ) .
. ,
H m -
( , [5], [6] ).
{ ( t ), t R N } -
u ~
Pu
N = 1 { ( t ), t R1}
u
H u , S u Vu
( M (t )
= 0 );
, .
B(s) =
~
Pu u
M ( t ) ( t + s ), s , t R N
. u
, | t | < H 0 , H 0 > 0
M ( 0 ) = 0 ,
u -
B ( t ) B ( 0 ) < A
ln | t |
1+
, A
> 0, > 0
. H u , Vu , S u ,
(t ) u , ( N + 1)
(t ) u ( ) g (t ) = u ,
( N + 1) N .
i (t ) = ( t ) ti ,
1 i, j N
(1)
t , ..., t
(k )
i = 1,..., N ,
k = 1, 2 ,...
ij (t ) = ( t ) ti t j ,
2
i = 1, ... , N , ij ( t ( s ) ) , 1 i j N
( , [11] ).
( t ) , (t ) ;
Y (t ) = ( ( t ) , ( t ) )
y12 + y 22 < u 2 ( t ) ( , ; )
(t ) ,
i ( t (s ) ) ,
,
( , [11] ).
D R N (t ), t R N
( t (s ) ),
.
B (t ) | t | < H 0 , H 0 > 0
;
+
Au ( ) I D
4B (t )
t1n1 ...
t NnN
N
4B (0)
<
## t1n1 ... t NnN
ln | t |
1+
, A
> 0, > 0,
= D , D . ( t )
, N
ni = 4 , ni = 0 , 1, ... , 4 ,
i = 1, ... , N
. ( , [8] )
~
~
lim Pu {u H u > } = lim Pu { u 2 Su2 N > } =
= lim Pu { u 22 ( N +2 ) Vu2
( N +2 )
> } = e
B ( 0)
= 2 , N u+ ( D )
( , [12] ).
,
i =1
0.
(t ), t R N
+
Au ( ) I I 0 ( I 0
= [ 0 , 1]n ) -
( , [3] ).
N = 2 u
2
## N = ( 2 B ( 0 ) ) N 2 ( N / 2 + 1 ) det (1) , > 0 ,
N + (D) , D = D U ,
N +2
= ( 2B ( 0 ) )
(1)
( z )
( N / 2 + 2 ) det , > 0 ,
2
(1)
= ( M i (0) j (0) )1 i ,
~
jN
, Pu , (t )
. , [ 0 , T ] ,
T
( , [13] ).
Au+ ( ) .
53
Au ( )
= { t : ( t ) = u }
. ( , [3] ).
( , [14] ) ; u
(
A ) ,
.
( , [15] ),
u A
u (
F - . ( , F , P ) , F ( x )
. > 0
F ( x + ) F ( x ) > 0 , x F (x) -
.
.
.
a1 , a 2 ,... ( ) p1 , p 2 ,...,
, p
, Fa ( x )
1
pa
pk ; ak -
kx
, I 0 , A -
( , x ] . Fa ( x ) -
) .
, F -
, Au ( ) Au ( )
( , [16] ).
.: [1] ., .,
, . ., ., 1969; [2] . .,
, . [1],
. 34178; [3] A d l e r R. J., The geometry of random fields, N. Y.,
1981; [4] . ., , ., 1964; [5] . ., ,
. ., ., 1955; [6] . ., .,
. 17. 2, 1962; [7] . ., .: IV , . 2, ,
1985, . 26971; [8] . ., . ., 1987, . 32, . 4, . 72233; [9] W i l s o n R. J., Adv.
Appl. Probab., 1988, v. 20, p. 75674; [10] . .,
. ., 1990, . 35, . 1, . 14853; [11] . ., . ., . .,
1969, . 14, . 2, . 30214; [12] . ., . ., 1979, . 24, . 3, . 59296; [13] . ., . ., 1974, . 19, . 3,
. 50113; [14] . ., . .,
.: , ., 1972, . 6289; [15] . ., .: , ., 1972,
. 90118; [16] . ., . .,
. , 1979, . 249, 2, . 29497; [17] . ., .: , , . , . 19,
., 1982, . 15599; [18] . .,
, ., 1970.
*
; atom
x .
( , F , P ) , F ( x )
. A F F { A}
= 0 , F -
A , , F { A}, A ( ).
.
. ( 0 , 1) r1 , r2 , ...
, ri -
.
rk [ 0 , 1] 2
F , [ 0 , 1]
54
. p = 1
F
.
p <1
,
[ F p Fa ] q = Fc
q = 1 p .
F = p Fa + qFc
(*)
Fa Fc . F ,
p = 1 Fc -
. ,
(*)- p = 0 . ;
(*) , p 0, q 0, p + q = 1 .
, .
.: [1] .,
, . ., . 2, ., 1969.
*
; atomic component , .
; atomic
measure , , -
, ( , A )
= ( U i ) U
, A , ( ) = 1 i ,
, i A , ( i ) > 0 A A
A i , ( A) = 0 ( A ) = ( i )
-:
. (
)
.
. .
1- ,
( ) .
- . - , ( B ) > 0 B A A B , A A , 0 < ( A ) < ( B ) , ( , A )
.
: A B A A A -
( A ) - [ 0 , ( B ) ]
.
.: [1] ., , . ., ., 1969.
; atomic distributions A , . A A ( , F , P )
, P ( A )
, A B A , P ( B )
=0
P ( A \ B )
=0
>0
i, j =
- , . .
- .
/
/ ;
acyclic / aperiodic Markov chain ,
, C ,
( , ; ). . . .- t
lim pij ( t ) = p j
t , t k , t k +1,..., t 1 ,
( k + 1) , P
t k +1, t k
P =
...
t , t k
( )
;
autocorrelogram o f r a n d o m p r o c e s s . , .
; autocorrelation function
, .
( )
;
autocorrelation o f r a n d o m p r o c e s s , t (t )
t k , t k +1 ...
1
...
...
...
t , t k +1
...
t k , t
t k +1, t
...
Pi , j , i, j .
( t k + 1), , ( t 1) - ( t k )
(t )
, -
( t k )
. F ( x0 , x1 ,..., xk ), ( t k ) ,
t , t k , t k +1, ..., t 1 =
, .
; autoinformation measure ,
pi - -
( t k + 1) ,, (t ) , k > 1
p j - ,
; attractor
D ( i ) D ( j )
( t k 1 ) , ..., ( t 1) -
(t )
C ,
cov [ ( i ), ( j )]
(i ) ( j ) ,
( )
; acyclic / aperiodic state o f a M a r k o v c h a i n ,
k = 0,
k = 1,
k = 2 , ...
D ( t ),
t , t 1 ,
(t , k ) =
t , t k , t k +1,..., t 1,
Pt k , t
( Pt k , t k Pt , t )1 / 2
. , F ( x0 , x1 , ... , xk )
(k + 1) ,
t 1, t k , t , t k +1, ... ( t k + 1) ,, ( t 1) - ( t ) ( t k )
- .
. . . . (t ) p
k p +1
(t , k ) 0 ,
k = 0 , 1,..., p
( t , k ) = 0 . -
t t + h . .
(t )
, (
. t - , h - )
.
.
.: [1] ., ., . , . ., . 1, ., 1974; [2] . ., ., ,
. ., ., 1973; [3] . ., .,
, .
., ., 1976.
, ; partial autocorrelation function , ,
,
, (t ) ,
t = 0 , 1,...
p -
; autocovariance function
, .
55
( )
;
autocovariance o f r a n d o m p r o c e s s , t
t+ h . M ( t
M t ) ( t + h M t + h ) . , , (
) . .
h - t - . .
.
, ;
partial autocovariance function .
; probabilistic automation ,
.
]
; automatical classification
n
C0 , C1 , ... , C m ( -
) , . C0 , C1 , ... , C m
; C 0
n , C j
, C j 1
; Cm n .
.: [1] ., ., , .
., ., 1977; [2] .,
, . ., ., 1988.
, ; scaling limit
of a probability distribution ,
.
,
; scaling limit of a random field ,
( X t Ak X k ) . . . ,
X 0 = a . M X t2 < , t 0
M Xt = a
Ak = k H , 0 H 1
M (Xt Xs) = | t s |
2
2H
. , -
H . 0,5 < H
X t X s
,
, H . . .
H = 0,5 , . . ,
H
= 1 , X t = a + X t , M X = 0 , M X 2 =
= 2 . . . , Ak = k H .
.: [1] . ., .: . , ., 1985, . 27477; [2] M a n d e l b r o d t B.
B., V a n N e s s J. W., SIAM. Rev, 1968, v. 10, 4, p. 42237;
[3] T a q q u M. S., Z. Wahr. und verv. Geb., 1979, v. 50, p. 5389.
; self-similarity , .
; self-similarity equations ,
.
( ) ; automorphism
o f a m e a s u r e s p a c e ( , A , P )
T , P
,
1
A) =
= P ( T A ) = P ( A ) . , -
A A
A A , T A A
P ( T
. .
( mod 0 ).
mod 0
. xn , < n <
.
, ,
X t , t 0 , k 0
; automorphism o f
m o d 0 , ( ) .
K K ; K
automorphism , K .
; autoregressive integrated moving average
process , ( ; ARIMA
process ), t = 0 , 1, ...
,
; selfsimilar probability distribution ,
.
## ; automodel / demistable / saling process ,
= Ak X t
(t ) ,
- . ,
( t k t )
d
[ , . . . .
( t ) d ].
56
. . . . (t )
Ak
, X k t
Y (t ) a1 Y ( t 1) ... a p Y ( t p ) = Z (t )
b1 Z ( t 1) ... bq Z ( t q )
C d1
C d2
( t 1) +
( z ) = z p a1 z p 1 ... a p
(*)
, (*)
Y (t ) = d ( t ) = (t )
(t ) =
b Y ( t )
=0
( t 2 ) ... + (1) ( t d ) , Z (t ) ,
d
t = 0, 1, ... ,
, b ai -
(
), a1 ,..., a p , b1 ,..., bq
1 ( z ) = ( z ) =
(t ) ( )
( , ; );
, a1 , ... , a p , b1 , ... , bq
.
. . . . .- . ( G. Box ) .
( G. Jenkins ) .
,
( , [1] ).
.: [1] ., .,
. , . ., . 12, ., 1974.
; autoregression model
( t ) + a1 ( t 1) + ... + a p ( t p ) = 2 Z ( t )
,
{ ( t ), t T } , T
,
## a1 ,..., a p , 0 < 2 < ,
{ Z ( t ), t T } ,
M Z (t )
s, t T
= 0, M Z ( s ) Z ( t ) = s t ,
. .-
.
. .
, .
. .
.
.: [1]
, 2 ., ., 1985; [2] ., .,
. , . ., . 1,
., 1974; [3] ., ., , . ., ., 1976.
; autoregressive process
(t )
.
., {Y ( t ) }
2 -
, (*) . .
f ( ) = ( 2 2 ) ( e i )
, , ( e ) . rk
= M ( t ) ( t k ) , rk = a1rk 1 + ...
+ a p rk p , k > 0 b
rk = 2
bb
+k
=0
. a k - k
= rk r0
= R p1 p
A = (a1 , ... , a p ) , p =
= ( 1 , ..., p )
= { 0 , 1, 2 ,...} -
=0
, (*) . p, q
d
R p
( ). , .
.: [1] G r e n a n d e r V., R o z e n b l a t t M., Statistical analysis of stationary time series, Stockh., 1956; [2] .,
, . ., ., 1964.
f r a c t i o n a l autoregressive process
;
,
;
autoregressive spectral estimator ,
; .
regression
Xn
{ X n , n = 0 , 1,...}
auto
X n1 ,
X n2 ,..., X nm . m
. X n -
X nk , k = 1,..., m
X n = 1 X n1 + ... + m X nm + n
(*)
(*)
, 1 ,..., m , i
a1 ,..., a p ,
,
( ), . .
(t ) = a1 ( t 1) + ... + a p ( t p ) + Y ( t )
p , Y (t ) -, ,
, 0, 2
. z
57
. (*) .
.
; rezolution bandwidth and calculation
bandwidth
,
,
. N
N ( x ) ,
N AN N 0
.
( ,
)
b = 1 2 AN .
N N .
.: [1] . .,
, ., 1982; [2] ., ., , . ., ., 1982; [3] . .,
, .,
1981; [4] B l a c k m a n R., T u k e y J., The measurement of
power spectra. From the point of view of communications engineering,
N. Y., [1959].
; separate redundancy , .
*
AN
bN (t ) = b ( AN1 t )
, AN , bN (t )
.
.
.
( , [4] ) N f N ( )
## ; decomposable characteristic function , .
/ ; reducible / decomposable Markov chain
.
, , ,
( ,
)
P1
,
:
1 ( M fN ( )) 2
Wl =
N D fN ( )
2N
(*)
( x ) dx
Wl
:
Wl
1
AN
1,33
AN
< 0 1 ;
1,86
AN
2 (2 + 1)
2
AN (2 2 + 2 + 1)
f ( ) -
( )
.
fN ( ) -
58
Q P2
P2
, 0 . ,
.
; decomposable
branching process
,
, . n -
(t ) = ( 1 ( t ) , ... , n ( t ) )
, k (t ) , t
k - . i i
; P1
= 1 AN , -
A A ;
A separating plan ,
N ( x) = AN ( AN x ), AN > 1, An ;
, 1 AN
. b
i + 1 (1 2 ... n ) (t )
= ( 1 ( t ) , ... , n ( t ) )
. . .- .
n .
( , ) (t ) - .
## Aii (t ) = M { i ( t ) | j ( 0 ) = ij , j = 1, ... , n } = e aii t ,
i = 1, 2 , ..., n
,
aii . aii - aii > 0
, N ( ,
[1] ); a n , bn > 0 , S n n
## Qi (t ) = P { 1 ( t ) + ... + n ( t ) > 0 | j ( 0 ) = ij , j = 1, ... , n }
.
f f c -
t . i -
aii 0 a11 ,..., ann p ( p 1 )
, t
1 p
Q1 ( t ) = q1t 2
q1
>0
( 1 + o (1)) ,
. a11
...
= ann = 0
( t ) Q1 ( t )
(t ) Qn (t )
P 1
< y1 , ... , n
<
A
(
t
)
Ann (t )
11
< yn | j ( t ) = ij ;
j (t )
1
( s1 ,..., sn ) = 1 1 +
sn
21n
. . . (t ) -
[2] [4]- .
[5]- .
.: [1] . ., ,
., 1971; [2] . ., . ., 1976, . 100,
. 42035; [3]
. ., . ., 1977, . 104,
. 15161; [4] O g u r a Y., J. Math. Kyoto Univ., Ser. A., 1975,
v. 15, 2, p. 251302; [5] . ., . .,
.: . . . . , . 23, .,
1985, . 367.
*
posability
self-decom-
, c ( 0 , 1)
f c , u
f (u ) = f (c u ) f c ( u ) ,
. , f c -
.
f ,
( , [1] ).
f 0
. S n bn
.: [1]
., , ., 1962.
; decomposable test , .
*
; law
of decomposability , .
; indecomposable distribution
- , R - ( n 2 )
. .-. - . .-
, , . .-
, . .-
.
. .
.
.: [1] . ., . .,
, ., 1972; [2] . .,
. ., . ., 1984, . 29, . 2,
. 34851; [3] . ., . ., . VI
. . . , ,
1962, . 42537.
n
t
( s1 ,..., sn )
0.
. ,
. .- .
>0
j =1
, f
Sn
an bn
N .
, L ,
; indecomposable
branching process
,
( )
n
(t ) = ( 1 ( t ) , ..., n ( t )) . . . , , k (t ) ,
t k - .
= M ( j ( t ) | k ( 0 ) =
. Aij (t )
= ik , k = 1,..., n ) , A (t ) = ( Aij ( t ))
A (t ) = A (1) . -
A (1) . ,
A (1) , t
Aij (t ) = ui j t + o ( t ) , , A (1) -
; ( u1 , ... , u n ) ,
(1 , ... , n ) , A (1) - -
. . . .
,
>1
<1
= 1
. (t ) -
59
, . . .
.
i t
## Qi (t ) = P{ 1 ( t ) + ... + n ( t ) > 0 | k (0) = ik , k = 1, ..., n }
,
1 t - .
t ,
(t )
(t )
P 1
< x1 , ... , n
< xn | k (0) = ik ,
Ain (t )
Ai1 (t )
k = 1, ..., n ;
k =1
k (t )
>0
,
x1 = x2 = ... = xn .
. . . .-
.
.: [1]
1971.
. ., , .,
; Baxter theorem
: 0 t T
(t ) t -
M (t ) = m (t ) t , s - b ( t , s ) =
= M [ (t ) M (t ) ] [ ( s ) M ( s ) ]
, 0 t T m(t ) ,
t s b ( t , s ) , (t ) N -
(
[ 0 , T ] N )
b ( t, s ) t
t=s
## [6] G i n e E., K l e i n R., Ann. Probab., 1975, v. 3, 4, p. 716
21; [7] . ., .: . .
, . 33, ., 1985, . 5357; [8] S l e p i a n D., IRE Trans.
Inform. Theory, 1958, v. IT4, 2, p. 6568; [9] V a r b e r g D.
E., Pacific J. Math., 1961, v. 11, 2, p. 75162.
## ; balanced block design
, .
p
p ; Banach
space o f s t a b l e t y p e p ,
p .
;
Banach space w i t h S a z o n o v p r o p e r t y , S -
.
m (t ) b ( t , s ) ,
, ,
, p < 1
D + (t ) = lim+
st
Lp ( ,
[2] ).
b (t , t ) b( s , t )
,
ts
.: [1] . ., . .,
1958, . 3, . 2, . 20105; [2] . ., . ., 1973, . 18, . 1, . 6677; [3] .
., , .,
1989.
b (t , t ) b( s , t )
D (t ) = lim
st
ts
2n
lim
k =1
kT
( k 1) T
n
n
2
l pn -
[ D (t ) D+ (t ) ] d t
containing
(*)
( , [1] ). , (t )
b ( t , s )
n
l p ; Banach space
. ., ( , [1] ), 1 p < 2 l p
= min ( t , s )
, (*)- T - ;
(*) . ( , [2] );
. . . ( , ., [3] [7] )
, 1 ( t )
2 ( t ) ( )
, ( ,
., [3] [9] ).
.: [1] B a x t e r G., Proc. Amer. Math. Soc., 1956, v. 7, p. 522
27; [2] L e v y P., Amer. J. Math., 1940, v. 62, p. 487550;
[3] . ., . ., 1961,
. 6, . 1, . 5766; [4] . ., .:
. . , . 5, 1971, . 7180; [5] . .,
. ., 1972, . 17, . 1, . 15360;
l pn
uniformly
>0
n n Bn B
n
, d ( Bn , l p )
< 1+
, l p ,
x = ( x1 , ..., xn ) n
:
1 p <
1 p
i =1
xi
= max { | xi | , 1 i n } ,
d ( E , F ) E F
61
d ( E , F ) = inf { || T || || T 1 || } ,
E F
.
. .
l2n
( ).
: . .- , .
l np
. . .
:
) . . l pn , ,
, , p < p . .
;
) . . l1n , ,
, , 1 < p 2 p . .
;
) . .
l pn
1 p
<2
, , , p . .
.
ln . .-
x
i
, . . B
xn n
, xn B . , 1 p 2 ,
p > 2 . . ; . .- 1 (
, p < 1 ) .
n p , .
xn n , xn B , n N
xn
B p . . . , 2 p < ,
, p < 2 . . .
n ,
. B p , ,
, c p B - M X k
= 1, 2 , ...
X d P
. , ,
. l PIP
cp
< ,
Xk
(*)
k =1
k =1
.
B p , , , (*)
p
p
B -
PIP
PIP; Banach space
, ( , A , P )
xn
);
. .-
.
.: [1] M a r e y B., P i s i e r G., Studia Math., 1976, v. 58,
fasc. 1, p. 4590.
X : B
M X k = 0 , k = 1, 2 , ... , n
( i i , i
w i t h PIP B
n ,
P { n = +1 } = P { n = 1 } = 1 2 ,
n N . B
k =1
k =1
.
Lr ( , , ) , 1 r < p
Xk
= max ( r , 2 )
p = min ( r , 2 )
,
. I p > 1 .
## ( PIP Pettis ntegral Property
).
.: [1] E d g a r G. A., Indiana Univ. Math. J., 1979, v. 28, 4,
p. 55979; [2] F r e m l i n D. H., T a l a g r a n d M., Math. Z.,
1979, Bd 168, S. 11749.
C [ 0 , 1] c0
p p ; Banach space o f c o t y p e
p , p .
p
p ; Banach space o f t y p e p
. 2 2
. :
. .- 2 2 , ,
( ) ( ).
p . . B . .- ,
xn
xn f n
;
p
p
.
, M exp ( i t f n ) =
62
= exp ( | t | p ) , t R1 . , 1 p 2 , ,
p > 2 . . ; . .-
( f n ) p , -
p < 1 . p . .- p
. . . 2 . . 2 . .-.
B . .- p , p1 < p
p1 . B - p
p1 p p1
, p < 2 p 2
> p p 2
.
.: [1] H o f f m a n n J r g e n s e n J., Aarhus Univ. Matem.
inst., preprint, 197273, 15; [2] M a u r e y B., P i s i e r G.,
Studia Math., 1976, v. 58, 1, p. 4590; [3] W o y c z y n s k i W.,
.: Probability un Banach spacez, N. Y., 1978.
; Bartlett estimator , ;
xi =
## S12 , ..., S k2 , 12 , ..., k2
2
; i S i i
i -
= 1, ..., k , N =
, i
i =1
M = N ln
. H 0 : 1
2
i i
i =1
= ... = k2
ln S i2
i =1
1
1 +
3 ( k 1)
i =1
1
> C ( k , 1 , ..., k , )
N
.
i
>3
Ck l
lk = x1k
ij
ni ,
i = 1, 2 , n1 n2 ,
n2
kl
## x2l , k = 1, ..., n1 , l = 1,..., n2 ,
n1 ,
l =1
L =
n1
2 -
. C ( k , 1 , ..., k , ) [2]- .
.: [1] B a r t l e t t M. S., Proc. Roy. Soc., 1937, V. A. 160,
p. 26882; [2] . ., . .,
, 3 ., ., 1983.
n1
(l
Q =
k =1
L )2
k =1
, kl . H 0 : 1
= 2 =
( )
n1 L
[ Q ( n1 1) ]1/ 2
>C
. H 0 , ( n1 1) -
t . n1
= n2
.
.: [1] S c h e f f e H., Ann. Math. Statist., 1943, v. 14, 1,
p. 3544; [2] . ., , ., 1966.
; principal component analysis ,
, 1) . . . (
) n x p
= U T x U
, min ( max )
x - .
:
p =1,
p = 2 .
j =1
( j ) max ,
U
2 (1 ) max , 2 ( 2 ) max v2 v1 .;
U2
U1
( k 1) -
ni
i - -
H 0 , -
xij ~ N ( i , i ) 2 ,
kl ( n1 n2 ) 1 / 2 ( n1 n2 ) 1/ 2 + n21 , l n1 ,
=
1
l > n1
n2 ,
## ; Bartle Dunford Schwartz theorem , .
; Bartlett lag
window , .
;
Bartlett test
j =1
( ) ; baricentre o f a
p r o b a b i l i t y m e a s u r e , ( ) -
j = 1, ..., ni .
## ; Banach Mazur distance ,
l np
; Bartlett Scheffe test
D ( x ) min , D ( x ) , x
U
D , D ( x ) -
63
, x
-
U TU = E
E U = 0,
E . .
2 ( j ) = j . N j U j -
j U j
S = ( N 1) X X
X,
( N n ) . . . ., ,
.
2) . . . ( ) p .
, n ( n
> p)
N
.
:
Q min ,
= X X T, = V V T, X ,
n
, V = X U ,
Q , (Q )
U
u1 , ..., u p
X X
S - ( N 1)
j ( x ) -
(
).
( , [3] ) .
. . . .
( , [4] ); . . . .- ( , [5] ).
.: [1] R a o C. R., Sankhya, ser. A, 1964, v. 26, p. 32958;
[2] G o w e r J. C., Biometrika, 1966, v. 53, 3/4, p. 32538;
[3] . ., . . . ., 1960, 3,
. 43239; [4] P e a r s o n K., Philos. Mag., 1901, v. 2, p. 55972;
[5] H o t e l l i n g H., J. Educ. Psych., 1933, v. 24, p. 498520;
[6] J o l l i f f e I. T., Principal Component Analysis, 1986; [7] . ., .:
- , ., 1974, . 189228;
[8] O k a m o t o M., .: Proceed. of the Second Intern. Symp. of
Mult. Analysis, N. Y. L., 1969, p. 67385; [9] A n d e r s o n T. W.,
Ann. Math. Stat., 1963, v. 34, 1, p. 12248.
; principal components method
, .
; Bayes formula , .
; BBGKY equations hierarchy / chain
64
, , .
; Bellman optimality principle , .
;
Bellman equation , ,
.
; Bellman Harris process ,
( G , h ) ,
. . .
[1]-
.
. . .- - .
G ( t ) . h ( s )
. Z ( t ) . . .- t
,
F ( t ; s ) = M [ s Z ( t ) | Z ( 0 ) = 1]
F(t ; s ) =
. . . .-
( ,
[2] ).
3) . . . ( ) f ( x ) j -
, .
; Bell numbers
h ( F ( t u ; s ) d G ( u ) + s (1 G ( t ))
0
.
, , ,
; .
.: [1] B e l l m a n R., H a r r i s T., Proc. Nat. Acad. Sci.
USA, 1948, v. 34, 12, p. 60104.
; Belyayev alternative , ;
; Benfords law , .
; Baire
algebra T
; T
f : T R -
. . .
- .
. .
. . .-
. G
.
.: [1] ., , . ., ., 1953.
; Baire
set , .
; Ber
function , .
; Bair measure 1)
, G
( , [1] ).
2) . . T . . . .
0 . .
( ) .
B0 ( T ) - U U = T
( U ) sup (U )
, . . , T -
= (T )
.
. .,
( , ) , .
.: [1] ., , . ., ., 1953,
. 21823.
; Baire class
; Baire
classififition , .
; Behrens Fisher problem
. x11 , ..., x1n1 , x21 , ..., x2 n2 ,
, xij ~ N ( i , i2 ) , i = 1 , 2 . ( 1 , 2 , 1 , 2 ) ,
2
j = 1, ..., ni
( X1 , X 2 ,
S12 ,
S 22
) ,
Xi =
ij
ni ,
1 j ni
Si2 =
(x
ij
Xi ) , i =1 , 2 .
1 j ni
. . . K ,
H 0 : 1 2 = { ( X 1 X 2 ,
S12
S 22
) K } -
( 0 , 1) , - ( ).
n1 n2 -
. . .- ( , [2] ).
. . .- ( , ., [3] [6] ) L
, -
H0
{ ( X 1 X 2 ) S1 ,
S12
S 22
L } -
12
22
.
.
.: [1] B e h r e n s W. U., Landwirtsh. Jahrb, 1929, Bd, 68,
6, S. 80737; [2] . .,
, ., 1966; [3] Selected papers in statistics and probability by A. Wald, N.Y. [a.o.], 1955, p. 66995;
[4] . ., . ., 1968,
. 13, . 3, . 56169; [5] W e l c h B. L., Biometrika, 1947, v. 34,
p. 2835; [6] L e e A. F. S., G u r l a n d J., J. Amer. Statist. Assoc.,
1975, v. 70, p. 93341.
; Burgs method o f e s t i m a t i o n o f p a r a m e t e r s o f a u t o r e g r e s s i v e p r o c e s s i -, i = 1, ..., p
a2 ( p) - p .
x1 , x2 , ..., xn p
( ,
). . ( J. Burg ) ( , [1] )
( , ,
) ,
a2 ( p) . . .
p - ( )
. . .-
( , [2] )
.
. .- .
, ., [3], [4]- .
[2], [5], [4]-
.
.: [1] Modern Spectrum analysis, N. Y., 1978, p. 3841; [2] U l r i c h T., B i s h o p T., Rey. Geophys. Space Phys., 1975, v. 13,
1, p. 183200; [3] S m y l i e D., C l a r k e G., U l r i c h T.,
Methods in Comput. Phys., 1973, v. 13, p. 391430; [4] . ., (
, , ), ., 1985; [5] . ., ,
. ., ., 1985.
; Burlings inequality ,
.
; Bernsteins inequality , . . ( 1911, , [1] ) ;
( , ).
M X j
= 0 M X 2j = b j , j = 1, ..., n ,
X 1 , ..., X n M X j
bj H
l 2
l ! 2 ( l > 2 , H , j - )
, S n
= X 1 + ... + X n
r > 0
P { | S n | > r } 2 exp{ r 2 2 ( Bn + Hr )} ,
(1)
65
Bn =
. X j
( M X j
= 0 , M X 2j = 2
Xj L,
j = 1,..., n ) (1)
P { | S n | > t n
} 2 exp { t 2 2 (1 + a 3 ) }
; a
= Lt
(2)
n .
. . (1)- . . (2)-
(2)
2
2
u 2 2
du =
t2 2
, 0 < <1
1 2 e
t
2 t
2
. 1967-
. .-
.
.: [1] . ., , 4 ., .
., 1946; [2] . ., Math. Ann., 1929, Bd 101,
S. 12635; [3] H o e f f d i n g W., J. Amer. Statist. Assoc. , 1963,
v. 58, 301, p. 1330; [4] . ., .
., 1968, . 13, . 2, . 26674; [5] . .,
. , 1968, . 3, . 6, 73139; [6] .
., . ., 1970, . 15, . 1, . 10607.
;
Bernstein theorem , .
; Bernulli
flow , .
; Bernoulli sequence , .
; Bernoulli automorphism ,
.
;
Bernoulli random walk . . .
.
.
. .-, ., . .,
x k h ( k , h > 0 ) ( ). t = 0
0 , t , 2 t , 3 t , ...
.
h p q = 1 p
, .
, (
) p - .
1.
.
i , i - ( )
. 1 , 2 , ..., n , ...
.
n t S n
= 1 + ... + n .
, . .-
. ( . .-
).
( ,
) ,
. .
.
. . ( , t = 1 , h = 1 ).
.
.
1 p q - , p = q = 1 / 2
, p q
. 1
, 2
3
. . , 1 , 2 , ..., n , ...
. i
.
( )
N 2 , 1 + ... + N < x N 2
0
).
66
. ., , : t
, x ( , 1,
N -
, 1 + ... + N
e s
ds
. 2 n N 2 n
M ( N 2n ) =
2 n +1 n
C2 n 1
2 2n
n :
M ( N 2n ) ~ 2
a = ( k + 1 2 ) ( k 0 , )
, q k - k 1 p .
. .,
a ( a > 0 ) . zt , x , t
x n ( n
) ,
x > a
zt , x = q zt +1, x 1 + p zt +1, x +1 ,
zt , a = 1 ,
0t n
z n, x = 0 ,
x > a
. p = q = 1 2
. ( A. Moivre ) . ( P. Laplas )
. :
2. , 200000
.
: . .-
( 2 ) .
, . .-
Tn n 1 2 - . , k , n k
T2 n
= 2 k p2 n, 2k
p2 n, 2 k ~
x (1 x )
, x
= xn, k = k n .
:
0 < < 1
P { Tn n < }
dx
x ( 1 x)
arcsin .
, 10000
9930
0,1
, , (
).
.
p > q + -; p < q - .
+
, ., p < q S
= max S i
0 i <
P { S + = x } = (1 p q ) ( p q ) x
.
.
. .- . .,
, . a
, a .
zt , x = 1
/2
sin (a + x )
( cos ) d ,
sin
(*)
n t x a
+ 1.
2
=a + x + 2
p = q =1 2 , t = 1 N , h = 1
, .,
. .
N
. T
-
=T
. n N
x=0, a
, t
=0 ,
N = , (*)
1
2
2
ez
dz =
2
2
ez
dz
.
X ( ) - min X ( )
0 v T
, .
.
. .
.
.: [1] .,
, . ., . 12, ., 1984.
; Bernoulli endomorphism ,
.
67
Bernoulli numbers
. -
. . :
x
=
x
e 1
n =0
ln ( x + 1) ( x + 1 2 ) ln x x + ln 2 +
xn
Bn
n!
, Bn . . .
= (1) n1 B2 n , n 1 . .
( , [1] ). n
. n 3 n -
Bn = 0 , . .- B0 = 1 , B1 = 1 / 2 ,
B2 = 1 / 6 ; B4 = 1 / 30 , B6 = 1 / 42 , B8 = 1 / 30 , B10 =
k=1
B2 k
x 2 k +1 .
2 k ( 2k 1)
. .-, ,
. h
,
## = 5 / 66 , B12 = 691 / 2730 . . .-
M =
Ck ( 21k 1) Bk h k k ;
k =0
n 1
k
n
n >1
Bk = 0 ,
= B h , > 1 , 1 = 1
k =0
.
n 90 B2 n , n 250
( , [2] ).
z - . . ( z ) =
k =1
B2 n = (1) n1
2 ( 2n )!
( 2n ) , n 1 .
(2 ) 2 n
. . , , ,
,
.
. .
, .:
2 2 n1 1
x 2 n1
dx =
ax
2n
e +1
a
2n
B2 n , a > 0, n 1 ;
, .:
tg z =
(1)
n 1
2 2 n ( 2 2 n 1 ) B2 n
n =1
sh z
ln
=
z
2 2 n 1 B2 n
n =1
z 2 n1
,
(2n)!
z 2n
,
n ( 2n )!
z <
z < .
. . :
n 1
x+n
f ( x + k) =
k =0
k =1
f (t ) d t
1
( f ( x + n ) f ( x )) +
2
B2 k
( f ( 2 k 1) ( x + n ) f ( 2 k 1) ( x )) + Rr ( x , n ) .
(2k )!
68
.
,
( , [3] ).
.: [1] ., , . .,
1986; [2] Tables of the higher mathematical functions, v. 2, Bloomington, 1935; [3]
.,
, . ., 2 ., ., 1975; [4] . .,
, 3 ., ., 1967; [5] ., , . .,
., 1965.
; Bernoulli distribution ,
.
; Bernoulli
scheme , .
;
Bernoulli trials ( ., )
,
. . .
.
p (
) , q = 1 p
( )
. n ,
.,
...
123
n 3
pp q ... q p =
1
424
3
n 3
= p3q n 3 . . .
. ., ,
, . n
. .- S n .
{ S n = k}
P { Sn = k } = C nk p k q n k
,
k = 0 , 1 , ..., n ;
(*)
Sn
. (*) n , P { S n = k } -
,
( , [1], [2] ). X 1 . { X n
q ... q p
1424
3
= k}
P {Y1 = k } = q k p ,
k = 0 , 1, ... . Y1
( ) p - . n
, m
. . .- , , n0 , n n0
n -
P{ m n p } 1
. Yr , r -
, , r -
( r + k ) -
{Yr = k }
## P {Yr = k } = Crk1+ k p r q k , k = 0 , 1, 2 , ...
. Yr
. . .- S n - n 1 , ..., n ; i
, P {i = 1} = P {M } = p , P {i = 0} = P {Q } = q ,
p + q = 1 . n = 1 + ... + n . ,
. .
( , ,
, ,
. ).
. .- 0 1 -
. , , p = q = 1 / 2
.
, ( 0 , 1) ,
. j ( )
0 1 , j
( ) / 2
j
= 1, 2 , ... .
- .
j=1
, P { i ( )
= 0} = P { i ( ) = 1} = 1 2 j ( ) -
. , -
1 - . .
, p = 1 / 2 . ( 0 , 1) -
, p ( p 1 / 2
) . . .
. . ( ,
). . .
.
.: [1] . ., , 6 .,
., 1988; [2] .,
, . ., . 12, ., 1984; [3] ., ,
, . ., ., 1963.
; Bernoulli theorem
. . . . ( J. Bernoulli ) Ars conjectandi ( )
.
. . . ( ,
) ;
. .
, (
) - n0
.
., . , p = 2 / 5 n 25550
P { 1 50 m n p 1 50} 0, 999
.
. ,
, n0
n0 > (1 + ) / 2 log 1 / + 1 /
,
m n p >
(1 P ) -
2 exp{ n 2 2} .
n
17665
( -
, n
6502
; :
n0 6498 ). 1 P
( ,
).
.: [1] ., , . .,
., 1986; [2] . ., , 4 .,
., 1924; [3] . ., , 4 ,
. ., 1946.
## ; Bernoulli randon variable 1) 0 1 , p
1 p ( 0 < p < 1 ) .
1 , ..., n . . .-, 1 + ... + n
.
2) 1 +1
. . . .-
.
; Bernoulli vector , .
; Bernoulli shift
. . . T
( Y , B , )
( I
i I
(Yi , Bi , i )
= Z + Z )
69
= (i , i I ) = ( i , i I ) -
, i
= i +1 .
= Z+
T
(
,
. ( Y ,
B,
), I = Z
(
)
) .
. .-
( ) ; ., . . , . . K . . .
( ) ( ) ,
( ) . .- . xn ,
< n < -
, ,
, xn
( , [1], [2] ),
( , [3] ) .
. .- ( , , ). T . .
t
{ S ,
, T
< t < }
= S .
1
. .- , .
.: [1] ., ,
, . ., ., 1978; [2] O r n s t e i n D.,
W e i s s B., Isr. J. Math., 1974, v. 17, 1, p. 94104; [3] . ., . ., , ., 1970.
;
Burr distribution , ,
. [1]- . . .
.
; Berry Esseen inequality
F G f , g ( F , G ) .
G
. . .- -
. ( , [1] ) . ( , [2] )
- .
:
T
( F, G ) A
f (t ) g (t )
T , q
dt
q
+B ,
t
T
(1)
= sup G ( x ) ,
x
A( ) =
( 2V ( ) )
V () =
B ( ) =
sin t
0
V ( )
,
2V ( )
(2)
dt
1 + cos ,
## 2V () > ( > 1,6996 )
.
A B
:
3,1
3,8
4,5
5,2
5,7
6,7
A( )
0,5930
0,4659
0,4169
0,3992
0,3956
0,3951
B()
4,4376
4,6812
5,1967
5,8610
6,3923
6,7236 .
A B (2) (1) V ( ) , -
A () B () ( ,
[3], [4] ). . . .
= 2 = 1; 2 ; 3 ; 4 ; 5
.
. .-
F ( x) = 1
1
, x0, >0, >0
(1 + x )
.
.: [1] B u r r I., Ann. Math. Statist., 1942, v. 13, 2, p. 21532.
70
, .,
,
. . . .-
,
( , , ).
.: [1] B e r r y A. C., Trans. Amer. Math. Soc., 1941, v. 49,
1, p. 12036; [2] E s s e n C.-G., Ark. Mat., Astr. och Fysik,
1942, Bd 28A, 9, S. 119; [3] Z o l o t a r e v V. M., Z. Wahr.
und verw. Geb., 1967, Bd 8, S. 33242; [4] v a n B e e k P.,
Z. Wahr. und verw. Geb., 1972, Bd 23, S. 18796.
; Berri Essen theorem M X i = 0 ,
M X 12 + ... + M X n2 = 1, = M X 1
+ ... + M X n
<
, X 1 , ..., X n -
S n
= X 1 + ... + X n
Fn -
-
,
( Fn , ) C ,
(1)
, C . . . ( ,
) , .
. ( , [1] [2]- ) - . X i
Sn
Sn =
= ( Y1 + ... + Yn )
n , M Yi = 0 , M Yi 2 = 1
; (1)
:
( Fn , ) C
n,
(2)
= M Y1 . (1) (2)- C -
( , [5] ). C
- . . [4]- :
(1) C 0,7975 , (2) C 0,7655
(
[4]
[3]
).
## C C0 = ( 10 + 3 ) 6 2 = 0,4097... . (1)- C - . (1), (2) . . .- . (2)-
:
( Fn , ) C1 min ( 3 , 3 ) n
C1 , n
1/ 2
. . ( ,
[2] ), .
- 0 1 ,
- 0 2
( , [3] ).
.: [1] B e r t r a n d J., Calcul des probabilits. P., 1889; [2] P o i n c a r H., Calcul des probabilits. 2 d., P., 1912; [3] ., ., , .
., ., 1972.
; Bessel
operator , .
; Bessel process a ( t , x) = ( 1) 2 x ,
b ( t , x ) = 1 .
L =
C 0 -
;
; ( x , y ) .
= min (1, n / 4) 3 =
= 3 x 2 F ( x ) ( x ) dx , F , Y1 .
.: [1] B e r r y A. C., Trans. Amer. Math. Soc., 1941, v. 49,
1, p. 12036; [2] E s s e n C.-G., Ark. Mat. Astr. och Fysik,
1942, Bd 28A, 9, S. 119; [3] E s s e e n C.-G., Skand. Actuar.,
1956, 34, p. 16070; [4] . C., .:
, ., 1982, . 10915;
[3] . .,
, ., 1986.
; Bertrand problem , .
;
( )
. .
. [1]-
.
( 1 2 , 1 3 , 1 4 ) :
x
1
2
d2
1 d
dx 2 + 2 x dx
. = d , . . d
w (t ) = {w1 ( t ) , ..., w d ( t ) }
w (t ) =
2
i (t )
=2
i =1
. .
; beta approximation
. . .
.
N , n M
:
P { = m } =
Cn Cn m
M N M ;
=
Cn
N
0,
max ( 0, M + n N )
m min ( M , n )
N 25 P { m }
. . ( , [1] )
P { m } I1 x ( n m + c , m c + 1)
, 0 t 1 , a > 0 , b > 0
71
1
B ( a, b )
I t ( a , b) = Ba ,b ( t ) =
mk = B ( m + k , n ) B ( m , n ) , k = 1, 2 , ...
z a 1 ( 1 z ) b 1 dz
; .-
,
m ( m + n)
mn ( m + n ) 2 ( m + n + 1)
( N 2)2
n =
N 1
c =
= ( m 1) ( m + n 2) , = 0 . m < 1 n < 1
, ,
m < 1 n < 1 ,
U
n M ( M 1)(n 1)
.
( N 1) [( N M )( N n) + n M N ]
;
beta distribution ( )
m = n =1 2
. .-
,
, m , n , -
B ( m , n )
x (1 x )
(1 x ) n 1 dx
. (1)- x = 1 (1 + t )
m , n ( t ) =
m 1
1 2 ,1 2 ( x ) =
(1)
, ( 0 , 1) - -
.-
. m = 1 n = 1 , . ( 0 , 1)
1
x m 1 (1 x ) n 1
B ( m, n)
F ( m , m , n ; it )
.: [1] . ., , , ., 1987.
B( m , n ) =
m,n ( x )
N ( n + M 1) 2 n M
,
N ( N 2)
m, n ( x ) =
n > 1 , m,n ( x ) -
n M ( N n) ( N M )
,
[( N M ) ( N n ) + n M N ] [ N (n + M 1) 2n M ]
x =
2 ( m n ) (m + n 2) . m > 1
1
t m 1
, 0<t <
B ( m , n ) (1 + t ) m + n 2
(2)
.
.
(1) (2)
I VI .
.- : 1
2 , m n
, ,
1 ( 1 + 2 ) m,n ( x )
. . -
, , .-
: . . ., F Fm, n =
= n m2 m n2
P { Fm , n < x } = Bm 2 ,
(1) m = 0,5 , n = 3,5 ; ( 2) m = 1, n = 3 ; (3) m = 1,5 , n = 2,5 ; ( 4) m = 2,
n = 2
( m + n = 4 ).
.- :
I x ( m , n ) Bm , n ( x ) =
1
=
B ( m, n )
.-
72
m 1
(1 y )
n 1
dy , 0 < x < 1
n 2
( m x (n + m x ) )
k ).
F , , . .
.
Bn m,
m +1 ( 1 p ) =
C nk p k (1 p ) n k
k =0
.
. ; ., .-
.
## .: [1] . ., . ., , 3 ., ., 1983; [2] ., -, . ., ., 1974.
; m u l t i d i m e n s i o n a l
beta distribution , .
; Bayes
formula ( )
. A1 , ... , An :
U Ai = , i j Ai I A j = ; ,
. i P ( Ai )
> 0.
P (B ) > 0 B
Ai
P ( Ai ) P ( B | Ai )
n
i = 1, n
(*)
P ( Ai ) P ( B | Ai )
. P ( Ai ) , Ai , P ( B | Ai ) , Ai ( P ( Ai )
> 0)
B . (*)
. ( T. Bayes ) , 1763-
.
, .
.: [1] o . ., , 2 ., ., 1974.
; Bayesian decision function /
rule
. . . .
; ( . . .- ) ,
- . , .
.: [1] ., , . .,
., 1975.
## ; Bayesian decision rule ,
.
; Bayesian
decision ( ). , ,
.
;
Bayesian test
. . . .
X , P , X , p ( x | ) .
H 0 : 0 H 1 : 1
; 0 1 , -
.
q 0 + q1 = 1 .
r ( ) = q0 M 0 ( X ) + q1 M 1 ( 1 ( X ) )
( ; )
, i
M i ( X ) =
= 0, 1
(x ) f
i ( x )
(dx ) ,
p ( x | ) i ( d ) .
i =1
( ) 0 1 0 ( )
1 ( ) H 0 H1 q 0 q1
f i (x) =
P ( Ai | B ) ,
P ( Ai | B ) =
( ) ( ) .
. . ( ; ) :
0 , q0 f 0 ( x) > q1 f 1 ( x) ,
(x ; ) =
1, q0 f 0 ( x) < q1 f 1 ( x)
, { n } { ( ; n )}
. .- ( ) . . , ( ; ) . .
.
. .-
.
. .-
; 0 1 - 0
1 , .
(x ) f
( x) (dx )
;
. X
f 0 ( ) f 1 ( )
.
. .
.
.: [1] . ., , .,
1984; [2] ., , .
., 2 ., ., 1979; [3] .,
, .: , . ., ., 1967;
[4] ., , . ., .,
1975; [5] . .,
, ., 1969.
; Bayesian measurement , ,
.
73
;
Bayesian principe -
.
. w ( , ) = c , k = 1 ,
.
( D , , w ) , ( D , , W ) ( X , P )
*Q .
; X X , P , ,
D ( X ) : X D
; Bayesian regression
. , .
; Bayesian
risk
w ( ( x) ; ) P (dx) .
W ( ( ) , ) =
.: [1] . .,
( ), ., 1984.
X -
P f =
= d P ( x) d -
Q
g (t ) = d Q( t ) d , X
Q X
. ,
. .- . , , ( )
, .
; Bayesian estimator
f (X ) =
q ( t ) f ( X ) (d t )
t
(*)
Q - . . .-
( D , , W ) Q
Q ( X ) ,
Q X
(*)
.
Q ( X ) = Q X Q X
.
. . .
D = { 1 , 2 },
= {1 , 2 },
w ( i , j ) = wij ,
.,
( w11 = w 22 = 0) . P X
{ = 1} { = 2 } ;
f 1 ( X ) f 2 ( X ) < a ( 1 q ) q ( 1 a )
= w12
,
n ( , )
. . .
( n x 2 + 2 )
( n 2 + 2 )
.
. . .- (
)
- .
, .
.: [1] . ., , .,
1984; [2] . ., . ., , ., 1979; [3] .,
, . ., ., 1975.
; Bayesian strategy
( ).
, , , .
, w ( , )
= c ( ) ,
2
74
- -
; e m p i r i c a l Bayesian
approach G
D , R
*Q =
= ( d ) ,
-
;
Bayesian approach , -
.,
= 2 ,
(w12 + w 21 ) , q { = 1}
, ( X )
a
L ( | d | )
, f q ,
- , Q X
; - ( , )
( ).
Q ( D , , w )
Q X
L(, d ) = d
q ( t ) ft ( X )
,
f (X )
q (t | X ) =
L ( , d )
tQ
(dt ) =
t q (t | X ) ( d t)
,
x
G
(k )
= ( x1 , ... , xk ) -
. x , X ( ., ) ; P ( | ) ,
X - , ,
- D
d .
x1 , ..., xk , X - X 1 , ..., X k
P ( | i ), i = 1 , ..., k
1 , ..., k G
.
, x
(k )
, X - ( )
PG ( ) =
P( | ) G(d )
.
G G
= {G ( , ) ,
}
, - PG ( , ) (k )
;
X
G
G ( , k ) - .
; . ( 1911, , [1] )
. .- .
G
( , [2] ) .
. .-
.
. . ( , [3] ) ( )
, G -
p k ( x , x ( k ) ) =
p ( x | ) G ( d )
(*)
RG ( k ) = M L ( , k ( X , X ( k ) )) =
...
Xk
(k )
.
.
( [4], [5] ) . , , ,
.
. .
G = G ( x ) k
= k ( x , x ( k ) ) -
; ( X
k +1
, X k +1 ) -
.
L ( , d ) G
k :
G(d
xi ) .
i =1
( k - ) k G -
.
;
.
G -
G - G X -
= PG2
. : PG1
G1 G2 G1 = G2 , G
.
1. ( 0 , ) -
0 ( ) > 0 (
) .
x -
. 0 1 :
, , 0 1 -
. , ( G )
-
g () , G . X
pG ( x ) =
) , PG
, p (x | ) , P ( , ) -
L ( , k ( x, x ( k ) )) P ( dx | )
,
RG ( k ) RG ( G ) , k -
( ) , p k ( x, x
G(d )
( g () ) d
x
X =x
2 pG ( max ( 0 , x ) < pG ( x ) ,
d1 , d 2 .
, G - X - . , pG (x)
( ., ) .
2. n ;
( ) . -
.
X , x n , G
75
x
pG ( y , n ) = P { X = y}
,
G (x) =
x + 1 pG ( x + 1, n + 1)
n +1
pG ( x , n )
.
G ( x )
(
; , [7] ), ,
. n , n 1
. n -
.
. . ( [5] [6] )
, , ,
.
, ,
. ,
( , ., [9] ).
k
.
, , O ( k
) - (
);
[10]- .
, . .
( , [11] ).
. x -
G (x)
( x , PG ) = inf G ( x) ( x , PG ) = sup G ( x)
G
; PG
= PG
G .
.
. .
.
, , ,
;
,
.
d . [12]-
76
( , ).
.: [1] G i n i C., Metron, 1949, v. 15, p. 13371; [2] F o r c i n a A. Int. Statist. Rev., 1982, v. 50, p. 6570; [3] . ., . . . . 1941, . 5, . 8594;
[4] R o b b i n s H., .: Proceedings Berkeley Symposium Mathematical Statistics and Probability, v. 1, Berk. Los Ang., 1956, p. 157
63 ( . . , 1964, . 8, 2, . 13340 ); [5] R o b b i n s H., Ann. Math. Statist., 1964, v. 35, 1, p. 120 ( . .
, 1966, . 10, 5, . 12240 ); [6] N e y m a n J.,
Rev. Inst. Int. Statist., 1962, v. 30, 1, p. 1127 ( . .
, 1964, . 8, 2, . 11332 ); [7] . .,
. , 1963, . 150, 4, . 73335; [8] S i n g h R. S.,
Ann. Statist., 1979, v. 7, 4, p. 890902; [9] . ., . . , 1983, 11, . 6773;
[10] . ., .: IV . . .
. . . ., , 1985,
. 2, . 78; [11] . ., .: .
, 1970. . . ., ., 1972, . 4855;
[12] . ., . . , 1983, 11,
. 4258.
; Bayesian approach t o s t a t i s t i c a l p r o b l e m G
RG ( ) =
R ( , ) G ( d )
. , . .- X
( ., ) P ( | ) -
( ) ( ) G
; R ( , ) -
( G
), RG ( )
= M L ( , ( x) ) , L( , ( x) ) .
RG ( ) - G = G (x)
, RG ( G )
.
, inf RG ( ) .
RG ( | x) =
L ( , (x ) ) p (x | ) G ( d )
pG ( x )
, p
= d P d
X [
(X , X ) ],
pG (x) =
p ( x | ) G ( d )
X - ( ) , L ( , )
.
. PG
, ,
RG ( ) =
G (
| x ) pG ( x ) ( d x ) .
,
, ,
G
> 0
RG ( G ) inf RG ( ) +
.
. H 0 : 0 ( d 0 ) H 1 : 1
=
= \ 0 ( d1 )
0 1 :
1, 1i ,
L (, di ) =
0 , i ,
,
i = 0, 1
= d i
H i H 1i
RG ( di | x) =
,
. G
.
; , , . .
.: [1]
.,
, . ., ., 1974; [2] H a r t i g a n J. A., Bayes theory,
N. Y. [a. o.], 1983; [3]
.,
, . ., ., 1975; [4] . .,
, ., 1984; [5]
. .,
.: , , 1984, . 10,
. 1353.
;
Bayesian deviation , , .
p ( x | ) G ( d ) pG ( x )
1i
.
, ,
( ).
.
. . (
)
( , ).
. .
,
( ,
,
). , ( .,
) d
,
d
RG ( d , ) = M { L ( , d ) | ( x ) = d } , d D
.
d 1974- . .
.
[5]- .
1 , 2 , ... ,
, G ( .,
, ) . .-
, . , G -
## ; equilibrium dynamical system , ( )
.
; equilibrium distribution ,
( ) .
; equilibrium potential ,
,
.
; equilibrium
statistical mechanics , .
,
;
;
equiprobable scheme f o r g r o u p allocation of
particles , , .
, ; equiprobable allocation of particles / uniform
allocation of particles , /
, .
( ) ; regeneration o f
a M a r k o v p r o c e s s , ; .
; renewal
input ,
77
; renewal
time
.
; renewal function F (t ) F - - n -
H (t ) =
(t ) =
n 0
P {X
t}
n 0
0) ;
n = X n X n1 , n 1
H (t ) , ( X n , n
F -.
P { n = 0} < 1 , H (t )
H (t ) = 1 +
H ( t s ) dF ( s )
0
( [ 0 , t ]
). H (t ) , ,
,
H( t + s) H( t) + H(s)
t , s 0 ,
lim
t dF (t ) . ,
; ) ( R R -
. . ), ) , )
( ), ) , ) .
.
.: [1] . ., . ., , . ., ., 1967; [2] . ., , .: . . . . , . 2, ., 1974, . 99128; [3] .,
, . ., . 2, ., 1984.
;
renewal process F (t ) ,
, , 1 , 2 , ... -
H ( t)
1
= ,
m
t
X n
, ( )
.
.
.
. .-
.
,
, . .- . . .-
( , )
, , .
. .-
: ) -
m2 =
t dF (t )
2
t =
t m H (t ) t m + m2 2m 2
.
, F ( t )
, a > 0
lim ( H ( t + a ) H ( t ) ) = a m
, F (t ) - { 0 , d , 2d , ...}
.
lim ( H ( t ) t m ) = m2 2m 2
.
.: [1] . ., . ., ,
. ., ., 1967; [2] ., , . ., . 2, ., 1984.
; renewal interval
; renewal theory
78
= 1 + 2 + ... + n , n 1 , X 0 = 0
. (t )
t ) = 1 + max ( n : X n t )
n 0
.
n = X n X n 1
( ) , t t
( ) . t
( X n , n
0)
,
.
. . . t
H (t )
= M t
;
:
H k (t ) = M ( t ) k ( k 2 ):
, .
H k (t ) =
I(X
C ki (1) i 1
i =1
k i ( t
s ) dH ( s ) .
(t )
(t ) ,
t ( X n , n
0)
( )
(t ) , (t ) (t )
(t ) = t t ,
lim t 1H ( t ) = 1 / m , m = M k
q ( t s ) dF ( s )
u
.
n n -
F (t ) -
= M 1
1 - ,
lim [ H ( k + 1) H (k )] = 1 / m
; k . . .
. . , : n
(1 F ( t ) ) dt
( 0 , ) - -
g (u )
( , ).
= 1 e t , t 0 , . .-
, P { t = n + 1 } =
F (t )
= ( t ) e
n! , n 0 , H (t ) = 1 + t , qu ( t ) = e
,
m = 1 / (t ) (t )
n
. .- ,
lim [ H ( t + h ) H ( t ) ] = h / m
## (t ) = 1 + max ( n : S n < t ) = min ( n : S n t ) ,
h > 0
, qu (t )
1
m
. . .- ,
k
P { ( t ) u , ( t ) } = qu + ( t )
lim qu ( t ) =
k =1
. . .
.
(t ) (t ) . (t ) (t )
H (t ) = M (t ) . . .
(t ) = t t 1
qu (t ) = 1 F ( t + u ) +
Sn =
.
.: [1] . ., . ., ,
. ., ., 1967; [2] ., , . ., . 2, 1984.
; ; ; s u p e r p o z i k
t i o n o f renewal process t , 1 k l
lim
1
m
g ( t u ) dH ( u ) =
1
m
g (k )
k =1
.
H (t )
( nk , n 0) ,
m H (t ) = t + M (t )
. .
n 0,
1 k l }
( n , n
0) .
t =
I ( n t )
dH ( u ) P{
> t u + }
, . .-
+ ... +
tl
. .- l
.
; renewal theorem -
. 1 , 2 , ... ,
,
, k
P{ ( t ) > } =
n0
t1
(*)
,
t (*)-
(t ) -
{ nk ,
g ( u ) du
, . n
, .
1
m
lim M ( t ) =
G ( t ) dt =
M12
,
2m
G (t ) =
P{
> } d
79
( k - t
x (t ) =
). , t
, H (t ) , F (t )
. .
.
.
,
k +2
. M 1
1 Me
lim inf
R (t ) =
1
m2
ln t
= o ( t k ) .
k +1
G ( u ) du + o t
t
M1k + 2 < k -
R (t ) =
1
m2
j t
y ( t ) dt ,
t dF (t ) .
:
(a ) F (t ) , F (t ) :
F ( n h + ) F ( n h ) < 1 ,
n0
h > 0 y (t ) [ 0 , ) -
> t } ,
, > 0 ,
c <
P {1
R (t ) < c e
var R ( t ) < c e t
;
renewal equation
t
x ( t s ) dF ( s )
, [ 0 , ) -
, x , y , F
.
[ 0 , t ] .
. .- .
y (t ) -
sup
[ n , n + 1]
y (t ) <
. k
## ; elementary renewal theorem
n0
(t , )
.
.: [1] . ., . ., ,
. ., ., 1967; [2] ., , . ., . 12, ., 1984; [3] . .,
, ., 1972; [4] S t o n e C., Trans. Amer. Math. Soc.,
1965, v. 120, 2, p. 32742; [5] T a k a c s L., Combinatorial
methods in the theory of stochastic processes, N. Y. [a. o.], 1967.
m -
P { 1 k + 1} + o ( t k 1 ) .
k> j
x (t ) = y ( t ) +
<
> 0 ,
1
m
lim x ( t ) =
M12
t
R (t ) = H ( t )
m
2m 2
i 1
y ( t s) d H ( s )
M12
t
+
+ o (1 ) .
m
2m 2
H (t ) =
sup
n 0 [ n , n + ]
y (t )
inf
[ n , n + ]
y (t ) 0 ;
(b) F (t )
, F
(t ) -
, y (t )
, [ 0 , ) - t
.
.: [1] . ., . ., ,
. ., ., 1967; [2] ., , . ., . 2, ., 1984; [3] . ., . ., . .,
. . ., 1983.
; white noise -
(t )
. . . ( )
B (t )
= 2 ( t ) , 2
, (t ) . . .
( .,
). (t ) . .
(t ) =
e i t d z ( )
(*)
e i t dz ( )
. .- .
80
( ) ,
M dz ( )
2
d ,
2
< <
Y (t ) =
.
(*) ,
(t )
, =
c ( t , s ) d (s) = c ( t ,
= ,
(t ) dz ( )
d (t )
[ d (t ) dz ( ) ],
c ( t , s ) ds <
a ( ) = a
( t ) d ( t ) .
, a
(t ) -
: (t )
(t ) = a +
= (t ) .
it
d z ( )
dz ( ) ,
Y (t ) = a ( t , Y ( t ) ) + ( t , Y ( t ) ) ( t )
Y (t ) ;
, ,
.
. .
(t ) -
Y (t )
Y ( s ) ,
s < t (u ) , u > t
P (z )
P ( d dt ) Y (t ) = (t )
Y (t ) =
(t ) =
(t
= ( a 2 ) d .
.: [1] . .,
, 3 ., ., 1987.
. ., -
;
f r a c t i o n a l white noise , .
;
white noise i n a f i n i t e b a n d w i d t h
( ,
) , , , f ( ) = f 0 = const ,
> , f ( ) = 0 (
B ( ) =
1
dz ( )
P ( i )
M dz ( )
; d i s c r e t e white noise ,
d Y (t ) = a ( t , Y ( t ) d t + ( t , Y ( t ) ) d (t )
(t ) d t
= ( , )
, =
s k )
, ~ ( ) , (t )
c ( t , s ) ( s ) ds =
, c ( t , s )
( ) d z ( )
( t ) ( t ) dt =
k)
. . ( k - -
... , 1 , 0 , 1 , ...
), (t ) (t )
.
f ( ) d = 2 f 0
sin
. . .
. .-
,
>
; prewhitening
. Yt -
81
, Z t
;
Zt =
cY
t r
, t
. .- ( , [1] [3] ).
. .- ,
, e ( t )
= 0 , 1, 2 , ... .
lim M [ e ( t ) e ( t ) ]2 = 0
r =0
h( ) =
e ( t ) 2
ei r
f ( )
(*)
( t ) = { 1 ( t ) , 2 ( t ) , ..., n ( t )}
r =0
,
( ,
f ( ) , Yt -
). cr ,
. .
p
h ( ) (
k (t ) +
).
, Z t h ( ) -
;
Bhattacharya Rao spherical distance ,
.
( )
;
bilinear model f o r r a n d o m p r o c e s s / bilinear
time s e r i e s m o d e l (t )
p
(t ) +
i =1
i =1
j =1
a (t i ) = e(t ) + b e(t i ) +
i
i =1
c i j ( t i ) e ( t j ) , t = 0 , 1 , 2 , ...
(*)
, e ( t )
: M e ( t )
= 0 , M e 2 ( t ) = 2 < ( ),
{ a i , 1 i p } , { b j , 1 j q } { c ij , 1 i r , 1
j s } . e ( t )
, . .- . (*)
p, q, r , s
BL ( p, q, r , s ) . i = 1, ..., r
j = 1, ..., s c ij = 0 , . .
( p , q )
.
. .-
,
. . .
. .- ,
82
i
ku u ( t
i ) = ek ( t ) +
i =1 u = 1
, (*)- f ( ) .
bkuj eu ( t j ) +
j =1 u =1
i =1
j =1
e ( t j ) ,
u = 1 =1
ij
ku
u (t i )
k = 1, ..., n
, e ( t )
= { e1 ( t ) , ...,
en ( t )} , t = 0 , 1 , 2 , ... : M e ( t )
,
i
{ aku
,
= 0 , G -
1 i p , k , u = 1, ..., n } , { bkuj , 1
ij
j q, k , u = 1, ..., n } ,
{ cku
, 1 i r, 1 j s ,
k , u , = 1, ..., n } .
## .: [1] G r a n d e r C. W. J., A n d e r s e n A. P., An introduction
to bilinear time series models, Gtt., 1978; [2] R a o T. S., J. Roy. Stat.
Soc., ser. B, 1981, v. 43, 2, p. 24455; [3] R a o T. S., G a b r M.
M., An introduction to bispectral analysis and bilinear time series models, B., 1984.
( X , I )
( X , I ) ; bicompact / compact set i n a t o p o logica l
s p a c e K X ,
. K , K
( . )
. K , ,
, K -
( , [1] ). . .
. - . ,
.
.-, . .
.: [1] .-., , . ., 2 .,
., 1981; [2] . .,
, . ., 1948.
; billiards , .
; bimodal distribution , ,
. . .
.
, , / .
; binary relations
statistics
, .
( ),
( ), ( ) .
. (
) , , . , , ,
.
,
- .
,
, . . .
( , [1] ). . .
.-
( , [2] ).
, .
, .
20-
( , [3], [4] ). A B
( A , B ) d ( A, B )
( A , B ) = 1 2 d ( A , B ) k ( k 1)
, k . ( , [5] )
.
- . . .-
, ,
.
.: [1] . ., - , ., 1979; [2] . ., .:
, .,
1985, . 169203; [3] . ., . .,
, 3 ., ., 1983; [4] ., , . ., ., 1975; [5] . ., , ., 1982; [6] ., ,
. ., ., 1978.
; binary branching process
, ( ), . . . .- t ( t ; s )
.
bt
( t ; s ) = 1 ( 1 s ) (1 s ) + 1
2
c1 c2 (t )
. ( t ) a ( t ) + b
= 1 ( t ) -
(c1 , c2 )
; , . . .-
c1 = 1, c2 = 0 c1 = 1, c2 = 1 . ,
. . . ( , ., [1] ).
. . .- . ( , [2], [3] ).
. . .- ;
. . .-
, ., -
.
. . .- (t )
; (t )
( t ) a ( a )
. ( t )
, a ( t ) ( t ) ;
>a
M ( t ) = 0 ,
M 2 ( t ) a 2 ,
. . .- .
(t ) ,
1,
sgn x =
1,
, Y (t )
x 0 ,
x < 0
= sgn ( t ) . . .-, Y (t )
(t )
( ,
); Y (t ) - (t ) -
( , [4] ).
(t ) . . .- B ( ) = M ( t + ) ( t )
= 0
( , [1] ).
.: [1] M a s r y E., SIAM J. Appl. Math., 1972, v. 23, 1,
p. 2833; [2] W i e n e r
N., Acta math., 1930, t. 55, 23,
p. 117258; [3] .,
, . ., ., 1963, . 194; [4] K e d e m B., Binary
time series, N. Y., Basel, 1980.
; binomial coefficient
C Nn
( x + y )N =
C Nn x n y N n ,
n=0
( t ; s ) = 1 e a t (1 s )
( e a t 1) ( 1 s ) + 1 ,
2a
a 0.
; binary random process ,
, b > 0 .
. . . .
.: [1] . ., , .,
1971.
C Nn =
N!
N ( N 1) ... ( N n + 1)
=
, 0 n N
n! ( N n )!
n!
83
. . N
n . . .
( nN ) .
pk = P { = k } = b ( k ; n, p) = Cnk p k q n k
( , ).
. .- . .,
C Nn = C NN n , C Nn ++11 = C Nn + C Nn +1 ,
N
n
N
= 2 N , C Nn =
n=0
C Nnkm , 1 m n .
k
m
k =0
. .-
. .,
CMm C Nn mM
,
C Nn
p ( M , N, m, n) =
0 m n N , m M N , n m N M
Bn =
1
M ( 1) ... ( n + 1) =
n!
n
N
P{ = N }
N =n
= u log 2 u ( 1 u )
. h ( u )
log 2 (1 u ) , = ( 2n + 1) / 2 N ,
C Nn 1 =
2 N h ( )
2 N
## 1) n = 5, p = 0,3 n = 10, p = 0,3 ;
e ( N , n ) ,
2) n = 20, p = 0, 1 p = 0,5 ;
3) n = 20, p = 0, 2 = np = 4 -
0 ( N , n) ( 16 n ( 1 ) ) 1 .
. .- . :
N!
=
,
n
,
n
,
...,
n
n
!
n
1
2
k
1 2! ... nk !
k >2,
n1 + ... + nk = N ,
q (
q 1 . . - ):
N
N
=
n q
(q
i =1
( q i 1)
N n
, q >1
( q i 1)
i =1
.
.: [1] . ., . ., , 3 ., ., 1983; [2] .,
, . .,
. 12, ., 1984.
## ; binomial distribution , k = 0 , 1, 2 ,..., n
84
p . .- ( ) , 0 p 1 . . .
.
1 , 2 , ..., 0 1 , p 1 p
,
( i n = 1
. . ).
i
: i - , i
= 1 , i = 0 .
,
. { i , i 1} -
1)
i =1
= 1 + ... + n , n 1 p . .
. . .- ( s )
( q
= 1 p) . (s)
= ( p s + q ) n -
b0 + b1s + ... + bn s . .
; bk
= b ( k , n, p ) .
. .- :
M = n p , D = M [ M ] 2 = n p (1 p ) ,
M ( M ) 2 = n p (1 p ) (1 2 p ) ;
. .-
As = (1 2 p )
n p (1 p ) ,
Es = (1 6 p (1 p )) n p (1 p ) ,
f ( z ) = ( 1 + p ( e i z 1) ) n
.
y
F( y ) = P{ y } =
[ y]
k
n
p k (1 p ) n k , 0 < y < n
k= 0
( )
,
,
. X n , n = 0 , 1, ... (X , d )
, ,
*
. X n , n
= 0 , 1,
... , (X , d )
,
, X n ,
n = 0 , 1, ...
, [ y ] , y - .
y
y n p + 0,5
+ Rn ( y, p )
F( y) =
n p (1 p )
, ,
Rn ( y , p ) = O ( n 1 2 ) .
n . .
F( y)
; common
probability space method
. { X n } ,
d ( X n* , X 0* ) 0
, Pn P0 ; Pn , X n , n
= 0 , 1, ... .
( , [1] ).
, ( Pn , P0 )
## ( Pn , P0 ) inf max [ , P { d ( X n* , X 0* ) > }]
1
B ( [ y ] + 1, n [ y ] )
t [ y ] (1 t ) n [ y ] 1 d t
. ( X , d )
( )
, B ( a , b ) .
n , p
(*)
. . . .
*
: (*)- X n
X 0 -
Pn P0
d ( X n* , X 0* ) 0
*
, b ( k ; n, p ) ,
{ X n }
( , [1] ).
. . . .- : { X n }
b (k ; n, p)
(n p)
k!
e n p ,
, n 0 < c y C ( c C )
, 0 < p < 1 p -
F( y) =
[ y]
k =0
k
k!
e + O ( n 2 ), > 0
= ( 2n [ y ] ) p ( 2 p ) .
. .- .
.: [1] . ., , 6 .,
., 1988; [2] .,
, . ., . 1, ., 1984; [3] . .,
. ., , 3 ., ., 1987;
[4] . ., . , 1953, . 8, . 3,
. 13542; [5] . ., . .,
, 3 ., ., 1983.
;
binomial sample , .
, X n
= Fn1 ( Y ) , n = 0 , 1, ... , Fn ( ) , X n -
, Y , [ 0 , 1] - .
. . . .
( , [2], [3], [4], [5], [6] ).
.: [1] . ., . .,
1956, . 1, . 3, . 289319; [2] . ., . ., 1956, . 1, . 2, . 177238; [3] .
., , ., 1961;
[4] S t r a s s e n V., Proc. 5th Berk. Symp. Math. Stat. Probab.,
1967, v. 2, 1, p. 31543; [5] K o m l o s J., M a j o r P., T u s n a d y G., Z. Warh. und verw. Geb., 1975, Bd 32, 1, S. 11131;
1976, Bd 34, 1, S. 3358; [6] B e r k e s I., P h i l i p p W., Ann.
Probab., 1979, v. 7, 1, p. 2954.
; one-step transition probability , .
; homogeneous chaos
n
, R , n 1
85
( , [1] ). . .
( )
.
.: [1] W i e n e r N., Amer. J. Math., 1938, v. 60, p. 897936;
[2] ., a , . ., ., 1987.
## ; homogeneous isotropic correlation function , .
; homogeneous channel , .
; homogeneous / stationary transition function , .
; homogeneous
correlation function , .
; homogeneous Markov process , .
; time homogeneous Markov chain
; homogeneous random field s
S = {s } X ( s )
, S G = { g } -
, s
G . . . . : n = 1, 2 , ...
g G n s1 , ..., s n g s1 , ... , g s n - , X ( s )
. . . ; s S , s1 S
M X (s)
<
M X ( s )
g G
= M X ( g s ) , M X ( s ) X ( s1 ) =
= M X ( g s ) X ( g s1 ) , X ( s )
. . . .
G
k
k R k
( R - Z
) . . . , . . . , ,
k
. (E , B ) -
. R -
, p n ( x , ) , n 0
G , R -
. . ., , .
. . .
;
. . .-
( , ., [1] [3] ). . . .
; , -
X = ( X n ) pn ( x , )
n - . E
= { 1, 2 , ...}
pij ( n , n + 1)
= p n ( i , { j})
n - . ( i, j E ) .
. . . . .
. . . .
.
; homogeneous measure , .
; homogeneous Poisson measure , .
,
;
R n y is homogeneous class of functions
n -
F : - f ( ) =
= f ( ) f F
- R2 . , F - 2 R2 ( n 1) ( n 2 ) . F -
N -
- -
.
.: [1] ,
., 1983.
86
R -
( ), (
)
( , ., [4] ),
. . .- ( , ).
.: [1] Y a g l o m A. M., .: Proeedings of 4th Berkeley
symposium mathematical statistics and probability, v. 2, Berk. Los.
Ang., 1961, p. 593622; [2] .,
, . ., ., 1970;
[3] . ., ,
., 1980; [4] . ., . .,
, . 2, ., 1967.
; homogeneous random process with independent increments ,
.
,
;
time homogeneous random process ,
.
; homogeneous generalized
field , .
; homogeneous and
geometric process , .
; homogeneous and isotropic geometric
process , .
; homgneous and isotropic random field
, G -
( ,
). M X (t )
< +
, m ( ) S1 S 2
m
[ ( S )
d ( ) ,
, cn
; (2)
.
. . . . R
R - .
2
. . . .- R -
:
X ( r, ) =
Z kl ( ),
(1)
Yk ( r ) Z kl ( d ) +
sin k
Yk ( r ) Z k2 ( d ) +
l = 1, 2 , (3)
.
( ., ). g G
M X (t ) = 0,
, X ( t )
Yn ( r ) d ( ) ,
cos k
k =1
X (t ) -
k =0
t R
M X (t ) t - M X ( t ) X ( s ) = B ( | t s | )
B(r ) =
= 2 n1 ( n / 2) n 2 ,
i j ]. (1)-
X (t ) ,
. (
) G
. . . . .-
,
.
B ( r ) :
( )
t s t s
M Z ml ( S1 ) Z ml11 ( S 2 ) = m 1 l 1 ( S1 I S 2 )
M X ( t ) X ( s ) = g M X ( g t ) X ( g s ) g ,
= { X1(t ) ,
... , X n ( t ) }
B ( r ) = M X ( t + r ) X ( t ) = ( Bi j ( r ) )
n J ( n 2) 2 ( x )
Yn ( x ) = 2( n 2) 2
( n 2) 2
2 x
, ( ) , [ 0 , + ) -
, X (t )
. X (t )
( , [1] ):
X (t ) = cn
+
h (m, n)
m=0
l =1
( r )
( n 2) / 2
Z ml ( d )
(2)
h ( m , n) = ( 2 m + n 2 )
( m + n 3 )!
( n 2 )! m!
r2
+ Bkk (r ) i j
.
Bl l ( r ) Bk k ( r ) , ,
.
Bk k ( r ) = M X k ( t + r ) X k ( t ) ,
( r , 1 , ..., n 2 , ) , t ,
S ml ( 1 , ..., n 2 , ) ,
ri r j
Bl l ( r ) = M X l ( t + r ) X l ( t ) ,
S ml ( 1 , ..., n 2 , )
I m + ( n 2 ) / 2 ( r )
## Bij ( r ) = [ Bll ( r ) Bkk ( r ) ]
X l ( t ) , X (t ) r , X k (t ) r -
. . . . .-
.
.: [1] . .,
, ., 1980; [2] . ., . ., , . 12, ., 196567; [3] . .,
. ., 1957, . 2, . 3, . 292338.
87
; joint
measurement , .
; joint distribution
. X 1 , ..., X n , ( , A , P ) -
;
Birkhoff Khinchin ergodic theorem
... , 1 , 0 , 1 , ...
M 0
, (X k , Bk ) -
## PX1 ,..., X n ( B1 , ..., Bn )
= lim
. . . ,
.
X 1 , ..., X n ,
n
. . R ( X 1 , ..., X n ) ( , ). X (t ) , t T
, t1 , ..., t n T X ( t1 ) , ..., X ( t n ) -
, n 2.
.: [1] . ., . ., , 3 ., ., 1987.
,
; joint probability density , .
; single server queueing
system ,
. . . .-
,
. , . . .-
z n +1 = f ( z n , n )
( z n ) , ( n ) .
n - wn
wn +1 = max { 0 , wn + n un }
, n , n -
, u n , n n + 1 - -
k = lim
k = m
k =0
1
2n + 1
k = n
, , I , { i } -
.
. ( [1] ),
. . ( [2] ) .
(t ) , < t <
M ( 0 ) < ( ).
.: [1] B i r k h o f f G., Proc. Nat. Acad. Sci. USA, 1931,
v. 17, p. 65660; [2] C h i n t s c h i n A., Math. Ann., 1932, Bd 107,
S. 48588.
; one-valued property
of a set relatively to a measure , ,
.
## , ; associated spectrum of a process , .
( )
; u n i o n o f events .
; combination w i t h o u t r e p e t i t i o n s
, .
( ) / ( )
/ ;
union / sum o f r a n d o m e v e n t s
A B
,
A U B sup ( A , B ) .
:
Ai ,
iJ ,
; Laplace first law / distribution ,
Ai
## ; single-channel queueing system .
88
1
n +1
1
n +1
X 1 , ..., X n
. . X (t )
M ( 0 | I ) = lim
. B1 B1 , ... , Bn Bn
<
i - ,
U A ,
i
i=J
, Ai
. ,
A ( A ) ,
A U = A,
AU = .
A A U B , B A U B . C
, A B C ,
, ., E - E c1
c2 -
AU BC,
i Dn D p+ f ( ci ) + ( 1) i D p f (ci ) = 0 , i = 1, 2
A U B A B
.
: A U B = B U A ;
:
:
A I ( BU C ) = ( A I B) U ( A IC ) ;
= A
.
.: [1] . ., . ., .
., , ., 1979;
[2] ., , .,
1969.
; one-dimensionl diffusion , . . .-
. . .- . ( , [1], [2] ) ,
. -
. . ( , [3], [4] ) . . .
( , [4], [5] ).
1
= ( t , , At , Px ) , E R
. . x D D -
, . . . E - , E
, , , .
E = [c1 , c2 ] , ( , )
. .- ( ) p ( x ) n ( x )
, x E . p ( x )
= Px { 1 = c2 } E - ,
, 1 , - I
= ( c1 , c2 ) - -
. n ( x ) I - D +p m ( x ) - ,
m ( x ) = M x 1 < ( D +p m ( x ) -
lim
y x
. .-
( , [4], [5] ). . .- . .-
[1], [2]- .
. .-
( , [4], [5] ). ,
. .
( , ; ; [4], [5] ).
, ; .
.: [1] F e l l e r W., Ann. Math., 1952, v. 55, p. 468519;
[2] F e l l e r W., Ann. Math., 1954, v. 60, p. 41736; [3] . ., . , 1955, . 105, . 20609; [4] . ., , ., 1963; [5] ., ., , . .,
., 1968; [6] . ., . . . -,
1960, . 9, . 14389.
; ;
; one-dimensional diffusion; c l a s s i f i c a t i o n
= ( c1 , c2 ) R1
E - X =
= { X t , , At , Px }
of
boundaries
. c1 . ) X
= lim X t Px { X = c1}
t
, c1
) Px { X = c1 , < }
, c1
; ) c , ( c1 , c )
, M x c , c1
[ , ; ) ) x E -, ) ( c1 , c )
m( y ) m( x)
p( y ) p ( x )
, c E ]. c2 .
; D p m ( x ) -
). , , n ( ci )
= n (c1 + 0 ) n ( c1 ) ,
2 = n ( c 2 ) n ( c 2 0) .
A U ( B UC ) = ( A U B ) U C ;
: A U A
, 1
= i ,
i = 1, 2 , 1 ( 2 ), c1 - ( c2 - )
c1 c2 , . .
.
1
c2 ( c1 ) -
. .- E R . ., c1 E
. n ( x ) E - I
c1
A f ( x )
A ,
Dn D +p
f ( x ) , x E . A
f
( , [4], [5] );
c1 E - ,
; c1 E X
E \ { c2 } - , c1
89
. : x x
c1
=0
( ,
; ).
E
. ( , [3] ).
.: [1] . ., , ., 1963;
[2] ., ., , . ., ., 1968; [3] F e l l e r W., Ann. Math., 1952,
v. 55, 2, p. 468599.
; one-dimensional random
process , .
; one-parameter
Euler numbers , .
; one-sample Students test , .
## ; one-sided Bernoulli shift , .
; one-sided generator , ( )
; one-sided distribution ,
.
; one-sided infinitely divisible distribution
[ r , ) ( ( , r ] )
r < . F ( x ) -
., . . . . .
.: [1] . ., . .,
, ., 1972; [2] ., , . ., .,
1975.
; one-sided Students
test , .
; bispectrum ,
.
; bispectral function
( , ,
).
; bispectral density , , M ( t ) = 0
( t ) ( , ) .
.: [1] B r i l l i n g e r D., Ann. Math. Stat., 1965, v. 36,
p. 135174; [2] B r i l l i n g e r D., R o s e n b l a t t M., .:
Spectral Analysis of Time Series, N. Y. L. Sydney, 1967, p. 153
88, 189232; [3] ., ,
. ., ., 1974; [4] S u b b a R a o T., G a b r M.,
An introduction to bispectral analysis and bilinear time series models,
N. Y. [a. o.], 1984.
; bit
2 ( . binary digit, ). . .
,
( . natural digit ) ( . hartley ) .
; Blackman Tukey method ( t ),
t = 0 , 1, ... ( ) t = 1, 2 , ..., T xt
*
f ( ) f T ( ) -
lext F = inf { x : x , F - } ,
f T* ( ) =
rext F = sup{ x : x , F - } .
>
. lext F
rext F
<
, F
. F , f ( t )
G ( x ) ,
k T 1
aT ( ) BT* ( ) cos
*
, BT ( )
(*)
=0
t +T
xt , xt ,
t =1
) -
aT ( ) , > kT , -
G x 0 ( x 0 )
, k T , T -
. lext F
>
( rext F
<
dG <
( x 1 d G < )
0
lext F =
x 1 dG ,
90
rext F =
dG .
( ., k T
= 0,1 T ). . . . [1]- -
## 20- 50 60- ( , ., [2] [4] ).
,
( T - 100
). T - . . .- ,
.
[ xt I T ( ) BT ( )
I T ( ) -
*
, f T - (*)
] ( , ., [5] ).
. . .-
( , ., [6] ).
.: [1] B l a c k m a n R. B., T u k e y J. W., The measurement
of power spectra, N. Y., 1959; [2] ., .,
, . ., . 12,
., 197172; [3] ., , . ., ., 1976; [4] ., .,
, . ., .,
1974; [5] . .,
, ., 1981; [6] . ., . ., . - . ., 1982, . 70,
9, . 24355.
; Blackwell theorem , , .
; block , .
( ) ; block decoding n
n
; - . . .-
,
. . .-,
, ,
. ,
. .
( , ( ) ).
.: [1] . ., . .,
, ., 1982.
; block code A
n . n
. .-
, R
= n 1 log 2 M . .-
, M . .- .
. . . A -
( ,
A < -
) . . A - k , . .
( ) ( n , k ) .
2
O ( n ) -.
, . .
.
.
. .-
.
.: [1] ., ,
. ., ., 1974; [2] - . ., .
. ., , , . ., ., 1979.
; block design
,
. -
. [1]-
. .,
, , , .
, ,
( ) .
,
.
,
. . .
. ., . .
yi j k = + i + j + i j k ,
i = 1, ..., t , j = 1, ..., b , k = 1, ..., K ij
(*)
,
yi j k , ( i , j ) k - ,
, i , j , j -
i - , ijk
. , ,
,
, , j .
( , ) . .- .
, , ,
. . , . .- ( ) ,
.
. , . .- ( , [2], [3] ).
k b
, t > k . ., t
r
, ( . ).
- r t = b k ( t 1) =
= r (k 1) , .
-
.
(*) i
k
c
i
= 0 ) -
. k1 , ..., k t 1
k
= ( k1 , ..., kt 1 )
91
D , ki , k i
. D - . .
.
, , D - - .
. .- ( . , ), b , k , t
r , , . . , :
1) m
; i -
ni ;
2) i - i ;
3) i - , j - , k -
p ij k = pki j .
m = 1 -. -
; - ,
( , [8] ).
, (*) . X
( i , j ) , i = 1 , ..., t , j = 1, ..., b ( ,
), i ,
, 1
T
= T 1 = 0 , T = (1 , ..., t ) , T =
= ( 1 , ..., s ) , 1 .
.
Z =
( i, j ) , ri =
(i , j ) ,
j
(i , j ) ,
kj =
r = (r1 , ..., rt ) ,
T
## K = diag (k1 , ..., kb ) , R = diag ( r1 , ..., rt ) min k i > 0 .
l T , , - , C ( ) -
( t 1) - , C ( ) = K ZK 1Z T , ,
l1T l2T k1
k2 , cov ( k1 , k 2 ) = l1TC ( ) l2 ,
C , C - g .
=k
, c , d C ( )
*
= c I + d 11
T
C ( ) - C ( ) S
, C ( ) S -
*
,
m
( (
i
i =1
92
) i ( ) ) 0
= 1, 2 , ...,
i ( ) , C ( ) - i - ( ).
*
: ( C ( )) S -,
*
, ( C )
( , ).
, k i = k ,
. - .
:
,
, . , ;
*
S - C ( ) *
. - .
.: [1] . ., , . ., 1958; [2] .,
, . ., ., 1980; [3] .,
, . ., ., 1970; [4] Partially
balanced design, .: Encyclopedia of Statistical Sciences, v. 6, N. Y.,
1985; [5] Nearly balanced design, .: Encyclopedia of Statistical
Sciences, v. 6, N. Y., 1985; [6] C l a t w o r t h y W. H., Nables of twoassociate-class partially balanced designs, Wash., 1973; [7] . ., , ., 1970; [8] . ., . .,
, ., 1979; [9] G i o v a g n o l i A.,
W y n n H. P., Proc. Berk. Conf. in Honor of J. Neyman and J.
Kiefer, 1985, v. 1, p. 41833.
; block scheme
, .
; block
struture , .
; block frequency , .
-
- ; Blumenthal Getoor Mc Kean theorem ,
;
, k j
; Bochner integral
( ) . , ( , A , ) ,
, B . f : B
(
) A - ,
.
lim
f n ( ) f ( ) d ( ) = 0
(*)
f n : B , f : B
. ,
, f n - B -
, (*) f -
. f
.
f ( ) d ( ) < +
( B ) = 1 1 B ... p B p
( B ) = 1 1 B ... q B
, B xt
= xt 1 , M xt , xt , at -
, a
.
. .- - .
.: [1] ., , . ., .,
1967.
. ( z ) = 0 ( z ) = 0
Z
: xt
: t R
f (t ) =
f (t )
j,
## ; Bochner Khinchin theorem
p, d , q
j = 1, ..., p , j , j = 1, ..., q 2
.
. . . [1]- . d - Wd , t
Rn
f (t ) - n
= (1 B ) d ( xt M xt )
. p q d -
. , R -
f ( 0 ) = 1
[2]- :
w d 1, t 1 w d , t
- .
. [1] [2]- . . . . . . ( , [3] )
.
( )
w d , t . ( p , q ) - -
. . .- R - ( ,
[5], [6] ).
., Z ( ) . ( F. Riesz )
. ( H. Herglotz ) . . . .
.
( ,
, ).
.: [1] B o c h n e r S., Vorlesungen ber Fouriersche integral, Lpz., 1932. ( . . .,
, . ., ., 1962 ); [2] B o c h n e r S., Math. Ann.,
1933, Bd 108, S. 378410; [3] K h i n c h i n e A., Math. Ann.,
1934, Bd 109, S. 60415 ( . . . , 1938,
. 5, . 4251 ); [4] ., , . ., . 2, ., 1984; [5] . ., . .,
, ., 1961; [6] .,
, . ., ., 1981; [8] .,
., , . , . 2,
., 1975.
; Box Jenkins approach / method
( )
. , ,
,
. ,
xt , t = 1, ..., n ( p, d , q )
(
) ; (1 B ) ( B ) ( xt
d
M xt ) = ( B ) at
.
d 1 , ( p , q ) -
w d , t ; k > q q
,
k > p , p
. p
q - ;
.
rq + i =
j rq j + i
i = 1, ..., p
j =1
.
,
C k = ( k + 1 k +1 + ... + q k q ) 1 +
j =1
2
j
,
C k , k = 1,..., q ,
w d , t = w d , t
w d , t j
j =1
. a
2
.
; ,
93
w d , t
. a t
. . . .
.
[1]- .
q = 0 ,
,
( .,
). q 0 ( , ., [5] ) j k
.
[3]- ( p , q )
( , [3] ). [4]- .
.: [1] ., ., . , . ., . 1, ., 1974; [2] S a i d S.,
D i c k e y D., Biometrika, 1984, v. 71, 3, p. 599607;
[3] A k a i k e H., A r a h a t o E., O z a k i T., Computer Science
monographs, 1975, 5; [4] A k a i k e H., N a k a g a v a T., Statistical Analysis and Control of Dynamic Systems, Dordrecht, 1988;
[5] F r i e d l a n d e r B., IEEE Trans. on Inform. Theory, 1982,
v. IT-28, 4, p. 63946.
## c ; Box Jenkins process , ( ).
; Box Wilson method
( ,
).
; Bogolyubov equations
hierarchy / chain
t
## ( n ) ( (q1 , p1 ) , ..., ( qn , pn ) ; t ) = ( nt ) ( (q1 , p1 ) , ..., (qn , pn ) ),
q j , p j Rd , 0 j h , h 0
, t
( = 0 )
. . . .
:
(n)
( ; t ) = { (n) ( , t ) , H (n) } +
t
+
## d qn+1 d pn+1 { ( n+1) ( , | qn +1 pn +1 ; t ) , Vqn+1 | )}, n 0.
(*)
H ( n ) ( ( q1 , p1 ) , ... , ( qn , pn ) ) =
1
2
|| p
||2 +
j =1
94
U ( || q
1 j1 , j2 n
j1
q j2 || )
U ( r ) , r 0 n , V ( qn + 1 | q1 , ..., qn )
U ( || q
n +1
q j || ) ,
j =1
qn+1 R
q1 , ..., qn R
, { , }
. { t } , ,
(*)
. (*)
[1]- . .
. ( ,
)
. (*)
- ( , [2], [4] ). ( , [5] ).
. . .
( , [6], [7] ).
.: [1] . .,
, . ., 1946, ; ,
. 2, ., 1970, . 90196; [2] G a l l a v o t t i G., L a n f o r d O.,
L e b o w i t z J., J. Math. Phys., 1972, v. 13, 11, p. 2898905;
[3] . ., . ., . ., , ., 1985; [4] . ., . .,
. . , 1974, . 19, 3, . 34463; [5] G u r e v i c h B. M., S u h o v Y u. M., Comm. Math. Phys., 1976,
v. 49, 1, p. 6396; 1977, v. 54, 1, p. 8196; 1977, v. 56, 3,
p. 22536; 1982, v. 84, 3, p. 33376; [6] S p o h n H., Rev. Mod.
Phys., 1980, v. 52, 3, p. 569615; [7] . ., . ., . ., .: . . ,
. 2, ., 1985, . 23584.
; Bolshevs approximation ,
.
## ; Boltzmann distribution r1 + ... + rn = r
, r1 , ..., rn
1 , ..., n
P { 1 = r1 , ... , n = rn } =
1
r!
r1! ... rn ! n r
(*)
.
r n r
n
i , i -
, (1 , ..., n ) . . .
1, 2 , ..., n r1 , ..., rn
(*) ;
.
.: [1] .,
, . ., . 1, ., 1984.
; Boltzmann statistics
( )
. . ( L. Boltzmann; 1868 71 )
;
.
,
N i
. i - Gi 6
h 3 , h
= 6 ,62 10 27
.
. Gi N i , N M
WB ( ... N i ... ) = N !
GiNi N i ! ,
1 i M
N =
, ,
. . .-
.
, , ,
,
WB - N ! .
. .- ,
,
N E -
. .- f ( , t )
( , [2], [3] ).
{ f n }, f n = f ( q1 , 1 , t ) f ( q 2 , 2 , t ) ... f ( qn , n , t ) , . . ( , [2] [4] ).
, . .-
. - . .-
,
,
( )
.
.: [1] ., , . .,
., 1956; [2] Nonequilibrium phenomena, Stud. Stat. Mech., 1983,
v. 10; [3] . ., . ., .: . . . . , . 14, ., 1977, . 539; [4] . ., [ .], .: .
. , . 2, .,
1985, . 233307; [5] ., ,
. ., ., 1981; [6] ., , . ., ., 1978.
; Bonferronis inequality n
A1 , ... , An r
P[r ] -
. ,
W -
;
:
ni = N i Gi = e( i )
kT
E =
k = 1, 38 10 16 erg grad ), T , .
.: [1] ., ., , . ., 2 ., ., 1980; [2] ., , . .,
., 1955.
;
Boltzmann equation . 1872- ( , [1] )
,
:
f ( q, , t )
t
+
f ( q, , t )
+
q
[ f (q, , t ) f (q, 1 , t ) f ( q, , t ) f ( q , 1 , t ) ]
| 1 | B ( 1 , ) d d1 ,
f ( q , , t ) q R 3
v R 3 t , S 2 ,
( , 1 ) 0 , B ( , ) , ,
= ( 1 , ) , 1 = 1 + ( 1 , ) .
,
[1] [4] ).
( ,
( 1 ) j r C jj r p j ,
(1)
j=r
pj =
i Ni ,
P[r ] =
1 i1 <...< i j n
IA
ik
k =1
, p0 = 1 .
n r Pr
:
Pr =
P[i] =
i=r
(1)
j r
C jj1r p j .
(2)
j=n
(1) (2) pr , pr +1 ,
..., pr + k 1 , pr + k , ..., p n ,
. ., (1) k = 1 k = 2 ,
## pr C r1+1 pr +1 P[r ] pr pr Cr1 pr +1 Pr pr
.
, r
=n
P { A } P I A .
i
i =1
i =1
, .
.: [1] B o n f e r r o n i C. E., Publ. Ist. Sup. Sci. Econ. Commun.
Firenze, 1936, t. 8, p. 162; [2] B o n f e r r o n i C. E., .: Volume
95
## di Studi in onore di prof. S. Ortu Carboni Genova, 1936; [3] ., , .
., ., 1984.
; Borel algebra , , .
;
Borel algebra
, .
. .- .
. , .
.: [1] ., , . ., ., 1953.
;
Borel set (
) . . .
.
,
. . . 1 -
, 1 . , .
.
G , F , G , F .
. .- - .
G
( . . ).
. . , .
.
, , .
.: [1] ., , . ., . 1, .,
1966; [2] ., . . .
. , . ., ., 1975.
;
Borel function T , f . a R
f
(( a , + )) T -
, g R
. T
. .
T - , f
( A ) -
. g f , T - . .- . g + f ,
g ( R ), g f , sup ( g , f ), inf ( g , f ) , g
1
x T ,
f = lim f n . .-. . .
n
. ( T , S ) , , T -
96
, T -
.
n
f R - g
. . , x R n
f ( x) = g ( x)
.
T . .- 1 -
; 1 . . .-
,
. . .-
.
. .-
.
.
,
( ). . .
; T Y
, A Y
f 1 ( A ) T
T Y , f , T Y
, B T - , f ( B )
, , Y - .
Y - ( ) . f
( ) ,
f ( B) Y .
.: [1] ., , . ., . 1,
., 1966; [2] ., . .
. . , . ., .,
1975.
,
; Borel strong law of large numbers
( , ) .
( , [1] )
. 1 , ..., n ,...
P { i = 0} = P { i = 1} =
S n
1
, i = 1, 2 ,... .
2
1 2
k =1
. . [1]- ,
Sn n 1 2
. .-. ( f n ) nN , T - . .
= Rn
. , R - ,
). : f , T -
1
. .
. g o ( T , S )
, f , ( , A R f
n .
. ( G. Hardy, 1914 ) . ( J. Littlewood )
,
lim
Sn n / 2
n ln n
<
1
2
. 1922- . .
; ,
P lim
n
Sn n / 2
1
=
= 1.
n ln ln n
2
, .
.: [1] B o r e l E., Rend. Circolo mat. Palermo, 1909, v. 27,
p. 24771; [2] .,
, , . ., ., 1963.
;
Borel mapping , .
, ; Borel field of sets , M M -
,
, .
## , ; Borel field of events , , ,
( )
( ) A .
; Borel
measurable function , .
; Borel
model ,
.
- P ( A ) ,
.
. . ,
.
. . .
. . . .- -
( , [1] ) ( , [2] ) .
.: [1] B o r e l E., Rend. Circolo mat. Palermo, 1909, v. 27,
p. 24771; [2] C a n t e l l i F. P., Atti Accad. naz. Lincei, 1917,
v. 26, p. 3945; [3] . .,
, ., 1987; [4] .,
, . ., ., 1962.
; Borel Cantelli Lvy
lemma , .
; Borel Tanner distribution m = k , k + 1, ... ( k
)
pm = P { = m } =
k
m m k 1 e m m k ,
( m k )!
## 0 < < 1, k > 0
( , )
. . . .
; Borel
measure T
. ( T ) = 1 , . .
. . .-
, , .
. . . .,
,
, ,
n
. . . R -
. . .
.: [1] ., , . ., ., 1953.
-; Borel zero-one law
, .
; Borel theorem , ,
## ; Borel Cantelli lemma :
{ An , n 1} P ( An )
,
P lim An = P
n
I UA
k =1 n = k
= 0.
A1 , A2 , ... , lim An n
0 -, 1 - . ,
, :
P(A )
n
: 1) k = 1 , = 0,3 = 0, 7 ;
2) k = 5 , = 0,3 = 0, 7 .
;
k - .
; empty blocks test F1 n1 F2
97
n2
0(1) ( n1 ,
n2 )
.
. . .
F1 F2 - .
( , ):
u = 20(1) ( n1 , n2 ) + 2 ( n1 + 1) + =
= 2 0( 2) ( n2 , n1 ) + 2 ( n2 + 1) + ,
, 0 1 , 0 ( n2 , n1 ) ( 2)
.
.: [1] ., , . .,
., 1967; [2] . ., .: .
. .
, . 26, ., 1988; [3] M o o d A. M.,
Ann. Math. Statist., 1940, v. 11, p. 36792.
## ; empty cells test
,
.
, x1 , x2 , ..., xn F ( x ) .
## z 0 = < z1 < ... < z N 1 < z N =
, F ( zk ) F ( zk 1 )
. 0
= 1 N , k = 1, 2 , ..., n
>C
## ( ) ( ); method of correction bias, jackknife method
, ,
,
,
,
.
. [1]- .
. . .-
. . . .-
, , . .,
, . . .-
. ,
.
. . .
.
x1 ,..., xn F ; , n ( x1 ,..., xn ) , . n
n i = n n ( n 1) in 1 ,
in 1
. -
.
n -
;
.
r
1 N -
0 ( n , N )
. 0
.
. . .- r = cr 0 0 + cr1 1 + ...
+ cr r r ; k k
. .
.: [1] . ., . ., . ., , ., 1976.
98
= n1 ( x1 , ..., xi 1 , xi +1 , ..., xn ) , xi
*n = n 1
ni
i =1
.
n n
M ( n ) = b1 n + b2 n 2 + o (n 3 )
*
, n -
, n
. . .
.
0 ( n , N )
i = 1, ..., n
.
. . . n -
( z k 1 , z k ] ,
. . . .-
, 0 C , 0
M ( *n ) = b2 n 2 + o (n 3 ) .
. . .- m ,
. n -
. . .
n* = ( n ( n 1) ) 1
ni
*n ]2
i =1
.
.: [1] M i l l e r R. G., The jackknife a review Biometrika
61, p. 117, 1974; [2] Q u e n o u i l l e M., Biometrika, 1956,
v. 43, 34, p. 35360; [3] T u k e y J., Ann. Math. Statist.,
1958, v. 29, 2, p. 614; [4] ., , . ., .,
1988.
; Boze Einstein model , .
; Boze Einstein statistics
- [ ( , , )
] . . ( S. Boze )
. ( A. Einstein ) 1924- . - .
n . - r
n
Cnr + r 1
C nn+r11
( ,
) r -
n C rn11 ( r > n) .
,
.
., { n p } ; n p
p ,
= 0 , 1, 2 ,... .
; boson
space , .
*
; partition ,
-
-
. Bi , Bi
i j , Bi I B j = , i, j = 1 , n ,
, ( i = 1, n ) ,
n
U1 B
= ,
i=
Bn = { B1 , ... , Bn }
. Bi - Bn
A (Bn )
Bn
n
2 :
## A (Bn ) = { { B1}, ..., {Bn }, {B1 U B2 }, ... , { Bn1 U Bn }, ..., , } .
n = 2 , B2 = { B, B } A ( B2 ) = { { B }, { B } ,
, } .
( , F , P ) , Bi F , i = 1, n , Bn =
. . . n p
= { B1 , ... , Bn }
V
. ,
- Bn .
i = pi2 2m Gi
, , Gi >> 1 .
{N i } ; N i
n p - .
,
W (Ni ) =
(G + N
i
i Ni
1)! N i !( Gi 1)!
ni = N i Gi = ( e
Ai =
i =1
.
, .
.: [1] . ., , ., 1980.
)
; partition o f
n u m b e r , .
E =
( )
.
A (Bn )
( i )
1)
K K ; K partition
{ T , t Z R }
t
1) t 0 T t ,
VT
t
= ( mod 0 ) ,
- ;
= 1, 38 10 16 erg grad
natural
2)
, , k ( k
),
T .
. . .-
, . . .-
,
.
.: [1] ., , . ., .,
1966; [2] . ., . ., . .,
. 2- ., .,
1997; [3] ., , . ., . 1, ., 1984; [4] . .,
, .: , . 2,
., 1970.
3)
T t = ( mod 0 ) ,
-. K . t < 0
. K .- K
.
/
; dizjoint familis of
measures -
. ( X , A ) .
G 99
=
( X X ) 0 ( X ) = 0
A , , ( { X , } ,
) .
( X X ) = 0
{ X , } ,
X I X
= =
{ ,
, A
()
.
. X [ 0 , 1] [ 0 , 1] , A
= [ 0 , 1] U [ 2 , 3 ]
. [ 0 , 1] , 0 x 1 , y =
, [ 2 , 3 ] ,
x = 2 , 0 y 1 .
X , { , [ 0 , 1] U
U [ 2 , 3 ]} , { , [ 0, 1]}
,
, n
R - , n
l
,
1- m
.
( , [1] ) . . n pnk , 0 < p nk < 1 , k = 1, ... , k n
. , ., ,
,
. (1)- Ynk , m = 1 ,
Z nk nk
k - , nk = nk ,
k = 1, ... , k n , ,
n pnk ; (2)-
zn = n .
. .-
; (1)- f nk k
n
( , [2] ).
cr r -
r=0
[1]- ( ) ,
. . .-.
X , c . - X - 2c
r =
kn
I (
nk
= r)
k =1
[ I ( A) , A ]. . .-
-. X
[ 0 , 1] - -
. . .- 2 - , X - . . .-
c - . , X - . . .
, , . . . .
.: [1] . ., . . ,
1984, . 113, 2, . 27375; [2] . .,
. ., , ., 1980; [3] . ., , .,
1979.
. . .-
[2] [6]- .
, n , k n
( , [3] ), 2
c
; decomposable statistic
Xn =
kn
f nk ( Ynk , Z nk )
(1)
k =1
, Ynk Z nk
, l m , f nk ,
R l + m
( ). Tnk = ( Ynk , Z nk ) , k = 1, ... , k n
Pn
nk = ( nk , nk ) , k = 1, ... , k n
- , ,
n =
kn
nk
= yn , n =
k =1
100
kn
k =1
nk
= zn .
(2)
. .- .
X n . .- ( f n k ( n k ), n k ) , k
= 1, ... , k n
, ,
, . .-
, Pn - . .-
, . .- .
.: [1] . ., . , 1970, . 192,
5, . 98789; [2] . ., . ., . ., .: . . . . ,
. 22, ., 1984; [3] . ., . ., . ., . . - , 1986, . 177; [4] . ., .: . . . . ,
. 26, ., 1988; [5] . ., .: . . . . , 1985, . 30, 1, . 17074; [6] H o l s t L.,
Ann. Probab., 1981, v. 9, 5, p. 81830.
;
separability condition , .
;
separation principle
, .
, w = (w t ) t 0 -
. { k , k
( M k
= ( Yt ) t 0
u = ( u t ) t 0 X s , 0 s t -
( Ft X ) ~ = (w
~ )
, X = ( X t ) t 0 w
t t 0
Ft
~
d X t = A ( t ) Yt d t + B ( t ) d w
t
1
n
1
2 P ( T )
1
2 P ( T )
l ( y) exp
L ( t,
( y T ( Y )) 2
2 P (T )
dy +
( y T ( Y )) 2
y, ut ) exp
2 P( T )
dy dt ,
( t ( Y ), P ( t ) ) t 0
A(t ) P (t ) ~
d wt ,
B (t )
2
A ( t ) P( t )
dP ( t )
,
= 2 a ( t ) P ( t ) + b 2 ( t )
dt
B( t )
w = (w t ) t 0
t
wt =
dX s A ( s ) s ( Y ) ds
;
B(s)
,
Y
(Y )
= ( t (Y ) )t 0
1
n
Mk
. . . . . . .
1933- . . ( , [6], . 99 100 ).
.
17- 18- . [1]-
, n
1
n
>0
X s , 0 s t -
Yt - Yt
.
.: [1] W o n h a m W. M., SIAM J. Control, 1968, v. 6, 2,
p. 31226.
; law of large number
(1)
n A ,
k , k = 1, 2 , ..., k - A - , , 1 0
p 1 p .
. , A
( , [2] ). p k , A
k - , k
p = ( p1 + p2 + ... + pn ) n
,
d i (Y ) = ( a ( t ) t ( Y ) + c ( t ) ut ) d t +
P { | n p | > } 0 ,
VT (u ) =
L ( t , Yt , ut ) d t
=M
0 < p < 1 p A -
.
( X 0 , Y0 ) ,
n p n
~ w -
, w
Vt (u ) = M l ( YT ) +
dYt = ( a ( t ) Yt + c ( t ) ut ) d t + B ( t ) d w t
< + )
1}
>0
= 1, 2 , 3 ,...
P{ | n p | > } 0
(2)
. , . . ( 1846 ) ;
. (2)-
. .
, . . .
.
: M n ( M i - ) p -, p - .
, k
( ) .
. . ( 1867 )
, 1 , 2 , ..., n , ...
101
>0
1
n
1
n
k =1
> 0
k =1
(3)
. . . M k
2
, -
= M k2 ( M k ) 2
, k - D k
1
n
k ,
. . . (3) ,
. . ( 1929 )
.
. . .-
.
. 1 , 2 , ..., n , ...
C1 , C 2 , ..., C n , ... ,
>0
P{ | n Cn | > } 0
k =1
= o ( n 2 ) . , . .
(4)
( { n C n }
. . . (
)
. . .- ( , [3] ):
; (4) Cn ,
{ k , k 1}
), 1 , 2 , ..., n , ...
n
1
D
k 0
2
n
k =1
lim P
n
1
n
>0
n
## 1,1 , ..., 1,k1 ,
2,1 , ..., 2, k2 ,
1
n
k =1
k =1
..
n,1 , ..., n, kn ,
< =1
. , ( , [3] ) . . . .
. . .-
[
(3)
2
;
; , ]. . . . . . . k .
k ,
k
: n,
k ,
0 ,
( n, kn - ,
).
. k1 = 1,
k 2 = 2 , ..., n, k = k / n
.
n = n,1 + ... + n,kn
k =1
n, k =
= m n . m ,
.
, 1 , 2 , ..., n , ... -
Cn
k M k Ln ,
k M k > Ln
? . . ( 1928 ).
n,k -
. n,k ,
n - M n M n
(3) .
102
>0, L>0
1+
<L
n ,k
n ,k
1
>1
. n
kn
P{ |
n,k
0,
n, k =
, Ln .
., ,
. . .-
: n
n, k
| > 1} 0
k =1
n
kn
~2
M
k =1
n, k
n -
. 1 , ..., n , ... ,
~
M n,k
, j i
. .
M n,k ,
j - n
kn
. n
k =1
, Cn
= M n
. . .- (3)
. { i }
( j i ) i -
i -
( ). . . . : n
. n n
1
M
n
n P { | 1 | > n } 0
(5)
.
(5) . ., n
. n
n t / n
- ), (5)
= e t -
, n - n
t
i , e -
, n -
.
. . .
. .,
n - n 1 , 2 , ..., n
: n
= 1 +
... + n ,
, 1 , 2
. n n
, n
1 1 2 x 3 2
e
x , x > 0,
p ( x ) = 2
0,
x0
. , i
, n n
( ) n [
, . . . , n n - o (1) ].
. . .- ( , )
, i j i j , .
. . ( 1907 )
P
n
( -
1 + ... + n
a = lim M k ; , n
1 (1 + x )
e
k a 0
k =1
k =1
a > 0
. . . .-
. . :
L , R , ( n) , n
D j < L ,
R ( i , j ) ( | i j | ) , { n }
(3) . . .
. { n } ,
lim
1
n
=0
j =1
, R j
= R (i , i + j ) .
. ,
. :
, . (
). ( , ).
d , ,
. ., { n } ,
M n ( x , x - )
,
1 + ... + n
n
>0
M 1 > 0
103
. . .
. ,
1
T
( t ) dt
( , [1] ).
, k , k 1 F (x ) ,
,
sup
, ( t )
( , ., [11] ).
,
.
. .
( 195155, , [10] ).
. .
. n, 1 , ..., n, n ,
## n2, 1 + ... + n2, n = n ,
>0
. f ( n,1 , ..., n , n )
. . . ; ,
n [ . ( 1925 ) ,
, , ].
.: [1] B e r n o u l l i J., Ars conjectandi, opus posthumum, Basilae, 1713 ( . . ., ,
., 1986); [2] P o i s s o n S.-D., Recherches sur la probabilit des
jugements en matire criminelle et en matire civile, prcedes des rgles gnrales du calcul des probabilits, P., 1837; [3] .
., . . ., . 2, . ., 1947; [4] . ., , 4 ., ., 1924; [5] . .,
, 4 ., . ., 1946; [6] .
., , 2 ., ., 1974;
[7] . ., . ., , . .,
1949; [8] ., , ., ., .,
1956; [9] .,
, . ., ., 1965; [10] . ., , .: .
. , ., 1955, . 54174; [11] ., , . ., ., 1962; [12] U s p e n s k y J. V., Introduction
to mathematical probability, N. Y. L., 1937.
; law of large numbers i n B a n a c h s p a c e s
. ,
, ,
. , 1
. . . .-
, 1
.
. . .-
.
. . . . . . .
104
< x <
Fn ( x) F ( x) n
0
1 ,
Fn ( x) =
1
n
I {
< x}
k =1
, I { k
< x} ,
< x}
{ k
. ,
( I { k < x } , < x < ) ,
.
. . .-
. ( , [2] ) . , { k , k
1}
( B, )
,
,
1
n
k =1
1 , , M 1
<
1
n
lim
= M1 ;
k =1
M 1 , 1 .
. . .-
, . . .- .
( )
. . .-
( , [3] [5] ).
. . .-
.
.: [1] G l i v e n k o V., Atti Accad. naz. Lincei., 1928, v. 8,
p. 67376; [2] M o u r i e r E., Ann. Inst. H. Poincar, 1953, t. 13,
p. 161244; [3] .,
, . ., ., 1965; [4] H o f f m a n J r g e n s e n J., P i s i e r G., Ann. Probab. 1976, v. 4, p. 58799; [5] T a y l o r R. L., Lect. Notes in Math., 1978, 672; [6] . ., . .,
, ., 1989.
; law of large numbers f o r n o n - s t a t i o n a ry random processes
1) ( t )
lim
1
T
(t ) d t
= ,
(1)
1
T
lim
( t ) =
(1)
t =1
,
( , ) 1
[
(1) (1)
]. t
, M ( t )
## < , 0 < t <
M ( t ) ( s ) = B (t , s )
t , s - , ., C <
T
> 0, S > 0
T S -
T S
1
TS
lim
T ,
S
B ( t , s ) dt ds
,
0 0
1
TS
T S
B ( t , s ) dt ds < C
0 0
, . . .
( , [1], [2] ).
. . . ,
,
( t ) , (
= Z ( + 0)
Z ( 0 ) , Z ( ) , ( t )
,
(1) (1)- 0 t T
S t T , T S
( , [3], [4] ).
. . .
( t )
( , [5], [6] ).
2) ., t ,
= lim
1
T
F ( x ) ,
M ( t ) d t
= 0 ( , [5], [6] ). ,
( t )
.
.: [1] K a w a t a T., Trans. Amer. Math. Soc., 1965, v. 118,
6, p. 276302; [2] K a w a t a T., .: Keio mathem. seminar
reports, 1, Yokohama, 1973, p. 123; [3] .,
, . ., ., 1962, . 49699; [4] .
., . ., 1959, . 4, . 3, . 291310;
[5] N a g a b h u s h a n a m K., B h a g a v a n C. S. K., Sankhya,
Ser. A., 1969, v. 31, p. 421424; [6] R a o M. M., .: Developments
in statistics, v. 1, N. Y., 1978, p. 171225; [7] ., ., , . ., .,
1969, . 99103.
; law of large numbers
f o r M a r k o v
p r o c e s s e s
-
. 1 , 2 , ... , i , i = 1, 2 , ...
( i , Ai ) .
(i , Ai ) = (, A ) 1 , 2 , ...
. : 1) 0 < ( ) <
>0
, ( A ) <
p ( , A )
< 1
, ,
X ( ) ( d ) < .
lim
= [ ( t ) M ( t )] ,
n = sup
1 k <
.
T T
B (t , s ) d t d s = 0
0 0
( , [7] ). ( t ) ,
= 0 , , , = 0 ,
= 0 F ( , ) - -
1
n
X ( ) = X ( ) ( d )
k
k =1
.
1) ; ,
.
,
:
M ( t ) = 0 = 0
2) - .
, A - , X ( ) , -
(2)
0 t
[ (2)-
1
T2
M ( t )
lim
= 0 -
, , , F ( )
, , ( t ) -
].
( , [3], [4] ); (t ) -
sup
P ( A | B) P( A) .
A Ak + n , BAn , P ( B ) >0
k - , k = X k ( k ) .
[3]- .
Xk ,
k =1
12
k
<
D k / k 2 < ,
k =1
105
lim
1
n
M k ) = 0
(*)
12
k)
<
12
k
n 2
k =1
=0
k =1
k =1
<
12
k
, k
k =1
( , ).
lim n1 n 2
=0
k =1
, (*)-
( , [2] ).
.: [1] . ., , . .,
., 1956; [2] S t a t u l e v i c i u s V., Proc. 2-nd Japan USSR
Sympos. on Probab. Theory, 1972, v. 1, p. 13852; [3] . .,
. .- . , 1984, . 3, . 5077.
; law of large numbers f o r s t a t i o n a r y
r a n d o m p r o c e s s e s 1) , ( t )
lim
T S
1
T S
(t ) d t
(1)
lim
T S
=0
1
T S
T 1
(t )
(2)
=0
= M (t )
, ( t )
, (*) .
(1
b ( )
, [
lim
k =1
T 1
1
T
lim
(1)
t =S
. [ (1) , (1)
], (
) .
. . .- . . ( , [1], [2] ).
,
]. . . .- . . , ( [4], [5] ); (2)
(2) ( t ) = 0
( , [3] ).
. . .- R
Z k
.
.: [1] . ., . ., 1933, . 40, . 12428;
[2] K h i n t c h i n e A., Math. Ann., 1934, Bd 109, S. 60415;
[3] ., , . ., ., 1956;
[4] S l u t s k i E. E., .: Actualits Sci. et Industr., 738, P., 1938,
p. 3355; [5] . ., .: , .,
1960.
## ; law of large numbers
f o r r a n d o m s e t s ,
; law of large numbers f o r
r a n d o m f i e l d s
. : , , 1
(
) . ., ,
X ( t ) R m
= 0 . -
, M X ( t )
B ( h )
= M X (t + h ) X (t )
B(h) =
...
i hk k
e k =1
F ( d ) ,
( t ) = 0 Z ( )
- ( , ., [3] ).
Bm - . X ( t )
F( ),
2) = M ( t ) , ,
[ , ]m
X (t ) =
b ( ) = M [ ( t + ) M ( t + ) ] [ (t ) M (t )]
... e
i t k k
k =1
z ( d )
, z ( ) , Bm - -
lim
1
T
106
b ( ) d
0
=0
(2)
M z ( S1 ) Z ( S 2 ) = F ( S1 I S 2 ) ,
S1 Bm ,
S 2 Bm . z { ( 0 )} - ,
lim
1
( 2 N + 1) m
|X j | N,
j =1,..., m
X (t )
= ( 0 , ... , 0)
. 1 M X (t ) - ( M X (t ) =
P { | S n n An | } 1
, ( 0 ) , R - 0
= 0 ) , M Z { (0)} =
= F {(0)} , F {(0)} = 0,
...
log log
j =1
1
|j |
( , [1], [2] ). X (t )
, X (t ) - ( ,
[3] ),
M | X ( t ) | ( ln + | X ( t ) | ) m1 < +
. . . .-, , .
[4] [5]-
.
.: [1] . ., . .,
1977, . 22, . 2, . 295319; [2] . ., .
. , 1981, . 25, . 2940; [3] S m y t h e R. T.,
Ann. Probab., 1973, v. 1, p. 16470; [4] . .,
, ., 1980; [5] . ., . . , 1984, . 30,
. 3438.
; strong law
of large numbers (
) ,
,
1
.
,
S n
(1)
= 1 + ... + n .
An - ,
S n n An 0
S n +1 (n + 1) An +1 , ...
>0
, , 1
, , .,
.
. . . . . [1]-
(
; ,
). ( 0 , 1) (
) ( , ). ., - ,
n ( )
2 n n ( ) -
n =1
0 1 1 2 . S n ( )
k ( )
k =1
n , S n ( ) n
. ,
S n -
( 1- ) 1 2
. .
, S n n - ( 0 , 1)
- 1 2 - . ,
- 10 0 , 1, ..., 9
. , 1 10 -
n ( 0 , 1)
- 1 10 - .
, r
r
(3)
n 1- , (1) . . . .- , .
, -
,
. (1) . . . .- , An - ,
n ln ln ln n 1 2
- 1 10 -
(2)
n 1- ,
(1) . . . .- , .
:
0
S n n An n ,
(2) . . . . .
1 , 2 , ..., n , ...
. . ., (1)
n 16
, (4)
An = 0 , An -
F ( d ) < +
, X (t )
(4)
.
. [2]- (
) . . . .-
.
Bn, k =
j M j
j =1
,
:
(B
n, 4
+ Bn2, 2 ) n 2 < .
n =1
107
. .
. ,
n 0 ( n )
P { | S n n An | > n } <
.
(5)
1 . , n -
S n n An n
. . . .- (
).
i k ri , k ,
< n 2 k +1 k
, Z k = 2
n
n -
(k )
(k )
, :
> 0
P{ | Z
m Zk | > } < ,
>0
(7)
m Z k , Z k - ( , [6] ). (7)- -
. .,
n = O ( n ln ln n ) n -
, (7) :
. . .
, . . .
. . . .-
exp {
k =0
max S n n An > n
nk < n nk +1
. 2
1 , (3) .
. (5) ,
.
.
. . . .- . . . .
. . . [3], [4]- An = M ( S n n )
|ik | = n
= 2 k
(5)
n =1
cn = sup | ri , k | , Cn =
. . . . (5)
, nk
> 0
DZk } <
Bn, 2 Cn = O ( n 2 ) .
. i -
. . . .-
:
. . ( 1930 )
,
( 1933 ) . (1)
D n n 2 <
(6)
= M ( Sn n )
(1) -
n =1
. . . .- An
. (6)
, bn
bn
n2
, . . . .-
D n
= bn n -
. (6) (
. . . .-
)
. m n , n - .
P{ |
m n | > n } . . . .- -
. ,
1 n m n < n .
, . . . .- -
.
108
D Z k = 2 2 k
(k )
. . . .-
( , [7] ). ., R ( n ) n
:
ln 2 n n | R ( n ) | , ( 0 + ... + n ) ( n +
+ 1) M 0 1 . . . . .,
Y = lim ( 0 + ... + n ) ( n + 1)
n
1 ( Y
0 -
. Y M 0 -
). . . . . .
. . . .- ( , [9] ).
. S n n - An -
.
.: [1] B o r e l E., Rend. Circolo mat. Palermo, 1909, v. 27,
p. 24771; [2] C a n t e l l i F. P., Atti Accad. naz. Lincei, 1917,
v. 26, p. 3945; [3] . .,
## , ., 1927; [4] . ., C. r. Acad. sci.,
1928, t. 186, p. 28587; [5] . ., C. r. Acad.
sci., 1930, t. 191, p. 91012; [6] . ., .
. . ., 1950, . 14, . 52336; [7] .,
, . ., ., 1956; [8] .
., , ., 1972; [9] ., , .
., ., 1965.
. . ( , [10] ) 1 , 2 , ...
. . . .-
. , . .
( , [11] ) , i
. . .
.
hk ( ), > 0 k = 1, 2 , ...
(t )
M ( t )
= M (t ) .
2) (t ) ,
(t ) [
(1) (2)-
, = M ( t ) ]
lim
, Z k (7)-
lim
hk ( ) , . ,
{ P (
>0
> n ) + exp ( 1 2n +1 hn ( ) )}
(8)
n =1
.
. . (7) . .
Z k Wk
,
. ,
- j ( (6) (8) ) . . . .-
.
[10]- ,
.
.: [10] . ., . ., 1972,
. 17, . 4, . 60918; [11] . ., .
., 1958, . 3, . 2, . 15365.
; strong law of
large numbers f o r s t a t i o n a r y r a n d o m p r o c e s s e s 1) (t )
lim
T S
1
T S
(t ) d t
(1)
lim
T S
1
T S
(t )
b ( s) = 0
s = 0
(3)
b ( s ) = M [ ( t + s ) M ( t + s ) ] [ ( t ) M (t )]
( ,
). ( t )
M (t ) < ,
(1) (2) 1 ,
, M (t )
<
[ ( t )
],
= M (t )
(3)-
[ ( t ) . . . .-
].
, , ( t ) ,
,
, [ ( t )
], (1) (2) 1 [ ( t )
]. [1]- ;
( , [2], [3] ) ( ) ( t ) . . . .-
, , , (3) 1 T 1 T
>0
(3) , . . . . ,
( , [4] ). [4], [5]- , 1 T ( log T ) / T , a > 3 (3)
a
( t )
T 1
b ( s ) ds = 0
Wk = max ( | m | m : 2 k < m 2k +1 ) .
. . . .-
1
T
= .
. . . .- ]
## log M exp ( hZ k ) I (Wk < ) = 2
h
, hk ( )
<
( t
, , )
. . . .- .
1) [ ( t )
(1)
t=S
1 . . . . .-
; ,
, . . . . .
( , [6], [7] ), C > 0 > 0 ,
s -
## b ( s ) < C ( log log s ) 2 -
(t )
109
( log log s )
.
2
b(s)
= ( n 1) 1
(t )
.
.: [1] B l a n c L a p i e r r e A., T o r t r a t A., C. r. Acad.
sci., Ser. A B, 1968, t. 267, 20, p. 74043; [2] L o e v e M.,
Rev. Sci., 1945, v. 83, p. 297303; [3] B l a n c - L a p i e r r e A.,
B r a r d R., Bull. Soc. Math. France, 1946, t. 74, p. 10215;
[4] . ., . ., 1964,
. 9, . 2, . 35865; [5] . ., .
., 1966, . 11, . 4, . 71520; [6] . .,
. ., 1973, . 18, . 2, . 38892;
[7] . ., . ., 1977,
. 22, . 2, . 295319.
; strong law of large numbers f o r r a n d o m
s e t s ,
.
## ; grand canonical ensemble
,
, ,
- ( , ). . . . . .
, .
,
( z, ) =
mn1
tr I z
a = n 1
k n = mn n < 1 .
xk ,
, ,
( z , )
) z 1
( ,
, ,
1 k n + k n ( , ) = z 1 .
. . . . : cn
lim cn2 ln [ n ( n mn ) 1 ] = 0
, cn ln det
## cn1 { ln det + ln [ ( n 1) mn ( Anmn1 ) 1 n ( n mn ) 1 ] }
m
= n ... ( n m + 1) ;
, An
T
( a1 a2 ) ( a1 a2 )
( a1 a2 ) T 1 1 ( a1 a2 )
n1 + n2 2 m m m
n1 + n2 2
n1 n2
, m
1 2 ,
M 1 = a1 , M 2 = a2 ,
1 = ( n1 + n2 2 ) 1
, z , I , mn , .
x1 , x2 , ..., xn ( N ( a , ) )
k =1
. p -
( xk a ) ( xk a ) T ,
k =1
, .
## . ; large dimensions effect -
n2
p=1
n1
( xk a1 ) ( xk a1 ) T +
k =1
( y p a2 ) ( y p a2 ) T ,
xi yi , 1 2
,
a1 = n11
0<c<
= n21
) - ,
, ( z ,
p lim [ ( z , ) M ( z , )] = 0 ,
n
( z )
mn
[ z
Im z 0
= M ( z , )
( 1 k n + k n z ( z ) ) ]1 + o (1)
p =1
110
a2 =
n2
yi .
i =1
, . . . .
( z ) = mn1
x,
i =1
lim mn n = c ,
n1
( a1 a2 ) T ( I +
+ ) 1 ( a1 a2 ) -
( a1 a2 ) T ( I + 1 ) 1 ( a1 a2 )
( n 1 + n 1 ) tr 1 ( I + 1 ) 1
1
, ,
1 k m + k m m1 tr ( I + 1 ) 1 = 1 , > 0,
km = m ( n1 + n2 2 ) 1
.
. . . .- .
A x = b , A = aij ,
## j = 1 , ..., m , i = 1, ..., n , b = ( b1 , ..., bn ) ,
x T = ( x1 , ..., xm ) T , A A
.
P { Fn G1 } 0 , P { ( Fn F ) n x G2 } 0
, (*)
.
. . ( ) G , H
x = [ I n + AT A ]1 ATb , > 0
Ax
=b
K (G , H ) =
. . . .- . X -
R ( G, H ) =
c R m b
n
( A T b, k ) ( c , k ) <
k =1
, ,
lim
sup
n k = 1, ..., n
n 1 k ( AT A ) < ,
lim m n 1 < 1 ,
.
A = X
1
p lim ( x , c) +
n
2
f ( z , y ) = n 1
[
k =1
+ k ( z ) (1 + f ( z , y ) ) ]
,
+ o (1)
k ( z ) K T ( z ) K ( z )
K ( z ) = n 1 2 A + z b c T n1 2 .
.: [1] . ., . ., 1987,
. 32, . 2, . 25265; [2] . ., , ., 1988; [3] . ., .
., . ., ,
., 1982.
; large
deviation f o r e m p i r i c a l d i s t r i b u t i o n f u n c t i o n
. Fn = Fn (t )
F = F ( t ) ( x1 , ..., xn ) , G1 , G2
. G1 G2
, n , x
dG
( t ) dH ( t )
dH
R (G , H ) = lim
K ( H + G , H )
K(H, H + G)
= lim
0
2
2
.
F
M1
M 2
G1 G2
ln P { Fn G1 } ~ n inf K ( H , F ) ,
HG1
ln P { ( Fn F ) n x G2 } ~ x 2 inf R ( H , F ) .
HG2
M1F M 2F .
y ( 1 + 2 f ( z , y ) ) + 2 1 +
n
1 1
; G (t ) H (t ) -
f ( z , y ) z = 0 dy = 0
z
1
2
, f ( z , y )
m
dG
( t ) dG ( t )
dH
, -
. var G = 0 , F ( G , H )
k k , A A -
ln
N ( aij , 2 )
lim n 1
(*)
= x ( n )
., F [ 0 , 1] - , M 2
inf K ( H , F ) = inf K ( H , F )
HG 0
HG
, G
G -
G - .
. . ( x1 , ..., xn ) .
(*) x <
ln n . .-
.
. .-
( ) Gi
.
, .
.: [1] . ., . ., 1957, . 42, 1, . 1144;
[2] . , . ., 1967, . 12,
. 4, . 63554; [3] . ., . .,
. . ., 1978, . 19, 5, . 9881004; 1980, . 21, 5,
. 1226; [4] G r o e n b o o m P., O o s t e r h o f f J., R u y m g a a r t F. H., Ann. Probab., 1979, v. 7, 4, p. 55386.
111
; large deviations f o r a r a n d o m w a l k
. S = ( S 0 , S1 , ... )
= ( s0 , s1 , ...)
,
as =
, , V s
aR
= ( a s0 , a s1 , ...) , a = { a s : s } .
x x
P {S x } 0
1. , inf g ( t ) > 0
B1 = { max ( s n ( t ) x g ( t ) ) > 0 }
t [ 0,1]
.
2. , inf g ( t ) < 0
B2 = { max ( s n ( t ) x g ( t ) ) < 0 }
t [ 0,1]
.
< 0 , 0 > 0
(1)
,
.
S n = 1 + ... + n , n 1 , S 0 = 0
( ) = X M e X1 < ,
(4)
, 1 2 . ,
, . .
(2)
( ) = sup ( ln ( ) )
. .
.
(2) . .
(1) ; S = S (n )
.
t ( a ( t ) / t ) = ( ) a ( t )
( S k , k n = n (x))
= n n .
(2) . . .
( )
. g (t ) , [ 0 , 1]
## < 0 < g + (0) ,
g (t ) < g + (t ) -
= 0 , D k = 1 (1)
. M k
sn ( k / n ) = S k
g (0) <
n,
k = 0, 1, ..., n
s n ( t ) [ 0 , 1]
t , - , .,
( 0 , 1)
( ) = 2 2 a ( t ) = t .
, 1
g
x g (t )
= { g ( t ) : g ( t ) g ( t ) < g + (t )}
g ( t )
, x , n
P { sn ( ) x } 0 .
(3)
g (t ) ,
x , n
P { sn ( ) x } 0 .
. .-
.
. .,
P { s n ( ) x } = [ P { max ( sn ( t ) x g + (t ) ) > 0} +
0 t 1
## + P { max ( x g (t ) sn ( t ) ) > 0}] ( 1 + o (1)) .
0 t 1
. . .
112
t - .
- , a ( t )
n
g - . g ( t )
P ( B1 ) ~ c n e
s n ( t ) -, .,
.
. a ( t ) -
n ( g )
(5)
a g ( t ) - x g ( t )
n -
. ., g (t ) a g ( t )
t g , q
t g g (t ) a g ( t ) -
,
= 1 2 1 ( q + 1) .
g (t ) a g ( t ) -
( q = ) , = 1 2 .
( t , g )
.
, a (t )
. (5) , 0 , ,
;
( t g , g ( t g ) ) .
P { S n = } > 0 P { B1 , S n = } , (
P { S n = } = 0 , P { B1 , S n } ,
).
=0
0 < t <1
= t (1 t ) .
(5) , , ,
n
(5)- n - n 1 e 1 g
, 1 = 1 /( q + 1) ( , [1]- ).
,
.
P ( B2 )
.
g m = min g ( t ) < 0 , h (t ) , ( 0 , 0 ) (1, g m )
g (t )
ln P ( B2 ) ~ n
, A A -
= z ( n ) . w ; y = y ( n ) ,
y = o ( z )
, z
P {1 > x } P {1 < x } x ( ., ) , . .
. . .
. ( 0 , 0 ) ( t , a ) s n (u ) ( ) -
<ut
.
( ) -
>>
ln n
P ( B1 )
P{1 > x n g1 ( t )} d t ,
(6)
< , r <2
, ( A ,
ln n )
. (4) ,
( A , n1 6 ) . , 1)
A - ,
2) (
A ).
A A y = o ( z ) ,
ln P { s n y A ) ~ ln w ( y A )
(8)
, ( A , z )
, . A -
ln ln n = o ( z ) ,
(A, z )
= o( y)
. (4)
, ( A , n )
P ( B2 ) ~ n P { 1 < x n g m }
M 1
2) ln w ( [ ( y A )] ) ln w ( y A ) ( 1 + o (1) )
ut
B , B , ( B ) .
.
, A 1)
g1 ( t ) = min g ( u )
2) y = o ( z ) w ( [ ( y A ) ]1 z ) = o ( w ( y A) ),
d ( u ) = a d (u ) -
. x
P { sn y A } ~ w ( y A ) , A A
1) y , A A ln w ( yA ) c y 2 ,
P ( B2 )
( , [2] ).
k (4) ,
. x P { s n x A} 0 .
, . , A z
:
A h (t ) .
P ( B1 ) ~ n
B
.
, A B g (t ) = 0
, . . ( A , z )
h( t ) d t
n
, u d ( u ) = 0 ,
k n
, , S n -
.
(3) C [ 0 , 1] -
( t , g )
A - .
= max S k ( , [4] ) -
= min { t : g ( t ) g m } > 0 . x c ln n ,
. , , S n
( t , b ) = t ( b / t ) + (1 t ) ( b /(1 t ) )
b (t )
P ( B2 ) ,
(6) (7)
( , [3] ).
g (t ) -
( t , b ) = sup ( t , ) 1 ( )
( , [5] ).
(7)
113
(8)- . ,
ln P { s n y A} ~
y2
inf
2 2
( x A P , x f
( g (t )) 2 dt , g A I C1 ,
(9)
C1 , [ 0 , 1] - .
(8) (9) y ~ c n , j , .
ln P { s n x A} -
. ,
x ( , [6] ),
1
ln P { s n x A} ~ n inf
g ( t ) dt ,
n
P {1
g A I C1 .
x } , P {1 < x }
( ) , A P { s n x A} (
f A ). , P ( A ) - , A
. , x
, P ( x A ) -
( x 0 , A
).
. . , , A . ,
. [
x ln P ( x A ) - ] ,
A ,
P ( x A ) - ,
) ;
. , ,
:
P { s n x A } = w ( x A ) ( 1 + o (1) ) + ( c1 ( A) + o (1) )
A = A ( g , g + ) = { g : g ( t ) < g ( t ) < g + ( t ); 0 t 1}
## n P { 1 > x } + ( c2 ( A) + o (1) ) n P {1 < x n };
, g ( t ) , g + ( t ) .
c1 ( A) , c2 ( A ) n - [ (7) (8)- .
( , ., [1] [3]- . )
. ( t ) = w ( t )
] x
ln n
( , [3] ).
(2)- . ( )
. k
( , [7] ).
.: [1] . ., . . ., 1964 , . 5, 2,
. 25389; 4, . 75067; [2] . ., . .
., 1962, . 3, 5, . 64595; [3] . .,
. , 1983, . 38, . 4, . 22754; [4] . .,
. ., 1967, . 12, . 4, . 63554; [5] . ., .: , , 1984, . 93124; [6] . .,
. ., 1981, . 26, . 1, . 7387;
[7] . ., . ., 1976,
. 21, . 2, . 23552; . 3, . 51226.
, W , C ( 0 , 1) - ,
A C ( 0 , 1)
ln W ( x A ) ~ x 2 ( A )
,
( A) =
( f ) =
( ( f )
), C
1
2
inf ( f ) ,
f AIC1
( f (t ) ) 2 dt
= f (t )
f ( 0 ) = 0 . A ( A0 ) = ( A ) , -
; large deviations f o r a r a n d o m p r o c e s s
.
. .
( ) . ( t )
. X , X
, P , P
A0 A , A
= C ( 0 , 1)
= ( f ) .
X .
X , , C ( 0 , 1) D ( 0 , 1) .
x A f = 0
X - (t )
( ) P ( x A ) -
. A P
( t )
114
( f ) = sup { ( f ) ln M e ( ) }, f X
X
( A0 ) = ( A )
(*)
1) + 2) ) A X
ln P ( x A ) ~ x 2 ( A )
( , [4], [5] ), ( A)
sn (t )
i nt
X i -
( X 1 , ..., X n
M Xi = 0 )
[ x n
x = x ( n ) ,
P { s n x A} n W ( x A )
] ( , [1] ).
s n (t ) ( )
. . P {
(n)
A}
(n )
( = x 1sn ( t ) ) ,
f (t ) = 0 , A .
, .
, ,
= D ( 0 , 1)
(n)
f A
lim k 1 (n) ln P { ( n ) A } = S ( A)
= inf ( f ) .
.
. . , .,
[2], [6] [9]-.
. .-
:
3) F ( f ) X - ,
lim k 1 ( n ) ln M e k ( n ) F (
(n ) . . ,
k (n ) S ( f ) -
(n )
,
- f
:
= sup { F ( f ) S ( f )}
f X
( n ) (t ) t ,
= n 1 - . z R1 , x R1
(n ) x
(n)
G ( t ; x, z ) = ln M e z ( (t + ) x ) ( n ) (t ) = x
(n )
( , [10], [11] ).
( , [2] ).
( t ) X =
, P
X - ( P
) .
(n)
H ( t ; x, ) = sup { z G ( t ; x, z ) }
z R1
G ( t ; x, z ) .
(n )
n S( f ) =
n1
H ( k n , f ( k n ) , f ( k n ) )
k= 0
## lim lim k 1 ( n ) ln P { ( ( n ) , f ) < } = S ( f ) .
0 n
, k ( n ) - , S ( f )
[ 0 , ) . S (A) = inf S ( f ) .
f A
:
0) s 0 ( s )
= { f : S ( f ) s}
;
1) A X
lim inf k 1 ( n ) ln P { ( n ) A} S ( A )
n
;
2) A X
lim sup k 1 ( n ) ln P { ( n ) A} S ( A )
n
, k ( n ) S ( f ) ,
(n )
, ( , [2] ). S ( A0 )
= S ( A)
[ (*) ], A X S
.
1), 2) :
. ( ,
[2] ).
x P ( x A) -
;
( , [12], [13] ).
.: [1] . ., . , 1983, . 38,
. 4, . 22754; [2] . .,
, .,
1986; [3] . . . , . ., ., 1978; [4] . .,
. ., . . ., 1978, . 19, 5, . 988
1004; [5] B a h a d u r R. R., Z a b e l l S. L., Ann. Probab., 1979,
v. 7, 4, p. 587621; [6] V a r a d h a n S. R. S., Com. Pure Appl.
Math., 1966, v. 19, 3, p. 261286; [7] . .,
. ., 1967, . 12, . 4, . 63554;
[8] . ., . ., 1976,
. 21, . 2, . 30923; [9] . ., . .,
, ., 1979; [10] S c h i l d e r M., TAMS,
1966, v. 125, 1, p. 6385; [11] . .,
. ., 1976, . 21, . 1, . 215; [12] . ., . . ., 1964, . 5, 2, . 25389; 4,
. 75067; [13] . ., . -
. . ., 1984, . 3, . 93124.
115
; large deviations
probabilities P { S n an > bn } , P { S n an < bn } ,
P { | S n an | > bn } , S n = 1 +
+ 2 + ... + n , { j } , {bn }
{an } , bn > 0 > 0 lim P { | S n an | bn1 >
, ,
n n ( ) ( ) ; D = { ( , d ) :
n 1 ln P { n n a } * ( a ) , a A ,
( )
.
i , i
= 1, 2 , ... ,
,
2 - , a n
=0
bn
= xn n
n xn .
( , [2] )
( , [3] ) ( , [2], [6],
[9] ) .
[7], [9]- . . .- ;
( , )
. : > 0 F (x)
= P { < x} , -
, ,
k s k
1+
( k! )
sk
, 0
2k
, k
= 3 , 4 , ... , 0
x < 1
(*)
{ 1 + ( x + 1) ( (1 x 1 ) 1 ) }
1 = (1 / 6 ) ( 2 / 6 )1/(1+ 2 ) ,
L (x) , = 0 , > 0
s k , ( x ) , ( , [8],
[9] ).
. . .-
. . . .
., (*) , x > 0
x -
x 2
( x )
P { x } exp
2
2
2 x + ( x )
= ( 1 + ) 1 .
( ).
1 , 2 , ..., n , ... j ( )
= ln M exp{ j } , j = 1, 2 , ...
116
; ( a )
= inf { a + ( ) :
[ c , d ] } ( , [9] ).
., , . ., .,
.: [1]
1962; [2]
. ., , ., 1972; [3] . .,
. ., , ., 1965; [4] . ., .:
, . 10, ., 1972, . 524; [5] . .,
. ., 1974, . 19, . 1, . 15254;
[6] . ., . ., 1965, . 10,
. 2, . 23154; [7] S t a t u l e v i c i u s V., Z. Wahr. und
verw. Geb., 1966, Bd 6, 2, S. 13344; [8] ., ., ., . . ., 1978,
. 18, 2, . 99116; [9] ., .,
, , 1989;
[10] P l a c h k y D., S t e i n b a c h J., Period. Math. Hungar.,
1975, v. 6, p. 34345.
; Brownian excursion ,
w e (t ) = ( 2 1 )1/ 2 w ( (1 t ) 1 + t 2 ) , 0 t 1
1
1 F ( x ) = (1 ( x ) ) exp{ L ( x )}
} A = { a : ( ) = a , D }
. : ( ) [ c , d ]
>}=0
[ c , d ], 0 c < d
( )
w e (t ) , 0 t 1 ,
1 = sup{ t 1 : w ( t ) = 0 } 1 = inf { t 1 : w ( t ) = 0 }
, t = 1 w (t ) - , w (t ) -
. . . ;
[1]- ( 2 ). . .
. .- [2] [7]- .
.: [1] ., .,
, . ., ., 1968; [2] D u r r e t t R. T.,
I g l e h a r t D. L., M i l l e r D. R., Ann. Probab., 1977, v. 5,
1, p. 11729; [3] D u r r e t t R. T., I g l e h a r t D. L.,
Ann. Probab., 1977, v. 5, 1, p. 13035; [4] V e r v a a t W.,
Ann. Probab., 1979, v. 7, 1, p. 14349; [5] G e t o o r R. K.,
S h a r p e M. J., Z. Wahr. und verw. Geb., 1979, Bd 47, H.1,
S. 83106; [6] K n i g h t F. B., Trans. Amer. Math. Soc., 1980,
v. 258, 1, p. 7786; [7] I m h o f J. P., Stud. sci. Math. Hung.,
1985, v. 20, 14, p. 110.
;
m u l t i p a r a m e t r i c Brownian motion ,
.
; ; Brownian excursion
, .
; Brownian motion process
( ) . 1827- . ( R. Brown ) .
- , ;
;
,
.
XX .
( L. Bachelir ), . ( A. Einstein ) .
( M. Smoluchowski ) . t
t ;
= 0 .
t 0
. [ 0 , t ]
, - ,
. ,
, t - .
,
- .
M t = 0 ,
M ( t + s t )
, 0
<
... < t n ,
. . c
. .,
c = k T 3 a
(*)
a , k , T
, .
. ( N. Wiener, 1929 ) . . .-
, { ( , t ); t 0}
( ( , F , P ) - ) :
1) ( ; 0 ) = 0 ;
2) 0 = t1 < t2 < ... < t n , ti +1 ti , i = 0 , 1, ... ,
n 1 ;
3) s, t 0 t + s t -
t1 , t2 , ..., tn
## pt1 , ...,tn ( x1 , ..., xn ) =
i =1
1 ( x xi 1 ) 2
1
2 (ti ti 1 ) 2 exp i
ti ti 1
2
x0 = 0 , x1 , ... , xn .
( ; t )
.
.: [1] ., ., , . ., . ., 1936; [2] .,
, . ., ., 1973; [3] .,
. . ., ., 1972;
[4] . . 2-, . , . . , . . , . . , ., 1985.
; m u l t i v a r i a t e Brownian motion process
, .
; Brownian bridge , .
;
Brownian meander
= f ( s ) t - .
,
f ( s ) = c s . c
## = t 0 < t1 < t2 <
w m ( t ) = (1 1 )1/ 2 w ( 1 + (1 1 ) ) , 0 t 1
( )
w m (t ) , 0 t 1 ;
1 = sup{ t 1 : w ( t ) = 0 } t = 1
w (t ) - . . .
.
.: [1] D u r r e t t R. T., I g l e h a r t D. L., M i l l e r D. R.,
Ann. Probab., 1977, v. 5, 4, p. 11729; [2] D u r r e t t R. T.,
I g l e h a r t D. L., M i l l e r D. R., Ann. Probab., 1977, v. 5, 1,
p. 13035.
;
brownian sheet , .
; Browns method of forecasting / prediction
. t . . .
t (k ) = a0 (t ) + a1 (t ) k + ... + a p (t ) k p
0 , s - ;
4) - ( ) ( , t )
.
a0 ( t ) , ..., a p ( t ) ,
t - .
.
( ) t 0
t = ( ; t ) -
t j
( j ) ) 2
j= 0
117
, - .
<1
. . { I k } -
U I T1 ( ,
,
.
).
.: [1] Stochastic geometry, geometric statistics, sterelogy, Lpz.,
1984; [2] .,
, . ., ., 1978.
t ( k ) - t +1 ( k ) - . ., p = 1 a0 ( t + 1)
a1 ( t + 1)
; Bunyakovskiis inequality
a0 ( t + 1) = ( 1 ) t +1 + a0 ( t ) + a1 ( t ) ,
M -
M M
a1 ( t ) = ( 1 2 ) t +1 ( 1 2 ) a1 ( t )
( 1 2 ) a0 ( t ) + a1 ( t )
.
.: [1] B r o w n R., Smoothing, forecasting and prediction of discrete time series, Englewood Cliffs (N. Y.), 1963; [2] .,
, . ., ., 1981.
; Brownian
sheet , .
; angular modulation ,
.
; angular frequency , .
; buffer memory , ( ).
; s t a n d a r d i z e d Boolean algebra
( M ) 2 M 2 M 2 .
## 1859- . . . . ( A. Cauchy ) (*) 1821- , (*)
. . .
, . ( H. Schvarz ) 1884- .
. ., , ,
a 2 + b 2 > 0 a b ,
a + b = 0 . (*)-
cov ( , )
; r a n d o m Boolean
.
function ,
; Boolean model
R n -, Tn . . . : Tn -
N = { ti }
{Di } . U =
ti Di { ti } { Di }
ti N
. N
, . .
. U
= {ti }
R n - . .-,
U I Tr . .-, Tr , R n - r ,
r n . U I T1 {I k } -
, { I k } . I k I - . U
. .-, - { I k , I k }
, { I k } -
( , ) .
118
D D ,
, .
(*)
( , ) 1
.
; Burgers equation , .
; Burkholder Gundy Davis inequalities
c p M [ , ] Tp / 2 M ( T* ) p C p M [ , ] Tp / 2 ,
,
p 1
= { t , t 0}
,
0 = 0 , [ , ] = ( [ , ]t , t 0 ) , -
, T
; t
= ( t* , t 0 )
= sup s . c p C p p -
s t
p > 1
, p = 1
## .: [1] B u r k h o l d e r D., D a v i s B., G u n d y R., Integral
inequalities for convex functions of operator on martingales, .: Proc
th
6 Berkeley Symp. Math. Statist. Probab., 1972, v. 2, p. 22340;
[2] . ., . ., , .,
1986.
;
bootstrap estimator ,
.
; bootstrap method
of estimation X n = ( x1 , ..., xn )
X n -
.
F , xi - , T (F ) , F -
[ ., T (F ) F - ,
]. Tn ( X n ) , X n T ( X )
. X n
n ( X n , F )
= Tn ( X n ) T ( F )
.
n
{ n ( X n , F )} D { n ( X n , F )} . n ( X n , F ) .
. . .- . n ( X n , F ) ( n -
) X n n ( X n , Fn )
. Fn , X n
*
, X n
## = ( x1* , ..., xn* ) , xi*
Fn i = 1, n
.
Tn - [ n ( X n , F )
]
n -
.
. , ,
*
. Fn xi j , i
= 1, N , j = 1, n
N *
X 1n
1*
*
, ..., x1*n ) ,
= ( x11
n ( X 1*n
, X N* n
Fn ) , ... , *N
= ( x *N 1 , ..., x *N n )
= n (
X N* n ,
Fn )
1* , ... , *N . . . .- .
Tn ,
(. )
.
.
.: [1] . ..,
, . ., ., 1988.
*
;
Boot-strap method
. . 1977-
. . ,
- .
- 20- 50-
. .
[8]- : Bootstrap
- . .
- .
; ,
.
, ,
; ,
. .
( ) -
, . .-
( B. Efron ).
,
. ,
. ,
. , (
) , , .,
. ,
,
. 1949-
. ( M. Quenuille ) ( , [1] ) . , ,
.
.
. .
( , [2], [3] )
(
) .
, ,
.
. 1969-
( , [4] ).
. . . ,
. . .
. . .-
.
.
1 , ..., n -
= P { i < x } . = ( 1 , ..., n )
F = ( F )
, F ( x )
119
. n
R ( , F ) = r ( , ) =
= ( )
R ( , F ) = M F L ( ( ) , F )
(1)
. L ( , F ) F
= ( F )
. . . (1)
R ( , F ) -
R ( ) = R ( * , F ) M F L ( ( * ) , F )
(2)
...
l ( ( z1 ,..., z n ), dF ( z1 ) ... dF ( z n ) )
, l ( , )
(7)
= L ( , F ), =
.
F - .
, F , ,
, n 1 , ..., n 1 / n
. , {1 , ..., n } -
. F , F
1 - 1 , n - n
*
, , F -
(1 + ... + n = n )
n ,
1* ,
... ,
n*
1 , ... , n
## P { 1* < x1 , ..., n* < xn | 1 , ..., n } = F ( x1 ) ... F ( xn )
(3)
.
. ( )
*
(5)
x dF (x) , s =
1
= (x )
=
( x ) dF ( x) n
P { ( * ) < x | 1 , ..., n } =
...
d F ( z1 ) ... d F ( z n ) = G ( x)
{ z : ( z ) < x }
P { ( ) < x }
...
dF ( z1 ) ... dF ( z n ) = GF (x)
L ( ( z1 , ..., z n ), F ) dF ( z1 ) ... dF ( z n ) =
L ( z , F ) dG ( z )
(6)
F . ,
F - F = { F , H } ,
H R p
p , F -
- , F = F
= ( Fn ) = t ( )
, - ( H ) , R ( )
120
i =1
, 0 <
. (2)
:
...
2
n
{ z : ( z ) < x }
R ( ) =
R ( ) - (7)-
(4)
= * , - -
.
, (6)
. ,
. , .
.
-
. . ( , [5] )
. n = ( x1 + ... + xn ) n
* = ( 1* , ... , n* ) F - n
1
, 0! = 1
1! ... n ! n
n!
1
*
m =
m
<;
i* ; ( i* , i =
i =1
= 1, ..., m ) , F = Fn ( 1 ,
..., n -
1 / n ) . m , n
P { | m* n | > (1 , ..., n ) } 0
> 0
. .
, t -
m , n
P { m ( m* n ) < t sn 1 , ..., n }
(t ) - ,
.
.
[6]- . . . . ,
, H n ( t )
= ( t n ) -
H n ( t ) H n ( F , t ) = PF {( *n ( * ) ( ) ) < t
n}
H n ( F , t )
,
H n ( F , t ) = PF { n ( ) ( ) < t
n}
.
, . . .
,
, .
.
.: [1] Q u e n o u i l l e M. N., Approximate tests of correlation in
time series // J. Roy, Statist. Soc.-Ser. B., 1949, 11, p. 6884; [2] Q u e n o u i l l e M. H., Notes on bias in estimation // Biometrika, 1956,
43, p. 353360; [3] T u k e y J. W., Bias and confidence in not
quite large samples / Ann. Math. Statist., 1958, 29, p. 614;
[4] . ., . ., ,
. .
. . . , 3, 1969; [5] B i c k e l P.
J., F r e e d m a n D. A., Some asymptotic theory for the bootstrap.,
Ann. Statist., 9, 6, p. 431443, 1981; [6] B e r a n R., Estimated
sampling distributions; the bootstrap and competitors, Ann. Statist.,
10, 1, p. 212-223, 1982; [7] .,
, . ., .,
1988; [8] : .-.: , 1978.
, .
; bootstrap distribution ,
.
## ; all possible regressions method
,
, .
.: [1] ., , .
., ., 1980; [2] , . .,
., 1986.
; Bienaime
Chebyshev inequality , .
;
Buffon ring , .
; Buffons needletossing problem
.
. 1733- , 1777-
( , [1] ). :
- 2 a O y
2 l ( l < a ) .
. x
, .
x
. = {( x , ) : 0 x l , 0 } .
x l sin .
C = {( x , ) : x l sin , ( x , ) }
2- . x ,
( 0 , 1)
( 0 , ) - ,
P (C ) =
l sin d
2l
a
(*)
.
(*) , (
). , n - m n P (C ) , 2 l a
~ mn
2l n
am
.
.
. .- :
,
121
; . .
,
.
.: [1] B u f f o n G., Essai darithmetique morale Supplementa
LHistoire Naturelle, 1777, t. 4; [2] ., .,
, . ., ., 1972; [3] . . , .,
1980.
; Buffon Sylvester problem
m -
; [1]- . ; . m
. Bi , i -
,
I B
i =1
i =1
Bi
. .
i =1
Bi
Bi
i =1
.
[2]- . , . . . .
.: [1] S y l v e s t e r J. J., Acta Math., 1890, v. 14, p. 185
205.
()( )
, ;
coupling o f r a n d o m p r o c e s s e s ,
,
- ( ).
( ), ,
.- . , ,
.
, ,
.
.
, ; .
.: [1] . ., , ., 1980; [2] . .,
, ., 1978; [3] . ., , , .:
.,
, . ., ., 1989.
; reward function ,
.
; attraction domain , .
;
domain of attraction o f a s t a b l e l a w , G . . F , An
Bn
> 0 , x - n
F n ( Bn x + An ) G ( x )
(*)
, n , n
. (*)
X i An
i =1
Bn
G ( x ) , X i F ( x )
.
. .-
. -
.
0 < 2 , . .- ,
0 <
<
|x|
F ( dx ) <
(*)- Bn Bn
= n1/ h ( n )
, h ( n ) .
G . .- , Bn
,
G
1/
Bn = a n
= n1 /
. (*)
( a > 0 ) Bn -
, F
.
F ( x ) - , 0 < 2
. .-
z
lim z 2 ( z )
d ( x) = ( 2 )
; ( z )
F ( z ) + F ( z ), z > 0 (
= 1
[3] ).
0 < < 2
F ( x ) = ( C1 + o ( 1) ) | x | h ( | x | ) , x ,
F ( x ) = 1 ( C2 + o ( 1 ) ) x h ( x ) , x
, h ( x )
0, C 2 0, C1 + C 2 > 0 .
= 2 ,
, C1
:
1) F ( x ) ;
2)
( 1 F ( x ) + F ( x ) ) x 2 ,
.
. .- ( , ., [3] ).
.: [1] . ., . ., ,
. ., 1949; [2] . ., . .,
, ., 1965;
[3] . ., . ., 1956, . 1,
. 3, . 35761.
123
; domain of attraction o f a s t a b l e d i s t r i b u t i o n o n a g r o u p
G , - G { n } G
{ xn } ,
Yn = n ( S n o xn )
(1)
n
G ,
o , S n
= X 1 o ... o X n -
, K , : K Aut (C0 )
= 0 K , 0 ,
C 0 - , , K -
( , [1] ). G g
exp : g G ,
G ( , , )
, g .
, o
o
( ., M ( G ) , M 1 (G ) )
G - ,
X 1 , ... , X n
A
g -
.
G Aut G
A -
. g - T .
M ( G ) -
,
EG
( E ) = ( ( E ) ) , (1)
n
n ( n ) xn
(2)
n
,
- n , xn , xn
. : D ( )
o
, , D ( )
( , [2], [3] ). ,
k , . .-
d
. ., R ( d ) { X n }
(d ) ,
= (U i , Yi ) ,
. X i X i
G
T = { t , t > 0}
U i SO ( d ) ,
( , ). G -
U i - .
: (d ) - -
, D ( ) , ,
. . M ( G )
, (1)
(2) n = 1 / n
. D ( ) DN ( ) ,
. . . (1) (2)
xn = e , . .
. G
. ( , , . .
. ) . .-
M 1 (G ) . ,
. G -
, G
,
D ( ) , ,
. .- ( , [1] ).
G C0 K
T C
, C 0
124
Yi , Yi R d R d - .
: 1)
n n -
SO ( d ) ;
## 2) P { | Yi | > x } = ( c + o (1) ) x h ( x ) , c > 0 , h ( x ) ,
x ( , [4] [6] ).
, .
.: [1] H a z o d W., Lecture Notes in Math., 1986, 1210,
p. 30452; [2] h k h l v Y u. S., In: Probability Measures on
Groups, X, N. Y., 1991, p. 23948; [3] H a z o d W., S c h e f f l e r H.P., J. Theor. Probab., 1993, v. 6, 1, p. 17586; [4] . ., . ., 1982, . 27, . 2,
. 34244; [5] . ., . . ., 1985,
. 25, 3, . 3952; [6] . ., . .
., 1990, . 30, 1, . 1429.
; domain of attraction o f n o p e r a t o r s t a b l e d i s t r i b u t i o n
R d - ,
d
R - { An }
R d - { an } , n
Yn = An ( S n an ) -
, S n =
= X 1 + ... + X n
. B -
, An n B
,
.
B , R d - d
. R - - ( a , C , M )
d
, a R ,
C , M . ,
= g p
, p R p -
. d n,
> 0 , M n, ,
n P { X , d n , ( y ) } M n , { [ y , )} = 0
lim sup
n S d 1
, X
mn,ln () = n M ( X , I {
b = a , +
. . . :
D N ( ) , ,
u ( 1 + u 2 ) 1 M (du ) .
K ( A ) = 0 , lim t { s B x : x A, s > t } = K ( A ) Fg ( )
t
, A , A
( , [1], [2] ).
, ,
.
S d 1 = { R d : = 1} .
2
j =1
an ( ) = 0
hn, ln ( n j ) , n j
ln ( )
, { n1 , ..., n d }
.
- . .-,
, ,
{ nj , j = 1, ..., d } ,
1
Rn : Rn* nj = ( an ( nj ) w (e j )) e j
. , D N ( ) , ,
S d 1
Fg ( ) D N ( g ) F p ( ) D N ( p ) -
g n ()
= Rn*
Rn*
:
1) X 1 an ( ) w ( g n ( ), M g n ( ))
;
2) lim
sup
S d 1
an ( ) w ( g n ( ) ) Rn*
= 0;
3) mn ( ) = mn, g n ( ) { n1 , ..., n d }
,
a 2 ) a 2 }
) ] M ( du )
u 3 (1 + u 2 ) 1 M (du ) +
, mn,
, D N ( )
w () = sup {w 0 :
2 1
Fg Fp , Rg R p -
an ( ) = sup { a 0 : n M ( X 1 ,
lim sup hn , ln ()
R p .
(1 + u
| u | >1
K ( A ) = M ( t x : x A , t 1) , A L .
| u | 1
I E ( t B x ) t 2 d t K ( dx ),
A L I Rp
2 1
u , [ (1 + u ,
a1 ( M , b ) = b +
an () w ( ln () ) }) ,
X,
= { 0 } , Rg R p =
L = { x R d : x = 1, t B x > 1 t > 1 } ,
## hn, ln ( ) = mn, ln ( ) a1 ( M ln , bln ) an ( ) w ( ln ( ) )
I E , E R d ,
d 1
I ( A) , A ,
= R . :
M (E ) =
(d n, , M n, ) , . l n () , S
, Rg R p B R p I R g
, , R d -
y > 0
, g , R g -
d 1
( y 2 w 2 ) M (dy ) + C
w2}
S d 1 - , a b =
= min ( a , b ) , C , C , M , M
Rn* - :
1
Rn* x =
x , nj
Rn* nj
nj ,
j =1
125
{ n1 , ..., nj } . Rn { S n } .
{ n1 , ..., n d } ,
( , [3] ).
, , , . , X 1
. .-, , ,
lim
sup
t S d 1
t P{
> t} M ( X1,
X1,
) =0
. a n ()
, An
( , [4] ). < 2
( , [4] ).
.: [1] J u r e k Z. J., . ., 1982,
. 27, . 2, . 396400; [2] H u d s o n W. W., M a s o n J. D.,
V e c h J. A., Ann. Prob., 1983, v. 11, 1, p. 17884;
[3] H a h n M. G., K l a s s M. J., Z. Wahr. und verw. Geb., 1985,
Bd 69, S. 479505; [4] H a h n M. G., K l a s s M. J., Lecture Notes
Math., 1981, 860, p. 187218.
JS
. M
= 2 2 . JS
, R -
+J S = ( 1 2 ( p 2 ) | X |2 ) + X
, a+
= max ( a , 0 ) -
.
X n JS
JS
n , + n
, -
n .
n > , > 0
:
n M n
= p 2 ( p 2) 2 4 n 1
n -
.
( 1 + (1) )
(1)- 1956- .
JS
JS
( , [1] ),
+
[2]- .
. ( )
;
attracting boundary , ; -
, , i
; Jessen Wintner theorem :
,
, ,
( , ). ( , [2], VIII .,
5 ), . . . . ( , [3] ).
.: [1] J e s s e n B., W i n t n e r A., Trans. Amer. Math. Soc.,
1935, v. 38, 1, p. 4888; [2] .,
, . ., . 2, ., 1984; [3] . ., . ., .
., 1975, . 20, . 4, . 71224; [4] ., , . ., ., 1979.
( 1 , ..., p )
(1 , ..., p ) .
.
N ( , G ) X
;
L ( , )
= T H ( )
, G , H
.
= X + G g ( X )
M L ( , ) = M ( L ( X , ) + ( X ) )
## ; James Stein estimator
, g : R R , N ( , G ) ,
2 E ,
R p g
, R - N ( , E )
J S
= ( 1 ( p 2)
JS
X -
( x) =
)X
< M x
= 2 , R p ,
ij
( x ) = f 1 ( x)
i , j =1
(1)
f ( x )
126
gi ( x )
2
+ g i ( x ) g j ( x) ,
xj
i , j =1
C = (Cij ) = G H G .
(2)
g ( x ) = ln f 2 ( x ) , f ( x ) > 0 .
, :
2
, E . p > 2
M JS
2 f
xi x j
= ( C 1/ 2 x ) ( x ) -
( x ) = ( ) ( C
1 / 2
, ( 1 , ..., p )
x) ,
.
, X -
. p > 2
, , = ( 2 p ) 2
= x
, 2 p
= ( 1 ( p 2 ) H 1G 1 X TG 1 X ) X
= G = E JS
.
JS H
. -
.
1) G ,
JS
G - n 1 ,
S . p > 2
( 1 ( p 2 ) ( n p + 3 ) X T S 1 X ) X
( x )
* * ; * algebra
## , A , A ; algebra, induced by set A , .
; algebraic decoding m
. ., n = q 1 , 2 t + 1
,
n 1
X - ( , [2] ).
j = 1, 2 , ..., 2 t
q - . .
j + i
yi , xi Fq
j =1
( ) X
T
X ( )
(1 2 ( p 2 ) | 0 |2 ) 0
, X
= ( X ij ), 0
.
3) X X i
, M X i
= i , M ( X i i ) 2 = 1 , M ( X i i )3 = 0 ,
M ( X i i )4 = k
a 4, 5 + 4 (k 3 ) 3
b ( a + | X | 2 ) ) X
yi xij = S j ,
j = 1, 2 , ..., 2 t
i=1
, i ( 0 , ) -
b p2,
ci i j = 0 ,
i=0
ij
, ,
## (2) ( , [5] [7] ). X
. ( ) .
.: [1] S t e i n C., .: Proc. 3d Berk. Sympos. Math. Statist.
Probab., v. 1, Berk., 1956, p. 197206; [2] J a m e s W., S t e i n C.,
.: Proc. 4th Berk. Sympos. Math. Statist. Probab., v. 1, Berk., 1961,
p. 36179; [3] S h i n o z a k i N., Ann. Statist., 1984, v. 12, 1,
p. 32235; [4] B r a n d w e i n A., S t r a w d e r m a n W., Ann.
Statist., 1980, v. 8, 2, p. 27984; [5] B e r g e r J., Ann. Statist.,
1980, v. 8, 3, p. 54571; [6] B r o w n L., Ann.Statist., 1980, v. 8,
3, p. 57285; [7] G h o s h M., P a r s i a n A., J. Multivar.
Anal., 1980, v. 10, 4, p. 55164.
( ) G ( ) -
2) yi =
, , Cij ( x )
p>2,
= ( 1
X - .
4) X -
Fq
, q m -
. (
## ; Boze R. C., Ray Chaudhuri D. K., Hocquenghen A. ) ,
t
.
( , ( )
) t - , ,
( ) .
, -
. .-
.
.: [1] ., , . ., ., 1971.
, ; algebra of events
1 < 2 X -
( , [4] ).
.
. . ,
.
. . .
.
X ,
127
. p > 3 , 0 < a
( 1 a | X |2) X
2 ( p 2 ) p M 0 | X | 2
| | ,
. . .
1) ;
k
. 2) R - { x = ( x1 , ..., xk ) R :
ai xi < bi , i = 1, ..., k , ai < bi + } -
, R - . 3) -
, . 4) ( .
P
. P ( E ) = 0 , E
= 1 , E . A A -
( A -
)
, P ( E )
.
.: [1] . ., , 2 ., ., 1974; [2] .,
, . ., ., 1969.
, -
; algebra of events
, , ,
, .
,
; algebra
of quasiloal observables ,
= Y * Y X 0 , -
. Y U , X
X U .
X Z 0 , X Z , -
U - ,
.
I C * U -
, X 0 ( X ) 0
( I ) = 1 . S
U * * .
. 1) ,
,
X ( ) C * . -
P (d )
2)
= ess sup X ( ) ( , ) -
LC ( , ) C
p ( )
p ( ) (d ) ,
( x) =
, , X X
X ( ) p ( ) ( d )
LC ( , ) .
( X * )* = X , ( X + Y )* = X * + Y *
3) H
B( H )
( X Y )* = Y * X * ,
C * . S , tr
H - , ( X ) = tr S X B( H ) -
( X )* = X *
X , Y , ...
, . .,
* . ,
, * . X
Y - , .,
( , ). , ( ) * -
, X ,
X = X .
*
X X
. . .
2
p( ) c ( mod )
= X
, ; algebra of observables -
X ( ) P (d )
(X ) =
128
U , C
.
*
C H - .
C *
*
. C
W * [ . 2) 3) ]. { X } U sup ( X )
= ( sup X ) , W *
U -
W *
2) 3) .-
.: [1] ., C * , .
., ., 1974; [2] ., .,
, .
., ., 1982.
,
; algebra of cylinders
= t
t T
n ( a , b ) = ( 2 ) 1
dy
cos [ y f (t ) ]
f (t ) d t ,
A , T , t
b - , n ( a , b ) 1 2
t T At
. i , i
. A
= {t , t T } : Ts E
A
t
t S
(*)
T - , Ts = {t , t S },
t
t S
A ,
t
At , t S -
t S
{ {t , t S } : t Et At ,
t S
t S } .
E (*) .
At , S - (*)
, A - , ,
.
, .
.: [1] ., , . ., ., 1969; [2] ., ,
. ., ., 1962.
; algebraic random equation
1 , ... , n
f (t ) = t + 1 t
n
n1
+ ... + n = 0
(*)
= 1 + i1 , 2 = 1 i1
= k + i k , 2 k = k i k , 2 k +1 = 1 ,, n =
= n2 k ,
1 ... n ,
l , l , l = 1, k , j , j = 1 , ..., n 2 k k , 0-
[ n 2 ] - .
1 , ... , n
p ( x1 , ... , xn ) , (*) . i -
,
p ( Re zi , Im zi ; i
( i
= 1, ... , n
= 1, ..., n )
i -
, i
p ( Re i , Im i ; i = 1, ..., n )
zi z j
i , j = 1, n ,
i > j
> arg z n , i
,
## arg 1 > ... > arg n ; arg z1 > ... >
z i
i = 1, n . (*)
(a, b)
hn (t ) = n t n1 (1 t 2 ) (1 t 2 n ) 1 .
.: [1] . . . ., . ., .,1999; [2] .,
, . ., ., 1965.
;
Jackknife method , (
) .
,
; sum of random number of random
variables , .
, ;
sheme of summation of random variables
,
S n = X 1 + ... + X n
. . .-
. (*) 1
,, 2 k 1
M n ( a , b) = 1 [ 1 hn2 ( t ) ]1 2 (1 t 2 ) 1d t ,
a
, S ,
= 1, n ( 0 , 1)
X j , j = 1, ..., n
.
. ,
.
( X 1 , ..., X n ) n -
( , , ),
(
, ). . .
, ,
.
p p -
; p summing operator
; degenerate distribution n . n -
. . . .
, ( ) r ,
n - . r . .-
. - .
. .
. . . ,
.
n ( a , b ) :
129
, x = 0 . .-. . .- L ( 0 ) .
, , ; nondegenerate distribution , .
; nondegenerating branching process , ; .
; degenerate / singular bivariate normal distribution , .
; degenerate / singular measure , .
; strictly stable distribution
, Bn > 0 -
{ t }t <
( ) ,
,
. ,
= ( t , At , Px ) , ( E , B )
, p ( t , x , ) ( , A )
;
G ( x, , , ) , 0 < 2 ,
).
t 0 , x E , B
, 0 <
{ t } At , t 0 , A = { A : A A , A I { t }
( -
At , t 0 } )
Px { +t | A } = p ( t , , )
Px ,
Px { + | A } = p ( , , ) ,
, , , 1)
( , [1] ), ,
: 1 , x > 0 ,
( 1 G ( x , , ) ) = C ( x , , ) (1 ) 2
= 1
, 1+
= (1 ) .
. . .-
.
.: [1] . ., , ., 1983.
; strong Markov property t = (t ), t 0
, { t }t >
130
Px
M x ( ) = M x ( M )
. . .- ; , , A ( t ), t 0
<s, >s
t < s t -
( , [1] ). A
.
G ( x , , ) = G ( x , , , 1) -
. (*)- . . .-
:
G ( x, , , ) = G ( x
(*)
. . . . ( , [1] )
0 , + 0 =
min ( 1, 2 1) , > 0
1 /
( - A
n ,
W - . W -
( , A ) s - s + t - t
1 , ..., n
## Z n = ( 1 + ... + n ) Bn1 , n = 1, 2 ,...
[2]-
(*)-
. . . .-
[3]- . . . .-
. . . .
.
.: [1] D o o b J. L., Trans. Amer. Math. Soc., 1945, v. 58,
p. 45573; [2] . ., . .,
. ., 1956, . 1, . 1, . 14955; [3] B l u m e n t h a l R., Trans. Amer. Math. Soc., 1957, v. 85, p. 5272; [4] . ., , ., 1959;
[5] . ., , .,
1975.
; strong Markov process
. . . . -
[1] [2]- ,
[ . . . . ( J. Doob ),
. . ,
. . . . ( J. Hunt ) ].
. . .-;
( , [1] ).
, : 1) = { t , At , Px }
E
2) , t 0
F ( x ) = M x f ( t ) , x E f E - , . . . ( , [1] ). A t
At + =
Au
u >t
.
.: [1] . ., . ., .
., 1956, . 1, . 1, . 14955; [2] B l u m e n t h a l R.,
Trans. Amer. Math. Soc., 1957, v. 85, p. 5372; [3] . ., . ., 1957, . 2, . 2, . 187
213; [4] . ., ,
., 1975.
; Johnson Welch transformation t
. X Y
, X
~ N ( , 1) , Y 2 n
= nX Y
. p - p
, T
p = T ( p)
1 +T
2n
, ( p) p ( , [1] ).
( Fv1 ( p )) ,
T < 0 ,
*p =
1
( Fv ( 1 p )) , T > 0 ,
1
, Fv ( p ) p . n
v n x3 ,
( sign T ) ( 1 x ) 2 ,
n ( 1 x 2 ) x 3 / 2 ,
~
2
2
1 / 2
)
, x = T ( T + 2n ) , p p = O (n
, n p p
= O ( n 3 / 2 ) .
## .: [1] J o h n s o n N. L., W e l c h B. L., Biometrika, 1940,
v. 31, p. 36989; [2] . ., . .,
t , ., ,
1975 ( ).
; pairwise independence ,
.
; paired correlation
coefficient , .
; attainable boundary , ;
## ; accessible / reachable state ,
; ,
.
; inattainable boundary , ;
,
; nonaccessible state o f a M a r k o v c h a i n
, ;
/ / / ; defect (t ) ,
(t ) t
t
( , ),
( t ) = t sup { s t : ( s ) (t )}
( ) M ( )
D ( ) . . . ,
> 0
{ : | ( ) M ( ) | }
D
P{ | M |
} 1/ 2 .
## - . ( I. Bienaym, 1853 ) . . ( 1866 ) . . .
, ,
. .
( ).
. . ;
,
M ,
P { } M
(
).
:
P{ | | } M | |r r ,
.
. . , { ( t ), ( t ) ; t
G (t ) = P { k
0}
. ,
t } M k = m
n =
, n 1
k =1
G* ( t ) =
(1 G( s ) ) ds / m
0
. . .
.: [1] . ., . ., , .
., ., 1967.
; Chebyshev inequality ,
.
132
P { | M | } M | M | r r , r 1
( r = 2 . .- ),
x - ,
f ( x)
P{ | | } M f ( ) f ( )
(*)
. (*)
. .,
P { } M e c e c ,
c>0
. ,
. . . . .- ,
( , [4] ).
,
. ., ,
- ,
P{ | | } ( 4 3 ) ( 2 2 ) ,
P { max X (i ) M X (i ) x } x 2 tr ( C 1B C 1 ) , x > 0
2 = D .
. .-
, . . .
, .
. .
.
P{ | 1 + ... + n ( M 1 + ... + M n ) | }
D (1 + ... + n ) 2
P { max | 1 + ... + k ( M 1 + ... + M k ) | }
1 k n
D (1 + ... + n ) 2
. . . ( ,
).
. .-
2
P{ | 1 + ... + n | } 2 exp
2
2 ( 1 + a / 3 )
, | i |
m 2 ai - ( ai X )
M X 1 a1
+ ... + M X n a n
, x 0 X
m! B 2 Lm2 2
(1)
= X 1 + ... + X n
p x = P{ | X | xB + M | X | } exp{ x 2 (2 + 2 xL / B ) }
. .-
.
. (X , | | ) X i
C , M i = 0 , 2 = D (1 + ... + n ),
a = C 2 ( , ). . .-
i
.
( , [5] ).
.: [1] . ., . ., 1867, . 2, . 19;
[2] . ., , 4 ., ., 1924;
[3] . ., , 2 ., ., 1974; [4] ., ., , . .,
., 1976; [5] . ., .: .
. . . , . 10, ., 1972, . 524.
( , [4] ). . ., (1)
ai = 0 m = 3
px cx 3 L B + exp { cx 2 } , x > 0
( , [5] ), c , c . (2)
, [4], [6].
, , .
.: [1] ., .,
, . ., ., 1976;
[2] T o n g Y. L., Probability inequalities in-multivariate distributions,
N. Y., 1980; [3] O l k i n I., P r a t t J. W., Ann. Math. Statist., 1958,
v. 29, 1, p. 22634; [4] . ., . .,
. ., 1985, . 30, . 1, . 12731;
[5] . ., . ., 1978, . 23,
. 3, . 63037; [6] . ., J. Multivar. Analysis,
1976, v. 6, 4, p. 47399.
; Chebyshev theorem , ,
.
## ; Chacon Jamison theorem ,
; .
; ;
; Chentcov distribution density estimator , , ;
; m u l t i d i m e n s i o n a l a n a l o g s o f
t h e Chebyshev inequality X A -
; Chernov boundary -
P{ X A} inf { ( X ) : ( x) 1, x A ;
( x) 0, x A }
. ., X
= ( X (1) , ..., X ( d ) )
. 1 2
h ( x) . 1 ( ) , 1
h ( x)
B -, C , ,
C 1 BC 1 ,
1 ( ) =
exp ( j h) p ( h 1 ) dh ,
133
j - s h ( x ) 1 ( s ) :
f , F
exp(s h) p(h ) dh ,
1 ( s ) =
a a + h F - , h > 0 .
F ( x ) =
( s ) = ln 1 ( s ) . , g ,
:
p g ( g = h 1 ) = exp( s h + ( s ) ) p ( h ) 1 ,
g -
M ( g 1 ) =
h p g ( g = h 1 ) dh =
d ( s )
,
ds
D ( g 1 ) =
d 2 ( s )
ds 2
.
p g :
p (h 1 ) dh =
exp ( ( s ) s h ) p g ( g = h 1 ) dh =
= exp ( ( s ) ) exp ( s h ) p g ( g = h 1 ) dh .
t
s 0 , t h
exp ( s h ) exp ( s t )
, ,
exp ( ( s) s t ) p g ( g = h 1 ) dh .
t
exp ( ( s) s t )
s 0 .
.: [1] .,
, . ., ., 1979.
; inversion formula f (t )
F ( a + h ) F ( a ) = lim
134
F -
1
2
e i t x f ( t ) dt
( . . ).
, [1] [3]-.
.: [1] ., , . .,
., 1979; [2] .,
, . ., . 2, ., 1984; [3] . .,
,
., 1987.
, ;
time reversal , .
; reversed Markov process
,
. t , t ( , ) . t
= t
.
-
t = t , t , , t
[ , ( h )
## = max { ( ) h , 0}, h > 0, h ]. , , , t
t
. t
t
.
.: [1] C h u n g K. L., W a l s h J. B., Acta. math., 1969,
t. 123, p. 22551; [2] S m y t h e R. T., W a l s h J. B., Invent.
math., 1973, t. 19, p. 11348; [3] N a g a s a w a M., Nagoya Math.
J., 1964, v. 24, p. 177204; [4] J e u l i n T., Z. Wahr. und verw.
Geb., 1978, Bd 42, H. 3, S. 22960.
; reversed Markov chain ,
p g ( g = h 1 ) dh < 1
1
2
1 e i t h i t a
e
f ( t ) dt
it
,
.
X = ( X 1 , X 2 , ... ) Y n = X n , n 0
Y
= ( ..., Y1 , Y0 ) -
( , [1] ); Y X -
. . .-.
. . .- . pij X
## = {1, 2 , ...} - i > 0
= ( 1 , 2 , ...) ( ) -
, Y
p i j = P {Y n = j Yn1 = i } = j pi j i
(*)
, i , j E , n 0 .
X . . . .
pij (*) ,
> 0 = p ji , X -
- X i
. pij
.
.: [1] ., , . ., .,
1956; [2] ., ., , . ., ., 1970; [3] . ., . ., , . 1, ., 1971.
; weight , / .
( )
; weighing o f s u p p o r t i n g p o i n t s
;
- - .
deviation , / .
, ; oblique
rotation o f f a c t o r i a l a x e s ,
.
## ; skew Wiener process , .
; method of
residues , ,
.
; exit rate / probability density ,
,
; weighted
design / strategy k
; exit
rate , , .
; output signal , .
; Chungs law of the
iterated logarithm ,
N .
; weight matrix
. . .
.
xnT =
D = x1 ,..., xN
i -
n -
, , 1 , +1 0
xn i
.
xn i i - n -
- 1 0
.
## yn = b0 + b1 xn1 + ... + bk xnk + n
, y n , n - ,
n = 1 , ..., N ; b0 , bi , i -
, i
= 1 , ... , k ; n , n - .
- ( ,
[1] ). ( ,
[2] ). . .- ( , [3], [4] ),
.
.: [1] Y a t e s F., J. Roy. Statist. Soc. Supp., 1935, v. 2,
p. 181247; [2] H o t e l l i n g H., Ann. Math. Statist., 1944,
v. 15, p. 297305; [3] . .,
, ., 1976; [4] B a n e r j e e K. S.,
Weighing designs, N. Y., 1975.
;
weighted least squares method / generalized least
squares method , .
; aggrgation of states o f a M a r k o v
c h a i n , ; .
;
aggregation function , ;
; multiple access channel
.
. .- ( ,
).
~
~
Y1 , Y2 , Y ( Y1 , Y2 , Y )
y y1 y 2 ) .
p ( ~
(M 1 , M 2 )
y1m1 Y1 N ,
y m2 2 Y2 N , 1 mk M k , k = 1, 2 ( Yk N , Yk
N - )
Y
C m1 m2 Y
, 1 mk
Mk ,
k = 1, 2 .
~
, C
~
y - ( ~
y Y N ) ,
m1m2
, y m1 y m2
( m1 m2 ) . m1
m2
135
Pe ( m1 , m2 ) =
~
y Cm1 m2
p(N ) ( ~
y | y m11 , y m2 2 )
~
y = (~
y1 , ..., ~
yN ) Y
## (*) ; , Ror, as. -
y k = ( y k1 , ..., y kN ) , k = 1, 2
p( N ) ( ~
y | y1 , y 2 ) =
p (~
y | y1n , y2 n ) .
n =1
. . Por
Pmax
1
Por =
M1 M 2
Pe ( m1 , m2 ) ,
Pmax = max Pe ( m1 , m2 )
m1 , m2
>0
M k
, k
= 1, 2 , Por. <
N , R
= ( R1 , R2 ) ( . )
. . .- Ror
R .
Rmax .
Ror . . . Ror
:
Ror = co
R ( p1 , p 2 ) ,
p1 ( y1 ), p2 ( y2 )
(*)
co , p1 , p2 , , Y1 , Y2
y | y1 , y 2 )
p ( ~
Y1 , Y2 , Y p1 ( y1 ) p 2 ( y 2 )
p( ~
y | y1 , y 2 )
~
R ( p1 , p2 ) = {( R1 , R2 ) : 0 R1 I ( Y1 ; Y | Y2 ) ,
~
~
0 R2 I ( Y2 ; Y | Y1 ) , R1 + R2 I ( Y1 , Y2 ; Y )}
, I .
, Rmax Ror -
( ).
. .- . ., L . .-
, . .
.
. .- . , ,
, . .- . ,
( ).
.
136
.
.: [1] . ., . .,
, ., 1982; [2] ., .,
, . ., ., 1985; [3] V a n d e r M e u l e n E. C.,
IEEE Trans. Inform. Theory 1977, v. 23, 1, p. 137.
## ; multiway contingence table , .
; multiple access
m1 , m2
N ( Rk )
-
, ,
. Ror, as. - . -
( , ). .-
:
, .
. .
.
.
.
.- .
( ),
, , ,
.
( ,
.)
.
. . .,
. -
( ),
,
( )
. . .-
.
.
( . ) . .
, .
.
, .
. .-
,
.,
,
, .
.- .
.: [1] ., , . ., ., 1979; [2] B e r t s e k a s D., G a l l a g e r R.,
Data Networks, N. Y., 1987.
; r a n d o m multiple access
, .
; multichannel queueing system , , ( , ,
) . . . .,
, ( ,
). . . .-
. - , . . .-
. . .- . . . .-
.
; multichannel queueing
system , .
system , .
; multiterminal channel .
K L
~
{ (X , SX ), (X , SX~ ), Q ( x, A ), V }
; multicomponent source - .
.
M U M X1 , ..., X M ( ) p ( x1 , ..., x M ) ,
xm X m . xm U
m - .
, p ,
N
p ( N ) ( x1 , ..., xM ) =
## ( X , SX ) = (X1 , SX1 ) ... (X K , SX K ) ;
L ,
~
~
~
(X , SX~ ) = (X1 , SX~ ) ... (X L , SX~ )
L
p (x
1n
, ..., x Mn )
i =1
p
xm
,
K ,
. . .-
( , ).
. .- ( K = 1 ),
( L = 1 ), ( K = L = 2 ). , .
.: [1] . ., . .,
, ., 1982; [2] ., .,
, . ., ., 1985.
(N )
, -
## = ( xm1 , ..., xmN ) X mN .
. . .- ( ,
, . );
.
. . . .
, ( ) .
.: [1] . ., . .,
, ., 1982; [2] ., .,
, . ., ., 1985.
, ( X , SX ) - V
;
multiple mixing , r ( , A , P )
: V , ,
T
:
, k Vk , k , -
lim P
n
( X k , SXk ) - , Vk ( X k , SXk ) - . , 1 k
X k . .- k - -
, X l , 1 l
L l -
.
. .-
Yk , Yk
p( ~
y1 , ..., ~
y L | y1 , ..., y K )
~
~
, y Y , y Y .
. .
. . ,
. .-
IT
i =0
A0 , ..., Ar ,
ki ( n )
Ai
P ( A ) = 0 ,
i
i =0
A - { ki (n) ,
## n = 1, 2 , ...} , i = 0, 1,..., r , lim
min
n 0 i < j r
k i ( n) k j ( n) = 0
.
. . . 1
; . . , , r 2
. r ( r + 1)
137
- - r - ( , ).
.: [1] . ., . , . ., 1949,
. 13, 4, . 32940; [2] . ., . .,
. ., , ., 1980.
; multiple stochastic
integral , .
; multiple Wiener integral ,
, f L
( R n+ )
( )
In ( f )
= ...
~ B ( mH 2 , m E 2)
, X
mH 2 m E 2 I ; mET mH
~ F (mH , mE )
, T
, mH 2 m E 2 II
.
, .
;
multidimensional Brownian process ,
.
f ( t1 , ..., t n ) dw t1 ... dw tn
. ,
f
- , . . . .
:
1) M I n ( f ) = 0 , n 1;
~
~
2) M I n ( f ) = M I n ( f ) , f , f -
3) M I n ( f ) I m ( g ) = 0 , m n ;
~
4) M I n ( f ) I n ( g ) = n! f , g~ ,
. . . .
. .- , ,
.
## ; multivariate normal distribution , .
; multivariate distribution s R s
. . ., , = ( 1 , ..., s )
1 ( ) , ..., s ( ) ,
, , f n -
( 1 , ..., s -
, , A , w - .
. . . , , .
.: [1] I t o K., J. Math. Soc. Jap., 1951, v. 3, p. 15769.
= R
). . .
(R n+ )
; multidimensional observation
n ( n > 1) -
- ;
5) L2 ( A ) =
n ( fn )
n=0
; multidimensional beta
distribution X
2 2
. , H ~ mH ,
E ~
m2 E
H E . T
= T (1 + T ) = H ( E + H )
=H E
x1 , ..., x s
.
. .- . . . R
f (t ) =
q ( ) =
B (mH
B ( mH
1
( mH
2 , mE 2 )
2 ) 1
i1 , ..., is
, 0t <
(1 ) ( m E
## = 1 ; ( xi1 , ..., xis ) R s
i1 , ..., is
( , ., / ).
2 ) 1
R -
s F ( x1 , ..., xs )
= p ( x1 ,..., xs )
x1 , ..., xs
0 1
, B ( a, b) .
138
## ( xi1 , ..., xis )
, ,
t ( mH 2 ) 1
2 , m E 2 ) ( 1 + t ) ( mE + mH )
p ( x1 ,..., x s )
A ( A , R
)
P { A} =
## p ( x1 ,..., xs ) dx1 ... dxs
, ) p x
p ( x1 , ..., x s ) 0 ,
,
,
. x1 ,..., x p (
Rs
. . i
= i ( )
## m < s i1 , ..., is . .- . , . . . 1 , ..., s
F ( x1 , ..., xs )
= F1 ( x1 ) ... Fs ( x s )
p ( x1 , ..., xs ) = p1 ( x1 ) ... ps ( xs ) , Fi (x ) pi ( x ) ,
f (1 ,..., s ) - . .-
; . . ,
M f ( 1 , ..., s ) =
## f ( x1 , ..., x s ) p ( x1 , ..., x s ) dx1 ... dx s
.
. .- t
(t )
= Me
i t x
= ( t1 , ..., t s )
- , t x
= t1 x1 + ... + t s x s . . .- M ...
ks
k1
...
k1
( s M s )
ks
M ( 1 M 1 )
, k1 + ... + k s . . .-
M =
= ( M 1 ,..., M s )
2- . i j
i , j M ( i M i )
( j M j ) = 0 , 1 , ..., s
( ). m , s - , . .
s
; . . R - m
.
1 , ..., s ,
, .-.
;
multivariate density , ; ,
. x1 , ..., x p
, ( )
( ) .
; ( )
; ( )
(
) . n
## { x i }1n = {( x1i , x2i , ..., x pi )}1n
i = 1, s , i ,
.
1 , ..., s
= ( x1 , ..., x p )
(1)
. . .
. . . .-
x , (1)
.
P ( x ) ( )
.
. . . :
. .
.; . . .; . . .-.
(1)
,
. :
;
; . : x
f ( x | , V )
f ( x | , V) =
1
( 2 )
p 2
12
exp ( x ) V -1 ( x ) (2)
2
N p ( , V ) ,
= ( 1 , ..., p ) T , x -
; multivariate statistical
analysis , -
l = M xl , l = 1, 2 , ..., p , V = vij
p
i , j =1
, x -
139
, vij
= M ( xi i ) ( x j j ) ,
x (
V = p ; ,
V = p < p ,
, p . ,
x
).
, (1)
N p ( , V ) , (2)- V
1
n
x i
(3)
i =1
( x i ) ( x i )
(4)
i =1
, , p
N p ( ,
1
- ,
V ) V
n
= nV
Q
( , [4] ) ,
|( n p 2) 2 exp 1 tr ( V 1 Q
)
|Q
,
p
( n 1) p 2 p ( p 1) 4
( n 1) 2
|V|
{(n j ) 2}
2
| V ; n) =
(Q
j =1
,
Q
.
0,
-,
| , ( | V
,
V
n
n 1
. (7) n1 n2
(1)
N p ( , V ) ; (3) (4) (5) ni S ni i -
, S n1 S n2
.
. . .- , , ,
,
.
.
1)
( ) ( , ,
. ) . .,
x p N p ( , V )
(1)
( 2)
, q p q x
x
.
~
-, V V
, :
(1)
(1)
( 2)
V =
V21 V22
= V11 V12 .
V
V
V
21
22
x
(2)
V11 V12
( 2)
( 2)
) N q (
(1)
), ) ( , [1], [2] ).
n N1 ( 0 , 2 p ) ,
n p
n p
T2 =
n ( ) S n1 ( )
p ( n 1)
p ( n 1)
1
[ (n1 1) S n1 + ( n2 1) S n2 ]
n1 + n2 2
S n1+ n2 =
+ B( x
n ( | S n | | V | 1) -
140
(7)
,
( p , n p ) ( p, n1 + n2 p 1) F
), T ( , [5] )
. ( ,
[1] ) S n
Sn =
n1 n2
( n1 n2 ) S n11+ n2 ( n1 n2 )
n1 + n2
= 1
V
n
n + n2 p 1
n1 + n2 p 1 ~ 2
T = 1
( n1 + n2 2 ) p
( n1 + n2 2 ) p
(6)
## M ( x (1) | x (2) ) = (1) + B(x ( 2) ( 2) )
(8)
,
B
,
:
=V
V
1
B
12 22
= V 11 V 12 V 221 V 21 ;
B N q ( p q ) ( B , VB ) -
n ( p q )
( VB V
).
,
( )
.
( )
( ) .
2)
.
[ (1)
], ., .
I. -
,
= *
; T , (6)
= .
II. ( , *
(4)
, V
j
j - ,
j = 1, 2 , ..., k .
V. x - p
p1 + p2 + ... + pm = p .
,
p1 , p2 , ..., pm
(1)
= ( 2) =
...
= (k ) =
nj
(x
U p , k 1, nk =
( j)
i
( j ) ) ( x (i j ) ( j ) )
j =1 i =1
k
nj
j =1
i =1
(x
( j)
i
) ( x (i j ) )
( j)
, xi ,
n j
j -
p -
( j ) ,
n = n1 + ... + nk (3)
## (1) = ( 2) = ... = ( k ) = V1 = ... = Vk = V ;
k
nj V
j
( n j 1) 2
(i )
x x (4)
.
, , , .
( , )
. ,
x - [ i -
p xi
x l , l = 1, 2 , ..., p
nj
j =1 i =1
(n k ) 2
( x (i j ) ) ( x (i j ) )
= ( x1i , ..., x pi ) T
[ l -
n xl
= ( xl 1 ,..., xl n )
].
( , [1], [2], [7] )
.
k 2
(
).
,
( x )
, ,
.
, , , :
(
);
;
.
( ) . r -
, ( , ) P ( x | r )
, , (1)
P (x) =
j =1
ni V
i
,
V
, V
i
],
i =1
nV
)
( 2)
=
; T
.
III.
( , )
(1)
x (1) , x ( 2) , ..., x ( m )
P ( x | r )
r=1
141
, r r -
(n)
( ). ( { x i }1 )
r , r k -
. , ,
, .
( ,
( . taxis + nomos ),
( , [2], [6] )
.
(
, i
xij
= 1 ,..., p ; j = 1, ..., n ) -
, ., -
ij
n
i , j =1
( , , ) , , ,
-
- .
( , [6] )
-
ij
n
i , j =1
( p ) ,
-
. ,
, ,
.
.
.
x ( . )
( ) q x
(1)
( . ) ( ) ( p q )
( 2)
;
(1)
F - x -
(1)
( 2)
q f (x ) .
,
, ,
.
( ) ( ) ,
xij ,
i = 1, ..., p , j = 1,..., n
ij ,
i , j = 1,..., n
.
142
, ,
( , ),
( )
.
.
, x = ( x1 , x2 , ..., x p ) ( x ) , z
= ( z1 , z 2 , ..., z m ) T
m << p
, n [ ekzo , + genos
; ] . [ (1)
] ,
, .
, (1) x -
,
.
. . .-
( ; ; . )
( ,
, , ).
.: [1] ., , . ., ., 1963; [2] . .,
., , . ., ., 1976; [3] . ., Bull. Int.
Stat. Inst., 1969, 43, p. 42541; [4] W i s h a r t J., Biometrika,
1928, v. 20A, p. 3252; [5] H o t e l l i n g H., Ann. Math. Stat.,
1931, v. 2, p. 36078; [6] K r u s k a l J. B., Psychometrika, 1964,
v. 29, p. 127.
; multidimensional scaling
( )
(
. ) ; . ,
, ( )
( , [1] ).
1952- ( , [2] )
.
.
. . - .
.
- ,
( , [3] ).
: . .
. .
. . .- .
.
,
, .
( , [4] ).
-
( , [5] )
. .
. .- ( , [6] ).
. .
.: [1] ., , .
., ., 1988; [2] . ., .: , . ., ., 1972, . 95
118; [3] S h e p a r d R. N., Psychometri, 1962, v. 27, 23,
p. 12540, 219-46; [4] C a r r o l l J. D., C h a n g J. J., Psychometri, 1970, v. 35, 3, p. 283319; [5] . ., , ., 1986;
[6] C o o m b s C. H., A theory of data, N. Y., 1964.
; multidimensional
random walk d ,
d
d Z , d > 1 .
R d , d > 1 . . . .
. ,
d
, R -
. . . .- 2 d
.
d 2 - , . . . . . . .-
( , [1] ):
d = 2 . . . (
). d = 3 0, 35 -
( d
> 3 ).
. . .- Z
( , [4] ). d
Z d+
- ,
Z +d 1
>3
.
. ( ., Z + Z
= 1 , Z +
d
= d ). 1- .
( ;
, [5] ). = 2
( , [5] ). ( , [5]
[3]- ).
Z d+ -
d 1 + K ( z )
,
K ( z ) z , | z | = 1
( , [5] ).
, , ( .,
2
. ). Z + ( , [5] ). -
( , [6] ). ,
, , ( ,
[7] ). -
, , - .
.
.: [1] ., , .
., ., 1969; [2] . .,
, ., 1972; [3] .,
., , . ., ., 1987; [4] . ., . ., . . . -, 1979, . 39, . 348; [5] . ., .: . . . . ,
. 13, ., 1976, . 535; [6] L e C a l l J. -F., Comm. Math. Phys.,
1986, v. 104, p. 471528; [7] C h a y e s J. T., C h a y e s L., Comm.
Math. Phys., 1986, v. 105, p. 22138; [8] B r i c m o n t J., F r o h l i c h J., Comm. Math. Phys., 1985, v. 98, p. 55378.
; multidimensional
random variable p ( p > 1) . ,
. . . i p -
= ( 1 , ..., p ) T . . .
Z + ( , -
) . . . .- , , , . .
.- .
i
Z +
( ) . , ,
d = 2 , 3
,
. . . . .
. . .- , . . . .-
,
143
## Pi1 , ...,i p = P { = U i1 , ..., i p }
, U i1 , ..., i p
= (U 1i1 , ...,U pi p )
U ji j , j .
Pi1, ..., i p
.
. . .-
.
.: [1] ., , . ., ., 1963.
; multidimensional random
process , .
;
model of structure of multivariate data
. n p X ; n , p
. ( p )
.
,
, : ()
,
() , -
, ; () ( . ),
;
() . .
( ) ,
,
; ()
.
() , X II = F ( X I ) +
,
, X I X II
( X II -
, X I - -
), .
, , ,
, ,
,
.
.
,
.
, , ( ), ,
.
. ,
,
144
( p > 1 ) ,
.
, ., .
, ,
. .
.
( ., ).
.
,
; ,
( ,
).
. .
, .,
. , .
.
.
. .,
,
,
.
., ,
.
. -
,
( , ).
,
( .,
) .
.: [1] T u k e y J. W., Ann. Math. Statist., 1962, v. 33, p. 167;
[2] T u k e y J. W., Exploratory data analysis, Reading, 1977; [3] L e b a r t L., M o r i n e a u A., W a r w i c k K. M., Multivariate descriptive statistical analysis, N. Y., 1984; [4] Interpreting multivariate
data, N. Y., 1981.
## ; visualization of multivariate data
; ,
( , ,
,
, . ) . ,
, .,
,
, . . . .
,
, . . . .
.
; multidimensional Wiener
process , m
, R - ,
X ( t ) = ( X 1 ( t ) , ..., X m ( t ) ), t 0 .
. . . .
M exp { i ( z , X ( t ) ) } = exp { t [ i ( a , z ) ( B z , z ) / 2 ] }
, a R
K K ; Y1 , ...,Yk ,
. , k - k - k -
, 1 k K . k - ( K 1)
mki , 1 i K ,
, B - . w ( t ) , t 0
. . .-
.
. . .- ,
( , [1] ). m = 2
( ). m = 3
P { lim w ( t ) = + } = 1
t +
> 1
( ln T ) / m1 / 2
P lim
t
T
inf w ( t ) 1 = 1 .
t >T
.: [1] ., .,
, . ., ., 1968; [2] . ., . ., , . 2, ., 1973.
## ; multiparameter Brownian motion , .
;
multiway channel
.
. . .
k .
nk ( ~
y1k , ..., ~
y n 1, k , mk1 , ..., mkN ) Yk k -
mki , i k
k - n - n k -
~
y1 ,..., ~
y n1 , k .
~
y ) ( y , ..., ~
1/ 2
n k K
N K
a = 0 , B = I ( I , R -
X ( t ) = ta + B1 / 2w ( t )
. N
). X( t ) , t 0 w ( t ) , t 0
1 k K , i k
1 mki M ki mki k - i -
R m - B .
B . . . . t 0 . . . w ( t ) , t 0
m
Y1 , ...,Yk p ( ~
y1 , ..., ~
y k / y1 , ..., y k )
1k
Nk
k - m1k , ..., m Kk
m 1k ,..., m Kk
= P { m ki = mki ,
i } ( -
Pe 1 Pe
k
mki - ). . .- R
R = { Rki } , > 0
Mki 2
N ( Rk i )
1 i, k K , i k Pe <
N . . .
R -
. R
.
. .- ( K = 2 )
. ( C. Shannon ) 1961- (
).
.: [1] V a n d e r M e u l e n E. C., A survey of multi-way
channels in information theory. 19611976, IEEE Trans. Inform.
Theory, 1977, v. 23, 1, p. 137; [2] .,
, . ., ., 1963,
. 62263.
; multimodal distribution ,
.
; Dudleys
metric , ; .
; Dudleys condition . { ( t ) , t T } T
; t T M (t )
= 0 , M 2 ( t ) < ,
T d ( t , s ) = [ M ( ( t ) ( s ) ) 2 ] 1/ 2 .
H ( , T , d ) , ( T , d ) -
.
1/ 2
j = 0 , 1, ..., n 1
(2)
( , [1] ). (2) k - k - ( )
. k -
S n -
k - .
(
).
,
, (2)
;
S n < S j , j = 0 , 1 , ..., n 1
(3)
k - k -
, k - S n - k -
( H ( , T , d ) )
Sn > S j ,
. .-. . . (t )
T - d
( , [1] ).
. . (t ) -
. (2) (3)
,
. F , .
.
( , ( ) , [2], [3] ).
.: [1] D u d l e y R. M., J. Funct. Anal., 1967, v. 1, 3,
p. 290330; [2] D u d l e y R. M., Lect. Notes Math., 1984, 1097,
p. 1141; [3] . ., . - ., .,
1985, . 5, . 327.
*
; embedded renewal process
,
( ,
).
1 , 2 , ... F - ,
,
S 0 = 0 , S n = 1 + 2 + ... + n ,
(1)
n , S n n > 0 . .,
I ( ) { S n I }
I
S1 , S 2 , ...
. n , n .
n > 0 ,
n > 0
146
. .
n -1,
{ S n }
16- .
o . (26, 16) .
[
; (2)
. ]
{ S n } n - .
S n n n = 135 - (
S 135 = 137 ). n
.
n , (1, 3, 5, 7, 9) (2, 4, 6, 8, 10)
, n n
( n = n n ) .
( S 0 , S1 , ... )
( ).
n = 31 ,
.
5 18 ,
o .
. T1 , ( 0 , )
, H 1
S H1 - . T1
. T1
, k
>n
=n
a -
n +1 , ..., k
,
T1 - T2
.
: k - k -
T1 + ... + Tk H 1 + ... + H k
. ,
( , )
.
, S n - {S n } .
,
.
. 1 , 2 ,...
ab a x
e ,
p( x) = a + b
a b b x
e ,
a + b
- ;
H 1
q = 1 F ( )
(
T1
; embedded Markov chain (t )
k k +1 , k 0 k
P
P { ( k + t ) A | A k } = P k { ( t ) A }
k
= ( k ) , k 0
(t )
. . . .- .
, ( k , k ; k 0 ) ,
(t )
= (t ) , (t ) = max { k : k t }
.
.: [1] . ., . ., . ., . , ., 1983; [2] . ., . ., ,
. 2., ., 1973.
; embedded branching
process 1)
. a b
ae
1 s
, ( )
.
.: [1] .,
, . 2, ., 1984.
i i
x>0
.
, i , -
b x
f ( s ) - . a = b ,
x<0
[ (0 , ) ( , 0 ) - ] -
. , H 1
;
, .
2) (t ) , t 0
.
. p ( x )
a e a x - . T1 -
; Ti H i -
H 1 -
= ( n ) , > 0 , n = 0 , 1, 2 , ... -
. . . . . ,
,
. . . .- ( , [1] ).
.
.: [1] A t h r e y a K. B., N e y P. E., Branching process, B.
Hdlb. N. Y., 1972.
; inclusionexclusion
method . ., A1 , ..., An
r
P (r ) =
f ( s ) -
2 f (s) = a + b
(a + b) 4ab s .
( b a ) b - .
(*)
pk =
2
(1 ) k r C kr p k
k =r
) ( b a ) b .
P { A j1 I ... I A jk }
## 1 j1 < ... < jk n
147
k r - r + t -
P ( r ) ( t -
) ( t - )
; ( , ).
. . . (*) .
.: [1] . .,
, ., 1977.
; circular
law . n
ij n 1 2
Hn =
ij ,
i , j = 1, ..., n , n -
; , M ij
0 < < , Re ij
>1
pij ( x) qij ( x) :
sup sup
n
= 2,
Im ij
= 0 , M ij
i , j =1, ..., n
>0
1 / (1 F ( x)) ,
( x ) = [ (1) (1 + x) + (1) (1 x ) ] / 2 ,
1 / F ( x) ,
x 1,
x < 1,
x 1.
0 < p <1 ,
x
,
(1 F ( x )) (1 F ( x ) + F ( x ) ) p , n
.
.-
p exp ( 1 /( p x ) ), x > 1,
P { Sn < x } q
q exp (1 / ( q x) ), x < 0.
, q = 1 p .
.: [1] D a r l i n g D. A., Trans. Amer. Math. Soc., 1952, v. 73,
1, p. 95107; [2] . .,
, ., 1986.
( ) ; support o f a m e a s u r e . T ( F ) = 1
2 +
i , j =1,...,n
< .
F T S -
x y
1 ,
p lim n ( x , y ) = ( x , y )
S ,
. t T , ,
S - , U
2 1 , x 2 + y 2 < 2 ,
( x, y)
=
x y
0 ,
x 2 + y 2 2 ,
2
F ( Re
x ) F ( Im k y ) ,
k =1
k , H n , y < 0
F ( y ) = 1 , y 0 F ( y ) = 0 .
.: [1] . ., . ., 1984,
. 29, . 4, . 66979.
; narrow
topology , , .
; narrow convergence , .
; topology of narrow convergence
, .
## ; Darling theorem 1 , ..., n F ( x )
, S n
( S ) = 1 , T
, .
148
, .
sup sup M ij
n ( x , y ) = n 1
## [ pij ( x ) + qij ( x) ] d x < ,
. ( . ,
= 1 + ... + n , n 1
(U )
> 0 . (
)
.
, ,
. .- . ,
,
. . .
( )
port d e s i g n
; sup , -
; carrier signal , .
; carrier frequency , .
,
; extension of measure
. R , , R -
. R A ( R )
, Y R , Y ( Y )
= ( Y ) ( ).
.
, ,
,
.
. A
= 0 Z Y Z A
= 0 ) ,
.
, ( ) . ( - ) ( Y U Z ) \ Z
; Y ,
Z Z
.
( ( Y U Z ) \ Z ) = ( Y ) .
; stable distribution
( ) .
.
X - Z X . :
1) , Z
- ;
2) 1 2 1 ( Z ) =
= 2 ( Z ) , Z
,
.
: , ,
,
- .
, X -
( Z j )1 j m , X ,
Z j , 1 j m
.
X -
, , .
( ) , ( ) . ( , [3],
. .- ).
.: [1] ., , . ., ., 1953;
[2] ., , .
., ., 1969; [3] . .,
, ., 1983.
,
; supporting point of a design , -
{ Ak , Bk > 0}
Y A , ( Y )
( , ( Z )
{Xk}
Z n = Bn1 ( X 1 + ... + X n An ) ,
n = 1, 2 , ...
S - . - . . .
( [1], [2] ).
. .- -
.
. .- , ,
0 < 2 . = 2
1
R ,
=0
0 < < 2 R ,
1
x < 0, H ( x ) = c2 x ,
H ( x )
= 0 H ( x ) = c1 | x | ,
x > 0 , ck 0 ,
c1 + c2 > 0 . . .- .
(
= 2 ; = 1 , c1 = c2 , = 0
1
, c1 c2 = 0 ) 2
G , , g
. G . .-
f . , = 1 .
. . ( [3] ):
log f ( t ) = i t | t | ( 1 i A ( t , , ) ) ,
()
A(t , , ) =
tg sign t ,
2
=
2
log | t | sign t ,
1 ,
= 1 ,
0 < 2 , | | 1 , 0 R1 .
. .- :
log f ( t ) = i t t
B ( t, , ) ,
(B)
B ( t , , ) =
exp i K ( ) sign t
2
+ i log | t | sign t ,
2
, 1 ,
= 1 ,
K ( ) = 1 + sign ( 1) -
; stable algorithm , .
; stable subspase , .
## ()- . (B) ()-
: A = B ; = 1 , A = B ,
149
A =
B , A =
. .- . . .- , ,
. .-
[4]- . S M
:
F S , , , b1 b2
B ; 1 ,
tg B K ( ) ,
2
2
A = B cos B K ( ) ,
2
A = B cos B K ( )
2
A = ctg
F ( b1 x ) F ( b2 x ) = F ( b x + a )
.
G ( x , , , , ) x -
b > 0 a .
. .- , = 0 = 1
. . .-
: x , , g ( x , , ) =
b1 + b2 = b , 0 < 2 , . .-
= g ( x , , ) ; x 0 , < 1 ,
g ( x, , 1 ) = g ( x, , 1) ; g ( x, , ) >
> 0 ; g ( x, , ) ;
g ( x, , ) 1 , g ( x 1/ , , ) < 1 , x > 0
.
S M -
= 0
. G . ., , 1, = 0 = 1,
= 0 ( ) C M
, An
. M . .- .
.
,
## log f ( t ) = | t | exp i sign t
2
0 <
2 , | | min 1,
= B K ( ) ,
C = B ;
1 , > 0 .
1
=1
= ( x 1 , , , 1)
: 1 2 ,
-
x g ( x , , , 1) = x g ( x
1 +
F ( a1 x + b1 ) F ( a2 x + b2 ) = F ( a x + b)
, F ( x )
, .
f ( t ) . .- ,
## a1 > 0 a2 > 0 b a > 0 ,
t t
t i t b
f
f
.
a = f a e
a
1 2
. .- :
1 , ..., n ,
n =
n
> 0 , n
k n ,
(*)
k =1
.
Fn ( x) n , {Fn ( x)} n
. .- . , . . F ( x )
, 1 , , 1 ),
= ( 1 + ).
. .- . .- ( = 1 )
( , ; ).
150
> 0 , 2 > 0 , b1 , b2
a > 0 b ,
12
B C = B
+ B2 .
arc tg
. .- . ( x , , , ) =
. R - . .
( , ),
( ,
) , . .-
. . - ( 20- 90- ) , ,
, . . ( , [4] ).
F (x) , .
(C)
( , , ) (B) C = B
b > 0 , a = 0 , F M
(*) , n
Fn ( x) , F ( x ) - .
, . .- . , .
.: [1] L e v y P., Calcul des probabilits, 1925; [2] L e v y P.,
Thorie de laddition des variables alatoires, 2 d., P., 1954; [3] . .,
, ., 1938; [4] . .,
, ., 1983; [5] .,
, . ., . 2, ., 1984.
; stable distribution i n a B a n a c h
s p a c e X
, :
b1 > 0 , b2 > 0 b > 0 a B ,
( b1 X ) ( b2 X ) = ( b X + a ) ,
.
(1)
<1
g ( x , , , ) -
x s g dx =
f ( x)
(dx) i w ( f , )}
(2)
S1
, f B , S1
= { x B : x = 1} ,
S1 - ,
f ( x) sign f ( x) (dx) , 1
tg 2
S1
( f , ) =
2
f ( x) ln f ( x) ( dx) ,
= 1.
S
1
1 (2)
. . ; . .-
, .
. . : B - . .
, B -
. . .-
,
,
. ,
.
.: [1] L i n d e W., Infinitely divisible and stable measures on
Banach spaces, Lpz., 1983; [2] W e r o n A., Lect. Notes in Math.,
1984, 1080, p. 30664.
; ; ; stable distribution; i n t e g r a l t r a n s f o r m s ,
g - .
, . .-
,
.
= 1 ( . .- )
(B)-
( Re s 0 ) :
sx
exp ( s s sign ( 1 ) ), 1;
gd x =
exp ( s + s log s ) ,
= 1;
(1 )
(1 s ) s 2
2
sin s
(1 s )
.
it
w k (t ) =
cos
cos
(k it )
(k it )
(1 i t )
(1 i t )
= 0 , 1 .
< Im t < 1 .
.: [1] . ., , ., 1983.
; stable distribution
o n a g r o u p G { t , t 0} . { t , t 0}
: t > 0
, k
( f ) = exp { i f ( x0 )
sin
b = b1 + b2 , 0 < < 2 , ,
. .- . = 2
. 0 < < 2 . .-
= 0.
( 1 < Re s < )
(1)
( X ) X ,
, , x < g ( x , , 1, , )
t ( A) = t A + X t ,
(*)
X t , G G - ,
{ t , t > 0} , G .
, T = { t , t
> 0}
, t , G
Aut (G )
t , s > 0 t s = t s t 0 t ( g ) e ( e , G ) g G . , G - ,
: E G
( E ) = ( 1 ( E )). f
A :
( f )( x ) = f ( ( x )) ( A) ( f ) = A ( ( f ))
( f ) ( A ) .
G
A { t , t 0}
G
> 0 ,
(*) , ,
. t > 0
X t = 0 , G
T X t G , t
151
. Aut (G ) ,
; 2) t , ,
Aut ( G ) - ; 3)
c ( 0 , 1) G X
, I
= c A + X , A { t , t 0}
, G ,
; 4) t T -
, ( A)
, , ,
R*+
() =
. 4)
G
. X = 0 ,
.
T ( , c ) , -
T ( , c ) .
( ( Z)
( ) .
.
G - M 1 (G ) -
, , T , { t = t ( ), t > 0} G -
,
1) t s = t + s , t - 2) t ( s ) = ts ,
## t , s > 0 . { t , t > 0} lim t = 0
t 0
1) 2) t , s 0
. t ,
t 0 2) , , T . { t , t 0} , ,
, ( t )
= c t , t 0
Aut (G ) c ( 0 , 1) c
. , { t , t
( , c)
0}
, ,
, c > 0 ( Z) { c , n Z }
n
{ c , n Z } ) ( , [2] ).
G , T = { t , t 0} , G -
, { t , t
0} ,
G -
= K ,
K , G K .
{ t , t 0} , T , K T .
, 0
C K (T ) = { x G : t ( x ) K K , t 0},
C (T ) = { x G : t ( x ) e, t 0}
, : ) C K (T ) G - C (T )
K - ;
) C (T ) G - ; ) t 0 t ( G \ Ck ( T ) )
= 0;
) C (T ) T , -
0} , t 0
( , [1], [2] ).
M (d )
{ t , t
H n ( , [3], [4] ).
, T
.
( , [7] ).
.: [1] H a z o d W., Lect. Notes in Math., 1982, v. 928, p. 183
208; [2] H a z o d W., Lect. Notes in Math., 1986, v. 1210, p. 304
52; [3] B a l d i P., Lect. Notes in Math., 1979, v. 706, p. 19;
[4] D r i s c h T., G a l l a r d o L., Lect. Notes in Math., 1984, v.
1064, p. 5680; [5] H a z o d W., N o b e l S., Lect. Notes in Math.,
1988, v. 1379, p. 90106; [6] H a z o d W., S i e b e r t E., Semigroup Forum, 1986, v. 33, p. 11143; [7] H a z o d W., S i e b e r t E., J. Theor. Probab., 1988, v. 1, p. 21126.
( , [5] ).
.
I = I ({t } ) = { Aut ( G ) : ( t ) = t , t 0}
t
( ) .
Z = Z ({ t }) =
= { ( , c) : Aut (G ), c > 0, ( t ) = t c , t 0}
t
( . ) (
) . , I
( , 1) Z .
( , c ) = c : Z R *+ = ( 0, )
. I Z :
1) Aut (G ) - Aut ( G ) R*+ -
152
t = t K ( , [6] ).
;
;
; s e r i e r e p r e s e n t a t i o n o f a stable distribution .
1 2 ,
g ( z , , ),
z 11 g ( z 1 , , ), 0 < 1
(z) =
, g . .- . ( z )
= 1 , = 0 ( z ) z = 1 ,
.
( , )
(z) =
z k 1 ;
> 0
.
(1)- ,
k =1
ak 0 < 1
ak =
(1) k 1
f ( x; , , , c) =
( 1 + k)
sin ( 1 + ) n
2
( 1 + k)
, x . .-
( , 0 , ), , [1]-.
.: [1] . ., , ., 1983.
; ; ;
stable distribution; s i m u l a t i o n o f r a n d o m v a r i a b l e s ,
.
., 0 < < 1 , = 1 , = 0 = 1
. .- Y ( ) , ( 0 , 1)
1 , 2
sin 1
sin
1
1 (1 )
log (1 2 )
(1 )
; stable function of
distribution , .
; index of
stable distribution ( , ).
(t ) -
tg 2 ,
( t, ) =
2
ln | t | ,
1 ,
= 1 .
e i t x exp { i t c | t |
.
.
1) = 2
;
2) = 1
;
3)
f ( x)
e 1 2 x , x > 0 ,
= 2 x 3
x0
0,
= 1 2 ;
4) = 0
.
> 0
f ( x ; , , , c ) f ( x; , , , c) =
c 1 f ( ( x y ) c 1 2 ; , , 0, 1) ,
1
= 1 x y
2
ln c ; 1, , 0, 1 , = 1
c f
c
=0, c = 1
f ( x ; , ) = f ( x ; , , 0, 1)
,
f ( x; , ) = f ( x; , ) .
(1)
(
) , , , , c , 1 1 ,
0 2 , c 0
- .
.: [1] C h a m b e r s J. M., M a l l o w s C. L., S t u c k B. W.,
J. Amer. Statist. Assos., 1976, v. 71, p. 34044; [2] . ., , ., 1983.
t
ln ( t ) = i t c | t | 1 + i
(t , a )
|
|
y
t
(1 + i
( t, ) ) }d t
|t |
. 1 2 a
.
g ( x , , ) -
sin (1 )
1
sin 1
1
2
( 0 < < 2 )
p ( 0 < p < )
.
.: [1]
, ., 1985.
;
stable process , ,
. . .-
: . . (t ) , ( 0 ) = 0
153
( , ) , C
> 0 , ( 0 , 2 ) [ 1, 1]
M exp{ i (t ) } =
## ; stable random set
= exp t i C | | 1 i
( , )
|
A R
1,
( , 1) =
2
2
M (t)
< , s ( 0 , ) , M (t )
A1 , ..., An
n A
ln | | .
- . . . .
.
= , t > 0 -
= = 0 (t ) - M exp { i (t ) } =
(t ) ,
s
n n -
( , ) = tg
; robust estimator ,
= exp { t C | | } , = 1 - .
## , ; stable distribution type
, .
; stable state ,
, x
:
=0
lim p ( t , x , x ) = 1 ,
t0
1/
1 ( k t ) k ( t )
. = 0 = 2
( t ) = t .
A1 U ... U An
q x = lim
t 0
1 p( t , x, x)
< ,
t
, .
.: [1] . ., , ., 1964.
. . .
; stable semigroup
{ Q ( t )} ,
t > 0 , t t - , G
. G ( , . VI, 1 ): X , X 1 , X 2 ,...
, G
S n = X 1 + ... + X n . G
> 0
,
d
S n = cn X + n
x - , = inf { t : t 0 , t x } , x -
(*)
, G ;
G (*) n = 0
, G
. t t - ( ) .
t 0 o Qt
). E
,
; E
( ,
).
( ),
## ; stability o f d e compositions of probability distributions
. .
. F , R -
Y = { w = ( F1 , F2 ,... ) : F j F, I w = F1 * F2 * ... F }
.
.: [1] .,
, . ., . 2, ., 1984.
G F
M -
M (G ) = { w : I w = G } F - F( , G ) =
= { F : L ( F , G ) } , L .
, t , 0 .
154
P { > t } = exp ( q x t ) ,
t 0 . q x = 0 . . ( =
Qt ( x ) = G ( t ( x t ) )
n cn
p ( t , x , x ) , x t x -
( , G ) = sup { ( F , G ) : F F ( , G ) } ,
( F , G ) = sup{ ( w , Y( G ) : w Y( F ) },
( w , w ) = sup { L ( F j , F j ) : j 1 },
,
F ( , ) -
( w , Y ( G )) = inf { ( w , w ) : w Y( G ) }
. , 0
(, G )0
(*)
. d ,
:
x 0 ( x ) 0 ,
. L F - . (*) ,
( , [2]
.- )
.
,
. . . , [1]- F = F1 * F2 ,
Fi F w -
.
.: [1] . ., , ., 1960; [2] . .,
, ., 1986.
; stability i n
p r o b a b i l i t y , ( )
.
( )
; stability o f q u e u e i n g s y s t e m s
.
.
.-
( u , f , ) , u
( Q ( r ) , Q ) ( d ( P ( r ) , P ) ).
f - F
= F (P)
P
Q
P = { P } Q = {Q }
,
( u
, ., , - , ).
., , F , r
P
Q
P ( r )
P Q ( r ) = F ( P ( r ) )
F (P) = Q
u = ( u1 , u 2 ,...) ,
= (1 , 2 ,... ) , R R l -
, U k -
lim P { k B } = F (B ) .
(r )
. , u k u k
F (B ) F
( B ) - .
(r)
.
( , ; , ;
, ;
).
. .
e
.- , ., k , k
( k + 1) -
s
, k k -
, x
= max ( 0 , x ) .
u = ( u1 , u 2 ,... ) ,
uk = ( ke , ks ) ,
X k = ks ke , = (1 , 2 ,... ) , k k - [ { vk } , k
vk +1 = ( vk + k )
],
w = ( w1 , w 2 ,... ) , { vk }
P {w1 B } = lim P { k B } ,
k
w k = sup
X i .
j 1 i = k j +1
V = { } v = f (u ) . u
Q
( ,
, , . ),
( , . ). U = {u }
, U V P Q
{( k
(r )e
,k
(r ) s
)} , r 1 -
( r ) .
, X = { X k }k 1 , M X 1
< 0;
X r X -
; r
(r )
M ( X 1 ) + M ( X1 )+ < ;
> x} 0
(r )
{ w k }, { n( r+)k }k 1
P { 1
(r )
>
r , n w .
155
d ( F1 , F2 ) = sup F1 ( x) F2 ( x)
k +1 = R ( (k + e ks i ke ) + ) , k 1
, ( F1 , F2 )
( F1 F2 ) ( x) , F1 F2 -
F(t ) d t ;
.
.- . X rj X j - ( .,
) (r )
. ., : X X
-
w = ( w1 , w 2 , ... )
.
( ) ( ., , , ;
) .
.-
( > 1 ), ,
e
( r )e
. ., {( k , k )} {( k
(r )s
, k
)}
( x) -
## M ( sj m ej ) < 0 ; M j ( (jr ) s ) C < ,
> 1 M ( X j ( r )+ )
X j ). (r )
: X X
, r - M X 1
< M X 1 < 0 ,
(r )
( W (r ) , W ) C ( ( F (r ) , F ) + ( Q(r ) , Q ) )
( X
= ( x1+ , ... , xm + ) -
, r - > 1
, F (x ) F
(r )
j
{ k } - .
## e = (1, 0 , ... , 0 ) m ; x = ( x1 , ..., xm )
i = (1, 1, ...,1)
; R ( x) , x
+
x x
F (x) = P { X 1 < x };
Q( x) =
< 0 , M ( X 1( r )+ ) C <
> 1 ,
( W , W ( r ) ) C1 [ ( ( 1e , 1s ) , ( 1( r )e , 1( r ) s ) ) ] 11/
, , ,
W ( x ) = P {w1 < x } .
r -
>0
(r )s
M exp ( j
) C < -
( W , W ( r ) ) C 2 ( ( 1e , 1s ), ( 1( r ) e , 1( r ) s ) )
d (W ( r ) , W ) C1[ d ( F ( r ) , F ) ]1 1 /
| ln (( 1e , 1s ), ( 1( r ) e , 1( r ) s ) ) |
.
r - > 0
(r )
M exp { X 1 } C <
d (W ( r ) , W ) C2 d ( F ( r ) , F ) ln d ( F ( r ) , F )
. ( d ) , , , ,
(r )
( ). X X
, , , ,
.
. m -
= ( ke , ks ) ( )
.
{ 1 , 2 , ...} , k = ( k ,1 , ..., k ,m )
,
156
. .
.
uk
. {( j , j )}
uk
= ( ke , ks ) k = ( )
. n -
q n
.
= ( q1 , q 2 , ...) ,
n { qn+ k }k 1 -
< ,
. M 1
## q k = I { ks > ke } + I { ks1 > ke 1 + ke } +
+ I { ks 2 > ke2 + ke1 + ke } + ...
. I ( B ) , B .
## qn( r ) , q ( r ) n , r {( (jr ) e , (jr ) s )} .
e
( r )e
. , { j } { j
;
{( (jr ) e ,
(jr ) s )}
r {( j , j )}
M 1( r ) s
< ; , j 0
M 1s
P 0s
e
k
k=0
= 0 = 0.
d X ( t ) = b ( t , X ) dt +
, k
s
{( k , k , k )}
.
. ,
ke
ks
uk = ( ke , ke , ks , ks ) ,
0 , 1s , 1s + 2s , ...
x - b ( t , 0 ) = 0 ,
r ( t , 0 ) = 0 x , (1)
.
.- ( , [1] ):
d X (t ) = grad x b ( t , 0 ) X dt +
s ( t ) = 1s + ... + s (t ) 1 , ( t ) 1s , 2s , ...
t ; q ( t ) , t
(
), q ( 0 ) = 0 ; { q ( x ), x 0}
t
0} .
1 ), ( k( r ) e , k( r ) s , k( r ) e , k( r ) s ) -
.
:
{ k = e ( x ) s ( k ) e ( k 1) + s ( k 1); k 1 }
, M
< 0 ; { e ( t ); s ( t ) } (r )
(r )
## { e ( t ) ; s ( t )} ; M [ e ( r ) (1) e ( r ) ( 0 )] M [ e (1) e ( 0)]
r
M [ s ( r ) (1) s ( r ) ( 0 ) ] M [ s (1) s ( 0 )]
, { q ( x) }
r { q ( x) }
.
.: [1] Handbuch der Bedienungstheorie, Bd1, B., 1983; [2] . ..,
, ., 1980; [3] . ., ,
., 1978.
( )
lity o f M a r k o v p r o c e s s e s
; stabi
. -
grad x r ( t , 0) X d w r ( t ) ;
(2)
r =1
( r
(1)
k 1 .
.
e ( t ) , t ,
{ q ( t + x ); x
. b ( t , x ) r ( t , x ) t -
X , b , r , R l - , w r -
, u = ( u1 , u 2 ,...) ,
( t, X ) d w ( t)
r
.
e
r =1
(r )
.- (
)
( , [1], [2] ). .,
X ( ) -
X ( 0 ) 0 sup | X ( t ) |
t> 0
.
.-
( , [1], [2] ).
.
{ 0 , 1, 2 , ...}
. P = { pi , j } ,
( ., 0
), q n = ( q n ( k )) k 0 , n
. :
P (r ) , r 1 ,
{ , ( r ) , r 1 } , r
sup
n
qn ( k ) qn
(r )
(k ) 0
k =0
i , j
pij( r )
pij
q0( r ) ( i )
q0 ( i )
.
.
.: [1] . ., ,
., 1969; [2] . .,
, . ., ., 1969; [3] . .,
, ., 1978; [4] . ., .:
. , ., 1984,
. 2227.
( )
; stability o f c h a r a c t e r i z a t i o n o f d i s t r i b u t i o n s
157
. . .
. .- . [1] ( 38 ) ,
. ( )
; , = F1 F2
( , )
.
. . . ,
.
( , [2] ).
.
.
, ,
= ( F1 F2 , ) 1 , 2
( Fk , k ) =
,
= O( 1
log 1/ ) .
. . . . . ( , [3] )
( , [4], [5] )
.
[6]- , . X , Y
, A X , B Y
J :X F
D = ( JA ) I B C = A
I ( J 1B )
(X , Y , F ,
A , B ) A B
. X X
. C
, x X
( x , A ) + ( J x, B ) 0 ( x, C ) 0
. D - ( , ) :
1) A - ;
2) B - ;
3) J , A
.
= { ( F1 , F2 )} , Fk F ( F , R1 -
), Y = F , J ( F1 , F2 ) =
= F1 F2 , A = X , B = {} , -
X
( ( F1 , F2 ) ( F1, F2) ) = L ( F1 , F1 ) + L ( F2 , F2 ) , = L
, L .
.: [1] L v y P., Thorie de laddition des variables alatoires,
2 d., P., 1954; [2] . ., . . . .,
158
## 1951, . 15, 3, . 20518; [3] . ., , ., 1960; [4] . .,
. ., 1986, . 31, . 3, . 43350; [5] .
., . ., . .,
, ., 1972; [6] . .,
. ., 1976, . 101, 3, . 41654.
( )
;
stability o f r e g e n e r a t i v e p r o c e s s e s
. V = ( V1 , V2 , ... )
, S 0
= 0 , S1 , S 2 ,...
, j
= S j S j 1 , j 1 , P { 1 1 } = 1 a ( k ) =
= P {1 = k} , k 1 . N 0 < < 1
, 1 k N } = 1
, { a ( )} N ( N , ) -
{ k : a ( k )
. V V , a a , N ( N , )
; :
)
M ( 1 ) g
M 1 g ,
,
n 1
n T
, d
, c
g , , N , - ;
)
## > 0 0 < g1 <
M exp ( 1 ) g1 , M exp ( 1 ) g1
, 0 < 0
c1 = c1 ( g1 , , N , )
## sup d (Vn , Vn ) inf { max d (Vn , Vn ) + c1 exp ( 0T )}
n 1
n T
, 0 c1
.
. : V
V - ; V V .
,
, .-
.
.: [1] . .,
, .,
1978.
( )
;
stabilit o f a s t a t i s t i c a l p r o c e d u r e
X
.
( .,
), . ( )
.
.- ,
; .,
.
( , [1] ). x
, tn
, -
= n 1 x s Styudent
. t n
. n - Styudent
t n
. n ,
n - Styudent t n -
,
t n .
. t n -
. .
n2 =
(x x)
i
. n
.
, , , n
2
n
, ( n 1) -
. ,
. .
,
.- : ( x )
x . F ( x ) -
( x ) - ,
, , F ( x )
x
.
.-
, . ,
X 1 ... X n
X = ( X ( k +1) + ... + X ( nk ) ) (n 2k ) ,
k = [ n ],
< 1/ 2
,
. .
( , ( ) ), ,
.
-
.
.
.- . 20- 1- -
, X -
,
X (1) , ..., X ( k ) ( )
( , X ( n k +1) , ..., X ( n ) - ) X ( k +1) - ( , X ( n1) - ) .
.: [1] ., ., , . ., ., 1973; [2] ., , . ., ., 1991.
( )
;
stability o f a s t o c h a s t i c m o d e l
.
. ( , ( ),
, ( ), ,
( ), , ). .
,
. , ,
( ,
( ) ;
).
( ) ;
;
; stability o f a
s t o c h a s t i c m o d e l;
q u a n t i t a t i v e e s t i m a t e s , , .
.- .
X Y
( X , ) (Y , ) , F : X Y
( ).
A X B Y ,
. ( A , B , F ) (
) .
X A B = Y ;
FA - ,
A X
B Y
C = A
I F 1 ( B ) ,
C X
( ) .
. A B -
.
159
1.
> 0 = ( ) > 0
( X ) ( X , A ) + ( FX , B ) < ( ), X X
( FX , FA ) <
, B X X .
2. > 0 = ( ) > 0
,
( X ) = ( X , A ) + ( FX , B ) < ( )
( X , C ) <
, > 0
= ( ) , ( X , C ) < ( X ) <
, X X
.
, X
Y ( ,
, ) , .
.
. .,
, ( ., , )
; ( ,
) ,
.
,
( .,
).
. .
., ( ., ,
) .
. .- 1- . 2
.
.
. ( )
,
( , ( ) ). ., , . .-
, , X Y
, , .
1 2- ( ) ( ) .- ,
.
160
. ,
, .
. .
.
. -
. ( ) (
( ) - )
. , ( ) -
[ , ( ) - ]
.
-
.
,
,
.
.: [1] Z o l o t a r e v V. M., Bull. Inst. Statist. Inst., 1977,
v. 47, 2, p. 382401; [2] . ., , ., 1986;
[3] . ., . ., , ., 1988.
( )
; stability o f d i f f e r e n t i a l e q u a t i o n s
w i t h r a n d o m p a r a m e t e r s
,
. .
F ( x, t , ) ; x , F R n , x
x -
,
xt = F ( x , t , ) , x ( t 0 ) = x0
. X
(1)
( x0 ,t0 )
( t , ) , (1)
F ( 0 , t , ) 0 , , x = 0
## (1) . > 0 > 0
, x0
sup P { | X ( x0 , t0 ) ( t , ) | }
t t0
, .
, t , X
( x0 , t0 )
( t , ) 0 , x = 0
.
x0 (2)
(x , t )
P sup X 0 0 ( t , )
t t0
, x = 0
.
p . ( p -
) .
,
. ,
( , [1], [2] ).
. . ,
. p p . , p p (
A > 0 ,
>0
M X
( x0 , t0 )
(t )
A | xk | p e t
)
.- .
.
p > 0 p .
dx dt = F ( y ( t , ), x ) ( F ( y , 0 ) 0 )
x = 0 . ( X ( t ), Y ( t ))
x = 0 .-
.
.: [1] . .,
, . ., ., 1969; [2] . .,
, ., 1969.
; parameter of stable ,
, , -
; stability theorem i n t h e q u e u e i n g t h e o r y
( ) ( , ( )
).
= { n }
( ) u = { u n } n +1
= f ( n , u n ) , n 1
. u n n R R
, k 1 , l 1 .
. .- .
I. ,
( ,
; ).
v , ( , ( )
).
II. -
( , ;
).
III. ( .
) ( , ;
).
,
- .
.: [1] . .,
, ., 1980; [2] . .,
. ., 1975, . 20, . 4, . 83447;
[3] . ., . ., 1976, . 101, 3, . 416
54; [4] . .,
, ., 1978; [5] . ., . ., 1975, . 20,
. 3, . 57183; [8] Handbuch der Bedienungsheorie, Bd 1, ., 1983.
;
;
; stability theorem; m e t r i c a p p r o a c h
l
. { n } R -
n +1 = f ( n , u n ) ,
n 1
(1)
{ u n } R k - .
d ( X , Y ) d ( FX , FY ) FX FY
. : f
l ( f ( u , ), f ( u , )) l ( , ) + k ( u , u )
, { u n } (1)- 1 { n }
d ( j , j ) d (1 , 1 ) +
j 1
d ( ui , ui ) ,
j = 2 , 3 , ...
(2)
i =1
,
l
l , k
R , R - .
, d
, (2)
. { u n } { u n } , (2)
, , .
( ., f ).
.
.: [1] Handbuch der Bedienungstheorie, Bd 1, B., 1983; [2] . ., . , 1976, . 101, 3, . 41654;
[3] . ., , ., 1986, . 416.
161
; ;
; stability theorem; t e s t
f u n c t i o n s m e t h o d
.
v = f (u ) ,
.
l
{ n } R - , 1
n +1 = f ( n , u n ) ,
n 1
(*)
, { u n } k
, R - ,
{ n( r ) }
1( r )
= 1, 2 , ...
(r )
{ u n }
(*)- . k , l ,
R k R l - ;
( X , Y ) = inf { > 0 : P { j ( X , Y ) > } < }
j
R - X Y ; j = k , l ; G ( x ), x 0
sup ( n , n( r ) ) < ,
n 1
inf { T C N + C1
T
. . ,
.
.: [1] . ., , ., 1978;
[2] . ., . ., 1975,
. 20, . 3, . 57183.
;
; ; stability theorem; r e n o v a t i o n s m e t h o d
(
).
, = { n } ( ) -
= { un }
n + 1 = f ( n , u n ) ,
n =1
, N
WN ( x , y ),
x , y Rl
x , y Rl
= ( N )
, ,
W N ( x , y ) N l ( x , y )
( u1 , u1( r ) )
WN ( f ( x , u1 ) , f ( y , u1( r ) ) ) WN ( x , y ) N ( x , y ), x , y R l
. :
{ I ( An )} , P ( A0 ) > 0 , w k + 1 =
= f ( w k , uk )
x, y
N ( x, y)
; {
)C <
(r )
( u1 , u1 ) 0 ( 1 , 1 ) 0 ; r
(r )
n 1
. .
(r )
< 1 ( )
(r )
} (*) -
; { An } {
; {w
(r )
(r )
} { (r ) } -
. , { I ( An )}
P ( A0 ) > 0 ,
sup ( n , n( r ) ) 0
: ( u1 , u1 )
(r )
} ( .,
) u
. : r -
162
P {{ n + k } B } P {{w k } B }
(r )
= {w n }
, n
, , { u
(*)
M N ( x , y ) 1
sup M G
n 1
.
An , [ n L , n ]
, N W N ( x , y )
1 G(n) } =
nT
, C1 C G - .
G( n ) < .
= ( N ( ) ) , N ( ) ,
1 ( )
lim inf P
r
>0
j =0
A(j r ) P
j =0
Aj
j ( u1 , ..., u L + j ) - , w
(r )
r w - .
. .- , .,
sup d
n 1
sup
n T + L
( n , n( r ) )
## min max sup d ( j , (jr ) ) ,
T
j T +L
d ( T ( u n T L , ..., u n 1 ), T (u ( r )
n T L
+ 2 sup P
n 1
A j + P
j = n T
n 1
n 1
j = n T
, ..., u
(r )
n 1
(r )
Aj
)) +
.
, ,
r
,
( , ( )( )
),
.
.: [1] . .,
, ., 1980; [2] . .,
.: , ., 1980,
. 5257.
; stopping
time , t - t 0 { : ( ) t } At ,
{ At , t 0} ( , A , P) = ( ) ( ) . t 0 t -
{ : ( ) < t } At
, { At + , t
As . -
s >t
{ At , t
0}
. .-
( ) = inf { t 0 : ( t ; ) }
, ( t ; ) ( M , B )
{ At , t
0}
.: [1] ., ,
. ., ., 1975; [2] . ., . .,
,
., 1982.
*
; stopping time
r e l a t i v e l y t o f l o w : , +
, { , F, P }
t } Ft
t , B B
( X , B) ,
(t )
Ft F ,
) F P ( , A F , P ( A ) = 0 B A B F , F P
)
F0 ;
) 0 s < t s t
Fs Ft ( Ft t -
);
) t 0 Ft =
IF
( Ft , t
s >t
) , {Ft }t 0
{ Ft }t 0 ( t ) , . .- ( t ) [ 0 , ] . ., ( t ) , . .-
.
, {Ft } . . . ,
t 0 { t } I A Ft A F
F ; F
( )
. F At Ft , At I { > t } , F
.
. .-
.
1) 1 2 . .-, 1 2 1 2
. .-.
## 2) n ( n = 1, 2 , ... ) . .-, lim n
B ( , inf { } = + ).
{ ( s ) B } , s
Ft
.
. { u n } { I ( An )} -
. Ft
. t
. .-, At +
. ,
t [ 0 , t ] -
d . d ( w1 , w1 )
0}
.
[ 0 , ) - ( t )
, { Ft }t 0 . . . . .-
lim n . .-.
3)
1 2
.
4)
F1
.-,
F F F 2
2 <
F 2
1 2 . .- 1 2
F 2 , F1 F 2 .
163
5) n . .- n ,
Ft =
F n ; n
F =
.
6) n . .- n , n
< , Ft =
>
IF
VF
; n ,
n <
F =
VF
<
7) 1 2 . .-, A F 2
I { 2 1} F1 , A I { 2 < 1} F
.
, . . t -
F
Ft =
Fs ).
; Dyson distribution
;
( . )
. . .
. . .
.
, ; debut
of a set , R + A - ,
DA ( ) = inf { t R + : ( t , ) A}
D A , , { t R + : ( t , ) A }
n . .- ,
n < n , - . . .
. . .
. { Ft } , { R + , B+ , P }
( , t ) = I{t > } ,
= [ 0 , ) , B+ R + -
, P , B+ - ,
, ( t , ) A );
, D A ( )
. ( )
= -
( At ) t 0
8)
. . . .-, A F *
A { } F ,
At - P
( At ) t 0 .
, .
.: [1] ., ,
. ., ., 1975.
; defect ,
{ } F ,
. ( ( t ) , ( t ) ; t 0 )
. , G ( t ) = P { k t } , M k = m
{ = } = F
n =
.
9) n . . n , . .-;
n n
G* ( t ) =
; a s y m p t o t i c deficiency , ( ) .
, , A F .
,
164
( 1 G (s) ) ds m
.
10) . .-, A . .,
= F
n 1
. .
A,
,
=
+ , A
. . F
k ,
. .-.
. ., A F .
k =1
n - ( n , , ),
( t ) ( t ) - t
( , A , P )
( t ) = t sup { s t : ( s ) ( t ) } .
= + . .
, R +
, . .-.
. .- :
*
t R +
s <t
R +
.: [1]
, . , ., 1985.
) ,
, ; defect o f a
random
w a l k / s p e n t w a i t i n g t i m e
. S n , n = 0 , 1, ...,
, g ( n )
.
d = g ( g ) S g 1
( )
, g = inf { k 1 : S k g ( k ) }
1- , .- , k n n 1 ( , ( )
.
1 , 2 , ...
).
.
,
1-
.-
. , d n = o( n ) , ,
S 0
= 0 , S1 = 1 ,
S 2 = 1 + 2 ,, S k = 1 + ... + k
g .
[ g
= S g g ( g )
( ) ]. g ( t ) = x =
## = const g , g , g ( x), ( x), ( x)
. k , ( x)
= g , ( x) = g
( x) -, .
P { ( x) > u , ( x ) > } x ( , [1] [3] ).
## a = M k < , 0 < = D k < , k 0 ,
, n , x
2
P { ( x) = n , ( x ) u , ( x ) } =
n x/a
x 2 a 3
1
x 2 a 3
1
a
P { 1*
u +
(t ) = ( 2 ) 1/ 2 exp ( t 2 2) ,
## .: [1] ., , . ., . 2, ., 1984; [2] . ., , 2 ., ., 1986; [3] . ., .
. . . 1955, . 19, . 24766; [4] . .,
. ., 1984, . 29, . 2, . 41011.
( ) ;
deficiency o f a t e s t ( ,
) , 1- 2-
d n = k n n ,
k n , n .
, ,
, n k n ;
d n .- d
= lim d n
, d
. d n d - ,
, ;
k n -
: 1
[ k n ] [ k n ] + 1
,
= 0 + t n 1 2 , t > 0
n, t -
n, t - ,
(
>0
lim n ( n, t n, t ) = r (t )
).
,
. ,
, u
= 1 (1 ), t ( , ) =
= I 1 2 ( 0 ) ( u + u ) , I () = M [ ln p ( X , ) ]2 ( ). ,
.
1* ( , [4] ).
d = 2 r ( t ( , ) ) ( u + u ) ( u )
t } d t + o
- ( R )
p ( x, ) . n
{ ( x) u, ( x) > } = { ( x u ) > u + }
, . .
.- , .
.,
> 0 = 0 -
= k n [k n ] .
n, t n, t n1 2 . , .-
. 2-
.
2- . 3-
r ( t ) 0 , d = 0 ,
, .
. [1]- . .-
20-
## 70- ( , [2] [4] ). [5]- n, t n, t
. ( )
.- [6]- .
.: [1] H o d g e s J., L e h m a n n E., Ann. Math. Statist.,
1970, v. 41, 3, p. 783801; [2] P f a n z a g l J., .: Developments in statistics, v. 3, N. Y. [a. o.], 1980, p. 197; [3] . ., . . . .-. , 1982, 5,
. 1826; 6, . 2330; [4] . ., .: Proceedings
of the International Congress of Mathematicians. Warszawa, 1624,
Aug., 1983, v. 2, Warsz., 1984, p. 106379; [5] . .,
. ., 1985, . 30, . 2, . 26988;
[6] . ., . .,
1989, . 34, . 3, . 491504.
165
; decoding
; l i s t decoding , .
()( )
; b l o c k decoding , -
, . . .
. , ., ,
.- ( , ( )
) .- .
( ) , L
( .- ), - ( > 0 )
( ).
. .,
X , :
~
P { X X } - ~
2
; M ( X X ) -
~
( X
).
.
.
. .
,
, .-
. , . : , , , . .
.-
( ) .
y1 , ..., ~
y n )
. ., ( ~
x1 , ..., ~
xm ) ( ) .- ( ~
, ,
, ( ~
y1 , ..., ~
y n ) ,
(~
x1 , ..., ~
xm ) ;
. .-
( ., ).
,
.- .
.
, .,
.
.: [1] . ., ., , . ., ., 1963; [2] ., , . ., .,
1974; [3] .,
, . ., ., 1966; [4] . ., . ., , ., 1982; [5] .,
, . ., ., 1966.
; s e q u e n t i a l decoding ,
( ).
; a l g e b r a i c decoding ,
.
; c o m p l e x i t y o f decoding ,
( ) .
166
()
; m a x i m u m l i k e l i h o o d decoding ; ~y
x ,
p( ~
y | x) ,
~
x = arg max p ( ~
y | x ) . x
,
. . .- , , ( , .- ).
.: [1] ., ,
. ., ., 1974.
; demodulation , .
; c o m p l e x demodulation ,
.
; demographic statistics
, ,
,
,
. . .
,
.
. . :
- , - .
, , -,
. . . .
, ,
.
. .-
. ( )
, ., ,
,
.
( )
( )
, .,
( ).
, ;
x -
( , , 1 5 ,
4; 10; 15 ). 1 .
.
,
( ) .
.
hx = K x / Px
(1)
,
K x , x - x + ( .,
) , Px
. .,
x - x + -
. .-
q x ;
x - x + -
K x , x
= lim k x = lim hx
0
. ., x ( x - x + x
) x x - .
x , x +
,
. .,
, ( ) ; , ,
. , (1)-
h
.
. , 0 1
0 2
, h h
( ) 0
h = 0 h1 + 0h 2 0h1 0h 2 =
Px .
.
.
= h1 + h 2 . -
K x = Kx
Tx
( ), ., , ;
Tx , ( x , x + )
. Tx -
ix + exp
(2)
hxi =
.
, , 1 q 0
.
.
, , t - ,
( t , t + 1) t + x - , ( x 1 ), , x 1 x
x - x + 1
.
; ,
.
t , t + 1
k xt , t +1 = ( K xt + K xt +1 ) 2 S xt +1
K xt , t x ,
x + , S xt +1 , t + 1 -
.
( )
ix + exp ix + d
0 i
hx
. 0 <
<
(3)
x+ - -
,
0 i
hx
= 1 ( 1 hxi ) K xi ( K 1x + K x2 )
.
- ,
0 i
hx
= 1 exp ( x k xi )
.
x , x +
k xt , t +1 -
( 1x + 2x + ) d d
( x , x + ) -
- Tx . . .-
hx = k x ( 1 + ( 2 ) k x )
.
; , (1) 1-
. (3)
fx =
x+ exp
x + d d ,
167
fx =
x + d
f x ,
, x .
k x -
f x = x [ 1 exp ( k x ) ]
kx ,
f x = x
.
,
, i -
i 1 .
,
, .
( )
. m
, , 10 , m
= 3, 4
5 ( ) , ( )
,
. . .
( .,
) .
,
. , ( )
.
.
( ) (
) .
( )
(
) , ., .
: ,
;
.
: .,
.
,
.
.
.
168
( , 1988 ) ,
( , 1955 ).
. .-
. K x
. ,
.
.
. .-
;
,
.
, , ,
.
,
. ( ., ) , ,
.
.: [1] . ., ,
., 1945; [2] . ., , ., 1971; [3] , 3 ., ., 1985; [4] , ., 1985; [5] S h r y o c k H. S., S i e g e l J. S., The methods and materials of demography,
4 ed., v. 12, Wash., 1980; [6] Model life tables for developing countries, N. Y., 1982.
; dendogram
,
.
; deile ,
= m 10 K ; m = 1, 2 , ..., 9 . .
, ,
. K 9 10 K1 10
.
## ; deimation , , p > 1 , X ( t ), t = 0, 1, ..., N 1
Y ( t ) = X ( t p ), t = 0, 1, ..., [( N 1) p ] , p > 1
, [ ]
.
Y (t )
( , ) ( 2 p )
- . -
- .
( 2 p ) 1 ( ) = 0
( ) 0
( , ). .-
( ).
p X (t )
## p X (0) , 0, ..., 0 , p X (), 0, ..., 0 , p X ( t ) , ... ;
t = 0, 1, ..., N 1
.
. . -
.- X ( t )
(2 )
( ) = 0 ,
( ) 0 .
. .-.
.: [1] ., ., , . ., ., 1982.
; deterministic choas
theory , .
; deterministic channel
. ,
. . Y - -
Y - , ( y ) Y
( y) Y
= log 2 M
, M
.
.
.
; deterministic
tact interval , .
; Davis inequalities ,
.
( )
; greatest lower bound o f
rando m
e v e n t s , ( ) / ( )
.
; exact endomorphism ( , A , P)
IT
A = N T ,
A , A T n A A -
( )
; least upper bound o f
r a n d o m
e v e n t s , ( ) / ( ) .
; exact band
, .
; bundleless measure , .
; value ,
, .
( ) ; value o f a
m o d e l , ,
( ) ; value o f a
g a m e , , .
;
value function ( f , ) = f ( x ) ( x ) dx
A f = g
A =
. ( g , ) = ( f , ) = I ( f ) , g ( x ) =
= ( x x0 ) f 0 ( x0 ) -
I ( f 0 ) - . g ( )
, ( x) - x I ( ) , x
[ I ( ) - ] .
. . , , ,
.
.: [1] . ., . .,
, 2 ., ., 1981; [2] .,
.,
, . ., ., 1972; [3] . . [ .],
, ., 1976.
; variate-difference method
Yt = f ( t ) + X t , t = 0 , 1, ... , f (t )
, X t -
, N , 0
1 . ( , A , P )
, X t
, . .-
; A -
M X t = 0 , D X t = 2 .
, N
. , X n , n 0
. .-.
. . . . .
[1]- K ( , K )
.
.: [1] . ., . . . ., 1961,
. 25, 4, . 499530; [2] . ., . .,
. ., , ., 1980.
; exact design
, .
. .
. .- , f (t )
r - t , r f (t ) = 0 ,
[ f ( t )
= f ( t ) f ( t 1) , r f (t ) = ( r 1 f ( t ) ) ]
M r Yt = 0 . , 2
Vr =
T r
t =1
( r Yt ) 2 ( T r ) C 2rr
169
, T .
T r
M Vr = +
2
( f (t ))
(T r )
t =1
C2rr
( D Vr - , .,
[1]- ). Vr ( 0 , 1)
.
[ , r
( m , q ) ,
0 m < q ]
, i , j
, x ; x , xt M xt . a t -
.
t < 1 , t q xt , a t
. a t -
raa ( k ) | k | > 0 , .
k =1
2 = n ( n + 2)
H q : f (t ) 0 ;
q
H q 1 : f (t ) = 0, q 1 f (t ) 0 ;
ra2a (k ) ( n k )
..
H m+1 : m + 2 f ( t ) = 0 , m +1 f ( t ) 0 ;
H m : m +1 f ( t ) = 0 .
(Vr Vr +1 ) Vr +1 .
.: [1] .,
, . ., ., 1976; [2] ., .,
, .
., ., 1976; [3] ., , . .,
., 1981.
,
; timevarying code ,
, ,
K p q 2 ( , [2] ).
. .- a t
a t . a t
, ( 1 )
Q = ( n 2 ) 2
Q = ( n 1) 2 , n - ).
= 0, 01 ; 0, 05
0,10 K
= 1, 63 ; 1, 36
1, 22 .
; B o x J e n k i n s diagnostic checking
q 0 ( )
. .-
, ,
( , [3] ).
. . ,
. . .-
( , [1] ).
( )
xt ,
= [ a1t , ..., a m t ]
; signed measure , .
; Dynkins formula , ; ,
.
t = 1, 2 , ..., n
.
( , ;
).
. .- , ( , [4], [5] ), ,
.
. . ( , [1] )
at = ( xt x )
( xt j x )
j =1
170
j =1
a t j , t = 1,..., n
m a t =
( , [4] ).
.: [1] ., ., . , . ., . 1, ., 1974; [2] K e n d a l l M., S t u a r t A., The advanced theory of statistics, 4 ed., v. 3,
L., 1983; [3] . ., ( , , ), .,
1985; [4] L u t k e o h l H., J. Time Series Analysis, 1985, v. 6,
1, p. 3552; [5] T s a y R., T i a o G., Biometrika, 1985, v. 72,
2, p. 299315.
; diagram method
-
.
,
.
.: [1] . ., . ., , . 2, ., 1967; [2] . ., . .,
. , 1974, . 29, . 3, .11159.
;
diagram expansion techniques
.
. .
( x ) , R d - , . R
nk
n1
( x1 ) ...
## ( xk ) dx1 ... dxk =
( n1 + ... + nn ) 2
x1 , ..., xk ,
[1] [3]- .
.: [1] ., .,
. , . ., ., 1984; [2] . ., . ., ,
., 1985; [2] . ., .: .
. . . , . 24, ., 1986, . 11186.
j 0 ], | | = 1 + ... + n , , , - n Tx
; , , a ( x + z , )
= a ( x, Tz ( ),
## x , z R ) - , a ( , [1], [2], [6], [8], [9], [13]
[15], [17] ).
(2)
K B = K B ( x , y , ) (2) :
K B ( x + z , y + z , ) = K B ( x , y , Tz ),
. . .
. n
p( x) , x R , n
. .-
p ( x ) log p ( x ) dx
Rn
.
. . , ,
.
. . ,
, .
.: [1] . ., . .,
. ., . 3- . . , 1958, . 3,
. 30020; [2] . ., . .,
, ., 1982.
, ; ; ; differential
operator with random coefficients / random differential operator; s p e c t r a l t h e o r y o f R n -
(2)
x , y , z Rn .
B ( 2) K B ( 0 , 0 , ) L ( )
, K B ,
trR B = M x { K B ( x , x , ) } =
= lim
; differential entropy -
= c 1 x ,
[ = (1 , ..., n ) , j ,
x j ( l ) , l . . .-
h( x) =
(1)
, x R n , Dx
( ) , i - ni
; xi (l ) ,
a ( x , ) Dx ,
A
, ., A - , B ( )
| | m
U z A U z = ATz ( )
A = A =
1
Ln
K B ( x , x , ) dx
(3)
|x j | L 2
. ( {Tx } ) trR B ( - -
). , (1) a -
, -
A . A
2
( A ) , L ( R ) - 2
. L ( ) - A ,
,
A u ( ) =
|a| m
a ( x , ) Dx
( Tx )
x =0
( A) , ,
2
, - L ( R
) - A -
, , {Tx } ( , [4] ).
, A .
171
N ( ) = lim NV ( ) | V |
(4)
V , R n - ,
NV ( ) , V -, - ,
A , |V | , V , V
, V , .
N ( ) . ( A )
N ( ) - .
= trR E , E , A -
, N ( )
. N ( )
( , [1] ).
+ N ( ) .
:
N ( ) = (2 ) n M x mes { : R n ,
(x , )
| | = m
< } + O ( ( n 1) m ) =
= c n m + O ( ( n1) m )
( , [1], [6], [11] ).
N ( ) - ,
;
( ,
., [10], [12] [14] , [16] [18] ).
(4)
( , [1], [2] ).
, V , (
0 )
( , [3] ).
- ,
( ) ,
, ind R A = trR N A trR N *
A
, A , A ,
N A N A* , ker A ker A* -
( trR
## . ., . ., 1978, . 106, . 10840, 45583;
[8] C a r m o n a R., Acta Applicandae Math., 1985, v. 4, 1,
p. 6591; [9] C a r m o n a R., Random Schrdinger operators, cole
ete de probabilits XIV, Saint Flour, 1984, p. 123; [10] D o n s k e r M. D., V a r a d h a n S. R. S., Communs Pure and Appl.
Math., 1975, v. 28, 4, p. 52565; [11] . .,
. ., . . . , 1986,
. 11, . 98117; [12] . ., . .,
., 1988, . 22, 2, . 73
74; [13] . ., .: . .
. . , . 25, ., 1987, . 367; [14] P a s t u r L. A., Sov. Sci.
Rev. C. Math. Phys., 1987, v. 6, p. 1112; [15] C y c o n H. L.,
F r o e s e R. G., K i r s c h W., S i m o n B., Schrdinger operators:
with applications to guantum mecha-nics and global geometry, B.
[a. o.], 1987, p. 319; [16] K i r s c h W., M a r t i n e l l i F.,
Communs. Math. Phys., 1983, v. 89, p. 2740; [17] M a r t i n e l l i F., S c o p p o l a E., Introduction to the mathematical theory of
Anderson localization, Bologna, 1987; [18] S i m o n B., J. Stat.
Phys., 1987, v. 46, 5/6, p. 91118.
, ; differentiation system
,
;
. , ( , S , ) ,
P , - .
( )
f ( x ) = lim
(Z n )
= P (x)
(Zn )
, P , - , , x , - ,
f , (
d
), ( Z n ) nN x ( d
), P f =
.
P -
:
1) Y S , Y - ,
Y ( P ) ;
2) ( )
(3)- ,
P ( x ) < +, x
).
( ,
)
. .- ,
( , )
( , [5], [6], [8], [9] ).
.: [1] . ., . ., 1977, . 104, 2, . 207
26; [2] . ., . ., : , ., 1977, . 93107; [3] . ., . , 1982, . 37, . 5, . 18586;
[4] . ., . ., . ., 1984, . 123,
4, . 46076; [5] . ., . . .
., 1978, . 42, 1, . 70103; [6] . ., . , 1973, . 28, . 1, . 364; [7] . .,
P .
1)
. 1)
.
.: [1] ., , . ., ., 1953;
[2] . ., . ., ,
, 2 ., ., 1967; [3] . .,
, ., 1984.
172
- ; P , -
## , ; differentiation of a random process ,
; diffuse measure , .
;
diffusion process ,
.
,
r
. , R
. . ,
r
At f (x) =
1
f
+
2
xi
bi ( t , x )
i =1
aij ( t , x)
i , j =1
2 f
xi x j
; c o n t r o l l e d
diffusion process , .
; diffusion process o n
a
g r o u p
= (t ) -;
G B ( B = B (G ) G -
)
2
{ s , t | 0 s t , ( s , t ) R } (C )
( G - C -
) , G - A
. bi ( t , x )
, ai , j ( t , x ) (
) .
( )
.
p ( s , x, t , y ) -
T ( s, t )
s
t =s
= lim
h0
, T ( s, t ) = T s , t
T ( s, t ) f ( g ) = s ,t ( g , f ) =
. . :
r
p( s, x, t , y )
+
x
+
1
2
i =1
aij ( s, x )
i , j =1
p ( s , x , t, y )
=
x
1
2
bi ( s , x )
i, j = 1
i =1
p ( s , x, t , y )
+
xi
2 p( s, x, t , y )
=0
xi y j
2
( ai j ( t , y ) p ( s , x , t , y ) )
yi y j
.
t = s ( y x ) ,
s < t ( x y ) .
[1], [2]-
,
,
.
.: [1] . ., , ., 1963;
[2] ., . ., , . ., ., 1968; [3] . ., . ., , ., 1965;
[4] . ., . ., , ., 1968.
; diffusion process w i t h f o r m r e f l e c t i o n t h e b o u n d a r y , ;
f ( h ) s, t ( g , d h ) ;
( ) ; A , s 0 , G - 2
C 0
.
T ( s , t )
u + Au = 0
s
( bi ( t , y ) p ( s , x , t , y )) +
yi
1
[ T ( s h, s ) E ]
h
, u
lim u ( s , g , t )
st
= f (g)
(1)
u ( s , g ; t ) = T ( s, t ) f ( g )
f C , (1)-
. .- A T ( s , t ) -
.
(1) :
1)
,
2
; 2) A C - C 0 ; 3)
C g
. .-
-
.
2
: A C0 d
A ( ) 2-
:
173
Am
i , j =1
m
st
[ m]
f C ( s , g ) H , d 0 h
b[i m ] ( s, g ) D gi + c ( s, g ) .
i =1
Am
lim u ( s , g ; t ) = f (g ) ,
a [ijm ] ( s , g ) Dgi D gj +
(2)
s0
G - aij ( s0 , h )
d 0[ m ] r ( h ) Gm - -
= p m ( A ) , p m , G -
. (3)
D (G ) m .
= lim Gm : u
D (Gm ) -
G = lim Gm ,
mN
g = lim gm ,
m N
m N
G =
= p m (u ) = u o pm m N
u ( s, t , h ) d
G -
s,t ( g , ) =
h . -
2-
## C0k (G ) = lim C0k ( Gm )
mN
D (G ) = lim D (Gm )
mN
( Gm ) mN
[m]
L ( u ) = 0 , L A + A +
(4)
( g m ) mN , C0k ( Gm )
k -
, ( D )1 i m , Gm
= pm (G )
gm
= d pm (g)
, N .
Am f (g ) =
m
A
,
; :
i ( g , h )
Bi[m ] = Am ( i ) , c = Am ,
= x i (h) x i ( g ) , ( x i (h) )1 i m , g G
N g p m ( h ) Gm , p m : G Gm ,
mN
aij[ m ] ( s , t )
= [ 0 , ) G
[m]
u
+ Am u [ m ] = 0
s
(3)
( ) u
[ m]
( s, g , t , h ) ,
u [ m ] ( s, g ; t ) =
Gm
174
i =1
( g, h )
f ( h) f ( g )
1 + ( g , h)
1 + ( g , h )
i ( h ) Di h ( g )
( s, g ; dh) . (5)
( g , h )
A + A -
, (4) s, t
.
. .-
.
. .- T ( s , t ) , G -
l g
G = lim Gm
m N
Gm \{ g }
( aij[ m ] , Bi[ m ] , c )1 i , j m, m N
aij[ m ] = Am ( i j ) ,
, A
u [ m ] ( s , g , t , h) f (h) d 0[ m ] h ,
, . . ( ) ( ) . . ., ,
A , D (G ) - .
. .- T ( s , t ) G
K - rg ,
. .,
( K ) . . ., ,
Dk (G ) - K , Dk (G ) , D (G ) - K Dk (G )
D ( G / K ) -
, G / K , G K
G .
K . .-,
,
.
(G , B ) -
( nk )1 k kn , n N
(6)
. Tn ( s, t ) , (6) , T ( s , t )
Am Am + Am ; Tn ( s, t ) Gm T ( s, t ) ( C )
,
f (g
Tn ( s , t ) f ( g ) =
) s(,nt) ( dh ) ,
(7)
## = n k ... n l , 0 s < t 1 , t n k s < t n k +1 , t n l t <
< tn l +1 , 1 k < l k n
## = t n 0 < t n1 < ... < t nkn = 1 , [ 0 , 1]
(n)
sup t n k = 0 ,
1,
0,
e ,
e .
T ( s , t ) u s, t
u (3) (4)
. (6)
bi( n ) ,
(n)
)1 i , j m, n N
. (6)
A
aij( n ) ( t nk ) =
. .-
1
2 t nk
bi( n ) ( t nk ) =
x (g) x
i
( g ) nk (dg ) ,
Ne
1
t nk
x ( g)
i
nk ( dg )
bi( n ) ( t nk ) =
(5) (2)- A
.
s(,nt) =
- R = F ( ) g
= F ( g ) k ( g ) K = k ( )
, r : C R , k : G K , C
( , [6] ).
.: [1] ., .
, . ., ., 1986; [2] ., , . ., ., 1981;
[3] ., ,
. ., ., 1965; [4] . ., .
., 1972, . 17, . 3, . 54957; [5] . .,
. ., 1972, . 17, . 4, . 80204; [6] .
., . ., 1973, . 18, . 1, . 4455.
; diffusion b r a n c h i n g process
G
. ,
. G
( ).
=0
x G
Z x ( t )
;
r .
H ( t , x , s () ) = M
Z x (t )
s (
x ,i ( t ) )
i =1
( e, g )
nk (dg ) ,
1 + (e, g)
(
) H ( 0, x, s ( ) ) = s ( x ) -
(e, g ) i
x ( g ) ( n ) ( t nk , dg ) ,
( e, g )
, st , TR , R - R
, x , 1 ( t ) , ..., x , Z x (t ) ( t ) , t
(6) A + A 2-
0 s t 1
= nk ... nl , C ( G ) - st = stR K
; t
nk (dg )
G \ Ne
1
( n ) ( t nk ) =
t nk
. , (7)
(s, t ) ,
G R r , > 0
, F ( s )
Ne
( n ) ( t nk ) =
, K
K -
= e ,
e(r ) =
( aij( n ) ,
G K R
, TK , K - K
G = R K . k( n ) k = ( (n ) ) -
n 1 k k
n
s - s , s
, TR , R - G / K - C -
n ,
lim
TK ( s , t ) T ( s , t ) -
, K ,
s(,nt)
( , [6] )
Tn ( s , t ) T ( s , t ) = TR ( s , t )
lim H ( t , x , s ( ) ) = 0
x G
=e
175
H
=
t
i =1
2H
+ ( F(H ) H )
xi2
.
M Z x ( t )
M Zx (t ) ~ K e
( ( m 1) 1 ) t
1 ( x) , x G , t ,
>0
, K
(*)
, m = F (1)
, 1
>0
, 1 ( x ) > 0 ,
r
i =1
2
+ d = 0 ,
xi2
; diffusion vector , .
; binary random process , .
; steepest ascent , ; dilatasion
lim ( x ) = 0
x G
(*) ( m 1) 1
;
dynamical forecasting , ,
;
; ; dynamic programming; s e n sitive
c r i t e r i a
, 1-
.
A ( x) , p ( y | x , a ) , -
, . . .
,
.
G ,
a A ( x ) , x X
G ( ,
).
.: [1] . ., , .,
1971; [2] I v a n o f f B. G., J. Multivar. Anal., 1981, v. 11, 3,
p. 289318.
( x) = M x
; diffusion coefficients
( ,
( ) ).
( )
) . . b ( t , x )
= (t , x ) (t , x )
(
)
( ) - , w ( t ) ,
, a ( t , x )
( t , x ) ( R d - ) ( R d1 -
d
R - ) .
. .- : t 0 t -
t + t - x ( t + t )
x ( t ) R d x ( t ) = x (
, ), o ( t )
( b ( t , x ) , ) t
- .
, , ,
.
.: [1] . ., , ., 1963.
176
t 1 r ( xt 1 , at ) ,
0 < <1
(1)
t =1
, M x , Px
, x X .
lim
( x) ( x)
, (
, X
. x
d x ( t ) = a ( t , x ( t ) ) d t + ( t, x ( t ) ) d w ( t )
d1 R
1 - .
d1
( 1 )n
, n
g 0 ( x , N ) = M x
g n +1 ( x , N ) =
1
N
0,
x X
n
= 1, 0 , 1, 2 , ... ( , [2] ).
r ( xt 1 , at ) , x X , N = 1, 2 , ...
t =1
1
N
tg
( x , t ) , n = 0 , 1, 2 , ... ,
t =1
, ,
g n ( x , N ) = M x
1
N
( , [3] ),
n = 1, 0 , 1, 2 , ... ,
C Nn + nt r ( xt 1 , at ) .
t =1
n ,
## lim [ g n+1 ( x , N ) g n+1 ( x , N ) ] 0 , x X
).
(1)
- .
:
(1) -
= 1
+ h2 ( x ) ( 1 ) 2 + ... , x X .
( x) = sup ( x ) , x X
;
s t o c h a s t i c dynamic programming ,
( x ) = h1 ( x ) (1 ) + h0 ( x ) + h1 ( x) (1 ) +
; Richardson dynamical number
+ h2 ( x ) ( 1 ) 2 + ...
{ h1 , h0 , ...}
, .
hm = hm n ,
m = 1, 0 , 1, ..., n . 1- -
( , [1] );
n 1 n . n 0
h1 , h0 , ..., hn n + 1 ;
a A1 ( x )
a A0 ( x )
## hm ( x) = sup P a ( hm hm1 ) ( x ), m = 1, 2 , ..., n,
a Am ( x )
(2)
x X ,
f (y) p(y | x, a) ,
yX
A1 ( x ) = A( x ) , Am ( x ) , Am 1 ( x ) , hm1 ( x) .
## (2) h1 , h0 , ..., hn2
( , [5] ).
.
. X = { x , y } , A ( x) = { a , b } , A ( y ) = c ,
p( y | , ) = 1,
p( x | , ) = 0 ,
;
dynamical system , .
; dynamical system w i t h p u r e p o i n t
s p e c t r u m
t
. {S } , ( , A , P )
t
( , ) {U }
. ,
{U t } .
1-
h1 ( x ) = sup P a h1 ( x),
Pa f ( x) =
( 1) .
.: [1] B l a c k w e l l D., Ann. Math. Statist., 1962, v. 33,
p. 71926; [2] V e i n o t t A. F., Ann. Math. Statist., 1969, v. 40,
p. 163560; [3] S l a d k y K., Kybernetika, 1974, v. 10, p. 35067;
[4] . ., . ., .
, 1982, . 37, . 6, . 21342; [5] D e k k e r R., H o r d i j k A.,
Math. Operation Research, 1988, v. 13, p. 395420.
= h1 ( x ) (1 ) + h0 ( x ) + h1 ( x) ( 1 ) +
.
{ h1 , h0 , h1 , ...}
, n 1 n ,
r ( x , a ) = r ( y, c ) = 1 ,
r ( x , b ) = 0 .
: ( x ) = a , ( y ) = c ( x ) = b , ( y ) = c .
( x) = ( x) = 1 + + 2 + ... = ( 1 ) 1 ,
( x) = + 2 + ... = (1 ) 1 1 .
, ,
.
. . . . . .
. .
, . . .
( ,
, ).
. .
(t ) =
a s cos ( s t + s ) +
bs sin ( s t + s )
, a s , bs , s
.
( , [2] ).
.: [1] . ., . ., . .,
, ., 1980; [2] . ., , ., 1981.
; ; ; dynamical system; s m a l l
s t o c h a s t i c p e r t u r b a t i o n s o f
:
1) &t = b ( t , t ) , t ;
2) &t = b ( t ) + & t ;
3) .
1) 2) , ,
( , [1], [2], [4], [5] ). 2)
177
, &t
= b ( t ), 0 = x
, ( , [3] ).
.: [1] . ., . .,
, ., 1979; [2] . .,
, .,
1986; [3] . ., . , 1983,
. 33, 6, c. 92931; [4] . .,
, .,
1987; [5] . ., ,
., 1969.
( )
; dynamical system o f s t a t i s t i c a l m e c h a n i c s
( )
M - { S t } .
M -
, , .
M - ( M M )
M - ( )
( M ) = 1 . ( )
M ( ) .
d
( , [1], [2] ). R U -
,
d = 1, 2
( U ) ( , [2] [6] ).
M -
.
. .- .
U -
. U
( M ) = 1 M M -
## .: [1] L a n f o r d O., L e b o w i t z J., L i e b E., J. Stat.
Phys., 1977, v. 16, 6, p. 45361; [2] F r i t z J., J. Stat. Phys.,
1983, v. 33, 4, p. 397412; [3] L a n f o r d O., Comm. Math.
Phys., 1968, v. 9, 3, p. 17691; 1969, v. 11, 4, p. 25792;
[4] D o b r u s h i n R. L., F r i t z J., Comm. Math. Phys., 1977,
v. 55, 3, p. 27592; [5] D o b r u s h i n R. L., F r i t z J., Comm.
Math. Phys., v. 57, 1, p. 6781; [6] M a r c h i o r o C., P e l l e g r i n o t t i A., P u l v i r e n t i M., Random fields. Rigorous results in
statistical mechanics and quantum field theory, v. 2, Amst. Oxf. N.
Y., 1981, p. 73346; [7] . ., . . ., .,
1974, . 29, 1, . 15258; [8] P r e s u t t i E., P u l v i r e n t i M.,
T i r o r r i B., Comm. Math. Phys., 1976, v. 47, 1, p. 8195;
[9] . ., . ., . .,
. , 1973, . 28, . 5, . 4582; [10] G u r e v i c h B. M.,
S u h o v Y u. M., Comm. Math. Phys., 1976, v. 49, 1, p. 6396;
1977, v. 54, 1, p. 8196; v. 56, 3, p. 22536; 1982, v. 84, 3,
p. 33376.
.
; ; ; Diophantine approximation; e r g o d i c
p r o b l e m s , ( )
.
; Dirac measure
, , .
; Dirichlet space , .
; Dirichlet
form , .
; s t o c h a s t i c Dirichlet
problem ,
; Dirichlet distribution
{ ( x1 , ... , xk ) R k : x1 + ... + xk = 1, xi > 0 }
,
f ( x1 , x2 , ..., xk ) =
1 ( , [7], [8] ).
U .
,
.
. .-
.
, M -
.
. U
. .
. .-
, t t = 0 S t
. -
.
178
(1 + ... + k ) 1 1
x1 ... xk k 1 ,
(1 ) ... ( k )
i > 0 , 1 i k
X 1 , ..., X k
. . . , , X
X i j
a0 j - ; 0 =
i =1
r1 , ..., rk
. .-
M ( X 1r1 ...
X krk )
k
( 0 )
+ ri )
i =1
( 0 + r0 )
( )
i
i =1
, r0
p( x , y ) ,
p1 ( xi ) =
Xi
i ( 0 i ) 02 ( 0 + 1 ) , X i
X j - i j , cov ( X i , X j ) =
, i 0
p2 ( y j ) =
= i j 02 ( 0 + 1) .
; discounting
i = 1, ... , m ,
(1)
p( x , y ) ,
i
j = 1, ... , n .
(2)
xiX
X Y p1 ( x)
## .: [1] ., , . ., ., 1974; [2] .,
, . ., ., 1967.
y j Y
i =1
p2 ( x )
{ X Y , p ( x, y )} , X Y .
X Y p ( x , y )
,
( , ).
xi X yi Y
; discount rate ,
, X Y
. , , .
. , ,
( ., ) X
( ) . , X
,
.
.: [1] . ., . .,
, ., 1928.
;
discret ansamble , m X = { x1 , ..., xm }
p ( x ) -
{ X , p ( x )}
xi X ,
i = 1, 2 , ..., m
( )
. p ( xi ) , P ( A )
A, AX
, p ( xi ) 0 ,
p ( xi ) = 1 , Pr ( A) =
i =1
xi A
A . =
.
Pr ( A) A .
X .
## X = {x1 ,..., xm } , Y = { y1 , ..., y n } .
, ( xi , y j )
X Y
X Y
xi X , y j Y , i = 1, ... , m , j = 1, ... , n .
X Y Y X .
X =Y
, X
.
. X 1 X 2 ... X n
( x (1) , x ( 2) , ..., x ( n ) )
(1)
( 2)
X 1 , x
, x
X 2 . . n
X - , X
.
X Y X Y .
( xi , y j )
X Y -
p ( xi , y j )
p ( x , y )
; xi X , y j Y . ,
(3)
( ) ; K a r h u n e n
L o v e disret decomposition ,
.
p ( xi )
p X
p ( xi , yi ) = p1 ( xi ) p2 ( yi )
; discrete white noise ,
,
(
2
-
), ( t ) -
= 0 , 1, 2 , ... .
. . . : = 0 B ( ) =
( t ) , t
= M (t + ) (t ) = 2
| | > 0
B( ) = 0
= 2
.
,
; the property of discrete
separability of data portion ,
f ( )
; discrete probability
distribution , .
*
; discrete Fourier transform :
a ( n1 , ..., nk ) =
N1 1
N k 1
j1 = 0
jk = 0
...
## x ( j1 , ..., jk ) WNn11 j1 ... WNnkk jk
179
. .- ,
, , ,
.
.: [1] . ., . .,
, 3 ., ., 1987.
( )
x ( j1 , ..., jk ) =
1
N1 ... N k
N1 1
N k 1
... a (n , ..., n ) W
k
n1 = 0
nk =0
( ),
n1 j1
N1
jl = 0 , ..., N 1,
l = 1, ..., k , Wn = exp { 2 i N };
... WNknk jk
nl = 0 , ..., N l 1,
k > 1
, k = 1 . . ., x a
, x . . .
.
A = WN N
## = ( a (1) , ..., a ( N 1)) T ,
, An
xN = ( x (1) , ..., x ( N 1 ) ) T , WN ,
( n + 1 ) -
( j + 1) - WNnj , n , j = 0 , ..., N 1 ,
( N N ) . . . .- ( , ,
).
*
;
discrete sourse , .
; digital modulation ,
.
; discrete
measure , .
; discrete distribution
{1}, { 2 },... A
( , A )
pi > 0 , i = 1, 2 , ... , pi = P ({i }) ,
. .
pi = 1 -
i =1
pi ;
A A
P ( A) =
.
. . ,
.
i j , Ai A j =
= P ( Ai ) 0 ,
X ( ) Ai ,
i= 1
X ( ) = xi , Ai . ( , A , P )
X ( )
. . .- .
. .- .
180
, .
*
; discrete random variable
, ( , F , P )
( ) . ( )
x1 , x2 , ..., xn , ... ,
( ) : R1 . ( ) , , n
{ : ( ) = xn } F
(1)
F
.
, ( ) (1) ,
( ) F , x R1
{ : ( ) < x } =
{ : ( ) = xi } F .
xi < x
, ( ) F
, ( , [1], 1, . 54 ),
x { : ( ) = x} F .
x1 , ..., xn ,... -
i: i A
P ( A ) = 1
,
; discrete sample of a process
, ( ) ( , F , P )
pi
A1 , A2 ,... A , pi
; discrete distribution
function , .
; discret renormalization group ,
, n -
P { : ( ) = xn } = pn , n = 1, 2,...
(2)
. (2)
.
P { : ( ) = xn } = pn , n = 1, 2 , ...
( ) -
.
( , , ,
, . ).
p n ( ) - xn
. ( ) -
:
( ) -
x1
x2
xn
[ , ]
P { ( ) = xi }
p1
p2
pn
.
[ , ] ( t )
( ) . . .-
.
( ) . . .
## < t < , cov ( ( t ) , (t ) ) = 0
pn 0 ,
, ( t ) + ( t )
pn = 1
, M ( t ) ( M ( t ) )
.
.: [1] . ., . ., .
., , ., 1979.
; disretization problem
.
{ ( t ), t (; + )} -
k , k
= 0 , 1, ...
= (k )} , ( t )
{ ( k )
.
M ( t ) = M ( k ) B ( ) =
= B ( ), t = 0 , 1, ...
.
( t )
( t ) ,
B ( ) , ( t ) .
( t ) f ( ) ( k )
f ( ) =
f ( + 2 j )
-
. < <
, ( t ) . ( t ) -
. Y ( t ) .
= 0 , D (t ) = 2
( t ) , t ( , ) M (t )
BY ( t )
2
2 2
n =1
1
4 n 2 2 (1 B (t ) )
exp
,
2
n
( ) 2
< t <
= o ( )
BY (t ) 12 . ( t ) - 2 h
2
,
,
fh ( )
j =
2
2h
1
3 2 2
exp 2 2 2 2
3
8 n h
n =1 n
< <
3 2
2 2 n
, ( t ) -
.
Y (t ) - (t )
(t) =
k =
sin ( t k )
( k )
sin ( t k )
(1)
*(t ) =
(t ) ( k )
k =
1949-
. . . ,
t -
.
M | ( t ) * ( t ) |2 = 0
k =1
.
( t )
B Y ( t ) 2 B (t )
(2)
. [ , ]
( t ) (1) , ,
. , (2)
, ( t )
, 2 - .
2 k 2
( 1) k exp
,
( ) 2
< t <
. ,
( ) < 1 B Y ( t ) 10 8 B ( t ) .
.: [1] . .,
, .1, ., 1966; [2] . .,
, ., 1981; [3] . ., . 1949, . 4, . 4, . 17378.
## ; discretizable random field , .
; disriminant analysis ( ) ( , , ) , .
181
~
QS ( ) =
X . k D1 , ..., Dk
,
, i - Pi ( X ) ,
{ d ( X ; , )} M
{ d ( X ; , )} ]2
[M
i
ij
i
j
j
ij
i
j
{ d ( X ; , )} + q D
{ d ( X ; , )}
q D
ij
i = 1, ..., k . Pi ( X )
ij
,
j = 1, 2 , ... , k , X jl D j ( { X jl } , l = 1, ... , n
wij
Di , { X jl },
l = 1, ... , n ). X -
D j ( )
S - ,
, . X .
. .-
k
Q(S ) =
cij q j PS ( i | j )
i , j =1, i j
, cij , q j PS ( i | j )
j - i - , j -
( )
S j - i -
.
cij ,
q j . , X
ln ( qi Pi ( X ) )
i = 1, 2 , ..., k j . , X i
d j(X ) =
= ln ( q j P j ( X ) ) .
d j ( X ) dij ( X ) =
= d i ( X ) d j ( X )
. , S ,
d j ( X ) = arg max d i ( X ) , S : X D j
1 i k
Qt ( S ) =
q j PS ( i | j )
i , j =1, i j
.
.
{ d ( X ; )} d i ( X ) = d i ( X ; )
,
, .:
182
qi
ij
d ij ( X ; i , j ) = d i ( X ; i ) d j ( X ; j ) , q i
( . ) X j1 , X j2 , ..., X jn j
q i = ni n .
, .,
j , wij
i
{ ( X ) } D
{ ( X ) }
cij , M
i
i
(X )
, X - i -
,
. , , k = 2 , .
. ( A. Wald )
. , , P1 ( X ) P2 ( X ) -
.
.: [1] F i s h e r R. A., Ann. of Eugenics, 1936, v. 7, p. 17988;
[2] . ., , . ., ., 1968.
; discriminant function
. X
= ( x1 , ..., x p ) T -
k
H 1 , ..., H k . k
g i ( X ) . .- , i = 1, ..., k
: l
= arg max g i ( X )
1 i k
, X H i .
i ( X )
= a ( X ) g i ( X ) + b ( X ) . .- -
g i ( X ) . . ,
,
a( X ) > 0
b(X ) ,
X -
b(X ) =
= g k ( X ) , a ( X ) 1 k 1 . .
. , , .,
.
, k
= 2 , g ( X ) = g1 ( X ) g 2 ( X )
. .- , X H 1 ,
, g ( X ) > 0 , g ( H ) 0 X H 2
. g ( X ) . .-
, H 2 H1
p
. . .-
( ) .
, . .- 1936-
. .
( , )
:
g ( X ) = X T S 1 ( X 1 X 2 )
= imin
1i
1
( X 1 X 2 ) T S 1 ( X 1 + X 2 ) + c ,
2
, , ; X i , H i
; c .
.: [1] F i s h e r R. A., Ann. Eugenics, 1936, v. 7, 17988;
[2] .,
, . ., ., 1963.
; discriminant model , .
,
.
. y1 ,..., y N
H i : y j = i ( x j , i ) + j , j = 1, ..., N
(*)
,
i
( ), i R
mi
i = 0, 1
, j
= 0 , D j = 2 , = ( x1 ,..., x N ) . 1 , ..., n ,
, M j
Ti ( , ) =
1i (
j =1
x j , i ) ) (dx)
.
,
. . x1 , ..., xn
i n = arg min
i
F ( y j i ( x j , i )) ,
i = 0, 1
j =1
xn +1 = arg max F ( 0 ( x , 0 n ) 1 ( x , 1n ) )
( (x
i
xn +1 ( n + 1) - . ( ,
).
.
min F ( 1 ( x , 1 ) 0 ( x , 0 ) ) ( dx ) , i = 0 , 1
i
* -
0 ( ) 1 ( )
.
: 0 ( ) 1 ( ) k k + d [ 1, 1]
1 ( x) =
k +d
m xm ,
m=0
F ( y ) = y 2 , 1 = :
n2 + m 1 . , l
m =1
Tk + d (x)
d
, ) 1i ( x j ,
1i
))
j =1
F ( ( x , )
x X
; design of discriminating
experiments , ,
i ( , i ) =
H i H 1 i -
F ( 2 , 2 )
.
1 i
, H 1i
L2 ( ) - i ( , i ) . H i
Ti ,
-
*
-
( , [2] ). k k + d r
, d = 1, 2 .
( , [2] ) 1 ( ., ) .
i
H i i , i = 1, ..., m
F0 i , ,
Ti ( , ) -
.
*
, H i
Ti -
i ( x , i ) = f i T ( x ) i
, ( )
i f i ( )
i
M
183
M (i )
i0 =
F (d ) ,
i
D0i =
( ) ( )
i
i
0
i T
F ( d i )
0
. (*)
0 i tr [ M (i ) ( ) ( D0i + i0 ( i0 ) T ) ]
.
( , ., [1] ).
,
, ( , [3] ). (*) 0 < < 1 , H 0
D ( ) - .
F , H 0
H 0 .
( ) , l ( D ( )) - ,
( , ) . ( , )
- , H 0
1
,
( , [4] ).
tr DF ( l ) DF ,
H 0
D ( ) . 1
,
.
.: [1] , .,
1983, .16; [2] . ., .: , ., 1975; [3] .,
. , . ., .,
1975; [4] M u l l e r - F u n k U., P u k e l s h e i m F., W i t t i n g H., Linear Algebra Appl., 1985, v. 67, p. 19034.
;
analysis of variance
. .
, . .-
, , .,
.
.
. . : , ,
( I ),
( II
)
; ( ) ,
.
. , . .
( , . ) . .,
, p
i - mi ( ) ;
m = m1 m2 ... m p - . , ,
p . q , ,
, - . ,
q ,
,
, ,
.
,
.
( , [3], [4] ).
.
1. Y
( ), = M Y . y -
. . .
. [1]-
. , . .-
.
, . .- ,
.
[2]- . . . ,
.
. .- ,
, . .- ,
( A )
. Yij , i = 1,..., q ( q = 4 ), j = 1, ... , n
184
, ij .
, yij i -
.
M Yij = + i ; , i - , ni , i -
q
, i
= 1, ..., q ,
= n.
i =1
Yi j = + i + i j , i = 1, ..., 4 , j = 1, ..., ni
(1)
, Yij , i - j - -
, ( D1 , D2 23% , D3 32% ,
D4 67% ), ( ) . . I . , i = 0 H 0
Yi k j = + i + k + i k + i k j ,
i = 1, ... , 4 ,
k = 1, 2 ,
y = 1, ..., nik
(2)
, nik A - i - -
1-
S =
i , i
H 1 : 1 ; H 2 : k ;
H 3 : ik
. , ,
( H 1 , H 2 ) ( H 3 ) .
. , ,
2
, -
,
.
,
.
.
( , [5], [7], [8] ); :
S1 , ... , S q , 1 , ... , q
S =
S i 2 2 ( )
i =1
, S1 , ... , S q 2
( i ) -
ij
i ) 2
= 1, ..., 4 -
,
4
n
i
=0
i =1
, .
:
1
n
= y =
nj
1
ni
i =1
yij y , i = 1, ..., 4
(3)
yij , n =
i =1 j =1
i = yi y =
: nik - , , . I II
1 , 2-
.
ni
(y
i =1 j =1
B - k - , i k j . . .- -
= ( Si i ) ( S j j )
F ( i , j ) .
.
2. 1- ( ., ) 0; 23; 32; 67 .
( ) . . , II
.
3. 1- ( 4 A ), ( 2
B ) .
. , F
i =1
ni
j =1
n ,
i
. .-
i = + i ) , S S
, -,
M S ( -
) ( ) M S
.
SS B =
n (y
i
y ) 2 ; B = 3, MS B = SS B B ,
(4)
i =1
SS R =
ni
i =1
j =1
(y
ij
yi ) 2 ;
R = n 4 ; MS R = SS R R
(5)
. MS R 2
2
: : M S R
= S 2 . (4) (5) -
S ST =
ni
(y
i =1
. H 0 : i
ij
y ) 2 ;
vT = n 1
(6)
j =1
0 -
;
F
F
,
185
F = MS B MS R
(7)
a2 = ( M S B M S R ) k ,
.
F ( B , R )
.
4. 1- n1
= ... = n4 = 9
( , [6],
[5] ):
I (3) :
## = 4,6284 ; 1 = 0,5321 ; 2 = 0,0032 ;
3 = 0,1193 ; 4 = 0,4158 .
(5) (6)
SS B = 4,2321 ; B = 3 ; MS B = 1,4107 ;
SS R = 14,0569 ; R = 32 ; MS R = 0,4393 ;
SST = 18,2890 ; T = 35
= 3,21
F ( 3; 32 ) ,
; F F
H 0 < 0,05 1- .
.
n1
n2
n3
n4
4,079
4,859
3,540
5,047
3,298
4,679
4,870
4,648
3,847
4,368
5,668
3,752
5,848
3,802
4,844
3,578
5,393
4,374
4,169
4,709
4,416
5,666
4,123
5,059
4,403
4,496
4,688
4,928
5,608
4,940
5,291
4,674
5,038
4,905
5,208
4,806
4,0963
4,6252
4,7477
5,0442
2 , 2
, a
k =
. i
a2 , ,
, 2
= 0 ,
( )
2
. a
186
i =1
2
i
i =1
n
i
i =1
( q = 4 ), H 0 -
;
;
m o d e l analysis of variance , II
; m i x e d
, ( ) . .
( , [1] ) ( )
.
, , .
, N
y = X +
U i i
(*)
i =1
.
d
X , U i , R , i R ,
mi
cov i = I mi , i = 1, ..., l .
M y = X , cov y =
), H 0
(7)
F .
( , [5], [7] ).
. .- [7]- .
[5]- .
. .-
,
.
. .-
( , [8], [9] ).
.: [1] F i s h e r R. A., Statistical methods for research workers,
13 ed., L., 1958; [2] S c h e f f e H., Ann. Math. Stat., 1956, v. 27,
p. 25171; [3] . ., ,
, . ., .: , .,
1981, . 4658; [4] . ., , .,
1971; [5] ., ., .
, . ., ., 1982; [6] C i s s i k J. H.,
J o h n s o n R. E., R o k o s c h D. K., J. Appl. Physiology, 1972,
v. 32, p. 15559; [7] ., , .
., ., 1980; [8] ., . ., , . ., ., 1983; [9] S c h r a d e r R. M., M c K e a n J. M., Commun. Statist., 1977, v. A6,
9, p. 87994.
a (
, q
.
(1) , i
, ij
q 1
dG
i
i =1
, Gi
= U i U iT , di = i2 -
i (*) ,
( , d1 , ..., d l )
.
.
( MINQUE ), y
,
, .
.
.: [1] F i s h e r R., Trans. Roy. Soc. Edinburgh, 1918, v. 52,
p. 399433; [2] ., , . .,
., 1980; [3] R a o C. R., K l e f f e J., Estimation of variance components and applications, Amst., 1988.
;
dispersion method i n t h e n u m b e r t h e o r y
+ = n
(1)
( )
,
.
. . . . 195861-
,
( ,
) . . .
( A. Weil ) .
(1)
D + = n
(2)
, , D - ( ) , D ( D ) , ( ) ( D ) ,
, D . F
.
D (D )
, A ( n , D )
. (2) ( )
A ( n , D )
D ( D )
. F S
= V
1 A ( n , D )
(3)
D ( D ) D + = n
. (3)- -
V =
V 2 D0 V
(4)
, D0 , (D) ,
V =
1 A ( n, D)
D ( D ) D + = n
(2) .
. .
,
2
1 A ( n, D ) =
D+ = n
D ( D )
1 , 2
. . .-
1 -
,
.
. (3)
(4)- (2) .
(2) (1)
.
. .-
= l
, l ,
, .
. . .- ( , [3] )
, , . .- . .,
: ( = x1 , x2 ,..., xk , k = const ,
= x y ) ; ( = p , = x y );
( a
D+ =n
S =
(5)- D (D) ,
(2)- D -
(5)
=p
= x +y ).
2
.: [1] . .,
, ., 1961; [2] . .,
. , 1965, . 20, . 2, . 89130; [3] . .,
. .,
, , 1974, . 522.
## ; variance ratio distribution , F .
; variance ,
;
D - (
, ). .
.
D = M ( M ) 2
(1)
187
D = M 2 ( M ) 2 ;
D ( c ) = c 2 D ,
= D
, .- .
M
( M < );
D , ( )
.
.
.
) x1 , x2 , ... P {
(
= xk } = pk , k = 1, 2 , ...
pk = 1 ),
(x
D =
( x M )
dF ( x)
(
);
) p(x)
( x M )
p ( x) dx
.
. .
. ., M M , M ( M )
.,
. .
.
,
. , .-
1 , ... , n
.- D ( 1 + ... + n ) :
D ( 1 + ... + n ) =
D
i =1
+ 2
cov ( ,
i
i<j
cov ( i , j ) = M { (i M i ) ( j M j ) }
188
(2)
. (2) , 1 , ..., n
. , , (2)
. , (2)-
cov ( i , j ) = 0 , 1 , ..., n - -
.
. . ,
,
. 1 , 2 , ..., n ,... , n D n 0 ,
>0
P { | n M n | > } 0
;
) F (x)
,
D =
D ( 1 + ... + n ) = D1 + ... + D n
M ) 2 p k
D =
1 , ..., n - ,
cov ( i , j ) = 0 . , -
c ,
., D ( )
( ) ( ,
), n - n M n - .
. .
.
-
= ( M )
D - ,
D . -
, , 0 1
, , .
.
-
= + a -
, a , . , F ( x ) - , ,
F (( x a ) / ) a
.
M
= 0 , D = 1
, M
= a , D = 2
. , ,
j)
~
a = M , =
~
D . -
.
.: [1] . ., , 6 .,
., 1988; [2] .,
, . ., . 12, ., 1984; [3] . .,
. ., . .,
, ., 1979; [4] ., , . ., 2 ., ., 1975.
( )
;
sample variance / variance o f e m p i r i c a l d i s t r i b u t i o n , .
; ;
; e s t i m a t o r o f variance
x1 , ..., xn
, ,
s2 =
1
n 1
( x j x )2
j=1
,
,
. , x
.: [1]
., 1967.
1
n
xj .
j =1
., , . .,
( ) ; variance o f
r a n d o m s e t
A - ,
D A = M 2 ( A, M A)
, M A , A .
.: [1] . ., . . . . .
. - , 1979, . 85, . 11328.
( )
; t u r b u l e n t e n e r g y dissipation
, ( )
.
, .
.: [1] . , ., 1998, . 188.
.: [1] . ., . .,
1968, . 2, 4, . 3143; [2] L a n f o r d O., R u e l l e D., Comm.
Math. Phys., 1969, v. 13, 3, p. 194215; [3] P r e s t o n C h.,
Random Fields, B. [a. o.], 1976; [4] R u e l l e D., Thermodynamic
; Dodge plan
, .
; likelihood function X
p ( x, ) - ( ); x X - , . ln p ( x , ) . , . . ( . . ) -
( ) .
0 ,
P0
d P
( x)
d P0
X ,
X = ( Y , Z ) , Z
. .
, Z z
Y - . .
.
. . . [1]- .
, ( )
.
; likelyhood ratio
( ) Yn
qn
pn
,
( X 1 , ..., X n )
, , X 1 , ..., X n , ... -
## ; dissipative Markov chain
( , ; ).
.: [1] . .,
, . ., ., 1969.
.
.
; disjoinct
spectral type of measure , ,
.
## ; sequence of disjoinct random variables , / .
; DLR condition
. [1] [2]-
.
pn = pn ( X 1 , ..., X n ) qn =
= qn ( X 1 , ... , X n ) , ( X 1 , ... , X n )
Yn =
qn ( X 1 , ... , X n )
pn ( X 1 , ... , X n )
. ,
pn -,
X 1 , ... , X n
pn -
. , , Yn
pn -, qn - ,
189
H 0 . . . . (1)-
. , Yn -
.
pn - .
.
i = { i }, i = 0 , 1 ( -
pn -, X n+1
) (1)
X 1 , ..., X n -
p ( x , 0 ) max { p ( x , 0 ) ; p ( x , 1 )} =
p n+1 p n ,
= min { 1; p ( x , 0 ) p ( x , 1 ) }
M ( Yn+1 | X 1 = x1 , ... , X n = xn ) =
q n+1 ( x1 , ..., xn , y )
pn+1 ( x1 , ..., xn , y )
pn+1 ( x1 , ..., xn , y )
dy .
pn ( x1 , ..., xn )
(*)
pn+1 ;
y - . qn+1 -
qn . , (*)- qn pn
,
. . .
- .
- ( x ) . . .
{ x : ( x ) < } -
( ; ) = M ( X )
(1) .
{ P ; }
,
M ( Yn+1 | X 1 , ... , X n ) = Yn
. . Yn .
, , .
.: [1] ., ,
. ., . 2, ., 1984; [2]
.
; m o n o t o n likelihood ratio , -
.
.
X P , , X
X n ) n . X
= ( X 1 , ...,
pn ( x ( n ) ; ) =
p ( xi , ) ,
1 i n
; likelihood ratio
f o r r a n d o m p r o c e s s s , ( ) .
; likelihood
ratio test, LRT , . . .
;
. . . . .
. ( , [1], [2] ).
X P
. X x
H 1 : 1 H 0 : 0
, 0 1 , - , 0 U 1
(n )
(n )
. . . . ( )
( x ) = sup p ( x , ) sup p ( x, )
0
(1)
(1)
190
(1)
n ( x (n ) ) = sup pn ( x ( n ) , ) sup pn ( x ( n ) , )
0
(2)
0 R k ,
r = k m > 0 m , Ln
= 2 ln n ( X (n) ) H 0 -
( 0 ) n
2
r - , . . .
. . . .- , , H 1n :
1n = { = 0 +
n ; 0 0 , R k \ 0 }
, , 0 , R
. Ln ,
2
r -
, ;
0 0 - .
.
,
( , [3] [6] ). . . .
## ( , [6] [9] ); , (2)
( , [10] [11] ).
(1) [ (2) ] :
( x ) = p ( x, 0 ) p ( x , ) ,
= 0 ( x) ,
{ P ; 0 }
. ,
. . . 0 .
. . .- ,
, ., ,
, C ( ) . ( , [6] ).
-
, , . . .
, , (
, . . .
). . . .-
, (
), . . . .
( ) ( ,
[3] ); (
). . . .
( , [12] ),
( ) ( , [13] ).
.: [1] N e y m a n J., P e a r s o n E. S., Biometrika, 1928,
pt 1, p. 175240; pt 2, p. 26394; [2] N e y m a n J., P e a r s o n E. S.,
Phil. Trans. Roy. Soc. London, 1933, v. A 231, p. 289337;
[3] W a l d
A., Trans. Amer. Math. Soc., 1943, v. 54, 3,
p. 42682; [4] W i l k s S. S., Ann. Math. Statist., 1938, v. 9, 1,
p. 6062; [5] ., , . .,
., 1967; [6] . ., .: .
. . . , . 17, ., 1979, . 356; [7] L e C a m L., .:
Proceedings of the 3-ed Berkeley symposium mathematical statistics
and probability, v. 1, Berk. Los. Ang., 1956, p .12956; [8] L e
C a m L., Univ. Calif. Publs. Statist., 1960, v. 3, 2, p. 3798;
[9] ., . , . ., ., 1975; [10] C h i b i s o v D.
M., v a n Z w e t t W. R., . ., 1984,
. 29, . 3, . 41739; [11] P f a n z a g l J., .: Developments in
statistics, N. Y., 1980, v. 3, p. 197; [12] B a h a d u r R. R., Some limit theorems in statistics, Phil., 1971 ( SIAM ); [13] .
., , ., 1984; [14] ., , . ., 2 ., ., 1979.
s t r o n g likelihood principle ,
; birthanddeath process { 0 , 1, 2 , ...} h 0
, h n n + 1 n 1
= ( x ) , { P ; }
, 0
; birth time
, .
;
-
; w e a k likelihood
principle , .
; likelihood equation
.
, n h + o( h ) n h + o ( h ) ; o ( h ) -
. n - n -
: ,
,
. . . .
.-
. . . .-
, . n = n + ,
n = n , . . . n ;
h 0 h + o ( h )
( ); h + o ( h ) ,
, h + o ( h ) . = 0 , .
= 0 ,
. .
n = ,
.
.: [1] .,
, . ., . 1, ., 1984; [2] . .,
. ., . .,
, ., 1982.
( )
; generating functional o f
a continuous convolution of measures sem i g r o u p G ,
D(G ) -
A f : = lim
t 0
f ( x) t (dx) f (e)
A , { t , t
0},
. A E (G ) [ g
, f D(G ) f g D (G ) ]
.
A . . , { t , t
0}
.
(n )
. . . t t
191
t 1 . .- A An
.
t(n )
t - n ,
f E (G ) n
, ,
.
2) X
,
An f A f ( , [1] [3] ).
. n , G
, k n , k n ,
, { t , t
0 } . . A
. :
1) n[ knt ] t , t 0 ( );
2) exp [ t k n ( n e ) ] t , t 0 (
);
3) E (G ) k n ( n e ) A ( . .- ).
,
( ,
[2], [4] ).
.: [1] H a z o d W., . ., 1995,
. 40, . 4, . 92934; [2] S i e b e r t E., Adv. Math., 1981, v. 39,
p. 11154; [3] K h o k h l o v Y u. S., in: Probability measures on
groups X, N. Y., 1991, p. 23947; [4] H a z o d
W., S c h e f f l e r H.-P., J. Theor. Probab., 1993, v. 6, 1, p. 17586.
; p r o b a b i l i t y
generating function , ( )
; e x p o n e n t i a l generating function , ( )
( )
; generating function o f a s e q u e n c e o f n u m b e r s , a0 , a1 , ..., a n ,...
F( z) =
zn
n=0
, z ,
. z -
. F ( z )
a n - , z
. a0 , a1 , ..., a n ,... ,
,
. . .
. 1) b0 , b1 , ..., bn , ...
;
g(z) =
n=0
192
bn
zn
n!
MXn
n=0
zn
n!
X .
3) X , ,
FX ( z ) = M z X =
P {X
= n} z n
n=0
, X
. X . .
X (t )
X (t ) = FX ( ei t )
.
. . . .,
P{ X = n } =
1 (n )
FX (0) ,
n!
M X = FX (1) ,
## D X = FX (1) + FX (1) FX2 (1)
M X ( X 1) ... ( X k + 1) = FX(k ) (1) .
4) X = ( X 1 , ..., X k )
.
## FX1, ..., X k ( z1 , ..., z k ) = M z1X 1 ... zkX k
X
.
. ( A. Moivre )
. ( P. Laplace )
, .
.: [1] ., ,
. ., ., 1963; [2] . .,
, ., 1978; [3] .,
, . ., . 12, .,
1984.
( )
; m o m e n t
generating function
X
. , . . s = 0 s
( s ) = M exp ( s X )
- . ( s ) s 0
, , . . mes > 0 .
. .
k = 0, 1, ...
M X k = (k ) ( 0 ) ,
, .
X 0 ( X 0 ) ,
. . s 0 ( , s 0 ) .
, , ( )
.
.: [1] .,
, . ., . 12, ., 1984.
( )
;
generating function o f a n i n t e g e r v a l u e d r a n d o m v a r i a b l e s . X ,
,
( z, X ) = M z X =
= e t A . ,
, .
.: [1] ., . ., . , . ., . 1, ., 1962; [2] .,
, . ., . 2,
., 1984.
( )
; generator o f a d i f f u s i o n p r o c e s s
k =0
L =
ai j ( x )
i, j
, -
a random vector
.
## / , ; generator / infinitesimal operator
. ,
. A
A f = lim
t 0
Tt ,
Tt f f
t
c( x)
xi
. .,
,
f (Xt )
( X s ) ds
( )
; generating function o f
bi ( x )
( X t , Px ) Px -
t R1
( )
; generating function o f a r a n d o m
v a r i a b l e , ( ) .
X : Tt Ts = Tt + s , t , s 0
.
. . ,
.
.: [1] ., ,
. ., . 1, ., 1984; [2] . ., , ., 1971.
, ( )
2
+
xi x j
( , ).
.: [1] . ., , ., 1963.
X . .
f X (t )
f X (t ) = ( e i t , X ) ,
Cb2
( ) ; generator o f a
p r o c e s s
z k P{ X = k }
X -
z , | z | 1 .
Tt
(*)
f X f - ,
f - (*)-
( D A ). Tt
, . . A
f ( X t , Px )
A - , A f = .
.
; generator
; random walk
; fulling
;
dominating vector , .
; sequence
of dominating vectors , .
,
;
dominated family of distributions ,
, , .
## ; Donsker Prokhorov invariance
principle ,
; Doeblin
condition
,
193
( ,
). 1937- . ( W. Doeblin ) .
(E , B )
p ( n, x, ) -
: ( )
B - m ,
, p ( m, , )
>0
1 .
.: [1] ., , . ., .,
1956; [2] ., , . ., ., 1969; [3] ., ,
. ., ., 1997.
; Doeblin
theorem ,
.: [1] .-., ,
. ., ., 1962; [2] B l u m e n t h a l R. M., G e t o o r R. K.,
Markov process and potential theory, N. Y. L., 1968; [3] S m y t h e R. T., W a l s h J. B., Invent. Math., 1973, v. 19, p. 11348;
[4] . ., . ., 1977, . 22, . 2,
. 26478; [5] M i t r o J. B., Z. Wahr. und verw. Geb., 1979, Bd 47,
H. 2, S. 13956, Bd 48, H. 1, S. 97114; [6] . .,
.: . ,
. 20, ., 1982, . 37178.
; dual predictable projection .
, .
; Doob inequalities * = { t* , t 0}
; Doeblin universal law
, .
P { T* a }
M T
<
;
; dual Markov process ,
( )
= ( t , At , P x )
( E , A ) - ,
E - .
= ( t , At , Px )
( d x ) p ( t , x ; d y ) = ( d y ) p ( t , y; d x )
(*)
,
(
( r ( x , ) << ( ) r ( x , )
<< ( )
> 0 , x E ),
. . . . ( ,
)
; . . .-
. ,
. . . . ,
. . .-,
~ ~
( , A ) -
Px , P x ~
;
, ,
.
194
p
M t
M ( T* ) p
p 1
*
, t
, p >1
= sup s , = ( t , t 0 )
s t
, T
T <.
( 0 = 0 ) , [ , ] =
= ( [ , ]t , t 0 ) = ( t ,
t 0 ) ,
).
(*) r r
1
M t , a > 0 ,
a2
P { T* a } <
1
1
M [ , ] T 2 M
a2
a
.
.: [1] ., , . ., .,
1956; [2] . ., . ., , ., 1986.
; Doob centering
tion , .
; Doob centering
tants { n }
func
cons-
M arctg ( n cn ) = 0 , n = 1, 2 ,...
cn . . . .-
. . . .- ,
{ n an } an
an = cn
{ ( t ), t T } ,
M arctg ( ( t ) f ( t ) ) = 0
f ( t )
.
.: [1] . ., , . .,
., 1956; [2] ., . . ., . 1,
., 1960.
; equalizing strategy , , .
,
; correction for continuity
; Doob Meyer decompozition X
X = M + A , M
, A .
. p
. ., Y
, Y Y = M + A (
A ,
).
. . .-
. . . . .
[1]- ; (
, ;
) . . [2]- . . . .
( , [3] [4] ).
.: [1] . ., , . .,
., 1956; [2] . ., , .
., ., 1973; [3] . ., . ., 1981, . 115,
2, . 16378; [4] L e n g l a r t
E., Lect. Notes Math., 1980,
v. 784, p. 50046.
## / / ; doubling / single redundancy , .
; doubling i n e n g i n e e r i n g
.
. .-
. .- ,
,
. .,
= 0 + 1 + ... + , n
, 1
,
.
/
; supporting point of a design ,
.
; site model , .
,
; forward Kolmogorovs
equation , ,
,
.
; rectangular distribution ,
.
, c = 0,5 - .
## n n p ( 0 < p < 1) = const
n ,
P { n x} =
Cnk p k (1 p ) nk =
k =0
x np + c
1
+ O
,
n p (1 p )
n
(1)
c . n n , n 1 p - .
P { n x } + P { n n n x 1} = 1
. (1)-
xnp +c
n p ( 1 p )
x n p + (1 c )
+
n p (1 p )
=1
.
z - ( z ) + ( z )
(2)
, c , (2)
, ,
x np +c
n p (1 p )
x n p + (1 c )
0
n p (1 p )
c = 0, 5 . (1) c = 0, 5
, .
.: [1] ., , .
., 2 ., ., 1975; [2] ., ., , . ., 2
., . 12, ., 198081.
*
; property ( regularty ) changing function -
( K a r a m a t a J. ) 1930-
.
U ( 0 , ) - , t
x A -
195
U (t x )
( x)
U (t )
( x ) = x ,
. , U
.
. . .- .
U (x) = x L ( x )
(2)
. U ( t x ) U ( t ) x -, , t
x > 0
L(t x)
1,
L(t )
1 F ( x )
(3)
), , (2)
.
U , , ( < < )
, , (2) ,
, L .
U ( x 1 ) , , , U .
.
F1 F2 ,
1
Li ( x)
x
1
( L1 ( x ) + L2 ( x ) ) .
x
~ x L ( x ) ,
1 F *r ( x ) ~ r x L ( x )
.
:
) lim U ( t )
t
U ( t ) ;
) U ( t ) = ( log t ) ;
) - U ( t ) = [ log log ( t + 1)] .
lim c ( t ) 0 ,
.
.
( 0 , ) - L ( -
1 Fi ( x ) =
1 G ( x ) ~
(1)
t +
lim ( z ) = 0 ,
z +
U ( t ) = c ( t ) exp
t0
(z)
z
dz
.
.: [1] .,
, . 2, ., 1984; [2] , . , ., 1985.
; proper estimator ,
.
;
Dvoretzky lemma , l pn -
; Dvoretzky procedure ,
.
. G = F1 F2 -
; Dvoretzky Rodgers
theorem , l np -
, ,
; Li -
; Edgeworth
serie
a n) n Fn ( x ) ;
f ( x) = ( x) +
(1)
bk , k + 2 ( k + 2) ( x ) + ... + bk , 3k (3k ) ( x )
nk
k =1
( sn M sn )
( s n
Fn ( x ) = ( x ) +
(*)
f (x) ,
+ Q1 ( x ) n
Qk (x) = ( x )
(2)
q1 + 2 q 2 + ... + k q k
= k
r =1
1
qr !
r ,
s = q1 + ... + qk .
dk ( x)
(x) =
.
d xk
.,
Q1 ( x ) = ( x )
bk , k + 2l n - , 3 , ..., k l + 3 -
= j j , 2
H k + 2 s 1 ( x ) ,
q r - , H
r
,
),
e x
(1)
r +2
( r + 2 )! r + 2
= 1 + ... + n , 1 , ..., n
1
+ ... + Qm ( x ) n m 2 + Rm ( x , n ) ,
D sn -
( x) =
1 2
1 1
10 2 ( 6)
( 4)
3 ( x )
4 ( x) +
n 4!
6!
1 1
35
280 3 (9)
( 5)
(7)
3 2 5 ( x ) +
3 4 ( x ) +
3 ( x ) + ... .
7!
9!
n
5!
bk , k + 2l .
## (*) . [1]- . . ( H. Cramr ) , , (*)
f (x ) -, ,
.
.: [1] E d g e w o r t h F. Y., Proc. Comb. Phil. Soc., 1905,
v. 20, p. 3665; [2] ., , . ., 2 ., ., 1975.
; Edgeworth Kramr expansion a , 2
k
k = M ( X i a ) , 3 k m + 2 , F (x ) ( X 1 + ... + X n
( 1 x2 ) ,
2
( 3 4 ) ( x 3 3 x) 3 ( x 5 10 x 3 15 ) .
72
24
= ( x)
1 1
f (x) = ( x ) 1 2
3 (3) ( x ) +
3!
n
Q2 ( x) =
, j ,
1 j - . ., . .
m + 2
r<m+3
M X j a
<
## lim sup M exp ( i t X 1 ) < 1 ,
Rm ( x , n )
( n(1+ x )
( 1 + x )r
n ( r 2)
(3)
, u 0 ( u ) 0 .
r = 3 F - R1 ( x , n ) <
< ( n ) n 1 2
. m = (1)
. . . .
. (1) . [2]- .
X j
R d - , d > 1 ( , [4] ) . .
( , [3] ). F
0 < 2 G (x) ( , [5] )
. . .- . . . .-
k
197
k =
.-
. n
x k d ( F ( x) G ( x) )
.
.: [1] E d g e w o r t h F., Trans. Camb. Philos. Soc., 1905,
v. 20, p. 3665; [2] C r a m r H., Camb. Tracts Math. And
Math. Phys., 1937, 36 ( . . , ., 1947 ); [3] . .,
, ., 1972; [4] . .,
. : .
, ., 1980, . 11621; [5] . ., .:
VI
, , 1962, . 4950.
; efficient estimator , . . .-
( - ). . . .
.
. . . ,
. ,
. . . .
, .
.: [1] ., , .
., 2 ., ., 1975; [2] . ., . ., , ., 1979;
[3] ., ,
. ., ., 1968.
( )
; s e c o n d o r d e r efficiency o f e s t i m a t o r s
.
[1]- . . 1 , ..., n ,
f ( x , ) - ,
, ()
. lim n n ()
n
= () ,
n ( )
,
lim ( n () n () )
.
, , ,
, .
, ,
[2] [6]-
. ,
198
## ( , [6] ). [7] [10]-
f ( T1 , ..., T p )
, Ti
2- .-
(
). 2-
.
.
2- .
.
R n ( n , ) = M
,
n ( n )
>0
= ( , + ) , + . 2-
. -
n
Rn ( n , ) = R0 ( ) n 1 p ( ) + o ( n 1 )
(1)
R0 ( ) = / 2 ( ) ,
= 2 / 2 1 / 2 ( ( + 1) 2 ) ,
p ( )
C () ,
0 < 1 . p [ , + ]
, (1)
.
(1) n : *n
> 0 , n
= M
ln f ( 1 , )
, n ()
n = n ( 1 , ..., n )
o ( n ) -
Rn ( *n , ) Rn ( n , ) n 1 ( ) + o (n 1 )
, n
q -
, q 0 , q 0 ,
q C ( ) , q n ,
q , , .
2- .
n = n + n 1 g ( n ) { | g ( n ) | < n1/ 2 }
(2)
, g C () , n -
. n ,
(2)- g
( + 1) M 3,1 ( )
g ( ) = T 1 ( ) 2
ln ( ) +
6 ( )
(3)
, ( ) > 0 ,
M i j () = M
ln f ( 1 , ) . (2), (3)
n, q .
n, p ()
2 1 ( )
( 2+ ) / 2
( )
( ) + p0 ( )
, p0 ( )
. ., f ( x, )
(4)
f ( x , )
= f ( x ) , f ( )
p0 () =
/ 2 ()
4 ( + 1) M 4,1 ()
24
2 ()
M 1, 4 ()
+ ( + 2 )
3
2
()
.
( ) = 2 () I ( 2 + ) 2 () , ( ) = q ( ) 2 ( )
( u ) du ,
() =
(u ) du
( ) d ,
= lim () ,
<
(u ) du ( ) d ,
= min ( , + ) ,
= max ( , + )
. : 1) , , ,
, = ;
2) , , , q
, max (, ) = . ., n , ,
,
+1
( 2 + ) / 2 (u ) exp
6
M 3,1 ( )
( )
d du
( , [11], [12] ).
. i
q ( , [12] ).
.: [1] F i s h e r R., Proc. Cambr. Philos. Soc., 1925, v. 22,
p. 70025; [2] L i n n i k Y. V., M i t r o f a n o v a N. M., Sankhya,
Ser. A, 1965, v. 27, 1, p. 7382; [3] . .,
. ., 1973, . 18, . 4, . 689702; [4] . ., . ., 1976, . 21, . 1, . 16
33; [5] . ., . . . ., 1981,
. 45, 3, . 50939; [6] P f a n z a g l J., W e f e l m e y e r W.,
J. Multivar Anal., 1978, v. 8, 1, p. 129; [7] R a o C., Sankhya,
Ser. A, 1963, v. 25, 2, p. 189206; [8] E f r o n B., Ann.
Statist., 1975, v. 3, 6, p. 1189242; [9] G h o s h J., S u b r a manyam
K., Sankhya, Ser. A, 1974, v. 36, 4, p. 32558;
[10] E g u c h i S., Ann. Statist., 1983, v. 11, 3, p. 793803;
[11] G h o s h J., S i n h a B., Ann. Statist., 1981, v. 9, 6,
p. 133438; [12] . ., . .,
1985, . 30, . 2, . 30938.
resource
efficiency
, .
( )
;
efficiency o f a s t a t i s t i c a l p r o c e d u r e
= lim () ,
, ( 1 , q = 1 ).
n, -
( )
. , 0 ( , )
+
( ) =
2 2 +
R ( n , , ) =
+ o (n 1 )
2
2na
, = cos
, < 2
= ( a, a ) ,
[ [3]- M 3,1 ( )
= p0 () = 0 ]. -
.
( ) .
, ,
.
.
; : ,
.
.: [1] . ., , .,
1984; [2] V o i n o v V. G., N i k u l i n M. S., Unbiased, Estimators
and Their Appliations, v. 1, Univariate Case, Dordrecht, 1993;
[3] G r e e n w o o d P. E., N i k u l i n M. S., A Guide to Chi-squared
Testing, N. Y., 1996.
, ;
efficiency / utility o f m e t e o r o l o g i c a l forecast ,
,
( )
.
- ,
. . , .
. .-, ,
( , [1] ).
. .- 0 = ( R R+ ) R
, R+ R
199
( , );
-
.
,
0 ,
0 . 0 = 0 , , .
, .
. .
,
, .
.: [1] . .,
, ., 1981; [2] . .,
, 1933, . 3, . 4, . 48999; [3] . ., . . . ., 1955, 4, . 33949;
[4] T h o m p s o n J. C., .: Weather forecasting and weather
forecasts; models, systems and users, v. 2, Boulder ( Colo.), 1976,
p. 52578.
## ; efficiency preservation coefficient , , .
; probability
. .
,
. .
.
.- ( ) :
.
.-
.
,
( p n -
)
.
.-
.- .
.
.
.-
,
. , .- ,
, ; .- . ,
,
.- .
.- .
200
,
( p, 0 p 1 ) ;
.
. , n -
k n ( A) n
[ k n ( A) , A n ]
, p .
n - k n ( A) n - p - , . .,
. 1 2 - .
n = 10 ,
.
;
4 5,
6- .
n = 100
, 40 60 ( ,
).
. , ,
, .- . , ,
.-
,
.
, .- , .
. .- . , , .- 0,003
( ).
.
: , - ( )
P . A P P ( A ) , A .
.: [1]
, 2 ., ., 1974.
. ., -
; probability i n q u a n t u m
p h y s i c s , ,
. ., , (
)
. ,
,
, , .
.
,
,
. ,
( , U ) - P ( d )
,
.
,
,
( , ).
,
( ,
; , ).
,
. ( N. Bohr )
20- 30-
( , [1] ). de facto .
. ( J. Neumann ) [2] ,
20- 60- .
H S , X ( , , );
S ( d ) ,
X ( ) .
S ( d ) S , X ( ) X
; ,
.
. .
( , [3] ),
, :
X ( )
H - . 2- X X ( ) X
, -
X ( ) X
f ( X ( ) ) f ( X ) .
. ( S. Kochen ) . ( E. Speker ) ,
.
, ,
-
X ,
X ( ) X , , . . ,
,
.
( , [4] ). X 1 , ..., X n Y1 , ..., Ym
X i Y j = Yj X i ,
i = 1 , ..., n ,
j = 1, ..., m
(1)
. , { X i } { Y j }
. (1)
X 1 , ..., X n Y1 , ..., Ym S
X 1 ( ) , ..., X n ( ) , Y1 ( ) , ..., Ym ( )
( , U ) S ( d )
,
tr S X i Y j =
X i ( ) Y j ( ) S (d ) M S X i Y j
n, m
S ( d ) S , X ( ) X ( ) , .
,
.
, .
., X 1 ( ) , X 2 ( ) , Y1 ( ), Y2 ( ) , [ 1, 1]
, ( , U , P )
;
M P X 1 Y1 + M P X 1Y2
+ M P X 2Y1 M P X 2Y2 2
(2)
. (2) tr S X i Y j
. 1 / 2 -
,
(2)- 2 2
. , ,
, .
, .
.: [1] ., , . 2, ., 1971;
[2] ., ,
. ., ., 1964; [3] B e l l J., Rev. Mod. Phys., 1966, v. 38,
p. 44752; [4] . ., , ., 1985.
, a
posteriori; a posteriori probability ,
.
201
a priori; a
priori probability , .
, ;
probability of connectedness o f a r a n d o m g r a p h
, .
; probability space ,
A (
) A - P ( )
( , A , P ) . . . . .
[1]- . - , . A
( ) . . .-, B A , A B,
P (B ) = 0 A A
. .-
. ( , A , P ) . .-,
A A N M , P ( M ) = 0 A U N
A
A -
P P ( A U N )
= P( A ) -
, A - . ( , A , P ) . . ( , A , P )
.
. .
: A f
E f
B , B E P ( f
( E ) A ,
( E )) = P ( f 1 ( B)) .
. .-
( , ) . . .- .
: T t1 , ..., t n
R
Pt1 , ..., tn
,
:
## 1) Pt1 , ..., tn ( I y1 , ..., yn ) = Pt
,..., t n
1
( y1 , ... , y n ) R
( I y 1 , ..., y n )
1, ..., n
1 , ..., n ,
## I y1 , ..., yn = { x = ( x1 ,..., xn ) : xi yi , i = 1, ... , n };
2) Pt1 , ..., tn ( I y1 , ..., yn1 , ) = Pt
R
,..., t n 1
( I y1 , ..., yn1 ) .
= { x = { xt }, t T , xt R1} -
A - P
202
, T t1 , ..., t n
n B
Pt1 ,..., tn ( B ) = P { t1 ( x) , ..., t n ( x) B }
t ( x ) = xt
A .
.: [1] . .,
, 2 ., ., 1974; [2] . ., . .,
, . ., 1949; [3] .,
, . ., ., 1969; [4] . ., . ., . ., , ., 1979.
*
( )
; probability space, g e n e r a t e d b y r a n d o m v a r i a b l e ( , F , P )
= ( )
( R1 , B , P ) ; R1
1
, B , R ( a , b ]
b}
= { x R1 : a < x
( ) , ;
P (B) = P { : ( ) B }, B B
P , B
( ).
.: [1] ., , . ., ., 1962.
; probability space method
, .
;
; error function , .
;
; transition probability ( E , B )
s x t E :
P { t E | s = x } = p ( s , x ; t , E ),
s t, x E , E B
p ( s, x ; t , E )
. ,
. .
. t
,
p ( s , x ; s + n , E ) n ,
. . . . . n . .
.
E
p ( s, x ; t , E ) =
p ( s, x ; t , y )
y E
. .- ;
p ( s , i ; t , j ) = pij ( s, t ), i E , j E . ,
. . , p ( s, x; t , E ) =
= p ( t s, x , E ) , pij ( s, t ) = pij ( t s ) .
s t -
pij ( t + s ) =
. .- (1), (2) .
. . , ,
, i
E \ R -
mi .
## pik ( s ) pkj ( t ) , i, j = 1, 2 , ..., N .
k =1
mi
;
( X , A ) ( Y , G )
. . p ( x , E ) , x X , E G
, 1)
p ( x , E ) E x A
mi = 1 +
; ,
; absorbtion probability i n a M a r k o v
c h a i n
, 1 ,
1 ( ). { X t }t = 0 , 1, 2 , ...
E , C
( ,
; ). , C
,
. C i . .
i ,
i(n ) ,
i E
(n)
, i , n - . .-;
1,
0,
i C ;
i E \ C
## p00 = pnn = 1 , pi ,i 1 = q pi ,i +1 = p , 0 < i < n , 0 < p < 1 ,
q = 1 p ,
. 0 n ,
R
i C ,
i E \ ( C U R ) .
(1)
. . i i R
pij j , i R
= i0
( i - )
( r i r n ) (1 r n ) ,
( n i ) n ,
p 1 2
p =1 2
i =
,
,
(r
=q p)
0 = 1 , i = q i 1 + p i +1 , i = 1, 2 , ..., n 1 , n = 0
1
q p
i n 1 r ,
1 r n
mi = i ( n i ) ,
R . E \ R
- ,
1,
0,
= { 1, 2 , ... , n 1 }
. (1), (2) 0 i
mi =
i(n ) = Pi { X t E \ C , 0 t < n , X n C } .
i =
( )
.
(1), (2)
(3), (4) .
( ). E = { 0 , 1 , ..., n } ,
. (3),
(4)
n 1
i =
(4)
n=0
i(0) =
(3)
mi = 0 , i E \ R
.: [1] .,
, . ., ., 1984; [2] . ., . ., 1980.
i =
pij m j , i R ,
jE
2) p ( x , ) x ( Y , G ) - .
(2)
j E
. R ( E )
, (2) (1)
. R , ,
, i R
R .
p 1 2 ,
p =1 2
. n
E -
0- .
R = {1, 2 , ... } (1), (2) (3), (4) :
1,
r i ,
i =
i (q p),
mi =
+ ,
p 1 2
p >1 2
p < 1 2,
p1 2,
i 0 ,
i 1 .
. .-
i j k
203
ij - i - j mij - . i k
( j )
. mij -
( , ;
).
(2) (3)
;
.
.: [1] ., , . ., . 1, ., 1984; [2] .,
., , . ., ., 1970.
; probabilistic combinatorial analisys ,
,
, ,
. .
,
.
.
. .- .
1. .
N -
( n m ) .
( N ; n, m) n = m ,
0 < 1 m N m ln m
2. .
( n m ) ( 0 , 1)
A
E nm = { per A = 0} ,
H nm = { A - m n + 1
}
, m n
m -
P { H nm | E nm } 1
, per A , A .
N n
N = N ( n ) > n2/3 + ,
( 0 , 1)
>0
> 0 , n ,
P { | per A M ( per A ) 1 | > } 1
, M ( per A ) , per A -
= n ln n + c n + o ( n ) P ( n , N ) =
= P { per A > 0 } , n P ( n , N )
. N
exp{ 2 e c } .
N = N r = n ln n + ( r 1) n ln ln n + n (n) + o (n)
n n P ( n , N r , r )
= P { ( A) r } ,
P ( n , N r , r ) 1 , ( A) , A ( 0 , 1) n
.
3. . S n , N = {1, 2 , ..., n }
n . N ( s, s ) = { i : s ( i ) = s( i ) , 1 i n } -
.
1 , 2
s s
. N ( s, s ) = s
= 1 , = 2 e , 0 < < 1 = e
,
= 1 (1 , 2 )
( e
, e ) ,
; 0 < < 1
, e
.
N
( n m ) ( 0 , 1)
. ( n m ) 0 , 1 , ..., s 1 , s > 1 ,
> 0 , m = s ,
.
204
s s .
i , si , 1 i k ,
M n ( s1 , ..., s k ) = { s : s s1 , ..., s sk , s S n } =
= per ( I ( 1 ... k ) ) ,
.
s S n , s1 , ..., s k S n
, - , A ( i ) = { s ( i ) = sv ( i ) }
, 1
k , 1 i n .
n ( s1 , ..., sk ) = | { i : A1 ( i ) U ... U Ak ( i ) }, 1 i n |
n k
=k
. , ., ( k n )
L ( k , n ) n k
L ( k , n ) = ( n ! )k e
( ) ( 1 + o (1) )
k2
.
4. .
n
n
.
( n ln n )
( n + 1) ( ( ~n + 1) ) . n ~
n ( n ) = 1
ln n -
= 1 2 ) .
d n , n
-
nl
n = 1 / l
(n)
~ ( n ) , n , , n
P { ( n ) = k } = (1 e1/ k ) exp
j 1 + o (1) .
k 1
j =1
( + 1) 1 x < y < 1
x y n
~ ( n )
P x
< y
n
k=1
(1) k
{ J k ( y , 1) J k ( x , 1) } ,
k!
P { xi + 1 = u n } =
xi m ,
...
dx
dx1
n ,
x1
xn
k 1 ,
< n .
Qn , n ,
1 / 2
n ln Qn ln 2 n ln 3 n
2
, ( 0 , 1) .
5. .
X = { x1 x2 ... xn } -
, n n 2 n
. n ,
1
( n ne )
n ( e 2 ) e 1
n ( 0 , 1) -
. n , n
P{ n < x 2 n }
k =
.
d ( xi ) ,
u
n
e u
( 1 + o (1) )
i = d ( xi ) 1 , 1 i n ; [ x1 1 x2 2 ... xn n ]
[ x1 1 x2 2 ... xn n ]
n n 2
; i - i
, 1 i n . ,
.
6. . n N -
( n2 )
n, N
J0 (m, n ) = 1 ,
J k ( m, n ) =
n = o ( n1/ 6 )
n ( 0 , 1) . l
u = dn
2 2
k x
( 1 2k 2x2 )
N = n ln n + cn , c , P ( n , N ) , n, N
2
k n k
k ,
n P ( n , N ) 1 .
P0 (n , N ) exp { e
2 c
P0 ( n , N ) n
} .
n , n, N
, n n n , = e 2 c
. ( n , N ) 1 n ,
( n , N )
.
7. . X n n n
. X
( X , )
x = ( x ) , x, x X x x , ( x , x)
. n ( x , ) ,
n u
= o ( n1/ 6 )
205
P{ n
n = u } = u eu
n = n [ er ln 2 e r ln ( e 1) ] , r e r = n ,
( X , ) - n n
1 / 2
ln n
n ln n
2
2
( 0 , 1) .
p n , , s n
x X
9. . X n
X = X 1 U ... U X k , X 1 , ..., X k
. n
; n
. n
P { ( n ( 2 n1 1 / 2 ) ) 2 n / 21 = x } =
k k 1
,
e k k!
P { pn = k }
n = u } = u eu
k = 1, 2 , ...
2
( 1 + o (1) )
r , x
. ( X , )
n / 6
n n
r
.
8. . X n k
( n , k ) -
. n Bn -
. Bn
. n ( n n ln n ) ( n ln n )
( 0 , 1)
. n (l ) l
r , r e
= m ,
n ( n ( l ) r l l ! ) / r l l ! ( 0 , 1)
## < i2 < ... < ik n
i1 , i2 , ..., ik k ( n (i1 ) i1 )
ik )
i1 , ..., ( n ( ik )
ik ) n -
1- ,
, l
= r l / l ! . n -
ex
( 1 + o (1) )
P { ( n ( 2 n1 1 / 2) ) 2 n / 2 1 } ~ ( ) ( ) ,
( x) =
e u
/2
du
.
X = X 1 U ... U X k , . n
; n
. n :
2
n
2 j
~
P n
= j
,
2
1 + 2C0
j = 0 , 1 , ...,
2 j ( j + 1)
n 1
~
P n
= j
,
2
2C1
j = 0 , 1 , ..., ,
C0 =
j =1
1
2
C1 =
j =0
1
.
2 j ( j + 1)
X n 2 1 ;
X 1 , X 2 , ... , . X = X 1 U ...
U X n n X
n
. | y |
o ( log 2 n ) y - n
P { n ( l ) > 0} 1 l ! / r l , r e r = n
. , n , n P { n
n1
0 , 2 n / 6 -
r ( n )
; n
. ( X , ) x X
. 1 i1
er ln 2 e r ln ( e 1) .
P { sn
x - ;
( 1 + o (1) )
= 1} 1
P { n [ log 2 n ] + y } = e 2
[ y ] + n
, [ log 2 n ] n
(1 + o (1) )
= { log 2 n } -
. n n -
log 2 n - .
10.
.
.
.
P{ n < x } ee
206
[ x ] +
n Gn ( p) p
, q
; p + q = 1 . Gn ( p )
Gn ( p ) , , ,
Gn ( p ) - . n ( p, k ) n ( q , k )
, Gn ( p )
Gn ( p ) k
n k
M n ( p, k ) = p 2 ,
k
2 ln n
ln (1 / q )
2 ln n
k > max
ln (1 / p)
, n
M n ( p , k ) = o (1) ,
M n ( q , k ) = o (1)
p n > n0 ( p )
2 ln n ln (1 / p ) -
Gn ( p) , 2 ln n ln (1 / q ) ( p + q
X 0 , X 1 , ...
## (Xn, X0) 0 (X n , X0) 0
.
( . ). 0 < p < 1 -
Gn ( p )
X -
. ,
X - 0 , 1 , ...
; [ z ] , z - .
## ; comparesion / ordering of probabilistic metrics
. X -
-
, -
n k
M n ( q , k ) = q 2
k
## .: [1] . ., ., 2 ., , 1962; [2] . .,
, ., 1961; [3] . ., , .,
1967; [4] . ., , .,
1974; [5] E l l i o t t P. D. T. A., Probabilistic number theory, v. 12,
N. Y., [a. o.], 197980.
= 1 ).
.: [1] ., .,
, . ., ., 1976; [2] . ., , ., 1978; [3] . ., , ., 1982; [4] . ., ,
., 1984.
; probabilistic methods i n n u m b e r t h e o r y
.
, . XX I . ( E. Borel ) . .
.
, , (
), . .
.
. ,
, ,
.
( n , 0 ) 0 , ( n , 0 )
/ 0
, p ,
- .
- - :
~ .
. ., 1 -
- . ,
. ., , L , ( , ; ), , , 0 ,
0 ( X , Y ) = sup { | f X ( t ) f Y ( t ) | : t R1}
1
4 r
2 ( X , Y ) = sup
| f X (t ) f Y (t ) |2 dt : r > 0
( f X , X ) :
L ~ ~ p ~ 0 ~ p 0 p
(*)
, pp -
:
X - X 0 , X 1 , ..., 0 , 1 , ...
n
( X n , n ) 0 ( X n , n ) 0
.
p ~
pp . (*) -
207
L pp pp , L pp pp ,
0 pp , pp 0 , 0 .
, X - , p
:
K X
( x , K )
, ( + 0 , K )
= 0 , K - ( , K )
pp X
,
( ) , (u ) ( + 0 ) = 0.
X
pp ( ) ; (u )
( + 0) = 0 .
,
## .: [1] . . . ., 1977, . 102, 3,
. 42534; [2] . .,
, ., 1986.
;
probabilistic metric
X -
. X X P -
,
(1)
(2)
(3)
- , , , , ,
( . .- ) . .-
.
. .
( , ( ),
( ), , , ).
. . .
[1] [2]
( . 1 )- . . .-
, ,
, , ,
; .
.: [1] . ., . .,
1983, . 28, . 26487; [2] . ., , ., 1986.
; ; ; probabilistic
metric; h o m o g e n i t y o f s
( Tc ( X ), Tc ( Y ) ) = c s ( X , Y )
, Tc
,
, a , b > 0 Ta Tb = Tb Ta = Ta b . Tc .
U , X 1 X 2 ... X n
( , ) = 0 P ( = ) = 1 ,
( , ) = ( , ) ,
( , ) ( , ) + ( , )
( )
( , )
, x, y U
Tc ( x
X - .
X - ( , ) < ,
y ) = Tc ( x) Tc ( y )
Tc (x) = c x
Tc (x) =
, ( , ) =
= x sign x ( x y = max ( x, y ) )
, ., (2) (3) ( , ) =
Tc .
, .
. .
. ( , ) -
; ; ; probabilistic
metric; r e g u l a r i t y o f -
P P , . .
,
(1)
. . .
( , ) = 0 P = P
.
( , ) (P , P )
- (
P , P
, ,
).
. . ( ,
,
. ).
208
X
.
, X , Y , Z X
(X
Z, Y
Z ) (X , Y )
(1)
. (1) :
X 1 , X 2 , Y1 , Y2 X
( X 1 X 2 , Y1 Y2 ) ( X 1 , Y1 ) + ( X 2 , Y2 )
(2)
. . , , , (1) X Y Z
.
(2)-
. , ,
.
, .
.,
( X Y = X + Y ) [ X Y =
= max ( X , Y ) ] .
.
.
X -
( , )
.
. .-
.
(1)- (2)- .
(1)-
(Z
X, Z
Y ) (X , Y)
. (1)
(1)- .
.: [1] . ., . ., 1976, . 101, 3,
. 41654; [2] . ., , ., 1986.
; ; ; probabilistic metric;
s t r u c t u r e o f
.
. . .-
,
. . .
.
.
X , (U , d )
,
.
I. . w ( X , Y ; t ) , P2 (0 , ) -
, P2
= { PXY } , X
.
w ( X , Y ; t ) : t , t1 , t 2
,
) w ( X , Y ; t ) = 0 P{ X
= Y} = 1
w ( X , Y ; t ) t - , -
w -
, (1) ,
- .
= 2 = .
(1) . .- :
( X , Y ) ( U = R ) ,
w ( X , Y ; t ) = (1 / 2) max ( | f X (u ) f Y (u ) | : | u | 1 / t ) ,
f X , X
;
- ,
w ( X , Y ; t ) = P{ d ( X , Y ) t } ;
,
:
w ( X , Y ; t ) = sup{ P { X A} P { Y At } : A B }
B , U -
At = { x : d ( x , y ) < t , y A} .
II. . Y = { g } , (U , g ) - .
( X , Y ; Y) = sup{ | M ( g ( X ) g ( Y ) ) | : g Y }
(2)
X - . .-. , s
( ), ,
, , . .-. Y
(U , d ) - ,
.
III. , , -
:
## 1. ( X , Y ) = M d p ( X , Y ) min (1, 1 / p ) , p > 0 ,
P2 - . .-.
2. ( X , Y ) =
g ( x)
PX PY ( dx)
(3)
( w PXY - , PX , PY
PX
X - . .-. ( X , Y )
= PY
);
, g ( x) 0 , U -
. , g ( x )
) w ( X , Y ; t ) = w ( Y , X ; t ) ;
= x
s ,
) w ( X , Y ; t1 ) w ( X , Y ; t1 + t2 ) ;
, U
) w ( X , Y ; t1 + t2 ) w ( X , Z ; t1 ) + w ( Z , Y ; t2 ) .
3. ( X , Y ) = sup
## 2 ( X , Y ) = sup { min ( w ( X , Y ; t ), t ) : t > 0}
. .- , g
(1)
g ( x , A ) (P
PY ) ( dx) : A A , (4)
209
. .-, A , U - . g ( x, A )
= I ( x A ) g ( A)
; probabilistic distance
. . .
d( X , Y )
. . .-
, . . :
( X , Y ) = sup{ | g ( A) | | PX ( A) PY ( A) | : A A}
P { X = Y } = 1 ,
, g ( A) , A - .
(3)-
.
4. (4)- , .
5. . .-
d(X , Y ) = 0
(*)
. . .- X Y PX , PY
d ( X , Y )
( X , Y ) = ( x , y ) ( PX PY ) ( dx) ( PX PY ) (dy )
. .- . . .-
(*) PX = PY
, , U U - , : x j U ,
d (X , Y) = 0
j R1
( x , x ) = 0 , ( x , y ) = ( y, x ) ,
i<j
; probability theory
( xi , x j ) i j 0 , 1 + ... + n = 0 .
. .-
( X , Y ) = 0 PX = PY
, , ,
M ( X , Y ) h ( X ) h (Y ) = 0 M h ( X ) = 0
.
. . S XY ( A , B )
P2 A A -
, , A , B -
.
## ( A, B ) = max sup inf d ( x, y ), sup inf d ( x, y )
y
B
x
A
yB
x A
A -
( X , Y ) = max { ( S XY ) ( SYX )}
( S XY ) = sup inf { ( A , B ), S XY ( A , B ) } .
AA BA
;
A = B
(5)
S XY ( A, B ) (5)
A { ( , x ) : x R1} .
.: [1] . ., . ., 1976, . 101 (143),
3, . 41654; [2] . ., .
., 1978, . 23, . 2, . 28494; [3] . .,
. ., 1983, . 28, . 2, . 26487; [4] . ., . ., 1984, . 29, . 4,
. 62553; [5] . ., Lect. Notes Math., 1983, Bd 982,
S. 17290.
210
( ).
n
k n ( A) / n p ( 0 p 1) .
A P ( A)
, -
S XY ( A , B ) = P ( X A) P ( Y B )
. . .
.
.
. S
A
, . S n
, n
, . k n ( A ) , A n
. k n ( A ) / n A
, . .
,
. . .- ,
.
.
, A
. .,
n
, 1 / 2 - .
,
51 52
; , 1 / 2 -
.
( )
:
, . A1 , ..., Ar :
. A1 , ..., Ar B A1 , ..., Ar
A1 , ..., Ar
A1 , ..., Ar .
A , - ;
A A :
B = A1 U A2 U ... U Ar =
P ( A) =
UA
i =1
C = A1 I A2 I ... I Ar =
IA
i =1
.
A
.
, - A -
: A , A
, . A , -
. ,
( ),
( ).
. .- P ( A )
. , A
= { }
A -
( A - ,
, ,
; , , ). P A A
( ) :
1. P ( A ) 0 ( P - ),
2. P ( ) = 1 ( P - ),
3. i j , Ai I A j = ,
U A
i
i =1
P( A )
i
i =1
( P - ).
A - P 1, 2, 3
, ( , A , P ) .
1933- . .
- . .-
.
( ) ,
P ( A ) . .
P
. ., , ;
A . P
p ( ) ,
:
p ( ) 0 ,
p ( ) = 1 .
p ( )
(1)
( .,
).
. p( ) - -
(1)
P ( A) =
N ( A)
,
N ( )
N ( A) , A ( A
) .
P .
, m ()
(
)
, A A
P ( A) =
m ( A)
m ( )
(2)
. (2) .
1 3 P - A
. , . .-
. . .
. . .-
.
A B P ( A | B)
P ( A | B ) = P ( A I B) P ( B) ,
P(B) > 0
(3)
.
.
P ( A | B) = P ( A )
(4)
, A B
( )
( , ). (3) (4)- A B
( ) :
P ( A I B) = P ( A ) P ( B)
(5)
(5) A B
P ( A I B ) . A B -
, , -
kn ( A I B )
k ( A ) kn ( B)
n
n
n
n
211
n - . , A B
.
, Ai Ai
P ( A1 I ... I Ar ) = P ( A1 ) P ( A2 ) ... P ( Ar )
, A1 , ... , Ar
.
.
,
,
- { Ai }
.
= ( )
( , A , P ) =
, { : ( ) B } A ,
A ,
( B - ) P (B) ( )
.
P (B) - F ( x)
= P { : ( ) < x } , x ( , )
) p ( x ) : B R
,
P ( B ) =
. , -
i .
S S1 , S 2 , ..., S n
(1)
( 2)
( n)
I Ak(nn )
p ( x) dx ;
p ( x ) 0
p ( x) dx = 1 .
; x1 , ..., xn , ... P { : ( ) = xk } = p k -
(6)
, S S1 , S 2 , ..., S n -
S .
P ( Ak(11) ), P ( Ak(22) | Ak(11) ) , ..., P ( Ak(nn ) | Ak(11) I Ak(22) I ... I Ak(nn11) ) (7)
(7) (3) (6)
.
:
) ,
(1)
( 2)
(n)
(7) , P ( Ak ), P ( Ak ) , ..., P ( Ak )
; ) S i S i 1 ( ) , (7)
## P( Ak(11) ), P( Ak(22) | Ak(11) ) , P ( Ak(33) | Ak(22) ) , ..., P ( Ak(nn ) | Ak(nn11) )
.
, .
P ( Ak(11) )
P ( Ak(22) | Ak(11) ), P ( Ak(33) | Ak(22) ) , ..., P ( Ak(nn ) | Ak(nn11) )
.
;
212
## C = Ak(11) I Ak(22) I ...
, ; C ,
.
. ( -
i j , Ai I A j = ( - ) -
Ak , Ak , ..., Ak
, ( ) , B
{ : ( ) B } , B R1 . { : ( ) B } -
; A1 , ..., Ar
## ( , , [1]; [2]; [3] ). Ai
. ( , A , P ) ,
P ( B ) =
Pk { = x k } .
xk B
p ( x) =
( x a )2
2 2
, x ( , )
(8)
e x , x 0, > 0
p ( x) =
0,
x<0
.
P { = k } = C nk p k q nk ,
k = 0 , 1 , ..., n
(9)
, , n
, p 0 p
1, p + q = 1.
.
( , ,
).
. 1 , ..., n P1 , ...,n { (1 , ..., n ) B }
1
, B , R - ,
B R n . B R1
I
i =1
{ i Bi } =
{ | S n a | > } -
P{ i B i }
i =1
, 1 , ..., n ,
.
,
, ., { a < 1 + ... + n < b } ,
( ) , .
. .-
(9) n -
1
2 n p (1 p )
n p)
ex
x = (m
n p (1 p ) .
,
. .- .
., ,
, ,
. ,
( , ,
, ,
).
1 , ..., n , ... M k = a ,
D k = 2 ,
;
Sn =
1 + ... + n
n
. > 0
{ | S n a | } n
1- , S n
a - . , S n - a -
0,
. , S n - a -
n - (
) n
;
.
.
( ),
.
n
.
,
. .
( ,
).
XX 20- ,
. ., 1
, 2
. ,
1 + ... + n , n -
1
( 1- )
n
. ,
n - ,
n1 , n - (
n
).
(
).
, , .
.
.
XX
( . ) .
. . .-
, , (t )
. t ,
, ., , , ( ) , t , . t ,
( ) . ( t1 ), ( t 2 ) , ..., ( t n )
; t1 , t 2 , ..., t n n -
. :
;
.
. t0 t1 ( t 0 < t1 )
( t ) t t 0 t -
, ( t1 )
( t 0 ) - (
).
.
t0
;
213
t0 ( )
t
> t 0
t0
.
,
.
.
;
.
;
M (t ) M ( t ) ( t + )
t - . ( t )
(t ) =
it
dZ ( )
, Z ( )
.
(
), .
,
( , ; , ;
).
,
. ( t ) s < t
( t ) -
( s ) - .
.
.
.
,
( ., , .
; , ). , ,
.
. . . 17 .
. .- . ( B. Pascal ), .
( P. Fermat ), . ( Ch. Huygens )
. . ( J. Bernoulli )
. .- ( 1713-
).
214
. . ( ) ( 18- 19-
) . ( A. Moivre ), . ( P. Laplace ), . ( C. Gauss ) . ( S. Poisson )
. . .
( , )
.
- ( 1812 ) ( 1837 )
;
. ( A. Legendre, 1806 ) . ( 1808 )
.
. .- ( 19- 2- )
. . , . . . .
. . . ( 18- )
; .
( L. Euler ), . ( N. Bernoulli ) . ( D. Bernoulli ) .
. .- . .- . . ,
. .- , . . . 19- . . . . .
. . . . . .-
;
. . . , ( 1867 ). ,
. . .
( 1901 ).
1907- . .
( ).
19-
[ . ( A. Qutelet ),
. ( F. Galton ) ] [
. ( L. Boltzmann ) ] . . , . .
. . . .- ( )
. . .- . .
( E. Borel ), . ( P. Lvy ), . ( M. Frchet ),
. ( R. Mises ), - .
( N. Wiener ), . ( W. Feller ), . ( J. Doob ),
. ( H. Cramr )
. .
; . . . .- , , ,
. .-
.
[ . . . . . . ]. ( 30- ) ( . . ) .
. . . . . ,
. . , . . . . . .-
.
: . ,
. 23, ., 197072; XIX .
. . . , ., 1978.
. B e r n o u l l i J., Ars conjectandi,
opus posthumum, Basileae, 1713 ( . . .,
, ., 1986 ); M o i v r e A. de, Doctrine
## of Chances, 3 ed., L., 1756 ( . N. Y., 1967 ); L a p l a c e
[P. S.], Thorie analytique des probabilits, 3 d., P., 1886; . ., . . ., . 23, . ., 194748; L i a p o u n o f f A., Nouvelle forme du thorme sur la limite de probabilit,
, 1901; . ., , 4 .,
., 1924; . ., , 4 .,
. ., 1946.
. . .,
, 6 ., ., 1988; . .,
, 2 ., ., 1986; . ., , , .,
1985; . ., , ., 1982; . ., ,
2 ., ., 1989; . ., . .,
, 3 ., ., 1987.
. . .,
, 2 ., ., 1974; ., , . ., ., 1962; .,
, . ., . 12, ., 1984.
; probability theory o n a l g e b r a i c
s t r u c t u r e s ,
;
. . .
, , , , .
,
.
,
.
( ,
).
( ., , ,
.).
. ( ., ) , ,
( ,
; ; ).
,
R1 -
.
, ,
.
( 1 , 2 , )
( .,
, ).
. . . 1
. R - ,
, .
R1 , -
.
( ,
).
. ,
( , ( ), ).
. . ,
.
1
R - ( , , , /
).
, , ,
,
.
.: [1] .,
, . ., ., 1965; [2] ., , . ., ., 1981;
[3] ., , . ., ., 1970; [4] R u z s a I. Z., S z e k e l i G. J.,
Algebraic Probability Theory, N. Y., 1988; [5] B l o o m W. R., H e y e r H., Harmonic analysis of probability measures on hypergroups,
B., 1955.
## ; semiinterquartil range / probable / deviation error , ,
. . . .- B
P{ | m | < B } = P{ | m | > B } = 1/ 2
, m ,
( - , m = M
). . . .
( ) :
( B / ) = 3/ 4 ,
( x ) , ( 0 , 1) .
, B
= 0 , 674496 .
; probability measure , , , ,
, A ( )
, P
,
215
P ( ) = 1 i j Ai I A j = ,
i =1
1)
i =1
U A = P ( A )
= { , } ,
( ).
A = { { } , { }, , },
P ({ } ) =
= P ({ } ) = 1 2 .
;
2) = { 0, 1, 2 , ...} , A , -
2 G . .- ( )
, 1 2 (*) ;
( ). 1 2 . .-
G
( ), 1 2
y G -
f ( y) =
) f 2 ( x ) d ( x)
P ( { k }) =
k
k!
e , > 0
, f1 f 2 1
( ).
2 .
3) = R ; A , R - ,
A A ,
1
P ( A) =
x2 2
dx
( );
x(t ) , A - , P ,
:
= (2 ) n
( t t
i
i 1 )
1 2
bn
a1
an
...
1
exp
2
i =1
n 0
## = t 0 < t1 < ... < t n < 1
( ).
.: [1] . ., , 2 . ., 1974; [2] . .,
, 6 ., ., 1988.
; probability measure o n a
g r o u p G
B . ,
, B , (G , B ) .
G , B .
1 2 , G ,
1 2 ( B) =
(Bx
1
) d 2 ( x) =
2 ( x 1 B ) d1 ( x) , B B ,
= S 1 S 2
. M (G )
e , e , G
.
( , A , P ) X 1 , X 2 : G ( G , B )
(
G ), X 1 X 2 PX1X 2
( xi xi 1 )
dx1 ... dxn ,
t i t i 1
, S 12
PX1X 2 , PX1 PX 2 ,
i =1
b1
. .- M (G ) -
. 1 2 S 12 , S 1 S 2
4) = C0 [ 0 , 1] , , [ 0 , 1] - -
f (yx
(*)
= PX1 PX 2 .
. .
, , ( ). ( ) , .
. .-
. , , . G . .
= , ,
.
. . S S -
- S - S
.
. G , ( X k ) ,
G -
= X 1 ... X n . S n G -
S n
S , , S -
PS
1 2 -
( ) ,
, G - 1 2 - . 1
, S n S PSn PS
216
. PS - ,
.
.
. G , , X 1 , X 2 , ...
G -
, X 1 -
X
k=2
. PS n
= , S n
.
.: [1] .,
, . ., ., 1965; [2] .,
, . ., ., 1981;
[3] P a r t h u s a r a t h y K. R., Probability measure on metric spaces,
N. Y., 1967.
;
arithmetic of probability distributions
( ) .
. . .- ( ) , I 0 ( ,
; )
.
. . .- :
( , , ;
. )
;
; ;
( , , ) . .
.: [1] . ., . .,
, ., 1972; [2] . .,
. ., . ., .:
. . . , . 12, ., 1975, . 542; [3] ., , . ., ., 1979; [4] . ., . ., 1986, . 31, . 1,
. 330.
,
;
arithmetics of probability distributions o n A b e l i a n g r o u p ,
, 1 2
- 1
2 -
. R , . ( H. Cramr ), . ( P. Lvy ),
. . , . . , . . , . . , . ( R. Cuppens ), . . ( , [1], [2] )
.
. , (
).
,
, ,
. ,
,
. . .-
.
G , M 1 (G ) ,
G - .
1
1
2 M (G ) , M (G ) -
= 1 2
, 1 M (G )
.
-
,
.
,
1
M (G ) I 0 - .
. .
1
: 1) M (G )
= m K , mK , -
,
, I 0 ; 2) I 0
( , [3] ).
d
, R -
,
I 0 . , T -
. , G
G T
. G
G T - .
, ,
( ).
, G , ( ) ,
. . , ,
, ( , [4] ).
R - -
, I 0 . ,
, . x G
x
, 2- ( , [5] ).
I 0 . , , I 0
, , I 0
I 0
. ,
. , , G
217
, ( ) . ; , I 0 , I 0
( , [4] ).
= e( F )
I 0 F
. F - .
I 0 . . ( F ) = { x1 , x2 } , G
= e ( F ) - I 0
x1 x2 , , ,
2- . F G -
, ( F ) { x0 , x1 , ..., xn } ,
{ x0 , x1 , ..., xn } , = e ( F )
I 0 ( , [6] ), 2 x0 = 0 . A G
{ xi }in=1
k1 x1 + ... + k n xn
( k i ) k1 = ... = k n
, A
=0
=0
F , F G
n
n m - , = e ( F ) I 0 .
F
G
, 2
; 2) G
I 0
R n + D ;
D , p > 2 p
.
.: [1] . ., . .,
, ., 1972; [2] .
., . ., 1986, . 31, . 1, . 330; [3] . ., ., . . .,
, 1965, . 9, 2, . 11546; [4] F e l d m a n G. M.,
F r y n t o v A. E., J. Multiv. Ann., 1983, v. 13, 1, p. 14866;
[5] . ., . . ., 1970, . 10, 3, . 53743;
[6] . ., ,
,1990.
; probability distribution , ( )
n o r m a l probability paper
,
-
.
M 1 (G ) I 0
. G 2
1
, M (G )
( E ) mG ( E ) , 0 <
<1
E , I ( mG , G -
).
,
, R -
, F e ( F ) I 0
, F , ( F )
.
.
G T - ,
, G R + D
, D
. G - F , e ( F ) I 0 .
. ,
I 0 .
I 0 ,
, I 0
.
: 1) I 0
218
100-, 8-
.
.
( ).
:
,
.
,
.
.: [1] ., . ., , . ., ., 1951;
[2] D i x o n W. J., M a s s e y F. J., Introduction statistical analysis,
N. Y. Toronto L., 1951.
,
; probability of misclassification
. . . . . . .
. X D1 , ..., Dk S -
: max ( d1 ( X ) , ..., d k ( X ))
= d j (X )
, S : X D j , d j ( X ) , j - . i - Di j -
D j . . .
P{ i | j } =
pi ( X ) d X
Rij
pi ( X ) , i -
## X - , Rij = { X : max (d1 ( X ) , ..., d k ( X )) =
= d j (X )) } .
pi ( X ) , , P { i | j } :
) , , ,
P { i | j } ; , . . .-
;
) P {i | j }
; . . .
.
, . . .-
( ) .
pi ( X ) , . . .-
( , ).
; probabilistic automation ,
,
. . . ( ) X ,
Y Z -: Z Y ,
( | z , x ) , z Z , x X . . .- . ,
x = ( x1 , ..., xn ) X n , z =
= ( z 0 , ..., z n )
= ( y1 , ..., y n )
( ) -
, A Z
P { z 0 A } = 0 ( A)
0 ( i ) = P{ z0 = i}
( a , j | a, i ) = P { zn = a, yn = j | zn 1 = a , xn = i }
.
X , . . . . .
. . .
,
. . . .- , . . y n -
, ( | z , x ) Z . . .-
.
. . . . . .
. .- ( , [1] ).
. . ,
.
.: [1] . ., , ,
1970; [2] ., , . ., ., 1963, . 43263; [3] R a b i n M. O., Inform.
and Control, 1963, v. 6, 3, p. 23045; [4] . ., . ., . ., , ., 1978; [3] . ., , , 1975.
; probability Borel measur
, .
; probability
generating function , .
; probabilistic number
theory ,
( , ).
. . . ,
N - .
, . . .- ,
.
( f ( m1 + m2 ) = f ( m1 ) + f ( m2 )
g ( m1 m2 )
m =
P { ( z n , y n ) A | z 0 , ..., z n 1 ; x1 , ..., xn } =
= ( A | z n 1 , xn ),
n 1
A Z Y A - 1 . xi , zi , yi i ( i - ) , ,
. X , Y , Z
. . . .; X Y Z
= g (m1 ) g (m2 ) ).
p N
p || m
( p ,
p || m , p m - ,
+1
m - )
f ( m)
219
g (m)
, :
f (m) =
An =
pn
f ( p )
p || m
sup
f
g ( m) =
g( p
).
, f ( p ) g ( p ) - , f (m) g (m) -
. .,
a ln m ( a ) .
h(m) , ,
. 1 ( m)
2 ( m) , m -
m
1 (m) h ( m ) 2 ( m) . .,
1,
2nt
( f (m) An ) 2 = n
m =1
1,
m m0 ( m ) 2 ( ln ln m ) ln m ,
## lim inf ( m ) = 1 lim sup ( m) ( ln m) 1 ln ln m = 1
m
(1)
f (m) , t > 0
- m n
f ( m) An < t Bn
.
, t , m n
p n
C .
, n0 , n n0 -
( m ) ln ln n < t
p|m
, m > 1 m - ( m)
1
n Bn2
f 2 ( p )
,
p
Bn2 =
n 3 / 2 C ln 1 n ,
(m) =
f ( p)
,
p
p || m
ln ln n
. 1917-
. ( G. Hardy ) . ( S. Ramanujan )
. 1979- . f n (m)
, Cn
> 0
E n ( h ) = n 1
h ( m)
m =1
. E n ( ) = ( 1 + o (1) ) ln ln n .
( n
). , , h ( m ) ,
, .
h ( m )
.
.
n = { 1, 2 , ..., n } , An , n - , Pn , n - [ Pn { ...} ,
m n , ]. { n , An , Pn }
h ( m ) , m n ,
E n (h) .
Pn { | h (m) C n | > Dn } = o (1)
Cn Dn
. , f (m)
.
220
Pn { | f n ( m ) C n | } = o (1)
, , ,
n = o (1)
p n
min { 1, ( f n ( p ) n ln p ) 2 }
= o (1 )
p
(2)
.
n p n , 1 m
( p)
, f n
(m)
:
f n( p ) (1) = 0 p m , f n( p ) (m) = f ( p )
( p)
, f n
(m) = f ( p )
n 1 ( [ n / p ] [ n / p +1 ] ) ~ p (1 p 1 )
.
f (m) =
f n( p ) (m)
pn
, (1) ,
f n( p ) ( m )
( (1)- 3 2
). ,
X p S n
= Xp
pn
X p - , ,
P { X p = f ( p ) } = p ( 1 p 1 ) , 0 .
XX 70- ,
f (m) , f (m) ln m -
. (2)
.
h ( m ) Pn { h ( m ) E } ; E . .
. Cn Dn -
Fn ( C n + Dn x ) = Pn { h ( m ) < C n + Dn x }
, (6) (5)
.
[ f
1
f ( p) 1
. . ( M. Kac )
p
: ., Bn
, Dn 0 -
.
Fn (C n + x ) -
Bn , ln n - . Fn ( An + Bn x)
, u = 0 , u -
Bn2
:
Fn ( x) , ,
| f ( p) | 1
| f ( p) | < 1
| f ( p ) | <1
( t ) = exp
f ( p) 0
(e
itu
1 itu ) u 2 dV (u )
.
.
Pn
ln p < x ln n , > 0 , 1
p || m
(3)
1
<
p
V ( u)
f 2 ( p )
p
p n ,
f ( p ) < u Bn
Cn = 0
1939- . ( P. Erdos ) . ( A. Wintner )
ln n
1- V ( ) = 0 , V ( + ) = 1
. ,
, (6)
, (5) .
( ). , F ( x )
f 2 ( p)
p
] (6)
. f (m)
Fn (C n + Dn x ) F (x) -
f ( p)
,
p
= f ( p)
f ( p )
V ( u ) ,
.
) h (m ) = f (m )
1
,
p
(4)
2 i
1+ i
1 i
ez
z exp
e i t u 1 u z
e du d z
, , .
Cn , XX 70- , Fn (C n + x) -
, . , Fn (C n + Dn x )
. Dn -
. C n
= An , Dn = Bn
Bn2
p n ,
| f ( p ) | > Bn
>0
f 0 ( p )
0
p
(5)
, A > 0 ,
.
. .,
f (m)
max
pn
( ; ,
) ,
Fn ( An + Bn x )
Dn , Dn = O ( ln A n)
1
2
u 2 / 2
du = (x)
(6)
.
(5) , Bn ln n -
. , Bn
f ( p) Bn1 = n = o (1)
Fn ( An + Bn x ) = ( x) + O ( n )
. ( n ) . ( A. Rnyi ) . ( P. Turn ) .
221
. x
x0
= o ( ln ln n )
Fn ( ln ln n + x ln ln n ) ,
x > 0 1 Fn ( ln ln n + x ln ln n )
e
Kn ( x)
| x| +1
( | x | ) 1 + O
ln ln n
= x ( ln ln n )
g ( p) 0
, P { ln | g ( m) |
. g ( m) , g (m) - g ( p )
= 0 , 1-
.
, , ,
.
. h(m) a Pn { h ( m)
= a } -
. f (m)
. ., Z - p(k )
## sup Pn { f (m) = k } p ( k ) = o (1)
, , , (4)
p ( k )
1
p
=0
+ rk (n)
f ( p) = k
k 0 ,
=1
rk ( n ) 0 .
.
. k = ln ln n +
+ o ( ln ln n ) ( m )
e ln ln n
(1 + o ( 1 ) )
k
2 ln ln n ln n
. k
Pn { ( m ) = k} .
. 1
: { n , An , Pn } , , A N
lim Pn { m A} , A N
n
. ,
. hk (m)
h ( m ) - ,
>0
## lim lim sup Pn { max
xkn
hk (m) h (m) } = 0
kZ
1=
p n,
k k
= 0 -
< ln | Cn | + xDn }
. , ,
Pn
hk ( m ) h ( m ) .
.,
f k (m) =
exp ( i t f ( p ) )
p
. Pn { f ( m) = k } ( x )
. .,
222
(7)
pn
Pn { (m) = k } =
, Gn (x) x
1
<
p
k Z
= o (1)
(7)
, , , d = 1 d = 2
f ( 2 ) .
.
.
k An
Bn Pn { f (m) = k }
Bn
F (x ) - Gn ( 0 ) - F ( 0 ) - -
f ( p ) K Bn -
## Gn (x ) = Pn { | C n g ( m ) |1/ Dn sgn g (m) < x }
it k
k Z
) g (m)
.
p (k) e
k - d , -
sup
x2
+ ( (1 + ) ln ( 1 + ) ) ln ln n ,
2
1 / 2
f ( p) = k
1
=
p
f ( m )
, x - ,
K n (x ) =
, ( m )
f ( p )
p m , p k
Pn
. f k ( m) f ( m) , , , (3)
( ).
- .
.
p - .
( , ( ) ) C[ 0 , 1]
A n , p r Q p -
D[ 0 , 1] . ., -
. Pn -
f (m) . Bn
Q n , p r Q n ( Q p ) = 1 / p . . . .-
yn (t ) = max { z ;
B z2
t Bn2
0 t 1,
},
=
= o (1) A A n
: Pn ( A) Q n ( A)
Wn ( m , t ) = Bn1 ( f y n (t ) ( m ) Ay n (t ) )
. D[ 0,1]
B -
f ( p ) -, . f
Pn ( B ) = Pn { Wn ( m , t ) B } , B B
. Pn
. , Pn ,
, , (5) .
.
f1 ( m + a1 ) + ... + f s ( m + a s )
.
,
,
ai
f i (m)
1 i s . ai - n - .
f1 (m + a1 ) + ...
+ f s ( m + a s ) ,
f ( P ( m ) ) , f ( P( pm ) ) ,
P ( m ) , m 1 , pm , m - f ( a m ) -,
am
( , [1] ).
. . .
( , [8] ).
. ) . Fn ( C n + Dn x )
x k dFn ( Cn + Dn x ) =
1
nDnk
( f ( m) C n ) k ,
{ n , An , Qn}
( p)
( m) , p r
. B
Pn
pr
f ( p ) ( m) B = Q n
f ( p ) ( m ) B + o (1 )
p r
.
.
f (m) Pn { f (m) B }
Pn { r (m) B } .
) . f n ( m)
g n ( m)
## Fn (x ) = Pn { f n (m) < x } Gn ( x) = Pn { g n (m) < x }
e i t x dFn (x) =
1
n
i t fn ( m)
m =1
| x|
k ( n ) =
f ( m ) f ( p ) (m) ,
m - p - ,
it
sgn k x dGn ( x ) =
1
n
g n ( m)
it
sgn k g n (m)
m =1
k = 0 , 1 ; 0i t = 0
m =1
k = 1, 2 , ...
( )
. , .,
, k ( n ) k -
. n t R
h ( m ) = hn, t ( m ) -
k -
(k 1)!! -
Fn (C n + Dn x) ( x )
.
) ( , [2] ).
, . . r = r ( n ) , ln r = o ( ln n ) , { , A n , Pn } { , A n , Q n }
,
= N , N , Q p ,
E n ( h ) - ;
h ( m ) 1
0- . 20- 60-
. E n ( h ) -
, .
Z (s) =
m =1
h(m)
=
ms
1 +
p
h( p)
h( p2 )
+
+ ...
2s
s
p
p
223
= + i
, s
>1.
h(m) =
2 i
m =1
ds
1
,
n + Z(s)
s
2
= 1 + 1 / ln n
= 1
Z ( s ) . ,
= n ( t ) ,
Mn (h) =
n i Z ( + i )
+ o (1)
(1 + i ) ( )
, ( s ) .
h ( m ) , o (1)
t [ T , T ] Fn ( x) Gn ( x)
.
E n ( h ) - h ( m )
x ( u ) = Eeu ( h )
x ( u ) = u 1 K ( u ) x ( ) d + o (1)
0
, K ( )
.
,
.
) .
Z - a
{ Z; a ( mod m )} , m = 1, 2 , ... . S
; S -
. P -
. h ( m ) , m N
h ( x ) , x S
. h ( x )
. ., f (m)
f ( x) =
f ( pk )
pk x
## . 20, 3, . 3952; [5] . ., . .
. ., 1964, . 28, . 30764; [6] . ., , ., 1974; [7] . .,
, ., 1971; [8] ., . . ., 1964, . 4, . 565603; [9] B a b u G. J.,
Probabilistic methods in the theory of arithmetic functions, Delhi,
1978; [10] E l l i o t t
P. D. T. A., Probabilistic number thery,
v. 12, N. Y. [a. o.], 197980; [11] E l l i o t t P. D. T. A., Arithmetic functions and integer products, N. Y. [a. o.], 1985; [12] G a l a m b o s J., Ann. Inst. Henri Poincar, Sect. B, 1970, v. 6, 4,
p. 281305; [13] H i l d e b r a n d A., Astrisque, 1987, 14748,
p. 95 106; [14] N a r k i e w i c z W., Teoria liczb, Warsz., 1977;
[15] P h i l i p p W., Mem. Amer. Math. Sos., 1971, 114;
[16] S c h w a r z W., Jahresber. Deutsch. math. Ver., 1977, Bd 78,
3, S. 14767; [17] .,
, . ., ., 1963.
, ; probabilistic solution of
differential equation
. 2-
. ,
.
, , ( , [1] [5] ).
,
( , [6], [7] ).
( , [8], [9] )
,
. [9]-, , ,
.
d
. R
d .
d
, R - b ( x ) , R
d1
- ( )
R - ( z ) , d1
. w t d1 . y R
=y
x S P ,
, , (3) .
Pn { f ( m) < u } P { x : f ( m) < u } -
, x0
. ,
S -
.
.: [1] . ., . , 1966, . 21,
. 1, . 51102; [2] . .,
, 2 ., , 1962; [3] . ., .:
, , 1974,
. 81118; [4] ., . . ., 1980,
; xt
. L
224
d xt = b ( xt ) dt + ( xt ) dw t , t > 0
y
DR
Lu =
i , j =1
xty
a ij ( x) u xi x j ( x) +
(1)
- D -
d
b ( x) u
i =1
a ij ( x) =
xi
( x)
(2)
1
( x) ( x)
2
. u ( x ) D -
,
L u + f = 0 D -,
(3)
u = D -
(4)
. G , G -
, G - .
n
{ }
, G . G ,
1
, y D
u( y) = M
f ( xty ) d t + ( x yy )
(5)
. : ) M (G )
x G , x H ;
) H - { j } , j 0 { n } n ( n
= 1 ... n
.
(5) (3), (4) . (3), (4)
- , , , (3), (4)
. u
(5) u
, , , u , (3), (4)
. .
y (1)-
(5)-
.
( , [14] )
( , [15] ), ( , [13] ) .
.: [1] P h i l i p s H. B., W i e n e r N., J. Math. and Phys.,
1923, v. 2, p. 10524; [2] C o u r a n t R., F r i e d r i c h s K., L e w y H., Math. Ann., 1928, Bd 100, S. 3274 ( . .
. , 1941, . 8, . 12560 ); [3] . .,
. . . . . , 1933, . 36372
( . . .:
, ., 1986, . 14148 ); [4] . ., Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, B., 1933 ( . .
, . ., 1936 );
[5] . ., Math. Ann., 1934, Bd 109, S. 42544;
[6] B a c h e l i e r L., Calcul des probabilities, P., 1912; [7] . ., Math. Ann., 1931, Bd 104, S. 41558, ( .
. . ., .: , ., 1986, . 60105 ); [8] I t K., Mem.
Amer. Math. Soc., 1951, v. 4; [9] . ., . .
., 1950, . 2, 4, . 3763; 1951, . 3, 3, . 31739; [10] . ., , ., 1963; [11] .,
., , . ., ., 1986; [12] S t r o o c k D.
W., V a r a d h a n S. P. S., Multidimensional diffusion processes, B.
[a. o.], 1979; [13] . .,
, ., 1977; [14] . .,
e . ., . , 1980, . 250, 3, . 521
24; [15] . ., . ., . .
. . , 1982, . 8, . 15368.
( )
; probabilistic characterization o f g r o u p s
, ,
.
1
. , R -
. , ,
,
.
M ( G ) - H
), G -
. G M (G ) , G ,
-
.
,
( , [1], [2] ).
.
G
. X 1 , X 2 , ... G , Yn
= X 1 X 2 ... X n . G , -
, , Yn X 1 , X 2 , ..., X n , ...
( , [1] ).
G
, G
x G
lim
n x n
=0
, G
{ n } ( ) ,
. G , ,
, G - .
.
.
. . .
,
.
,
( , ).
1
, R -
. ,
225
,
.
, ,
( )
; .
; .,
, ,
,
( , ;
).
.
G
. G
( , [1], [3], [4] ).
1
, M (G ) -,
.
,
. SO ( 2 ) SU (2) , .
.
, .
.: [1] ., , . ., ., 1981; [2] . .,
. , 1970, . 195; . 127477; [3] .
., . , 1970, . 192, . 72335; [4] . ., . , 1972, . 203, . 52427.
; probabilistic coding , ( )
( ) .
. . .- ( ) . . . .-
.
Y ,
X
. X Y - ( , )
F ( x , y ) -
= x ,
Y F ( y | x ) -
. X
.
X Y
F ( y | x)
M [ Y | X = x ] = Y (x) -
226
Y ( x ) ,
. . .- F ( y | x ) - .
= F ( y | x ) ,
Py (x)
F ( y | x ) , X Y F ( y | x) , Py
y - -
.
, . . .
, ,
Y ( x ) - .
X Y
( , , ;
); . . .-
. . . .-
n ( .,
n ) Y .
R = ( r1 , ..., rn )
, ri i - ri
0,
= 1 . X
i =1
( ., n ) , X j -
i -
P i | j - ij
= nij
ij
i =1
, ; probabilistic weather
forecast ( )
. X
, nij , Y - i - X - j -
. Y
, , Y
; X -
Y -
.
( ij ) ,
. . .
;
, . . . .-
,
, ,
.
. . .- ( , ) .
Y , 1,
0 . P { | X = x } = M ( Y | X = x )
.
. . .-
,
,
. . . .
.
.: [1] . ., . .,
, ., 1983.
, ;
; ; probabilistic weather
forecast evolution
( ). . . .-
.
.
.
. . .-
. ( ) ; ( )
; ,
,
; ,
. ,
.
,
;
,
.
:
.
, .
,
.
., n
( n 1 , ..., n ).
R = ( r1 , r2 , ... , rn ) , r j 0,
rj = 1
j =1
( r j ,
j ). ( ., j ) D
, d i
=1
= (d1 , ..., d n )
= 0 .
j i d j
( R , D )
,
S ( R , D )
= Si ( R )
.
. . . .-
,
PS ( R , D ) = 1
= 1
1
2
1
( R D, R D) =
2
n
(r
d j )2
j=1
.
.
;
R
.
( ., : ,
) D = ( 0 , 0,1) R1 = ( 0,4, 0,1, 0,5 )
R2
## = ( 0,1, 0,4, 0,5 ) -
RPS ( R , D ) = 1 ( R D, R D ) ( N 1)
, r j
k =1
rk ; d j =
dk .
k =1
, RPS , PS .
.: [1] . ., . ., , ., 1983; [2] M u r p h y A. H., E p s t e i n E. S., J. Appl. Met., 1967, v. 6, 5, p. 74855.
;
random / stochastic process , .
, ; large deviations
probabilities , .
; redundancy
. . . : 1)
; 2)
;
3)
;
4)
,
. X = { x }
X 0
, X 0 X .
227
: , , . : ( )
. . .
, . . . ., -
, . . . .
.
. .
. . .
.: [1] , ., 1983;
[2] . ., . ., - , ., 1975;
[3] , ., 1985; [4] .
., ,
., 1974.
multiple
redundancy
, .
( , [4], [7] ).
.-, .,
( , [1], [2] ). K. to [3]- a E .-
. , = { a }
E \ { a } ( t
t - t
>0
{ a }
E \ { } Pa ,
A ( t ) , a . ,
~
.- { Y } -
,
; t e m p o r a r y redundancy
, .
o p t i m a l redundancy
, .
; g e n e r a l redundancy ,
.
; ; structured redundancy , .
( )
; B r o w n i a n excursion , .
( )
; excursion
o f a M a r k o v p r o c e s s = ( t , At , Px ) , t 0
[ , ]
t ( t = ,
) t -
: ) D [ 0 , ]
= ( l )( D ) < [ l , [ 0 , ] - ], { Y
: ( A ( ) , Y ) D } -
;
) Di
( Di ) .
. ( , [5] );
.-
( , [5], [6] ). [5]- .- .
.: [1] ., .,
, . ., ., 1968; [2] .,
, . ., ., 1972;
[3] I t K., Proc. Sixth Berk. Symp. Math. Statist. and Probab.,
1972, v. 3, p. 22540; [4] . ., .
., 1971, . 16, . 3, . 40936; [5] M a i s o n n e u v e B.,
Ann. Probab., 1975, v. 3, 3, p. 399411; [6] G e t o o r R. K.,
S h a r p e M. J., Adv. in Math., 1982, v. 45, p. 259309; [7] B l u m e n t h a l R. M., Z. Wahr. und verw. Geb., 1983, Bd 63, 4,
S. 43344.
( ) ; excursion o f a
W i e n e r p r o c e s s 0- w ( t )
.
Z , w ( t ) , t 0 .
( ,
). , ( E , B ) - -
, B . { t : t } -
, , Z1 ti Z
[ 0, ) - Z - Z n , n = 1, 2 , ...
{ ti }
( , ) ( , ) -
, Z 2 ti Z U Z1
, t : [ , ) E
; .-
.
.-
= X t + : [ 0 , ) E
Yt
en ( t ) =
w ( an + t (bn an ) )
.-
,
228
.,
bn an
Z n = ( an , bn )
0 t 1 , n = 1 , 2 , ...
= sign w ( t ) . Z
, en n
P { n = 1} = P { n =
1} = 1 2 . en . t Z n n
2y2
p ( 0 , 0; t, y ) =
2 t 3 ( 1 t ) 3
0 < t <1 ,
2(t s)
e( y + x)
2 t ( t 1)
. . . ( , , ).
. . .-
,
. . .-
. , . . .-
y>0,
1 s
p ( s , x ; t , y ) =
1 t
[e ( y x )
e y
32
y
x 2 ( t s )
2 (t s )
] ey
( )
, T ( , [1] ).
(*) . . .- x1 , ..., xn -
2 ( 1 t ) + x 2 2 (1 s )
## 0 < s < t < 1 , x > 0, y > 0
.
(an , bn ) { ti } -
. .-
t
p(a, b) =
a ( b a )3
, 0<a<t <b<
a (t a)
p2 ( b ) =
1
b
t
bt
(*)
;
exponential family of distributions
p( x, ) =
d Pw
=
d
= C ( ) h ( x ) exp { C1 ( ) 1 ( x) +
,
... + C m ( ) m ( x) }
(X , U )
P = { P } ,
(*)
xX , ,
## ( ) > 0 , C1 ( ) , ..., C m ( ) , - , h ( x ) 0 , 1 ( x) , ... , m ( x) , X -
.
m . . .- (*)
. (*) , , X -
1 ( x ) , ... , m ( x) -
C1 ( ) , ..., C m ( ) - .
( 1 , ..., m )
..., m . . .- -
m . . .- . = { ( 1 , ..., m ) : }
.
R
( xi ) , j = 1, ..., m .
. . .-
; experiment error , , -
= C j ( ) , j = 1,
, t j ( x1 , ..., xn )
p ( x ; a , 2 ) = ( 2 2 ) 1 / 2 exp{ ( x a ) 2 2 2 }
.
.: [1] .,
, . ., ., 1972; [2] ., .,
, . ., ., 1968.
= ( t1 , ..., t m ) , ,
. n - -
. . . .
.,
p1 ( a ) =
i =1
1
2
p ( xi ; ) = R ( T ( x1 , ..., xn ) ; ) r ( x1 , ..., xn )
i =1
= (a, 2) ,
m = 2 , 1 ( x) = x 2 ,
C1 ( ) = 1 2 2 , 2 ( x) = x , C2 ( ) = a 2 .
,
. . . . 1-
. . .
.
. . .- XX 30-
. . ( G. Darmois ),
. ( B. Koopman ) . ( E. Pitman )
. . . .-
. ( R. Ficher ) . . . .
.
.: [1] . ., . ., . ., , ., 1972;
[2] . ., , ., 1966; [3] ., , . ., 2 ., ., 1979; [4] . ., , ., 1972;
[5] B a r n d o r f f - N i e l s e n O., Information and exponential
families in statistical theory, N. Y., 1978.
;
exponential autoregressive model / EXPAR proses , .
;
exponential autoregressive process / EXPAR proses , , ,
.
; exponential inequality
, .
229
( )
; exponential transformation o f r a n d o m p a t h
:
= c cos ,
w
. ( c cos )
exp ( l c cos ) ; , ,
c .
.: [1] . ., . ., , 2 ., ., 1982; [2] . ., .
. . . , 1981, . 21, 6, . 143544.
## ; exponential generating function , .
; exponential distribution
A+ ( z , ) = exp
= f (t )
( ).
St f ( t ) }
= f = inf { t 0 :
S 0 = 0,
S1 , S 2 , ... ,
, ( , [1],
[2] )
1 z M e i X1 = ( 1 M ( z ( 0 + ) e i ( 0 + ) ; ( 0 + ) < ) )
*
; ( 0 + ) ( 0 + )
. (1) (3) ( x ) . . ( x ) , ,
( ( x )
= 0, Im < 0
( , [1], [2] ):
1 M ( z ( 0 + ) e i ( 0 + ) ; ( 0 + ) < ) = A+ ( z , ),
e i u d u M ( z ( u ) e i ( u ) ; ( u ) < ) =
230
<
),
k 1 P { X k > 0}
; 0 M X 1
(1)
( , [1] ).
A, B, C ( , [4] ), t 0
M X 1* > 0 ,
A ( t ) P { ( 0 + ) > t } B ( t )
= 0, 0 < D X 1 < ,
A ( t ) + B ( t ) P { ( 0 +) > t } C ( t )
,
( t ) = P{ X1 > t } , ( t ) =
( u ) du .
X 1 -
M ( 0 + ) < , n ( , [1] )
. | z | < 1, Im
(3)
. ( 0 + )
( x), ( x ) ,
f ( t ) = x = const f , f -
( 1 M ( z e i ; * < ) )
S1 = X 1 , ...,
, X 1 ,..., X k
k =1
; M X 1
) ,
= 1 z M e i X1 ( , ).
f = S f ( )
zk
M e i Sk ; S k > 0
k
k =1
. S t , t 0
S k = X 1 + ... + X k
A ( z , ) =
, .
( ) ; excess / overshoot o f a w a l k ,
( ).
(2)
p p ; exponential p stability , ( ) -
A (z, )
1 +
,
A
+(z, )
G ( t ) lim P { (u ) < t } =
u
1
M ( 0 + )
P{ ( 0 + ) > x }dx
0
. , u ( 0 + )
. G ( t ) , ( ) -
P{ ( u ) = n , ( u ) < x } u = u ( n )
( , [3], [5] ). u 0
( , [6], [10] ):
k +2
k +1
M X1
M X1,
k +1
k+2
k +2
M k (u )
M X1
M X 12 ,
k +1
M k (u )
M X 1 > 0,
s 2
. . . H 0 : 2 0 -
M X1 0
, X i -
,
A . .-
( , [3], [13] ),
( , [6], [7] ), ( , [9] ) .
. (u ) - n
[8], [11], [12]- .
.
.
d.: [1] .,
, . ., . 2, ., 1984; [2] . .,
, .,
1972; [3] . ., . . ., 1962, . 3, 5,
. 64594; 1964, . 5, 2, . 25389; [4] . .,
. ., 1964, . 9, . 3, . 498515; 1972, . 17, . 1,
. 14347; [5] . ., . .,
1984, . 29, . 2, . 41011; [6] . .,
. ., 1973, . 18, . 2, . 35057; 1976, . 21, . 3,
. 48696; [7] . ., . .,
. ., 1969, . 14, . 3, . 43144; [8] . ., . ., 1984, . 29 . 2, . 248
63; [9] . ., . . . ., 1969,
. 33, 4, . 861900; [10] L o r d e n G., Ann. Math. Statist.,
1970, v. 41, 2, p. 52027; [11] K e s t e n H., Ann. Probab.,
1974, 2, p. 35586; [12] J a c o d J., Ann. Inst. H. Poincar B,
1971, v. 7, 2, p. 83129; [13] W a l k H., Stoch. Proc. Appl.,
1989, v. 32, 2, p. 289304.
( )
; excess o f a d i s t r i b u t i o n
. .-
. .-
( , [1] ). . , , . ,
. .- ( )
,
(
) .
.: [1] ., , .
., 2 ., ., 1975.
; coefficient of excess ,
; excessive function ,
;
( ,
). ( E , B ) , p ( x, ),
xE , B
. 0 f
f B
P f f ,
P f () =
f ( y ) p ( , dy )
f ( .,
1 , 2 , ... f 0 f i = f (i )
{ f1 , f 2 , ...}
Pf f
pij f j fi , i 1
, pij , i j
).
. .- : (x)
( x ) , (x) ( (x) )
x E B
( ) . . .- . 0 , B v
R =
. .- .
n0
. .-
.
. . f 0 (
n
2 .
2 = 0 . 2 > 0 , 2 < 0
<
( , [2] ) -
.
. . . .,
E , B
. ( E , B )
p ( t , x , dy ) , t 0 , x E
2 =
1
n s4
(X
i =1
X )4 3
.
,
X 1 , ..., X n ,
, 4 ,
; excessive function f o r a M a r k o v c h a i n
) lim P f
2 = 44 3
f,
0 f ( B -
B - )
t 0 Pt f f
231
lim Pt f = f
## d.: [1] ., , . ., ., 1997;
[2] D e l l a c h e r i e C., M e y e r P.-A., Probabilits et potentiel,
Ch. IXXI: Thorie discrte du potentiel, P., 1983.
t 0
Pt f ( ) =
( ) ;
extrapolation o f a r a n d o m p r o c e s s ,
f ( y ) p ( t , , dy )
f
. . .
.: [1] ., , . ., ., 1997;
[2] D e l l a c h e r i e C., M e y e r P.-A., Probabilits et potential,
Ch. IXXI: Thorie discrt du potentiel, P., 1983.
; excessive function f o r a p a r t o f t h e W i e n e r p r o c e s s
, ;
; excessive measure
, -
. ( E , B )
p ( x , ), x E , B
x ( k +1) = x ( k ) + k sk ,
k
(*)
, 0 , 1 , ...
; s0 , s1 , ... ,
n ( );
x ( 0) X . (*)
g ( x ) = M y ( x ) -
[1] [3] );
3) -
p ( x, ) (dx )
( ., 1, 2 , ...
= ( { i } )
, {1 , 2 , ... } ,
i 1
).
. . .
. .- . ., ( E , B )
t 0 , xE
. . B
. . . .
:
pij , i j
P ( )
x X R n . y ( x )
pij i j , j 1
P ,
M y ( x ) - ;
( , [1] ).
. . . , , ,
( ) .
. . . , ,
:
1) -
(
);
2)
( : , , , D ,
Q , A . ). ( ,
. B -
; design of extremal experiments
; excessive measure f o r a M a r k o v c h a i n
; , .
p ( t , x, dy ),
, Pt , t 0 t 0 Pt
Pt ( ) =
p ( t , x , ) ( dx)
232
(
; , [2], [3] );
4)
( ,
[2] [4] ). . . .
( , [2], [3], [6] )
( , [2] [5] ).
,
k > n x ( k ) x ( 0) , ... , x ( k 1)
.
20- 60-
.
[ ( n + 1) ] y ( x
(i )
( , ; , )
.
, , ,
,
.
.
x ( k ) s k
N ( N > n) [ X
y ( x )
m = m ( ( nn ) )
m = m ( (1n ) ) F ( m ) = 1 F ( m ) = 1 2n
. . . .-
: F ( u n ) = 1 1 / n
u n n F ( u1 ) = 1
u -.
u n - 1 , ..., n
u1 - -
x ( k ) - ,
( x) - x -
1- . ( n 1) / n 1 / n -
. - ( , [1] ).
].
(k )
( x ) )
. ,
s k ,
x -
( k + 1)
. x
.
(k )
x
k -
- { z j },
y ( z j ), j = 1, 2 , ...
,
; x
j = 1, 2 , ...
( k + 1)
y ( z j ) < y ( z j 1 )
= z j .
.
2
[ x
(k )
1( n ) , ..., n( n ) F ( n ) ( x)
lim ( F
(n)
## ; distribution of extreme values 1 , ..., n
( n n ) (1 n ) ;
.
1 F ( x ) ,
Fn n (x ) = P { ( n n ) < x } = F n ( x ),
F1n ( x ) = P { (1n ) < x } = 1 (1 F ( x )) n .
F (n) ( 0 ) = 1 1 n .
( x ) ) = ( x ) = exp ( e
n
a x
lim n ( 1 F ( y / a n ) ) = e y
, a
= lim an , F ( n ) (1 / an ) = 1
n
1 / n e ( , [2] );
exp ( r /( x + r ) ) k , x r ,
lim ( F ( n ) ( x) ) n = ( x) =
0,
x < r
n
+ ( a1 , a 2 , ..., a n ) T 2 n
]
,
.
.: [1] ,
., 1983; [2] . ., . ., , ., 1987; [3] . ., . ., , ., 1965; [4] . ., . ., . ., , 2 .,
., 1976; [5] . ., ,
., 1979.
lim
1 F ( n ) ( x)
= ck ,
1 F ( n) ( c x )
## inf { x : F ( n ) ( x) > 0} = r < ;
, c > 0 , k
>0;
lim ( F ( n ) ( x) ) n = (x) =
exp ( ( (w x ) / w ) k ) , x w ,
1,
x >w
lim
x 0
1 F ( n) ( c x + w )
= ck ,
1 F (n) ( x + w )
## sup{ x : F ( n ) ( x) < 1} = w > 0
, c > 0 , k
>0.
( x ) , ( x ) , ( x )
.
F( n n ) ( x ) = 1 F(1n ) ( x )
233
,
.
.: [1] ., ,
. ., ., 1965; [2] G n e d e n k o B., Ann. Math., v. 44,
3, p. 42353.
; extremal statistical
problem X 1 , X 2 , ..., X n X, y Y , f n : X n Y R1
( ). ,
.
, , .
.: [1] ., , . ., ., 1953;
[2] ., , .
., ., 1969.
*
f n - y ( f n ) Y -
## ; equivalent random variables ( , F , )
, X 1 , X 2 , ..., X n y -
P { : ( ) ( ) } = 0
## y ( f n ) = arg min f n ( X 1 , X 2 , ..., X n ; y )
y Y
.
, .
( C. Gauss, 1795 )
. . .- . . . . .
. . .- .
.: [1] . ., .:
, ., 1982, . 412.
;
; equivalent partitions , .
; equivalent measures
( ) . , E ,
S , E -
( ) ( )
. ( ) ( ) . . .-,
## F ( x ) = P { : ( ) < x } = P { : ( ) < x } = F (x)
1
x - ( x R ) .
= ( )
= ( )
= . =
, ( ) ( ) . . .
.
. 1) ( ) 0-
, ( ) -
p ( x) = p ( x)
x - ( x R1 )
( ) = ( )
( ) ,
) M , S -
. M -
, ;
( << << ).
F = F , = . ( )
x
: P { : ( ) = x } = 0 . ,
M - ,
M -
;
.
= P { : ( ) = 0} = 0
, , ,
<<
d
( x) = 0
x E :
d
P { : ( ) = ( )} = P { : ( ) = ( )} =
, ( ) ( ) . . . .
2) ( ) , 0- P { : ( )
= 0 }<1
( P { : ( ) = 0} = 1
, ). ( ) - -
.
- . .,
, .
,
1
. ., R -
1
234
{..., x2 , x1 , x0
= 0, x1 , x2 , ...}
P0 = P { ( ) = 0},
P j = P { ( ) = x j } = P { ( ) = x j }, j = 1, 2 , ...
; 0 < x1
P0 + 2
## < x2 < ...
= 1.
( ) = ( ) . , = ,
P { : ( ) ( )} > 0, ( ) ( ) . . . .
3) , , .
1
= =
g (x)
( x R ) B , g ( ) g ( )
g ( )
= =
= g ( ) .
.
.
( , . 1 )
= ;
= , = . = , = .
,
2
. ,
. P {
= 0} = 1
.
.: [1] ., .
., 1999; [2] ., . ., 1962.
, ;
equivalently conversing sequence of random variables , .
; equivalent charges , ,
.
; equivariant estimator
;
elementary event .
( , A , P ) ,
.
, - , . .
: , ,
, . . A {}
.
, , , ,
, .
## ; space of elementary events , ,
; elementary measure ( ) , ( )
.
( X , G , S , ) ( ) ,
, X
g X = X g A = A g G ; - G , P ( A )
= Pg ( g A )
A A g G - .
( g X )
= g ( X ) g G - ,
( X ) G
, X X - .
G - L , g G - L ( , )
= L ( g , g ) .
. . .
.
.: [1] ., , . .,
., 1975; [2] ., , .
., 2 ., ., 1979.
aliasing ,
- / / .
; elementary set , ( ) .
; elementary probability , .
G ( G )
. , Y , X - G ( -
) .
Y ( Z ) = ( Z I Y ),
. { P , } ( X , A)
, G , X - ,
, G
, S , X
G , S -
Z S
Y , S G
( G ) .
( ) .
Y (
) ( ), Y . :
Z Y g g G ( g ( Z ) Z ) = 0
( Z )
= 0 (Y \ Z ) = 0
, , .
( )
.
. , R , n 1
n
, R -
,
.
.: [1] . .,
, ., 1983.
; Elfing equivalence
theorem , .
235
; elliptic
law n n = ij
;
( ij , ji ), i j , i , j = 1, ..., n
. , n = 1, 2 , ...
( ij , ji ), i j , i , j = 1, ..., n
M ij = 0,
M ij
= n 1 , M ij ji = n 1 ,
i j,
, n
ij
0 < <1
n
i , j =1
ij , ji - -
= 1 , ..., 4 , > 1
pij ( x s ), s
sup
sup
i , j =1, ..., n
>0
ij ( xs )
dx
< , s =1 ,
..., 4 ,
s =1
sup
sup
i , j = 1, ..., n
M ij n
2+
<
. x y
lim n ( x , y ) = ( x , y )
1 ,
n ( x , y ) = n 1
F ( Re k x ) F ( Im k y ) ,
k=1
y < 0 F ( y ) = 1 , y 0 F ( y ) = 0 , k , n
, ( x, y )
p ( x , y ) = 1[ 1 ( a 2 + b 2 ) 2 ]1
, , x y
2
2 2
(b x a y ) ( 1 a + b )
2
2 2
+ (a x + b y) ( 1 + a + b )
< a + b2
2
a = Re , b = Im .
.: [1] . ., . , 1985, . 40, . 1,
. 67104.
;
;
; M o n t e C a r l o s o l u t i o n m e t h o d f o r a n elliptic equation
u ( ) - ,
236
u ( x ) = M x F (W ) , , W
x , F
. ., u ( x ) = 0, x G
= ( x ) ; W (t )
, F (W ) =
= (W ( ) ) , G
; u ( x ) |G
. , , , , -,
.
( ).
, F (W ) -
.
,
( , [6] ).
.
, .,
.
.: [1] . ., [ .], , ., 1980; [2] . .,
. ., . ., , ., 1984;
[3] ., .: ,
. ., ., 1959, . 275301; [4] M u l l e r M., Ann. Math.
Statist., 1956, v. 27, 3, p. 56989; [5] . ., . ., . ., .: ,
., 1982, . 6982; [6] . ., . ., . , 1984, . 275, 4, . 80205.
c ; emigration
i n b r a n c h i n g p r o c e s s , ,
.
; empirical Bayesian decision function / rule
,
## ; empirical Bayesian estimator
, ( ) , .
.
f ( x, ) = x e x !, x = 0 , 1, ... -
,
,
:
G ( x) =
( x + 1) fG ( x + 1)
,
fG ( x )
x ,
, 1
fG ( y) =
f ( y , ) G ( ) ,
G . , G ,
k
x1 , ..., xk -
. . . N
. . . .-
.
f G ( )
. k f G ( x + 1)
f G (x) ( .,
fG ( y ) - x1 , ..., xk
).
G ( x) = ( x + 1) fG ( x + 1) fG ( x)
; . . . .
.
.: [1] ., , 1966, . 10, . 5, . 122
40; [2] ., , . .,
., 1975.
; empirical Bayesian approach
, .
## ; empirical covariance matrix , .
;
empirical moment , .
( ) ; empirical mean value o f a
random
s e t , ( )
.
; empirical
orthogonal functions , X ( r ) ,
N x ( rk ),
k = 1, ..., N
= 1, ..., N
b = bik ,
## bik = b ( ri , rk ) = x (ri ) x (rk ) ,
, x ( r )
. , . . .
j ( rk )
N
x ( rk ) =
a j j ( rk ) , k = 1, ... , N
j =1
{ x ( rk ), k
= 1, ..., N }
j (rk )
. , j ( rk )
= j ( rk ) , (*)
m < N
. a j
; a j - :
. . . .
, k G ( x)
{ j (rk )}, k
2 ... N 0 .
1,
a j ak = j jk , jk =
0,
j = k
j k
, (*)
,
2 ( m ) = m +1 + ... + N
.
. . . . [5] . . [1]
. ,
r1 , ..., rN N ( , [2], [3] ).
, , m - N - 2
( m ) . .,
N 10
2 (1)
j =1
jj
j=1
50%- (
), (3) 10%- -
. b - j
. a j -
N x ( rk )
, . . .-
. ,
. . . j ( rk ) j
.
. . . G
x ( r ) .
(
, ).
x ( r ) B ( r1 , r2 )
X ( r ) , , j ( r ) j , G , B ( r1 , r2 )
bi k j ( rk ) = j j ( ri ) ,
i , j = 1 , ..., N
k =1
k =1
( j ( rk ) ) 2 = 1,
j = 1 , ..., N
. , (*)
.
237
.: [1] . ., . . . .,
1960, 3, . 43239; [2] , ., 1970; [3] . ., , 1980, 4, . 11319; [4] . ., . ., . .
, 1982, . 18, 5, . 45159; [5] L o r e n z E. N., MIT Statist. Forecasting Proj. Sci. Rep., 1, Camb.
( Mass )., 1956.
; empirical measure , , ( )
; empirical distribution ,
. 1 , ..., n -
(1) Fn* ( x )
.
,
Fn* ( x)
x1 , ..., x n
, 1
. xi , i
## < 2 < ... < n
= 1, ... , n
1 n
1 , ..., n
( xi
, ., x
, x
l n - ). , x1 , ..., xn
< xi +1 . , xk < x xk +1
x < x
k - . , x k < x x k +1 ,
k = 1, 2 , ..., n 1 ,
, xi
(1)
Fn* ( x ) F (x) -
F (x)
,
k
= Fn* ( x), x (, + )
n
F (x) = P { < x }
0,
= k n,
1,
x (1) ,
( k ) < x ( k +1) , 1 k n 1
x > (n) .
, x1 , ..., x n
x k
< xk +1
1 n - . 1 , ..., n -
Fn* ( x )
.
Fn* ( x )
; x x1 x < x
,
x > xn
< x n -
;
xk < x xk +1
Fn* ( x ) =
k
n
k
Fn* ( x ) =
n
(2) . ,
1 , ..., n . . x
.
( . ) . n
Fn* ( x ) . x -
M Fn* ( x) = F (x) ,
x1 , ..., xn n ,
- xi
.
.
x1 , ..., xn
1 , ..., n
D Fn* ( x) = F ( x ) [ 1 F ( x) ] n
## P{ Fn* ( x) = k n } = C nk [ F ( x)]k [1 F ( x) ]nk
,
k n .
P
x n Fn* ( x )
F ( x ) -
. , Fn* ( x ) F (x ) -
. n
.
, i , i - ( n -
. . . F (x ) -
1, 2 , ... , n - .
x1 , ..., x n
1 x
Dn = sup Fn* ( x) F ( x)
) i -
( i = 1, ... , n ) , i -
P { lim Dn = 0 } = 1
n
238
).
## Dn . . Fn* ( x ) F (x) . . . ( 1933 ) F (x )
:
+
lim P { n Dn < z } = K ( z ) =
(1) k e 2 k
2 2
, z>0.
n =
F ( x ) , F0 ( x )
Dn ( ,
, ).
. .- ( ) , .,
1
D =
n
*
k =1
1
n
(1)
( xk M ) ,
r
k
k =1
D =
1
n
(x
M )
k =1
. .-
.
.
.: [1] . ., . ., , 3 ., ., 1983; [2] . ., , . ., ., 1960;
[3] . ., , ., 1984.
## ; empirical distribution function / e. d. f. , .
;
empirical process , ( )
( 3)
( 2)
( 3)
( 2)
( x (3) ) y 2 ( x (3) ) dV
( x ( 2) ) y3 ( x ( 2) ) dS ,
(*)
## = ( x , y , z ), x( 2) = ( x, y ), G(1) ( x (3) ) , G( 2) ( x (3) )
) , X
y1 , y 2 , y3 ,
dV = dx dy dz dS = dx dy .
(*) -
, r ,
y1 , y 2 , y3
( x (3) ) y1 ( x (3) ) dV +
+
G ( x
k =1
( 3)
X =
m *r =
dX dt
= y3 ( x, y, t ) -
.
X , ,
X -
:
xk ,
y2 = y2 ( x , y , z , t )
= y1 ( x, y, z , t ),
y3
1
M =
n
*
, X -
, , y1 =
curve , .
## ; empirical influence function
, ;
(
).
., X - , , -
( ) . X X -
,
y1 , y 2 , y3 ,
.
, y1 , y 2 , y3
,
( X - y1 , y 2 , y3
)
. y1 , y 2 , y3
. . . .
. . .- - ,
, ,
- , ,
. ( )
, (*)
. . . .
.
.: [1] . ., .
., 1975.
;
e x a c t endomorphism , .
; endomorphism
m o d 0 , ( ) .
239
( ) ; endomorphism
o f a m e a s u r e s p a c e ( , A , P )
.
, T : :
1) T = ;
1
2) A A T A A , P ( T A ) = P ( A ) , T
( , A , P )
.
1) T A
; entropy power
p (x) = p ( x1 , ..., xn ), xi R,
. . . . ( C. Shannon )
.
.
. . N ( )
N ( ) =
( mod 0 ) ;
. X n , n 0 .
;
energy norm , ; .
;
energy spectrum , , .
; energy
, ;
; energy
level , .
, ; energy of a particle ,
,
## ; transverse correlation function , .
; Engset
formula N , , n k
pk -
pk =
( N k )! k
p0 ,
k!
k
0k n
, dt
, dt
, . .
( T. Engset ) 1918- ;
.
.: [1] Handbuch der Bedienungstheorie, unter der Leitung von B.
W. Gnedenko und D. Knig. Bd 2, B., 1984.
; pulse width
modulation , .
; negative outlier , , .
240
i = 1, ..., n
1
2
exp h ( )
( 2 e ) n / 2 n
, h ( ) , -
h ( ) =
p ( x) log p ( x ) dx ,
p (x ) - .
ri j , i, j = 1 , ..., n
N ( ) det ri j ;
, ,
N ( ) M ( M )2
; , ,
. ,
. .- .
, N ( ) N ( )
( ) ,
N ( + ) N ( ) + N ( ) ;
, ,
.
.: [1] ., , . ., ., 1963, . 243332; [2] B l a c m a n N. M., IEEE Trans. Inform. Theory, 1965, v. IT-11, 2,
p. 26771; [3] C o s t a M. H. M., C o v e r T. M., IEEE Trans.
Inform. Theory, 1984, v. IT-30, 6, p. 83739.
;
entropy criterion , ().
; entropy
p k = P { = xk }, k = 1, 2 , ...
(X = { xk , k = 1, 2 , ... } )
H ( ) =
log pk
( 0 log 0 = 0 ). ( )
( )
2 e . N = X
, .-
H ( ) log N ,
H ( ) = log N . H ( ) 0
H ( ) = 0 , ,
pk -
H ( ) =
pi ln pi r
i =1
H (
p1 + ... + p r
1
. p1 = ... = pr =
r
.
f (x ), x X
H ( f / ( ) ) H ( ) .
( )
entropy o f a p a r t i t i o n
q j = P { = y j }, j = 1, 2, ... , Y = { y j ; j = 1, 2 , ...}
.
.: [1] ., , . ., ., 1969; [2] ., , . ., ., 1974; [3] .
., . ., , ., 1982;
[4] ., ,
. ., ., 1963.
; a l g o r i t h m i c entropy / Kolmogorov
complexity entropy , .
pk = P { = xk }, k = 1, 2 , ..., X = { xk ; k = 1, 2 , ...}
pk | j = P { = xk | = j } .
qj
, Pi D
.
:
) H ( | ) H ( ) , ,
,
; ) H ( ) = H ( | ) + H ( ) = H ( | ) + H ( ) .
k = ..., 1, 0 , 1, ...}
= { k ,
H ( ) = lim n 1H ( ( n ) )
n
(n )
(*)
= ( 1 , ..., n ) . , (*) -
H ( ) = lim H ( n | n1 )
n
.
( , S ) ,
,
.
H ( d | d )
H ( d | d ) =
log
d
( d )
d
E (D ) =
( Di ) ln ( Di )
( - ),
D = { D1 , D2 , ... } , Di , D (
pk | j log pk | j
( , )
H ( | ) ,
H ( | ) =
)-.
.- ,
,
.
H ( ) , H W ( ) H ( )
p + ... + pr
ln 1
= ln r
r
, d / d
.
.
.-
H W ( ) , W -
Di ,
i j
Di I D j = .
, E (D ) = .
; d i f f e r e n t i a l entropy ,
.
( ) ; entropy o f a
d y n a m i c a l s y s t e m ,
. ( , A , P ) T
h (T ) . = ( ) ,
( , A , P ) ,
,
n ( ) = ( T n ) , ,
## n = 0 , 1, ..., H n ( 0n1 ) , 0n 1 = ( 0 , ..., n1 ) n 1
. n H n ( 0
) n
h ( T , ) ; h ( T , )
{ n , n 0 }
( , [3] ) n
241
( 1 , ..., n )
- . ,
h (T ) = sup h ( T , ) ,
- .
( )
, h (T ) .
. -
= const
, .- .
, 0n1
Tn = T 1 ... T n1 ,
Ai0 T
## Ai1 I ... I T n+1 Ain1 ; Ai1 , ..., Ain 1 ,
.
. . . ( , [2] ) . .-
. .
( , [1] ) .
.-
.
.- [2]-
; h (T )
= h (T , )
, n = 1, 2 , ...
A ( , T
T
( . ) ).
T , ,
- A Tn - , n Z .
( , [4], [5] ),
.
T ( Y , B , )
,
h (T ) =
( yi ) ln ( yi )
, y i , ;
h (T ) = .
h (T ) =
i pij ln pij
i, j
, { i } , pi j
.
. : .
- ( ) .
242
:
1) h (T ) 0- + - ( 0 , )
;
2) k 0 h ( T k ) = k h (T ) ; { T t , t 0}
, t
h (T )
= t h (T ) ( , [7], [11] ); T ,
h ( T 1 ) = h (T ) , h ( T t ) =
= | t | h ( T 1 ) , < t < . h (T 1 )
.
3) . .- ;
4) . ( , , ) .-
; .- ;
5) T P , T
P1 P2 - aP1 + ( 1 a ) P2
, h ( T )
, hi , Pi
= ah1 + ( 1 a )h2
.-,
i = 1, 2 .
.- ( , [6] ),
( , [7] ),
( , [8] ) .
.: [1] . ., .: .
, ., 1987, . 8691;
[2]
. ., .: . , ., 1959, . 124, 4, . 76871;
[3] . ., , ., 1960; [4]
. , . , 1959, . 124, 5, . 98083;
[5] . ., . , 1959, . 125, 6, . 1200
02; [6] . ., . , 1963, . 148,
4, . 77981; [7] C o n z e J. P., Z. Wahr. und verw. Geb.,
1972, Bd 25, 1, S. 1130; [8] K r e n g e l U., Z. Wahr. und
verw. Geb., 1967, Bd 17, 3, S. 16181; [9] . .,
. , 1967, . 22, . 5, . 356; [10]
., , . .,
., 1969; [11] . ., . ., . ., , ., 1980; [12] .,
., , . .,
., 1988.
( ) ,
; t o p o l o g i c a l entropy o f a d y n a m i c a l
s y s t e m , ( )
.
( ) ; entropy o f c o n t i n u o u s r a n d o m
v a r i a b l e s , .
( )
; entropy o f a r a n d o m w a l k o n a
g r o u p , .
; r e l a t i v e entropy , ,
.
( ) D
D ; entropy o f m e a s u r a b l e o f D p a r t i t i o n ,
( X , ) D
.-, .
D
, . H (D ) =
, ( , ) - p ( , ) W
, -
~
~
x X :
~
W = { p ( , ) : M ( , ) } ,
(C ) log ( C ) ,
D -
. , , .
;
p r e f i x entropy , .
( ) ; entropy o f f i n i t e
o b j e c t , .
( ) ; entropy o f
s t a t i o n a r y p r o c e s s
= { k , k = ..., 1, 0 , 1, ...}
E ( ) = lim n 1E ( n )
(*)
n = {1 , ..., n } .
, (*)
E ( ) = lim E ( n | n 1 )
n
.
, .
;
c o n d i t i o n a l entropy , ,
, ( ) .
; entropy
, / .
~
(X X, S X S X~ )
p ( ) -
( , ) W - . W
( W -
),
H W ( ) = inf I ( , )
; I ( , ) , - -
(1)
(2)
x ) .
. ( x, ~
(1) ( W (2)
) (
) .
.: [1] ., , .:
, . ., .,
1963, . 243332; [2] .,
, . ., ., 1974; [3] .,
, . ., ., 1969.
; epidemic process
. ,
, ,
,
,
.
.
,
,
.
/
/ ; epsilon
entropy / entropy
. . ( C. Shan-
non ) . X ,
P ( ) = p ( ) , ( X, S X )
~
x ) , S X S X~
( X, S X~ ) , ( x , ~
( )
>0
~
H ( ) = inf I ( , )
(X, S X )
~
(X, S X~ )
>0
W W ; W entropy
.
( x, ~x ) , x X ,
H (D) =
p ( ) -. p ( , )
(1)
~
, I ( , ) , - -
X X -
( , ) , P ( )
= p ( )
~
M ( , )
(
(2)
).
~
~)
, ( X, S X ) ( X , S X
( x , ~
x )
( , )
243
0,
( x , ~x ) =
1,
x=~
x,
x~
x
) . - . . H W ( ) (1)
,
,
=0
, P~ W .
H 0 ( ) = H ( ) .
. . .
. . Q C
= ( 1 , ..., n )
, M j = 0 , j = 1, ..., n
. . ( H ( C , Q ) ) . . ., C
.,
( x , ~x ) =
log max ( M 2j , 1)
j =1
,
n
min ( , M
2
j
) =
j =1
.
,
. - .- .
= ( 1 , ..., n ) j , j = 1 , ..., n
,
~
X = X,
( x , ~x ) = n 1
~
x ),
, ,
( , ; ).
E = { 1, 2 , ..., } . . . X
pij ( n ) -
i E , i - , ( 1 , 2 , ... )
x = ( x1 , ... , xn ) ,
= lim pij ( n )
n
( . . .-
i =1
~
x = (~
x1 , ... , ~
xn ) X
). n ,
pij ( n ) j
1- ( i , j )- i , j E
0,
i ( xi , ~xi ) =
1,
H ( )
xi = ~
xi ,
x ~
x.
i
= 1 + log + ( 1 ) log ( 1 ) .
H ( ) - .
.
; ergodic Markov chain
P { j = 1} = 1 2 ,
(x ,
. . .- . .
. . . .
.: [1] . ., . .,
, ., 1982; [2] . ., .
, 1956, . 108, 3, . 38588; [3] . .,
. ., . , 1959, . 14, . 386;
[4] ., ,
. ., ., 1963, . 243332; [5] B e r g e r T., Rate distortion
theory, Englewood Cliffs (N. J.), 1971.
1
2
- C
2 - -
(xj ~
x j )2
j =1
H ( ) =
. . . C -
.,
( x , ~x ) = ( 1 ( x , ~x ) , ..., m ( x , ~x ) )
, = ( 1 , ..., m ), i > 0 , (2)
~
M i ( , ) i ,
i = 1 , ..., m
(3)
~
(2) (3) , , ( , ) -
. ( , ) P~ - W
, PX ( )
= p ( )
, -
W ( H W ( ) -
244
, . . . ( , [2] ).
. , .,
[3]- .
.: [1] .,
, . ., . 1, ., 1984; [2] . ., .
, 1977, . 234, 2, . 31619; [3] .,
, . ., ., 1989; [4] . ., . ., 1985, . 30, 2, . 23040.
( )
; ergodic problems o f
D i o p h a n t i n e a p p r o x i m a t i o n
. , .
, . ( E. Borel ), . .
.
q - [ 0 , 1]
T1 : x { q x } , { } ,
. T1
; 0 , 1 , ..., q 1 [ 0 , 1]
q -
.
[ 0 , 1]
{ 1 x }, x 0 ,
T2 x =
x = 0
0,
1
ln 2
dx
,
1+ x
A A
.
, T2 .
i - ( , , . 5,
. 811 ) ai ( ) , qi ( ) i - , ,
., ( )
:
lim
- :
0 ij < 1 , 1 i m , 1 j n . mn mn
E mn . Amn , mn Pmn
. ( mn , Amn , Pmn )
n
. a R - ,
T : ( a1 , ..., a m ) ( mod 1)
.
.
,
A
M ( A) =
m , n , mn , ;
1
( a1 ( ) + ... + an ( ) ) = ,
n
T = T a : mn 1m
, i ,
ai , mod 1 .
A A1m
Pmn ( T 1 A ) = 1m ( A )
; a1 a 2
Ta1 Ta 2 ,
A1 , A2 A1m
Pmn { T a11 A1
I T a 21 A2 } = P1m ( A1 ) P1m ( A2 ) .
2
1
,
lim
ln q n ( ) =
n n
12 ln 2
-, .
( , [5] ) ( ,
[2] ) - .
p q , , -
1. S , a 0 , a R n
, a S
A ( a ) A1m .
1m
(1)
{ A (a)}
a S
( a1 , ..., a n ) A ( a ) ( mod 1) , a S
(2)
p q < f(q) q
mn .
, f ( x ) -
(1) S -
, mn (2)
; ( ) .
;
i1 a1 + ... + ik ak
< fi (a ) , 1 i n
a
a
a = max ( | a1 | , ..., | ak | ) .
.
2. mn N ( Q , ) (2)
a =
= ( a1 , ..., ak ) R k -
( )
( ). .
( ,
[3] ).
= (a1 , ..., an ) S
= max ( | a1 | , ..., | an | ) Q . S -
, mn
N ( Q, ) = ( Q ) + O ( ( 1 2 ( Q ) ) ( ln ( Q ) )3 2 + )
,
>0
245
(Q ) =
P1m ( A ( a ) ) .
. { S
aS ,
max ( | a1| , ..., | an | ) Q
, p - ,
,
.
.: [1] ., , . ., 1969; [2] . ., . .,
. ., , ., 1980; [3] . ., , .,
1977; [4] . ., , 4 ., ., 1978;
[5] G a l a m b o s J., Representations of real numbers by infinite series, B., 1976; [6] S c h w e i g e r F., Acta arithmetica, 1966, v. 11,
p. 45160.
;
ergodic theory XX 2030-
. ,
. ( L. Boltzmann ) .
( J. Gibbs ) . . .- .
,
( ,
) .
. . .- .
.
, , ,
, .-
. .,
( ) ,
. , - , .,
. , ,
. .,
,
Pn = a0 n r + ar 1 n + ... + ar .
(1)
a0 , a1 , ..., ar 1
,
{P ( n ) } - r
( , [1] ).
, , . ( ,
[1], [2] ).
,
.
X 0 ( ) , { S (t ) } ,
. X t ( )
= X 0 ( S t )
246
(t )
,
{ X t ( ) }
. . .
. X 0 ,
X t ( < t < )
X0
- ,
, { S
(t )
( ,
).
, .
. .,
( , A ) , { S ( t ) } ,
.
{ S
(t )
} - -
, ,
.
1. { S ( t ) }
CA
t , < t <
.
t
P ( S C )
= P (C )
(t )
{ S } -
. ,
A , -
, { S ( t ) } - ,
P .
; , .,
.
, { S
(t )
} -
.
. P t
,
* t
( S ) P
( ( S * ) t P ) ( C ) = P ( S t C ) , C A .
* t
( S ) P P0
,
P0 -
.
.
2. . { S ( t ) } P0
.
1
, f L ( , A , P0 )
(
) ,
1
T
lim
f ( S t ) dt = lim
= lim
1
2T
1
T
t -
f ( S t ) dt =
f ( ) d P ( )
0
3. . { S (t ) }
, P0
Q 0 P0 - -
d Q0
* t
. Q t = ( S ) Q 0
d P0
P0 -
qt ( ) = q0 ( S t ) .
lim
f ( ) d Q t ( ) = lim
t
= lim
f ( ) q ( ) d P ( ) =
t
f ( ) q ( S ) d P
0
f ( ) d P ( )
0
(t )
, {S } ( , ). Q
. ,
P0 . . . .
.
4. . . . . K ( , [3] ) .
. K K
. K
,
( , ).
= { ( s ), < s < }
0 = f ( ) = f ({ ( s ) , < s < }) ,
t
= f ({ ( s + t ) , < s < })
( ), K . K
.
. K
.
5. .
.
X 0 = f ( )
,
1
DX t
( Xt t M X0 )
f ( S u ) du
t M X 0 = M X t , DX t X t -
,
.
, q0 ( )
f ( S ) dt .
P0 -
X t ( ) =
.
.
X t
= f ( S t )
. . .-
X 0
.
6.
. . .-
b f (t ) = M f ( S t ) f ( ) ( M f ( ) ) 2
.
, , ( ) . , ,
.
b f (t ) - . .- .
.
7. .
.
{ S t }
. , , ,
, ., , .
,
,
.- .
. .-
.
.
.
( H. Poincar ) .
. ( J. Hadamard ), .
( G. Birkhoff ), . ( E. Hopf ) ( [2], [4] ).
XX 60- . .
( , [3] )
K .
.
. . . .-
,
, . . , K
. . . . .
( , [1], [2] ) . K
247
.
( , [6] ). ,
. , K
-
K .
.
. .-
.
.
,
.
( x1 , ..., xd ) d
, t ( s )
, N t ( )
= (t+ ) t , ,
t+ , B .
T
= [ 0, )
( x1 + c1 , x2 + c2 x1 , ..., xd + cd , x1 + ... + cd , d 1 xd 1 )
. .-
, cij ( , [1] ).
R
( [ 0 , ) , B+ N ) ( X , B ) ( B+ , [ 0 , )
t , t T
t t ,
.
N
,
Z ,
. , , ,
.
.
.
.: [1] . ., . ., . .,
, ., 1980; [2] .
. , . 2, ., 1985; [3] . ., .: . , ., 1987,
. 8691; [4] . .,
, . ., 1950; [5] C ., .
, 1959, . 124, 4, . 76871; [6] .,
, ,
. ., ., 1978; [7] .,
, . 12, . ., ., 1987.
( )
; ergodic
class / s e t i n a M a r k o v c h a i n ,
;
, t , t T
: ( t , ) t ( )
).
( x1 , ..., xd )
St ( ) =
1
t
t 1
k=0
, ,
St ( ) =
1
t
du
M . , . .
S ( )
= lim S t ( )
S ( )
= M ( ) -
( )
.
; , ,
0 , 1 , ... M < ,
; ergodic theorems
= t + s ( ) t T
.
t , t T t , t T
lim S t ( ) = M ( | G )
(*)
-
. ( , A , P ) [
, G ,
, N ,
t = t 0 .
T = { 0 , 1, 2 , ... } ] [ T = [ 0 , ) ],
( X , B ) ( t ) , t T
; N ,
t T
.
, t , t T
.
s T s
248
1 M
<
(*) p .
.
S ( ) - M ( ) - G
. G -
.
.
; ;
,
.
. .-
. , N 0 -
).
. ., X , ij
, t , t 0 + t , t 0 -
i - j
- . M
<
{ ij } -
. i j
d = 1 ,
ij
t d t < ,
## lim pij ( n ) = f jj1
lim S ( ) = M
dt |G
t
.
. .-
, , < 1 cij < n 1
M ( | G )
lim M t ,
i j X E - ,
lim S t ( ) S t ( )
ij (
lim M t
M t
. .- .
,
.
.: [1] ., , . ., ., 1969; [2] . ., ,
., 1981.
; ergodic theorems f o r M a r k o v
p r o c e s s e s 1) . .
.
, ., ,
,
.
. .
( 1906 07 ). . . ( 1936 ),
. ( W. Doeblin )
- . . .-
,
.
. . ( , [5], [2] )
. pij ( n )
X n i j .
lim n 1
pij ( n ) j cij n ;
pij ( m ) = ij
m =1
, , j ,
ij
).
E ( , [12] );
.
; cij
.
X = ( X n , Px ) ( E , B )
. .- .
( M. Frechet ), . , . . , . . ,
. ( K. Yosida ) . ( Sh. Kakutani )
( , [1] ).
( , ) ,
p ( n , x , ), B , n 1 X
Pn ( ) =
p ( n , x , ) ( dx ) .
, ( )
<
( , [4] ).
; , X
m B ( E )
T ,
B ( E )
:
( , ;
x E , n , p ( n , x, ) 0
, d , i , f ii , i
P - -
.
X (E ) = -
). i ,
## lim pii ( n d ) = d fii1
n 0- ,
ij = 0 ( , ;
, B -
P m T < 1 ,
= sup f
, P , B ( E ) -
P f () =
f ( y ) p ( , dy ) .
249
. . ( , [4] ). E - E k B -
. .- ( , [2] ). X { j } , -
: ) Ek -
= 1 ; ) X
, x E k p ( x, E k )
Ek - X k [ ( Ek , Bk )
, Bk , B
] k
xD = E \
UE
p ( n , x, D ) 0 n
x D
. , X k
,
p ( n , x , ) k ( )
n x E k ( X k
).
, .,
. ,
lim n 1
p ( m , x, ) =
m =1
( x, E
Ek - ,
p ( x , ) , x Ek , Bk , X k ;
lim
) k ( )
x E B ,
( x , Ek ) = lim p ( n, x , Ek )
n
Ek .
. . .
( , [3], [11] ) ,
P
. . . . .- ( , ).
. .-
.
( , [3], [4], [10], [14] ).
. .
. ,
. .-
. .- ,
( , [7], [13] ). ,
. . ,
( , [7] )
. . ( , ).
,
.
. .-
( , [11], [12] ).
[9]- .
.
2) . .
( , ).
.
1
n
f ( Xm ) =
m=0
f ( j)
f ( j ), j E
.
.
.: [1] . ., , . .,
., 1956; [2] - , ,
. ., ., 1964; [3] F o g u e l S. R., The erqodic theory of
Markov processes, N. Y. [a. o], 1969; [4] R e v u z D., Markov
chains, Amst. [a. o], 1975; [5] K o l m o g o r o f f A., . .,
1936, . 1, . 60710; [6] . .,
. ., 1960, . 5, . 2, . 196214; [7] D u f l o M.,
Revuz
D., Ann. Inst. H. Poincar. B, 1969, t. 5, p. 22344;
[8] . ., . , 1977, . 234, 2, . 316
19; [9] N u m m e l i n E., Z. Wahr. und verw. Geb., 1978, Bd 43,
S. 30918; [10] . ., . ., 1981,
. 26, . 3, . 496509; [11] . ., . ,
1982, . 263, . 55458; [12] . ., .
., 1985, . 30, . 3, . 23040, 47885; [13] . ., , ., 1989; [14] . ., . ., 1986, . 31, . 4,
. 64150; [15] .,
, . ., ., 1989.
; ergodic theorems
for stationary random process and homog e n o u s r a n d o m f i e l d s .
1.
; . T , P (T ) , T -
, ( , A , P ) ;
, X (t ), t T , T -
p 1 M
( X (t) ) < ,
p
t T . Lp -
X ( ) ( )
M * ( X ) L p ( , A , P ) ,
{ n } P ( t )
lim
X ( yt )
n ( dt )
= M *(X )
(1)
L ( , A , P ) y T
; { n } X ( ) p
, . L - M ( X ) L - , 1 r < p .
r
250
L - ,
X ( ) M ( | X ( x) | p ) <
p
, L -
, 1 p < . X t1 , t 2 ( t ) = X ( t , t1 , t 2 ),
An ( B) = ( An I B) / ( An ) ; (1)
lim
t , t1 , t 2 T , M * ( X t1 , t2 ) = M * ( X ) . X ( )
*
= M( X ( t ) | J X )
, M ( X )
, J X , X ( )
A - P = mod 0 .
X ( ) , L2 ( , A , P ) -
X ( t ), t T
H X
( An )
X (t ) -
JX
t T .
[2] [1]
- . ,
, X ( )
X ( ) J X
( ),
*
M ( X ) = M X , M
M X (t )
-
. . ( L. Boltsmann ) . .
( J. W. Gibbs )
,
. X = R
Z M ( X )
( X ( t ) | J )
M ( X ) = M
X
.
. .- An : An ( B ) =
= An I B An R - ( Z - )
, An
An
= [0, n]
= [ n , n ] , B , B R (
B Z ) ( ).
. .- ( , ).
2. ;
. { n } P (T )
T -
; :
1) { n } T -
;
2) { n } L - , M ( X ( t ) ) <
p
( p 1 ) X (t ) .
T An , n
= 1, 2 , ...
0 < ( An ) < T - ,
An
{ An }
.
( , [23] ).
( mean ergodi theorems
). { n }
lim
JX = { : H X ,
( X (t ) | J ) ,
U X = , t T } . M * ( X ) = M
X
X ( y t ) ( dt ) = M ( X )
{ An } ,
U t X ( y ) = X ( t y ), t , y T U t .
( An An x )
=0
( An )
( , [19] ); R -
r ( S n ) S n An
{ An } ( , [5] ). ( X
=R
Z , An
= [ 0, n ]
[ n , n ] ) . . ( , [12] )
. . ( , [6] ) ( X
= Rm
Z , m 1, An
m
) . X
,
( An ) { An }
; . .
( , [22],
. . ). [13]
[19]- . .-
.
3. ;
.
M ( X ( t ) ) <
X (t ) , t T y T
(1) [ (2) ] , { n }
( { An } ) T -
;
( poinwise ergodic theorems ). q P (t ) , q -
T - , n q
k - , n =
q -
1
n
, n = 1, 2 , ... -
k =1
( , [21] ); q X ( ) { q }
( , [18] ). { An }
## sup ( ( An An1 ) ( An ) ) <
1 n <
251
, { An }
( , [8], [21] [23] ); ., ( S n ) S n
m
, R -
.
, [12] ( , [3] ) . .
. [9] [11]-
.
4. .
. [24]- lim
lim
n
1
n +1
k =0
.
.
k =0
f , g L ( , A , m ) , g
0.
lim
g ( )
, . .-
( , A , m ) ,
1
; ergodic theorems f o r W i e n e r
p r o c e s s e s -
k=0
f ( k )
f ( k )
. 1, . 11382.
. .
## [4]- . .- L ( , A , m ) - ( , [17] ), [20]- . . . .
l ( A)
l ( s t : w ( s ) A)
=
l (s t :w (s) B)
l(B)
, l , w ( s )
, A , B , l (B )
< .
.: [1] ., .,
, . ., ., 1968.
; ergodic random process
(t ) ,
. [7]- . .- L ( , A , m)
[ ( t )] m ( )
L ( , A , m)
( m
S < 1 S -
## ( , [15] ). . .- [10]- [10], [15], [16] [22]
.
,
. .- m
; R -
( , [14] ).
.: [1] A l a o g l u L., B i r k h o f f G., Ann. Math., 1940,
v. 41, p. 293309; [2] B i r k h o f f G., Duke Math. J., 1939, v. 5,
p. 1920; [3] C a l d e r o n A., Ann. Math., 1953, v. 58, p. 18291;
[4] C h a c o n R., O r n s t e i n D., Ill. J. Math., 1960, v. 4, p. 153
60; [5] D a y M., Trans Amer. Math. Soc., 1942, v. 51, p. 399412;
[6] D u n f o r d N., Duke Math. J., 1939, v. 5, p. 63546; [7] D u nf o r d N., S c h w a r t z J., J. Ration. Mech. Anal., 1956, v. 5,
p. 12978; [8] E m e r s o n W. R., Amer. J. Math., 1974, v. 96,
p. 472-87; [9] E m e r s o n W., G r e e n l e a f F., Adv. Math.,
1974, v. 14, p. 15372; [10] K r e n g e l U., Ergodic theorem, B. N.
Y., 1985; [11] V e r s h i k A. M., Selecta Math. Sovietica, 1982,
v. 2, p. 33150; [12] W i e n e r N., Duke Math. J., 1939, v. 5, p. 1
18; [13] U r b a n i k K., Studia Math., 1958, v. 16, p. 268334;
[14] . ., . ., 1977,
. 22, . 295319; [15] ., .,
. , . ., ., 1962; [16] . ., . ., . .,
, ., 1980; [17] .,
, . ., ., 1969; [18] . .,
. ., 1965, . 10, . 55157; [19]
. ., . . ., 1962, . 2, 1,
. 195213; [20] . ., .
., 1972, . 17, . 38083; [21] . .,
. , 1967, . 176, . 79093; [22] . ., . . . -, 1972, . 26, . 95-132;
[23] . ., ,
252
()
s T [ ( t + s ) ] -
T ),
M [ ( t )] - ( (t ) [ ( t )] - ).
20- ,
; , ,
( , ., [1] ).
t ;
.
, [ ( t )] , ,
, ,
[ ( t )] M [ ( t )]
t - , (t ) . . . .
[ ( t )]
( T ).
[ ( t )] t 0
( t 0 ) ; . . . M ( t )
= const
, (t )
(
T
1 ). [ ( t )] = ( t 0 ) ( )
.
( ,
T )
( , ,
);
1
1-
. (t )
;
(t ) ( 1
) M (t )
<
, ; , .,
,
(t ) -
T 1 m -
; , m
( )
= M [ ( t ) ] -
. ,
, ; (t )
( , ,
- ), ,
( )
, ( t ) M (t ) F ( )
( , [2], [3], [4], t , [5], [6] ).
(t ) (t ) - -
[7],
[8]- ; .
( 0 s T T
[ ( t )] -
[ ( t + s ) ] -
, ( t 0 )
) [9]- .
( )
(t )
.
(t )
- ( t 0 ) ( t1 ) ;
(t )
, (t )
, Y ( t )
= ( t ) ( t + )
( -
) 1- .
B ( ) = M ( (t + ) ( t ) )
b ( ) = M [ ( t + ) M ( t + ) ] [ ( t ) M ( t ) ]
f ( )
T
F ( )
2- . ( x0 , ..., xn 1 ) n
,
M [ ( t 0 ), ( t 0 + t1 ) , ..., ( t 0 + t n 1 ) ] <
[ ( t ) ] = [ ( t0 ), ( t0 + t1 ) , ..., ( t0 + tn 1 ) ]
(
## [ ( t ) ] = lim n [ (t0 ), (t0 + t1 ) , ..., ( t0 + tn 1 ) ]
n
)
,
.
,
( ) .
(t ) ,
m ( ) = ( 2T ) 1
T
[ ( t
,
(t ) - T 1 (t ) - -
(t )
## . [10], [11]-, . . [12]-
, (t )
+ s ), ( t 0 + t1 + s ) , ..., ( t 0 + t n 1 + s ) ] ds
, , ( , ., [5] ). : .
.: [1] . ., . ., . .,
, ., 1980; [2] M a r u y a m a G., Mem.
Fac. Sci. Kyusyu Univ., Ser. A, 1949, v. 4, 1, p. 45106; [3] ., ,
. ., ., 1961; [4] . ., . . ., 1950,
. 2, 2, . 2547; [5] . ., , ., 1963; [6] ., .,
. , . ., ., 1969; [7] P a r z e n E.,
Ann. Math. Statist., 1958, v. 29, 1, p. 299301; [8] . .,
, ., 1964; [9] . ., ,
, 2 ., 1989; [10] N e u m a n n J.,
Ann. Math., 1932, v. 33, 4, p. 587642; [11] N e u m a n n J.,
, . ,
., . 1, ., 1987, . 762; [12] . ., .
, 1949, . 4, . 2, . 57128.
ergodicity
( , A , P)
, .
A A
, P ( A ) > 0 , P ( \ A ) > 0
.
.
.
253
{ S
(t )
} X = { X ( t ) , t R d Z d } -
, . , Y ( x ) L ( , A , P )
T
1
Td
Y( S
...
( t1 ... td )
x ) dt1 ... dt d M Y
( Z ).
.
.: [1] . ., . ., . .,
, ., 1980.
; Erlang formula n k
pk
, :
pk =
1
k !
i=0
1
i!
, 0 k n .
pn - . 1917- .
.
, .
; Erlang distribution
( n ) n n 1 n x
x e
,
( n)
p( x) =
>0
, [ 0 , ) - , , , n , n ,
(n)
n = 0, 1, 2, ...
(n) =
t n 1 e t d t ,
( Re n > 0 ) ,
( n + 1) = n ( n ), ( n + 1) = n!,
(1 ) = ( 2 ) = 1, (1 / 2 ) =
. . 1 1 n
. . . .
. ( A. Erlang ) .
n . . n , n
; n = 1 . ., > 0 . . .-
f ( z ) = ( 1 iz / n )
.
254
## = 1 , (1) : n = 1, (2) n = 3, (3) n = 10
. .-
.
.: [1] .,
, . ., 2 ., ., 1971.
; erosin
, -
; Esseen
theorem , .
;
confidence set , , ,
( ,
, , ,
).
, ; confidence probability , , ,
.
; confidence interval
. .
. . .-
. ,
. .-
. . .
. R
, P ,
, ( ) ,
1
. ( x ) ( x ) R - ,
1 ( x ) < ( x ) , x R n .
p ( ) = P { ( ) < < ( s ) } ,
, ( ( ), ( ) )
, ( 0 < < 1) , ( )
( ) ,
. p ( ) - , ( , )
. .
. P = inf p()
. p ( )
( , ) , ( , )
( ) .
. . .
( . ) .
- ,
C ( X ) [ C ( x ) , xX ]
, . .- , -
~
P { | n | < } = p ( )
u ( ) -
C ( X )
PC ( )
PC ( ) = PC [ u ( ) , ] ,
n
n
. ,
PC ( )
u ( ) C ( X )
, .
. -
~
, n - , , -
( n -
). , (
n - 20 30 )
.
, .
.: [1] . ., . ., . ., ,
., 1979; [2] . ., , ., 1964;
[3] . ., - , 2 ., ., 1962;
[4] ., , ., 1979.
; confidence estimation -
PC ( ) - C ( X )
. PC ( ) = P { C ( X)
u () | } = const P { X X | } = const = 1
. PC ( )
, , PC = inf PC ( )
; (
).
. .-
. .,
0,5 1 ,
Pc [ u ( ) , ]
.
, u ( ) - u U
. X X
.
( x )
.
,
PC ( u , ) , u U , .
p( x | ) , x X ,
C , , u - u ( ) -
. , X - .
. . X - C ( X ) ,
u ( )
.
U , u ( ) , , C ( X )
xX U -
; , u U
{ C ( X ) u }
.
PC ( u , ) =
p ( x | ) d ( x ), u U ,
{C ( X ) u }
- C ( X )
u
.
: C -
C
PC [ u ( ) , ] ,
,
PC ( u , ) PC ( u , ) ,
u U ,
.
1 ,
. .
255
. .- u ( ) -
.
, M X i
X 1 , ..., X n , n 2
= 1 , D X i = 2
, .
u ( ) = 1 .
X =
. T
1
n
Xi
i =1
s2 =
1
n 1
(X
X )2
i =1
= n ( X 1 ) s -
( n 1) - 1
2 > 0 )
( | 1 | < ,
t -
. .- ( ,
[1] [3] ).
.: [1] ., ., , . ., . 12, ., 1983; [2]
.,
., , . .,
., 1973; [3] . ., . ., ,
., 1981.
; confidence band o f l e v e l
, 0 < < 1 , ( x ) = T f ( x)
A X
inf
{ X ts
R P
n < 1 < X + t s
n}
t - . 1 C ,
= ( 1 , 2 ) -
PC ( 1 , 2 ) = P { | T | < t }
.
,
.
.
.: [1] ., , . .,
., 1967; [2] .,
, . ., 2 ., ., 1979; [3] ., , . ., ., 1964; [4] . ., . ., 1965, . 10, . 1,
. 18792.
## ; confidence limit / bound , .
;
confidence region , .
;
confidence bound , , -
D ( y ) R A , D ( y )
P { ( ( x , ( x ) ), x A ) D ( y ) }
(1)
= ( y1 , ..., y N )T
P = P ( y F ) F =
, y
= ( f ( x1 ) , ..., f ( xN ) )T , xi X , i = 1, ..., N .
(1) ,
. . ,
. B D ( y ) D ( y ) \ B
,
.
, P F - , V -
= F T V 1F ( ) ~
, . V M
2
( 0 , M
) .
( x ) = f T ( x ) , P { U R P } ,
( x ) = inf u f ( x ), ( x ) = sup u f ( x )
u U
u U
T
(2)
D ( y ) = { ( x , ( x ) + z ) , x A , z ( x ) }
, .
; confidence band
D ( y ) = {( x , ( x ) + z ) , x A , ( x ) z ( x )} (3)
. F - -
. ., k
n F n Fn -
p p ,
, ( 0 , 1)
P { F n ( x ) F ( x ) F n ( x ) x - }
0 + 1 x V
. F n Fn
, F
. F
256
= { u : 2u T M u k } , R - -
( x ) = ( x ) = k
, b > 0, x
s 2 = ( N p ) 1
i =1
1 + b ( x x )2
= ( x1 + ... + x N ) N .
( yi ( xi ) ) 2
,
, s , k F .
, [1].
(3)-
( x ) = ( x ) =
= sup f T ( x ) :
max
i = 1 , ..., p
T
f ( zi ) , i k
, i
(4)
> 0 , zi X -
, k , ( x )
( x)
M ( x) ( x) =
= 2 f T ( x ) M 1 f ( x1 ) .
.,
( x ) = 0 + 1 x (1) + ... + m x ( m) ,
( x (1) , ..., x ( m ) ) = x R m
x1 , ..., x N
( x )
, k z1 , ..., z m + 1
m
R - , V
= I
X i :
M X i = 0, D X i = 1,
M X i X j = p, 1 i < j m + 1 .
R - m
( ) .
( , [2] ) ( z i ) ,
(4)
.
( x ) w ( x ) dx ,
w ( x ) 0 , x R
(3) .
.: [1] ., ,
. ., ., 1980; [2] . ., . .,
, .,
1973; [3] N a i m a n n D., Ann. Statist., 1984, v. 12, 4,
p. 1199214.
; confidence zone , .
; reliability optimization
.
. .
. :
( . ),
( ) , .
.
(
)
. ,
.
: ., , , .
( )
.
.
,
. (
. )
( ) .
.
,
;
,
.
. .-
: (
)
(
) .
.: [1] , ., 1983;
[2] . , ., 1985;
[3] . ., , ., 1969;
[4] . .,
, ., 1981; [5] . ., . ., , ., 1971.
; mathematical theory
of reliability
. . . .-: 1)
; 2)
,
; 3) ,
( , [1] ) .
. . .-
. , .
.
257
. . .- ( ) X = { x }
. t
( ) (t )
. ( )
( ) X 0 , X - . ( ) ( t ) X 0
.
. ( )
( ),
.
,
.
, ( ) .
n . i - xi
, xi
= 1 , xi = 0 .
. =
( x1 , ..., xn )
= ( x1 , ..., xn )
= 1 , = 0 .
.
,
:
= min xi =
1 i n
x .
i
i =1
i Fi ( t ) = P { i t } , pi
= P { xi = 1} = Fi ( t ) , Fi (t ) = 1 Fi ( t )
. , t
F (t ) = h ( F1 (t ) , ..., Fn ( t )) .
. ,
,
.
.
( , ) ( ; , [4] ).
, ., , ,
( , ,
).
,
, .
( , ). ( )
,
.
( , ).
. -
. t = 0 . t ,
, .
, .
U i Vi
x1 , x2 , x3 , A B , B C
F ( s ) G ( s ) -
A C
. - ,
. ,
.
.
,
.
T = m + m1 ( 1 )
= max ( x1 x2 , x2 x3 , x1 x3 ) = x1 x2 + x2 x3 + x1 x3 2 x1 x2 x3
; m1
.
i - xi , i - .
xi xi , i = 1, ..., n ( x1 , ..., xn ) ( x1 , ..., xn )
, .
xi - pi
= P { xi = 1}
.
h ( p1 , ..., p n ) = P { = 1}
. h ( p1 , ..., p n )
. i -
258
, m
= M U i = M min (U i , Vi ), = P { Vi U i }
, .
( ).
.
( , ).
. . .-
. . .- .
, , ( ., )
. ,
.
. . .-
.
( )
.
. 1) n
.
,
, . i
mi
= 1, ..., n . mk = min mi
, , ij = 0 .
, pij
(k j )
, i
1 i n
k
.
( ) . N .
. :
B
C .
, ., T -
, r -
. t S (t ) -. [ N , B, ( r ) ]
N ,
r - . [ N , B, ( r ) ]
+ ( N r + 1) t r
S ( tr )
t1
= t1 + ... + tr 1 +
t1 , ... , t r
## < t 2 < ... < t r . [ N , B, ( r ) ]
= 1 e t
= ( r 1 ) / S ( t r ) .
[ N , B, T ] , N T ,
.
,
,
( , [1], [5],
[6] ).
, , ( )
.
xij i -
j - , ij i -
, x j
= { xij , ij , i = 1,..., k j }
. i - , ij = 1 ; i -
pij 1,
i =1
pij
F ( xij ) =
## F ( xij ) F ( xij 0 ) = pij > 0 F ( )
, F ( ) x j -
ki
[ F ( x ) ] [1 F ( x ) ]
ij
ij
ij
1 ij
C( xj, ) > 0
i =1
. : )
, , ; )
, F ( s ) G ( s ) - [ G ( s ) ]; )
.
x1 , ..., x m , ,
. F ( s )
F ( s ) = 1
F ( t )
>0
1
us
D ( u )
N (u )
, D ( u )
(*)
= D (u ) D (u 0),
D ( u ) ij = 1 xij u xij -
; N ( u ) , xij
, i
u xij -
## = 1, ..., k j , j = 1, ..., m . (*) F ( s )
.
x1 , ..., x m ), ), )
, m
(*)
F ( s )
( , [7] [8] ).
. ,
( .,
, . ). ( ) , ,
.
. . . . -
259
.
.
.
. , 0
( )
F0 ( t ) = P { R t } R . 0
r ( 0 , ) 0 , r ( 0 , 0 ) = 1 .
1 = 1 ( t )
0,
1 ( s ) ) ds = R
. 1
F ( t , 1 ) = P
= F0
. , r ( 0 , )
F0 ( s ) .
. . .-
( , [1] ).
.: [1] , ., 1983;
[2] . ., . ., . .,
, ., 1965; [3] ., ., , . ., ., 1984; [4]
, ., 1985; [5] K a l b f l e i s c h J., P r e n t i c e R. L., The statistical analysis of failure time data, N. Y. [a. o.],
1980; [6] L a w l e s s J. E., Statistical models and methods for lifetime data, N. Y. [a. o], 1982; [7] . ., . .
, 1985, 4, . 4559; [8] .
., . ., .: , ., 1985, . 317; [9] , . 8, ., 1990.
; reliability
,
, , ; .
; confidence coefficient , , .
; reliability function : 1) . .
R = R ( p1 , ..., pn )
; R
= ( p1 , ..., pn ) , R = 1 ( 1 p1 ) ...
260
(u ) du
n - .
. . E ( R )
, Pe ( n , R ) n R
r ( 0 , 1 ( s ) ) ds
p1 , ..., pn -
P (t ) = exp
E ( R ) = lim sup
log Pe ( n , R )
n
n
r ( , (s) ) ds R =
0
P (t ) .
. . P ( t )
,
, ; reliability function, r e l i a b i l i t y c o e f f i c i e n t n -
r (
... ( 1 p n ) .
. .- ( ., ) .
2) . . (0 , t )
( ) n . E ( R ) -
R C , C
, Rcr . 0 R < Rcr
Rcr
E ( R ) ( , ( ) ,
).
.: [1] ., ,
. ., ., 1974; [2] ., . , . ., ., 1965; [3] .,
., , . ., ., 1985.
## ; reliability indices of system .
.
, , ,
.
, ,
, , ,
.
( ) . .
. .
( )
t0 .
, P ( t 0 )
= P { t 0 } .
M -.
t
, P { t }
=.
.
.
.
- .
. .
.
, , ,
.
,
. ,
;
. .
.
.
.
.: [1] . .
27.002-83, ., 1983; [2] . , ., 1985; [3] , .,
1983.
; reliability index
.
i - X i ,
P { X i = 1} = pi = M X i , i = 1 , ..., n .
X = ( X 1 , ..., X n ) . i -
i -
. . .
P { ( X ) = 1 } = h = M (X)
, M X , X ,
.
. .-
. .- :
h = h (p) .
p1 = p 2 = ... = p n = p
h(p)
. h ( p ) - (
h ( p ) - )
. ,
. . p -
h .
h ( p ) = pi h ( 1i , p ) + ( 1 pi ) h ( 0 i , p ) , i = 1 , ..., n
( , [1] ).
.: [1] ., ., , . ., .,
1984.
/ ; confidence level ,
, , .
; ; ; reliability theory;
analytical statistical method /combined sim u l a t i o n m e t h o d 1 , ..., k f .
1 , ..., k
, f M1 , ..., M k . f , M i
( , ,
).
;
.
.
. ,
( ),
( ) .
( ,
, ) .
.
; .
261
.
.
- .
.: [1] . ., , ., 1975; [2] . .,
, ., 1980; [3] . .,
, ., 1982; [4] . ., . .,
, ., 1988.
; Euclidean
quantum field theory
d
, R ( d 2 )
d
M
( , R
). M -
,
,
.
R d - , M d - d
: T S (R ),
S (R d ) [ S (R d ) , R d -
], ( T ) = ( T , ) R d -
S (R d ) A
( ) .
( , SO .;
, [2] ).
( , [3] ).
## [3] O s t e r w a l d e r K., S c h r a d e r R., Helv. Phys. Acta,
1973, v. 46, p. 277302.
; Euclidean
field , .
; Euclidean
shape , .
x; Euclidean approach , P [ ]2 ,
.
; evolutional censoring ,
( ).
; evolutional stochastic differential equation
( . )
d u ( t , ) = A ( u ( t , ) , t , ) dN ( t ) +
+ B ( u ( t , ), t , ) dM ( t )
t > t0 ,
d u (t , x ) =
1 ... n =
S n ( x1 , ..., xn )
( x ) dx ... dx
i
d
1
d
n
i =1
( Rd )n
i S (R d ) ,
S n ( x1 , ..., xn ) ( )
( )
d n
( M ) , n
= 1, 2 , ...
- .
.: [1] ., P [ ]2
## , . ., ., 1976; [2] ., ., . , . ., ., 1984;
262
1
u xx ( t , x ) dt + u x ( t , x ) d w ( t ) ,
2
( t , x ) (0, T ] R1 ,
u (0 , x ) = u 0 ( x ) , x R1
R -
; . ( E. Nelson,
, [1] ) .
u ( t 0 , ) = u 0 ( )
, A B
H , N , M
, , (1)
. H A B .
. . . .-
. .,
S (R d ) - ,
( F A )
(1)
, w , u 0 , L (R )
1
, W2 (R ) , W2 (R )
. . . .
1
. W2 ( R ) A :
1 2
, B: =
2 x 2
x
, W2 (R )
.
. . . . . . . .
( , [1] [3] ),
,
. ., 2- . . . .-
( , [4] ).
. . . .-
. , M
. :
, (1)
.
. . . . .-
.
H , H H , V
( ,
). : ) V H V ; ) V , H -
; ) c , V -
= ( , ) H
V ,
V ( , - - )
, H V , V
, . , a) )
V H .
W pm (G ) ( = V )
L2 ( G ) ( = H ) , G ,
d
R - , d p
2 ( d m p ) G -
. , ( , A , { At }, P )
E
( E = E )
w (t )
. P
, L ( E , H ) ,
E - H - ,
; , S
= V [ 0 , T ] -
A : S V B : S L 2 ( E , H ) .
,
v V . ,
du ( t , ) = A ( u ( t , ) , t , ) dt +
+ B ( u ( t , ), t , ) d w ( t ) , t T ,
(2)
u ( 0 , ) = u0 ( )
(3)
. . . . K , > 0
1 , 2 , 3 V , ( t , ) [ 0 , T ] :
A1 ) A - : A (1 + 2 ) R1 -
;
A2 ) ( A, B) - :
2 (1 2 ) ( A (1 ) A ( 2 )) +
K 1 2
B (1 ) B ( 2 )
2
A3 ) ( A, B) - :
2 A ( ) +
B ( )
K( 1 +
);
A4 ) A - :
A ( )
K(1+
).
A1 A4 u 0 L ( ; A0 ; H ) ,
(2) (3) .
L2 ( [ 0 , T ] ; P , V ) I L2 ( ; C ( [ 0 , T ] ; H )) ( , [1], [2] ).
( .,
, ,
) .
. . . . .-
, ( A + 1 / 2 B B )
.
. . . .-
. . .,
A , B , .
, . . . . ,
. ., :
du (t ) = u ( t ) dt +
u ( t ) (1 u ( t )) d w ( t )
.
( ,
[5] ). , (2) (3)
. ; A5 ) V -
H - .
A B
( , [6] ).
. . . .-
, - .
. . . .- ,
, . . . .-
( , [6] ).
, . . . .- (1) . . . . . ,
A2 , A4 , A5 .
. . . .-
. ,
.
. A3 . ,
.
B . . . .-
( , [7] ).
. . . .- ( , [8] ).
.: [1] P a r d o u x E., quations aux drivs partiells stochastiques nonlineaires monotones. Thse doct. Sci. Univ. Paris Sud., P.,
1975; [2] . ., . ., .:
. , . 14, .,
1979, . 71146; [3] . ., . , ., 1983; [4]
. ., . ., , ., 1983; [5] V i o t M., Solutions faibles
dequations aux derivees partielles stochastic nonlineaires. These doct.
Sci. Univ. Pierre et Marie Curie, P., 1976; [6] G r i g e l i o n i s B.,
M i k u l e v i c i u s R., Lect. Notes Contr. and Inf. Sci., 1985, v.
69; [7] C u r t a i n R. F., P r i t c h a r d A. J., Lect. Notes Contr.
nd Inf. Sci., 1978, v. 8; [8] . ., .
, 1982, . 37, . 6, . 15785.
;
evolutionary spectral function ,
.
263
## ; evolutionary spectral representation (t ), < t <
. ( t ) -
. . . :
(t ) =
ei t A ( t ; ) d Z ( ) ,
(1)
Z ( )
, ,
,
M dZ ( ) d Z ( ) = 0, M dZ ( )
= dF ( ), (2)
A ( t ; ) t -
A ( t ; dF ( ) < ,
(3)
2
<
) t - .
. ( . . .- ; , [1], [2] )
A ( t ; ) -
f ( ) = F ( ) ]
( , ., [2], [4] ).
.: [1] P r i e s t l e y M. B., J. Roy. Statist. Soc., Ser. B , 1965 ,
v. 27, 2, p. 20437; [2] P r i e s t l e y M. B., Spectral analysis and
time series, v. 2, Ch.11, L. [a. o.], 1981; [3] ., ., , . ., ., 1972; [4] M a n d r e k a r V., Proc. Amer.
Math. Soc., 1972, v. 32, 1, p. 28084.
; evolutionary
spectral measure ,
; Euler numbers
n 1,
1 k n (*)
E n, k ;
## E0,0 = 1, En,0 = E0,k = 0 k > n En, k = 0
. (*) . ( L. Euler ) 1755-
. . .
:
E n, k =
(1) m Cnm+1 ( k m ) n .
m=0
A(t ; ) =
it
. .- :
dKt ( ) ,
1 x
=
1 xe t (1 x )
dK t ( )
En, k = k En 1, k + ( n k + 1) En 1, k 1 ,
( t - M (t )
F ( )
t - = 0 -
. . . .- (t ) [ dZ ( ) (2) , A ( t ; ) (3)
t - (1) ] .
(t ) , , , t - A ( t ; )
F ( )
xk
k =0
tn
.
n!
(1 , 2 , ..., n ) - k
i 1 < i ,
( 0
i = 1 , ..., n
= 0 ).
k = n 2 + x n 12
En , k
n!
= A ( t ; ) F ( ) (t )
. , , ,
, (t ) (1) . . .-
.
,
( ., ; , [3] ) .
[ ., (t )
264
n=0
n, k
. . , ,
. E n,k / n! 1, 2 , ..., n
e i (t s ) A ( t ; ) A ( s ; ) dF ( )
. F ( t ; )
x -
B ( t ; s ) = M ( t ) ( s )
B (t ; s ) =
ex
/2
.
. . En -
:
k =0
.: [1]
, ., 1977.
x 2 k
( 2 k )!
n=0
En
xn
.
n!
. .,
## ; Einstein Smoluchowski model , .
( ), ; numerical simulation o f a r a n d o m p h e n o m e n o n
. -
,
, ,
, ,
.
.: [1] . ., ,
., 1973; [2] . .,
, 2 ., ., 1975.
; negligible set
. ( X , S , ) , Y
X - .
( Y ) = 0 , Y
*
( , S , ) ,
- . X
, X - x X
U ( x )
, Y I U ( x )
( Y X ) X - ,
Y X X
.
. . . , , .
( , [3] )
. . . .,
E
E - E -
.
.: [1] ., .
. .
. , . ., ., 1977;
[2] ., , . ., ., 1953; [3] . ., , .,
1984.
;
indifference zone
H 0
H 1
. . .- H 0 - ,
. . ., ,
,
., > 0 , 0 +
[ ( 0 , 0 + ) ] ,
H 0 : 0
.
.
.: [1]
., , .
., 2 ., ., 1979.
;
negation / complement of an event ,
.
;
reflecting barrier / boundary ,
.
;
reflecting boundary , ;
; bound
model , .
( ) ; communicating states o f a M a r k o v c h a i n , ; .
; complementarity observables , .
A
A ; elementary event, f a v o r a b l e
t o A A
. .,
= { GG, G, G, } , A = {
}
. A . . .-
{G G} { G } .
; maximum
likelihood method ,
.
265
/
; maximum likelihood principle
,
; ,
.
x X
,
p( x, )
## ; most stringent test , ,
.
/ ;
most accurate confidence set ,
,
,
.
X , ( X, B, P ) ,
). . . . .-
1 2
( x ), ,
.
.
. . . . ,
C ( X )
( x ) .
H 0 : 0 H 1 : 1
. . . .
;
sup p ( x ; )
sup p ( x ; )
H 1 H 0 [ H 0 : =
= 0
H 1 : = 1
. . . .
p ( x ; 1 ) p ( x ; 0 )
].
, . . . .-
,
( ) ( , , ).
. . . .
. , X x1 x2
, p ( x1 ; )
h ( x1 , x2 ) p ( x2 ; )
, -
. X Y
x y , X Y -
p X ( x; ) pY ( y ; )
.
x y ,
,
p X ( x ; ) h ( x , y ) pY ( y ; ) ,
.
.: [1] ., ., ,
. ., ., 1978; [2] . ., .,
, . ., ., 1973.
266
= { C ( X )} P
, -
inf P { C ( X ) } = P
. - C ( X ) . . . . , C ( X )
P { C 0 ( X )} P { C ( X ) },
. , , .
.: [1] . ., , .,
1984.
; most accurate invariant confidence set
.
X , ( X , B , P ) ,
,
= { C ( X )} P,
0 < P < 1 ,
. X - X g
G = { g } { P }
. g G C ( g X )
= gC ( X ) , - C ( X )
G
, g , - g
G
= {g}
. -
C ( X )
P { C0 ( X ) } P { C ( X )},
, - C0 ( X )
.
, , .
.: [1] . ., , .,
1984.
## ; most accurate bound / limit
.
, , .
, ; worst distribution , , .
; least favorable
distribution
.
( X, DX , P ) , X
d ( )
( D , DD )
, ( , D , t ), t T
. L ( , d )
d
.
( t , d ) =
L ( , d ( x)) d P ( x) d ()
d
sup ( t , d ) = ( t * , d )
t*
d ( ) . . . . d
( ) ( * , d )
t
. , , . . . , ,
,
; , d
,
## inf sup ( t , d ) = sup ( t , d * )
dD
tT
tT
d .
,
. . . . X X Q
H1 X -
H 0 = { P , } H 0
. ,
p ( x ) =
= d P ( x) d ( x) q (x ) = d Q ( x) d ( x ) , ( x ) ,
( X, DX ) , { t , t T } ,
( , D ) .
t T H 0 H t , X
p ( x ) = d ( )
.
, H1 - H t
. t , . . .
{ t , t T } t*
t*
X f * ( x)
t
q (x) , H t*
{ H t , t T } H1
. , , , .
.: [1] ., , .
., 2 ., ., 1979; [2] ., , . ., ., 1975.
## ; most powerful test
H 0 . , H1 ,
1-
. H1
H 0
;
1- ; H1
, H 0 .
( H1
H 0
ft ( x ) =
. . . . : H
H1 H 0 -
tT
. H1 -
, { t , t T }
t T
).
= P { H 0 | H 1} 1 - . ,
= P { H 1 | H 0 } ( , = 0,05 ;
= 0,01 )
- .
1- 2-
(
) . . .. . . .-
; . . .-.
H 0 H1
, . . .-
.
.: [1] . ., , .
., 2 ., ., 1979; [2] . ., .:
, . 8, . 1988, . 12942; [3] . ., . ., . .,
, ., 1979; [4] N e y m a n J., P e a r s o n E., Phil. Trans. Roy. Soc. London., Ser. A,
1933, v. 231, p. 289337.
267
## ; minimal excessive majorant , ( )
( yi X i ) 2 = min
i =1
, X i ,
;
weighted least squares / generalized least squares
method , .
; least squares method -
X i - . S () 1 , ..., p -
;
. .
. . . . . ( C. Gauss, 1974 95 ) .
( A. Legendre, 1805 06 ) . .
( A. Cauchy, 1936 ) . . . . . . . ( 1958 ) . . . .-
. . ( K. Pearson, 1896 )
. . . .
. . . ( 1898 ) . . ( 1946 ) . .
. .-
. . ( A. Aitken, 1935 )
. .
( R. Fisher, 1922 25 )
. . . ( 1946 ), .
. ( J. Durbin, M. Kendall, 1951 ), . ( W. Kruskal, 1961 ) . . . .- ( )
. . ( S. Rao, 1962 )
. . ( A. Hoerl, 1962 ) . . . . . ( Ch. Stein, 1956 )
. .
.
. . . . .
, n Y = ( y1 ,..., y n )
Y = X + , X = ( xij ) p
( n p ) . , p n (
T
, A = X X
);
= (1 ,..., p ) T ,
;
n .
, ( 1 , ... , n )
. S () -
,
; S
= 0 , ,
( X T X ) = X TY
. , X
, , A
= X T X
= ( X T X ) 1 X TY .
Y = ( y1 , ... , yn ) T .
F ( x ) - . .,
F ( x ) -
y1 , ... , y n ,
, ( ,
, );
p
: 1)
j -
j =1
j =1
j ,
## j = 1, ... , p ; 2) det ( cov )
= min
,
); 3) cov
= min ,
. . . .
;
4) tr (cov ) = M ( M ) T ( M ) = min ,
M = .
. . . .
. ,
2 = D 1 = Y X
( ),
S2 =
1
T . S 2 , 2 - .
n p
F ( x )
F ( x ) = P { 1 < x }, F ( x ) = 1 F ( x ) -
. . . .
. Y - .
. . . .-
S ( ) = Y X
268
= ( Y X )T ( Y X ) =
~ N p ( , 2 ( X T X ) 1 , n
1
( X T X )1 2 ( ) ~ N ( 0 , 1) , n .
,
.
2
i , i N ( 0 , )
: 1) . . . .
,
( T , 2 ) = ( 1 , ..., p , 2 )
N n ( 0 , ) , ,
n - .
2
, = 0
( , ) -
, W
; 2)
S 2
; 3)
, . . . .
N p (, 2 ( X T X ) 1 n
, n . n
2
S
: F (x ) .
n
,
j = 1, ... , p
r ( j , S 2 ) 0
, ,
.
:
, ( k p ) . G
G H
= G
. . . .
,
.
, , X
, ( , , ).
P
= 2 ( X T X ) 1 - ,
A = X T X (
= H = ( h1 , ..., hk ) T
. H . . . .
h1 , ... , hk
h1 , ..., hk
) ]
i j . , h h
j
, hi - i j hi
= 0
. , , . . . .- , ,
.
; ( n n ) . W
:
( Y X ) T W ( Y X ) = min .
, ,
cov =
1
. W =
, . . . .
. .,
n
~ N p ( , ( X T 1 X ) 1 )
1
T 01 [
np
.
n
yi = + i ,
i = 1, ... , n , 1 , ..., n
( ,
,
, M i = 0 ; M i 0
,
).
D i = 2 d i , d i , 2
. Y
= 1 + -
= ( y1 , ..., yn ) T ,
( 1 n ) . 1 = (1,...,1) -
, Y
= ( 1 , ..., n ) T . . . . .-
S ( ) =
w (y )
i
= min , w i = di1
i =1
, :
= w 1
= 0 ; i , j = 1, ..., k ,
S2 =
( , S 2 ) ( , 2 ) . . . . .
: M ( hi hi ) ( h j h j )
= Y X ( -
).
. G
= 01
w =
w i yi ,
i =1
i =1
2 :
S2 =
1
n 1
w i ( yi ) 2
i =1
n - 1 , ..., n
~ N ( , 2 w )
, -
.
,
i N ( 0 , 2 d i ) , S 2 ( ) S , ,
( n 1) S 2 2 (n 1)
2 , ( ) S
( n 1) - t
.
269
1. 20
. ( ) ( ni
yi - ,
= n = 20 ).
D =
ij xi x j
i , j =1
40
41
42
43
44
46
ni
=
.
X = 1 = ( 1,...,1) , = diag
n y n
= 41,50
= 0,95
19-
t = 2,09 ,
, = 41,50
t S = t
ni ( yi ) 2
( 12 , ..., n2 )
1,
(1) -
i =1
yi i2
i2 ;
i = 1
= 2
1
= y =
n
i =1
. . .
. . , ,
( ) X X ( )
=
= const p
( , ). Q (c ) =
T
n ( n 1) = 0,80
i =1
= 0,95
42,30
40,70
. ,
.
D N ( 0, 1)
.
x (t )
, -
, w i = ni , ,
W = diag ( n1 ,..., nk )
n : ( )
39
. ,
i=n
yi
2. y (t ) = x ( t ) + ( t )
, x ( t )
= { R p :( ) T X T X ( ) c }, c > 0 . 2
i , i N ( 0 , )
c > 0 :
,
h ( s , t )
(t ) , x (t )
P { Q ( c 2 ) } = P { 2p c },
2
. y (t ), t1 , ..., t n
p p -
. -
.
Y = X + , Y =
; 2
2
S - :
P { Q ( cS 2 ) } = P { Fp , n p c },
## = ( y ( t1 ) ,..., y ( tn ))T , X = ( x ( t1 ) , ..., x ( tn ) ) T , = ( ( t1 ) , ...,
( tn ) ) T , 0
= ( ij ) , ij = k ( ti , t j ) .
. . . .- , S ( )
= ( Y X )T
1 ( Y X ) = min = ( X T 1 X ) 1
T
X Y
ij
. -
,
,
n
n
ij xi y j
ij xi x j
=
i , j = 1
i , j =1
270
(1)
F p ,n p p, n p F
.
3.
a1 , ... , a n
, Ai -
yi - : yi = Ai ai .
, M yi
= 0
. ,
M yi = + ai = ( + a ) + ( ai a ),
a =
1
n
a
i =1
, :
Y = X + , = ( 1 , 2 ) T , 1 = + a ,
1
a1 a
X =
2 = ,
1
1
...
a2 a ... an a
i ( 0 , ) ,
. 1 , 2
X T X .
:
2 =
i =1
1
1 = y =
n
y i ( ai a )
yi ,
i =1
( ai a ) 2
i =1
D 2 = 2
(a a )
i
P ( k +1) = [ I K ( k +1) X k +1 ] P ( k )
2
X k , X k - ; K
P { Q ( C 2 ) } = P { 22 c } = 1 exp ( c 2 ) .
2
:
[ y
2 ( ai a )]2
Q (c ) = { R 2 : n ( ) 2 + 2n a ( ) ( ) +
+ ( ) 2
ai2 c }
. . . . A
jj
= X T 1Y
cov = P = A1 .
1
A = X X
, ,
.
T
( ti ) + i , i = 1, ..., n
, 1 , ..., p
t ( t , t1 , ..., t n ). X
= ( i ( ti )) -
. , ,
, i (t ) - ( ., 1 ( t ) = 1,
2 ( t ) = t , 3 ( t ) = t 2 , ... ). , j ( t ) [ ., j ( t )
i =1
. . . .-
, ( p 1) .
j =1
= P { F2, n 2 c } .
, , ( , )
(k )
( , )
.
. . . . ( . , )
. . . .-
yi =
i =1
P { Q (c S )}
= diag ( 12 , ... , n2 )
K ( k + 1) = P ( k ) X kT+1 [ X k +1 Pk X kT+1 + k2 ]1 ;
i =1
1
S2 =
n2
P -
. , =
( k + 1) = ( k ) + K ( k +1) [ y k +1 X k +1 ( k ) ];
( 1 - 0,01
).
.
( , ).
. . . .-
.
. A
-
( ) 2
( 2 ) 2
Q (c ) = R 2 : 1 1 + 2
c ,
D1
D 2
D 1 =
, -
. A
A = A + I ,
(k )
1 2 :
, A - -
(0)
( 0)
(k )
.
P
1 2 , ,
, . -
= cos ( j 1) t , j = 1, ..., p ].
, , j ( t ) ,
.
,
j ( t ) , j = 1, ... , p -
271
. p
,
( , )
, .,
.
4.
, y
2
( m ) , ( km / s ).
= a 2 + b y + c
. , :
10
[ y a
S ( a , b, c ) =
2
i
. Y
,
(2) .
( )
.
5. t
= exp ( t )
. , -
, ln ln -
( )
. . . . .-
,
,
D ( ln ) 2 D . ,
bi c ]2 = min
ln i w i
i =1
yi
4,28
3,53
3,16
2,98
2,60
2,23
2,05
1,67
1,49
1,12
88,7
98,0
100,0
102,8
106,7
115,8
118,9
130,2
138,9
146,3
f i S = 0
## = 0,0007; b = 0,205 ; c = 16,84
(0)
,
-
, X
G2 ( ) = 0 + 1 + 2 2 }
( 0 ) = 0 , j , k = 0 , 1, 2 , ..., j k ,
i =1
,
. ,
G0 , G1 , G2 .
. . . .
. n Y
F (0)
Y F ( ( 0) ) =
( ) X +
= ( y1 , ..., yn )T -
yi = f i ( 1 , ..., p ) + i , i = 1, ..., n
, i
p
n
, f i F : R R
( ,
). . . . .- j -
j - ,
:
2
i =1
272
yi f i ( ( 0) ) =
j =1
f i (0)
( ) X j + i , i = 1, ..., n ,
j
= ( 1 , ..., n ) T
(3)
X =
= ( X 1 , ..., X p ) T -
. . . .-
S ( ) = Y F ( )
= F ( ( 0) ) X + . ,
,
X :
,
:
10
= (0) (2)
X X 1 , ..., X p -
{ G0 ( ) = 1 ; G1 ( ) = a0 + a1 ;
G ( ) G
.
.
.
-
. {1, , }
= i2
[ yi f i ( 1 , ..., p ) ]2 = min .
(1) = ( 0) + X . ,
D i
~
p
(n)
, : n
,
~
(0)
.
,
1 2
.
(3) ,
( ) ( ,
).
,
,
.
.: [1] . ., - ,
2 ., ., 1962; [2] S e a l H. L., Biometrika, 1967, v. 54, 12,
p. 124; [3] . ., , ., 1981; [4] . .,
, . ., ., 1968; [5] ., ,
, . ., ., 1977;
[6] , ., 1975; [7] A r n o l d S. F., The theory of linear models and multivariate analysis,
N. Y., 1981; [8] . ., ,
., 1989; [9] . ., ,
.: , . 3, ., 1982.
## ; least squars method
w i t h e s t i m a t e d c o v a r i a n c e m a t r i x
( ) .
Y = X +
= ( 1 , ..., n ) T
M = 0 , cov =
. ,
4, p. 71323; [5] . ., . . -
, 1968, . 104, . 189214; [6] . ., .
. . , 1979, . 85, . 13757; [7] . ., . , 1979, . 247, 3, . 56569;
[8] . ., , .,
1981; [9] . . ., . ., ., 1978.
## ; constrained least squares
method . ( ) , , ., ,
.
, n Y
. . . .
( ) = ( X X ) X Y
T
Y = X +
(1)
. ,
,
1
T 1
(2)
( ) -
( , [1] ). (2)
(1)
. ,
( , [2] [5] ).
. . . .
( , [6] ) (2) .
(
)
( k + 1) = [ X T ( ( k ) ) 1 X ]1 X T ( ( k ) ) 1Y
,
(k )
(k )
= ( 1 , ..., n )T
, M i
= 0 , D i = 2 ,
. ,
.
(
[1]- ; , [7] ). ( ,
[8] ).
(2)
-
;
( , ., [9] ).
.: [1] Z e l l n e r A., J. Amer. Statist. Assoc., 1962, v. 57,
p. 34868; [2] J a m e s G. S., Biometrika, 1951, v. 38, p. 32429;
[3] R a o C. R., Proc. 5-th Berk. Symp. Math. Stat. Probab., 1967,
v. 1, p. 35572; [4] H a n n a n E. J., Econometrica, 1976, v. 44,
R p
= ( 1 , ... , p )
.
,
= ( 1 , ... , p ) T .
. ,
H = b
(2)
, H , m p , ( m p )
. , b
= ( b1 , ..., bm ) T .
p
, R -
(2) , m .
, (2) , -
= (1 , ... , p ) T
= + ( X T X ) 1 H T S ( b H )
= Y X ( k )
(1)
, X , p n , (n p )
i = 1, ... , n ,
( ) = ( X X ) X Y
T 1
= ( y1 , ..., yn )T
(3)
S = [ H ( X T X ) 1 H T ]1
= ( X T X ) 1 X TY , . . . .
( ) ( , [1] [3] ).
(2)
.
cov = V [ I H T SH ( X T X ) 1 ] ,
V
= 2 ( X T X ) 1 = cov . 2
273
s2 =
1
( Y X ) T ( Y X )
n p + m
.
( X H )
[5]- .
.
H b
,
( H )i bi , i = 1, ... , m .
(4)
(4) , -
. . . . - , ,
( H )i = bi , (4)
. -
,
( , [5] ).
M [ ( ) T ( ) ] =
j =1
D j +
[ M (
j ) ]2
j =1
, (4) -
. . . .
( , [5] ).
( , [6], [7] ).
(4) ,
( , [8], [9] ).
.: [1] C h i p m a n J. S., R a o M. M., Econometrika, 1964,
v. 32, 12, p. 198209; [2] L i e w C. K., J. Amer. Statist. Assoc., 1976, v. 71, 355, p. 74651; [3] . ., , . ., .. 1968;
[4] . ., , .,
1981; [5] ., ,
, . ., ., 1977; [6] .,
, . ., ., 1979; [7] .,
., , . ., ., 1986; [8] . , 1982; [9] . ., , ., 1989.
; simplest point process
, .
; most selective confidence set , .
; nearest neighbour algorithm ;
274
; nearest mutual ancestor ,
. . . . . . . . . .-
[1], [2]- .
.: [1] F l e i s c h m a n n K., P r e h n U., Math. Nachr., 1974,
Bd 64, S. 35762; [2] . ., .
., 1975, . 20, . 3, . 61423; [3] . ., . ., .: . .
. , . 23, .,
1985, . 367.
;
best linear unbiased estimator, BLUE ,
; best linear prediction ,
;
;
noise immunity
.
,
.,
. .- / ( )
.
.: [1] ., . .,
, . ., ., 1982; [2] ., .,
, . ., ., 1969; [3] . .,
, 2 ., ., 1965.
; robust estimator
, .
; nuisance parameter
,
. .,
. .-;
. .-. . .- ; ,
. . .-
, .,
yi = a + i , i = 1, ..., n a
1 , ..., n . . ( ) .
. .-
. -
( , ., ) ; ,
, . .
( ) .
. .- (
) .
. .- ,
. .- , . .- . .-
( , ., C ( ) ).
[5] B a s u
p. 113.
. . . [1]- .
.: [1] H o t e l l i n g H., Ann. Math. Statist., 1940, v. 11, 3,
p. 27183; [2] . ., , ., 1966; [3] . .,
. , ., 1982, . 21732; [4] B a s u D., J. Amer. Statist. Assoc., 1977, v. 72, 358, p. 35566;
## D., J. Statist. Plann. and Inference, 1978, v. 2, 1,
/ / ; key
renewal theorem , .
; null
hypothesis , .
/ /
/
; r a n d o m permutation substision ,
; stochastic
curve integral ,
; factor
, -
; factor
analysis ,
x = + +
(1)
, x
( p 1) , = M ( x ) ,
M [( x ) ( x ) T ] = ,
( p m )
, ,
xT =
1
N
xi S =
i =1
1
N 1
( xi x ) ( xi x ) T
i =1
~~
= T + S
~
~
, ( p m) , ( p p )
],
( S ; S ) - . -
~2
12
: [ tr ( S S )]
~ 1
tr ( SS
~
) ln | SS 1 | p .
~
) = inf ( S ;
(S;
)
( p m) x ( m << p ) , M ( T ) = 0 , M ( T ) = I
A p
( , , ,
), ( p 1) (
: S p A p
, A p ( p p ) ,
),
M ( ) = 0 , M ( T ) = 0 , M ( T ) = , . . .- (1)
, S p ( p p )
( S, ) =
f ( gi ) ,
i = k +1
= T + .
(2)
, x
-
. m > 1
[ (1 2 ) m ( m 1)] ,
, , (1)- -
= T , - = T T
(2) . ,
.
( , [1] )
. . .
. .
( ),
,
,
, .
x1 , x 2 , ..., x n
.
,
276
k + pm
f ( x) ,
, x = 1
~
H = S H G,
H TS H = I
(3)
, G g1 ... g p ( p p ) , H
,
f ( g k ) > f ( g k + p m )
= S H (I G )
1
1
12
, G1
## g1 ... g k g k + p m+1 ... g p
(m m) , H1 (3)
~
( p m)
. -
( x - )
N -
). 2 m > p . . .
(
) ( , [4], [5] ).
, - (3)
. , -
~
, p -
.
- . x ~ N p ( , )
-
k + p m
( g i1 + ln g i 1)
y = ( y1 , ..., y n ) ,
. , y = c + s + l , c = F x ,
M y = M s = M l = M c. n r F
s , l - , M x s T = 0,
M x l T = 0, M s l T = 0 . y = t + l , y = c + u
= s + c, u = s + l;
t , u
, t
.
,
C = M y y , Cc = M c c = F F ,
1
2
k + p m
Ct = M t t ,
S 2 = M s s ,
E2 = M l l ,
U 2 = M u u = S 2 + E 2 ;
(1 g i ) 2
i = k +1
x 0
V = diag C ,
0 -
T S 1
) - .
T S 1 ( x0 x ) (1
. ., , , .
, ,
, , . . . , , ,
. ( , .,
[1] [3] ).
.: [1] ., , .
., ., 1972; [2] ., , .
., ., 1974; [3] ., , .
., ., 1980; [4] A n d e r s o n T. W., R u b i n H., Proc. 3-rd
Berk. Symp. Math. Statist. and Probab., 1956, v. 5, p. 11150;
[5] T u m u r a Y., S a t o M., TRU Math., 1980, v. 16, 2,
p. 12131; [6] . ., .:
,
., 1980, . 20832.
;
; ;
i d e n t i f i c a t i o n o f a m o d e l o f factor analysis
-
= + -
, ( p m)
,
m T .
, .
## ; ; ; factor analysis models
, ,
( ), ( ), ( ),
( ).
.
H 2 = diag Cc ,
T 2 = diag Ct = H 2 + S 2 ,
= 0 , M x x = I .
x ,
i = k +1
( .
). , M x
C0 = C V .
D
.
. . w = (w1 , ..., w n )
= (w Cc w ) ( w D 2 w )
. .
D2 = V 2 ,
D 2 = U 2 , D 2 = H 2 , ,
, , .
c = ( w C w ) ( w V 2w )
.
. ,
,
, .
.: [1] M c D o n a l d P. P., R o d e r i c h P., Brit. J. Math
and Statist. Psychol., 1970, v. 23, 1, p. 121.
;
factor set , .
; factor experiment
.
,
( , , . .-
,
,
277
).
( , ),
, , ,
, ( ,
.,
).
. i - si , li1 , ..., lis
( , ) , Fi
lik ,
, i -
k - ;
. . .- ;
( , ).
. .- . ( ) . . .
. . .-
. . .- .
,
,
. . .-
( , .,
), ,
,
( , )
.
.
:
yijk = + x1i + x2 j + ijk ,
yijk , k - ,
x1i
x2 j
, ijk , , , ,
2
,
.
, . n
:
y = + 1 x1 + ... + n xn + .
y -
.
:
## yijk = + x1i + x2 j + x1i x2 j + ijk .
278
. , , , . .- .
. . . .-
. n
n
2
( ). . .
, ( ) . . . ( , , [1] [3] ),
.
. .- . ,
. .-
(
( .
), .
.
, . . ,
, ,
.
,
. . .-
. ., n
n
2 - ( n + 1) -
( , 2
2
n p
n p
. .- ,
n +1
). -
. ,
; ,
,
.
. .- - ( ., , ,
),
. .-
. ., ,
,
.
( )
-
( , ,
). . .-
. A ,
, B A
.
. .-
( ,
, ,
).
. .
. . , , xi ni
= N
## . i ( j ) A = { i1 , ..., ir } A ( j1 , ..., j r ) , x i j - xi1 , ..., xir
j1 , ..., j r -
A ( j1 , ..., jr ) =
k =1
ik
( jk ) , . .
, .
. .-
- . s k ( N , k , s, t )
. r
, 2r + 1
, . r
= 1
.
( ,
):
, ( - ) (
. ,
.
,
.
( ., ) .
.: [1] . ., , ., 1976; [2] , ., 1983, . 1014; [3] . .,
. .,
, ., 1987; [4] R a g h a v a r a o D., Constructions and
Combinatorial problems in design of experiments, N. Y. [a. o.], 1971;
[5] R a k t o e B. L., H e d a y a t A., F e d e r e r W. T., Factorial
designs, N. Y. [a. o.], 1981.
; factor
model , .
; factorial moment
. , X m ( m 1 )
[m ]
= X ( X 1) ... ( X m + 1)
) D
. . , .
( , ,
).
,
. ., 0 , 1 2 EG ( 4 , 3 )
( , )
. . .- ,
. .- . ., . .
- ,
DX = M X [ 2] +
## + M X [1] (M X [1] ) 2 . . .-, ,
X ( z ) -
: M X
[m ]
= [m ] (1),
x1 + 2 x2 + 2 x3 = 0 ( mod 3 )
( m ) ( z ) , ( z ) m - . -
. .
.
, ; .
.: [1] . ., .,
, . ., ., 1966.
x1 + x2 + 2 x3 = 0 ( mod 3 )
D = {xui } =
x1
x 2 x3
x4
0
1
2
0
1
2
0
1
2
0 0 0
0 1
1
0 2
2
1 1
2
,
1 2
0
1 0 1
2 2 1
2 0 2
2 1 0
## ; rotation of factorial axes
u = 1, ..., 9
i = 1, ..., 4.
xu 3 xu 4 , 3
xu1 + 1 xu 2 + 1
, ,
:
0 1 2
1 2 0
2 0 1
0 2 1
1 0 2.
2 1 0
( p m) - ( ,
; ) m T
; ( p
m m )
( )
, 1
; T , ( m 1)
(
(m 1) ). T
, . . . ,
. . . .
, . , . [3], [4]-
279
.
.: [1] ., , .
., ., 1972; [2] ., , . .,
., 1980; [3] A r c h e r C. O., J e n n r i c h R. I., Psychometrika,
1973, v. 38, 4, p. 58192; [4] J e n n r i c h R. I., Psychometrika,
1973, v. 38, 4, p. 593604.
( t ) = min { k : S k t } , ( t ) = S (t ) t ,
. n t < 0
.
. .,
Yn =
; factorial semiinvariant / cumulant
. ( z ) . ln (1 + z ) Z
m! - m
. . .
( , ).
.: [1] . ., ., , . ., ., 1966.
; factorization identites
(
)
.
1 , 2 , ...
,
f ( ) = M e i 1 , S 0 = 0,
Sn =
j =1
.-
| z | 1,
( Im = 0 )
1 z f ( )
1 z f ( ) = Az+ ( ) Az ( )
(1)
,
Az ( ) , Im > 0
1 z f ( )
. (1)
Az ( ) = exp
e ix dx
n =1
zn
P { S n < x } , | z | < 1
n
,
P { S n = 0} = 0 , n = 1, 2 , ... .
; n
## = const > 0 , t = const > 0 .
S n
[ Az ( ) ] .
( , [2] ):
P {Yn d x, n = k }
280
0+ = 0 = 0 ,
P { ( t ) dy , ( t ) = l } .
(2)
. .- . (2) ,
.
.
Az , , , ,
; ., t 0
n ( t ) t
n ( t ) ( S k t k , 1 k n
)
n t .
n ( t )
n ( t )
Az
.
n ( + ) n -
.
. | z |
< 1,
< 1 | z | , Im 0
(1 z )
zn M (
exp ( i Yn ) ) =
n=0
Az + (0)
Az + ( )
. .- :
1.
n=0
0k n
n+ = max { k n : S k = Yn } ,
(t ) = S (t ) t .
.
( )
Az+ ( )
Y0 = 0 , Yn = max S k , Y = Y ,
n = min { k n : S k = Yn } ,
(t ) = min { k 1 : S k t } ,
(n , t )
( 0, 0 ) , (1, S1 ) , (2 , S 2 ) ,
n = min { k n : S k = Y n}
min S k ,
0k n
z n M ( exp ( i Yn ) ; n = n ) =
1
.
Az + ( )
z n M ( n exp ( iYn ) )
n=0
2.
z n M ( n exp ( iY n ) ) =
(1 z ) (1 z f ( ))
n = 0
(3)
3. = 1 (3)-
Az+ ( 0 )
z n M exp ( iYn ) =
(1 z ) Az+ ( )
n=0
1. | z | 1 , Im 0
1 M ( z ( 0) e i ( 0) ; ( 0 ) < ) = Az + ( ) .
M ( z ( + 0) ; Y > 0 ) = 1 exp
M ( 0 ) =
n =1
P { Y = 0}
zn
P { S n > 0} ;
n
P{ Y < } = 1 , P {Y = 0} = 0 ,
n =1
6. M exp ( i Y ) =
P { S n > 0}
< .
n
3. , ( ( ) = f ( i ) -
x > 0
M ( z ( x ) e ( z ) ( x ) ; ( x) < ) = e ( z ) x .
M ( z ( 0) e ( z ) ( 0) ; (0) < ) = 1 .
P { n = n } = P { Yn = 0} =
P {1 = x } = c p x1 ( c > 0, p 0, x = 1, 2 , ...) .
P{ ( x ) = n} =
P { n = k } = P { k = k } P {Ynk > 0} .
n 1 ( 1 2 P { S n > 0} ) . | r | <
n =1
<
k - ),
n
( + 0)
< =
lim P < = lim P n
arcsin
n
n
n n
. | z |
< 1,
x
P{ Sn = x } +
n
p x
x 1
P{ S n = x}
P { S n = x 1} ,
+
1 p n
n
f ( n , x )
Az + ( )
d
P { S n < x } , S n - . p = 0
dx
( 1 1) ,
M ( e
(4)
,
x = 0
M ( z ( + 0 ) e i ( + 0 ) ; ( + 0 ) < )
i 1
; 1
< 0}
x
P{ S n = x} .
n
M ( e
i 1
; 1
> 0}
, 1 z f ( )
Az + ( )
P { ( x ) = n} =
e i x d x M ( z ( x ) e i ( x ) , ( x ) < ) =
x
1 d x
f (n, x) +
f ( n , x) ,
dx n
n
P{ ( x ) = n} =
Im 0 , Im > 0
## ( c > 0, > 0, x > 0) ;
) 1 -
( 2 n)!
;
22 n (n!) 2
( , M1 = 0 , D1
4. ( x ) (x ) :
a) P {1 x } = ce x
P {Yn = 0}
;
2n 1
( ).
2.
.,
( , Y ) -
= er
Im = 0
.
7. k ,
M ( 0 )
< + )
( + ) > 1
(z) , ( ) = 1 z ,
,
0 < z < 1 inf ( ) .
A1+ (0)
,
A + ( )
P{ ( 0 ) = n } =
2. M 1 = 0 , D 1 = 2 < , M ( 0 ) -
0 < Re
5. :
8. r =
(4) -
. .- :
z n
= exp
M exp ( i max ( 0, S n ) ) .
n = 1 n
4.
Az+ ( ) Az+ ( ) .
( , [2] ). 1 e
281
( , [8] ):
Q0 ( z , ) =
z n M ( e i Sn ; N > n ) ,
( s ) = M e i t 1 , ( s ) = ln M e i s (1) , Im s = 0 ,
) 1 z ( s ) = + ( s, z ) ( s, z ) 0 ( s, z ) ,
n=0
Q1 ( z , ) = M ( z N e i S N ; S N a ) ,
Q2 ( z , ) = M ( z N e i S N ; S N b ) ,
N
## = min { n : S n ( a, b ) }, a > 0 , b > 0 .
. . . , , . ( ) .
( , , ) .
.
( . ) . . ,
.
. . ( , [3] ) . ( , [3], [4] ).
. .- ,
. . . .
( t ) ( , [5], [6] ).
z ( z ( ) ) ,
( ) = ln M exp ( (1) ) .
( , [7] ).
.: [1] .,
, . ., . 2, ., 1984; [2] . .,
, .,
1972; [3] . ., . . ., 1962, . 3, 5,
. 64594; [4] . ., . . ., 1969, . 10, 6,
. 133463; [5] . ., . .,
1966, . 11, . 4, . 65670; [6] . ., .
., 1979, . 24, . 3, . 47585, . 4, . 873-79; [7]
. ., . . . ., 1969, . 33, 4,
. 861900; [8] K e m p e r m a n
J. H. B., Ann. Math. Statist.,
1963, v. 34, 4, p. 116893.
; factorization method
.
k , k 1
Sn =
k =1
( t ) . . .
282
( s, z ) = exp m
zk
P{ S k < x } ,
k
e isx dx
zk
P { S k = 0}
k
0 ( s, z ) = exp
k =1
k =1
) [ ( s ) ]1 = f + ( s, ) f ( s, ) ,
f + ( s, ) =
= exp ( e i s x 1) d x ( e t t 1 P { ( t ) > x} d t ,
0
f ( s, ) = exp ( e i s x 1) dx
(e
0
t P { ( t ) x } dt .
t 1
S n ( t ) . .,
S n+ = sup S m , + (t ) =
0m n
= sup ( t ) ,
0u t
M e i s Sn = + ( 0, z ) (1 z ) + ( s, z )
n=0
M ei s
(t )
dt = f + ( s, )
.
.: [1]
. ., . .,
1966, . 11, . 4, . 65670; [2] . .,
, ., 1972; [3] , 2 .,
., 1985.
; factorization theorem
, .
; interaction
of factors D - . . ( ,
)
(
) . N D F1 , ..., Fr ( r 1)
( r )
z = { z1 , ..., z N }T R N ,
N
= 0,
i= 1
z F1 , ..., Fr
z i2
0.
i =1
( r 2 ) - ,
( r 2 ) -
z - D
F1 , ..., Fr
.
( . ) .
, .- .-
.
( ) ; factor
quotient o f a d y n a m i c a l s y s t e m , , .
; Fano
algorithm , ( ).
;
Fano inquality
,
. X , Y N ( ) .
= P { X Y } , ( ) H ( X | Y )
H ( X | Y ) log 2 ( N 1) + h ( ) ,
h ( )
= log 2 (1 ) log 2 (1 ) .
. . .
.: [1] ., ,
. ., ., 1974; [2] . ., .
., , ., 1982.
; utility function . : M
, f , M - .
M - U ,
a f b U ( a ) U (b)
(1)
, (1)- U
.
U , V { M , f } ( , ), g ( x )
, U ( a )
= g (V ( a ) ) .
{ X , A }
(1) , M - P
U (P) =
u ( x ) P( dx )
(2)
( u , X - ), u
, , . . (2) ( , ).
. . .
( , [1] ): ,
- . . .
. . .
.: [1] B e r n o u l l i D., Specimen theoriae novae de mensura
sortis, Comment. acad. sci. imper. Petrop., 1738, t. 5, p. 17592
( . .: Econometrica, 1954, v. 22, 1 ); [2] .,
., ,
. ., ., 1970; [3] .,
, . ., ., 1978.
; utility theory ,
.
, ,
. .
,
( ,
. ),
. ,
.
. .- .
( , [1] ).
( . ) -
( , 4 ). . .- .
. ( , [2] ).
1. .
M -
T M 2 . ( a , b ) T , a f b
( a , b - ) ( a, b ) T , b
fa
b , a - ). f
( a , b -
,
( a f b a f b ) ( a
f b, b f c a f c )
. a f b b f a , a ~ b
( a , b - ). a f b ( a f b
a ~ b ). f f
. { M , f }
. ~ M
M - f f
.
2. . M - U ,
a f b U ( a ) U (b)
,
. U .
283
U , V , { M ,
f}
, g ( x ) , U ( a )
H M , a f b
a, b M a
f h f b h H
, f , . 1: M - f ,
, , M - f
, .
.
## { ( x, y ) f 0 ( x1 , y1 ) x > x1 x = x1 y > y1}
,
{ ( x , y ) }
,
0
, f
,
.
3. . M , [0, 1] a , b M
a + (1 ) b M
1 a + 0 b = a,
a + (1 ) b = (1 ) b + a
[ 0, 1]
( a + ( 1 ) b ) + (1 ) b = a + ( 1 ) b .
. .- 2-:
:
) [0, 1]
c M
f b a + (1 ) c f ab + (1 ) c ;
2 0 ( ) , [ 0, 1]
( f b, b f c ) a + (1 ) c f b, b f a + (1 ) c ;
f -
, U , [ 0, 1]
U ( a + (1 ) b ) = U ( a ) + (1 ) U ( b ) .
(1)
4. . { X , A } , A M
=P ,
X - . Px , x
,
x
1 2
= g (V ( a ) ) .
10 (
. 2- 3
; P , f
f y Px f Py
f X -
. P
, P
284
{ P ( A ) = 1, y p x x A} Py p P
,
U (P) =
u ( x ) P ( dx )
, u , X -
.
, u
, U
(
, . 5, 7, 8 ).
2 3 . .-
.
, ,
. , ,
.
,
,
.
.
.
., ,
(1)
0
1 2
, , (1)
: a M -
; (1)
a - , (1 ) b -
.
.
5. .
. F F
; f , F - , U
1 : U
.
2 : A n
Fn ( A ) F ( A ) , U ( Fn ) U ( F )
. .
4: F ,
F - U 2 ,
U (F ) =
u ( x ) F ( dx) .
(2)
1 , (2)-
u .
U
, (2)- u
. .,
, U
F ( x ) G ( x ) dx , -
, (2)- u
( ).
6. . F
. , .
F ( x ) , F . :
B1 ( 1 - ):
F , G F x -
F ( x) G( x)
, F
(3)
fG;
B2 ( ):
F , G F m x m F ( x ) G ( x ) ,
F f G . [
F ( m x) = 1 F ( m + x + 0)
Q , U ( F ) = U ( G )
F , G
U (( F + Q ) 2 ) = U ( ( G + Q ) 2 )
. .
5. F , U
A 2, B 1 C ,
u ,
U (F ) =
U (Ex ) = x
( m,
) -
,
, .
,
. mF
F2
, F
, U ( F )
= f ( mF , F )
f ( x, y ) x > 0 ,
f ( x , y ) y < 0 ( B1 B2
(5)
( E x , x - ),
U ( F ) = u 1
( u ( x ) F ( dx ) ),
(6)
u , u - . F -
(4)- .
( ). F
, u ( x ) = exp{ c x }, c > 0
. (6) :
U (F ) =
1
ln
c
( e
cx
F ( dx ) .
, C, 1
U (F G ) = U (F ) + U (G )
. F m ,
m . B1 B2
(4)
A 1 , u .
,
x -
x > 0 - , F m ].
( u ( x ) F ( dx ) ) .
, . , .
, F G
,
( + ) 2
F o G ( ). o
. :
U ( F ) = U ( G ) , U ( F o G ) = U ( F ) .
. 6: F
o ,
F F - mF ; F -
). G F
, (3) , U ( F ) < U ( G ) .
U A 1, B 1 ; v
(2) , u
, F (2)-
, B1 , , ,
u , B2 , ,
, u
.
7. .
, ,
. C
:
, U ( F )
= ( mF ) . (5) , U ( F ) = mF . 6-
, F
, ., mF F -
. 6 :
F ,
U A1 ,
U ( F ) = const .
285
8. .
.
. ,
, ., n :
U (F) =
... u ( x , ..., x
1
## ) F ( dx1 ) ... F ( dxn ) .
(7)
(7)- .
, F -
n , j , j -
u ( x1 , ..., xn ) . (7)
(1 , ..., n ) . ,
u ( x1 , ..., xn ) = ( u ( x1 ) + ... + u ( xn ) ) n
(7) (2) , u ( z )
z .
X X (
, , 4 ) u ( x, y ) ;
x y - .
X -
x f y u ( x, y ) w ( y )
, w . ,
,
. ,
.
, w ( y ) = u ( y , y ) u ( x, y )
y - x - , u ( y , y )
y ,
.
( X , )
P ( , [4] )
P f Q
u ( x,
y ) P ( dx ) Q ( dy )
w ( y ) Q ( dy )
P f Q
u ( x, y ) P (dx) Q (dy)
u ( x, y) Q ( dx ) Q( dy )
P P y -
u ( x,
g ( y) =
y ) P ( dx ) . ., P = F
286
u ( x , y) F ( dx ) = 0
(8)
F .
F ( , 4 ) U (5), 1, 1 : U ( F ) = U ( G ) ,
U ( ( F + G ) 2 ) = U ( F ) ( )
. U ( F )
(8) , u ( x, y )
, x -
u ( x, x ) = 0.
.: [1] B e r n o u l l i D., Comment. acad. sci. imper. Petrop.,
1738, t. 5, p. 17592 ( . .: Econometrica, 1954, v. 22,
1 ); [2] ., .,
, . ., ., 1970; [3] D e b r e u G.,
Theory of value, N. Y. L., 1959; [4] ., , . ., ., 1978; [5] . ., . ., . .,
- , ., 1980; [6] Handbook of mathematical psychology, v. 13, N. Y.
L., 196365; [7] A l l a i s M., Econometrica, 1953, v. 21, 4, p.
50346; [8] . ., .: , ., 1983, . 181212.
; phase ,
,
,
,
, , < <
X ( t ) Y ( t )
## FXY ( ) = arg f XY ( ) = arc tg [ Im f XY ( ) Re f XY ( ) ]
, f XY ( ) , X ( t )
Y ( t ) . , FXY ( )
X ( t )
Y ( t )
.
, .
.: [1] ., .
, . ., ., 1980; [2] .,
., , . .,
. 2, ., 1972; [3] ., ,
. ., ., 1974.
; phase
diagram ( .,
)
.
.
w ( y ) 0 ,
( , 5 ).
;
state space o f a M a r k o v c h a i n o r a M a r k o v
p r o c e s s
( ).
;
state space o f a r a n d o m / s t o c h a s t i c p r o c e s s ,
(, A , P) ( t ) = ( t , ) , t T
( T )
, ( M , B ) , ( t , )
t - t T
( , A ) ( M , B )
.
.
. . , ,
, ( ., M , B
, M - ).
.: [1]
, 2 ., ., 1985.
; phase transition
.
. .-
. U
(1 , 2 ) . , X
A x A X
U A ( x A )
- -
. U
P , (1 , 2 ) ,
- (
- ) P
- 0 -
( 0
), U
( (1 , 2 ) ) U
. . , .
X . .
.
- .
, . .-
,
.
0 , U
0
,
U 0 - U . .
. . .
, ., . . . ,
. .-
. , . .
. .-
X Y ,
.
. .
, . . . ., . . .- ( ., )
.
,
, (
) . .-
. . . .
. .- .
. = ( ) , -
. .
.
.
, - . .- .
. .- ,
,
., .
. .- ,
( , ),
.- .
.: [1] . ., . .,
. , ., 1985;
[2] ., , . ., ., 1971; [3] ., . , . ., ., 1971; [4] . .,
, ., 1980; [5] L i g g e t t, T h o m a s M.,
Interacting particle system, N. Y. [a. o.], 1985.
;
phase modulation , ,
.
; phase density , .
; faza spectrum , .
; phase frequency characteristic , .
; phase shift / displacement , , .
/
; phase
modulated oscillation
X ( t ) = A sin ( 0 t + a ( t ) )
(*)
, 0 ,
a ( t )
0 - .
. . .
( , ., [1] [4] ); a ( t )
, . (*) a ( t )
287
( ; , [4], [5] ).
X ( t ) ,
( X ( t )
Feller solution
equation
lim M X ( t + ) X ( t ) = B0 ( )
f 0 ( )
( , )
a ( t )
( , ., [1] [8] ). ., a ( t )
,
M [ X ( t + ) X ( t ) ] 2 = D ( )
,
B0 ( ) = 0,5 A 2 e D ( )
cos 0 ,
f 0 ( ) = 0,25 A 2 {G ( 0 ) + G ( + 0 ) }
d t = ( t ) dWt + b ( t ) dt
G( ) =
2 } cos d .
a ( t ) D ( ) = 2 | |
( , [9] ), ,
A 2 2
8
b - ):
1) b ,
2) 2 ,
3) ( ) 2
.
1) (*)-
. 2) 3)
. . .
.: [1] . ., . . ., 1973,
. 37, 3, . 691708; [2] . ., . .,
. , 1979, . 245, 1, . 1820; [3] . ., . . . ., . 50, 2, . 21141.
; Feller transition function
1
1
+
.
( 0 ) 2 + 4 4
( + 0 ) 2 + 4 4
, a ( t )
( ,
).
.: [1] .,
, . ., . 2, ., 1962; [2] . .,
, . 1, 2 ., ., 1974;
[3] . ., ,
. ., . 1, ., 1972; [4] P a p o u l i s A., Probability random
variables and stochastic processes, 2 ed., N. Y. [a. o.], 1984; [5] P a p o u l i s A., IEEE Trans. Acoust., Speech, Signal Proc., 1983,
v. 31, 1, p. 96105; [6] P a p o u l i s A., Trans. 9-th Prague Conf.
Inform. Theory, Statist. Decis. Funct., Rand. Processes, 1983, V. B,
p. 10511; [7] . ., , 4 ., .,
1962; [8] Z a d e h L. A., Proc. IRE, 1951, v. 39, 4, p. 42528;
[9] W i e n e r N., W i n t n e r A., Nature, 1958, v. 181, 4608,
p. 56162.
; Fefferman inequality , .
288
; Feller Markov chain ,
exp{ D ( )
(*)
. . . (
, ,
f0 ( ) =
;
of a stochastic differential
( t , At , Px ) ,
x R d Px
- ). B0 ( )
;
Feller process (E , B )
, E , B
, p ( t , x, ) , t 0 , x E ,
B , , t 0
Pt f ( ) =
f ( y) p ( t , , dy)
Pt : f Pt f ,
t 0 ,
f E -
- .
. [1]- .
. .- . . . ( , ).
(
E ). ., E , ( E , B ) - p ( t , x , )
, E -
, t 0 p ( t , x, U ) 1 , U
x ,
-, ,
( ,
[2] ). . .
.
:
Pt , t 0 -
E -
( , [3] ). ., E
Pt
E - ,
## = 1 k T , k , k = 1,38 10 16 erq / grad ,
.
,
( ,
[2];
).
. .- , .
,
,
( , [4] ). .
.: [1] F e l l e r W., Ann. Math., 1952, v. 55, p. 468519;
[2] . ., , ., 1963;
[3] W a l s h J. B., Ann. Math. Statist., 1970, v. 41, p. 167283;
[4] . ., . ., 1981, . 26, . 3,
. 496509; [5] . ., . , 1982, . 263,
3, . 55458.
; Feller semigroup , .
; Fermi Dirac statistics
( 1 2 , 3 2 , ...) ,
## h = 1,05 10 27 erq.san . . ( E. Fermi ) 1926- . ( P. Dirac )
: . . .-
. . .
.- ( ) -
.
. . .- { n p } ,
n p , p ( n p
= 1)
( n p
= 0) .
, i
= Pi 2 2m Gi
Gi >> 1 . { N i } , N i n p - .
,
W ( Ni ) = i
Gi !
N i ! ( Gi N i )!
.
, E
N
i
N
i
ni = N i Gi = ( ( i ) + 1) 1 , ,
T .
.: , . - .,
, . 1, ., 1984.
;
fermion space , .
; Fernique
theorem , ;
/ ; c h a n n e l fading
,
(
) ( ). .
;
,
( ., ) . .
. .
. ,
( )
. (
), ,
. [ 0, 2 ] ,
. . . w ( x )
w ( x )
= ( x 2 ) exp{ x 2 2 } (
) w ( x )
= ( x 2 ) exp{ ( x 2 +
+ a 2 ) 2 } I 0 ( a 2 ) ( ) .
.-
,
, , .
.: [1] ., , . ., ., 1973.
; Feynman Kas formula ,
.
; difference pseudomoment , .
## , ; difference function of meteorologic
losses , .
; difference scale , .
; outliers
detection
289
( , [1], [2] )
, x o x ( ( t )) , i ( t ) t , , [ 0, 1] -
i ~ N ( 0, 2 )
yi = T f ( xi ) + i ,
i = 1, ..., N
.
yi , f ( ) -
T
ei* = | yi T f ( xi ) | i1 ( 1 f T ( xi ) M 1 f ( xi ) ) 1 2
M =
yi -
, D - ,
D -
; ( ,
[3], [4] ).
D
d 0 ( x, y ) = inf { > 0 : ,
( i , yi -
.
( D , d ) , .
f ( xi ) f ( xi ) ). . .
i =1
. . . . .-
, yi
, hi :
xo y
sup { | ln ( ( ( t ) ( s ) ) ( t s ) ) | : t s } }
.
D - [3]- . A D ,
, ,
A -
w x [ 0, ) , w x [1 , 1) , w x ( ) 0
x A -
hi = f T ( xi ) M 1 f ( xi )
w x (T ) = sup { | x ( s ) x ( t ) |: s, t T } ,
.
.: [1] . ., , ., 1987, . 224
42; [2] ., .,
, . ., 2 ., . 12, ., 1986; [3] A t k i n s o n A.
C., Plots, transformations and regression, Oxf., 1985.
w x ( ) = sup { min { | x ( t ) x ( t1 ) | , | x ( t 2 ) x ( t ) | } :
( ) 2
2 ; space o f r i g h t continuous functions with left limits / spac e o f C A D L A G f u n c t i o n s
, [ a, b) -
, ( a, b] - [ a, b ]
. , , 2-
.-
-
. .
2- .
2- .-
: , D = D [ 0, 1] ,
[ 0, 1) - , ( 0, 1] -
, [ 0, 1]
.
= sup { | x ( t ) | : t [ 0, 1] }
, D - C [ 0, 1] -.
D
d ( x, y ) = inf { x o y + i : }
290
t1 t t 2 , t 2 t1 , t1 , t , t 2 [ 0, 1] } .
D -
. . C [ 0, 1] ( )
. D,
.
B
( )
B
( , [5] ),
( , [6] ). , C ( )
.
2- .- [2]- .
.: [1] . ., . , 1955, . 104,
. 3, . 36467; [2] . ., .
., 1956, . 1, . 3, . 289319; [3] . .,
. ., 1956, . 1, . 2, . 23947 ( .:
, ., 1986, .
38493 ); [4] . ., . ., 1956, . 1, . 2, . 177238; [5] D u d l e y R. M., III. J.
Math., 1966, v. 10, 1, p. 10926; [6]
.,
, . ., ., 1977; [7] . ., . , 1972, . 27, . 1, . 341;
[8] . ., . .,
.
S S ; S space
, ,
; space
, .
,
; space with topology ,
.
, ; space of elementary events , .
, ( ) ; decision space ,
, .
## , ; space of continuous functions X
0 C ( X )
f1 , ..., f n > 0
:
f i ( x) d ( x)
f i ( x) d 0 ( x) < .
C ( X ) -
X ( , [2] ).
C ( X ) - ( G )
, , , (G) -
C ( X ) -;
(G) . C ( X ) -
.- .
X -
C ( X ) .
X - ( ) ,
, C ( X ) ( , [1], [2] ).
K , C ( K )
= sup { | x ( t ) | : t K }
K -
( , [3] ).
C ( K ) -
( C ( K ),
) -
.
K = [ 0, 1] . 2-
.
C ( K ) - .
C [ 0, 1] ,
, , ( , [4] ).
.: [1] ., , . ., .,
1975; [2] K e l l e y J. L., N a m i o k a I., Linear topological spaces,
N. Y. [a. o.], 1963; [3] . ., . ., , , 2 ., ., 1967; [4] . ., , . ., ., 1961; [5] , 2 ., 1972.
, ; space
of measure
. ( X , G ) , G
, C ( X ) , X - ,
f = sup f ( x )
x X
. ( X , G ) , C ( X ) -
C ( X ) G ( S ) -
. , C ( X ) -
( f ) =
f ( x ) d ( x )
( , [1] ).
C ( X ) - , -
,
*
M C ( X )
; M - .
. M
C ( X ) - . M - M - . ( X , G )
, M -
( n ) - , M ( , [1] ).
( X , G ) ,
M - ( P, Q)
.
- . P M
, P -
- .
X ,
M - -
.
.: [1] . ., . ., 1940, . 8, 2,
. 30748; 1941, . 9, 3, . 563628; 1943, . 13, 23, . 169
238; [2] ., ., .
, . ., . 1, ., 1962; [3] .
., . ., 1950, . 1, . 2, . 177238;
[4]
., ,
. ., ., 1977; [5] ., .
, . ., ., 1977.
, ;
state space o f a M a r k o v c h a i n ,
,
; state space o f a r a n d o m p r o c e s s
, .
## ; spatial statistical structure , .
; branching process ,
,
291
, ,
. ., ( ) i (t )
, . . .- ( t + 1) -
( t + 1)
:
( t + 1) = 1 ( t ) + ... + ( z (t )) ( t ) .
(n) = n
## = n + c , c = const > 0 ; ( n) = max ( 0, n C ) , C = const > 0
; ( n)
.
.: [1] . ., . .,
. ., 1974, . 19, . 1, . 1525; [2] . ., . ., 1974, . 19, . 2,
. 31939; [3] . ., . ., 1975,
. 20, . 2, . 43340; [4] . ., .
., 1977, . 22, . 3, . 48297.
; fiducial probability -
. . , , ,
, . ,
, (5)
. (5)
. . .
( A. P. Dempster, 1963 ), . . . ( D. A. S. Fraser,
1961 ), ( 1961 ) .
.: [1] . .,
, ., 1968; [2] F i s h e r R., Proc. Camb. Philos. Soc.,
1930, v. 26, p. 52835; [3] F r a z e r D., Biometrika, 1961, v. 48,
p. 26180; [4] . ., . , 1970, . 191,
4, . 76365.
; fiducial interval F ( | x ) = 1 F ( x | )
,
, x ,
F ( x | ) - . x x
(
) .
. . ( R. A. Fisher ) .
. T
, g ( T , ) , T -
F ( | x)
, - .
. . .- 1 ( x ) 2 ( x ) -
P { g (T , ) < } = F ( )
(1)
. 1
. .
inf { F ( 2 | x ) F ( 1 | x ) } = 1
x
g (T , ) <
> h (T , )
(2)
. (1)
P { > h ( T , ) ] = F ( )
. T0 , T
(3)
P { > h ( T0 , ) ] } = F ( )
(4)
- .
., N ( , 1) n
x . y
P x <
=
n
= ( x ) ~ N ( 0, 1 n )
Px -. -
F ( x | ) -
=
= 1 F ( x | ) ( x
*
, x - F ( | x )
= 5,832 ,
P > 5,832
= ( )
n
; fiducal distribution P = { P , } x - .
( , [1] ) , x
1
dy = ( ) .
N y 0,
n
. . F ( | x)
.
- , , .
,
.
.: [1] F i s h e r R., Proc. Camb. Philos. Soc., 1930, v. 26,
p. 52835; [2] . ., , ., 1973.
(5)
).
. . ( ,
[2] [4] ). g G - X -
. x - P g x -
292
Pg () ,
. : X D
( , ( g x )
= g ( x ) x g - )
L ( d )
( ,
Lg ( g d ) = L ( d ) , d g -
) . G , P :
0 P0
; filter phase
,
.
, .
( ) ; filtering o f s e m i m a r t i n g a l e , .
( )
; s i g n a l filtering
. . s (t )
x , g G , g x
P .
D , - [ B ()
s ( ) s ( t ) -
B ( X )
. s ( ) - x ( )
; , G
B B ()
].
G -
D - ( g ) ( B ) ( g ( B ) )
. . .
L ( ( x ) ) d P ( x )
P* =
= { Px* : x X } .
G X - , . .-
P * = { Px* : x X } S ( x )
P { S ( x ) } = Px* { S ( x ) }
-
[ S ( x ) - S ( x ) g G
g S ( g x) ].
. . . [1]- .
. ., 1, 2, 3, ... ,
n ,
1, 2, 3, ... , ..., 3, 2, 1
, ,
,
. t - t > 0 t -
.
( ,
; ).
.: [1] L e v y P., Ann. sci. cole norm. supr., 1951, t. 68,
p. 32781; [2] - , .
. ., ., 1964.
h ( t , ) x ( ) d
h ( t, ) M x ( ) x ( u ) d ,
u [ t 0 , t1 ]
t0
.
x( ) s ( )
, .-
. .
.-
( , [2] ).
.: [1] ., , , . ., . 1, ., 1972; [2] . ., . ., , ., 1974.
of
random
.
<
t1
M x (u ) s (t ) =
, h ( t , )
t0
## .: [1] F i s h e r R., Proc. Camb. Philos. Soc., 1930, v. 26,
p. 52835; [2] F l a s e r D., Biometrika, 1961, v. 48, p. 26180;
[3] . ., . , 1970, . 191, 4,
. 76365; [4] . ., , ., 1973.
..., 3, 2, 1 n
t1
s ( t ) =
.
.
u [ t 0 , t1 ]
M | s ( t ) s ( t ) |2 s ( t )
L ( ) d ( ) L ( ) d ( )
x (u ),
)
; filtering
p r o c e s s e s , ;
; filtering equation
for filtering density of a diffusion proc e s s
( t , x )
:
( t, x ) = ( t, x )
( t , x ) dx ,
( t , x ) 2-
. , ,
. ( t , x )
293
( t, x )
. . -
.
.: [1] . ., . ., , ., 1974; [2] . ., , ., 1983.
; filtering equation f o r d i f f u s i o n p r o c e s s e s
t [ f ] = M [ f ( ( t ) ) | Y ( s ), s t ]
; final probability o f a b r a n c h i n g p r o c e s s , ; .
; final reward
, .
; final type
i n a b r a n c h i n g p r o c e s s
( ), Y ( s ) ( -
( , ).
,
.
), f . t [ f ] -
; finitary flow
, (t )
,
. . . . .
[1]- .
.: [1] . ., . ., , ., 1974.
; gain of a filter , .
; filtered
Poisson process , .
; final
probability E
i j -
pij ( n ) - n
j E
i -
= 1 p j . . .- -
.
. ., , , E 1-
,
( , ; ).
. .-
- 1
; E
, .
p j 1
i j
.
( ).
E = {0, 1, 2 , ...} , p00 = 1 , pi ,i +1 = p , pi , i 1 = 1 p , 0 < p < 1 ,
; finiterange potential , ( ) .
; finitary
boundary , ;
.
; Fisk integral , .
; Fisk Stratonovich integral , .
; Fisher approximation ,
.
;
Fisher transformation
2
. n n -
. n ,
2
, { n } , 2
M n
x R, x :
2 n
2
P n
x = ( x ) 2 ( x )
3
2n
= {1, 2 , ...} .
1 / 2 , . .- p0 = 1 , p j = 0 ,
1
,
n
= ( x) + O
j 1 . p > 1 / 2 , i = 0, 1, 2 , ...
i
. .- .
, .
294
lim pij ( n ) = 0 ,
j 1
1
2n
2
1
1
+ ( 5 ) ( x ) + ( 3) ( x )
+ O 3 2 =
9
2n
n
, R
1 p
,
lim pi 0 ( n ) =
n
p
= n, D n2 = 2n -
. , n - ,
i = 1, 2 , ... , E = C U R , C = { 0}
p
, .
( x)
(1)
( x ) = ( x ), r ( x ) = ( r +1) ( x ).
.
, , n -
2 n2 -
2n 1 1 - .
, n x R , x
P 2 n2 2n 1
1
x = ( x ) 2 ( x )
1
2n
i , i
+
n
(2)
. . .
. , , > 0 .
m
P { m } = P { 22m + 2 2 }
; (2)
:
4 ( m + 1) 1 ] =
= [ 4 m + 3 2 ).
(3)
25
X =
1
n
( X 1 , Y1 ) ,
( X 2 , Y2 ) , , ( X n , Yn )
, , ,
:
P { X i < x, Yi < y } =
( x 1 ) 1
( x 2 ) 2
2 1 2
( u 2 2 u + 2 ) 2 (1 2 )
du d ,
1 = M X i , 12 = D X i , 2 = M Yi , 22 = D Yi ,
1
1 2
M ( X i 1 ) ( Yi 2 ) .
X i Yi
Xi , Y =
i =1
1
n
Y .
i
i =1
2 n 3
( 1 2 ) ( n1) 2 ( 1 r 2 ) ( n4) 2
(
n
2
)
n + m 1 ( 2 r )m
, | r | < 1,
pn ( r ) =
2
m!
m=0
|r|1
0,
. . [2]-
r - z
z =
1+ r
1
ln
= arg th r
1 r
2
, , , n - ; n -
z , :
Mz =
1+
1
+
ln
1
2 ( n 3)
2
Dz =
3 2
+ ...
1
4
(
n
3
)
1
1+
,
ln
2
1
1
2
1
2 6 2 + 3 4
+ ...
.
1
n 3
2( n 3 )
n
3
6 ( n 3)2
z H 0 : = 0
, , H 0
z z ; Fisher z transformation
( Yi Y ) 2
i =1
. [1]- , r n 3
4x +1 2
.
.: [1] . ., . .,
1963, . 8, . 2, . 12955; [2] . .,
, . ., ., 1977;
[3] G r e e n w o o d P. E., N i k u l i n M. S., A guide to chisquare
testing, N. Y., 1996.
. (3)
( X i X )2
i =1
= ( x) + O
( X i X ) ( Yi Y )
i =1
r =
1 1 ( 3)
1
1
( x ) ( 3) ( x )
+ O 3 2 =
6 12
n
2n
P { m } 1 [ 4
| | 1 .
z
.
.: [1] F i s h e r R. A., Biometrika, 1915, v. 10, p. 50721;
[2] F i s h e r R. A., Metron, 1921, v. 1, 4, p. 332;
[3] . ., . ., , 3 ., ., 1983; [4] . ., , . .,
., 1977.
; Fisher exact test , ,
.
295
F F ; Fisher F distribution , F , , ,
pab ( x ) =
1
( a b ) a 2 x b 21 [ 1 + ( a b) x ]( a +b )
B[ a 2 , b 2 ]
, ( 0 , ) -
, a, b
>0
F = 22 s12 12 s 22
1
= 2
m 1 n 1
. F .- ( . F .
).
, B ( a, b )
. a
x
>2
= ( a 2) b a (b + 2)
b > 2
. . F .-
b ( b 2) -, b > 4 2b ( a + b 2 ) a ( b 2 ) 2
2
( b 4 ) - . . F -. 2-
( VI ). . F .- a 2 b 2
X 1
X 2
= b X 1 a X 2 -
Fa ,b
. Fa ,b
Ba ,b ( x )
:
( a b) x
.
P { Fa ,b < x } = Ba 2, b 2
1 + (a b) x
. F .
.
a = m b = n ,
Fm n =
m2
n2
(*)
F m n
2
F , m n
, m n
.
. F .
2
. X 1 , ..., X m Y1 , ..., Yn , ( a1 , 1 ) ( a2 , 2 )
.
s12 =
1
m 1
(X
X) ,
1
n 1
(Y
Y)
s22 =
, 1
, X
b = 10; (3) a = b = 10 .
F F , ,
, , ,
.
. F .-
. m = 1 (*) Fmn - n - . . F .- .
. F .- .
( , [1] ) , , ,
z ,
1
ln F . z
2
. , . F . .
( G. Snedecor, 1937 ) .
. F .-
F z =
.
, , z .
.: [1] F i s h e r R., Proc. Intern. Math. Congr. Toronto, 1928,
v. 2, p. 80513; [2] ., .,
, . ., ., 1966; [3] ., , . ., ., 1980; [4] . .,
. ., , 3 .,
., 1983.
z z ; Fisher z distribution
296
F : (1) a = b = 2 ; (2) a = 4,
m, Y =
Y
j
p( x) =
= 2 m1m1 / 2 m2m2 / 2
( ( m1 + m2 ) / 2) e m1x
( m1 / 2 ) ( m2 / 2 ) ( m1e 2 x + m2 ) ( m1 + m2 ) / 2
(, ) -
m1
1 , m2 1
. . z .-
:
f (t ) = [ m2 m1 ]it
( ( m1 + i t ) 2 ) ( ( m2 it ) 2 )
.
( m1 2 ) ( m2 2 )
(1 m2 1 m1 ) 2 (1 m2 + 1 m1 ) 2 .
F
m1 m2 F ,
1
ln F 2
m1 m2 . z .-. -
; Fisher Yates test ,
.
## ; physical realizability condition , .
; physical spectrum density ,
( )
.
z =
F
. [1]- 1924-
. ,
. z .
. . z .
z ,
F .
-
.
## .: [1] F i s h e r R., Proc. Intern. Math. Congr. Toronto, 1928,
v. 2, p. 80513; [2] ., ., , . ., ., 1966.
;
FKG inequality ., ; Fock space
. H ( )
H s,an ( s)
(a ) ( n ;
, [1] ).
; Fisher Irwin test ,
,
f ij , i = 1, 2 , j = 1, 2 -
f i. =
f ij ,
f. j =
f ij , i, j = 1, 2 -
. H 0 : pij
= pi. p. j
Fs , a ( H ) = n = 0 H s, an ,
. H = L2 ( , ) [ ( , ) -
f11
P { f ij | f i. , f . j , i , j = 1, 2}
H 0 -
( , [1], [2], [3] ).
[2], [6]- .
.: [1] ., ., , . ., ., 1973; [2] . ., . ., , 3 ., ., 1983;
[3] G i b b o n s J. D., .: Encyclopedia of statistical scienes, v. 3,
N. Y., 1982, p. 11821; [4] L i e b e r m a n G. J., O w e n D. B., Tables of the hypergeometric probability distribution, Stanford ( Calif. ),
1961; [5] V i l a p l a n a J. P., Table of hypergeometri probability
distribution, pt. 2, Panama, 1976; [6] ., .,
, . ., .,
1985; [7] P r a t t J. W., G i b b o n s J. D., Concepts of nonparametric
theory, N. Y., 1981; [8] F i s h e r R. A., J. Roy. Statist. Soc., 1935,
v. 98, p. 3954; [9] Y a t e s F., J. Roy. Statist. Soc., 1934, v. 1,
p. 21735; [10] I r w i n J. O., Metron., 1935, v. 12, 2, p. 8394.
Fs ,a ( L2 ( , ) ) xi , i = 1 , ..., n
], . .
, p , . H 0 -
H s, a0 = C 1 ,
## = { f 0 , f1 ( x) , ..., f n ( x1 , ..., xn ) , ... }
.
= f0
f n ( x1 , ..., xn ) d ( x1 ) ... d ( xn ) .
n = 1 n
.: [1] . ., , 2
., ., 1986; [2] ., .,
, . ., . 1, ., 1977.
## ; Fock representation o f c o m mutative and anticommutative relations
. H L
H - f -
297
( f L) , { a ( f ), f L }
,
( , H -
):
a ( f1 ) a ( f 2 ) m a ( f 2 ) a ( f1 ) = 0,
a ( f1 ) a ( f 2 ) m a ( f 2 ) a ( f1 ) = ( f1 , f 2 ) E ,
j j ( ti
[ i L -
j =
( s ) ( a ) ]
, L -
:
i =1
= n ( i f ) ( 1 ... i ... n )
i
a ( f ) 1 = 0 , 1 Fs ,a ( L ) .
; Fokker Planck equation , , , ,
, .
;
Fokker Planck Kolmogorov equation ,
, .
:
p
yi =
j =0
298
j ti
+ i =
t
i
j 1 ( t i
) j 2 ( t i )
( ti ) = 1 .
(1)
:
1) j =
y
i
( ti ) ;
i =1
2) D ( j ) = D ( i ) = 2 ;
3) s 2 = 2 = ( n p 1) 1
y
i =1
2
i
j =1
2
j
, p -
0 , ..., p
,
p +1 =
yi p +1 ( ti ) ,
i =1
( 2 ) p +1 =
; Forsythe
method / algorithm
,
.
yi -, i = 1 , ..., n -
i =1
i =1
## (1) . . (2) . L (1) . .-, L
( , [2] ).
.: [1] . ., ,
2 ., ., 1986; [2] . ., . .,
.
, ., 1961.
( , [1] ),
j (2) :
(2)
i =1
s ,a
j =
ti 2j 1 ( ti ) ,
a ( f ) ( 1 ... n ) s ,a =
1
n
j j ( t ) = ( t j ) j 1 ( t ) j j 2 ( t ) ,
{ a ( f ), f L } , L , L - s ,a
1 ( t ) = 1 t
. Fs ,a ( L )
(1 ... n )
0 ( t ) =
. H a ( f ) , f L
(2)
. ,
, a ( f ), a ( f ) -
i =1
( ) , (1)
k = j,
, 0 k, j p
k j,
1,
0,
(t ) (t )
j
( , ) , L - , -
(1)
j j ( t )
(1)
, + -
M i = 0 , D i = 2 ,
) + i ,
j =0
n p 1 2
1
( ) p
2p +1
n p2
n p2
.
.
.: [1] F o r s y t h e G., J. Soc. Industr. Appl. Math., 1957,
v. 5, p. 7488; [2] ., , .
., ., 1970.
## ; Forsythe relation / equation ,
.
; Fortet
Kharkevich Rozanov spectrum ,
.
## ; Fortet Kac theorem ,
.
; Fortet Mourier metric
( , ; ). (U , d )
, X U - -
= { f } U -
Gq
, :
| f ( x) f ( y) |
L p ( f ) = sup
:
d ( x, y ) [ max (1, d ( x, c ), d ( y , c ) )] p 1
x y, x, y U 1, p 1, 0 < q
= sup{ | f ( x ) | : x U } q
, c , U - .
## ( X , Y ; Gqp ) = sup{ | E ( f ( X ) f ( Y )) | : f Gqp }
. . .- : 1) ( X , Y ;
, 2) ( X , Y ;
G11 )
G1 )
( , [2] ).
q < , . . . X - . X n X c U
mn = M d p ( X n , c ) < , n = 0, 1, ... ,
( X n , X 0 ; Gp ) 0 , n mn m0
( X n , X 0 ) 0 (
).
## .: [1] F o r t e r R., M o u r i e r E., Ann. Sci. cole norm.
supr., 1953, v. 70, 3, p. 26685; [2] D u d l e y R. M., Aarhus
Univ. Lect. Notes Ser., 1976, v. 45; [3] R a c h e v S. T., Publ. Inst.
Statist. Univ. Paris, 1982, v. 27, 1, p. 2747; [4] . ., . ., 1983, . 28, . 2,
. 26487.
; Foster
criterion , .
; fractal , .
.
( . , ,
);
. .
. [1]-,
: . ,
,
. ( . )
,
, ,
( . ) ( , [2]-, ).
;
, ,
, , . M ( . )
: x U n ( x )
, - .
[ ., , ,
rn - ,
D , y - ( y x ) rn -
g n : U n D ; g n (U n I M ) , ( , , )
- ]
. .
, . ,
, , ,
, .
( ., ) ,
. .
, :
. ( , [1] )
. ,
. ,
fraction ( .
) ( )
fractus .
. .- , , , , .
, .
. , (
; , [4]-
): .
; .
,
. , ,
.
. ( K. Weierstass ) . - ( B. Van
de Wairden ) , . ( G. Peano ), .
( H. Koch ) . ( D. Hilbert ) , .
( W. Sierpinski ) .
-
.
;
.
( .
) .
299
, , ,
[2], [3].
, ., , ,
, , .-. .
. ,
, . .
4
10 , 10
. -
, .
. ( ., - .-;
). ,
, ,
. , ;
:
?
,
.
.- , .
( , , [4] ).
.: [1] M a n d e l b r o t B. B., On fractal geometry and a few of
the mathematical questions it has raised. Proc. Intern. Congr. Math.,
Warszawa. 1983, v. 2, Warszawa, 1984, p. 166175; [2] . ., . ., , . ., .,
1993; [3] C a r l e s o n L., G a m e l i n T. W., Complex dynamics,
N. Y., 1993; [4] F a l c o n e r K. J., The geometry of a fractal sets,
N. Y. [a. .], 1985.
; fractional brawnian motion , .
( ) ; fundamental lemma o f
m a t h e m a t i c a l s t a t i s t i c s ,
.
; fundamental matrix o f a M a r k o v c h a i n
, ;
; functional redundancy , .
; path integral , .
; d i s c r e t e Fourier transform , .
-
{ A , }
( ) =
A ( g ) ( dg )
(*)
( ,
G ).
.
G , G - . . ( ) :
1) . . ( )
;
2) A , A ( g ) = I g G , ( )
= I -
, I , H - .
3) . .- { ( ) , }
;
4) 1 2 1 2 , . .
1 ( ) 2 ( ) -
5) ( B ) = ( B 1 )
( B
= { g G : g 1 B } ),
( ) - ; ,
, , ( ) -
;
6) G - n - ,
, , n ( ) ( ) ,
.
G ( ., ),
, . -
. (*)
.
G ,
, G . ,
G - - . . G
( ) =
( g ) ( dg )
G -
( ) . ( ) , G
G
.
1) 6) ( ) -
Fourier transform
. .,
tion on a
;
of a probability distribug r o u p G -
300
( 1 ) = 1 ,
1 G
( ) 1 ;
1 ( ) 2 ( ) =
= 1 ( ) 2 ( ) . , ( ) -
.
, G
= R R ,
1
t ( x ) = e , x G = R1 , t G = R1
itx
. . -
,
( t ) =
R1
exp { i t x } ( dx ) .
.: [1] .,
, . ., ., 1965; [2] .,
, . ., ., 1981.
; f i n i t e Fourier transform
, .
; f a s t Fourier transform
, .
; Fourier Stiltjes
transform , .
;
Gelfand integral B -
. ( , B , ) , B , , , B -
X , - B -
B -
B -
~ 2
D (T ) = F : p 2 ( F ) = ( n + 1)! f
< ,
n +1
~
f n , f n - .
F D ( T ) . . .
x ( X )
X - x
, x X
x ( x ) =
. x
.
, . . -
.
.: [1] . ., . -. . .
. . . i. . . ., 1936, . 13, . 1,
. 3540; [1] ., ., , . ., 2 ., ., 1962.
( )
; l a g w i n d o w generator , .
( ) ; generator o f a p r o c e s s
; . , , , .
;
extended stochastic integral ( , . )
- , w
. . . .- .
( ,
[1], [2] ). ( , A , P ) , A = {w } , H =
= ( Ft )
F =
I n ( f n ( , t1 , ..., t n ))
n=0
, f n ( )
L2 ( R +n+1 ,
n +1
) , I n -
302
n +1 (
f n ( t , t1 , ..., tn ) )
n=0
, T ( F ) = 0 , T (F )
x ( x ) d
. H - F
. -
= L2 ( R + )
T (F) =
= p 2 ( F ) .
H - ,
F D ( T ) T ( F )
d w t - .
F T (F ) H - L2 ( ) - .
T D .
F H , D ( D )
DFt
2
H
dt <
, F D ( T )
M T (F )
= F
2
H
+ M
DFt ( s ) DFs ( t ) ds dt .
T D :
D ( D ) , F D ( T ) , M D , F = M T ( F )
;
, ,
L2 (R + ) -
.
. . . , , ,
( , F , P ) ,
F L2 (R + ) - -
,
.
. . . P F
( , [3] ). . . .
D ( T ) -
.
, T
. : F = wT I [ 0,T ] .
T ( F ) = wT2 T .
( . . .; , [4],
[5] ). F H , { ei } , L2 (R + ) -
. F dw t
F , ei I i ( ei )
D Ft
d t < , DFt ( s ) -
, . . . , T ( f ) + tr D F - . . . .
- .
. . .
.
.: [1] . ., . .,
1975, . 20, . 2, . 22337; [2] . ., . ., .: -
. ( 2530 1974 ), . 1, ,
1975, . 12367; [3] . ., . ., 1987, . 32, . 1, . 11424; [4] N u a l a r t D., Z a k a i M.,
Probab. Theory and Related Fields, 1986, v. 73, 2, p. 25580;
[5] ., . . ., 1984, . 24, 1,
. 121130; [6] . ., . .,
. , 1973, . 208, 3, . 51215.
~
Y , Y1 ,
~
~
y1 , ~
y 2 | y ) , Y , Y1 ,
Y2 p ( ~
~
~
~
Y2 , ~
y1 Y1 , ~
y 2 Y2 , y Y . . .
K = 1 , L = 2
( , )
. ( M = 3 ):
,
.
1- A1 = { 1, 2 , 3}
2- B1 = {1, 3} ,
B2 = { 2 , 3} , C1 = {1} , C2 = { 2} . ,
,
. . .-
y1 , ~
y 2 | y ) - ,
p ( ~
p( ~
y1 | y ) p ( ~
y 2 | y ) (
(U , Y , Y1 , Y2 ) -
(U
U Y + 2
); ,
p (u , y , ~
y1 , ~
y2 ) = p ( u , y ) p ( ~
y1 , ~
y 2 | y ) -
y1 , ~
y 2 | y ) . .-
, p ( ~
R =
= { ( R1 , R2 , R3 )} , . . . .
. . .
p1 ( ~
y1 | y ) p2 ( ~
y 2 | y ) q ( ~
y2 | ~
y1 ) ,
~
~
~
y1 Y1 , ~
y 2 Y2
p( ~
y2 | y ) =
q ( ~y
~
y1
|~
y1 ) p ( ~
y1 | y )
, . .
( . ) ( ) . . .-
y Y ,
~
~
y1 Y1
p1 ( ~
y1 | y ) = 0 p1 ( ~
y1 | y ) = 1 , . . .
. .- .
., . .- (
. .- ). . . ,
. .
.
.: [1] . ., . .,
, ., 1982; [2] ., .,
, . ., ., 1985; [3] C o v e r T., IEEE Trans.
Inform. Theory, 1972, v. 18, 1, p. 214.
## ; benefical aging distribution ,
.
; abyusted range , .
; Gibbs phenomenon (
).
. .
( , , A1 = {1, 2 } ,
) . f ( x )
B1 = C1 = { 1} , B2 = C 2 = { 2} )
, a f ( x0 0 )
(
, , A1 = { 1, 3} , B1 = { 1, 3} , C1 = { 1} , B2 = { 3} ,
C2 = { 2} ).
s n ( x ) -
{ x : 0 < | x x0 | < h }
~
0 R3 I (U ; Y2 ) ,
~
~
0 R1 + R3 min { I ( Y ; Y1 ) , I ( Y ; Y1 | U ) + I (U ; Y2 ) } ,
~ ~
( I ) ( U , Y , Y1 , Y2 )
f ( x0 + 0 ) b . A < a b < B
A = lim s n ( x),
n
. .- R
( ) .
. .- R - . ., . .-
( R1 , R3 ) ,
f ( x) - x0
x x0 0
B =
lim s n ( x ) ,
n
x x0 + 0
x0 s n ( x) . . . . .
,
, .
. .
.
(
), .
303
.: [1] . ., , . 1,
., 1977, . 958; [2] ., ., , . ., ., 1982.
; Gibbs limit state , -
( ) . -
.
-
;
Gibbs distribution
.
d R d ( )
N . . N
X 1 , ..., X N
## p ( x1 , ..., x N ) = exp { H ( x1 , ... , x N ) } Z N ,
(1)
, , U
.
, ,
.
- . . . ,
N -
pN ,
, H ( x1 , ..., x N ) , x1 , ..., x N - -
, Z N ,
,
p - 1- ;
= N = const
, - . . U
e N
N,
=
N!
N =0
e N
, N
N!
(4)
, N -
.
, ,
H
.
, H ( x ),
N
. . . , -
x [ 0 , )
, ,
H ( x1 , ..., xN ) =
1
2
( | x x
i
(2)
|)
i j
. ,
H ( x1 , ..., xN ) =
1
2
(|x x
i
i j
|) +
U (x )
i
(3)
H , U ( x ), x ,
, . ,
e
, dist ( x , ) 0 U ( x ) 0 .
H ( x1 , ..., xN ) =
## k =1 1 i1 <i2 < ...ik N
( ) .
, ,
N
- ( x1 , ..., x N )
N { x1 , ..., x N }
. , . .
.
.
, ,
N N
304
N . .
.
. .
(
. . ), . ., ( . . )
.
N - H . . . (1) , H ( x1 , ..., x N ) = H
.
, -
=N
= H - , -
U , -
,
. e . . . .
( . . ) .
- t T . .-
. U
= { U A (xA ), A T }
, T
( X , B , m ) . .
p ( xt , t ) = exp ( H ( xt , t ) ) Z
( )
,
H ( xt , t ) =
A ( xt
, t )
Z .
. ., , U -
. . .
. .- . U ,
.
.: [1] ., . , . ., ., 1971; [2] D o b r u s h i n R. L., T i r o z z i B., Comm. Math. Phyz., 1977, v. 54, 2, p. 17392; [3] E l l i s R., Entropy. Large deviations and statistical mechanics, N. Y.,
1985; [4] G e o r g i i H.-Q., Canonical Gibbs measures, B. N. Y.,
1979.
; Gibbs
postulate , .
; Gibbs density , .
; Gibbs finite state , .
## ; Gibbs random field
.
. . .
. T
, ( X , B , m ) , m .
U = {U A ,
H ( x) =
, R U { } - , x A X
- . , T
A - . T x X T
H ( x | xT \ ) =
U A ( xA )
(1)
A T , A , | A |<
, x A , x A - -
= Z d d
X m
= { 1, 1} , T = Z d ,
U A 0 A = {t }
A = { s , t } , | s t | = 1 ,
. X
. T = Z , ,
,
. . . .
;
,
p ( x | xT \ ) d
. ., X
, ,
.
Z . . .- . U {t , t 1} ( xt , xt 1 ) = ln pt ( xt | xt 1 ) , pt ( xt | xt 1 )
,
A { t , t 1} , U A 0 .
= Zd , X = R
H ( x) =
(4)
U s , t x s xt
s , t Zd
U A , | A | ,
A (xA )
.
,
A T , | A | < } A T A
AT
U s, t R
|U
sup
t Z d
s ,t
|<
s Z d
.
.
U
= {U ( A )}
, A , R
. x R -
( xT \ ) =
exp { H ( x | xT \ ) } m ( dx )
(2)
, m , m | | . , m
H ( x | xR d \ ) =
UA
A x: A
(3)
, R x = x I . ,
. x , xT \
. . . ,
, - -
p ( x | xT \ ) = exp{ H ( x | xT \ ) } Z ( xT \ )
, (1) (2)
, (3)- .
xT \ .
= { X ( t ), t T \ } = { X ( t ), t } - -
T X T \
X
m (3)- , { X t , t T }
.
( . ) .
. . .
1 .
d
, , , Z
X U A ( x A )
. X
X - ,
.
305
sup | U ( x A ) | <
: 0 A
V - , PV ( | xW ) W Z d \ V
(5)
xA
. U . . .- Zd
, X - S (U )
( )
. ,
. U . . . , n ,
xZd \V , n = 1, 2 , ...
n
(
) - .
. . .- S (U ) U
. . ,
. . .-
, , .,
, .
, , .,
S (U ) U .
d = 1 . . .-
## ( diam A ) sup U ( x A ) <
(6)
xA
A: 0 A
.
(5) , (6)
, ,
.
( d .
) ( , [1], [2] ):
sup
t Zd
s, t
< 1.
(7)
st
s, t Z , s t
d
## s,t = sup var ( g{t} ( | ~xZd \{t} ), q{s} ( | xZ d \{s} ) ) ,
var ( , )
x , x
s ~
.
. . .-
. - U = { U }
- . -
U , x -
,
,
, .
. . .
. ., (5) (7) : PV
306
d
, V Z
lim
sup var ( PV ( | xW ), PV ) = 0.
dist (V , W ) x
W
, sup | U ( x A ) | -
xA
(8)
diam A
, (8)-
. (6) ,
.
,
. . . S (U )
( ) d . , . -
( ., ) . .
,
(4)
. ,
V Z
xt R ,
t V
, c > 0
u s ,t xs xt c
s , t V
(x )
t
(9)
t V
, t Z
,
. . . . 0 -
. . . .
. (9)
. ., . d = 1, 2
( xs xt ) 2
s , t Z d : | s t | =1
. . . . d = 3 ,
,
.
.: [1] . ., . .,
1968, . 13, . 2, . 20129; 1970, . 15, . 3, . 46997; [2] . ., . ., 1968, . 2, . 4, . 31
43; [3] . ., . ., . , ., 1985; [4] ., , .
., ., 1977; [5] . ., , .,
1980; [6] D o b r u s i n R. L., Gaussian random Fields Gibbsian point
of view, .: Multicomponent random systems, N. Y. Basel, 1980,
p. 11959; [7] P r e s t o n C h., Random Fields, B., 1976; [8] R u e l l e D., Thermodynamic formalism, Reading ( Mass. ), 1978.
; input / arrival
flow / stream . . . , t n R
;
, , ( t n t n +1 ). t n -
n-
. .- n -
.
. .
. . .-
. .-, . .- .
, . .
.
. , .
.: [1] . .,
, ., 1972; [2] . ., . ., , 2
., ., 1987.
; arbitrary input / arrival flow / stream
with lack of memory ,
.
. .-
.
, .
.: [1] . .,
, ., 1963; [2] . ., , ., 1978.
,
; regular input / arrival flow / stream with
lack of memory , .
,
; singular input / arrival flow / stream
with lack of memory ,
; recurrent input / arrival
flow / stream , n - -
z n
;
entrance distribution , ;
P { z1 < t } = F0 ( t ) , P { z n < t } = F ( t ) , n 2
= F (t ) = 1 e t , t 0 -
. . (
). , .
.: [1] .,
, ., 1965; [2] . ., , ., 1978.
; input / arrival
flow / stream with lack of memory
[ 0 , t )
, t 0 ;
{ (t ), t 0 }
. , ti , t j , 1 t i
<tj n
( t j ) ( ti ) , . . . ( t , t + s ) k t
,
. .-, s > 0 .
, t < s t , s
( s ) ( t ) ( s ) ( t ) -
; (t ) .
,
{ xn } .
; entrance law
, .
; input signal
. . . .
.
( ),
: (t ) ,
, .
t n = z1 + z 2 + ... + z n , n 1 .
. F0 ( t )
; entrance rejets
, .
;
waiting time
.
. .
. . .- :
, ,
( , ).
. .-
M , M , - M
= ( M + )
, ,
.
; expected utility , .
; expected reward criterion
,
( )
; power o f a s t a t i s t i c a l t e s t H1 H 0 , H1 , H 0
. H 0 H1 ( H 0
,
: H 0 : 0 , H 1 : 1 = \ 0 ),
H 0 - H1 - . . ( ) -
307
( = 1 = 0 U 1 )
.
:
H 1 : 1 = \ 0 H 0 :
0
. inf ( ) -, ( )
1
. ( , , ).
.: [1] . ., , .
., 2 ., ., 1979; [2] ., ., , . ., ., 1971; [3] . ., , . ., ., 1960;
[4]
., , .
., ., 1975.
,
; power function of a test
.
( X, B , P ) , X
x
H1 X
P -
H 0
H1 - H 0 -
( x ) d P ( x) ,
= 0 U 1 ,
(*)
. (*)- ,
P , , . . ( )
; gain , .
; strictly unimodal
distribution , / .
; strong Feller property o f
a M a r k o v p r o c e s s e s ( X , B) ( X ,
B )
P ( t , x , ) , t > 0
f B ( X )
P ( t , x, dy ) f ( y ) C ( X )
, B ( X )
, C ( X ) .
H 0 = { P , 0 }
( ) =
F (x) = Tt f (x) =
P H 1 = { P , 1 = \ 0 }
( )
,
; strong likelihood principle ,
.
.
; strong Feller
transition function ,
## ; strong Feller process . E , B
. ( E , B )
, H1 H 0
H 0
X - . f
.
. ( J. Neyman ) . ( E. Pearson ) H1 H 0 -
. .
: 1 , 0
0 <
<1
() ,
. . () -
,
H 0
.
.: [1] . ., ,
. ., 2 ., ., 1979; [2] ., , . ., 2 ., ., 1975; [2]
. ., , .,
., 1960.
; power spectrum , .
308
t0
p ( t , x , ),
Pt f ( ) =
>0
xE,
f ( y) P ( t ,
, dy )
.
. . .- ( , [2] ). . . .-
. . .-
( ,
)
. . .-
( , [2] ). . . .- [1]-
.
. . .- ,
.
( , [3] ).
.: [1] . ., . .,
1960, . 5, . 1, . 728; [2] . ., , ., 1963; [3] T u o m i n e n P., T w e e d i e R., Proc. London
Math. Soc., 1979, v. 38, p. 89114.
( )
; strong harmonizability o f a
r a n d o m p r o c e s s ,
.
; strong infinitezimal operator / generator ,
.
## ; strongly measurable mapping
.
( , A , ) , B . f : B
A - , f
.
f n : B 0 A ,
( \ 0 ) = 0
lim
f n ( )
f ( ) = 0 , f
(
) .
f : B , , ,
[
0 A , ( \ 0 ) = 0 f ( 0 ) , B - ] B
f 1 ( ) A .
f : B , ,
, , ( )
( ).
.: [1] ., , . .,
., 1967.
## ; strong random linear operator , .
/
/ ; strongly unimodal distribution , / .
; strong
convergence , .
; gain , .
; Haar measure
G E
B - , G - ,
s G { s g : g A}
A B
, ( s A )
= ( A ) , s A
. f ( p ), p E
{ s g : g A}
B
f ( p) (d p) =
f (s p) (d p)
. . ( . . ) . . .-,
G - K -
( K )
= ( K 1 ) , . .-, K 1 ,
{ g 1 : g K } . T
~ T -
. . ;
~ = c , c
. .-,
. T - ( T - ) , ( t ) > 0, tT , ( e ) = 1 , e , T
. ( t t )
- : X (U )
. ( A ) = ( s A ) ,
, ;
ring of sets , .
,
; ring of sets , , , .
;
smooth measure X
= ( t ) ( t ) , t , t T
U
U
= lim (U ) .
( , )
.
. X
, , ,
, .
.: [1] . ., . ., 1961, . 55, . 1,
. 35100.
( ) ; smoothing inequality
. . .- :
( P, Q ) c ( P , Q ) + C ( ),
T - . .-,
( p g ) = ( g ) ( p ), g T , pA , A
, c , P Q
T .
, { ,
.: [1] .,
, . ., ., 1950; [2] . .,
, ., 1965.
; event
, ,
.
*
A B
A B ; event A implies
event B A
B
A B B A . A , B
.
B = A U B A = A I B , A B ; A A ;
B C A C ; A , A A .
.: [1] . ., . 5-
., .: 1969; [2] ., , ., 1969.
310
> 0} -
, 0
,
C ( ) , , P Q - 0
. ,
( x ) = ( x / ) . - P
Q
.
. .- .
,
, ( x )
= ( x / ) , 0
1. . .:
( P, ) c1 ( P , ) + c2
c1 c2 , P .
2. . .:
a) ( P, Q ) ( P , Q ) + c log 1 / ,
c , P Q ;
S ( ) +
) ( P, Q ) 1 ( P , Q ) + c2 (1 + ) ,
P , Q , q ( x ) h -
= sup { h 1
h
q ( x + h ) q ( x ) dx } <
3. s . .: 0 < s 2 ,
4. . .
. ., f ( x ) g ( x ) , x 0
f ( x ) 0 , g ( x ) 0 P
( P, ) f ( ( P , ) ) + g ( )
.
.: [1] B e r g s t o m H., Skand. aktuarietidskr., 1945, v. 28,
1/2, p. 10627; [2] E s s e e n C. G., Acta math., 1945, t. 77,
1, p. 3125; [3] S a z o n o v V. V., SANKHYA, ser. A, 1968,
v. 30, pt. 2, p. 181204; [4] . ., .
., 1970, . 15, . 4, . 64765; [5] . .,
. ., 1975, . 20, . 1, . 312; [6] . ., . ., 1976, . 21, . 2,
. 40610.
, -
( ) .
;
Hamburger theorem , .
; Hamiltonian strategy ,
.
,
; Hahn theorem
on signed
, . X -
S . X
A B { A , B } :
1) Y S Y I A S ( Y I A ) 0,
2) Y S Y I B S ( Y I B ) 0.
A , ,
B,
. , { A , B }
X
. | | ( Y )
= + (Y ) + (Y ) , Y S
( Y )
( Y ), Y S
.: [1] ., , . ., ., 1953;
[2] . ., . .,
, 5 ., ., 1981.
; hanning window , .
; Hunt process
.
E , ,
X = ( X t , , At , Px ), t 0
E
= E U { } , E -
, .
( )
1 2 ... X n E
{
= < }
, X ,
= lim n .
n
; smoothing distribution ,
= + ,
-
. S .
c , P Q .
; smoothing parameter
, ,
( , X { A , B } -
| | ,
s ( P, Q ) s ( P , Q ) + c ,
0 <
Y S
). .
, c1 c 2 .
+ (Y ) = ( Y I A ) , (Y ) = ( Y I B ),
( , [1], [2] ).
. .-
( , [1],
).
.: [1] . ., ,
. ., ., 1962; [2] B l u m e n t h a l R. M., G e t o o r R. K.,
Markov processes and potential theory, N. Y. L., 1968.
; Hardy
entropy , .
; metod of harmonic decomposition , .
; harmonic function f o r a M a r k o v p r o c e s s ,
;
harmonic function f o r a M a r k o v c h a i n ;
311
; ( E , B )
= ( X n , Px )
p ( x, ) , x E , B X
. E - f
( 0 f
) B E -
f ( y ) p ( x , dy ) =
M x f ( X1 )
; harmonizable correlation function ,
.
; harmonizable random
field t = ( t1 , ..., t n ) n R n
f ( x)
f
, M x , Px
,
( t n Z
)
X (t ) =
[ ., 1, 2, ...
, f f i
= f (i )
, pij , i
j , i 1 ]. . .-
( x ) , x E , ( x )
B
Px .
M x f ( X )
f ( x)
<
n
X (t ) , n ,
t Z n { k :
M X (t ) X ( s ) B (t , s )
= inf { t 0 : X ( t ) U } , X
( ; ,
).
. n
. .
( ) . .-.
; harmonic interpolation
n
0 +
2k
cos k x + 2 k 1 sin k x )
k=1
; harmonic averaging ,
312
F ( dk , dk )
(2)
n n
, Z
n , n
F ( , ) = M Z ( ) Z ( ) ,
Z () =
dZ ( k ) .
.: , .
## ; harmonizable random process t R1
1
t Z ,
(t ) =
it
dZ ( )
(1)
(t )
1
1
, t R = ( , ) , t Z -
= [ , ] , Z ( ) -
. . . .- 1940- .
( M. Love ) [1]-
. 496 99- . . , (t )
. . .-
( , ).
i ( t k s k)
X (t ) .
. .
U ,
U , E -
B ( t , s ) (2)
M x { f ( X ); < } f ( x )
,
..., n } -
. (1)-
. . .-
, (1)
. , f
f ( x ) X
k i < , i = 1,
, Z ( d k ) , n n -
E -
X = ( X t , , At , Px ) , t 0 -
E - ,
(1)
Z ( dk )
t R n k = ( k1 , ..., k n ) n ,
pij f j = f i
f . .- ,
it k
{ f1 , f 2 , ... } -
F ( , ) = M Z ( ) Z ( ),
Z ( ) = dZ ( )
, ,
(2)
V ( F ) = sup
F ( i , j ) <
i =1
j =1
. . [3]- , Y (t )
(3)
, -
1 , ..., n 1 , ..., n
. (1)-
[ F ( d , d )
].
(1), (2) (3)-
M ( t ) ( s ) = B ( t, s ) =
i ( t s )
F ( d , d ) (4)
H (t ) . . .- (
Y (t ) H
; , ). .
. [5]- , ,
, . . .- H
- H -
H - .
. . . ( . . . )
:
lim
, . (t ) -
. [ F ( , ) ; [1]-
, , [2] ].
= ( , )
, t R
. . . .
. . [3]-
( , , [4] [8]- ). . . (3)
F ( , )
lim
ab
<
i 0 t
( t ) d t = Z ( 0 + 0 ) Z ( 0 0 )
1
T
B( t + , t) d t =
e i F ( d , d )
## ( , [3], [6], [8] ). . . .
, , ,
0 ( t ) = ( t ) M ( t )
= 0 , = 0
~
. . . . B ( )
~
F () =
W ( F ) = sup
B ( t , s ) - (4) (t ) - . . . , F ( d , d )
1
T
dF ( ) = F ( d ,
d )
(5)
F ( )
,
1 , ..., n 1 , ..., n
.
. . .- .-
. . . .
. . . ; ( -
i =1
j =1
F ( i , j )
, a1 , ..., a n
b1 , ..., bn . (5)
, (2) Z ( )
( , ., [9], . 347 48 [5] ). (5)
(4) ,
; (t )
B ( t , s ) - (4) (t ) (1)
(
). (t ) (1) (2) (5) ,
. . . ; ( )
. . [10]- ,
;
. [6] [8]- (
) , ,
( , (t ) -
; ).
) R ,
n 2 .
, , [6] [8]- .
.: [1] ., , . ., ., 1962;
[2] H u r d H. L., IEEE Trans. on Inform. Theory, 1973, v. 19, 3,
p. 31620; [3] . ., . .,
1959, . 4, . 3, . 291310; [4] B o c h n e r S., .: Proc. Third
Berk. Symp. Math. Statist. and Probab., v. 2, Berk. Los Ang., 1956,
p. 727; [5] M i a m e e A. G., S a l e h i H., Indiana Univ. Math.
J., 1978, v. 27, 1, p. 3750; [6] R a o M. M., Lenseignement
math., 1982, t. 28, 34, p. 295351; [7] R a o M. M., Proc. Nat.
Acad. Sci. USA, 1984, v. 81, p. 461112; [8] R a o M. M., .:
Handbook of Statistics, v. 5, Amst., 1985, p. 279310; [9] ., . ., . , . ., . 1, ., 1962; [10] . .,
. ., 1963, . 8, . 2, . 18489.
; harmonization conditions
B ( t , s ) = M ( t ) ( s )
1
, (t ) , t R
1
t Z
= ( , ) (
= { 0, 1, 2, ... } )
313
. t R ,
( , ).
( (t )
= h ( t ) 0 ( t ) , 0 ( t ) F0 ( ) ( t )
, . . , ., [2] ).
F ( , ) =
(t ) , ,
B ( t, s ) =
i ( t s )
d 2 F ( , )
, F ( , ) , R
(1)
2
( , ., [3] ). B ( t , s ) R -
F ( , ) - N < N -
, (1)
, n ( t1 , s1 ) , ..., ( t n , s n )
c1 , ..., cn
3) Z (u ) ,
, F0 ( u )
,
h ( t, u ) =
c
( x, y ) R
B (t j , s j )
h ( t , u ) dZ ( u )
.
,
.
4) Z (u )
M Z (u ) Z (v ) = B Z (u , v )
2
R - , h(t ) (2)
exp ( i t j x i s j y )
d 2 F ( , ) =
(2)
j =1
R 2 - , B ( t , s )
f ( , ) -
(t )
d F ( , ) = f ( , ) d d .
2
, h(t ) -
h (t ) =
h ( t u ) dZ ( u )
. 1) B ( t , s ) B ( t , s )
(t ) =
.
(1) (2)
( , ., [3], [10]
[11] ).
. .-.
dK u ( )
K u ( ) - - u -
it
(t ) =
j =1
sup
2) K ( )
M d Z ( u ) dZ ( v ) = ( u v ) dF0 ( u ) dv
-
## ( , [1], [3] ). (1) (
) . ( ,
) ( , [3], [4] ).
. .- ( ,
., [5] [7] ). . .- ( [8]-
. . ) , , . , [5] [8]- - ( [9]-, 6- )
K ( ) K ( ) dF ( ) .
it
dK ( )
314
= dK ( ) dK ( )
exp [ i ( u v ) ] d
B Z (u , v ) .
.: [1] ., , . ., ., 1962,
. 49699; [2] R a o M., Lenseignement math., 1982, t. 28, p. 295
351; [3] H u r d H. L., IEEE Trans. on Inform. Theory, 1973,
v. IT 19, p. 31620; [4] . ., .
., 1963, . 8, . 2, . 18489; [5] B o c h n e r S.,
Bull. Amer. Math. Soc., 1934, v. 40, p. 27176; [6] C r a m e r H.,
Trans. Amer. Math. Soc., 1939, v. 46, p.191201; [7] D o m i n g u e s A. G., Duke Math. J., 1940, v. 6, p. 24655; [8] E b e r l e i n W. F., Duke Math. J., 1955, v. 22, p. 46568; [9] R u d i n W.,
Fourier analysis on groups, N. Y. L., 1962; [10] . .,
. ., 1959, . 4, . 3, . 291310;
[11] C a m b a n i s S., L i u B., Inform. and Control, 1970, v. 17,
p. 183202.
## ; Harris recurrent Markov chain , .
; Harrison seasonal model
.
:
{a
s (t L + j ) = 1 +
cos ( k j ) + bk sin ( k j ) } ,
(*)
ak =
bk =
2
L
2
L
s ( t L + j ) cos ( k j ) ,
,
.
.: [1] ., , . ., ., 1962.
( )
; product o f a m e a s u r a b l e s p a c e s , ( ) .
( ) ; product m e a s u r e . ( X t , S t , t ) t T ,
, t T X t S t t ( X t ) = 1 ( ,
j =1
s ( t L + j ) sin ( k j ) ,
j =1
j = 2 ( j 1) L .
(*)-
s ( t L + j ) .
.: [1] H a r r i s o n P., Appl. Statist., 1965, v. 14, p. 10239;
[2] ., , . ., ., 1981.
; hartley
t ) . , t T Yt S t t T
t - Yt = X t .
,
. : 1H . = 1 lg 2 bit
3,32 bit .
;
Hartman Wintner Strassen theorem ,
, A
A =
A A ,
(*)
A ,
P P , , A - , : (*)
A A
P ( A) =
( A )
. A - .
.- ,
( , A , P )
( , A , P ) - X , ,
X .
t T
p rt , t T -
t T
, A , A ,
of
t T
( )
, =
( , A , P ) . , ,
. ( S t ) t T -
probability
.
t T
( ) ; product o f a
f a m i l y o f a l g e b r a s , ( ) ; product
s p a c e s ( , A , P ) ,
t T
t T
t T
t T
(Y )
t
t T
.
( t ) t T
. ,
t T
X , S ,
t
t T
t T
t T
, ( X t , S t , t ) t T
X , S
( X t , S t ) t T
t T
t T
t ( t ) t T
t T
, t , t T
315
t . ., R n
t T
n
( ,
n - ). ,
{ 0, 1}
~
; Hellinger integral P P
:
<< Q , P << Q Q
nN
n N n { 0, 1}
, n ({ 0 } )
= n ({1}) = 1 2 .
. ,
. ,
.
.: [1] ., , . ., ., 1953;
[2] ., , .
., ., 1969; [3] ., .,
. , . ., . 1, ., 1962.
; Hausdorff pseudometric ,
.
G X
G , { x : g ( x ) > 0}, g G .
( X , G )
X -
G - .
.: [1] ., , . ., .
., 1937; [2] . ., . , 1976,
. 31, . 2, . 368.
## ; Helly theorem : F ( x ), F1 ( x ), F2 ( x ) , ... -
~ 1
dP
dQ
dQ
; Q , P P
.
, .
; Hellinger metric ( , A ) P ( )
( P, Q ) = (1 2 )
[ ., , V
2 ( P , Q ) = (1 2 )
g ( x ) dF ( x ) g ( x ) dF ( x )
n
(*)
. , (*) { Fn }
].
dP dQ
(*)
. - dx
, , P p ( x )
, (*) p q P Q
2 ( P , Q ) = (1 2 )
p ( x)
q ( x)
) dx
2
. . .
H ( P, Q ) =
= (P + Q) 2
. . V - . . .
n g ( x )
dV
dQ
dV
, V , - , P
Q -
, { Fn ( x ) } F ( x ) - ,
dP
dV
## ; instantaneous availability index
. ,
, .
dP
dQ
. , , Q
; .
; Hausdorff structure o f a p r o b a b i l i t y
m e t r i c , ; .
;
Hausdorff theorem
~
( d P ) ( d P )1
dP dQ
dV
dV dV
2
( P, Q )
=1 2
.
. . :
1) 0 ( P, Q ) 1 , ( P, Q ) = 0 P
2)
( P, Q ) = 1 P
F - .
.: [1] H e l l y E., Sitzungsber. Math. Naturwiss. KI. Keiser.
Akad. Wiss., 1912, Bd 121, S. 26597.
(P Q) ;
316
= 1 H ( P, Q )
=Q;
3) = 1 2 , P = P1 P2 , Q = Q1 Q 2 ,
1 2 ( P, Q ) = (1 2 ( P1 , Q1 ) ) ( 1 2 ( P2 , Q 2 ) )
,
def
( x, y ) yI ( x = 0 ) = 0 ( x, y ) , 0 ,
Pi , Q i P ( i ) , i = 1, 2 .
1 2 ( x, y ) = (1 2 ) ( x y ) 2 ,
. .- .
4) . .
:
2 ( P, Q) ( P , Q )
2 ( P, Q ) ,
2
1 2 ( x, y ) ( x, y ) 8 1 2 ( x, y ) .
1
. .
,
( , ).
, ., , P Q
, .
, , ( ) ( )
~
P P H
~
H ( ) 1-
2- .
( P, Q) = ( 1 2 ; P, Q ) - :
2
( P, Q ) ( ; P , Q ) 8 ( P , Q ) ,
1
( ; P, Q ) = 0 P = Q
. . .
P - ( Q P )
:
~
~
Er ( P, P ) = 1 ( P, P )
P n = P1 ... Pn
~
~
~
P n = P1 ... Pn
~
( 1 2 ) e 2 n Er ( P n , P n ) e n
,
~
~
= log H ( P, P ) 2 ( P, P ) .
.: [1] ., . .,
, . ., ., 1994.
; Hellinger distance , . .
( ; P, Q ) =
( pV , qV ) d V ,
0 < <1
(1 ) y x y
Q << P H ( ; P, Q ) 1 ,
QP
H ( ; P, Q ) 0 ,
QP
H ( ; P, Q ) = 0 , ( 0, 1) ,
QP
H ( ; P, Q ) = 0 , ( 0, 1) .
P n , Q n , n 1
( x , y ) = x
, V , ( , A )
, , P , Q
) ( Q n ) ( P n ) - n ,
P n ( A n ) 0 Q n ( An ) 0 , A n A ; ) ( Q n ) ( P n ) -
n k
A1 , A2 , ... , k P nk ( Ak ) 1
Q
, ( Q << P ) Q -
H ( ; P, Q ) = 1 2 ( ; P, Q )
~
Er ( P, P ) = inf { ( ) + ( ) : }
1 2
.
. .
nk
( Ak ) 0 . :
0 n
## ( Q ) ( P ) lim lim inf H ( ; P n , Q n ) = 0 ,
n
0 n
[ P << V ,
Q << V ;
,
.,
( Q ) ( P ) lim inf H ( ; P, Q ) = 0
( ; P, Q ) = M V ( pV , qV ) = 1 M V p V q1V ,
( Q ) ( P n ) lim inf H ( ; P, Q ) = 0
, M V , V
( 0, 1)
V = ( P + Q ) 2 ] pV = d P d V , qV = d Q d V . . .
.
:
( x, y ) 0 , x, y 0 ,
( 0, 1) ,
n
317
;
Hellinger process -
. ( , A , A = ( At ) t 0 ) ,
A = ( At ) t 0 ( , )
~
( , A ) P , P , ( , A ) - ~
~
~
; Pt = P | At Pt = P | At , P P At -
Q , P
loc
loc
<< Q, P << Q,
~
t 0 Pt , Pt , Q t Pt << Q t ,
~
Pt << Qt , Q t = Q | At .
zt =
H ( ) t
~
d Pt
z~t =
,
d Qt
d Pt
,
d Qt
Yt ( ) = z t z~t1
= M QYt , Pt Pt -
H ( )t
d Pt
dQ
t
~ 1
dP
t
d Qt =
dQ
t
~
( d Pt ) ( d Pt )1
= 0 A ( )
, Y ( ) = M ( ) A( ) . ( 0, 1)
h( ) 0 = 0 h ( ) = ( h ( ) t ) t 0
A ( ) 0
, ( Q
), h ( )
= I h ( ) ,
~
~
= I , = { z _ > 0} U [ [ 0 ] ] , = { z > 0} U [ [ 0 ] ]
A ( ) = Y ( ) h ( ) ,
Y ( )
s_
d hs ( ) .
~
h ( ) P P ( 0, 1) c
. z~ z z~
Q ,
( z , z~ ) ( z , z~ )
, :
h ( )t =
(1 ) 1
c
c
2, z , z
z
1
c
c
, z , z~
z - z~_
1
c
c
, z~ , z~
z~ 2
x
y
+ 1 +
, 1 +
~
z
z
_
_
318
~
1) Q = ( P + P ) 2 ,
z + z~ = 2
h ( ) =
(1 ) 1
z_
1
+ ~
z_
z c, z c
x
x
z
+ 1 +
, 1
~ .
z
z
_
_
~
d Pt
~ loc
2) P << P Z t =
,
d Pt
h ( ) =
(1 )
2
1
c
c
Z , Z
Z 2
x
x
Z
1+
+ + (1 ) 1 +
Z
Z
1 1
1
h = 2 Zc, Zc
8 Z
2
1
2
1+
x
Z
Z
.
. .-
.
1. A =
VAt , = ( )
~
{ t } At , t 0 ) ; P << P ~
~
P
, P0 << P0 h ( )
0,
0 .
A ( )t =
= x + (1 ) y x y1 .
Y ( ) = ( Yt ( ) t ) t 0 Yt ( ) = z t z~ 1 D
M ( )
z c , z~ c , z c z~ c - -
; ( x, y )
; :
z~ c , z~ c , z c z~ c - -
z c, z c ,
t( z , z~ ) .
~ loc
2. P << P ,
~
: P << P
~
, P { h ( 1 2 ) < } = 1 ,
~
P P ,
~
P { h (1 2 ) < } = 0 .
.: [1] L i p t s e r R. S h., S h i r y a e v A. N., On the problem
of predictable criteria of contiguity. Proc 4th USSR Japan Symp.,
Lect, Notes Math., 1983, 1021, p. 386418; [2] .,
. ., ,
. ., ., 1994.
; hamming window , .
; Hannan Quinn riterion ,
;
; Hermitian positive function , .
## ; Hermitian random matrix n
H n
= ij
,
T
H n
- . . . .
H n - , H n 1 ... n 1
- . . . . H n -
## ( H n i I ) ai = 0 , ( ai , ai ) = 1 , arg a1i = ci , i = 1, ..., n
, ci
, 0 ci
2 , i = 1, ..., n.
n -
,
, B ,
, An = (a1 , ..., an ) . . . . H n - ,
, i , i
; L , B - .
U n H n U n
H n
U n - , An
. . . H n -
P { An L B } =
( dU
## | arg u1i = ci , i = 1, ..., n )
.
( , [1] ).
.: [1] . ., , .,
1980.
, ;
function
(
) . K
, G
( G - + ). K
( X i ) i I
K -
UX
i I
(X )
i
i I
, : K G .
( I I - ). , ( ) , G
,
.
G R - ,
. . . .- - . ,
(
+ ).
, . . . .
. , , . ( , )
. - ( , )
. .-. . .
.
.: [1] . ., . ., . . .
, 3 ., ., 1987; [2] ., , .
., ., 1953; [3] . ., . .,
, 2 ., ., 1977.
; countable probabilistic
automation , .
; countable / denumerable Markov chain
1) . . .
.
2) . . .
. . . .
. . ( , [1],
[2] ).
1936 37- . . ( ,
[4], [5] ).
. . .
,
( , [6], [10] ).
( )
( ) pij , i j
E - . E
, , , { 0, 1, ..., N } { 1, ..., N }
, N ; E , , { 0, 1, ... } {1, 2 , ... }
. ( )
pi = P { X 0 = i } , i E ,
n
= { X 0 = i0 , ..., X n = in }
P ( n ) = pi 0 pi0i1 ... pin1in
{ X 0 , X 1 , ... } . ,
## P { X n = in | n 1 } = pin 1in , P { X n = in n1 } = pin1 in
P { n 1 } 0 , n
pij P = pij
E -
319
. pij ( n ) n i
pij ( m + n ) =
pik ( m ) pkj ( n ), m, n 1
k E
pij ( n ) P n .
P n n = 1, 2 , ... , , , ,
n
n
P - .
. . ,
( , ;
).
. . . . , , , ( , ., [6] ),
, ,
( , [7] [10] ).
1. { 0, 1 , ..., N }
. , 0 N
i ( 0 < i
pi , i +1
= p ( p ( 0, 1) )
pi , i 1
= q =1 p
< N)
i + 1
## , 1907, . 1, 3, . 6180; [3] . .,
, 4 ., ., 1924; [4] . .,
, ., 1986, . 17378;
[5] . ., , ., 1986, . 18397; [6] . .,
, . ., 1949; [7] .,
, . ., . 1,
., 1984; [8] . ., . .,
, . 1, ., 1971; [9] . .,
. .,
, , ., 1979; [10]
. ., , . ., ., 1969; [11] ., .,
., , . ., ., 1987.
; algebra of countable type , -
, , .
; ennumerative problems i n
c o m b i n a t o r i a l a n a l y s i s , .
; heteroscedastic regression ,
.
i 1 -
;
Heisenberg model
. , 1 j N 1 ,
; {x s } ( X -
i - lim pij ( n ) = 0 .
n
## i = lim pi 0 ( n ) i = lim piN ( n )
n
i + i = 1 . p = q = 1 2 , i = 1 i N 1 .
p 1 2 , i = ( N i ) ( N 1) 1 ,
= q p 1 ( 0 i N ) .
2 ( ).
{ 0, 1, ... } . . .- pi 0 = qi
pi , i +1 = 1 qi , qi ( 0, 1 ) , i 0 pij
= 0 . i
i + 1 -
0
, n - i 1
,
i (
, ).
, , ,
( , ), pi
= 1 qi .
j = lim pij ( n ) ,
=1
= 0 p0 p1 ... p j 1 , j 1
.
.: [1] . ., . .-. - -,
1906, . 15, 4, . 135 56; [2] . ., . .
320
) S R
:
, U
h xt , A = { t } ,
U A ( x A ) = I x s x t , A = { s , t }, s t = 1 , s , t Z d ,
A -
0,
. .- ( Z
) h
=0
I > 0 ( )
, ,
-
.
. . XYZ
. X
= S 1 R 2
XY . XY 2
. Z
: XY
, ,
, , -
.
. . .
.: [1] F r o h l i c h J., S i m o n B., S p e n c e r T., Comm.
Math. Phys., 1976, v. 50, 1, p. 7985; [2] B r i c m o n t J., F o n t a i n e J., L a n d a u L. J., Comm. Math. Phys., 1977, v. 56,
3, p. 28196; [3] F r o h l i c h J., S p e n c e r T., Comm. Math.
Phys., 1981, v. 81, 4, p. 527602.
; redundancy .
X - X { X k } ,
< k <
X = ( ..., X 1 , X 0 , X 1 ,... ) ,
U - .
= 1 H (U ) H max
, H (U )
H max X
,
.
.
.
. .
. , ,
,
.
.: [1] ., ,
. ., ., 1974; [1] . ., .
., , ., 1982.
;
decision function X D -
. ( D , A ) -
x - ( | x )
( ) . . .
x ( | x ) D -
= ( x ) . . .,
. . .- , ( | x )
d
= d ( x)
, -
. ,
. .- ;
. .- X - D -
.
. .
( , , , ).
.: [1] ., ,
.: , ., 1967, . 300522.
;
; Bayes decision function ,
; m i n i m a l decision function , .
;
;
; m i n i m a l c o m p l e t e c l a s s o f decision
function , .
;
u n i f o r m l y b e s t decision function
. , .
; o p t i m a l decision function , ( ) .
;
n o n r a n d o m i z e d decision function ,
.
;
r a n d o m i z e d decision function , .
;
; ; c o m p l e t e c l a s s o f
decision function
,
. K
( , - ). K
* K * , *
R ( , )
R ( , ) ,
K K K
, R ( , ) ,
,
. , ;
;
decision criterion , , , .
; decision rule , , -
; i n v a r i a n t decision
function ,
Hi ,
( , d )
( , A , P )
; u n b i a s e d decision function , ( ) -
; a d m i s s i b l e decision
function , .
i = 0 , ..., k
{ At } ,
At A ( ), d , 0 , ..., k (
) (
). ., ,
. . .
321
( , d ) . .- w ( , , d )
,
inf
( , d )
w(,
, d ) q ( d ) =
w (
, , d ) q (d )
, ( , d ) . . q ( d ) ,
( ) . .-
.
. .
( ).
.: [1] . .,
, ., 1976.
## ; geometric probability . . . 18-
( , ). 19-
,
.
.
.
3 - (
) A - P -
. . ( J. Bertrand )
:
1) - B C
. . B C
. , P ( A )
( , .,
( ) ). . [6]-
.
. .- ,
( , ).
.
. ,
. . .-
.
( )
.
. , , ,
.
.: [1] ., ., , . ., ., 1972; [2] .,
, . ., ., 1983;
[3] B a d d e l e y A., Adv. Appl. Probab., 1977, v. 9, p. 82460;
[4] L i t t l e D., Adv. Appl. Probab., 1974, v. 6, p. 10330;
[5] Geometrical probability and biological structures, B. [a. o.], 1978;
[6] R e n y i A., Wahrschein-lichkeitchnung, 2 Aufl., B., 1966.
;
geometrically ergodic Markov chain ,
.
## ; geometric distribution m = 0 , 1, 2 , ...
=1 3.
pm = P { = m } = p ( 1 p ) m , 0 < p < 1
2)
.
P( A ) = 1 4 .
3) [ 0, 1] -
.
[ 0, 2 ] - .
P ( A) = 1 2 .
( , ).
(1 p ) p (1 p ) p - ; -
As = (2 p)
1 p ,
Eks =
= 6 + p 2 (1 p ) - .
p
,
p .
:
1)
dp d
2
1 p2
2) 1 p dp d ;
3) (2 ) 1 dp d .
.
( H. Poincar ) . ,
( ; , ). 3) . -
322
p = 0, 2 p = 0,5
. . pm
. . . ,
; p
.
. . :
. . , m n
, ,
P { m + n | m } = P { n }
. , . . . .- .
.: [1] .,
, . ., . 1, ., 1984; [2] . .,
. ., . .,
, ., 1979.
;
geometric process J . U
, .
n
J : R - d , R -
n
, R - , . . .- U .
U
. .- . U = R R . .
( ),
. . . . .
. .-; J -
n
, J - - .
, ,
,
( , ),
. H
H . R n -
, , , J
U = R n - R d - ,
( ) - ,
R d . ., -
= R n [ 0, ) -
. , J
= R2
[ 0, ) [ 0, 2 ) ,
. : O x O x 1
R [ 0, ) - d
. . .- R
, . . - (
, d
), R - .
d
R - 1 = L P ,
d
L , R - , . .- , P
; P . . .-
. .-
( ).
. .-
P ( ,
).
. .
,
. .,
; ,
( ) .
. .- .
R d - . .
. .- .
, = L m , L ,
. 1) 1
R n - , , m
, [ 0, ) - -
. 1 -
.
. .-
n
. R -
, n
R - , x
, . .- ;
.
( ) ,
.
J - . .- ( ) .
. .- .
N ( B ) B J
. B1 , ..., Bk J
. .- 1 - . . .
1 .
( ., 1 -
,
), . .- 1) - . ( ., ,
1 - k ) . .-
( ).
2)
( , [1] ).
2) 2
. 2 - -
k ( B1 ... Bk ) = M [ N ( B1 ) ... N ( Bk ) ] k -
. . 2 - . 2 -
. . .-
- .
323
k
, . .-
3)
- .
.: [1] Stochastic geometric statistics. Stereology, Lpz., 1984.
; i s o t r o p i c geometric
process , .
( )
( ); s t a t i o n a r y ( h o m o g e n e o u s )
geometric process , .
, ;
geometrization of the statistical theory . P ( ) .
{ P , } [ (
) - A ] . , ,
. ( , )
: P ( )
P ( d ) ( , ) .
, ,
( (i ) , A (i ) )
{ P(i ) , } , i = 1, 2 12 21
,
P( 2) 21
{ P(1) ,
P(1)
P(1)
12
P( 2) ,
} { P( 2) , }
. - . .,
.
-
. . .,
.
.
.
( ! ) ( )
. ( )
.
d.: [1] . .,
, ., 1972; [2] C e n c o v
N. N., Oper.
Forsch. Statist., Ser. Statist., 1978, v. 9, 2, p. 26776; [3] A m a r i
S h u n I c h i, Lect. Notes in Statist., 1985, v. 28; [4] Geometrization of statistical theory Lancaster, 1987.
; g r o s s - e r r o r sensitivity , .
324
; hydrodynamical limit ,
.
; hydrodynamical
limit approach / propagation of chaos
.
. . . . [4]-
,
,
, .,
.
. . .- , ,
. 0 ,
- - .
t = 0 P0
, , P0 P0
. ,
P0 - ( P0 ) ,
. , P0
Pt t ( )
. 0 Pt
Pt
, .
.: [1] . ., . ., . .,
.: . .
, . 2, ., 1985, . 23584; [2] B o l d r i g h i n i C., D o b r u s h i n R. L., S u h o v Y u. M., J. Stat.
Phys., 1983, v. 31, p. 577615; [3] Nonequilibrium phonemena II.
From stochastics to hydrodynamics, Amst. N. Y., 1984, p. 123294;
[4] M o r r e y C., Comm. Pure Appl. Math., 1955, v. 8, p. 279326;
[5] S p o h n H., Rev. Modern Phys., 1980, v. 52, p. 569615;
[6] S z n i t m a n A. S., Probab. Theory and Related Fields, 1986,
v. 71, p. 581613.
; hydrodynamical equation , .
; hydrodynamic equations
. . .- ,
:
3
u ( t , x ) dt +
u u
x j =
j =1
= v u + p ( t , x ) + f ( t , x ), div u = 0,
u
= (u1 , u 2 , u 3 ) , p , f -
, v > 0 , R
. -, ,
= ( x1 , x 2 , x 3 ) , u ( t , x ) = 0 , t = 0 u ( t , x ) = u 0 ( x ) . u 0 ( x ) =
= u0 ( x, ) , u ( t , x, ) .
u ( t , x, ) P
.
; Hilbert transformation , : x
## ; Hilbert random function
, .
; Hille Iosida theorem
.
X , A , X
.
. A Tt
X :
) A D A , X - ;
) f A f = g
gX
> 0
f D A ;
f DA ,
>0
f Af
X + , X - (
+
+
). f A f X , > 0 f X
, f X
Tt f X
.
.: [1] ., ,
. ., ., 1951; [2] ., ,
. ., ., 1967; [3] . ., ,
., 1963; [4] ., . ., . , . ., . 1, ., 1962.
## ; hypergeometric distribution m = 0, 1, 2 , ..., n
pm = P { = m } = C Mm C NnmM C Nn
(*)
( , )
, M , N n M
N ,n N .
n = 50, n = 10, M = 10 M = 25
. ., , ,
(*) N ,
M , N M
, n
m
. , (*)
max ( 0, M + n N ) m min ( n, M )
. (*) m > n m
Cn
= 0 m 0
= 0 n
, , p m
m
.
1 - . M N = p , (*)
## pm = C nm ANm p ANn qm ANn
,
Anm = C nm m! , p + q = 1 .
p N ,
pm C nm p m q nm
. . .- N -
,
= n p - . . .- 2 = n p q +
+ ( N n ) ( N 1) n p q
.
N . .-
.
( ) , ,
, N
.
:
6n M
p0 exp
.
3
(
2
N
1
)
+
1
. .-
(z) =
ANn M
ANn
j =0
AMj ANj
ANj M + j n
zj
.
j!
F ( , , , z )
,
= n , = M , = N M n + 1
( . . ).
. .
.
(*)
.
.: [1] L i e b e r m a n G., O w e n D., Tables of the hypergeometric probability distribution, Stanford, 1961; [2] . .,
. , . .,
., 1966; [3] . ., . .,
, 3 ., ., 1983.
325
; hypergroup
K , K K -, K - B ( K )
## ; piecewise linear aggregate ,
( x, y ) a x o y , o
## ; piecewise linear process
.
:
1) x, y , z K x o ( y o z )
= ( x o y ) o z )
( x , x K );
2) x, y K supp ( x o y ) ;
3) K - K - x x , x = x
( x o y )
= y o x x, y K ;
4) e K , e o x = x o e = x
x K ;
5) x, y K
e supp ( x o y )
x = y ;
6) K K - K - ,
R( K )
( x, y )
supp ( x o y ) .
x, y K x o y = y o x , K
; 1) 4)
, K .
[1]-
.-
( , [2] ). .-
-
.
.-
.
, , . . ,
.- ( , [4], [5]- ). .-
.
.-
.- .
.-
.
.: [1] J e w e t t R. I., Adv. Math., 1975, v. 18, 1, p. 1
101; [2] S p e c t o r R., Lect. Notes in Math., 1975, v. 497,
p. 64373; [3] D e l s a r t e J., C. r. Acad. sci. Paris, 1938, t. 206,
p. 17882; [4] . ., .: . . , . 26, ., 1985, . 57106; [5] . .,
.: . , . 24, .,
1986, . 165205; [6] H e y e r H., Lect. Notes in Math., 1984,
v. 1064, p. 481550.
; hyper Greek Latin cube
, .
## ; hyper Greek Latin
square , .
326
X ( t )
. X ( t )
X ( t ) = ( ( t ), 1 ( t ), 2 ( t ) , ..., k ( t ))
, ( t )
[ ( t ) , X ( t )
], i ( t )
, k
= | ( t ) | [ ( t ) ]
. X ( t )
-,
( 1 ( t ) , ..., k ( t ) ) -
. . . .-
, ( t ) 0
( t ) = 0 j ( t ) = ij , ij
. j ( t ) t
, ij
. ( t )
- , , 0
. . . .
; . . .-
.
. . .- ,
,
, .
, ,
. . . . . . .-
( ) .
,
, . . .- .
.
.: [1] . ., . ., . ., , ., 1973;
[2] . ., , ., 1978; [3] . ., [ .], , .
, 1988.
; histogram
. .
.
x1 , ..., xk +1 k
( , )
X 1 , ..., X n
; [ xi , xi +1 )
mi , i = 1, ..., k hi = ( mi n ) 100%
; x1 , ..., xk +1
mi ( xi +1 xi ) hi ( xi +1 xi ) ,
[ xi , xi +1 ] ;
, , .
.
[ xi , xi +1 )
mi hi - .
, ,
. k 1 + log 2 n
.
., 1000 .
.: [1] . .,
, ., 1972; [2] . .,
. , .,
1970; [3] A b o u J a o u d e S., Ann. Inst. H. Poincar. B., 1976,
v. 12, p. 21331.
; Hitsuda
Skorohod construction ,
.
; Hodges estimator ,
.
; Hall theorem , .
; Holt Winters model . , t
X t k , . . .-
,
.
22-27
27-32
32-37
37-42
42-47
47-52
100
130
400
170
100
100
, X t + k
= { a0 ( t ) + a1 ( t ) k } s ( t + k L ) -
, s ( t ) , L
( )
. .
( .,
, L = 12 ).
a0 ( t ), a1 ( t ), s ( t ) :
a0 ( t ) = 1
xt
+ [ (1 1 ) { a0 ( t 1) + a1 ( t 1) }] ,
s (t L )
a1 ( t ) = 2 { a0 ( t ) a0 ( t 1) } + (1 2 ) a1 ( t 1) ,
s ( t ) = 3
xt
+ (1 3 ) s ( t L ) ,
a0 ( t )
1 , 2 , 3 .
.: [1] H o l t C., Carnegie Inst. Tech. Res. Mem., 1957,
2; [2] W i n t e r s P., Manag. Sci., 1960, v. 6, p. 32442;
[3] ., , . ., ., 1981.
1000 ,
6 ; 5 .
. .
. .
.
.
.
. :
2
; homoscedastic regression
- , .
;
Hotelling test , T 2 , H 0 : , p
N ( , B )
L .-
(
= ( 1 , ..., p ) T - H 0
) n 1/ 3
. -
~ n1/ 3 ; m
B - 0
= ( 10 , ..., p 0 ) T
.-
. . . , p
n 1 m + 2 k ~ n m / ( m +2 )
X 1 , ..., X n , n 1 p , -
( , [1] ). C ( , [2] ).
N ( , B )
( , [3] ) L . - ( ., , . .
).
T 2 = n ( X 0 ) T S 1 ( X 0 ) ,
327
1
X =
n
i =1
1
Xi , S =
n 1
(X
X ) ( X i X )T
i =1
B .
F =
p( x) =
( ( n + 1) 2 ) x k
21
( 1 + x n ) ( n+1)
( ( n k + 1) 2 ) ( k 2 ) n
k 2
, ( 0, ) - ,
n k
, n k
1 . k = 1 . T 2 . , k > 1
n p
T2
p ( n 1)
p n p
F
n ( 0 ) T B 1 ( 0 )
2
. k Y ,
S =
T T , . , H 1 : 0 -
= 0 p N ( , B ) ,
H 0 :
X 1 , ..., X n F
, , H 0 ,
p n p
.
. . .
T
i Zi
i =1
, Z i Y -
Y ,
T 2 = Y T S 1 Y n -
. T . ( Y ,
). k = 1 ,
F .
. .- , F F ( p, n p ) , H 0
, F ( p, n p ) , F -
1
n
T2 =
Y2
n2 n
= t n2 ,
t n n - .
L ( , B ) = L ( X 1 , ..., X n ; , B ) =
1
| B | n 2
exp
np 2
( 2 )
2
(X
i =1
) T B 1 ( X i )
X 1 , ..., X n -
0 H 0 : =
= 0
. H 1 :
## = ( X 1 , ..., X n ) = sup L ( 0 , B ) / sup L ( , B )
B
, B
n = 10
T F
:
2n
= ( n 1) ( T + n 1) = ( n p ) ( p F + n p ) .
2
H 0 : = 0 . .
( , , ).
.: [1] ., , . ., ., 1963; [2] . ., , . .,
1968.
T 2
T 2 ; Hotelling T 2 distribution
: (1) k = 5;
( 2) k = 6; (3) k = 7.
, Y - ( , ) -
,
n ,
. T
. .
. T . t
, .
X 1 , ..., X n
,
T 2 = n ( X ) T S 1 ( X )
n 1 . T .-
328
X =
1
n
1
n 1
i =1
(X
X ) ( X i X )T .
x0
= 0, xn = n 1
i =1
( t
t =0
. F -
xt ( )
; , ., [3], [4] ). ,
xt ,
n k +1 2
T
nk
k n k + 1 F
( , [3], [5] ). Vn , , c n
, T .
.: [1] H o t e l l i n g H., Ann. Math. Statist., 1931, v. 2, p. 360
78; [2] ., , . ., ., 1963.
2
T
T 2 ; Hotelling T 2 statistic ,
T 2 .
; Hoeffding measure , .
; Hlders inequality
. [1]- . : , r > 1 ,
1 r + 1 s = 1 ,
12
, Vn ~ n
n
( ),
c = const , H 0,5 - (
H - 0,74 - ). H
(
)
.
xt ( ) . .-
( , [6] ). ,
( )
B ( k ) = 2 f ( 0 ) <
k =
[ B ( k ) , xt , f ( ) ], H
M | | ( M | |r )1 r ( M | | s )1 s
.
. .- ,
M | | ( M 2 )1 2 ( M 2 )1 2 .
= 0, 5
( , [3] ). . . [7],
[8]- , yt
yt +1 yt xt -,
D ( ) = M ( yt yt ) 2 = C | |2 H , 1 2 < H < 1
## .: [1] H o l d e r O., Nachr. Ges. Wiss. Gtt., 1889, 2,
S. 3847; [2] ., , . ., .,
1962.
.
xt
; Hurst
phenomenon
B ( k ) = 0,5 C { | k + 1 |2 H 2 | k |2 H + | k 1 |2 H }
( ,
xt ) . . ( , [1], [2] ) ; , , , ,
, ,
.-
690 ( )
.
xt , t = 0, 1, ..., n ( ) ( . ; R / S
) :
Vn = Rn S n =
max
0u n
t =0
xt u xn min
0 u n
n 1
x
t =0
2
t
( xn ) 2
t =0
xt u x n
,
H - ( , , [9] ).
, [10],
[4]. [3]- . .
( .,
,
) . . .
## [11], [12]- (1 B ) xt = at , 0 < < 1 2
, B xt
= xt 1 , a t
a ,
, ,
f ( ) =
= ( a2 2 )[ 4 sin 2 2 ] , H = + 1 2 .
. . xt .
329
( , ., [13] ), xt
= yt + mt
, yt
Myt = 0,
My
k =
t +k
(k ) = 0 ,
k =
yt - , mt
. . .
.: [1] H u r s t H. E., Trans. Amer. Soc. Civil Engrs., 1951,
v. 116, p. 770808; [2] H u r s t H. E., Proc. Inst. Civil Engrs., 1956,
pt 1, v. 5, p. 51990; [3] M a n d e l b r o t B., T a q q u M., Bull.
Intern. Statist. Inst., 1979, v. 48, book 2, p. 69104; [4] H e l -
## l a n d K., V a n A t t a C., J. Fluid Mech., 1978, v. 85, pt 3,
p. 57389; [5] F e l l e r W., Ann. Math. Statist., 1951, v. 22, p. 427
32; [6] L l o y d E. H., Adv. in Hydrosci., 1967, v. 4, p. 281339;
[7] M a n d e l b r o t B., C. r. Acad. sci., 1965, t. 260, 12,
p. 327477; [8] M a n d e l b r o t B., V a n N e s s J. W., SIAM
Rev., 1968, v. 10, p. 42237; [9] D a v i e s R., H a r t e S., Biometrika, 1987, v. 7, 1, p. 95101; [10] M a n d e l b r o t B., W a l l i s S., Water Resources Res., 1969, v. 5, 2, p. 22867;
[11] H o s k i n g J., Biometrika, 1981, v. 68, 1, p. 16576;
[12] A n d e l J., Kybernetika, 1986, v. 22, 2, p. 10523;
[13] K l e m e s V., Water Resources Res., 1974, v. 10, 4,
p. 67588.
## ; pure Bayes strategy ,
, .
; pure
strategy , , ; .
; pure state
, .
; chaos
(
) ,
(
)
.
. ,
x t
t (x) - . x ,
x
, ,
( ) ,
(
) . , [ , x y t ( x ) t ( y )
( ., ) ],
, t ( x ) -
t -
( , t (x) t ( y ) ),
t -
( | ln | )
. (
) ( - ) ,
( | ln | )
( t ( y ) - ), ,
, . x
,
,
.
, , ,
. ., t (x )
. .
( , [1] ).
, ,
.
-
, ?
(
).
,
(
)
,
( ,
)
( ), , ,
, ( ),
(
),
( ) ( , [2], [3] ).
, , , t
( ., )
f ( ( x ) ) g ( x ) d
t
f , g
( )
. , .
,
;
. ,
. (
) ; :
0 , t
(
),
t .
( , [3] ).
( ,
).
( , [4] ).
, ; ( ) ,
.
( , ) ( s
W W
s
). W W ,
,
331
,
, ,
.
,
, ,
. 3 2
(
) ;
. ( .
s
. ) W W -
, (
) ; .
,
, ,
( ! ) ( , [5] ).
.
.: [1] . ., [ ], ., 1976; [2] ., ,
, . ., ., 1978; [3] ., , . ., ., 1979; [4] . .,
. ., , . ., ., 1993;
[5] P a l i s J., T a k e n s F., Hyperbolicity and sensitive chaotic
dynamics at homoclinic bifurcations, Cambridge, 1993.
; h o m o g e n e o u s chaos , .
; characteristic transform
, M .
; characteristic functional ,
F ( g ) =
exp ( i g ( x ) ) dF ( x ) , g T
, T
, T , T - .
F - ( B - )
T - g -
. . . . . [1]-
.
T , B .
1) F ( 0 ) = 1 F ,
n
2) F , T
,
( T - { f n }
x T f n ( x ) f ( x ) f T f -
);
3) T - g , g1 , g 2
F ( g ) 1 ,
F ( g1 ) F ( g 2 ) 2 [ 1 Re F ( g1 g 2 ) ] ;
4) F ( g ) = F ( g ) ;
5) . .
;
6) . .
. .- ;
7) K , T - , { F : F K } . .- T
;
8) { Fn } F - , { Fn }
T - -
( ) F - .
T - 1) 2)
, , T -
,
( , [2] ).
T -
. .
. B -
- . ,
,
( , [4], [5] ).
. .-
. ., [6]
. .
.
. .- , [7].
.: [1] . ., .
, ., 1985, . 17879; [2] M o u r i e r E., C. r. Acad.
sci., 1950, t. 231, p. 2829; [3] G r o s s L., Mem. Amer. Math.
Soc., 1963, v. 46, p. 162; [4] . ., .
., 1958, . 3, . 2, . 20105; [5] . .,
.: . VI
( 1960 ), , 1962, . 45562;
[6] . ., . ., 1973,
. 18, . 1, . 6677; [7] . .,
, ., 1984.
( )
; characteristic functional o f a
r a n d o m m e a s u r e , .
; characteristic function ,
332
( z R )
cl F ( g k g l ) 0
k , l =1
c1 , c2 , ..., cn
T g1 , g 2 , ..., g n
z -
( z ) = M ei z = M ( cos z + i sin z )
(1)
; -
3. [ a, b ]
. .- :
. ( z )
.
(z) =
(z) =
e i z x dF ( x )
(2)
(z) =
( z ) . .- F ( x )
exp ( i t x ) d ( x )
(2)
, ( B )
= P { B } , B ,
.
. .:
R -
( z ) = e z
.
+
(z) =
izx
(3)
p ( x ) dx
p ( x) =
, - . .-
(4)
p ( x) =
cos z xk pk ,
a2 + ( x )2
( z ) = ei z
1 sin z x
pk .
k=
( | e
izx
| =1 )
. .- (2) (
) .
1. > 0 . .-
( z ) = exp [ ( e i z 1 ) ]
.
2.
. .-:
( z ) = ei z p + q = 1 + p ( e i z 1)
, P {
< x < +
k=
iz
( z ) = ea | z| ;
= M cos z + i M sin z
M sin z =
, . .:
e i z xk P { = xk } =
= 0 , = 1.
a
, a > 0,
a2 + x2
k=
M cos z =
.
6. :
( z ) = M ei z =
; a
(z) =
= 1 -
5. > 0
( z ) . .-
( ) .
, P { = xk } = pk ,
. .
pk 0 ,
( z ) 2
( z ) = exp i z a
(z) =
sin a z
az
4. ( a, ) . .
;
, . .
izx
e i zb e i z a
1
.
ba
iz
[ a, a ] -
F ( x) ,
k = 1, 2 , ... ;
= 1} = p , P { = 0 } = 1 p = q .
a| z|
.
. . .-
.
10. , z -
| ( z ) |
(0) = 1 .
20. a b , a + b
. .- a + b ( z )
= ei zb ( a z ) .
30. ( z ) = ( z ) .
40. ( z ) .
50. n 1 , ..., n
. .- . .-
( z ) , ..., n ( z ) - :
1
333
m1 = s1 ,
1 + ... + n ( z ) = M ei z ( 1 + ... + n ) =
= M (e
i z1
i z 2
... e
i z n
) =
m2 = s 2 + s12 ,
i ( z )
m3 = s3 + 3 s1 s 2 + s1 ,
i =1
## m4 = s 4 + 3s 22 + 4s1 s3 + 6s12 s2 + s14 ,
1 , ..., n
,
1 + K + n ( z ) = [ 1 ( z ) ]n .
s1 = m1 = M ,
## 60. z1 , ..., z n , c1 , ..., cn
n ,
n
k=
r=
1 1
M | |
, k
k -
(z) =
k =0
(0) = i M ,
3 M | |n
.-
k .
( a ; ) ( )
s1 = a, s2 = 2 .
( i z )k
(i z )n
M k +
n ( z)
k!
n!
z 0
n ( z ) 0 .
, k 3 s k = 0;
, | n ( z ) |
s3 = m3 3m1 m 2 + 2m13 ,
..........................................................................
( z ) . .- k
(k )
s 2 = m2 m12 = D ,
s4 = m 4 3m 22 4m1 m 3 + 12m12 m 2 6m 44 ,
( z k z r ) ck cr 0 .
70. n 1 ,
n
..................................................
,
.
. . ,
. . :
F( y ) F ( x ) =
f ( z ) = ln ( z ) z = 0 k
( k
1 M | | k < + ) ;
(k )
f (z ) - k
( 0 ) - 1 / i k - - k
1
i
[ ln ( z ) ](zk =) 0
1
ik
f (k ) ( 0 ) =
1 dk
ln ( z )
.
k
k
i d z
z = 0
k ( k ) k
k
.
1
f ( 0 ) = M ,
i
s2 =
i z x
e
iz
i z y
( z ) dz .
(*)
F( y ) F ( x ) =
1
lim
2 u 0
2 2
e i z x e i z y
( z ) e z u dz ,
iz
x< y.
, . .,
,
( z ) = M cos z .
( z ) . . -
.
.
1. ( 0 ) = 1 ( z ) ( z R1 ) -
1
M = f ( 0 ) ,
i
D = f ( 0 )
s1 =
lim
. (*)
:
() . k s k ,
s =
1
2
z1 , ..., z n 1 , ..., n ,
1
f ( 0 ) = D
i2
, M D ,
.
m k , k :
n = 1, 2 , ...
n
(z
z j ) i j 0
i , j =1
,
( ).
mk = M k , k = 1, 2 , ... .
2. , , ( z ) ( z R1 ) :
334
2) ( 0 ) = 1 ;
1) ( z ) 0 ;
3) z , ( z ) 0
, ( z ) . .- (
1920 ).
3. ( z ) . . expP ( z ) ( P ( z ) ), P ( z )
).
4. 1 , ..., n
, M n
= 0, D n2 = 1, FS~ ( x )
~
+ ... + n
Sn = 1
n
, ( z ) i
n
( 0 ; 1)
1}
.
. .- . .-. , . .-
. .- .
f (t ) . . f1 ( t )
= f ( x) e
i a t
=e
iat
( a
) f 2 ( t )
. .- .
. .- e i a t , . .-
. .- .
, . .
f (t ) = f1 ( t ) f 2 ( t )
f1 ( t ) f 2 ( t ) . .-
, . . . . . f1 ( t ) f 2 ( t ) f (t ) . .- ( ) .
. . ,
. . . . . ., F ( x )
, . . . .-.
. .- . .
( , [1] ).
) . .
. .- .
, .
) . .- .
n = ( 1 , 2 , ..., n ) ( ) . . z
= ( z1 , ..., z n )
( z ) = M exp [ i ( z1 1 + ... + z n n ) ]
. -
, n
. .
( z1 , ..., zn ) =
; a n a l y t i c characteristic function ,
.
## ; characteristic functions method
. . . n { FS~ ( x ), n
., , .
., . 2, ., 1984; [4] . ., . .,
. . .
, 3 ., ., 1987; [5] . .,
, ., 1983.
( zk )
k =1
.
.: [1] ., , . .,
., 1979; [2] ., , .,1962; [3] -
. 19- . .
( , [1] ). . . .- , ,
P -
.
P
f ( t, P ) =
it x
P ( dx )
.
. . . (
, ) ,
. :
1) P
f ( t , P ) .
2) P1 , P2 , ..., Pn
f ( t , P1 * P2 * ... * Pn ) = f ( t , P1 ) f ( t , P2 ) ... f ( t , Pn )
.
3) Pn n P w
( Pn
P ) , f ( t , Pn ) f ( t , P )
t R
| t | T .
4) f ( t , Pn ) t = 0
g (t ) , g ( t )
= f ( t, P )
P Pn
P
( ).
- . ., P
( ) t f ( t , P )
, P -
f ( t , P ) - , .
. . .
( , , ).
( , ,
).
335
. . .-
( .,
).
. . .
, ,
, (
).
f ( , P ) =
q ( , x ) P ( dx )
,
. . .- .
.: [1] . ., , ., 1948, . 179
218, 22150; [2] ., , . .
., ., 1979; [3] .,
, . ., ., 1981.
,
,
; characteristic index,
parameter of stable , , .
( ) ; characteristic operator
. ,
= ( t , At , Px ) E - . U U
U .
x A
M x f ( xU ) f ( x )
A f ( x ) = lim
U x
M xU
x E , f DA [
DA ( x ) A x ,
DA ( x ) ].
xE
.
. . , .
. . : x0 f
, A f ( x0 )
0.
. . -
: A f ( x ) x
f . . . : f
336
. .
, (
):
A f (x) =
1
2
2 f
( x) +
xi x j
ij ( x )
i, j
b ( x ) x
i
( x),
aij ( x ) ( ),
.: [1] . ., , ., 1963;
[2] . ., . ., , . 2, ., 1973.
; characterization theorems
.
1. X :
1) X 1 ,
X 2 , X 3
2) X - p ( x ) , x = ( x1 , x2 , x3 )
2
x1 + x 2 + x3 . X
p( x ) =
1
1
exp
( x12 + x22 + x32 ) ,
3/ 2 2
2
( 2 )
2
> 0 (
).
n
2. X R
X = ( X 1 , ..., X n ) .
,
X =
1
n
j=1
, U x
, .
x , f DA ( x ) ,
0 . R d -
bi ( x ) .
.
, q
.
f ( , P ) 1) 4) -
DA =
A f ( x0 )
s2 =
1
n
(X
X )2
j =1
. ,
, X .
3. X k R
, a1 , ..., a n , b1 , ..., bn
. Y1
= a1 X 1 + ... + an X n
Y2
= b1 X 1 + ... + bn X n
, X
.
. , n = 2 , a1 = a 2 = b1 = 1 b2 = 1 , Y1
= X 1 + X 2 , Y2 = X 1 X 2
. ai b j -
, Y1 Y2 -
X .
.
X R Q1 ( X ) Q2 ( X ) . .- .
.: [1] . ., . ., . ., , ., 1972.
## ; characterization theorems f o r p r o b a b i lity distributions on Abelian groups
. . .- ( ,
[1] )
, , .
G , G
, CG , G
(G ) , G - , s (G ) ,
G - , I (G ) , G -
. G K
K ;
T .
f n : G G f n ( g ) = ng ( n ) f n (G )
=G
(n)
G
= G , G .
G -,
( 2)
. X + Y X Y ,
, ,
, I B (G ) (G ) = E g
( , [2] ). :
CG 2 ,
G ,
X Y , Y , I (G ) (G ) ,
X + Y X Y
( , [2] ).
, X j , j
L1 = 1 X 1 + ... + m X m
= 1 X 1 + ... + m X m
= 1, 2 , ..., m R1 -
L2 =
( j , j - )
, X j
(
).
. X j , j = 1, 2 , ..., m G , j , { a j }mj=1 ,
{ b j }mj=1
[ , j -
G ( j ) { 0} ] L1 = a1 X 1 + ... + a m X m L2 = b1 X 1 + ...
+ bm X m
. G
G Rn + D ,
n0
(1)
, D
( . ) , j -
j (G ) . G ( 2 )
= { 0}
, j -
X Y , X + Y X Y
. G 3 -
, G . G -
, j - -
B (G )
. ,
I B (G ) = I (G ) I B (G ) . k I B (G ) -,
( 2)
, , K
= K .
B (G ) ( )
I B (G ) (G )
B (G )
I B (G ) (G ) = B (G )
G :
K G G / K
T 2 - ( , [2] ).
= k (G ) , K G
G / K
.
, X Y (
)
X + Y X Y ,
X Y ( ).
. , X Y
G -,
,
CG 2
, , ,
j1
= j2 = G ,
j1 j2 -
j -
( , [3] ).
. G ,
G , i X j ,
j = 1, 2 , ..., m G
m
{ a j } j =1 { b j } j =1
= b1 X 1 + ... + bm X m
j - j I ( G ) ( G ) ( , [3] ).
, L1
= a1 X 1 + ... + a m X m
L2
L1 L2 , . , ,
. X j , j = 1, 2 , ..., m G -
j ,
m
, { a j } j =1 , { b j } j =1 , G
337
. L1
+ am X m
L2 = b1 X 1 + ... + bm X m
= a1 X 1 + ... +
j - j (G )
, G ,
( p)
G
= { 0} , p ( , [3] ).
Aut ( G ) , X
. X j , j
= 1, ..., s i , G
, X j , j
, { a j }mj= 0
(2) a0 X 1 a1 X 1 + ...
+ am X m , X j
( ). I A ( G ) = I ( G ) I A ( G ) .
A M (G ) G -
,
j , j Aut ( G ) .
. G
L1 L2
j - j I ( G ) ( G ) ?
, R , n > 1 ( ,
[1] ).
.
(I) , G :
n
## () Z ( 2 m1 ) ... Z ( 2 mn ), 0 m1 < ... < mn ,
R1 - -
= 1, 2 , ..., m
I A ( G ) s ( G ) = A (G )
(3)
, G
(1) , , G ( p ) = { 0}
, p ( , [4] ).
A M (G ) (3) ,
, .
G :
p CG p
( , [5] ).
U (G ) G -
## () 2 Z ( 2 m1 ) ... Z ( 2 mn ), 0 m1 < ... < mn ,
, ,
j - ;
(II) , G Z ( 3 ) G , G () ()- ;
g G T -
. ( G ) U ( G ) . ( G )
= U ( G ) T
j - , j
, CG
j ,
G
( , [6] ).
n g = g 0 G -
j1 j2 j1 = j 2 = mZ (3)
, j - ( j )
( j ) Z ( 3 ) ,
Z ( n ) n , 2 , 2
( 2 ).
. G , .
j , G
X j j Aut ( G ) , L1
= X 1 + ... + X s
L2
= 1 ( X 1 ) + ...
+ s ( X s ) j -
j I ( X ) ( G ) .
A = { a j }mj= 0 ,
A (G ) G -
; - : X j , j = 1, 2, ..., m , m 2 ,
G -
, a0 X 1 a1 X 1 + ... + a m X m
. M (G )
A =
{ a j }mj= 0
## a02 = a12 + ... + am2
.
338
(2)
, g 0
.
.: [1] . ., . ., . ., , ., 1972;
[2] . ., . ., 1986,
. 31, . 1, . 4758; . ., 1996, . 41,
. 4, . 90106; [3] . ., . , 1988,
. 301, 3, . 55860; [4] . ., . . .,
1989, . 41, 8, . 111218; . . ., 1990, . 42, 1,
. 13942; [5] . .,
,
., 1990; [6] . ., Studia Math., 1982, v. 73,
p. 8186; [7] . ., . , 1969, . 6, 3,
. 30107; [8] .,
, . ., ., 1981.
, ; characterization of
distributions
.
. .- . 1)
x 0, t > 0
P { x + t | x } = P { t }
( ),
,
P { t } = e t , t 0,
> 0 .
2) , +
, -
. ( )
( ) .
, .
.: [1] . ., . ., . ., , ., 1972.
;
exclusion method , ,
, .
; outer measure
, .
, ; error
, ,
, ; , , .
;
probable error , .
( ) ; error o f e x p e r i m e n t , ,
.
, a
, b
= M ( X i a )
( ), i
- -
, ; , . , , 1 , ..., n
, M i
, ,
, X i
p ( X 1 , ..., X n | a, 2 )
p ( X 1 , ..., X n | a, 2 ) =
1
; s y s t e m a t i c error , .
x1 , x2 , ..., xn .
= a + i , i = 1, ..., n
, X 1 , ..., X n
. . .- . ( C. Gauss )
.
. ( W. Gosset; Student, , 1908 )
. ( R. Fisher, 1925 ) .
. .
.
( ) a
= D1 = ... = D n
.
, , , .
, ,
, b -
, ( b = 0 )
; m e a n r o o t s q u a r e error
, .
; s t a n error , .
; r a n d o m
error , .
; error integral , .
;
theory of errors , ,
( 2 )
1
exp
2
2
( X i a )2
i =1
.
. .-
. . .-
2
a
.
T =
X,X
i
2
i
. T
Xn =
1
n
S n
i =1
, a
1
n 1
(X
X n ) 2
i =1
Xn
S n
, X n a
, xi
.
.
x1 , ..., xn - , X 1 , ..., X n
.
X i
X i = a + b + i , i = 1, 2, ..., n
=0
.
1 , ..., n
dard
n 1
2
n
S n2 -
( n 1) 2 .
n ( X n a ) S n
( n 1) - , a
339
( ) .
. . .- ; .
.
.: [1] . ., , . ., ., 1960; [2] . ., -
, 2 ., ., 1962; [3] . .,
, .: , . 4, .,
1984.
; vector of
errors , .
; linear approximation , ( )
a (t ) B(t
1
[ r Y (t ) . , k (t )
. ], B (t ) (t )
. . . . . ,
.
.: [1] . ., , ., 1980.
;
sequence of linearly independent random variables
, .
; s t o c h a s t i c
, .
linear algebra
; linear filter ,
2
L ( )
2
( t ) L ( ) t
:
( ) =
a (t s ) ( s )
a ( t s ) ( s ) ds ,
a (t ) ( r k ) ,
340
a(t )
( )
it
a ( t ) dt
( )
.
(t )
(t ) =
Y (t ) =
it
( d )
a ( t1 ) B ( t2 t1 ) a T ( t2 ) < ,
it
( ) ( d ) ,
L2
. FY ( ) (t )
F
dFY ( )
= ( ) dF ( )
( ) . , ,
, fY ( )
,
= ( ) f ( ) T ( ) .
(t ) , Y (t ) ,
fY ( ) = ( )
f ( ) , ( ) , arg ( )
, ( )
.
a ( t ) = 0 , t < 0 , . .
( ) . ,
,
M ( t ) = 0, M (t )
= s , M ( t ) ( s ) = 0, t s
(t )
Y (t ) =
Y (t ) = A ( ( t ) ) =
i t
t =
s =
t1 , t 2
Y ( t )
1 ( ) ,..., n ( ) . { n ( )}
( ) =
( , F , P )
Y (t ) = A ( ( t ) ) =
ai - -
a (t )
( , ( )
) , ( ) ( )
ai i ( ) = 0 } = 1
i =1
t1 ) a ( t 2 ) dt1 dt 2 <
## ; linearly independent random variables
P{ :
a (t s ) ( s )
s =
, ,
. Y (t ) -
. F
Y (t ) , ,
, F
ln f ( ) d >
( )
ln f ( )
d
>
1 + 2
( )
. [ 0, p ] ,
a (t ) . .- (t )
(t ) =
t+ p
a (t s ) ( s )
s=t
p
,
1
f ( ) =
2
i k
, 3
., ., 1987; [3] . .,
, ., 1963; [4] . ., , ., 1980;
[5] . ., , ., 1981; [6] .,
, . ., ., 1976; [7] .,
, . ., ., 1974; [8] ., ., ,
. ., ., 1982; [9] ., .,
, . ., ., 1978;
[10] Handbook of statistics, v. 5, Time series in the time domains,
Amst. [a. o.], 1985.
,
linear function
k= 0
estimable
y = + , y RN , RN , R p ,
.
[ 0, q ] ,
M = 0, cov = 2 I
. .- Y (t )
(t ) ,
l - , a y , M a y
(t ) q
. .,
= l T .
p N
T
, l ( -
( t ) + b1 ( t 1) + ... + bq ( t q ) = ( t )
2
Q( z ) = 1 + b1 z + ... + bq z
f ( ) = 2 2 Q ( e i )
T
. l
= 0
= y ,
T
M l T T = l T ,
D l T = 2l T ( T ) l
A , A -
g , AA A = A . .,
y j = 0 + 1 + j , j = 1, ..., N y j = 0 + 2 +
( t ) + b1 ( t 1) + ... + bq ( t q ) =
T
). l l
.
1- .
:
(t k ) .
k =0
Q ( z ) , ,
( q, p )
, ( q, p ) -
( . ) .
. . - ( , [4] [8] ). .,
(t ) = ( t 1) ( t 1) + 1 ( t 1),
Y (t ) = H ( t ) ( t ) + 2 ( t )
. Y (t ) -
(t )
( , [7] ).
, [9], [10].
.: [1] . ., . ., , 3 ., ., 1987; [2] . ., [ . ] -
+ j , j = n + 1, ..., 2n l11 + l 2 2 l1 + l 2
= 0 (
).
.: [1] ., , . ., .,
1980; [2] ., , .
., ., 1980.
; linear
hypothesis
2
: n N n ( a, I )
( I ) s < n . R
s
a r s .
r s .
. .- .
= ( 1 , ..., n ) ,
, M i
= a i , i = 1,
i = s + 1 , ..., n
Di = ,
a1 , ..., a s ,
..., s, M i
= 0,
i = 1, ..., n
. a1
= ... = ar = 0 ,
341
r < s < n H 0 .
. 1 , ..., n 1 , ..., m , n + m 2
, N1 ( a, ) N 2 ( b, )
, a , b ,
= b = 0 . .-,
b = b0 . . .
. H 0 : a
a0 b0 a = a0 ,
.: [1] ., , .
., 2 ., ., 1979.
## ; linear filter of a random process , ( )
.
( )
;
; linear interpolation o f a
r e g r e s s i o n f u n c t i o n
. . .
.
Y = ( Y1 , ..., Yn ) T
X = ( X 1 , ..., X m ) T
f ( x) =
M ( Y | X ) -
= ( f1 ( x ) , ..., f n ( x ) ) , x = ( x1 , ..., xm ) R m , Y - X -
M (Y | X ) = f ( X )
.
Y = f ( X ) f (x) -
A
A M ( X X ) = M ( YX ) , A M .
,
T
D = M ( X X T ) ,
(
) A
= M( Y X ) T D 1 .
,
, , ,
. A X
f ( X ) = M ( Y | X ) -
. f ( X ) . .
Y = A X : 1) M = 0
( n . ); 2) M ( X )
=0
( M -
). Y . .
Y = AX + , AX
M = 0 .
, , , ,
.
; linear code
; linear correlation X Y ,
Y X
X Y
, | | - , .
. .
.
; linear
test
.
X
Y . f ( x )
## ; linear rank statistic ,
, Y
Y = B X
Y - , B ,
( m n ) .
M - . = Y Y = Y B X
B
.
z = ( z1 , ..., z n ) R n
## ; linear regression experiment , .
;
linear regression T
. X = ( X 1 , ..., X m ) Y =
= ( Y1, ..., Yn ) T . f ( X )
z T M ( T ) z = z T M { ( Y Y ) ( Y Y ) T } z
= M ( Y | X ) , f ( X ) = a + AX -
, Y - X -
, a
= M Y A M X , A = aij
, ( n m ) .
. Y - Y
, ,
. M X M Y ,
z M { ( Y Y ) ( Y Y ) } z z M { ( Y Y ) ( Y Y ) } z
. .- f ( X )
z R
, B
= A Y = B X
342
m n . , Y - X -
: f ( X )
= AX
. ,
. .
M (Y | X ) = A X
, , M ( Y | A )
= a + AX
(1)
,
.
Y , yi , i = 1, ..., n n
Z = (1, X 1 , ..., X m ) T , B = a : A
,
M (Y | X ) = M (Y | Z ) = B Z
(2)
( (2) ) . ( B A a ). (1) , M X
= Om
, Ok
M Y = On
= (1
o , ... , o ) . ,
424
3
k
Y - X - ,
Y Y , Y
= A X
= Y Y : 1) M ( | X ) =
= M = On ; 2) M ( X T | X ) = M ( X T ) = Om n ,
Om n m n
( ,
Y = Y +
).
A A -
A M ( X X T ) = M ( Y X T ) . X
D = M( X X T )
, A A
= M ( Y X ) D 1
T
.
. . X Y
-
.
, , , ( ) .
## ; linear regular stationary process ,
.
; linear system , .
;
linear estimator ,
(
) . . . .-
,
. , . . .- ,
, , , ,
.- ,
. , ( )
. . .-
. ., Y = X +
( )
= ( X X ) 1 X Y , Y
; X , n p ( p - ) xi ,
i = 1, ..., n , k = 1, ..., p
, Y p
.
Qk p , k = 1, ..., p
M = 0, M ( ) = 2 I ( I )
n .
.: [1] ., ,
. ., 2 ., ., 1975; [2] . ., , . ., ., 1968; [3] .,
, . ., ., 1975; [4] ., , . ., ., 1980.
linear m i n i m u m v a r i a n c e estimator
R ( Y , , 2w ) T
l - ;
w
( . . . ).
. . . l
=0
l -
T
, ,
Tw 1 = Tw 1Y .
; a d m i s s i b l e linear
estimator , .
; linear stochastic differential equation
, .
;
; ; linear stochastic differential equation; s t a b i l i t y o f s o l u t i o n s t
R n - . . . . ,
d X (t ) = B( t ) X ( t ) dt +
r ( t ) X ( t ) dw r ( t )
(*)
r =1
, B
r , n n , X (t ) , R n - ,
w r (t ) . ,
, (*) . , p . . . .- p
343
. , (*)
p (
) , x - p
V ( t , x ) , k i
>0
( , ) X i ,
k1 | x | V ( t , x ) k 2 | x | ;
p
L V ( t , x ) k3 | x | ;
2V
xi x j
V
k 4 | x | p 1 ;
xi
= K + f
, p > 0 -
p . t
.
n
. . . .- R -
| x | = 1
. n = 2
( , [4] ). n
( , [1] ).
.: [1] . ., , ., 1969.
; linear stochastic differential parabolic equation o f s e c o n d o r d e r
, . . . . .- , .,
.
d u ( t, x ) =
b (t , x ) u
i
i =1
l =1
i =1
il (t ,
xi
i, j = 1
ij
( t , x ) u xi x j ( t , x ) +
(t , x ) + c (t , x ) u (t , x ) d t +
x ) u xi (t , x ) + hl (t , x ) u (t , x ) dw l (t )
aij ( 1 2 )
il
jl
, w (t ) ,
.
344
i = 1, 2 , ..., m .
k4 | x | p 2
[ L (*) ].
t - (*) -
;
;
; linear equation; M o n t e C a r l o
s o l u t i o n o f -
, .
; K . X i :
1) n = K n + f
;
Xi
;
2)
;
3)
.
.: [1] . .,
, ., 1975; [2] . ., . .,
. ., , ., 1984.
; Markov line-wise process
( t ) = { X ( t ), Y ( t ) } , burada X ( t ) = t t k birinci tip s t k - ( t k t < t k +1 )
, Y (t )
( t k , t k +1 ) E = {i }
. t k
( t k +1 0 )
( 0, j )
= ( y, i ) Pij -
. . . .-
.
. . . .
.
.: [1] . .,
, .: . VI . . ,
, 1962, . 30923.
; chi divergence , .
; chi square test , 2 ,
0 : = ( 1 ,..., k )
,
. ( K. Pearson, 1903 ).
n ; k E1 , ..., E k ,
, (
i j
Ei I E j = ,
i, j = 1, k ). i , Ei
, i = 1, k . i ni
= ( 1 , ..., k )
Pk 1 ( x ) = P { k21 x } =
n!
P { 1 = n1 , ..., k = nk } =
p1n1 p2n2 ... pknk ,
n1! n2 ! ... nk !
e z 2 dz
.
, : n E1 ,..., E k k ,
cov ( i , j ) = n pi p j .
i j
pi( 0) = pi(n0) n -
n pi i
. pi
k
> 0,
## p1 + ... + p k = 1 , 1 + ... + k = n , k < n , M i = n pi ,
n k , 0- .
( 0)
min n pin
=1
[2]- X 2 -
) H 0 ,
p1
p1( 0) ,
..., pk
k 2
( , [23] ).
pi i :
( k 1 ) -
pi = P ( Ei ) , i = 1, k .
D i = n pi (1 pi ) ,
2 ( k 1)
1
2
( ( k 1) 2)
pk( 0)
pi( 0)
pi( 0)
i =1
X2 =
( i n pi( 0) ) 2 n pi( 0) = n +
i =1
2
i
n pi( 0)
.
2
H 0 X x
: X
H 0 , X
<x
H 0
. x - , H 0
,
- ( 0 < < 0,5 ) . , x - X
## > 30 , min n pin(0) 5
. k
P { X x | H0 }
(*)
(0)
( 0)
sup P { X 2 x | H 0 } Pk 1 ( x ) ;
| x| <
min n pin( 0) . - . .-
i
. [11]- . [10]-
. k > 3 k n -
( 0)
pin
, i
= 1, 2 , ..., r
r + 2 , ..., k n
pin( 0)
pin( 0)
pin( 0)
mi , i = r + 1,
= 1, 2, ..., r
0 ). H : pi = pi(n0)
(U
; m1 , m2 , ..., mr
( , i
. - x - X
( , k n - ),
, . ,
x x - ; (*) ,
.
: n n pi
0 , k ;
, n , k pi
P {X 2 < x | H }
0
( ) -
i =1
x k +1
sup
2 ( k 1)
| x| <
X 2 n
mi ) 2 mi + k2 r 1
i =1
, U i , mi
, U 1 , ...,U r ,
k2 r 1 . X 2
5 r k min n
i
pin( 0)
<5
(0)
n pin
i 5- r
.
345
X 1 , ..., X n
F ( x, ) - ( | x | < )
T
= ( 1 , ..., m )
{ F ( x ; )} ,
H 0
. - . .
. x0 < x1 < ... < xk ,
( x0
= , xk = + ) k ( k > m )
( x0 , x1 ] , ..., ( xk 1 , xk ] ,
i = 1, 2 , ..., k
pi ( ) = P { X 1 ( xi 1 , xi ] | H 0 } > 0 ,
p1 ( ) + p2 ( ) + ... + pk ( ) = 1 X 1 , ..., X n
= ( 1 , ..., k )
. -
X 2( ) =
n pi ( ) ] 2 n pi ( )
i =1
. H 0 X ( n )
, n
,
~
X 2 ( n ) = min X 2 ( ) .
pi -
( pi () > 0 ) , pi ( ) u
2
- ( i = 1, 2 , ..., k ; u ,
pi ( ) u
= 1, 2 , ..., m )
m - , [19]-
H 0 ,
~
X ( n ) n
( k m 1)
, . - . . H 0 ( , [1], [6], [8],
## [14], [17], [23] ).
X 2 ()
X 1 , X 2 , ..., X n n [
n , L () =
f ( X ; ) ,
i
f ( x; ) =
i =1
F ( x; ) x
H 0 , X 2 ( n )
n
],
## 12 + 22 + ... + k2 m 1 + 1 k2 m + ... + m k21
( , [18] ),
346
1 ,..., k 1 M 1
= 0 , D 1 = 1
, 1 , ..., m
0 1 ,
( , [18] ). . . ( , [18] )
, . - .
.-
( , )
.
. [19], . . [18]- ,
,
,
.
,
.
[3]-: ., ,
;
[13] [16]- , [15]- ,
; [5], [7] [9], [20]-
n -
## ; [5], [12], [21]-
.
, ., [2] [5], [9], [14],
[22], [23]- .
.: [1] ., .,
, . ., ., 1973; [2] . .,
. ., 1971, . 16, . 1, . 320;
[3] . ., . ., 1973, . 18,
. 3, . 58392; [4] ., . ., .:
, . 10, ., 1987,
. 4971; [5] D r o s t F. C., Centrum voor Wiskunde en Informatica,
1988, v. 48; [6] ., ,
. ., 2 ., ., 1975; [7] . ., . ., . ., 1974, . 19, . 4,
. 88688; [8] L a u t e r H., P i n c u s R., Mathematisch statistische
Datenanalyse, B., 1989; [9] M c C u l l o c h C. E., Communs Statist.
P. A. Theory and Meth., 1985, 14 (3), p. 593603; [10] Y a r n o l d J. K., JASA, 1970, v. 65, 330, p. 86486; [11] C o c h r a n W. G., Ann. Math. Statis., 1952, v. 23, 3, p. 31545;
[12] . ., . ., 1973,
. 18, . 3, . 67576; [13] . ., .
., 1979, . 24, . 2, . 38589; [14] . ..,
. ., 1977, . 22, . 2, . 37578;
[15] . ., ., .
., 1978, . 23, . 3, . 48194; [16] . .,
. ., .
, 1975, . 1, . 10409; [17] W a t s o n G. S., Biomtrics,
1959, v. 15, p. 44067; [18] C h e r n o f f H., L e h m a n n E. L.,
Ann. Math. Statist., 1954, v. 25, p. 57986; [19] F i s h e r R. A.,
J. Roy. Statist. Soc., 1924, v. 87, p. 44250; [20] D u d l e y R. M.,
.: Banach center publication, v. 5, Warsz., 1979, p. 7587;
[21] M o o r e D. C., JASA, 1977, v. 72, 357, p. 13137;
[22] K a l l e n b e r g W. C. M., O o s t e r h o f f J., S c h r i e v e r B.
F., JASA, 1985, v. 80, 392, p. 95968; [23] G r e e n w o o d P.
E., N i k u l i n M. S., A guide to chi square testing, N. Y., 1996.
## ; chi square distribution , 2 , ( , )
p( x) =
n2
1
e x 2 x n 21
( n 2)
, ( 0, ) - ;
n ,
.
, , ,
F ) . ,
. . .
.
. . . , .
. . .-
. n - : .,
( n2 n )
2n
.
n
P { n2 < x } ( 2 x ) 2n 1 )
, ( x )
. , .
.: [1] ., , .
., 2 ., ., 1975; [2] ., .,
, . ., ., 1966; [3] L a n c a s t e r H.,
The chi-square distribution, N. Y., [a. o.], 1969; [4] .
., . ., , 3
., ., 1983.
1)
2)
; chi distribution , , n
p( x) =
2
1
x n 1 e x 2 , n > 0
2 n 21 ( n 2 )
( , ), ( 0, ) - ,
.
: 1) (1) n = 2 ;
(3) n = 6 2) (1) n = 10 ; (2) n = 20 ; (3) n = 30 .
(2) n
=4;
. . .
; III
. . . .-
( ) . . . .- f ( t )
= (1 2 i t ) n 2 ; -
, n 2 n - .
. . .- .
n - . . .- ,
X 1 , ..., X n ,
n2 =
= X 12 + ... + X n2 . . . .-
.
. . . . .,
,
( 1 , ..., n )
n2
## - : (1) n = 1; ( 2) n = 3; (3) n = 10.
f (t ) =
1
( n 2 )
k =0
( i 2t ) k
( (n + k ) 2 )
k!
; :
m k = 2 k / 2 ( ( n + k ) 2 ) ( n 2 )
( ; ,
347
. ,
,
( ( n + 1) 2 )
2 ( ( n + 1) 2 ) ( n 2 ) n + ( 2 2 )
( n 2 )
. X 1 , ..., X n ,
n
2
i
i =1
n - . .
; n = 2 . . , n = 3
.
;
queueing discipline ( , )
. . . .
. ., ,
, .
( , . ). . .
. . .-: ( ), (
), (
) .
; queueing system ( , )
m ( 1 m ) ; . .- .
, , . . ;
.
, . .
. ,
,
( , . ). . .
( , ;
).
. . 20- 50-
.
. : 1, 2 , ..., m ; i
i + 1, ..., m
. .
.
.
.
( ), ,
( ).
.
- . .-
. .,
.
348
-
( . )
: .,
.
( , ,
, , ,
). . . . .-
: ., ; ;
; ;
, .
.: [1] . ., . .,
, 2 ., ., 1987; [2] .,
, .
., 2 ., ., 1971; [3] . ., , ., 1972; [4] ., , . ., ., 1979;
[5] ., , .
., ., 1979.
; o p e n queueing system ,
.
; s i n g l e s e r v e r queueing
system , .
; s i n g l e - c h a n n e l
queueing system , .
; m u l t i l i n e a r queueing
system , .
; m u l t i c h a n n e l
queueing system , .
; m u l t i c a s c a d e
queueing system , .
; ; ; s t a b i l i t y o f a
queueing system , ( )
.
,
; s i n g l e - s e r v e r
queueing system
.
M G 1 (
; ,
) . .
( 1932 ).
, ( s ) = M exp{ s } ,
. = < 1
( s ) = (1 ) 1 (1 ( s ) ) , Re s 0
s
= M .
, ( s ) = M exp { s } ,
,
.
( GI M m ,
G I Ek m , M G 1 , Ek G 1 )
. ., G I M m -
( s )
( s ) = ( s + ( s ) ) Re s 0 , s > 0 .
M G 1 , G I M 1 , E k G 1 ,
G I Ek 1 -
k
pk
k m 1 pk = A k
, A ,
(1 ) m x
dF ( x )
,
, F ( x )
= a < m
. G I G 1
( , ).
( )
.
. .-
.
. . .- .
.
.: [1] . ., , ., 1966; [2] . .,
, ., 1972; [3] , . ., ., 1984; [4] . ., . ., . ., , ., 1982.
M M m k -
,
; m u l t i s e r v e r queueing system
( , )
. G I G m . .- (
).
. , n - wn
## min { wn1 , ..., wnm } , wni , n -
n - i -
. { wn1 , ..., wnm }
.
a ,
. < m a ,
; wn - n
.
ma
wn
a -
m a -
.
G I G m , , .,
,
k w -
: , m
k Z
, k
= (1 )
k 1
=1,
(1 ) - . . .-
.
.: [1] . ., . .,
, 2 ., ., 1987; [2] . ., , ., 1972; [3] C o o p e r R. B., Introduction to queueing
theory, 2 ed., L., 1981.
; queueing loss / balk / blocking system ,
( ) (
). . .-
. M M m 0
. ( A. Erlang ) . ( C. Palm )
( , ) ,
,
. ( ,
).
( , ,
; ).
( , )
.
; ,
. ;
; ;
.
. .- :
( )
. . .-
349
. . .
. ,
( , ; )
( , [2] ).
. .-
. .,
. .
. .
, , .
.: [1] . .,
, ., 1963; [2] . .,
. ., . .,
, ., 1968; [3] . .,
, ., 1980.
; c l o s e d queueing system
, .
; p a r t i a l l y
a v a i l a b l e queueing system ,
.
; f u l l y a c c e s i b l e
queueing system , .
; queueing theory ,
,
. . . . . .- : ,
,
,
:
( , [1] ).
. . . . ( A. Erlang ) 20- 20-
,
. . . 1932 33-
.
. . 1931- , . . . 1934- . . . . . .
( 1955; , [1] ).
. . .-
, ,
. ( , ) ;
( , ) ;
.
350
,
.
(
). m - ,
( m
=1
) ( m
>1
) .
.
, r
.
( ). ( r = ) ( r = 0 ) .
. 20- 50-
. ( D. Kendall )
.
A| B | m | r , m r , A , B
. A | B | m | , ,
A B m . A = GI ( general independent ) ,
B = G ( general )
. A | B | M , E k ,
D ,
( ) , , . , Z n -
n -
.
M M m r
. (t ) t
( ) . (t ) k > m k = , k m + r
k = 0 , k m k = k ,
k > m
k = m ,
, . r < =
, t k
pk (t ) t k
. ,
. N
N m + r - .
. r
<m
= / ;
(t )
m - .
; = m
, . < m (t )
( , ,
).
( 1953- . ) . . . : ,
.
. . 20- 30-
M G 1 .
. . . :
1. .
, ,
.
( ,
)
,
.
( , )
.
.
-
. ,
(
).
:
.
.
.
.
( , )
.
.
, . ,
x -
F ( z | x ) .
, . . . .
( , , ).
2. . ,
, . ,
. . .- .
.
, . .
G I | G | 1
. . ( ) , . . .
.
.
( , , ). . . .
.
.
( .,
)
.
. . .
( , ;
,
).
3. .
( )
.
. ., M G
. m M M m
. (
,
) .
. . .
( , )
( )
,
( ,
).
4. .
. . ( 1955 ) . , ,
, . ( , . )
( , ).
. . .- ,
. : .
. . 1932-
( M | G | 1
- );
L = w , L
( ), ,
w ; . .
:
351
, ( , . )
- .
5. , .
, ,
. :
pik
, ,
i - { f k }
(f
f i ) pik , > 0 -
. i -
f k pik < . -
( 1956 ).
( M | G | m | r )
. x(t )
,
f ( x ) | x |
f ( x ) ,
t 0 M f ( x ( t ) ) -
( ,
; ).
( ) .
,
. , .
6. . , , ( ) . .,
. , ;
.
7. . . .- . ;
, , .
.: [1] . .,
, ., 1963; [2] . ., . ., , 2
., ., 1987; [3] . .,
, ., 1966; [4] . ., , ., 1980; [5] . ., [ .,] , .,
1973; [6] . ., . ., . ., , ., 1982; [7] ., , . ., ., 1979.
( ,
) . ,
.
1.
( , )
.
,
.
( , [1] ),
( , [2], [3] ).
.
2. ,
( )
. ( , [4] ),
.
,
.
( .,
)
. 1- ,
,
.
, .
: (
), (
, )
.
.
.
. . .- , ,
.
, (t ) =
= { e ( t ), r ( t ), s ( t ); t 0}
, ,
q (t ) = e ( t ) r ( t ) s ( t ) 0 . e (t )
t ,
r (t ) t ,
s (t ) . ,
, q (t ) (t )
= r ( t ) e ( t ) .
e, r , s
( . )
, .
(t ) -
;
; asymptotic methods
of queueing theory
e, r , s
352
. ,
, .
, ( , [5] ).
3.
( ) . ,
, ,
.
(
) . ,
.
-
. ,
( , [6] ), ( , [7] ), ( , [8] ), ( , [9] ),
( , [5] ).
4. .
( ) ( ) ( , [10] [11] ). .,
q (t ) = ( q1 ( t ) , ..., q N ( t ) ) 1, ..., d
N =
q (0)
i
i =1
1
q ( u N )
N
( u ) = ( 1 ( u ) ,..., d ( u ) )
. (u )
,
q (t )
.
, ,
. .
.: [1] K i n g m a n J. F. C., Proc. Camb. Phil. Soc., 1961,
v. 57, p. 90204; [2] . ..,
, ., 1972; [3] . .., . ., 1972, . 17, . 3,
. 45868; [4] . ., . . ., 1963, . 3,
1, . 199205; [5] . ..,
, ., 1980; [6] F r a n k e n P.,
Operationsforschung und Math. Stat., 1970, 2, S. 923; [7] . ., . ., 1975, . 20,
. 4, . 83447; [8] . .., .
., 1977, . 22, . 1, . 89105; [9] . .,
.: , ., 1980,
. 5256; [10] D a i J. G., Ann. Appl. Probab., 1995, v. 5, p. 4977;
[11] . ., . ., , 1992, . 28, . 326.
n
) (t )
: ( t )
= ( ( t ); 1 ( t ) , ..., n ( t ) ) ,
(t ) , i (t ) , i -
; i ( t ) = 0 . . . ( D. R. Cox,
1955 ) , . .
( , ).
; ; ; queueing theory; i n s e n s i t i v i t y p r o b l e m / i n v a r i a n c e p r o b l e m , , ,
,
.
( , [1] ). . .
( 1961 ).
.
.: [1] , . ., ., 1984.
; statistical simulation of
queueing systems , ( )
; queueing network -
( , ) .
( , ) ; ; .
. . .- .
. . ,
. . .-
,
.
.
N ( )
n . j -
G j ( x ) = P { j < x }, j = 1, ..., N
(k )
j },
k = 1, 2 , ... ( ) -
;
; ;
queueing theory; s u p p l e m e n t a r y v a r i a b l e s
m e t h o d ( t ) = ( ( t ); 1 ( t ), 2 ( t ),...)
, (t ) ( ), i (t ) , t
, . , n , , G
. .,
M G n (
353
= ij
. i -
ij
j - .
, G
= ( G1 , ..., G N )
.
. .-
, G j ( x )
q ( t )
jx
= [ q2 ( t ) , ..., q N ( t ) ]
, q j ( t ) ,
, q1 ( t )
= 1 e
j - t
= n q2 ( t ) ... q N ( t ) .
lim P { q ( t ) = k } = pn ( k ) ,
k = ( k 2 , ..., k N ) ,
(1)
pn ( k ) = c
( / )
i
ki
= n,
i =1
i =1
c , i ,
( , [1] [4] ).
. .- G
= 1, 2, ...
(k )
, . .- 1 , k
j , j 2 ij
. j -
j1 .
N 1 ( 2- N - ) . , G
.
, q1 ( t ) = .
Gj ( x) = 1 e
jx
, q (t )
.
j 2
1 1 > j j
(2)
),
,
. . . , . 20- 60- , 70-
;
,
, , . .
. .
( ,
[10] [13] ).
.: [1] ., ,
. ., ., 1979; [2] ., , . ., ., 1979; [3] K e l l y F. P., Reversibility and stochastic Networks, N. Y., 1979; [4] . .,
. ., .: . .
. . , . 21, ., 1983, . 3119; [5] G n e d e n k o B., K o n i g D.,
Handbuch der Bedienungstheorie, Bd 2, B., 1984; [6] F r a n k e n P.,
[a. o.], Queues and point processes, B., 1981; [7] G o o d m a n J.,
M a s s e y W., J. Appl. Probab., 1984, v. 21, p. 86069; [8] . ..,
, ., 1980; [9] . ..,
, 2 ., ., 1966; [10] . ..,
. ., 1986, . 31, . 3, . 47490; 1987, . 31, . 2,
. 28298; [11] . ., . ., .:
. . . . , . 26, ., 1988, . 396;
[12] . ., . ., . ., , ., 1989; [13] D a i I. G.,
Ann. Appl. Probab., 1955, v. 5, p. 4977.
; Khinchin inequality f o r i n d e p e n d e n t r a n d o m f u n c t i o n
L p .
f k p > 2
lim P { q ( t ) = k } = p ( k )
(3)
p(k ) = c
sup f k
Lp
( t ) dt = 0
.
N
( / )
i
1 i
ki
c ( , [1] [4] ).
(1) (3)
( , [1] [4] ). (1), (3)
G j ( x ) -
, (2) 1 M 1
> j j , j 2
i =2
( , [9] ).
( , [8], [9] ).
. .-
(
ck f k
k= 0
Lp
k =1
1/ 2
ck2
N , ck - .
. ,
n , pn ( k ) p ( k ) ,
354
< ,
2
k
< ,
k =1
rk = sign sin 2 k t
k rk ( t )
f (t ) =
, p > 0
k =1
Ap
k =1
1/ 2
ck2
1/ p
| f ( t ) | dt
Bp
k =1
1/ 2
ck2
, B p
= ( O p ) , p .
. . [1]- . A1 -
1
2 - .
. .- ( , [4] ).
0 < p , q < C ( p, q ) , B
xk
C ( p, q)
k rk ( t )
k =1
k rk ( t )
k =1
Lp
Lp
. . .- :
, G I 0
, G I 0 F1 , F2 ...
( G Fk - ).
.: [1] . ., . . . , 1937, . 1,
. 1, . 617; [2] . ., . ., , ., 1972.
; Khinchin theorem i n t h e
q u e u e i n g t h e o r y 1)
( ) -
a k2 + bk2 <
= lim
( ak cos k t + bk sin k t )
k =1
L p - ( p
<
) .
## .: [1] K h i n c h i n A., Math. Z., 1923, Bd. 18, S. 10916;
[2] K a r l i n S., Trans. Amer. Math. Soc., 1949, v. 66, p. 4464;
[3] . ., . ., 1966, . 21, . 6,
. 382; [4] . ., ,
. ., ., 1973; [5] .,
, . ., . 1, ., 1965.
; Khinchin formula , ,
, .
; Khinchin spectral function , .
; Khinchin
theorem X nj S n = X n1 + ... + X nn
. S n An
- ; An . . .-
. . ( , [3] ).
, , .
.: [1] . ., , . ., 1938; [2] . ., . .,
, . ., 1949; [3] . ., . ., 1936, . 1(43), . 91730.
( )
( ); Khinchin factorization theorem
. H = H 1 H 2 H k
( k = 1, 2 ) Ea
H E a , H
. G
. ( [1]-, ,
[2] ) , F F = G F1 F2 ...
k =1
, 1
1
P { ( ) > 0}
. . ( +
);
2) . . ,
; ;
3)
. .: ,
.
.: [1] . ., . .,
, 2 ., ., 1987.
; Khinchin theorem i n t h e t h e o r y o f s t a t i o n a r y r a n d o m p r o c e s s e s ,
, ( t ) ,
< t <
B ( )
= M ( t + ) ( t ) -
B ( ) =
e i dF ( )
(1)
- ; ( t ) F ( ) ,
. . . 1934- . .
( , [1] ) . . ( N. Wiener; , [2] )
1930- x ( t ), < t <
,
lim
1
2T
B ( )
x ( t + ) x ( t ) dt
= B ( )
(2)
- ; ,
B ( ) (1) (
M ( t + ) ( t ) - ; , x ( t ) ( t )
355
). .
, F ( 2 ) F (1 )
[ 1 , 2 ]
x ( t ) (
) .
, B ( ) | |
, . .
, B ( ) -
f ( ) f ( ) d ( )
d x ( t )
. . . [3]- 1914- .
( , [4], [5] ).
. .- ( t ) ,
t = 0, 1, 2 , ... . [7]- .
(1)-
.
.: [1] . ., . , 1938, . 5,
. 4251; [2] .,
, . ., ., 1963, . 4; [3] E i n s t e i n A.,
Arch. Sci. Phys. Natur., 1914, Ser. 4, t. 37, p. 25456; [4] . ., . ., 1985, . 21, . 4, . 101
07; [5] . ., .: . 1982
1983, ., 1986, . 2556; [6] . .,
, ., 1981, . 14546;
[7] W o l d H., A study in the analysis of stationary time series.
Uppsala, 1938.
; Khinchin Kolmogorov
theorem , .
; partial coherence , .
; partial correlation
function k kk -;
k - kk
- . X t ( p, q )
, ( p, q )
;
( B )( X t MX t ) = ( B ) At
, ( B )
, B X t
(*)
= 1 1 B ... p B p
= 1 1 B ... q B q -
= X t 1 , M X t , X t
2
, At , a
. k ( )
X t kk , k
356
= 1, 2, ...
. . .
=0
, X t
, k
>p
kk
= 0 .
p 0 , q 0 , k > p , . . . ( B )
.
. . . ,
.
.: [1] ., ., .
, . , . 12, ., 1974.
; partial correlation coefficient
. , ,
X 1 , ..., X n R n -
## X 1 34 ... n X 2 34 ... n X 1 X 2 , X 3 , ..., X n . X 1 X 2 . . . 12 34... n
, Y1
= X 1 X 1 34 ... n Y2 = X 2 X 2 34 ... n
12 34 ... n =
M { ( Y1 M Y1 ) ( Y2 M Y2 ) }
DY1 DY2
, 1 12 34 ... n
1.
. . .
= ij
ij , X i X j , ij | | ij
,
12 3 ... n =
k = 1, 2 , ... -
, ( B )
( , [1] ). (*)- q
., n
=3
12
.
11 22
12 3 =
12 33 13 23
2
2
( 1 13
) (1 23
)
X 1 , ..., X n X i X j
. . . . . . . 12 34 ... n
X 1 X 2 ( ) 12 -
. 12 34 ... n 12 X 1 X 2
X 3 , ..., X n
. X 1 , ..., X n -
, . . .- .
. . .-
r12 34 ... n =
R12
R11 R22
, Rij , rij
rij - R = rij
P { ( t1 ) = ... = ( t n ) = 1} =
.
, r12 34 ... n
( ( N n + 1) 2 )
( 1 x 2 ) ( N n 2) 2 ,
( ( N n ) 2 )
. t 0 p ( t ) 1
p ( t )
r
1 r2
, r
= r1234 ... n
p (t ) = exp
( 0, ]
min ( t , x )
m
(
dx
)
1 ex
, m , ( 0, ] -
,
( N n) .
.: [1] ., , .
., 2 ., ., 1975; [2] ., ., , . ., ., 1973.
/ / ; marginal distribution , .
/ ; arithmetic of special semigroups
n
. . . . R -
. . . .-
20- 60- . .
( , [1] ) . . . . . : I ( )
, I 0
n
( R -
; , ;
), N . .
. p P . (t ) ,
t 0 , 0 1
0 = t 0 < t1 < ... < t n
= P { ( t ) = 1} p -
. p P
( N ). . . .
N n
ti 1 ) = 1}
. P -
.
I
1 < x < 1
t =
i =1
P { ( t
; I 0
= { e t : 0} ,
e t -
; N G P - .
: 1) : n
; n
SO ( n + 1)
R - -
SO ( n ) -
; . 2)
,
: ;
; .
3)
: ,
; ; , .
.: [1] Stochastic analysis, L. [a. o.], 1973; [2] U r b a n i k K.,
Stud. ath., 1964, v. 23, 3, p. 21745; [3] . ., . ., 1986, . 31, . 1, . 3
30; [4] . ., . ., 1986,
. 31, . 3, . 43350; [5] R u z s a I., S z e k e l y G., Algebraic
probability theory, 1988.
; special semimartingale , .
; control sequence ,
.
; control o f t h e
systems with distributed parameters
. .-
. ,
. ., ,
.-.
.-
; . .,
.
d ( t , x ) = [ L ( t , x ) + l ( t , x ) u ( t , x )] dt + b ( t , x ) dw t ,
( t , x ) |D = 0 , ( 0, x) = ( x )
[ L , w ,
( x ) , l b , t R + ,
x D R n ].
u .- .
.: [1] K u s h n e r H. J., SIAM J. Control, 1968, v. 6, 4,
p. 596614; [2] .-., . , 1973, . 28,
. 4, . 1546; [3] B e n s o u s s a n A., J. Franklin Inst., 1983,
v. 315, p. 387404; [4] . .,
, ., 1985; [5] ..,
, .
., ., 1987.
## ; controlled diffusion process d R d
,
.
. . . dxt = b ( t , s + t , xt ) dt ,
t 0 ( t )
xt R
w ti i
d
i A
{w ti + h
w ti
; ( , t , x )
i = 1, ... , d1 ) , t 0 ( , A , P) -
{ At } - d1 (
= ( xi , i = 1, ..., d ) R d
( d d1 ) , b ( , t , x )
d . , b - , t , x -
, , t -
x -
b i ( , t , 0 )
ij ( , t , 0)
A - ,
{ At } - t = t ( ) ,
t 0 ,
; A
d
. s 0 , x R , A
dxt = b ( t , s + t , xt ) dt + ( t , s + t , xt ) dw t ,
x0 = x
(1)
(
, s, x
). xt
) . A
.
C ( [ 0, ), R d ) R d - , [ 0, ) -
. [0, ) t
d
. C ( [ 0, ), R ) - x[ 0 , )
C ( [ 0, ), R d ) , s
. N t
x s ( x[ 0, ) -
N t -
C ( [ 0, ), R )
) s
wt =
( At ) -
( , t , x ) , b ( , t , x )
= 1, ..., d1 }
t , h 0 -
) . A -
:
. . .- .
( , A , P) , { At , t 0} , A -
(w ti ,
, i
. a A , t 0 , x
, At
, w t , ..., w t 1 , w t t ,
A -
= t ( x[ 0, ) )
{N t } - t = t ( x[ 0, ) )
(1) { At } -
,
= t ( x[ 0, ) )
, ( s , x )
. ( s , x )
AT ( s, x)
358
t ( xt )
A M ( s, x) ( s , x )
. [ 0, t ]
xr t -
m , t , x - ].
( M ) = .
( s, x ) = sup M s, x
, t . AT ( s, x )
t
e t f ( s + t , xt ) dt + ( s + , x ) e
0
[ A M ( s, x ) ] (1)-
. , s 0 , x R , AT ( s, x ) d
(1)
xt , s , x . ,
t ( ) = t ( x[0,, s, x) ( ) ) AT ( s, x ) A , s, x
, xt
, s, x
= xt
( ) .
, s,
, , xt
C ( t , x )
0 , f ( t, x )
sup ( L + f ) = 0
L =
( s, x ) =
a
a ( , s, x ) = ( aij ( , s, x ) ) = ( , s, x ) ( , s, x ) 2
f t ( s + t , xt ) dt + g ( s + Q , x Q ) e
,
, s , x ,
.
( s, x ) -
( s, x ) = sup ( s, x )
(2)
( s, x )
( s, x )
. 0 ,
. (2)- A AT ( s , x )
T ( s , x )
, -
( M ( s, x ) )
. A M ( s, x )
AT ( s, x ) A ( M ) ( s, )
( T ) ( s , x ) ( s , x )
= [ -
, ., , b, c, f , g ( , x ) ,
, ( s, x )
- x -
, t , x -
K (1 + | x | )
c, f , g
, K
= 0 ( s, x )
(4)- t0 =
= 0 ( s 0 + t , xt )
,
( s0 , x0 ) - -
= { t0 } ( s0 , x0 ) -
( M ) ( s0 , x ) = ( s0 , x0 ) .
= ( , T ) R d , T , , b , c , f , g (4) ( , [1] ).
, Q
. . .
. ., A
( , [1] ), T
t
.
. . . ,
, ,
g - Q - , -
()
. ( s, x )
(A M ( s, x ) )
c ( s , x )
xi
i, j 1- d -
( s + r , xr , s , x ) dr ,
= M s , x
+
+ aij ( , s, x )
xi x j
s
+ bi ( , s, x )
( s + t , xt , s , x ) - Q -
r
(4)
. A, s 0 , x R d
, s, x
t Q = min ( t , Q ) , ( s + , x ) e t -
[ 0, ) R - g ( t , x ) -
t , s , x =
, s, x
, . (3)-
.
. A [ 0 , ) R
(3)
, ( s , x ) = Ms , x
Q
e t f t ( s + t , xt ) dt + g ( s + Q , x Q ) e Q
359
. . .- ,
,
d
( C [ 0, ), R , N ) , .
.: [1] . ., , ., 1977; [2] F l e m i n g W. H., R i s h e l R. W.,
Deterministic and stochastic optimal control, B. [a. o], 1975;
[3] . ., . ., 1975, . 98, 3, . 45093;
[4] . ., . .,
, ., 1974; [5] W o n h a m W. M., SIAM J. Control,
1968, v. 6, 2, p. 31226; [6] D a v i s M. H. A., SIAM J. Control,
1976, v. 14, 1, p. 17688.
; controlled Markov
process
,
. , ,
. . . .-
( )
.
,
. . . . (
) . . .
. . .- ;
( ,
, ). . . .-
( , );
,
( , [4] ). . . .-
( , [5], [6] );
( ,
). . . .- [7]-
.
. . .- ( )
. . .- (
) .
.: [1] . ., . .,
, ., 1977; [2] . .,
, ., 1977; [3] .,
.,
, . ., ., 1978; [4] ., . . ., 1978, . 18, . 1, . 14767;
[5] . ., . ., 1971, . 86 ( 128 ), . 61121;
[6] B e n s o u s s a n A., L i o n s J. L., Lect. Notes in Math.,
1975, v. 134, p. 522; [7] C h i t a s h v i l i R. J., Lect. Notes in
Math., 1983, v. 1021, p. 7392.
## ; controlled Markov chain
,
. . . . ,
; . . .-
360
Px - (
M x - ) . . .- x
. . . .
.
. . . .
. . .: 1)
X , 2) x X
( ) A ( x ) , 3) a A( x ) -
x y
p ( y | x, a) . ,
( x ) = M x
r ( xt 1 , at ) t 1
(1)
t =1
[ ( 0, 1] ],
g ( x ) = lim
1
M x
n
r (x
t 1 ,
at )
(2)
t =1
, r ( x, a ) , a A( x ) ,
x X
r ( xt 1 , at , xt )
p ( xt | xt 1 , at )
]. - ( g )
( )
[ g ; g
(2)- ].
:
Pa f ( x ) =
f ( y ) p( y
x, a )
y X
Ta f ( x ) = r ( x , a ) + P a f ( x ) , a A( x )
P f ( x ) = sup P a f ( x ) ,
a A ( x )
T f ( x ) = sup T f ( x ) , x X .
a A( x )
< 1 (1) = T
( ) . (2)
g
g = Pg , h + g = T1h
(3)
( h
X ) (3)- ,
h g .
x A( x ) ,
x X
. g ,
,
.
X A
. . .-
( , [2], [3], [5], [7] );
.
, ,
; .
.: [1] .-.,
, . ., ., 1964; [2] . .,
Trans. 4-th Prague Conf. Inform. Theory , 1967, . 131203;
[3] . ., . .,
, ., 1975; [4] ., .,
, . ., ., 1977;
[5] . ., . ., 1980,
. 25, . 1, . 7182; [6] . ., . .,
. , 1982, . 37, . 6, . 21342; [7] . ., . ., . , 1984, . 275,
. 80609.
: 1)
( X , X ) ; 2)
;
controlled object , ,
( A , A ) ; 3) x0 X , 0 < t1 <
## < t 2 < ... < t n
> 0 )
x1 , x2 , ..., xn X ( n = 0, 1, 2 , ... )
( t1
/
;
controlled semi Markov process / s e m i M a r k o v
decision
p r o c e s s ,
x (t ) , t 0 ,
a1 , a 2 , ...
## 0 = t 0 < t1 < t 2 < ... x ( t n 1 ) an , ,
= t n t n 1 x ( t n ) . . . . . [1]- .
. , , M
. . .
. . = 1 . . . .-
.
.: [1] J e w e l l W. S., Operations Research, 1963, v. 11,
p. 93871; [2] ., .,
, . ., ., 1977; [3] D e n a r d o E. V.,
F o x B. L., SIAM J. Appl. Math., 1968, v. 16, p. 46887;
[4] R o s s S. M., J. Appl. Probab., 1970, v. 7, p. 64956; [5] . ., . . - , 1970, . 111,
. 20823; [6] . ., . ., 1981, . 26, . 80815.
, .
## ; controlled jump process ,
. . . .- . ,
, ( . . .- )
. ( )
a A t 0 x
q ( t , x0 t1 x1 ... t n xn , a ) ; 4)
t n +1
=t
t n + 1 ( n + 1) - -
xn +1 - Q ( t , x0 t x1 ...
t n xn , a ) , X , X - . , , q . x0 t1 x1 t 2 x2 ... t n
= { x ( t ) } :
t n t < t n +1 , n = 0, 1, 2 , ...
x ( t ) = xn
(*)
= 0 t n -
+ ). At , t > 0 = { }
( t 0
; x ( s ) , s
<t
A
t
. A - , { A }t = 0
t
= ( t, ) , t > 0 ,
. t n ( ) < t
tn + 1 ( )
( t , )
= ( x0 t1 x1
... t n xn , t ) -
.
, x X . . .,
, { x ( t ) }t 0 ,
(*)
Px { x0 = x } = 1 ,
Px { t n +1 > t x0 t1 x1 ... t n xn } =
tn
## = exp q ( s, x0 t1 x1 ... t n xn , ( x0 t1 ... xn , s ) ) ds
Px { xn + 1 | x0 t1 x1 ... t n xn t n+1 } =
## = Q( | t n +1 , x0 t1 ... xn , ( x0 t1 ... xn , t n +1 ) ) , n = 0, 1, 2, ...
. . . . ( ),
q Q x0 t1 ... xn xn
, . . .,
t - , . . .
. { x ( t ) }t 0 ( t , )
= ( t , x(t ) )
,
( t , ) = ( t , x ( t ))
.
361
( x ) = M x
( s , x ( s ), ( s ) ) ds
0
r ( t , x ( t ), ( t ) ) dt
( ) . . .
,
, X , A (
r t - ),
M x , Px , r ,
0 . .,
( x ) = ( 0, x ) = sup ( x ) , x X
t 0
(t , x) .
( t , x ) = sup [ r ( t , x , a ) + q ( t , x, a )
t
a A
[ ( t , y ) ( t , x ) ] Q ( dy | t , x , a ) ( t , x, a ) ( t , x )]
X
( ) .
, , ,
, . . . . . . .-
, .
. . .-
.
.: [1] . ., . .,
, ., 1977; [2] . .,
. ., 1980, . 25, . 24770; [3] D u y n S c h o u t e n F. A. v a n d e r, Markov decision process, Amst., 1983.
;
controlled random sequence ,
; controlled random
process ; ( ) . . . .-
( , ,
, ).
; controlled discrete time
random process , ,
( ) . , ( ) ( ) .
.
362
. ( R. Bellman )
( , [1] ); ,
( ) ( , [2] ).
.
. . .- .
[3]- . .
. , [5] [6] [7]- . .
.- . .,
. . . . . .-
( , [8] );
.
. . ( , [9], [10] )
. . ( , [11] ) .
. . .- [13]-
. 70-
. . .
.
. . .,
( ). Z = { X , A, A ( h ) h H , p }
: 1)
( X , X ) ; 2)
( )
( A , A ) ; 3) H = U H t , t = 0, 1, ...
A ( h ) , h H
( H0
), Dt +1
=X
= { ( h , a ) : a A ( h ) , h H t } , H t A
H t +1 = Dt +1 X ;
h H t t
ht
ht = x0 a1 x1 ... at xt , xi , xt X ,
ai +1 A ( x0 a1 ... xi ), 0 i < t ,
(1)
a A ( ht ) t + 1 at +1 - ; 4)
p ( / ha ) , X , h H , a A ( h ) h a Dt +1 ,
t = 0, 1, 2 , ... ( X , X ) - ,
X Dt +1 - ha .
x X .
ht , t 0 at +1 -
; , , ( | h ) ,
A , h H , h H
A ( h ) - A H (t ) -
( t 0 ) h ( A , A ) - . H (t ) ( h ) t ( h ) . [14], [20]-
, h
; -
.
t h A ( h ) , h H t
[
, ,
t ( h ) A ( h ) t : H t A ].
, , H - (
) ; ( Z , ) , (3)
. { n }n 1
,
. { t } t 1 ht t ( ht )
t ( | h ) = 1 ( t ( h ) ) , h H t , t 1 ;
- . H (1) h -
w ( x ) = lim M x n ( h )
n
(
, , , n - ). :
. x X -
H - Px
, , :
Px { at +1 B | x0 a1 ... xt } = ( B | x0 a1 ... xt )
( Px )
## Px { xt +1 | x0 a1 ... xt at +1} = p ( | x0 a1 ... xt at +1 ) . (2)
( Px )
B A ,
X , t = 0, 1, 2 , ...
Px { x0 = x } = 1
{xt } , t = 0, 1, 2 , ... Px ,
x
Z . . .-.
Px M x .
p
xt +1 = f ( ht , at +1 , t +1 ) , t = 0 , 1, 2 , ...
. ( , [19] ).
A ( ht ) = At ( xt ) , p ( | ht at +1 ) = pt ( | xt at +1 ) , t 0
t 1 at
t =1
( , ).
.
(3)
x X H
Px , M x
hH
. (2) x - h H n , n > 0 ,
Ph { ( x0 a1 ... xn ) = h } = 1
, t n .
( )
w ( h ) = sup w ( h ) , h H
. H t - w w
wt wt
; w0 . w ( h ) - h - .
a A( x) , x X
,
].
t ( | ht ) = t ( | xt ) , t 0 ,
= ( | x) ,
x X , .
( ) ,
{ xt }t 0 ( ) .
, x X
w ( x )
= w( x )
h H w ( h ) = w ( h ) ,
( . )
,
[ A t ( x ) = Ax , pt ( | xa ) = p ( | x A) ,
w ( x ) w( x ) g ( x ) ,
x X
X -
, g
( g
= const = > 0
).
, ,
, .
hN = hN ( h ) -
( N ) .
w ( x ) = M x ( h )
r (x
w ( x ) = w ( x ) , x X -
, { t }t 1 -
, t ( | x )
1
n
w ( x ) = lim M x
(3)
363
R ( h ),
( h ) =
(4)
U ta f ( h ) =
t =1
Rt t .
Rt
0 ( Rt 0 )
Ut f (h) =
a
t - , -
( ) . (4)
n 0
:
wn ( hn ) = M h
a A ( h ) , h H t 1 ,
Tt a f ( x ) =
[ r ( x a y ) + ( x a y ) f ( y ) ] p ( dy | xa ) ,
t
h Hn .
U t f ( h ) = sup U ta f ( h ) ,
Z (4)-
Rt ( ht ) =
i ( xi 1 ai xi ) rt ( xt 1 at xt ) , t 1
i =1
, , i
>0.
rt ( x a y ) t ( x a y ) , x a y X A X
t
a A( h )
U t f ( h ) =
i =1
i ( xi 1ai xi ) n ( xn ) , n = 0, 1, 2 , ...
n ( x ) , x X , t
= 0 t = n
a
t
f ( h ) ( da | h ),
h H t 1
; (4)
U t U t
( ) ; , t 1 .
wn ( hn ) =
a At ( x ), x X
t >n
Rt (ht ) ,
wn ( h ) = sup wn ( h ) ,
[ Rt ( hay ) + f ( hay ) ] p ( dy | ha )
f ( hay ) p ( dy | ha ),
Tt f ( x ) = sup Tt f ( x ) ,
a At ( x )
Tt f ( x ) =
f ( x ) t ( da | x ), x X
, (
a
= T a Tt = T t - ).
( Z , ) .
Tt
( ) Z
, rt = r t =
(
)
:
t - . n ( x ) = ( x ) = w ( x )
n - . , , = const
;
,
=1
<1
, ( Z , )
Z , t 1
r (a x y ) , 1 t N
rt ( x a y ) = R( x ) , t = N + 1
0,
t > N +1
,
N
. X - R .
(4) H t - H t 1 - , X X - .
t = 1, 2 , ...
364
wt 1 ( h ) = U t wt ( h ),
h H t 1 , t = 1, 2 , ...
wt 1 ( h ) = U t wt ( h ),
t 1 ( x ) = Tt t ( x ), x X , t = 1, 2 , ... ,
(5) (6)
(7)
( x) = T ( x) , x X
, ,
t 1 ( x ) = T t ( x ) , x X , 1 t N
(8)
.
X
. ,
, N ,
, t = 1, 2 , ..., N
[ (5) wN
= , (6)
wN = 0 , (7) N = 0 , (8) N = R ].
.
.
( )
wt 1 ( h ) = U t wt ( h ) ,
h H t 1 , t = 1, 2 , ... ,
wt 1 ( h ) = U t wt ( h ) ,
t 1 ( x ) = Ttt ( x ) , x X , t = 1, 2 , ... ,
,
wt1 ( h ) = U t wt ( h ) ,
h H t 1 , t = 1, 2 , ...
wt1 ( h ) = U t wt ( h ) ,
t1 ( x ) = Tt t ( x ), x X , t = 1, 2 , ... .
.
- ,
; (4)
,
(9)
lim M h w n ( hn ) 0, w ( h ) , h H
n
lim M h wn ( hn ) = 0 , h H
, . (9) -
= .
, ,
.
X , A
w - ( ) > 0
( )
. :
1) U t Tt ( ) ; 2) h A ( h ) , h H t 1 ( ) a
= t ( h ) x At ( x ) , x X
= t ( x ) ,
> 0 a A ( h ) h H t 1
U t wt ( h )
a At ( x ) x X
Tt a t ( x ) .
, . U t Tt
.
.
( )
: 1) X A
; 2) A Dt , t 1 ;
3) p , t = 1, 2 , ...
, X - f
F ( ha ) =
f ( x ) p ( dx | ha ) , ha Dt
Dt - ; 4) H - ( ); 5) (4)
Rt -
.
t rt , t , rt .
wt , wt t
(
, ), U t , U t Tt ,
t ;
( , ) ( [19],
[23]- ).
: 1) X A ; 2) Dt , t 1
p ( | ) ;
3)
; 3 ) (4)
Rt ;
3 ) t rt
. U t , U t Tt
,
, ,
t , , -
x - (
) ( H t 1 X -
), . (
) ( ,
[14], [20], [24] ); X , A , A ( h ) , P t
, , Rt rt
(
{ z : f ( z ) > c } ).
U t , U t Tt
, wt , wt t
( , ), H t 1 X - -
t
( h x -
).
> 0
.
K .
: 1) w K
w ( ) ; 2) K
( . )
; 3) w
K . K
365
, , , (
x0 x1 ht )
.
. . .
. .,
,
[ 1) ]
,
.
[25], [26]- .
X A -
.
( ) .
. . .- t xt
( yt , zt ) ,
( B | ht )
yt , z t
A ( ht )
., , .
., ., 1977; [19] . ., . ., , ., 1977; [20] .,
., : , . ., ., 1985; [21] . .,
, ., 1981; [22] W h i t t l e P., Optimization
over time, v. 12, N. Y., 198283; [23] S c h a l M., Proc. 6-th Conf.
Probab. Theory, Bus., 1981, p. 20519; [24] . .,
. ., . , 1982, . 37, . 6, .
21342; [25] . ., . ., Statistics
and control of stochastic processes: Steklov seminar 1984, N. Y.,
1985, p. 69101.
; controllability
. x , n , u , n , A (t ), B (t ) , n n n r
; x
F ( t , ) B ( ) B
y0 a1 ... yt . ( y , )
, z 0
. t z t
366
( ) F T ( t , ) d
( F ).
A , B ,
( yt , zt ) - ( yt , t )
.
. .
. . .- ,
( , [21] ).
. . .-
. ( ) , , ,
, ,
.
, , ;
.
.: [1] ., , .
., ., 1960; [2] B e l l m a n R., J. Math. Mech., 1957, v. 6,
p. 67984; [3] .-.,
, . ., ., 1964; [4] B l a c k w e l l D.,
Ann. Math. Statist., 1962, v. 33, p. 71926; [5] B l a c k w e l l D.,
Ann. Math. Statist., 1965, v. 36, p. 22635; [6] B l a c k w e l l D.,
Proc. 5-th Berk. Symp., 1967, v. 1, p. 41518; [7] S t r a u c h R. E.,
Ann. Math. Statist., 1966, v. 37, p. 87190; [8] D u b i n s L. E.,
S a v a g e L. J., How to gamble if you must, N. Y., 1965; [9] . ., Trans. 3-rd Prague Conf. Inform. Theory, 1964,
p. 65781; [10] . ., Trans. 4-th Prague Conf. Inform.
Theory, 1967, p. 131203; [11] . .,
. ., 1965, . 10, . 318; [12] . .,
. , 1964, . 155, . 74750; [13] H i n d e r e r K.,
Foundations of non-stationary dynamic programming, B., 1970;
[14] B l a c k w e l l D., F r e e d m a n D., O r k i n M., Ann.
Probab., 1974, v. 2, p. 92641; [15] H o r d i j k A., Dynamic
programming and Markov potential theory, Amst., 1974; [16] S t r i e b e l C h., Optimal control of discrete time stochastic systems, B.,
1975; [17] . ., . .,
, ., 1975; [18] .,
= A ( t ) x + B ( t ) u [ s , t ]
, ,
rang [ B A B A 2 B ... A n 1 B ] = n
. ( A , B )
d x ( t ) = Ax( t ) dt + B dw ( t )
x (t )
t
> 0 ,
, , ( A , B )
. .
.
.: [1] K a l m a n R. E., Bol. Soc. Mat. Mexicana, 1960, v. 5,
p. 10219; [2] . ., . .,
, ., 1974; [3] . . .,
, . ., ., 1984.
; ideal gas
-
(
). . . > 0
( -
z -
- ).
p R
( -
), R - . ( )
( 2 1 ) d
exp
d 2
. . . ( ,
( ) ): ( d p )
, p R . . .
. .-
. - , . .-
K ( , [1] ) ,
B ( , [2] ).
M ,
, t = 0 S t t
;
. (3) . .
2 = M ( X s Y s ) . 2 ( X , Y ) = 0
FX
Ds , m ,
f ( m ) ( x ) f ( m ) ( y ) x y , x , y R1
> 0 ( d p )
.
. . ,
( , [3] ).
. . ; R ( , [2], [4], [5] ), . . .
. .- ( , [6],
[7] ). , , , . .
.
.: [1] . ., . ., .
., 1971, . 5, . 3, . 1921; [2] A i z e n m a n M., G o l d s t e i n S., L e b o w i t z J., Comm. Math. Phys.,
1975, v. 39, 4, p. 289301; [3] L a n f o r d O., L e b o w i t z J.,
.: Lecture Notes in Physics, v. 38, Berk. [a. .], 1975, p. 14477;
[4] . ., . ., 1972, . 6, . 1,
. 4150; [5] . ., . ., .:
. , . 14, .,
1979, . 147254; [6] Comm. Math. Phys., 1985, v. 101, p. 36382;
[7] M a l y s h e v V. A., N i k o l a e v I., J. Stat. Phys., 1984,
v. 35, 3-4, p. 37579; [8] B u n i m o v i c h L. A., S i n a i Ya. G.,
Comm. Math. Phys., 1981, v. 78, 4, p. 47997.
= FY , FX ,
X .
3. m 0 < 1 , s = m + > 0 .
, R - f ;
s ( X , Y ) = sup { M ( f ( X ) f ( Y ) ) : f Ds }
s . .-. s 1
:
s( X, Y ) =
1
(s)
( x t )
s 1
d ( FX ( t ) FY ( t ) ) dx ,
.
4. f X , X
. s
3 ( X , Y ) = sup { | t | s f X ( t ) f Y ( t ) : R1 }
s . .-.
5. L ( X , Y ) ,
2
2
r
4 ( X , Y ) = sup
L ( xX , xY ) dx : r > 0
; ideal metric s -
. .-.
X - ( X , Y )
FX1 ( x )
, X -
,
(X
Z ) (X , Y )
Z, Y
(1)
Tc : X X, c > 0 s
(Tc X , Tc Y ) = c s ( X , Y ).
(2)
## = sup{ u : FX ( u ) x } , FX 4 ( X , Y ) = ( FX1 , FY1 )
.
. .- .
6. 5 ( X , Y ) = sup{ | x | s FX ( x ) FY ( x ) : x R1} , s 0
Tc X = cX
Y = max ( X , Y )
s . .-.
7. 6 ( X , Y ) = sup { | t | s W X ( t ) WY ( t ) : t R1 } , s 0
. .- .
.
Tc X = c X X Y = X + Y
. .-, W X , X -
. . ,
. . ( ; ).
8.
. . . .,
X Y = X +Y ,
1. ( X , Y ) = M X Y
0 < 1 ,
. .-.
2. 2 ( X , Y ) = sup { | M ( ( X + x ) s ( Y + x ) s ) |: x R1 },
s . .- ( s 1 ). 2
M ( X k Y k ) = 0, k = 1, ..., s 1
(3)
Tc X = | x |c sign x X
,
Y = X Y
Y = max ( X , Y ) Tc X = c X
, s
= 1 . .-.
. .-
( , )
.
367
X 1 , ..., X n , M X 1 = 0 , M X 12 = 1 , , Y1 , ..., Yn
,
. S n
= ( X 1 + ... + X n )
n , S n* = (Y1 + ... + Yn )
n -
. S n n - . s > 2 s
s ( S n , Y1 ) = s (( X 1 + ... + X n )
n , ( Y1 + ... + Yn )
n) =
= n s 2 s ( X 1 + ... + X n , Y1 + ... + Yn )
n s 2
( X k , Yk ) = n13 n s ( X 1 , Y1 )
(4)
k =1
.
s = ( X 1 , Y1 )
M X
, (3)
< , (4) -
n s ( S n , Y1 ) -
.
X - ,
-
= 0 . .
: a
c > 0
( Tc ( X
a ) , Tc ( Y
a ) = (X , Y )
X , Y X.
. .-
, , U n = T1 C n X n a n ,
Vn = T1 C n Yn
an -
X n , Yn
3) q ( w ( u c ) ) c q ( w ( u ) ) , c > 0 ;
4) q ( cw ) c q ( w ) ,
1
, R .
s
= d + ( + )
. .-.
.: [1] . ., . .,
1983, . 28, . 2, . 26487; [2] . .,
, ., 1986,
. 1.
; idempotent measure o n a g r o u p =
-, ( ) .
G
G -
( , [1] ). ,
, ., ,
(
)
- .
. .-
.
G G - . .-
G - (
) .
( ) ( , [2] ).
.: [1] ., - , . ., ., 1981; [2] ., , . ., .,
1950.
, .
Tc s = 0
## ; ruin problem / gamblers ruin problem
. .-. Tc ,
X , Y X Tc ( X
Tc
Y ) = ( Tc X ) ( Tc Y ) , c > 0 ,
. .-
. .,
Tc
. . .
, R
X Y - ,
( X , Y ) = q{ x ( X
TX U , Y
TX U ) }
q ( w )
w (u ) , u R1 W
w, w1 , w2 W .
1) q ( w ) 0 , q ( 0 ) = 0 ;
2) q ( w1 ) q ( w1 + w2 ) q ( w1 ) + q ( w2 ) ;
368
X 1 , X 2 , ...
S n = X 1 + ... + X n ,
= ( a , b) = min { n : S n ( a , b )}
S1 , S 2 , ... ( a , b ) ,
a < 0 < b . X j 0 , P { < } = 1 . . . P { S
P { S A , = n },
a }, P { S b }
P { S n B , > n },
A ( , a ) U [ b, ) ,
(*)
B ( a, b)
. ( , ,
), (
), , - .- .
- ,
.
, ,
p , q
=1 p
. X j j - ,
P { X j = 1} = 1 P { X j = 1} = p
, S n n .
a , b
. { S a }
, { S
b}
. :
P { S a } =
1 ( p q )b
, pq;
1 ( p q )a +b
P { S a } =
b
, p = q =1 2
a+b
. ,
b a , p = q = 1 2 ,
a
a + b 1 ( q p )a
M =
, pq;
q p
q p 1 ( q p ) a + b
M = a b, p = q.
2n
P { S a, = n } =
p(na) 2 q(n+a) 2
a+b
a 1
cos
n 1
a+b
k =1
P { > n } =
( 4 pq ) ( n +1)
a+b
p
q
b2
sin
sin
k
a+b
a + b 1
sin
ak
a+b
k
k
sin
a+b
a+b
k
1 2 pq cos
a+b
cos n
k =1
q
kb
+
a+b
p
a 2
sin
ka
.
a+b
( , [3] ) . ( A. Moivre ), .
( N. Bernoulli ), . ( J. Lagrange ), . ( P. Laplace ) ,
.
(*) .
X j - ,
M(e
i X j
, Xj
< 0)
M ( e
i X j
( X j - e
, Xj
> 0)
- ),
(*)
( , [4] ). .,
## P { X 1 x } = C e x , C > 0 , > 0 , x > 0
X 1 -
p1 0,
k = 1, 2 , ...
. (*)
, . , X j
(*) a , b
( , [1], [2] ).
.
.: [1] . ., . .,
1962, . 7, . 4, . 393409; [2] . ., .
., 1979, . 24, . 3, . 47585; . 4, . 87379; [3] T a k a c s L., J. Amer. Statist. Assoc., 1969, v. 64, 327, p. 889906;
[4] K e m p e r m a n J. H. B., Ann. Math. Statist., 1963, v. 34,
4, p. 116893.
*
; distance between two random variables
= ( ) = ( )
r ( , ) = sup
x R1
P { : ( ) < x } P { : ( ) < x } =
= sup F ( x ) F ( x)
x R1
r ( , ) ; -
P { X 1 = k } = C p1k 1 , C > 0 ,
; F ( x ) F ( x ) , ( )
( )
r ( , ) r ( F , F )
1
. r R -
. ,
d
1) r ( , ) 0 r ( , ) = 0 = ;
2) r ( , ) = r ( , );
3) r ( , ) r ( , ) + r ( , ).
, , .
.: [1] ., ,
., 1999.
*
( ) ; double sequence o f r a n d o m v a r i a b l e s { X k , n }; k = 1, 2 , ..., n ; n = 1, 2 , ...
. n - ( )
, . . ( ,
, ).
.: [1] ., , .
., . 1, 2., ., 1984.
stationary random process , ( )
.
369
## ; doubly stochastic matrix
pij
1- ,
i -
ij
. ( ) W
(1)
Q ( ) ( , [4] ).
( ) , (W , U ) -
= 1,
j -
( )
( d ) N ( ) -
W
ij
= 1.
. . .
( , ; ), ,
- ,
,
.
.: [1] .,
, . ., . 1, ., 1984.
*
(
) ; doubly stochastic Poisson process
( , [1] ) . , . . . .-
( ) W ( ) .
N ( ) . . . .- ( ) ( ) -,
P{ N ( B ) = k } =
[ ( B )]k ( B )
e
( d ) =
k!
[ ( B)]k ( B )
e
k!
= M
, ,
( [1], [2], [3] ).
( , F , P ) , R = ( , + ) ;
.
.
N ( ) , t
B , R - ;
M ;
M
B = { N ( ) : N ( A1 ) = r1 , ..., N ( Ak ) = rk } , k 1 ,
r1 0 , ..., rk 0 ,
F - P M -
. , N ( ) ,
( ) = M N ( ) W ,
U
C = { ( ) : ( Ai ) = xi , i = 1, ..., k } ,
Ai B , Ai I A j = , i j , x 0 , i, j = 1 , ..., k , k 1
, W -
.
B B
[ ( B ) ]k ( B )
e
, k = 0 , 1, 2 , ...
k!
+
+1
t e t , t 0 , ( +1) = t e t dt ,
f (t ) = ( + 1)
0
0,
t < 0 , > 1, > 0
, G ( 0 )
; Q ( ) ,
, N ( )
0
. . . ., ( t ) = ( 0, t ] = G ( t ) , t 0 ; -
( ) , G (t )
A1 , ..., Ak B , Ai I A j = , i j
P{ N ( B ) = k } =
M -
(1)
( ) W ; N ( B1 ) , ...,
P{ N ( t ) = n} =
, B B , | B | , B
( n + + 1)
( n + 1 ) ( + 1 ) + G (t )
+1
N ( t )
= N ( 0, t ] , t 0 -
.
. . . .-
.
. . . .- .
1. ( ) . . . .,
1
, , ( x )
. 2. ,
, . . . .- ,
f *( s) =
. , <
= B
. -
G( t )
, n 0
+ G ( t)
N ( Bn ) , n 1 , Bi B , Bi I B j = , i j , i, j = 1, ..., n
( B )
=0
1
1 ln g * ( s )
g*( s ) =
f 0* ( s ) = g 0* ( s ) f * ( s )
sx
d G ( x ) , G ( x )
370
G (0) < 1 ,
g 0* ( s ) =
e sx dG0 ( x ) G0 ( x )
( 1 , 2 ) -
. 3.
Me
s 1 ( 0 )
= g 0 ( s ) , Me
s [ 1 (1) 1 ( 0 ) ]
= g (s)
.
4. F ( x ) ,
, . . . .- ,
>0
K(z) , z > 0 ,
z dK ( z ) < +
, s 0
f *( s ) = + s +
(1 e sz ) dK ( z )
(*)
. (t ) :
0 . , (*)
K ( + ) K ( 0 ) < . K = 0 , . K > 0 ,
K ( z ) K (0)
K
(t ) 0 -
G (z) =
; (t )
. (
, 1 2 ) -
1 K - ,
( t ) = 0 G ( x )
.
.: [1] . ., . ., . ., . . ., 1983; [2] S n y d e r D. L., Random point process, L. etc: Wiley and Sons, 1975;
[3] G r a n d e l l J., Doubly stochastic Poisson process. Lect. Notes.
Math., 1976; [4] K e r s t a n J., M a t t e s K., M e c k e J.,
Unbergrenzt teilbare Punktprozesse, Berlin: Akad. Verlag, 1974.
; two-armed
bandit , .
( ). t = 1, 2 , ...
( 2 2 ) ,
, .
. . [1]-
.
( 1 , 2 ) ( , ) - ,
( M , ) - , M
> ,
1 2 - (
). [2]- .
,
.
( )
( , [3] )
. [4]- m n
( ) , ( m n )
.
( m n ) ,
( 2 2 )
.
,
.
.: [1] T h o m p s o n W., Biometria, 1933, v. 25, p. 28594;
[2] F e l d m a n D., Ann. Math. Statist., 1962, v. 33, p. 84756;
[3] G i t t i n s J., Multi-armed bandit allocation indices, N. Y., 1988;
[4] . ., . ., , ., 1982; [5] B e r r y D., F r i s t e d t B., Bandit problems, L. N. Y., 1985.
;
; dual Markov process .
## ; binary symmetric channel ,
j -
, F j
1 p
1 p
= 1, 2 .
. 1 p p
, ( F1 , F2 )
. . . . . . . .
; j
.
. ( ,
). u
.
, .
, : 0 1 . ( 1 , 2 ) ,
( )
C = 1 + p log p + (1 p ) log (1 p )
.
( )
binary digit o f a n i n f o r m a t i o n
;
quantity
, , .
371
## ; Laplase second law / distribution ,
.
; bivariate / two dimensional normal distribution X 1 X 2 ( )
(t ) = exp it T m
, t
1 T
t Ct
2
= ( t1 , t 2 ) , m = (m1 , m2 )
C =
12
1 2
1 2
22
X 1 X 2 .
| | < 1 , . . .- ,
f ( x1 , x2 ) =
1
2 1 2 1
exp
2
2(1 )
( x m )2
2 ( x1 m1 ) ( x2 m2 )
( x2 m2 ) 2
1 21
1 2
22
1
( ).
X 2 = x2 X 1 -
m1 = m 2 = 0 , 1) 1 = 2 = 1 ,
f ( x1 | X 2 = x2 ) =
1 2 (1 2 )
1
1
exp
x1 m1
2
2 12 (1 2 )
. | |
. . .- , [1].
( x 2 m2 )
2
F ( x1 , x2 ; ) -
F ( x1 , x2 ; ) = ( x ) ( y ) +
= 1 -
m = 0 , 12 = 22 = 1 1 2 -
f ( x , x ; ) d .
1
f ( x1 , x2 ; ) , F ( x1 , x2 ; ) - ,
( x)
372
k ( x) k (
y)k
k ( x ) = ( 2 ) 2 H k 1 ( x ) exp ( x 2 2 ) k! , k = 1, 2 , ...,
F ( x1 , x2 ; ) = P { 1 < x1 ; 2 < x2 } =
k =1
F ( x1 , x2 ; )
= ( x1 ) ( x2 ) +
= 0 ; 2) 1 = 1 , 2 = 2 ,
= 0 ; 3) 1 = 1 , 2 = 2 , = 0,8
.
H k ( x ) .
.: [1] . ., . .,
,
., 1962.
T 2
T 2 ; two-sample T 2 statistic
T .
; two-sample Student
test , .
## ; two-sided Bernoulli shift , .
;
double exponential distribution ,
.
## ; two-sided confidence interval , .
;
two-sided hypothesis , H : ( 1 , 2 ) ,
1 < 2 . :
? ,
( ).
K : ( 1 , 2 )
H . .-
.
.: [1] ., , .
., 2 ., ., 1979.
## ; two-sided Markov process , .
; two-sided Student
test , .
; double
exponential distribution , , ,
,
p ( x ) = e | x | / 2
, ( , + ) - ,
,
## >, < <
( , ).
. . .
,
, . . .
1 2 ,
+ 1 2 .
. . .- ; ., 2
-, 2 -,
Es = 6 3 -
2 it
f (t ) = e
( t + ) = 0 2
,
- .
. . . .
.
.: [1] .,
, . ., . 2, ., 1984.
; bimodal distribution ,
.
/ ; climate
(
, ,
) . .-
, (
)
( ), (
) (
).
,
, . ,
, .
. .-
. .- ,
, , ,
.
.
.
, . .
, ,
. . , .
.: [1] . ., . ., , ., 1979; [2] . ., ,
., 1982; [3] , . ., ., 1977; [4] The global climate, Camb.,
1984.
; climatic
norm / long-rang averaged value
. . . ,
;
= 0,5 ; (2) = 1 ; (3) = 2
.
= 0 (1)
373
. .- . . ., , 30 100
. . .-
, 1931 60-
.
.: [1] , ., 1957.
;
climat forecast , , .
; climatic system , / .
; climatic time series
, , . . . .-
( )
. - . . . 250 . , 100 , 30
.
12
; stochastic model of climate / climatic stochastic models ,
.
## , ; boundary hitting time / first
passage / time for a boundary g = inf { t > 0 :
t g (t ) } ;
t , 0 = 0 , g ( t )
( ), g (0 ) > 0 . . . . .
0 = 0 , k = 1 + ... + k
0 , 1 , ..., n
= inf { k 1 : k x }
g (t ) = x = const . . . . . | z | < 1 ,
Im < 0
. ( x )
M ( z ( 0 + ) ; ( 0 +) < ) = 1 A+ ( z , 0 )
14
( ) 10 10
. . . . .
.: [1] . ., . ., , ., 1982.
( )
/
; first arrival time ,
i x
(1)
d x M ( z ( x ) ; ( x ) < ) = 1 A+ ( z , 0 ) A+ ( z , )
( , [1] [3] ),
A+ ( z , ) = exp
1 z M e
i 1
k =1
zk
M ( ei k , k > 0 )
k
( , ). (1) , ( 0 + ) (
/
/ ; first
passage time / hitting time
), , ,
. E , , = ( t ,
0 , At , Px ) , t 0
. A E . . . TA = inf { t 0 ;
t A } , t = 0 A =
= inf { t > 0 ; t A} (
0 = inf ). A
( , [1] ). A
. . . E \ A
.
, . . .- T( a , b ) -;
k 1 P { k > 0} ;
k =1
M 1 0 ( , [1], [2] ).
( 0 + ) , : ) ( 0 + ) ( 0 < < 1 )
= lim
## / ; first exit time ,
,
.
.: [1] G e t o o r R. K., Lect. Notes in Math., 1975, v. 440.
;
first exit time / hitting time ,
374
> 0}
k =1
; ) ( 0 + )
( a, b ) R , T( a , ) a
.
.: [1] G e t o o r R. K., Lect. Notes in Math., 1975, v. 440.
P{
( P { k > 0} )
k=1
( , [6] ). H ( x )
= M ( x )
. k
H ( x) =
P {
x} .
k =1
= M 1 > 0 , ( x ) x 1 / a
x H ( x ) / x 1 / a ( , [1] [3] ).
{ ( x ) n } = { max k x }
1 k n
, x
= x(n)
P { ( x) n } =
e t
dt + o (1)
x / n
, o (1) x - .
, , > 0
, | | M exp ( 1 ) <
. x 1 -
,
P { ( x ) = n} =
x n ( x / n )
e
( 1 n, x n)
n3 / 2
, ( )
(2)
= sup { ln M e 1 }
, ( 1 n , x n )
, 1 n , x n
( , [4], [5] ).
M1
= 0 , 2 = D1 , 0 < 2 < , x
## . 2, . 13771; [6] . ., . ., 1971, . 16, . 4, . 593613; [7] . .,
. ., 1984, . 29, . 2, . 41011; [8] . ., . . ., 1975, . 15, 1, . 2366;
[9] . ., . . . ., 1969, . 33,
4, . 861900; [10] . ., . .,
. . ., 1977, . 29, 4, . 46471; [11] . ., . , 1980, . 254, 5, . 104244;
[12] E r d o s P., K a c M., Bull. Amer. Math. Soc., 1946, v. 52,
4, p. 292302; [13] . ., . , 1997,
. 353, 6, . 71113.
## ; first significant digit law
; , d = 1, 2, 3 , ..., 9
lg
d +1
- ( lg d
).
( , 1.)
1.
n 0 ,
n ( x / n ) ~ x 2 (2n 2 ) 0
(2)-
P { ( x ) = n } ~ C x n3 / 2
.
( , [7], [8] ).
f (t ) ,
x g ( t ) = x f ( t x ) . . . .
d
lg
d +1
## 0,3010 0,1761 0,1249
0,0969
0,0792
0,0669 0,0580
0,0512
0,0458
. . . .- [2]
[3]- ,
.
- . . . .- . , 0 < 1 [0, 1] -
x .
## = M1 > 0 , 0 < = D1 < , y = a t y = f (t )
z b = f ( z ) < a ,
x g
k b 0 = 0 , 1 = 1 b , ... , 1 = 1 b , ... ,
k = 1 + ... + k k b . . . .
( x a z ) - . x a z ( a b ) ,
= 10 lg
x a z 2 ( a b ) 3 ( , [1],
[3] ).
y1 < z < y 2 , ,
P { x g < y1 x } 0 ,
P { xg > y 2 x } 0
(3)
.
(3) .
. . . . ( , [9], [10] ). ( [11],
[13] ), .
.: [1] ., , . ., . 2, ., 1984; [2] . ., , ., 1972;
[3] . ., , ., 1976; [4] . ., . . ., 1962, . 3, 5, . 64594;
[5] . ., . ., 1960, . 5,
d - .
d +1
d
= { } [0, 1] -
*
10
d lg
d +1
- . ,
d
* -
.
1.
( ln m ) 2
1
exp
3
, > 0
p ( ) = t 2
2 2
0
,
0
.
2
lg m lg e lg e
d -
375
pd = lg
; illustrative variable ,
d +1
+ 2
exp ( 2 k 2 2 2 lg 2 e )
d
k =1
sin [ 2 k ( lg ( d + 1) m lg e ) ] sin [ 2k ( lg d m lg e ) ]
2k
.
., m = 0, 46 , = 1 ( [4]- , . 246 ), p d
= lg
d +1
+ Rd ;
d
Rd - 2- .
2.
d
Rd
d
Rd
1,18710-2
3,00710-3
3,14610-3
-4,39110-3
3,67410-3
2,45010-3
1,27510-3
3,25410-4
3,76710-4
, ; simulation of a random
phenomenon
, ,
. . . .-
.
-
( , ( ) ).
; immigration i n a b r a n c h i n g p r o c e s s ,
= 1, 2 , ..., N 2 n
. N , ., N = 100
2. n
lg
d +1
- . ,
d
## 2n = 10{ n lg 2} 10[ n lg 2 ] { n } , n = 1, 2 , ..., N
N [ 0, 1] -
, [ z ] , z
, { z } .
.: [1] .,
, . ., . 2, ., 1984; [2] R a i m i R. A., Amer.
Math. Monthly, 1976, v. 83, 7, p. 52137; [3] H i l l T. P., Statist. Sci., 1995, v. 10, 4, p. 35463; [4] ., , . ., 2 ., ., 1975.
; null hypothesis
. . ., , H 0 . ,
. . , (
. .- ) > 0 ( )
( ) .
/ ( )
( ) ( ) ; initial distribution o f a M a r k o v c h a i n ( p r o c e s s )
( ) s X s ,
s = min { t : t T } , T
( ) . . / . .
( ) , { X t1 , ..., X tn } , ti T
.
376
; impulse
noise .
; impulse phase modulation process ,
.
; impulse renewal process
, .
; impulse response function
( )
h(u ) , L y ( t ) x ( t )
y (t ) = L x (t ) =
h ( u ) x ( t u ) du
. x ( t )
,
L ( ., ) , . . .
h ( u ) : h ( u ) u < 0 h ( u )
=0
y ( t ) -
(
). . . . ,
; ., x ( t ) - x ( t ) . . . ( u ) - .
h ( u ) . . .-
H ( ) -. , .
; impulse
noise , , ( , , . )
.
. .- ;
-
. ,
,
( , [1] ). . .
( ,
[2] ). .
.: [1] .,
, . ., . 1, ., 1961; [2] . ., , 2 ., . 2, .,
1975.
impulsive process w i t h t h e m o d u l a t e d
b y i n t e r v a l s , .
;
tacts
,
; impulse process with
constant tact by intervals ,
.
; impulse Poisson process ,
.
; impulse response function
( , ) ; { t n , n = 0 , 1, ... } {A n } .
A n = ( An , Bn ) - , An , n - , Bn , ,
(t ) =
An ( ( t t n ) Bn ) .
(2)
n =
, Bn
= const
, -
, ( t )
; An
, t n
, ( t )
; An , n - Bn , ( t )
, ,
.
. . . { t n , n
= 0 , 1, ...}
.
( t ) . . .- , -
, .
{ t n } (
; impulse random process
, ( t )
( ., , , )
. . . .-
(t ) =
( t tn , A n )
(1)
n =
, ( t , A ) A
= ( A1 , ..., Ak )
; A ,
t n , n = ... 1, 0 , 1 ...
( , )
, .
(t ) =
(t t
n
, An )
(1)
n =
. . .- ;
, n - n . (1)
( ., [1] ) . . .- ; (1) . . .-
.
. . .- (
), , ( t )
(
); { A n , n = 0, 1, 2 , ...}
tn
- )
(
; , ., [2] )
. { A n ; n = 0, 1, ...} t n
,
. . . . . . . . . .- , t n +1 t n
( , ),
, ( t )
- . . .- ,
,
T0 ,
T0 ,
,
t n - t n = n T0 + n .
n = = const
) ,
( t ) . . .
.
( ., = 0 ), { A n }
, ( t )
( {A n }
, ) ; , T0
377
, ( t )
. n , n
= 0 , 1, ... ,
{A n } -
, , . . .-
,
( t n
n T0 -
,
).
-
( . . .- ; , ., [1] )
.
. . .-
.
.: [1] . ., . ., , ., 1973; [2] . ., , 2 ., ., 1982;
[3] . ., , 2 ., . 1, ., 1974; [4] . ., .:
, 2 ., . 1., .; [5] ., , .
., . 12, 196162; [6] ., , .
., ., 1974; [7] . ., . ., p . ., . , ., 1983.
( ) ; failure
, .
, , .
; failure hunting ( ) .
. ,
,
.
,
, - .
.
.: [1] , ., 1985.
## ; average operating age
to failure
.
, , ,
.
; fault tree
( ,
). . .
.
: 1, 2 3, 4
. . .
. . . .
378
.: [1] ., .,
, . ., ., 1984.
, ; failure
of element , .
, ; failure of
system , .
/
; probability of
breakdown free / failure free operation / survival
function , .
1
1 ; distribution function of failure rate
i n t h e r e j i m 1 , .
; failureless
. . .
, , , .
; gamma percent age operating before failure
( )
t -, P (t )
. ,
, .
## ; average operating age before failure
X - M X -.
( ) , ,
; index o f a r a n d o m v a r i a b l e,
index o f a d i s t r i b u t i o n
f (t ) ,
## ( ) = sup { h : min ( | f ( t ) | : | t | h ) > 0}
.
. [1]- C r ( )
Br ( )
0 < r
< ( )
( , [2] ).
.: [1] . ., . , 1970, . 194,
5, . 101012; [2] . .,
, ., 1986.
( ) ; indicator o f e v e n t , A
A,
A
1,
I A ( ) =
0,
; indicator variable , .
;
indicator metric
## ; pointwise ergodic theorem , .
;
persistant forecast , , .
; infinitesimal matrix ,
P -, t 0 ,
: f A
D A -,
lim t 1 ( P t f f )
t 0
, f A f . ,
. .
.
2)
, ( )
. .-
Pt f ( x ) = M x f ( t )
K ( X , Y ) i( X , Y )
; ( t , Px ) . .-
, ; indicator of set A
0, A,
I A ( ) =
1, A
I A IB A B , I A = I B A = B ,
I AB = 0 A B = ,
I An =
An
I U An = I A1 + ( 1 I A1 ) I A2 + ( 1 I A1 ) ( 1 I A2 ) I A3 + ... ,
I lim inf An = lim inf I An ,
I lim sup An = lim sup I An , I lim An = lim I An .
A
M I A ( ) = 1 P { A } + 0 P { A} = P { A}
.
. . .
. .
M x f ( ) f ( x ) = M x
A f ( ) d t
t
f D A (
## I inf At = inf I At , I sup At = sup I At ,
,
t 0
; , Fx ( ) < , f
I = 0 , I = 1 , I A + I A = 1
An
A f ( x ) = lim t 1 ( M x f ( t ) f ( x ) )
). t
. , - - 0 1 -
. ,
, x - ( M x , Px
, - I A ( )
I I An =
, 1)
i ( X , Y ) = M I ( X Y ) = P{ X Y }
, I . . .
. ( )
. . .,
,
,
i ( T ( X ) , T ( Y ) ) = i ( X , Y ) . i ..- K
## , , ; generator / infinitesimal operator ,
). . . 2-
.
( t , Ps , x ) . .
At f ( x ) = lim ( t t ) 1 ( M t , x f ( t ) f ( x ) )
t t
;
.
.: [1] . ., , ., 1963;
[2] ., ., , . ., ., 1968; [3] . ., . ., , . 2 , ., 1973.
379
3) . . . . ;
infinitesimal operator / generator of a conv o l u t i o n o f m e a s u r e s G t - { S t , t
0}
= e
=0
.
G U G
( ) K
lim sup nk ( G \ KU ) = 0
.
, { t , t 0 } G
, lim t
nk ( ) u u
lim sup
n 1 k k
n
n 1 k k
n
, { nk } K -
, e , G e -
.
.: [1] .,
, . ., 1965; [2] . ., .
., 1972, . 17, . 3, . 54957.
t 0
.
t > 0 t
St f ( x ) : =
; informant
f ( x y ) t ( dy )
, f G b
. .
.
,
t
{ P , t } P ( d )
S ( G ) -.
; t -
G -
Su ( G ) { S t , t > 0 }
, - . ( ) ( -
. S t , t > 0
- . , { P }
t 0
Su ( G ) -
D N f - , { t , t 0} G .
A f : = N f ( e ) . . . , N f ( x )
= A ( Z x f ),
(1)
, ..., ( N ) ) .
t
- ( d )
N f ( x ) : = lim t 1 ( St f ( x ) f ( x ) )
. .- N
Z x , G b
x S ( G )
p ( ; t ) .
P ( ) - . p ( ; t )
t = ( t1 , ..., t m )
,
.
.-
:
. N
A
S t , t > 0 ( , t )
.
, , ;
.
.: [1] ., , . .,
., 1967; [2] ., , . ., ., 1981.
, ; infinitesimal system
of measures G nk , k = 1, ..., k n , n 1 -
p ( ; t )
grad t ln p ( ; t ) =
1
p ( ; t )
p ( ; t )
, ...,
t1
p ( ; t )
t n
p ( ; t )
.
. . -
,
. ,
: ,
: G
U
lim sup nk ( G \ U ) = 0 .
n 1 k k
n
nk ( ) , , nk
, { nk } , ,
T - u -
380
p ( (1) , ..., ( N ) ; t ) =
pk ( ( k ) ; t )
k =1
. :
## grad ln p ( (1) , ..., ( N ) ; t ) =
grad ln pk ( ( k ) ; t ) .
k =1
.- .
,
, ,
j , k
Mt
ln p ( ; t )
=
tk
Mt
Zn
.
. .- -
. ., ( Z , Z )
ln p ( ; t )
p ( ; t ) d = 0 ,
t k
PZ~ n -
= { ( Z j , Z j ),
j = 1, 2 , ... } ~
~
n ~n
( Z , Z ) = ( ( Z1 , Z1 ) , ... , ( Z n , Z n ) ) -
2 ln p ( ; t )
ln p ln p
= I jk ( t ) = M t
t j
t k
t j tk
~
~
I ( Z n : Z n ) ( Z n , Z n )
. . .
~
~
Z n = Z n , I ( Z n : Z n ) = H ( Z n ) Z n
.
.
-
, : , > 0
I jk ( t )
j ; k =1
(1)
, ...,
(N )
) t = ( ) ,
, ,
.
grad ln p ( ; t ) = 0
t = -
,
( )
. t =
*
n0 ( , )
N ,
*N
.
.: [1] ., , . .,
., 1967.
; information sequence ,
; information stability
.
lim P
n
, i
~
Zn; Z n
>0
~
iZ n ;Z~ n ( Z n : Z n )
1
~
I (Z n : Z n )
, ,
~n
( Z , Z ) -
< =1
H n ( X n ) X n n ( ,
).
~n
, I ( Z : Z ) -
( : )
iZ n ; Z~ n ( ; ) = log a Z n ; Z~ n ( ; ) , a Z n ; Z~ n ( ; ) ,
~
~
( Z n ; Z n ) PZ n ; Z~ n - Z n Z n -
(*)
,
~
( x n , ~x n ) , x n X n , ~x n X n -
. (*) ,
~
PX n ( ) = p n ( ) ( X n , X n ) -
lim
> <
U n ; ( X n , S n ) , X
~
n
P n ( ) = pn ( ) X , ( X n , S ~ n )
X
X
log P ( Z n )
H (Z n )
n
n
~
M n ( X n , X n ) n , n 0
, .- ,
det I jk ( t ) 0
## 0 < I (Zn : Z n ) <
n > n0 ( , )
~
I (X n : X n )
=1
H n ( X n )
, U n
.
( Q n , V n ) ;
~
(Y n , SY n ) (Y n , SY~ n ) ,
Q ( yn , A ) ,
y Y n ,
A SY~ n .
(Y n , SY n ) - V n - n
. C ( Q , V ) ( Q , V )
. A S
~
Y n
~
P { Y n A | Y n } = Q (Y n , A)
381
, Y
~n
I =
~n
. ( Y , Y )
n
lim
~
I (Y n : Y n )
=1
n
C (Q , V n )
n
## ; informational correlation coefficient
R( X , Y ) =
1 e 2 I ( X :Y )
, X Y , I ( X : Y ) .
. . . R ( X , Y ) - X Y
I ( X : Y ) .
. . . R - -
:
R = 0 , , X Y . X Y
,
- , R = ; ,
I = 2 1 ln (1 2 ) .
. . .
. R -
R
1 e 2 I ,
382
j =1
nif
n
ln
n nij
ni. n. j
, n
, s t
, nij , ( i, j ) ,
t
ni. =
, ( Q , A )
; informated redundancy . . .- ( , )
i =1
.
. .
.
, . .-
.
.: [1] .,
, . ., ., 1967; [2] . .,
. , 1959, . 14, . 6, . 3104; [3] . ., , 1960, . 7, . 1
201; [4] ., , .
., ., 1960.
*
ij
n. j =
j =1
ij
i =1
, . . .- .
, I
.
.: [1] L i n f o o t E., Information and Control, 1957, v. 1, 1,
p. 8589; [2] ., ,
. ., ., 1967.
; information matrix , N
N
p
y = + , y R , R , R , M = 0 ,
cov - cov = W 1 ( det 0 )
= TW ;
. . .-
( , .,
,
).
; ; information matrix;
F i s h e r i n f o r m a t i o n t
. P ( d )
p ( ; t ) - t
= ( t1 , ..., t m )
( , )
. .- t =
I jk () =
ln p ( ; t ) ln p ( ; t )
t j
t k
p ( ; ) d
(1)
t =
j , k = 1, ..., m . t
. .
.
I() . .-
jk
( ) dt j dt k =
(2)
j,k
, { P }
.
P =
(d p
)2 p j ;
p j = P ( j ) , j
. (2)
. ,
.
I ( ) Q = P f
Q
. .-
. . . , z1 , ..., z m
I Qj , k z j z k
j, k
P
j ,k
z j zk
; information distance ( , A )
P Q
J ( P, Q ) . . .-
. . .-
:
j, k
. . . . I
(i )
( ) -
( P, Q ) =
pi ( ; t ) . .-,
p ( (1) , ..., ( N ) ; t ) =
p (
i
(i )
(i )
; t)
S ( P, Q ) = 2 arc cos
I N () =
N I () . . .
. t
( ) = ( (1) , ..., ( N ) )
D
[ N I () ]1
. -
.
m
j ( )
j ] [ k ( ) k ] I j k ( ) m N 1
(3)
j,k = 1
,
.
. , ., :
M M I () m N 1 + o ( N 1 ) ,
(4)
(3)- M
V .
dV =
(1)
det I ( ) d 1 ... d m
; det I ()
=0
, , , . , . , . .
( , [1] ).
.: [1] F i s h e r R. A., Proc. Cambr. Phil. Soc., 1925, v. 22,
p. 70025; [2]
.., , . ., ., 1974; [3] . .,
, .,
1972.
( ) ; R i e m a n n
information metric , .
P ( d ) Q ( d )
(2)
.
P Q
. .- ( , ).
. , , , .
,
: )
, )
( ., ) . ,
, ) ) .
, )
) J ,
, , ,
. [ .,
; , (2) ].
, A
, P Q
- . (4)
.
det I () = 0 -
() . ., N
i =1
I N ()
P ( d ) Q ( d )
l ( ) = ln
dP
( ) ,
dQ
(3)
I (P : Q ) =
P ( d )
P ( d ) = Q ( d )
ln
P (d )
Q ( d )
Q ( d ) (4)
Q - P -
( 0 ln 0 = 0 );
, ,
. (
383
) I ( P : Q ) ) ) -
. ., -
. P - Q - ,
N 2- ( P , ) N - -
, 1- n - ,
0 a0 N a1 < 1
N b1 1
I ( Q : P )
n -
. , ., , , , I ( P : Q ) I ( Q : P ) . P Q - -
.
ln N = ln N ~ N I PQ ,
I PQ = ln min
1
( R0,5 , R0, 25 ) (P, Q )
8
= { 1 , 2 }
R ( 1 ) = , R ( 2 ) = 1 ,
, R
0 Q 1.
) ,
ln N ~ N I (P : Q)
. 0 < b0
( P, Q )
. . .-,
. (4)
) I ( P : Q ) -
Q - P -
.
.
, ,
:
l ( ) [ R ( d ) + Q ( d ) ] =
[ P ( d ) ] [ Q ( d ) ] ( 1 ) =
## = min max { I ( R : P ) , I (R : Q)}
R
[ln P (d ) ln Q ( d ) ][R ( d ) Q ( d ) ] = 0 ,
I ( R : P ) = I ( R : Q ) + I ( Q :P ) .
.
)
J ( P, Q ) J ( P , Q )
(5)
; , ( , A ) -
cap ( , P ) ( , [4] ).
- P Q P
=Q
. , )
J . ( P, Q )
, J ( P, Q )
.: [1] ., , .
., ., 1967; [2] . .,
, ., 1972; [3] ., ., , . ., ., 1985; [4] A m a r i S h u n i c h i, Differential-gometrical methods in statistics, B.,
1985; [5] M o r o z o v a E. A., C e n c o v N. N., . Probability
theory and mathematical statistics, v. 2 (Vilnius, 1985), Utrecht, 1987,
p. 287310.
+ ( z ) = sup [ P ] ( A ) z Q( A ) : A A ,
(z ) = inf [ P ] ( A) z Q( A ) : A A
0 z
(3) l ( )
P ( Q ) . (1) (2)
, ,
+
( , [3] ) ( z ) ( z )
= + ( z ) + z 1 , .:
1
I ( P, Q ) =
( ) ( z ) z 1dz +
(+)
( z ) z 1dz .
J . . ( , A ) cap ( , P )
J .
J (5) , (1)
384
c( I )
. c ( J )
= 0, 5 .
;
information measure
.
. .- .
. .-
, A
I ( A ) , A ,
,
I ( A )
. , . .
I ( A ) - A f
P ( A )
= p - ,
I ( A ) = f ( p ) . f ( p )
[ 0 p
1 f ( p ) 0 :
] [ f ( p ) :
].
x1 , ..., xn p1 , ..., pn
X . . I ( Ai )
; Ai = { X = xi } . . . X . .
H ( x )
pi log 2 pi , ,
.
. .- . .,
If ( P Q) =
- , P
= ( p1 , ... , pn ) ,
Q = ( q1 , ..., qn ) , f ( t )
= t log t
; K u l l b a c k
L e i b l e r S a n o v information , .
f ( . f -
. f ( t )
( )
; information
f ( pi qi )
; information density , .
f ( p ) = log 2 p
, p ( x ), q ( x ) , P Q .
.: [1] A c z e l J., D a r o c z y Z., On measures of information
and their characterizations, N. Y., 1975; [2] C s i s z a r I., Trans. 7th
Praque Conf. Inform. Theory, Statist. Decis Funct., Random Process.
Eur. Meet. Statist., 1974, 1978, v. B, p. 7386.
f ( t ) = ( t 1) 2 2
; a p r i o r information , .
; amount of information
.
X Y ( , A , P ) -
f ( t ) = | t 1 |
, ( X , SX ) (Y , SY )
.
,
. 1
PX ( A ) , A SX , PY ( B ) , B SY , -
H ( p1 , ..., pn ) =
1
log 2
, N ( p1 , ..., p n ) , p1 , ..., p n . H 0
.
, ,
.
1
2
( p
( 1) /( 1)
> 0, > 0, , 1, 1
.
. .-
. . I f ( P Q )
I f (P Q ) =
I(X : Y) =
a XY ( x , y ) log a XY ( x, y ) PX ( dx ) PY ( dy )
X Y
( , , 2 e
), a XY , PXY PX PY -
## H 0 ( p1 , ..., pn ) = log 2 N ( p1 , ..., p n )
( ) . I ( X : Y ) ,
PXY PX PY -
. , H 1 = H
. = 0 -
H (X ; , ) =
; PXY ( C ) , C SX SY
q ( x ) f ( p ( x ) q ( x ) ) d ( x )
. PXY
PX PY ,
I ( X : Y )
X Y . .
I ( X :Y )
I(X : Y) =
pij log
i, j
pij
pi q j
, { pi } , {q j } { pij } , X ,
Y - ( X , Y ) . .,
I ( X : X ) = H ( X ) , X .
( X , Y )
p XY ( x, y ) , . .
I (X : Y ) =
XY
( x , y ) log
p XY ( x , y )
dx dy
p X ( x) pY ( y )
(1)
385
, p X ( x) pY ( y ) , X Y
. ,
I ( X : Y ) = sup I ( ( X ) : ( Y )) ,
(2)
( ) ( ) .
X , Y Z
I( X : Y | Z) =
r
k
i, j
pij|k log
pij | k
pi | k q j | k
, { pi | k } , { q j | k } ,
{ pij | k } X , Y ( X , Y ) Z - , { rk } , Z
. I ( X : Y | Z ) (1) (2)-
. . .-
:
1) I ( X : Y ) = I ( Y : X ) 0 ; , X Y
;
2) X Y . .
I ( X : Y ) = H ( X ) + H (Y ) H ( X , Y ) =
= H ( X ) H ( X | Y ) = H (Y ) H (Y | X )
; X Y
I ( X : Y ) = h ( X ) + h (Y ) h ( X , Y ) =
= h ( X ) h ( X | Y ) = h (Y ) h (Y | X )
, . . ;
3)
I (( X , Y ) : Z ) = I ( X : Z ) + I ( Y : Z | X )
I (( X , Z ) : Y ) + I ( X : Z ) = I ( X : ( Z , Y )) + I ( Y : Z )
.
. .-
. .- .
X Y n
( X , Y ) ,
I( X : Y ) =
DX Y
1
log
,
2
D X DY
D X , DY D XY , X , Y ( X , Y )
. ,
I( X : Y ) =
1
log (1 r 2 ) ,
2
r , X Y - . . .
.
386
.: [1] .,
, . ., ., 1974; [2] . .,
. , 1959, . 14, . 6, . 3104; [3] . .,
. ., , ., 1982;
[4] ., .:
, . ., ., 1963,
. 243332.
; n o n p a r a m e t r i c Fisher information
, .
; information theory
, ,
,
.
. ., ,
, , , . . .
. .- . ( C. Shannon ) . ,
1948- ,
.
- : , .
, ,
,
.
. .-
, . , . .-
; , . , ( , , )
,
.
. .-
;
.
. . .- . . ( , [7] )
.
.
. . ,
, p ( ) , (X , SX ) , ( , A , P ) X . ., X
= { X (t ),
a t b } ,
X ( t ) (X 0 , SX 0 ) .
X
n
( X , S
Xn
= ( X 1 , ... , X n ) ( X n
) , ( X 0 , SX 0 )
n ) n , X k k
; k = 1, ... , n .
. (
),
, ( , A , P) -
~
~
, ( X , SX~ ) X k
~
. X
, ,
( X 0 , SX 0 )
.
. , X
X
- , , , -
. . .- , ,
~
~
( X , X ) (X X , SX SX~ ) -
W
. W
, , m
~
~
( x, x ) , x X , ~x X
j = 1, 2 , ..., m = ( 1 , ... , m )
~
. , ( X , X )
j ,
W ,
~
M j (X , X ) j ,
( , m
=1
(1)
).
. ( Q , V ) (Y , SY ) , (Y , SY~ ) ,
A SY~ y Y
~
(Y , SY~ )
Q ( y , A ) , y Y , A SY~ (Y , SY )
V . (Y , SY ) , (Y , SY~ )
; Q ( y , A )
, ,
y -; V
.
~
P { Y A | Y } = Q (Y , A)
(2)
Y - V . V ,
, ( y ) = ( 1 ( y ) , ..., l ( y )) , y Y
= ( 1 , ... , i ) ;
, Y - V ,
M j (Y ) j ,
j = 1, ... , l
(3)
.
( Y
) V , ,
- , . Y
, Y
Y Y - ).
Y - , (2)
Y - . V -
,
( .,
,
).
,
( ), 0
(Y , S
Y0
~
) , (Y 0 , SY~ 0 )
~
~
Y = { Y ( t ) } , Y = {Y ( t )} .
~
~
~
Y = ( Y1 , ..., Yn , ...) Y = ( Y1 , ..., Yn , ... ) -
n - n = 1, 2 , ... Y ,
~n
~
~
(
Y
,
...,
Y
)
Y = 1
Y = ( Y1 , ... , Yn )
n
j = 1, 2, ..., m
, ,
; (1) , - .
( Q, V ) -
( Q, V ) , ; A SY~
( , A , P )
, (Y , SY )
~
(Y , SY~ ) ,
, ; ,
, . ,
~
~
~
Y n = ( Y1 , ..., Yn ) Y n = ( Y1 , ... , Yn ) .
,
.
( ) Y - x - f ( x )
, x X ,
~
~
X - , Y - ( ~
y ) ,
f ( x ) , x X , , ,
- [ ,
f ( x) , -
y )
]. , f ( x ) ( ~
, x X
= f ( x ) ; ~y
x =(~
y)
, ~
y
387
. .- ;
.
. p ( ) , W ( Q, V ) ,
f ( ) ( )
, X ,
~ ~
Y , Y , X ( ) , X ~
p ( ) , ( X , X )
~
W - , ( Y , Y ) ( Q , V )
,
, H W ( p ) , , (5)
, .
( )
, , .
.
t
lim
Y = f (X ) ,
~
~
X = (Y )
~ ~
Y , X
, Y - X
Y - Y - ,
,
,
.
. .- .
, p ( )
, W , ( Q, V ) . ,
,
, f ( )
( ) ,
( Q , V ) W
. p ( ) , W , ( Q , V )
,
. ,
:
.
. [15]- .
I ( ; ) ;
= ( Q , V ) = sup I ( Y : Y ) [
~
( Q, V ) ( Y , Y )
= inf I ( X : X ) W [ [1]
H ( p ) = H W ( p ) ];
]; HW ( p )
( X , X ) ,
~
( X , X ) W - ; X -
p ( ) .
: p ( ) W ( Q , V )
(5)
.
(5) .
388
C ( Q t , V t ) <
{ ( Qt , V t ), t = 1, 2 , ...} .
t
~t
X - p ( ) , ( X , X )
t
W
lim
~
I( X t : X t )
=1
HW t ( p t )
t
~t
( X , X ) t
, p ( ) , H
Wt
( p t ) <
W t
.
; .,
HW t ( p t )
C ( Q t , V t ) ,
.
V , (3) - + , (1)
- + ,
>0
W . (
t
) : p ( )
W t
( Q t , V t ) ; t ( ) p t ( , )
t - , t
H W t ( p t )
lim
HW ( p ) C ( Q , V )
~
I( Y t : Y t )
=1
t
C (Q , V t )
(4)
. (4) X , Y ,
~t
(Q , V ) ( Y , Y ) -
C ( Qt , V t )
>1
HW t ( p t )
, > 0
t0 , t t 0 t
p t ( ) , ( Q t , Vt )
t
W .
p ( ) -
.
t
, W - W
t
V - V .
.
,
( )
.
.
U
X
= { X k , k = ..., 1, 0, 1, ...}
; - ( X k ) M X
. X
= { X k , k = ..., 1,
0, + 1, ...} ( -
~0
0
( X = X ). , -
.
. , L = LN
( , [4] ) ; N = [ L L ] ( [ x ] , x
~L
= ( X 1 , ..., X L )
). X
, Pe
Pe = L1
Xl }
l =1
, . : H (U ) , C
, L -
Pe log ( M 1) + h ( Pe ) H (U ) 1C
h ( x ) = x log 2 x (1 x)
log 2 (1 x ) .
,
H (U )
1C -
( )
, L
,
L .
RN =
~
Pe, x L = Px L { X L X L }
Pe =
P{ X
log 2 M
N
LN
xL
= x L } Pe, x L = P { X L X L }
, P L {} , X
= x L X
.
: N R < C R - , Rn
R x L X
x -
Pe, x L exp { NE ( R ) }
( P e
), , E ( R ) ,
R - ( ,
). , ,
R < C R - N
.
, : R > C R -
N P e 1 .
, Y
P {X
X 0
X L = ( X 1 , ... , X L )
;
. ,
M = 2 , R N = ; -
=Y
, Q ( y, { y })
=1
= log 2 S
,
, S .
,
.
. , , , , , .
,
. ,
,
,
.
,
,
. ., k + 1 X
(1 )
, ... , f j ( X
( j)
) X
- Y
, ( j
(0)
= 1, ... , k )
( j)
= f j ( X ( j ) )
( Y
(1)
, ..., Y
(k )
) X ( 0)
.
LN , R N M
U H (U ) -
389
, , ,
.
, t t < t
t
. ,
,
,
.
,
.
.: [1] ., .,
, . ., ., 1969; [2] ., , .
., ., 1967; [3] .,
, . ., ., 1974; [4] . .,
. , 1959, . 14, . 6, . 3104; [5] . ., . ., , .,
1982; [6] . ., .:
, ., 1987, . 2958; [7] . .,
.: , ., 1987, . 213
23; [8] Key papers in the development of information theory, N. Y.,
1974; [9] . .,
-, 2 ., . 2, ., 1975; [10] . .,
, 1960, . 7, . 3202; [11] ., , . ., ., 1960;
[12] ., .
, . ., ., 1965; [13] . .,
, 2 ., ., 1965; [14] ., .,
.
, . ., ., 1985; [15] .,
, . ., ., 1963,
. 243332.
( )
; informativity o f s e t o f v a r i a b l e s q ( ) x = ( x1 , ... , x p )
p
z = ( z1 , ..., z q ) ;
I p ( z ) .
.- .
.
z y
.
z
. , p < q , z
x
.
; reflection
principle w (t ) - a
, a w ( t + s ) , s 0 . , w ( 0 ) = 0 ,
a > 0 m ( t ) = max w ( s ) , t 0 .
0 s t
P { m ( t ) > a } = P { < t } .
(1)
( , ). . .- w ( )
a - (
):
P { w ( t ) > a < t } = P {w ( t ) < a < t } = 1 2 .
(2)
## ; rate of information transmission
. ,
~
~
Y = { Y ( t ) , < t < } Y = { Y ( t ) , < t < } -
R =
~
lim ( T t ) 1 I ( YtT : Yt T )
(*)
T t
~T
, I (Yt : Yt ) ,
~
Y Y ( t , T ]
. (*) , .,
w ( t ) > a x < t ,
.
, ., . . .-
.
.: [1] ., ,
. ., ., 1974; [2] . ., .
., , ., 1982; [3] . .,
, 1960, . 7, . 1201.
(1) (3) ,
; amount of
information , .
390
## 2P {w ( t ) > a } = P { < t } P {w ( t ) > a < t } .
(3)
-
P { m* ( t ) > a } = 2P {w ( t ) > a }
. . . m (t )
p ( t , y ) = 2 ( t , y ) =
2 y2
e
t
2t
y0
, ( t , x ) w (t )
p - ) (U 1 ) ( u ) -
. w (t )
[ m ( t ) < a ]
.
. .- :
1) ( ),
k ( s, u ) = exp ( i ( s, u ) ) , ( , )
, :
m (t ) - a
2)
p ( t , x ) = ( t , x ) ( t , 2a x ), x < a ,
. .- w (t ) , m (t )
= min w ( s )
0 s t
c < m (t ) ,
m ( t ) < a ]
, :
k ( s, u ) =
p ( t, x ) =
[ ( t , x 2k ( a + c ) ) ( t , 2 a x 2k ( a + c ) )] ,
c < x < a
c a
p ( t, x ) =
[ ( t , x 2k ( a + c )) + ( t , 2a x 2k (a + c ) )],
k =
c < x < a .
.: [1] .,
, . ., ., 1972; [2] ., .,
, . ., ., 1968;
[3] . ., . .,
, ., 1967.
; integral renewal theorem
, .
## ; integral transform {P}
( {F } n
R
{p} ) n
( n = 1) ,
k ( s, u ) = J ( s , u )( s u )
1/ 2
, u > 0 , J
;
5) ( ) (n
s 0,
k ( s, u ) = 0 ,
= 1) , k ( s, u ) = 0 , u 0 ,
## k ( s, u ) = cos s u ( sin s u ) , u > 0 .
. . ,
U ( P1 o P2 )( s ) = 1 ( s ) 2 ( s )
.
. .- - ( , [1], [2] ).
. .
( , , , ) .
.: [1] ., ,
. ., . ., 1948; [2] . ., . ., 1957, . 2, . 4, . 44469; [3] .,
., , . ,
. 12, ., 196970.
;
probability integral , ,
x
erf ( x ) =
e t dt , | x | <
.
. . ,
( x) =
1
2
t 2 2
dt =
1
[ 1 + erf ( x
2
2 )]
;
. 0, 1
= erf ( t
P { | |
t} =
2 ) . :
( x ) = H ( x ) = ( x ) = erf ( x ) ,
s = ( s1 , ... , s n ) C n - { } -
k ( s , u ) P( du )
4)
{P} o ,
, U {P} - { } - -
(s ) = U P ( s ) =
k ( s, u ) = u s , u > 0 ;
m (t ) a > 0 c < 0
3) ( n = 1) , k ( s , u ) = 0 , u 0 ,
p ( t , x ) = ( t , x ) + ( t , 2a x ), x < a .
k =
exp ( ( s , u )) ;
R n -
Erf ( x ) =
(*)
U ,
Erf c ( x ) =
k , R C - ( . .-
). k
(*)
( ., ). P
( F
( x) =
erf ( x ) ,
Erf ( x ) =
e t dt ,
dt 1 =
Erf ( x
2).
391
.: [1] ., ., ., .
, , , . ., 3 ., ., 1974.
. [ P1 ] [ P2 ] ,
; integral geometry
, .
.
. ( J. Sylvester ), .
( H. Poincar ), . ( W. Blaschke )
. . .-
( , ,
. ) .
R
M n , R - k
. k , n -
. ., g M 2
P1 P2 .
P1 , P2 , [ P1 ]
[ P2 ] [ ] G .
G \ ( [ P1 ] U [ P2 ] ) -
;
( ), . A , A G ( :
). [ ] ,
.
a, b, c . -
I[ c ] ( g ) = I [ c ] I [ a ] ( g ) + I [ c ] I [ b ] ( g ) ;
dg = dp d , ( p, ) g -
I[ a ] ( g ) = I [ a ] I [ b ] ( g ) + I [ a ] I [ c ] ( g ) ;
I [b ] ( g ) = I [b ] I [ a ] ( g ) + I [b ] I [ c ] ( g ),
R - Tn - ,
( k
= 0 ).
., T2
m ( dg )
= dp ( d ) , ,
[ 0, 2 ] - ( , ).
,
T2 .
I A ( g ) , A G . ( )
(
).
:
2 I[ a ] I [ b ] ( g ) = I [ a ] ( g ) + I [ b ] ( g ) I [ c ] ( g ) .
R - F
H B , F - H
, H
( , , H = M 2 H = A ,
), H , R - .
. .-
( , ). , , ( , [1] ).
. .-
( , [4] ), ( , [4] ).
G
= { P1 , P2 } ; [ ]
1- , , ( 1- [c] -
## [ a ] I [b] - ; [a ], [b], [c]
). 1-
,
, ,
.
. G ,
( ) . , , [ P ] =
= { g G : g P R 2 }
( ) .
, [ P ]
392
1.
D , { i } ,
D - . i
2 I i ( g ) = I [ ai ] ( g ) + I [ ci ] ( g ) 1
ij [ ij ]c ,
, [ ai ], [bi ], [ci ] , i
. i
2I D ( g ) =
, ., i+j
ij+ [ ij ]c
. 3-
I [ di ] ( g ) N
, N , [ d i ] , D
.
, { [ d i ] }
, [ d i]
= { [ d i ]} U { [d i] } , { d i} -
. [ d i]
r -,
c ,
2I D ( g ) =
[ di ] ( g )
I
i
=N
, r
[ di]c
(g) + r N
2.
. , d i -
( ). ,
2 ID ( g ) =
[ di ] ( g )
I
i
[ di]c
, g ij .
(g) .
, ,
( , [3], [4] ).
1 , ... , n
. G [ 1 ] , ..., [ n ] .
A{ i } [ i ] , G -
( ) . A { i }
Br { i }
. G m ( ) P R
m ( [ P ] )
=0
, m ( )
ij
i< j
= ij
m ( ) .
{Pi } - , cij ( B ) Pj g ij G
cij (B) = I B ( i , j ) + I B ( i, j ) I B ( i , j ) I B ( i , j )
.
3
, {Pi } , { j } , cij ( B ) ,
. ij , Pi
. cij ( B )
, R - , -
( B ) m ( [ ij ] )
, { Pi , Pj }
I n ( i , j ) i, j I n ( g ) -
. : G -
m ( ) B Br { i }
1
m (B) =
2
3.
\ ( [ Pi ] U [ Pj ] ) -
- ; g ij
++
( 2 ). , ., i j
, R - , R - .
.: [1] ., . , 1938, . 5, . 97
149; [2] .,
, . ., ., 1983.
## ; integral limit theorem
.
,
,
.
( ) ,
. . . .
393
,
. . . .-
( , [1] )
, . . .
,
.
.: [1] . ., . , 1952, . 7,
. 3, . 112.
( )
; integral
scale o f c o r r e l a t i o n , .
( )
; integral structure o f a p r o b a b i l i s t i c
m e t r i c , , .
, ; path integral ,
.
; B o c h n e r integrability
, .
( ) ; intensity o f f l o w , .
;
instantaneous intensity , .
( )
; intensity
o f s t r a i g h t l i n e s p r o c e s s , ,
.
## , ; hazard function / hazard
rate function / failure rate function
F (t )
( t ) = f ( t ) F (t )
; intensity
1) .-
-. t t - .
. ( , ).
.
(t ) .
( a, b )
b
( t ) dt
## k , k k ( t1 , ..., t k ) dt1 ... dt k ( t1 , t1 + dt1 ) , ... ,
t
. (*)
t
F ( t ) = exp ( s ) ds
0
. F (t ) ,
F (t ) = 1 e t , > 0 , (t )
( t ) dt ( t , t + dt )
( )
; integrated spectrum o f a n o n s t a t i o n a r y p r o c e s s ,
f ( t ) = dF ( t ) dt , F ( t ) = 1 F ( t ) . -
; P e t t i s integrability ,
,
; integration
of random process , ; -
(*)
;
rose of intensites R n -
H ( ) .
n
R - Tn
.
m L ,
H ( ) =
S
cos m ( d )
n 1
, m , ( n 1) ,
L , R1 - , S n1 R n - ( n 1 )
.
H ( ) , , R n -
Tn -
. H , m .
.
, .
.: [1] S t o y a n D., M e c k e J., Stochastische Geometrie, B.,
1983; [2] S c h n e i d e r R., W e i l W., .: Convexity and its
applications, Basel, 1983, p. 296317.
394
;
intensity measure , .
( t k , t k + dt k )
; k ( t1 , ... , t k ) , k .
2) ( t ) = F ( t ) [1 F ( t ) ]
; F (t )
,
; intensity of a fibre process
, .
, ; intensity of process , ,
.
( ) ( ); interdecile range , .
; interdecile range , , .
; interference channel
.
. .
( K = 2) , ( L = 2 ) , ( M
= 2)
( A1 = {1}, A2 = { 2 }) .
( B1 = {1}, B2 = { 2})
(C1 = {1}, C 2 = { 2} ) .
. . ;
( ., ) .
.: [1] ., ., , .
., ., 1985; [2] . ., . ., .:
. . .
. , 1978, . 15, . 12362.
; interval estimator ,
( )
( , ). . . .-, , ( ), ( ),
;
. . .-.
.: [1] . ., , .,
1984; [2] ., .,
, . ., ., 1973.
; interval data
statistics 1
, R - , ,
. . . .
( ) ( , [1] )
( , [2] ).
. . .- ,
, , ( , [3] [6] ).
,
. ,
,
.
; . . .- -
( ) (
,
) ( , [7] ). , ,
( , [9],
[10] )
, ,
.
, . . .-
( , [8] ).
. . .- . ., :
,
, ,
.
( , [9], [10] ).
.
. . .-
.
. . .-
,
.
. . .- ,
.
.
.: [1] . ., - , ., 1979; [2] . ., .
, 1981; [3] . .,
, ., 1987;
[4] . ., . ., , . , 1989; [5] . .,
, ., 1991; [6] ( 92 ), . 12, .,
1992; [7] . ., .: . . . . . , 1990, . 89
99; [8] . ., .:
. . . . . , 1993, . 14958;
[9] . ., .:
. . . . . , 1995, . 11424;
[10] 11.01183. .
, ., 1984.
;
interval scale , .
; invariant ring , .
;
invariant test
. : g - g 0 = 0
, H 1 : \ 0 H 0 : 0
395
G
. ( x )
, ( x )
. .-
. T ( x )
,
T ( x ) . . . x - T ( x )
.
G - ,
. .
.
, . .- . ( g x ) = ( x )
g - , ( x )
.
.
,
.
. . ,
gS ( x ) = S ( gx) .
. .- .
. .- .
( , [1], [2] )
inf M g ( x ) M 0 ( x ) sup M g ( x )
g
- ,
.
G
.
.: [1] ., , .
., 2 ., ., 1979; [2] K i e f e r J., Ann. Math. Statist., 1957,
v. 28, 3, p. 573601.
; invariant measure 1) ( , A )
T . . A - ,
A A
( A )
= ( T 1 A ). T
f ( x ) d ( x )
f (T x ) d ( x )
.
, .
2) x X , A A , ( X , A ) , P ( x, A )
. .
A - ,
A A
( A) =
396
; invariant distribution T
X (t )
( , A ) P = P 0 ( A ) ;
X (t ) P ( t , x , A ) A A
t T -
P 0( A ) =
P ( t , x , A ) P ( dx )
0
(*)
. P
, -
P ( t , x , A ) Pt ( A )
= P0 ( A)
= P { X ( t ) A} =
X ( t ) -
.
(*) . .
. X (t ) ,
. . .
. .-
, y1 y 2
( , [1] ).
.: [1] . ., , ., 1986, . 18397; [2] . .,
. ., , 3 ., ., 1987.
; family of
invariant distribution , .
; invariant estimator
( , ) .
.
## ; invarian statistical structure ,
; invariant statistic
;
. x
G g
, x a g x , a g , t ( x )
t ( g x) = t ( x)
.
.: [1] . ., ,
., 1973.
P ( x , A ) d ( x ) .
, ( )
.
;
invariant measure , .
; invariance i n t h e s t a t i s t i c a l
g a m e / d e c i s i o n t h e o r y
. D , X
. , X
G
e -; , g 2 g1
x - g 2 ( g1 x) - , g
g - .
, G , g X X -
X .
g G , g ,
A
P ( g X A ) = P g ( X A )
(1)
, P .
- , g = g g
P
G . (1) ,
M ( g X ) = M g ( X )
.
( D , , W ) , ( X , P )
: P G - , w D , g G D
,
w ( , ) = w ( , g )
(2)
G , G
.
g - g
.
D , G g G . , X -
g G D
G
G .
g , g g ( ). , , .
( g X ) = g ( X ) ,
( X )
. (X )
.
.
.
.
G , , . , D -
, g
*
( g X )
= g
. ,
= g * ( X ) .
g g = e ,
,
( gX ) = ( X ) -
. { 1 } { 2 }
g i = i , i = 1, 2 ( , [2] ).
( ) ( ) ,
.
.: [1] , .
( )
;
invariance o f a s t a t i s t i c a l p r o c e d u r e
. , G ,
, D - . ( g x ) = g ( x ) , ( x )
, ; g , G .
R ( ) = R ( | ) = M L ( ( x ) | )
;
L ( g d | g ) = L ( d | )
. - G
.
.
, . . G = ,
. 1:
c : X s ( x )
= c ( x ) 1 x
. 2: s ( x )
c ( x ) . 3:
x X x
=x
, d - d
=d
. 4: L ( d / ) d
d + - . ,
D d , d n
. 5: G -
n , g -
lim sup n ( B g ) n ( B ) = 0
, .
,
lim inf
R( | )
(d ) = R ( 0 )
, 0 .
- - , *
D Px
397
L ( | )
, - L ( | )
. ( )
L ( | )
w ( ) ,
L ( | ) P
(t ) w (t ) = O ( (t ))
P ,
,
P ( B ) =
L( | ) =
P ( B ) ( d )
, ( ) ( ) -
( ) w ( )
.
1 , 2 , ... ,
L ( | ) (d ) .
.: [1] . ., . ., .
., 1975, . 20, . 2, . 30931; [2] W e s l e r ., Ann.
Math. Statist., 1959, v. 30, 1, p. 120; [3] . .,
. . -. . . . ., 1977; 2,
. 5866.
; invariance principle i n B a n a c h s p a c e s
M1 = 0 , D1 = 1 , k t < k + 1
(t ) =
( B,
+ ( t k ) k +1
. . , (t )
= o ( ( t ln ln t )1/ 2 )
, (*) ( , [1] ).
( )
,
P { lim sup ( t ) ( 2 t ln ln t )1 2 = 1} = 1 .
) - ,
; , i -
i =1
. { i } -
>2
M 1
<
, (t )
, .
{ i }
= o ( t ) (*) .
, { i } -
t >0
{ i } -
lim P max
n
k n
i =1
>0
(i i ) > n = 0
(*)
, { i } , .
. .
. , (*) ,
, ( , [1],
[2] ). , ,
. .- ( ,
( )
, [1], [2] ).
.: [1] D u d l e y R. M., P h i l i p p W., Z. Wahr. und verw.
Geb., 1983, Bd 62, H. 4, S. 50952; [2] . C., .
. - ., 1985, . 5, . 327; [3] . .,
. ., . ., 1980, . 25,
. 4, . 73444.
; insensitivity /
invariance problem in theory ,
;
; s t r o n g /
almost
s u r e invariance principle
398
12
Me
t 1
<
( t ) = ln t
(*) ( , [2] ). . ( ,
[1] ) ( , [3] ) .
.
.: [1] S t r a s s e n V., Proc. 5-th Berkeley Sympos. Math.
Statist. Probab., 1967, v. 2, 1, p. 31543; [2] K o m l o s J., M a j o r P., T u s n a d y G., Z. Wahr. und verw. Geb., 1976, Bd 34,
1, S. 3358; [3] B e r k e s I., P h i l i p p W., Ann. Probab.,
1979, v. 7, 1, p. 2954.
; invariance principle f o r
s t o c h a s t i c / r a n d o m p r o c e s s
. . .
. { X (j n ) , j
= 1, ... , mn }
n = 1, 2 , ... X jn , , n j -
mn
DX
M X (jn ) = 0 ,
( n)
j
= 1.
j =1
tk( n ) =
DX
(n)
j
s n
= s n ( )
jk
sn ( tk( n ) ) =
(n)
j
j k
( n)
, t k
t t k( n+)1
( ), Pn
C [ 0, 1] - s n -
( j
; W - w -
.
C [ 0, 1] n
Pn W
(1)
(n )
)
(n
j
= j
M {( X
(n) 2
j ) ;
|X
( n)
j
| > } n
0 ;
>0
= X j n1/ 2
mn
=n
, X 1 , X 2 , ...
, , . [1]- . . . ,
{ X (jn ) } -
. . .
. (1)
f ( )
P { f ( sn ) < u } P { f ( w ) < u }
. - . .-
{ X
(n)
j
} , , -
. .
(1) (2)- . .- . .- ( , ;
).
.: [1] D o n s k e r M., Mem. Amer. Math. Soc., 1951, v. 6, p.
112; [2] . ., . .,
1956, . 1, . 2, . 177238; [3] .,
, . ., ., 1977.
;
; ; invariance principle; r a t e o f c o n v e r g e n c e s n
< , 1 , 2 , ...
Pn w
W
n = ( Pn , W ) .
L( n )
mn
(j n )
>2
. . ( , [1] ) (4)-
<
(3)
## . (1), (2), (3) (n )
.
. .
n Pn
W . .,
n ( B ) P { s n B } P {w B } ,
n ( f , u ) P { f ( sn ) < u } P { f ( w ) < u }
,
B , C [ 0, 1] , f ( ) ,
C [ 0, 1] - . B ,
P {w ( B ) ( ) } C B , > 0
, n ( B )
n ( B ) (1 + CB ) n
, (B )
.
( )
(4)
, B B
, f ( )
f ( x ) f ( y ) C1 ( f ) x y ,
x , y C [ 0, 1]
, f ( ) C 2 ( f )
,
n ( f , u ) ( 1 + C1 ( f ) C 2 ( f ) ) n
(5)
.
(4) (5) (1), (2) (3) n ( B ) n ( f , u )
. ., B ,
j =1
## n C ( ) ( L(n ) )1 ( +1) , > 2
t 1
. { j } -
W -
.
C [ 0 , 1] Pn
(2)
n = O ( n 1 2 ln n )
(2)
f ( x) = x (1) (2)-
=n,
,
n
. t M e
.
(1) .
. . ( , [2] ).
(n )
n = o ( n ( 2 ) 2 ( + 1) )
j=1
, X
.- ). mn
n > 2 M 1
mn
(1)
## n ( B ) C ( B, ) ( L(n ) )1/ ln ( 1 + ( L(n ) )1/ ) , > 2
.
399
f ( x ) max x ( t )
;
irregular point ,
0 t 1
,
.
f ( x) =
g ( x ( t ),
t ) dt
.
B = { x C [ 0, 1] : g ( t ) x ( t ) g + ( t ) , t [ 0 , 1]}
g ( t + h ) g ( t ) K h ,
mn = n ,
j = n (j n ) j = 1, ... , n ,
3
n ( B ) Cabs ( K + 1) M 1 / n1/ 2
;
rose of direction , .
; direction statistic ,
; directed set , ( ) .
,
; sign
method of correlation analysis , .
.
.: [1] . ., . .,
1956, . 1, . 2, . 177238; [2] K o m l o s J., M a j o r P., T u s n a d y G., Z. Wahr. und verw. Geb., 1976, Bd 34, 1, S. 3358;
[3] . ., . , 1983, . 38, . 4,
. 22754; [4] . ., . -
,1985, . 5, . 2744.
; inversion
, -
; sign
test ( n ; p = 1 2 )
H 0 . H 0
,
P { k | n, 1 2 } =
; Ionescu Tulea theorem
.
( S n , An ) , pn ( x0 , x1 , ... , dxn+1 )
( S 0 S1 ... S n , A0 A1 ... An )
( Sn +1 ,
, ) . [1]- . . .-
( , A ) = ( S 0 S1 ... , A0 A1 ... )
, x S 0 P ( x , d )
= I 0, 5 ( n k , k + 1) ,
k = 0 , 1, ... , n ;
I z ( a, b ) =
1
B(a, b)
a 1
(1 t ) b 1 dt ,
B ( a , b ) . . .-
0 < 0, 5 H 0
min { , n } m
0 z 1,
, , m
## , ..., p n 1 ( x0 , x1 , ... , xn 1 , dxn )
x -
A0 . . . .-
, ,
t = 0 , 1, 2 , ... , , .,
.
P ( x , d ) - S 0 - ( dx ) - -
,
.
.: [1] I o n e s c u T u l c e a C. T., Atti Acad. Naz. Lincei
Rend., 1949, v. 7, p. 20811; [2] ., , . ., ., 1969.
400
= m ( , n )
. .-
1 x ( d x0 ) p0 ( x0 , dx1 ) p1 ( x0 , x1 , dx2 ) ,
P ( x , d )
i
n
, n
( S 0 ... S n )
C (1 2 )
i=0
An +1 ) , n = 0 , 1, 2 , ... ( -
- .
0 t t + h 1
Cni (1 2) n 2 ,
i=0
m +1
i
n
(1 2 ) n > 2
i =0
. . .- 1 , ..., n
,
x
P { i < x } = P { i > x }
H 0
.
. .
( ) ,
i
i =1
1,
0 ,
( x) =
x > 0,
x<0
, , H 0 ,
( n ; p
= 0, 5 ) -
.
. . 1 , ..., n
0 -
H 0
; 1
= 1 0
, ... , n
= n 0
, .
.: [1] ., , .
., .: , ., 1967, . 300522.
; Ito formula ,
, ( t ) t 0 ,
2
.
.: [1] . ., . ., , [3 .], ., 1983; [2] ., , . ., 2 ., ., 1979; [3]
. ., , . .,
., 1960; [4] . ., - .
.,
, 3 ., ., 1969.
( )
r a n d o m v a r i a b l e s
, .
,
( X 1 , ..., X n ) , k = 1, k = 1, ... , n
( X 1 , ... , X n )
(1 X 1 , ... , n X n ) ,
.
( X )
. - . . , ( X )
. - . . . . - . .-
f C ( R ) ( f ( t ) ) t 0 t
f ( t ) = f ( 0 ) +
f (
) d s +
+
+
1
2
f (
) d c, c
[ f ( )
s
f ( s ) f ( s ) ( s s )].
s t
, 1 , d c
, d ,
c, c
,
c
. ,
c2
, m c ,
= m + A -
m -, A
.
c, c
t ) .
.
,
.
( , [1] ).
.: [1] ., , 1959, . 3, 5, . 13141;
[2] D e l l a c h e r i e C., M e y e r P. A., Probabilites et potentil,
pt. 2, ch. 58. Theorie des martingales, P., 1980.
; sign correlation
function , .
; loss function ( ) D -
; Ito representation R d - , ,
( t ), t 0
:
(t ) = c (t ) +
D - ,
-
( ) d ( d D )
( )
L ( , d ) . . .-:
( R k )
(D=)
L ( , d ) = d
H i : i , i = 1, ... , k 0 1
0, i , d = d i , i = 1,..., k ;
L (, d ) =
1,
; d i , H i .
x ( ) ( dt , dx ) +
0 | x| 1
x ( dt, dx ) ,
(*)
0 | x | >1
c ( t ) -
( R+ Rd , B ( R+ Rd ) )
( , dt , dx )
( t ) :
(, A) =
I ( ( t ) 0 ) I (( t , ( t )) A )
t0
( t ) = ( t ) ( t 0 ) , ( A) = M ( , A) .
401
- . [2]- ; , (*) .
.: [1] I t o K., Japan. J. Math., 1942, v. 18, 2, p. 261301;
[2] . ., . ., , 2 ., ., 1977.
; Ito process
t
t = 0 + a ( s, ) ds +
0
b ( s, ) dw ,
s
t [ 0, T ]
(*)
, w A
= ( At )
, a b , A
a ( , ) L1 ( [ 0 , T ] ) ,
b ( , ) L2 ( [ 0, T ] ) -
. A
, t [ 0, T ] w (t )
At
h>0,
s [ 0, T h ]
w ( s + h ) w ( s ) As . , (*)
.
.: [1] . ., . ., , ., 1974.
## ; Ito stochastic integral ,
.
;
Ito Nishio theorem , ;
; Ito Wentzel formula
,
.
f ( t , x )
d f ( t , x ) = I ( t , x ) dt + H ( t , x ) dw ( t )
, w ( t ) , { At }
, I ( t , x ) H ( t , x )
( ) . ( t ) , { At }
d ( t ) = b( t ) dt + ( t ) dw ( t )
.
f , T , H
df ( t , ( t )) = I ( t , ( t ) ) dt +
H ( t , ( t ) ) dw ( t ) + b ( t )
l
f ( t , ( t )) +
xi
1
2
il ( t ) jl ( t )
+ f ( t, ( t ) ) +
2 i , l , j ...
xi x j
402
i, l
H l ( t , ( t )) dt +
xi
il ( t )
f ( t , ( t ) ) dw l ( t )
xi
i, l
il
(t )
.
.
.: [1] . ., . .,
1965, . 10, . 2, . 39093; [2] . ., , ., 1938, . 208.
;
hierarchical classification procedures
N , D
.
N N 1
. -
. :
, . .
, , .
. , ,
, . .
.
k i j
i + j ( ),
, ni n j
d k , i + j = ( ni d ki + n j d kj ) ( ni + n j )
, d ki d kj , k i
j . . . .-
.
.: [1] ., ., , .
., ., 1977.
; needle
, .
; Ising model
. d . .- d
t Z
, {x s }
{ 1, + 1} .
U :
U { s , t } ( x s , xt ) = I x s xt ,
U { s} ( x s ) = h x s ,
|s t | = 1 ,
s, t Z d ,
U . h
. { y s }
U . .-
:
U{s , t } ( ys , yt ) = 4 I y s yt , | s t | = 1 ,
U{s } ( ys ) = ( 4 dI + h ) y s ,
s, t Z d
. ,
.
, d = 1 . .- -
y s , yt { 0 , 1} .
I > 0 . ., I < 0 . . .
. . .
. .- [1]- ,
d = 2 f d ( , h = 0)
. ,
,
( , [10] ).
cr ( d ) .
. . .-
d 2 .
f 2
cr ( 2 ) = ln ( 2 + 1) 2-
> 2 f d ( , h = 0 )
cr ( d ) .
d ( , h ) h
( , [2] ), d 2 , >> 1 h = 0
. d = 1 f1 ( , h)
> 0 , < h <
. d
.
. .-
. h 0 h = 0, < cr ( d )
.
h,
.
h,
lim
h 0
> cr ( d ) h = 0 -
. ,
x0
= x0
> 0,
x0
+1 , + .
( , [3]
[6] ) .
. d = 2 ,
h = 0
( , [7], [8] ). d
3 , h = 0
> r ( d ) cr ( d )
. .,
,
xt
> 0, t (1) 0 ,
## < 0, t (1) < 0
. d
4 , r ( 3 ) > cr ( 3 )
r ( d ) = cr ( d ) . r ( d ) -
-
( . ) . . .-
. .
.
.: [1] O n s a g e r L., Phys. Rev., 1944, v. 65, 2, p. 117
49; [2] I s a k o v S. N., Comm. Math. Phys., 1984, v. 95, 4,
42743; [3] P e i e r l s R., Proc. Camb. Phil. Soc., 1936, v. 32,
3, p. 47781; [4] . ., .
., 1965, . 10, . 2, . 20930; [5] G r i f f i t h s R., Phys.
Rev., 1964, v. 136A, 2, p. 43739; [6] . .,
, , 1980; [7] A i z e n m a n M., Comm. Math.
Phys., 1980, v. 73, 1, p. 8394; [8] H i g u c h i Y., .: Random
fields, v. 1, Amst. N. Y. Oxf., 1981, p. 51734; [9] . ., . ., 1972, . 17, . 4,
. 61939; [10] D o b r u s h i n R. L., S h l o s m a n S. B., Sov.
Sci. Rev., 1985, v. C5, p. 53195.
; rezolution , .
,
; m e t r i c isomorphism of dynamical systems
.
( , A , P) (1 , A1 , P1 ) .
: 1 ,
1) = 1 ;
2) 1 ;
3) P P1 ,
P ( 1 A1 ) = P1 ( A1 ) A1 A1 ;
4) ( , A , P) - {T t } (
) (1 , A1 , P1 ) -
{T1t } , t
P - (
403
t - ) T t = T1t , {T t }
{T1t }
. ,
mod 0 ; mod 0
t
, , 1 , {T } {T1 }
.
## 1), 3), 4) , {T1t } {T t } (
) .
, ( , [1] ).
( , A , P) , t
t
{T } t
{T1 } ,
; isotropic random space ,
.
; isotropic random set
.
; isotropic random field
. ,
.
g M X ( t ) = MX ( gt ) ,
M X ( t ) X ( s ) = M X ( gt ) X ( gs )
t R
. . . .
t ( ) = ( T t ) , . , ,
, t -
A - . ,
, ,
( , [2]
[4] ). {T t }
Yt
t
, {T } -
{T1t } -
t Yt ( )
.
, .
, ,
, , K , ;
( ,
[5], [8] ).
( ., ) -
,
.
. . . . ,
.
( , [6] ),
( , [7], [8] );
( ,
[8] [10] ).
.: [1] . ., , 1962, . 147,
4, 797800; [2] . ., . . -,
1963, 1, . 2632; 1965, 13, . 6872; [3] K r i e g e r W.,
Trans. Amer. Math. Soc., 1970, v. 149, 2, p. 45364; [4] K r e n g e l U., J. Math. Anal. and Appl., 1971, v. 35, 3, p. 61120;
[5] O r n s t e i n D., Adv. Math., 1970, v. 5, 3, p. 34964;
[6] N e u m a n n J., Ann. Math., 1932, v. 33, 3, p. 587642;
[7] O r n s t e i n D., Adv. Math., 1970, v. 4, 3, p. 33752;
[8] ., , , . ., ., 1978; [9] . .,
. . . ., 1962, . 26, 4, . 51330;
[10] . ., . , 1962, . 144, 2,
. 25557.
X (t ) =
X (t ) ,
h ( m, n )
l
m(r )
S ml ( 1 , ... , n2 , )
m=0 m=0
, (r , 1 , ... , n 2 , ) , t
l
, S m ( 1 , ..., n2 , ) , m
l
, h ( m , n ) , S m ( r )
,
## M X ml ( r ) X ml11 ( s ) = bm ( r , s ) mm1 ll1
, i
(*)
, bm ( r , s )
m 0
h ( m,
n ) bm ( r , s ) < + , bm ( 0, s ) = 0
m=0
- . . . .
X ( r, ) =
{ X
1
k
( r ) cos k + X k2 ( r ) sin k }
k =0
, X k ( r ) , l
= 1, 2 (*)
.
, . . .- .
.: [1] . .,
, ., 1980.
; isotropic random process
, K ( z ) | z |
2
k
k =1
404
, R - X ( t )
; X ( t ) - R -
. K ( z )
X ( t ) = { w1 ( t ) , ..., w d ( t ) }
2
K ( z) = c | z |
, c >0 , 0<
<2
= | z |2 d
,
; Jirina process [ 0 , ) ,
t > 0 ,
(
) .
. , [1]- . , , .
.: [1] J i r i n a M., . . ., 1958, . 8, 2,
. 292313.
; Jordan decomposition , .
;
sufficient estimator
M ( g~, ( )) M ( g , ( )) ,
.
. . .
. . . .
.
.: [1] ., , .
., 2 ., ., 1975; [2] .,
, . ., ., 1975.
.
, . .- , . . , . .-
( , [2] ). P T . .- ,
.
, P
. P
;
sufficient statistics
. ,
. .- -
. .
.
. . .
P = {P }, x ( X, A )
, M , P
. . M
< ,
( x ) -
( ) - ( t )
M ( T ) =
(*)
P - ( ) , , t ( T , L )
T P .
P , P ,
, T
( ,
[2] ): T , , P
. .- , p ( x ; ) = d P d -
p ( x; ) = R (T ( x ); ) r ( x )
.
. .- ( , [2] ) : g = g ( x )
= ( )
( g , ) ,
~
g
; ( x; )
, T ( ) I T ( ) . I T ( ) I ( )
,
, ,
, T . . ( , [1] ). . .-
.
P , P ,
f ( y; ) ,
I ( ) = M ( ln p ( x; ) / ) 2
= g~ ( t ) ,
406
p ( x; ) =
f ( y ; ) ( y , ..., y ) = x
1
i =1
, , f ( y; ) - , P - . .-
( , ).
, .
P . .- , T1 = h ( T2 )
< T2 . , ( X, A ) , P
, T1
, . . .
. . T (x)
m
T =
t ( y ) , ..., t
1
i =1
i =1
( yi )
.
. . . .,
: , T x ,
T P . .-. . .- (*) .
. .
. . . .
, . ( J. Neymann ) 1935- .
. ( P. Halmos ) . ( L. Savage )
1949- . . .
. . ( H. Scheff ) .
. .- . .
.
.: [1] . ., . .,
, ., 1979; [2] . .,
. ., . ., , ., 1972; [3] ., , . ., 2 ., ., 1979; [4] .,
, . ., ., 1991; [5] B a h a d u r R. R., Ann. Math. Statist., 1955, v. 26, 3, p. 49097;
[6] F i s h e r R. A., Philos. Trans. Roy. Soc. London. Ser. A, 1922,
v. 222, p. 30968.
; m i n i m a l sufficient
statistic , .
; n e c e s s a r y sufficient statistic , .
;
sufficient topology
.
,
. T Tk
( T , T ) . .-; . . .-
T - . . .
, .
. .- .
.: [1] . ., . ., , 1969, . 12, . 767; [2] .
., , ,
1989.
; sufficiency i n t h e g a m e t h e o r y
. P X
, , S ,
, w (u ) , R -
.
, :
= (X )
( )
- S
= M ( / S )
M w ( S ) M w ( ) .
S , .
.: [1] ., , .
., 2 ., ., 1979; [2] ., , . ., ., 1975; [3] . .,
. ( ), ., 1984.
; Kahanes contraction principle
: { X k }
B - , { k }
X k B - ,
k =1
X
k
k =1
n
B - .
X k , n 1
k =1
B - ,
k X k , n 1 B -
k =1
. . . .
,
[1]- . . .
.
.: [1] .-., , .
., ., 1973; [2] H o f f m a n n - J r g e n s e n J., Studia
Math., 1974, v. 52, 2, p. 15989.
; gauge model o f s t a t i s t i c a l m e c h a n i c s
,
.
G d
d Z
( ) -
= g1
, G g
S .
( , ),
S - .
, .
.
w ( , ) = w ( ) w ( , ) -
g = { g } , ( ) , Z d - -
.
- ( . ) S ( g )
S(g) =
f (g
(*)
p : p
( R )
.
407
, Z ( ) p , -, g p
, f G -
, f ( g )
= f ( g 1 )
, G - -
. (*)-
-
d d 0 =
1
exp { S ( g ) }
Z
, d
dx
) Ys , 0 s t
t (Y ) .
= {Yt , t 0} w
~ = {w
~ , t 0}
, w
t
Y
dYt = A ( t ) t dt + B( t ) dt
t
[ a(s)
% Z
(1)
(2)
. (1) (2)
cov ( 0 , Y0 )
( Y0 MY0 ),
cov (Y0 , Y0 )
2
cov ( 0 , Y0 )
P0 = cov ( 0 , 0 )
,
cov ( Y0 , Y0 )
0 ( Y ) = M 0 +
, e
( ; , [2] ).
= { x , x Z d }
Pt A ( t )
[ d Yt A ( t ) t ( Y ) d t ]
B2 (t )
dPt
P 2 A2 ( t )
= 2 a (t ) Pt + b 2 ( t ) t 2
dt
B (t )
, , G
t 0
d t ( Y ) = a ( t ) t ( Y ) d t +
f ( g ) = ( ( g ) + ( g ) ) e
inf B 2 ( t ) > 0
, ;
(
) , Z . = lim
+ A 2 ( s ) ] ds < ,
, G -
(*)
( ; , [1] ). (*)- f
>0 ,
(3)
0
= 0,
0
, x , G g U g
Pt = M ( t t ( y ) ) 2 .
M (
;
, [1] ).
(
)
Y
~
w w
0 , Y0 ~ , (1) (2)
, w , w
g ( x1 ) g 1 ( x2 ), = { x1 , x2 }
(
x u ( x ) x ,
= { x , x Z d } G - Z d -
. ,
, .
.: [1] ., , . ., .,
1985; [2] W i l s o n
K., Phys. Rev., D., 1974, v. 10, 8,
p. 244559; [3] O s t e r w a l d e r K., S e i l e r E., Ann. Phys.,
1978, v. 110, 2, p. 44071.
; Kalman
filter w = {w ( t ) , t 0}
d t = a ( t ) t dt + b ( t ) d w t
= { t , t 0}
t ( t
408
(3)
t ( Y )
Pt - .
, .
.: [1] . ., . ., . ,
. ., 1961, 1, . 12341; [2] . ., .
., , ., 1974.
; Kalman Busy method , ;
; channel w i t h a d d i t i v e n o i s e
. .
~
Y = Y + Z , Y , Z
Y -
( ). , Z
( Z
) .- .
. . .
., ( ) . Y
= Y1 + Y2 + Z
, Y1 Y2
, Z
Y1 Y2 -
.
.: [1] ., ,
. ., ., 1974; [2] . ., .
., , ., 1982.
; a s y m m e t r i c channel , .
.:
~
( X , SX ), ( X , SX~ ) ( -
); A SX~ SX
x X ( X , S X~ )
Q ( x , A ), x X , A SX~ ( X , SX ) -
; a s y n c h r o n o u s channel , .
V
. Q ( x, A )
; h o m o channel , .
; m u l t i p l e a c c e s s channel ,
geneous
; m u l t i t e r m i n a l
.
channel ,
;
m u l t i w a y channel , .
; d e t e r m i n i s t i c
.
channel ,
channel
; b r o a d c a s t
.
- ; b i n a r y s y m m e t r i c channel , .
; i n t e r f e r e n c e
.
channel ,
; q u a n t u m c o m m u n i c a t i o n channel
, .
( ) ; G a u s s i a n channel , .
; n o n a n t i s i p a t i v channel ,
.
;
; channel c a p a c i t y
,
.
, , .
; d e gr a d e d channel , .
; ; c o m m u n i c a t i o n channel ,
. .
.
; V
. ( , A , P )
X X
~
( X , S X ) ( X , S X~ )
~
A SX~ P {X A | X = x } = Q ( x, A )
X - V
~
, X X ( Q, V ) , .
( X , S X ) ( X , S X~ ) [ a , b ]
(Y , SY ) , (Y , SY~ )
. .- .
( X , S X ) = Y [ a , b ] , SY [ a , b ] ,
~
~
( X , SX~ ) = Y [a, b ], SY~ [ a , b ]
[ a , b ]
.- ;
~
~
Y = {Y ( t ), t [a, b] } Y = {Y ( t ) , t [ a , b ]} ,
~
(Y , SY ), (Y , SY~ )
, Y ( t ) Y ( t ) t
.
, .-
, (Y , SY ) (Y , SY~ )
= (Yk , Yk + 1 , ..., Yn )
~n
~ ~
~
Yk = (Yk , Yk +1 , ..., Yn ) ; Y j
~
Y j , j
n
Yk
= const . .
( , ) t .- .
. .- Q V - ( , .,
, ,
409
. ). .-
, .
( , ).
.- ( , ., ,
, ).
.: [1] ., , . ., ., 1967; [2] ., , . ., ., 1974; [3] . ., . , 1959, . 14, . 6, . 3104;
[4] . ., . .,
, ., 1982; [5] .,
, . ., ., 1960; [6] .,
. , . ., ., 1965;
[7] ., ., , . .,
1985; [8] ., , .
., 1963, . 243332.
; s e m i d e t e r m i n i s t i c channel , .
; z e r o - e r r o r
.
channel
;
s y m m e t r i c channel , .
ronous
channel
; s y n c h , .
; chanell w i t h s y n c h r o n i z a t i o n
.
, .
(~
y1 , ..., ~
y n )
n = n c c = 1, ~
yj = yj;
j < i
j > i c = 1,
~
yi +1 = yi ,
yi = a , ~
~
y j c = y j
,
( y1 , ..., y n ) i - ,
( a
) .
. t ,
t = 0 , 1 , ... yi p,
p p+ -
: ; ;
p(a ) a , , yi
t + 1 . .-
(
) ,
.
. : , ,
. .
.: [1] . ., , .
., ., 1975; [2] . ., . ,
1965, . 163, 4, . 84548; [3] . ., . ., . ., , .,
410
## 1986; [4] . ., , 1967, . 3, 4 . 1836; [5] . .,
, 1970, . 6, 3, . 4349.
; f i n i t e - m e m o r y
channel t
> 0 [ t , t + ]
( t m, t + ] , ,
m -
, ( ) . .
,
, Y = ( ... , Y1 , Y0 , Y1 , ...)
~
~ ~ ~
Y = ( ... , Y1 , Y0 , Y1 , ...)
Yi
~
Yi
Y Y
. ,
:
1) m ,
yi = yi Y , i = t m , ..., t + n 1
~
~
yt , ..., ~
yt + n 1 Y
~
~
P { Yt = ~
yt , ..., Yt + n 1 = ~
yt + n 1 | Yk = y k , k = ..., 1, 0, 1,...} =
~
~
= P {Yt = ~yt , ..., Yt + n 1 = ~yt + n 1 | Yk = y k , k = ..., 1, 0, 1, ...}
;
~
~
j+m<k
2)
yi , ..., ~
yj Y
~
~
~
y k , ..., y n Y yl Y , l = ... , 1, 0 , 1, ...
~
~
P {Yi = ~
yi , ..., Y j = ~
yj;
~ ~
~
~
Yk = y k , ..., Yn = ~
y n | Yl = ~
yl , l = ... , 1, 0,1, ...} =
~
~
~ ~
~
~
= P { Y = y , ..., Y = y | Y = y , l = ... , 1, 0,1, ...}
i
~
~
~
P { Yk = ~
y k , ..., Yn = ~
y n | Yl = ~
yl , l = ..., 1, 0, 1, ...}
.
m
( ) , m = 0 , ,
.
.: [1] .,
, . ., ., 1967; [2] ., , . ., ., 1960; [3] .
., . , 1956, . 11, . 1, . 1775.
;
s t a t i o n a r y channel , .
; ; ; n e t w o r k o f
channel , .
; channel w i t h f e e d b a c k ,
; .
.-
- .
.-
. .-
. ,
.
.: [1] ., , . ., ., 1963, . 46487; [2] .,
, . ., ., 1972.
; m e m o r y l e s s channel
.
~
~ ~
Y = ( Y1 , Y2 , ... ) Y = ( Y1 , Y2 , ... )
.-
~
~
P { Y1 = ~
y1 , ..., Yn = ~
y n | Y1 = y1 , ..., Yn = y n } =
; channel w i t h f i n i t e n u m b e r o f s t a t e s
t
. .-
.
,
~
~
= P { Y1 = ~y1 | Y1 = y1} ... P {Yn = ~yn | Yn = yn }
~
n yi Y , ~
yi Y , i
,
, Yi Yi , i, j
= 1, 2 , ...
{ p s0 , s0 S } .
. Y Y
q ( y, s' ; ~
y , s" ) k .-
~
y s" -
, y ( k 1)
.
,
. P { Yi
~
~
y Y , s ' , s" S
Y Y
Y Y
.
S , .- ,
q ( y, s'; ~
y , s" ) , y Y ,
= 1, ..., n
= ~yi | Yi = yi }
i - , . (
)
.
~
q(y, ~
y ) , y Y , ~
y Y
y)
, q ( y , ~
s ' .
~
= P {Yi = ~y | Yi = y } , i = 1, 2 , ... .
{ p s0 , s0 S } 0 .-
n
y n , s n ) . Qn ( y , s0 ; ~
.: [1] ., ,
. . ., ., 1974; [2] ., , . ., ., 1963, . 243332.
;
canonical variable ,
sn 1S
Qn ( y n , s0 ; ~
y n , sn ) =
q ( y n , s n 1 ; ~
y n , s n ) Qn 1 ( y n 1 , s0 ; ~
y n 1 , s n 1 )
; canonical correlation
,
. . . 1 , ..., s s +1 , ..., s + t , s t
Q1 ( y , s0 ; ~
y , s1 ) = q ( y, s0 ; ~
y , s1 ) ,
y n = ( y1 , ..., y n ), y n 1 = ( y1 , ..., y n 1 ), yi Y ,
1 , ..., s s +1 , ..., s + t :
~
y n = (~
y1 , ..., ~
yn ) , ~
y n 1 = ( ~
y1 , ..., ~
y n 1 ) ,
1) i ,
~
~
yi Y , i = 1, ..., n : s k S , k = 0 , 1, ..., n .
Qn ( y n , s0 ; ~
yn) =
1- , 2) i
Qn ( y n , s0 ; ~
y n , sn )
sn S
. .- n
n
Qn ( y , ~
yn )
~
= P {Y n = ~y n | Y n = y n }
n
Qn ( y , ~
yn) =
n
p s0 Qn ( y , s 0 ; ~
yn)
n
s0 S
Y n = (Y1 , ..., Yn )
~
~
~
Y n = ( Y1 , ..., Yn ) .- n
.
.: [1] ., , . . ., .,
1974; [2] .,
, . ., ., 1967.
, 3)
,
4)
.
s = 1 , . . 1 2 , ..., 1+ t
.
. . .
,
.
, .
.: [1] H o t e l l i n g N., Biometrica, 1936, v. 28, p. 32177;
[2] .,
411
, . ., ., 1963; [3] ., .,
, .
., ., 1976.
; canonical
correlation coefficients 1 , ..., s s +1 , ..., s + t ,
s t
U = 11 + ... + s s
V = 1 s +1 + ... + t s + t
; U
V
s +1 , ..., s + t
, 1 , ..., s
= MV = 0 MU 2 = MV 2 = 1
. . . .
11
21
1
, 12
= 21 1- 2-
.
r - r -
(r )
(r )
V
U
, 1- U
(r )
(r )
V
V - r 1 . U
( r ) = ( 1( r ) , ... , s( r ) ) T ,
( r ) = ( 1( r ) , ..., t( r ) ) T
11
21
= r
12
22
=0
; canonical correlation
analysis ,
c = M ( x x T ) x T = ( x1T , x T2 ) T
x1
= ( x1 , ... , xn ) T
x 2
= ( xn + 1 , ..., xn + m ) T
. . . .-
M ( )
2
M ( ) M ( )
412
= x1Ta
= x T2 b
a T c12 b
T
, x
( c ).
## aT = ( a1 , ..., an ) bT = (b1 , ..., bn )
, (1)- (1)
( - 1-
). (1)
= aT c12 b
( a T c11a 1)
( a c11 a ) ( b c22 b )
(1)
## ( bT c22b 1) max (2)
,
. (2)- ,
a=
1
c11
c12 b ,
1
1
c11
c12 c22
c21 a = 2 a ,
; 11 22
1 , ..., s s +1 , ..., s + t
, c11
c22 , c12
b =
, c21
12
=0
22
... s > 0
( ,
). MU
1
c22
c21 a
(3)
= c12T . (3)-
1
1
c22
c21 c11
c12 b = 2 b
(4)
## (1) (2)- i = i2 = aiT c12 bi -
2
. 1
22 ... 2n
= 12 - , a1
b1 1
. , ,
( , ).
min (n , m) - . , (4)
. , (3) . . . .-
.
.: [1] H o t e l l i n g N., Biometrika, 1936, v. 28, p. 32177;
[2] ., ., , . ., ., 1976;
[3] V a n d e n W o l l e n b e r g A. L., Psychometrika, 1977,
v. 42, 2, p. 20719; [4] . ., , ., 1982; [5] . ., , 1984, 10,
. 5056.
;
canonical correlations and variables , ;
.
() ; canonical measurement , ,
.
;
canonical parameter , , ,
.
; canonical distribution . ,
- .
; canonical spectral equation ( x )
.
n = ij( n ) ij( n )
,
ij(n ) ,
i j , i , j = 1, ..., n
K n ( u , , z)
K ( u , , z ) ,
. . .-
0 z 1, t < x ( 0 x 1 )
G ( x, z , t ) L
k
>0
0 z
1,
xk d G (x , z , t )
t ( , )
.
.: [1] . ., ,
., 1980.
; canonical and mean value parameterization ,
. . .- - ,
n
ij( n )
xdP
{ ij( n )
< n},
{ Ps , s Dom R } , d Ps / d
x <
## > 0 , i n u < ( i +1) n , j n < ( j + 1) n
K n ( u, , z ) = n
y2
d P { ij( n ) ij( n ) < y } ;
1 + y2
K ( u , , z ) , 0 u 1, 0 1 u k
, z ,
p ( ; s ) = p0 ( ) exp [ s i qi ( ) ( s) ]
, s q1 ( ) , ..., q n ( ) , -
, ( s ) ,
( s ) = ln
n ( x ) = n 1
F (
x),
k =1
0 , y 0 ,
F ( y) =
1, y < 0
k , n . 1
lim n ( x ) = ( x )
( x ) ,
( x ) -
(1 + itx ) 1 d ( x ) = lim
a0
x d Ga ( x, z , t ) d z
(1)
( ) exp [ s i qi ( ) ] { d }
(2)
n
Dom (2) s R
.
Dom , , (s )
( , [1] ), , dim Dom
Dom = .
dim Dom = n
<n
1( ) = 1 ,
q1 ( ) , ..., qn ( ) mod ,
,
. (1) t = ( t1 , ..., t n ) ( s )
t j ( ) =
( )P { d } = ( s j ) ( s )
s =
(3)
, t
. (s )
, Ga ( x , z , t ) , x -
s t int Dom -
t < ) , -
.
: Pt { }
(0 x
1, 0 z 1 ,
Ga ( x , z , t ) = P { [1 + t 2 a ( Ga ( , , t ) , z )]1 < x }
;
a (Ga ( , , t ), z )
( , [1] ).
= ( t1 , ..., t n )
q ( ) ,
(2)
q ( ) = (q1 ( ) , ..., qn ( ))
413
(3)
q ( ) - ( , [2],
[3] ).
0 Dom ( : s s = s )
P ( X , L ) -
. P, Q P
, t
p ( , s ) = p ( , 0) exp[ s i qi ( , 0 ) I (P0 : Ps )]
,
P0 -
qi ( ; 0 ) = ( s ) ln p ( ; s )
s=0
(4)
qi ( ) -
, 1 2 0 1
.
, R -
.
( 0 , 1) - . .-
2i
i =1
( 0 , 1) - ,
u
= 3 + 2 3 .
.: [1] .,
, . ., . 2, ., 1984.
414
d ( x1 , x2 ) H ( d x1 , d x2 )
(1)
XX
,
P Q - , ( X , L ) ( X , L ) - H
. ,
(1)
.
. . [1]- .
P ,
Q , d ( x1 , x2 ) ,
x1 x2
, .
:
( P, Q ) = sup
f ( x) (P (dx) Q (dx)) ,
(2)
f ( x) f ( y ) d ( x1 , x2 ) , x, y X -
. (2) . . [1]-
. - (2) ,
( , [4], [2] ).
- .,
d,
P ( A) = Q ( A a ) , a R
. .-.
A 0 B A + B -
( P, Q ) = inf
2i 1 2 2 i 1
i =1
: u
()
, i
f
L
, X -
. X
2 i / 2 2i
, X -
i =1
= (3/ 2)
d X - , d - -
; Cantors distribution ( 0 , 1)
=3
. ( X , L ) ,
Ps -
.
.
. . . -
.
.: [1] ., , .
., 2 ., ., 1979; [2] . ., , ., 1972; [3] B a r n d o r f f N i e l s e n O., Information and exponential families in statistical
theory, N. Y., 1979; [4] A m a r i S., Differential-geometrical methods
in statistics, N. Y. L., 1985; [5] . .,
, . ., 1943; [6] ., ., ,
. ., ., 1965.
; Kantorovich metric
R n -
, ( P, Q )
= a .
d ( x, y ) = I ( x y ) ( I ),
. ,
, :
H . .,
( , [3] ).
. . . ,
.
.: [1] . ., . , 1942, . 37,
78, . 22729; [2] . , .,
1984, . 29, . 4, . 62553; [3] D o b r u s c h i n R. L., S h l o s m a n S. B., Progr. in Phys., 1985, v. 10, p. 34770; [4] S z u l g a A., ., 1982, . 27, . 2, c. 40105.
; Kantorovich distance , .
;
Kantorovich theorem , .
; Caratheodory
theorem , , .
;
Carleman criterion ,
.
; Carleman condition , .
; Karhunen theorem
: m ( S ), S B
( A , B , m ) , { ( t ; a )} ,
t T
, L ( m) , a A
, B ( t , s )
T
= {t }
= M X (t ) X ( s)
X (t )
B ( t, s ) =
( t;
( s < 0 s > n ,
bs = 0 ), (1)- ,
T A . . .-
X (t ) =
bk Z ( t k ) , t = 0 , 1, ... ,
k =0
= ts ( Z (t )
). | | > n B ( ) = 0 , X (t )
M Z (t ) Z ( s )
.
2) X (t ) , < t <
, f ( ) , g ( )
,
2
( L
g( )
= f ( ) . b (t ) ,
) ; X (t )
B ( ) -
,
M X (t ) X ( s ) = B ( t s ) =
a ) ( s; a) m ( da )
g ( s )
b ( t u ) b ( s u ) du .
(4)
(1)
(4) (1)- , A = ( , ) ,
m ; ,
, X (t )
X (t ) = ( t ; a ) Z ( da )
X (t ) =
(2)
b(t u ) d Z (u )
, Z (da) ,
A - , M Z ( S1 ) Z ( S 2 ) = m ( S1 I S 2 ) .
, X (t ) (2) ,
B ( t , s ) (1)
. Z (S ) X (t ) , t T
H X ,
, , { ( t ; a ) , t T }
M d Z ( u ) d Z ( ) = ( u ) d u d ,
Z (u ) .
,
. f ( )
( 1 + 2 ) 1 log f ( ) d >
(5)
L2 (m) - .
,
( , [6] ) g ( ) ,
. ( , [1], [2]
[4] ). ( , [2], [3] ).
, f ( ) -
. .- .
. 1) X (t ) , t = 0 , 1, ...
| | n
M X ( t + ) X ( t ) = B( ) = 0 .
n
b b k , = 0, 1, ..., n ,
B ( ) = k =0 k +
>n
0,
(3)
b0 , b1 , ..., bn
( , [5] ). B ( )
= B ( ) , (3)
MX (t ) X ( s ) = B ( t s ) =
t j bs j
j =
, t
= 0 , 1, ...
b (t ) ; (5) .
. .- ( , .,
( ) ).
.: [1] K a r h u n e n K., Ann. Acad. Sci. Fennicae. Ser. A,
1947, Bd1, 37, S. 379; [2] . ., .
., , . 1, ., 1971, . 2924; [3] . ., . .,
, 2 ., ., 1977, . 26971; [4] R a o M. M., .: Handbook of Statistics, v. 5, Amst., 1985, p. 279310; [5] ., , . .,
., 1976, . 25253; [6] ., .,
, . ., ., 1964, . 32.
; Karhunen Loeve expansion , , ,
. . . a t b
415
,
X (t )
X (t ) =
. . .
k k ( t ) Z k
k =1
, , k (t ) , k = 1, 2 , ... ,
b
B ( t, s ) ( s ) d s = ( t ) ,
atb
(2)
B ( t, s ) = M X ( t ) X ( s ) ,
], k , k
X (t ) -
= 1, 2, ... , k (t )
,
b
Z k = ( k ) 1/ 2
X ( t ) k ( t ) d t ,
k = 1, 2 , ...
M Zk Z j
= kj . . . .
,
;
- ( , ., [1] [6] ), ,
. (1) (2)
a t b X ( t ) X ( t ) ,
t T , T
, d s T -
m ( d s ) , T - . ., X ( t ) R
, d s
m ( s ) d s = m ( s1 , ... , s n ) d s1 ... d s n
. X ( t )
, Z k , k
X (t ) =
= 1, 2 , ...
, (1)-
1 ( , ., [7],
[8] ).
. . .- ( , ., [9]
[13] ).
( , ).
. 1) B ( t , s ) = min ( t , s ) , a = 0 , b = 1 .
(2) (0)
= 0 , (1) = 1
( t ) + ( t ) = 0 ,
## ( Ak cos ak t Z k(1) + Bk sin bk t Z k( 2) )
k =0
, A k Bk ( p)
, M Z k
ak tg T ak = ,
( T ) ( T ) = 0
( t ) + [ ( 2 ) / ]
( t ) = 0 . ,
Z (jq ) = pq kj , ak bk , k = 1, 2 , ...
bk ctg T bk =
X (t ) =
k =1
; d i s r e t e
Karhunen Loeve decomposition
.
,
. x , n
. :
n
sin ( k 1 2 ) t
Zk .
( k 1 2 ) t
x =
(3)
X ( t ) (3) .
( N. Wiener ) 20- 30-
( , ., [14] ).
2) B (t , s ) = exp [ ( t s )] , a = T , b = T , (2) ( T ) + ( T ) = 0 ,
416
(2)
[17]- , [15] [18]- .
.: [1] K o s a m b i D. D., J. Indian Math. Soc., 1943, v. 7,
p. 7688; [2] K a r h u n e n K., Ann. Acad. Sci. Fennicae. Ser. A,
1946, Bd 1, 34, p. 37; [3] L o e v e M., Rev. Sci., 1946, v. 84,
p. 195206; [4] ., , . ., .,
1962, . 499502; [5] . ., . . .
., 1953, . 17, 5, . 40120; [6] . ., .
. .- , 1954, . 24, . 342; [7] .
., . ., ,
2 ., ., 1977, . 25758; [8] . ., .
., , . 1, ., 1971, . 27579; [9] . .,
, 3 ., ., 1962; [10] . ., . ., , . ., ., 1960; [11] ., , . ., . 1, ., 1961;
[12] ., , . ., ., 1963; [13] ., , . ., . 1, ., 1972;
[14] ., ., , . ., ., 1964, . 21529; [15] . ., . . . . , 1973,
. 9, 1, . 3446; [16] . ., .
. , 1977, . 16, . 13541; [17] .
., . ., . , 1957, . 12, . 1, . 3
52; [18] S l e p i a n D., K a d o t a T. T., SIAM J. Appl. Math.,
1969, v. 17, p. 110217.
. . . :
.
[9] [13], [15], [16]- .
f ( ) X ( t )
yi i = y
i =1
= ( 1 , ..., n ) , y = ( y1 , ... , yn )T .
n
, det 0 . , -
, x - n
. - . ,
, y
:
yi = i x ,
i = 1, ... , n .
, y x
. y
, x - .
y m ,
m < n . ( x - ) y - ,
x =
yi i +
i =1
bi i .
i=m + 1
x =
( yi bi ) i
i = m +1
. m
x -
:
n
2 (m) = M { x }2 =
M{( y
bi ) 2 } .
2 ( m ) - . bi
:
bi = M yi = iT M x ,
iT i
i = m +1
, .
i = i i ,
.
n
, opt ( m)
( ., - ) .
. . .- . ,
, .
( ) . . .- .
,
,
.
; transition
probability i n a M a r k o v c h a i n , ,
; transition rate / density ,
( )
; transition function
.
. (E , B ) . p ( s , x ; t , ) ,
## ; categorical weather forecast
i = m +1
2(m) =
. 7, 1, . 513; [3] . .,
, 1976, . 12, 1, . 515.
i = m+1
.: [1] .,
, . ., ., 1979.
;
V A U
, A . B - U , V
( U ).
n ( ,
[1], [2] ) ( n ) ( n )
(
) . .-
. . .-
( , [3] ),
.
. .- . [1]- .
.: [1] ., , . ., ., 1970;
[2] . ., , 1971,
## s < t ( , ) , x E , B x - < t < u,
x E , B
, - s
p ( s , x ; t , dy ) p ( t , y ; u , ) = p ( s , x ; u , )
(*)
, p ( s , x ; t , ) -
. p ( s, x; t , E )
1 , . .
. . . ., . . . lim p ( s, x ; t , E ) 1 ,
ts
. . . . .
t , ( , A , P ) - , (E , B ) -
.
P { t | s } = p ( s, s ; t , )
P
, p ( s , x ; t , )
. . t
, .
(E , B ) ,
. .- . . ( , [1] [3] ). ,
. .- .
. .
[4], [5]- ( , )
.
417
E , . .
P . .-
i j n -
P ( s , t ) = pij ( s , t )
; i , j E , pij ( s , t )
= p ( s , i ; t , { j }) .
. . , , I ( , ) ( ., ) s , t
.
s
. . p ( s , x; t , ) Pt , s
## pij (n) -. E C1 , ..., C k R
, . .
<t
Pt s f ( x ) =
P1
...............
p ( s , x ; t , d y ) f ( y)
Pk
Q
Pt ( ) =
s
, P1 , ..., Pk C1 , ..., C k
( d x) p ( s, x; t , )
.
. . s , t ,
p ( s , x ; t , ) p ( t s , x ; ) , ,
. . . . . Pt
; Pt
s
Pt Put
Pus
= Pt s -
(*)
, Pt Ps
= Pt + s
; Pt .
E , B , E - .
U
x U
lim p ( s , x ; t , U ) = 1 , . .
ts
, Q .
. .
.
; transition
density
p ( s , x ; t , ) -
()
f ( s , x ; t , y ) -, , p ( s , x ; t , ) s < t
p( s, x; t , ) =
;
infinitezimal transition rate , ( )
-
[ s < t T - ,
x E , y E , B , ( E , B ) ].
. . . . .- Pt
( , ), . . . . ( , . . ) . . . - .
. .- ( , ; ).
.: [1] ., , . ., ., 1969; [2] . .,
, 2 ., ., 1974; [3] .
., . ., 1980, . 25, . 2, . 38993;
[4] . ., , ., 1963; [5] . ., . ., ,
. 2, ., 1973.
f ( s , x ; t , y ) ( dy ) ,
s < t , x E , B ;
(1)
.,
f (s, x ; t , y ) ( dy ) 1,
s < t , x E .
(2)
, : 1) . . ; 2) . .
( x , y ) - B B ; 3) . .
f ( s , x; u , y ) =
f ( s , x; t , z )
f ( t , z ; u , y ) (dz )
(3)
x E , y E
s < t < u,
1) 3) y - ].
; transition
matrix
1) 3) (2)
f (1)
. . . f ( t , x , y ) = f ( s , x ; s + t , y ) -
P = pij
. . .- i
pij
0,
ij
= 1
- .
418
; Kemeny
median ,
( , , . )
{C} . A1 , A2 , ..., An { C }
d ( Ai , A ) = min
A{C }
i =1
( )
d ( A , A)
i =1
A {C } ,
d ( Ai , A ) , { C } - Ai A
. . . ( , [1] ). . . ( , [2] ).
.: [1] ., ., . , ., ., 1972; [2] . ., .: , ., 1985, . 5892.
P .
C P ( ) ,
K A C P ( K )
d ( A , B) =
f ij ( A ) f ij ( B )
i , j =1
,
( , [1] ). . .-
, . ( , [2] ). . .
.
.: [1] ., ., . , . ., ., 1972; [2] . ., - ,
., 1979.
; Campbell measure
,
.
() =
({ a } ) ( )
a
, S
= { a : ( { a } ) > 0 } R1
, a ( ) , a
() =
a S
A = A I B M
,
A = { ( a , ( ) ) : a R 1 , ( ) M , a S } R 1 M , B ,
1
A , f (c ) , c
MP
f (c) (d a) =
= ( a , ) A
f ( c ) CP ( d c )
, .
. .
.
.: [1] ., ., .,
, . ., ., 1982.
;
Campbell theorem . ( ) , ,
( A , A) - B ( B ) < .
A f (a) , a A B A
f (a) (d a) =
B
f (a) (d a)
.
, .
.: [1] ., ., ., , . ., ., 1982.
; Kendall rank c o r r e l a t i o n c o e f f i c i e n t
X Y ( )
( X 1 , Y1 ) , ..., ( X n , Yn ) ,
. , . .
,
({ a }) ( a ( ) ) ( )
. R -
, a S ( )
C P ( K ) K -
( 0 , 1)
C ; f ij (C ) = 0
K = A I B M , C P ( K ) = M P ( B ) ,
. . .
B . . .-
X C {C}
f ij (C ) = 1 , , , xi x j
( M , M) - -
f ij (C )
## ; Kemeny distance X = {x1 , x2 , ..., xk } A
( k k )
, M , M -,
{ ( ) : ( B ) = k , B B } M
R - , M , ( R , B ) -
= 2 S ( r1 , ..., rn ) n ( n 1)
, ri , Y - ,
( X , Y ) , X - i - ,
S = 2 N n ( n 1) 2 , N ,
419
j > i r j > ri .
1 1 .
. . .
( M. Kendall, , [1] ).
. . .
, M = 0
D = 2 (2n + 5) 9n ( n 1) .
( 4 n 10 ) ( , [3] ). n > 10 -
> u / 2
R (t ) =
. . , ,
. X , Y
- ,
. . M
= 2 1arc sin
.
, ,
.
.: [1] ., , . ., .,
1975; [2] . ., , . ., ., 1960; [3] . ., .
., , 3 ., ., 1 983.
## ; KEPSTR Kolmogorov equation Power Series Time Responce
( t ), t = 0, 1, ...
-
## exp { 0 + 1 ( z ) + ... } = a0 + a1 z + ...
{ k } , k
= 0 , 1, ... ,
(1)
= 0 , 1, ... ,
( t ) -
z < 1.
, z ,
{ k }
. . .
( , [1] [3] ). (1)
.
f ( ) (t )
.
f ( ) =
2 | 270,927 | 537,818 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2023-14 | latest | en | 0.200918 |
https://community.smartsheet.com/discussion/201/weeks-days-remaining-formula | 1,708,649,628,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947473871.23/warc/CC-MAIN-20240222225655-20240223015655-00069.warc.gz | 185,301,313 | 89,860 | #### Welcome to the Smartsheet Forum Archives
The posts in this forum are no longer monitored for accuracy and their content may no longer be current. If there's a discussion here that interests you and you'd like to find (or create) a more current version, please Visit the Current Forums.
# Weeks/Days Remaining Formula
edited 12/09/19
How can I create a column that displays "x Weeks and x Days remaining" until a task's start date? I know about RYG Balls and what I can do with conditional formatting, but those just aren't sufficient for my needs.
In Excel/Sheets I use the MOD function to do this using the following formula:
=IF(([Start Date]-TODAY())>0,(INT(([Start Date]-TODAY())/7)&" weeks"&IF(MOD([Start Date]-TODAY(),7)=0,"",", "&MOD([Start Date]-TODAY(),7)&" days")),"")
but MOD does not seem to be supported in SmartSheet.
Any ideas?
• ✭✭✭✭✭✭
We use a version of the following formula to display the number of weeks and days to a task's start date (perhaps obviously, our column name is [Start On]) is:
=INT(([Start On]1 - TODAY()) / 7) + "w, " + (([Start On]1 - TODAY()) - (7 * (INT(([Start On]1 - TODAY()) / 7))) + "d")
You'll have to handle things which are overdue, or if you want to consider business weeks vs. calendar weeks, but you get the idea, I'm sure. Also, if you save the sheet, close it, and do not open it until it is a new day, you'll immediately have an available "save" button, because it's a new day (today() in the formula gets re-evaluated and changes the result of the formula). Unless you save it, this is always going to happen. It's not a big deal, but you might wonder why every morning you open the sheet up, and then close it without making changes, you're prompted to save your changes.
• That is perfect. Thank you very much.
• John, have you calculated the weeks remaining if the date in the future pans over calendar years?
This discussion has been closed. | 497 | 1,918 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2024-10 | latest | en | 0.922346 |
https://it.mathworks.com/matlabcentral/answers/522942-matlab-onramp-8-1-obtaining-help | 1,623,497,649,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623487582767.0/warc/CC-MAIN-20210612103920-20210612133920-00147.warc.gz | 324,435,012 | 26,198 | # MATLAB Onramp: 8.1 Obtaining help
59 views (last 30 days)
Jigyasu Chand on 4 May 2020
Answered: Anand Gopal on 19 Jan 2021
Hi folks!
I just started exploring MATLAB and I am stuck at the following place. In the course "MATLAB Onramp" which is one of the 'Getting Started' courses, I am not able to get the solution of '8. Obtaining help - Further Practice'. The task is to create a matrix that
• Contains random integers in the range from 1 to 20,
• Has 5 rows, and
• Has 7 columns.
matrix with normally distributed numbers (instead of uniformly distributed numbers).
-J
Guillaume on 4 May 2020
I've just discovered that there are several versions of the Onramp course (R2018b, R2019a, and R2019b). The last two have slightly different presentation for the assignment (the first one doesn't seem to work properly) but in neither of them do I see that assignment under further practice. Which version are you using (look at the url)? Perhaps it is also language specific.
The assignment doesn't make sense and certainly won't be answered by the doc. You could rescale and recentre a normal distribution and possibly round it to integers but it still wouldn't be restricted to a range. And you could hardly call the result a normal distribution anymore.
You should report it as a bug to mathworks.
Jigyasu Chand on 8 May 2020
That seems fair. Thanks a lot!
-J
Abubakr Eltayeb on 14 Aug 2020
x = randn([5 7])
Anand Gopal on 19 Jan 2021
x = randi([1,20],5,7) | 393 | 1,460 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2021-25 | latest | en | 0.952288 |
https://www.gamedev.net/topic/647996-solved-tessellation-on-icosphere/ | 1,493,524,882,000,000,000 | text/html | crawl-data/CC-MAIN-2017-17/segments/1492917124297.82/warc/CC-MAIN-20170423031204-00139-ip-10-145-167-34.ec2.internal.warc.gz | 886,324,995 | 39,925 | • FEATURED
View more
View more
View more
### Image of the Day Submit
IOTD | Top Screenshots
### The latest, straight to your Inbox.
Subscribe to GameDev.net Direct to receive the latest updates and exclusive content.
# [SOLVED] Tessellation on IcoSphere
Old topic!
Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.
5 replies to this topic
### #1chrisendymion Members
Posted 19 September 2013 - 02:18 AM
Hello ;) Before, sorry for my poor english...
I'm working on a procedural engine with some results on generating a full planet.
But I have a "crack" problem with the tessellation (based on frustum and distance LOD).
The icosphere is generated by code with X subdivisions. Let's go with 0 subdivision for theory..
I'm using this algorythm for adaptive tessellation :
Every vertex's distance is calculated and the minimum is used for defining the tessellation factor on the edge.
So, in theory, there will be every time the same factor for the same edge, and in result, no cracks !
But, there is a problem.. Which vertex for which edges ?
I need to address tessellation factor for each 3 edges in a hard order.
Edge 1 from triangle A must be the same edge shared by triangle B in the same order (egdge 1).
(in the exemple here, vertex variable is the tessellation factor calculated from distance to camera)
TRIANGLE A
_output.Edges[0] = min(_vertex2, _vertex3);
_output.Edges[1] = min(_vertex1, _vertex3);
_output.Edges[2] = min(_vertex1, _vertex2);
TRIANGLE B
_output.Edges[0] = min(_vertex2, _vertex3);
_output.Edges[1] = min(_vertex1, _vertex3);
_output.Edges[2] = min(_vertex1, _vertex2);
So, if the edge shared by A and B is edge[1], A.vertex1 must be the same vertex as B.vertex3....
If not, there will be differents tessellation factors and cracks...
Whith a flat grid, this is working very well (every edges shared are in the same order) :
But for an icosphere.. It's not possible (view eclated from upside) :
The order is wrong......
I'm using 3 points patch list, I tried with 6 points patch list (neighbors), but it's the same problem...
I'm not sure to have correctly explained the problem.
My general question is : How to do tessellation on icosphere without cracks ?
Is there a better algorythm ?
I'm far from an expert in graphics..
Thank you very much for any help ;)
Here some screenshots from actual rendering results :
Edited by chrisendymion, 23 September 2013 - 06:55 AM.
### #2unbird Members
Posted 20 September 2013 - 04:32 PM
Note sure if I follow your problem right, but if you got your tesselation factor calculation right, the geometry shouldn't matter. Hold on, I grab my tesselation test scene and feed it a icosahedron icosphere.
What SDK and especially what shader compiler version are you using ? I once had a bad bug with the June 2010 SDK compiler in conjunction with tesselation factors.
Update: Hmmm, maybe I understand better now. The link you gave uses quad tesselation. For triangles one needs to know this
And jep, my tesselator has no trouble with an icosphere (had to make one first).
Edit: By the way, nice screenshots!
Edited by unbird, 20 September 2013 - 05:30 PM.
### #3chrisendymion Members
Posted 23 September 2013 - 02:00 AM
Thank you Unbird for your help.. And thank you for your time testing this with your engine !
I'm sorry, I didn't understand posts from the link you gave me...
Ok, it's helpfull to know that edgeX is opposite to vertexX. But what else ?
You said in the other post :
As long as neighboring patches share the same control points, and the control point to tessellation factor algorithm will produce the same output for the given input control points, then the order doesn't make any difference.
But as the picture I posted before (ico from upside), I cannot produce the same control points without altering the order, which will break the cull back face. Or I'm missing something ?
I thought about using a 4 points patch list, the fourth point is the order to give to the edges.
It almost works, but creates strange artifacts.. Perhaps the bug you said about the June 2010 SDK (I'm using it) ?
Which SDK are you using ?
Using the 4 points for the edge's order :
_output.Edges[0] = floor(min(vertex1, vertex2));
_output.Edges[1] = floor(min(vertex0, vertex2));
_output.Edges[2] = floor(min(vertex0, vertex1));
if (inputpatch[3].Model_VertexPosition.x == 1.0 && inputpatch[3].Model_VertexPosition.y == 3.0 && inputpatch[3].Model_VertexPosition.x == 2.0)
{
_output.Edges[2] = floor(min(vertex1, vertex2));
_output.Edges[1] = floor(min(vertex0, vertex2));
_output.Edges[0] = floor(min(vertex0, vertex1));
}
Results (other edges are working correctly) :
Using Edges[0] or Edges[2] with 1 as value does not generate artifacts (but one edge tessellation is logicaly missing) :
So artifacts appear only if the three egdes are > 1, if one edge is set to 1, it works.
Is that a bug ? I'm totaly lost...
[UPDATE] Yeah, forget the bug's part.. If found this, changes made it to works !
Now with my idea of passing a fourth point, which says the order of edges, it's working with 0 subdivision.. But it will be hard to do the job with the subdivision.
No better algo ? I feel complicate unnecessarily.
Edited by chrisendymion, 23 September 2013 - 06:01 AM.
### #4chrisendymion Members
Posted 23 September 2013 - 06:43 AM
Oh.............................
Nevermind... Forget all my posts....
All the troubles were caused by the SDK June 2010 !!!
All is working very well now !! ;)
Thank you again for your help !!
If someone has the same problem, you need to add this in the hull shader file :
#if D3DX_VERSION == 0xa2b
#pragma ruledisable 0x0802405f
#endif
All this drawing and posting time just for a little bug....
### #5unbird Members
Posted 23 September 2013 - 06:57 AM
Glad to hear it's working. And thanks for that link. I did not know one could "patch" the compiler this way. It's also good to know the bug is actually documented.
All this drawing and posting time just for a little bug....
Tell me about it. It isn't little if it's not your fault - since you look for compiler bugs last (if at all). And waste time.
### #6chrisendymion Members
Posted 23 September 2013 - 07:00 AM | 1,638 | 6,337 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.40625 | 3 | CC-MAIN-2017-17 | longest | en | 0.837784 |
http://english.stackexchange.com/questions/46887/is-spatial-contiguity-a-pleonasm | 1,469,293,487,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257823133.4/warc/CC-MAIN-20160723071023-00201-ip-10-185-27-174.ec2.internal.warc.gz | 90,240,815 | 17,231 | # Is “spatial contiguity” a pleonasm?
I used the terms "spatial contiguity" to emphasise the relation between two objects as opposed to synchronism, i.e. chronological contiguity. I then questioned myself whether or not that would constitute a pleonasm.
What do you think?
-
If you talking about both space and time in the same passage then it sounds ok. You're using 'contiguity' with the spatial and temporal modifiers to emphasize the difference.
Otherwise if you're just talking about one kind or the other, you'd use just 'contiguity' for space and 'continuity' for time (or the respective adjectives).
-
I like your answer, though i doubting whether continuity is the chronological equivalent of contiguity. A tunnel is a space continuum too. I think contiguity denotes juxtaposition. Anyway the first part of your response answers my question I think. – Benjamin Nov 1 '11 at 12:33
Yes, I sense that 'continuity' might not be the perfect parallel for time. Maybe 'successive'? – Mitch Nov 1 '11 at 12:36
It's only a convention that we normally use contiguous in spatial contexts, and continuous in temporal ones. And whilst a continuous line is often used "metaphorically" (of a ruling dynasty, for example), it usually refers to an unbroken succession in one spatial dimension. The distinction between the two words isn't that clear-cut, so as you say, there's no reason OP shouldn't explicitly clarify his intended meaning in context. – FumbleFingers Nov 1 '11 at 13:09
@FumbleFIngers: did you mean "in one temporal dimension"? – Mitch Nov 1 '11 at 13:12
@Mitch: No, I meant that usually a "continuous line" is just that - a line in one spatial dimension (or two, if it's not straight). But normally, "continuous" in other contexts refers to in the temporal dimension. Mind you, I usually think of "continuum" in the context of space-time continuum, where it's four-dimensional. – FumbleFingers Nov 1 '11 at 14:16
The differentiation between contiguity and continuity is that in the first case you are referring to the adjacency of two distinct entities. In the second you are referring to the extension (in time or space) of a single entity. So whether you really needed to specify 'spatial' would depend on the context. It might or might not be necessary or helpful to include it.
- | 547 | 2,301 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.90625 | 3 | CC-MAIN-2016-30 | latest | en | 0.926014 |
https://books.google.gr/books?qtid=e97e78a0&dq=editions:UOM39015063875382&id=vNc2AAAAMAAJ&hl=el&output=html&lr=&sa=N&start=10 | 1,632,038,779,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780056752.16/warc/CC-MAIN-20210919065755-20210919095755-00366.warc.gz | 210,949,342 | 6,246 | Αναζήτηση Εικόνες Χάρτες Play YouTube Ειδήσεις Gmail Drive Περισσότερα »
Είσοδος
Βιβλία Βιβλία
... second and third places ; observing to increase the second place by 5, if the shillings be odd, and the third place by 1, when the farthings exceed 12, and by 2 when they exceed 37.
The Youth's Assistant in Theoretick and Practical Arithmetic - Σελίδα 65
των Zadock Thompson - 1826 - 164 σελίδες
Πλήρης προβολή - Σχετικά με αυτό το βιβλίο
## The Mercantile Arithmetic, Adapted to the Commerce of the United States, in ...
Michael Walsh - 1828 - 315 σελίδες
...decimal of a hogshead. Ans. ,2341+ CASE III. To fold the decimal of any number of shillings, pence, and RULE Write half the greatest even number of shillings for the first decimal figure, and lei the farthings, in the given pence and farthings, possess the second and third places; observing...
## Adams's New Arithmetic: Arithmetic, in which the Principles of Operating by ...
Daniel Adams - 1828 - 264 σελίδες
...being more than 12, but not exceeding 36,) is '024 £ ., and the whole is '874 & . the Ans. Wherefore, to reduce shillings,, pence and farthings to the decimal of a pound by inspection,-^ Call every tivo shillings one tenth of a pound; every odd shilling, fice hundredths; and the number...
## Adams's New Arithmetic: Arithmetic, in which the Principles of Operating by ...
Daniel Adams - 1828 - 264 σελίδες
...being more than 12, but not exceeding 36,) is '024 £ ., and the whole is '874 £ . the Ans. Wherefore, to reduce shillings, pence and farthings to the decimal of a pound by inspection, — Call every two shillings one tenth of a pound ; every odd shilling, five hundredth^ ; and the number...
## The Improved Arithmetic: Newly Arranged and Clearly Illustrated, Both ...
Daniel Parker - 1828 - 348 σελίδες
...the value of ,9554 of a pound ? Ans. 19s. Id. l,184cr. CONTRACTIONS OF DECIMALS. PROBLEM I. RULE. 1. Write half the greatest even number of shillings for the first decimal figure. If the shillings be odd, add 5 in the second place of decimals. 2. Let the farthings in the given pence...
## A Short System of Practical Arithmetic: Compiled from the Best Authorities ...
William Kinne - 1829
...111. To reduce any number of shillings, pence and farthings by inspection to the decimal of a pound. RULE. — Write half the greatest even number of shillings for the first decimal figure, and let the farthings in the given pence and farthings possess the second and third places ; observing...
## Daboll's Schoolmaster's Assistant: Improved and Enl. ...
Nathan Daboll - 1829 - 240 σελίδες
...decimal of any number of shillings, pence and farthings, (to three places) by INSPECTION. RULE. — 1. Write half the greatest even number of shillings for the first decimal figure. 2. Let the farthings in the given pence and farthings postees tli» second and third places; observing...
## Arithmetic: In which the Principles of Operating by Numbers are Analytically ...
Daniel Adams - 1830 - 264 σελίδες
...more than 12, but not exceeding 36,) is K)24 £ ., and the .whole is '874 £ . the Ans. Wherefore, to reduce shillings, pence and farthings to the decimal of a pound by inspection, — Call every two shillings one tenth of a pound; every odd shUKng, five hundredths; and the number...
## Adams's New Arithmetic: Arithmetic, in which the Principles of Operating by ...
Daniel Adams - 1830 - 264 σελίδες
...being more than 12, but not exceeding 36,) is '024 £ ., and the whole is '874 £ . the Ans. Wherefore, to reduce shillings, pence and farthings to the decimal of a pound by inspection, — Call every two shillings tenth of a pound ; every odd shUfing, five hundredths ; and number of...
## Arithmetic: In which the Principles of Operating by Numbers are Analytically ...
Daniel Adams - 1830 - 264 σελίδες
...a league ? 21. Reduce 10 s. 9£ d. to the fraction of a pound. IT 76. There is a method of reducing shillings, pence and farthings to the decimal of a pound, by inspection, more simple and concise thau the foregoing. The reasoning in lelation to it is as follows : i•Vf...
## Mercantile Arith
Michael Walsh - 1831
...hogshead. Ans. ,2341+ CASE III. To find the decimal of any number of shillings, pence, and farthings, by inspection. RULE. Write half the greatest even number of shillings for the first decimal figure, and let the farthings, in the given pence and farthings, possess the second and third places ; observing... | 1,188 | 4,432 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2021-39 | latest | en | 0.661793 |
http://hyperphysics.phy-astr.gsu.edu/hbase/perpx.html | 1,516,539,247,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084890582.77/warc/CC-MAIN-20180121120038-20180121140038-00280.warc.gz | 186,113,859 | 1,537 | # Perpendicular Axis Theorem
For a planar object, the moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia of two perpendicular axes through the same point in the plane of the object. The utility of this theorem goes beyond that of calculating moments of strictly planar objects. It is a valuable tool in the building up of the moments of inertia of three dimensional objects such as cylinders by breaking them up into planar disks and summing the moments of inertia of the composite disks.
### Show the development of the relationship.
Index
Moment of inertia concepts
HyperPhysics***** Mechanics R Nave
Go Back
# Perpendicular Axis Theorem
The perpendicular axis theorem for planar objects can be demonstrated by looking at the contribution to the three axis moments of inertia from an arbitrary mass element. From the point mass moment, the contributions to each of the axis moments of inertia are
Index
Moment of inertia concepts
HyperPhysics***** Mechanics R Nave
Go Back | 203 | 1,030 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2018-05 | longest | en | 0.862123 |
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1856 | 1,603,337,304,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107878879.33/warc/CC-MAIN-20201022024236-20201022054236-00001.warc.gz | 176,619,680 | 10,412 | NTNUJAVA Virtual Physics LaboratoryEnjoy the fun of physics with simulations! Backup site http://enjoy.phy.ntnu.edu.tw/ntnujava/
October 22, 2020, 10:34:29 am
You cannot always have happiness but you can always give happiness. ..."Mother Teresa(1910-1997, Roman Catholic Missionary, 1979 Nobel Peace Prize)"
Pages: [1] Go Down
Author Topic: X(t), V(t) and a(t) plots : drag x to view v(t)/a(t) or set constant a (Read 4653 times) 0 Members and 1 Guest are viewing this topic. Click to toggle author information(expand message area).
ahmedelshfie
Moderator
Hero Member
Offline
Posts: 954
« Embed this message on: June 24, 2010, 07:46:25 pm » posted from:SAO PAULO,SAO PAULO,BRAZIL
This following applet is X(t), V(t) and a(t) plots : drag x to view v(t)/a(t) or set constant a
Created by prof Hwang Modified by Ahmed
Original project X(t), V(t) and a(t) plots : drag x to view v(t)/a(t) or set constant a
This applet hope to help you understand relation betwen x(t), v(t) and a(t).
$v(t)=limit_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}= \frac{dx(t)}{dt}$, and $a(t)=\frac{d v(t)}{dt}$
The applet will draw x(t),v(t) and a(t) when you drag the tiger horizontally =>to change it's position.
You can also set up acceleration or velocity with slider and click accelerate to star the acceleration.
The step is for you to move one time step at a time.
Embed a running copy of this simulation
Embed a running copy link(show simulation in a popuped window)
Full screen applet or Problem viewing java?Add http://www.phy.ntnu.edu.tw/ to exception site list | 455 | 1,569 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.125 | 3 | CC-MAIN-2020-45 | latest | en | 0.796394 |
https://www.meetyoucarbide.com/fr/how-to-calculate-the-surface-roughness-in-ball-end-milling/ | 1,718,493,991,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861618.0/warc/CC-MAIN-20240615221637-20240616011637-00744.warc.gz | 811,348,060 | 52,994 | The surface roughness of a part is a technical requirement that measures the surface processing quality of the part. It significantly impacts the part’s fit, wear resistance, corrosion resistance, and sealing performance. The factors that affect surface roughness mainly include the workpiece material, cutting parameters, machine tool performance, and tool material and geometry parameters.
During the actual machining process, the cutting depth, feed rate, and spindle speed are predetermined and kept constant throughout the cutting process. Therefore, it is essential to optimize the combination of factors affecting surface roughness to obtain the optimal surface quality value. This article begins with the calculation formula of surface roughness and its relationship with chip thickness. It further explores the relationship between surface roughness, cutting depth, and feed rate. Additionally, it examines the impact of various factors on surface roughness through experimentation.
# Mechanism of Surface Roughness Generation
## Mechanism of Residual Height Generation
In curved surface machining, the residual height is mainly formed by the tool moving along the tool path and leaving material on the surface of the workpiece unremoved. As shown in Figure 1, the following parameters are defined: P as the tool contact point, R as the radius of the curved surface, θ as the angle between two radius lines, and n as the normal vector at point P. The stepover distance is represented by d, and it is closely related to the residual height h. Based on Figure 2(a), we can derive the following relationship:
In the equation: r represents the tool radius, and kh represents the normal curvature of the machining surface along the cutting feed direction.
When using the sectional plane method to generate tool paths, calculating the normal curvature (kh) can be challenging. In practical machining, an approximation is often used, where a plane approximates the surface between two adjacent tool paths, as shown in Figure 2(b). The stepover distance is considered the normal distance between the sectional planes. In this case, the residual height (h) can be described by the following equation:
## 1.2Calculation of Surface Roughness
Due to the presence of residual height, the surface of the part after mechanical machining will have many uneven peaks and valleys. This microscopic geometric shape is known as surface roughness, as shown in Figure 3. The parameter Ra is defined as the surface roughness, which is given by:
In the equation, L represents the sampling length.
Zooming in on Figure 3, we obtain Figure 4. When h’ is less than Y et, we can deduce:
6
When h” is greater than Y et, we can deduce:
In the equation, E represents the area of the region. Since y_a needs to ensure that the area above and below the central line is equal, i.e.,
In equation (6), p’ and p” are weighting factors. p is closely related to the chip thickness h. After a series of derivations, we can obtain
the expression of the sampling area is as follows
In the expression:
Substituting equations (4) and (5) into equation (8), we obtain:
After substituting equation (7) into equation (9) and simplifying through calculations, the relationship between the sampling area of surface roughness and the chip thickness is obtained as follows:
According to the above equation, it can be seen that there is a very simple relationship between surface roughness and chip thickness. When milling with a ball-end cutter, the feed per tooth is constant, while the chip thickness varies continuously based on the cutting depth and feed rate.
# Experimental Data and Analysis
## Experimental Conditions
Under steady-state cutting conditions, by varying the cutting depth and feed rate, the surface roughness values are measured for different parameter combinations. The micro-topography of the machined surfaces is observed using a three-dimensional profilometer, and the influence of cutting parameters on surface roughness is analyzed.
The experiment is conducted on the edge part shown in Figure 5, using a FANUC precision machining center machine. The workpiece material is 45# steel, and a high-speed steel milling cutter with a diameter of 12.5mm is selected as the cutting tool. The spindle speed is set at 800 r/min, and the cutting depth varies from 1mm to 6mm. Different feed rates are used for cutting at depths of 1mm, 2mm, 4mm, and 6mm, as illustrated in Figure 6.
## Data Measurement
After completing the machining of the part, measurement points are selected on the curved section of the part shown in Figure 5. For each set of experimental conditions, data at these measurement points are measured twice, and the average value is taken as the experimental value. The experimental data are presented in Table 1
## Data Analysis
From the experimental data, it can be observed that when machining the part using a ball-end cutter and keeping the feed rate constant, the surface roughness increases with an increase in cutting depth (see Figure 7). At lower cutting depths, the surface roughness values are smaller, but excessively small cutting depths result in longer cutting times and lower processing efficiency.
Although there is a certain difference between the experimental values and theoretical values in this study, they are relatively close. Hence, the provided calculation formula in this study can be adopted. For the selected workpiece in this study, the optimum surface roughness is achieved when the cutting depth is 2mm, and the feed rate is 700mm/min.
# 3conclusion
The study investigated the influence of various machining parameters on surface roughness during the milling process of the workpiece. The theoretical impact of surface roughness on the surface quality of the workpiece was explored, and a theoretical calculation formula for surface roughness was derived based on its generation mechanism.
Using the trial machining method and different combinations of parameter data, the surface roughness of the machined parts was measured using a three-dimensional profilometer. The calculated theoretical values from the formula were then compared with the experimental values.
The research demonstrated that both the calculation formula and the machining method are feasible and effective in predicting and controlling surface roughness during the milling process. | 1,260 | 6,411 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.203125 | 3 | CC-MAIN-2024-26 | latest | en | 0.928647 |
http://www.dewassoc.com/support/msdos/what_are_binary.htm | 1,516,211,846,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084886952.14/warc/CC-MAIN-20180117173312-20180117193312-00312.warc.gz | 426,008,151 | 3,314 | What are binary, octal, and hexadecimal notations?
Binary Notation
All data in modern computers is stored as series of bits. A bit can take on one of two values. The two values are generally represented as the numbers 0 and 1. The most basic form of representing computer data, then, is to represent a piece of data as a string of 1's and 0's, one for each bit. What you end up with is a binary, or base-2 number; this is binary notation. For example, the number 42 would be represented in binary as
101010
Interpreting Binary Notation
As with normal decimal (base-10) notation each digit moving from right to left represents an increasing order of magnitude (or power of ten). With decimal notation each succeeding digit's contribution is ten times greater than the previous digit. Increasing the first digit by one increases the number represented by one, increasing the second digit by one increases the number by ten, the third digit increases the number by 100, and so on. The number 111 is one less than 112, ten less than 121, and one hundred less than the number 211.
The concept is the same with binary notation except that each digit is a power of two greater than the preceding digit rather than a power of ten. Instead of 1's, 10's, 100's, and 1000's digits, binary numbers have 1's, 2's, 4's, and 8's. Thus, the number two in binary would be represented as a 0 in the ones place and a 1 in the twos place, i.e., 10. Three would be 11, a one in the ones place and a 1 in the twos place. No numeral greater than 1 is ever used in binary notation.
Octal and Hexadecimal Notation
Since binary notation can be cumbersome, two more compact notations are often used, octal and hexadecimal. Octal notation represents data as a base-8 number. Each digit in an octal number represents three bits. Similarly, hexadecimal notation uses base-16 numbers, representing four bits with each digit. Octal numbers use only the digits 0-7, while hexadecimal numbers use all ten base-10 digits (0-9) and the letters a-f (represent the numbers 10-15). The number 42 is written in octal as
52
and in hexadecimal as
2a
It can sometimes be difficult to tell whether data is being represented as octal, or hexadecimal (especially if a hexadecimal number doesn't use one of the digits 8-f), so one convention that is often used to distinguish these is to put `0x` in front of hexadecimal numbers. Thus, you will often see
0x2a
as another less ambiguous way of representing the number 42 in hexadecimal. An example of this usage can be seen here: Character set comparison chart.
Note: The term binary when used in phrases such as "binary" or "binary attachment" has a related but slightly different meaning than the one discussed here.
What is a binary file?
A binary file is one in which the eighth bit of each byte is used for data. Computers and programs can read binary files, but people cannot. Executable files, compiled programs, WordPerfect documents, SAS and SPSS system files, and spreadsheets are all examples of binary files.
Files that contain machine-specific codes (i.e., processor-specific microcode) are binary files. However, not all binary files contain processor-specific codes. Some binary files contain text or data in a non-ASCII format that is unrelated to the microcode used by the processor. For example, most graphics files, all compressed files, and many other file types use all eight bits per byte, so are termed "binary".
What is a bit?
A bit is a binary digit, the smallest increment of data on a machine. A bit can hold only one of two values: 0 or 1.
Because bits are so small, you rarely work with information one bit at a time. Bits are usually assembled into a group of 8 to form a byte. A byte contains enough information to store a character, like "h".
What is a byte?
Byte is an abbreviation for "binary term". A single byte is composed of 8 consecutive bits capable of storing a single character. | 908 | 3,945 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.5625 | 5 | CC-MAIN-2018-05 | longest | en | 0.919083 |
https://www.slideserve.com/zonta/jeopardy | 1,532,336,493,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676595531.70/warc/CC-MAIN-20180723071245-20180723091245-00546.warc.gz | 993,732,756 | 14,354 | Jeopardy
1 / 42
# Jeopardy - PowerPoint PPT Presentation
Jeopardy. Geometry Unit 3 Review. Unit 3 Review jeopardy. Ocean Front Properties - 100. The following number sentence demonstrates what property? a + b = b + a. NEXT. Ocean Front Properties - 100. The following number sentence demonstrates what property? a + b = b + a
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
## PowerPoint Slideshow about 'Jeopardy' - zonta
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
### Jeopardy
Geometry Unit 3 Review
Ocean Front Properties - 100
• The following number sentence demonstrates what property?
• a + b = b + a
NEXT
Ocean Front Properties - 100
• The following number sentence demonstrates what property?
• a + b = b + a
• Commutative Property
HOME
Ocean Front Properties - 200
• Simplify the following and identify the property that you used.
• 5(x – 2)
NEXT
Ocean Front Properties - 200
• Simplify the following and identify the property that you used.
• 5(x – 2) = 5x – 10
• Distributive Property
HOME
Ocean Front Properties - 300
• The following number sentence demonstrates what property?
• a(bc) = (ab)c
NEXT
Ocean Front Properties - 300
• The following number sentence demonstrates what property?
• a(bc) = (ab)c
• Associative Property
HOME
Ocean Front Properties - 400
• The following illustrates what property?
• If a = b and b = c, then a = c.
NEXT
Ocean Front Properties - 400
• The following illustrates what property?
• If a = b and b = c, then a = c.
• Transitive Property
HOME
Ocean Front Properties - 500
• Give one number from each of the following sets:
• Real Numbers (R)
• Whole Numbers (W)
• Integers (Z)
NEXT
Ocean Front Properties - 500
• Give one number from each of the following sets: Sample Answers
• Real Numbers (R): 4, 3.625, ½
• Whole Numbers (W): 0,1,2,3…
• Integers (Z): …,-2, -1, 0, 1, 2,…
HOME
What Did You Say? - 100
• If a pentagon is circumscribed about a circle, which shape is on the outside?
NEXT
What Did You Say? - 100
• If a pentagon is circumscribed about a circle, which shape is on the outside?
• Pentagon
HOME
What Did You Say? - 200
• State whether the following polygon is circumscribed about the circle, inscribed in a circle, or neither.
NEXT
What Did You Say? - 200
• State whether the following polygon is circumscribed about the circle, inscribed in a circle, or neither.
HOME
What Did You Say? - 300
• Name a point of tangency in the following drawing.
You must indicate whether the letter is capital or lowercase.
NEXT
What Did You Say? - 300
• Name a point of tangency in the following drawing.
A, B, or C
HOME
What Did You Say? - 400
• If a tangent line where drawn through the given point on the circle, would its slope be positive, negative, zero, or undefined? Explain.
NEXT
What Did You Say? - 400
• If a tangent line where drawn through the given point on the circle, would its slope be positive, negative, zero, or undefined? Explain.
Positive
HOME
What Did You Say? - 500
• If the circumference of a circle is 15, find the diameter, and radius.
NEXT
What Did You Say? - 500
• If the circumference of a circle is 15, find the diameter, and radius.
• Diameter – 15; Radius – 7.5
HOME
F, D, & C, OH MY! - 100
• Convert the following sentence into geometric notation:
• “The distance from point A to B is equal to the distance from point C to D.”
NEXT
F, D, & C, OH MY! - 100
• Convert the following sentence into geometric notation:
• “The distance from point A to B is equal to the distance from point C to D.”
• AB = CD
HOME
F, D, & C, OH MY! - 200
• What can we conclude about the segments if the following statement is TRUE.
• AB = CD
• Write answer using geometric notation.
NEXT
F, D, & C, OH MY! - 200
• What can we conclude about the segments if the following statement is TRUE.
• AB = CD
HOME
F, D, & C, OH MY! - 300
• Give the perimeter of the following regular hexagon:
6
NEXT
F, D, & C, OH MY! - 300
• Give the perimeter of the following regular hexagon:
6
P = 36
HOME
F, D, & C, OH MY! - 400
• A line that is coplanar with a circle and intersects the circle in only one point is called a __________.
NEXT
F, D, & C, OH MY! - 400
• A line that is coplanar with a circle and intersects the circle in only one point is called a __________.
• Tangent Line
HOME
F, D, & C, OH MY! - 500
• Construct a copy of (I will draw on your paper).
• Construct a bisector of .
• What do we know to be true about the point of intersection of the bisector with ?
NEXT
F, D, & C, OH MY! - 500
HOME
According To My Calculations - 100
• A B C D E
• Use the number line to answer the following question.
• What is BD?
NEXT
According To My Calculations - 100
• A B C D E
• Use the number line to answer the following question.
• What is BD?
• BD = 11
HOME
According To My Calculations - 200
• A B C D E
• Use the number line to answer the following question.
• Segment congruent to .
NEXT
According To My Calculations - 200
• A B C D E
• Use the number line to answer the following question.
• Segment congruent to .
HOME
According To My Calculations - 300
• Give a correct explanation/demonstration of how Pi is derived:
• Hint:
NEXT
According To My Calculations - 300
• Give a correct explanation/demonstration of how Pi is derived:
HOME
According To My Calculations - 400
• Find the circumference of a semi-circle with r = 6.
NEXT
According To My Calculations - 400
• Find the circumference of a semi-circle with r = 6.
• 6
HOME
According To My Calculations - 500
• Calculate the distance a car travels in 10 revolutions of its 20-inch tires. Answer in feet.
NEXT
According To My Calculations - 500
• Calculate the distance a car travels in 10 revolutions of its 20-inch tires. | 1,686 | 6,213 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.28125 | 4 | CC-MAIN-2018-30 | latest | en | 0.773922 |
https://www.physicsforums.com/threads/einstein-theorem-questions-about-why-b1co-0modp-and-boc1-0modp.160866/ | 1,702,027,271,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100739.50/warc/CC-MAIN-20231208081124-20231208111124-00187.warc.gz | 1,056,358,244 | 15,227 | # Einstein Theorem: Questions about Why b1co=0modp and boc1=0modp
• catcherintherye
#### catcherintherye
Then the proof goes on to consider coefficient of x viz a1=boc1 + b1co, and we know that a1=0modp and b1co=0modp so boc1=0modp...
...2 questions why is b1co necessarily =0modp ? and secondly why does it follow that boc1=0modp? ...
... i mean sure we know the sum b1co + boc1 =0modp but surely this doesn't imply b1co=0modp e.g 9=0mod3 but 9=5+4 and it is not true 5=0mod3
Last edited:
btw does anyone know how to get all the mathematical symbols and stuff cos I'm getting really bored of typin g everything out long hand!
I don't know what you're trying to prove here, but the way you've worded it seems to suggest that you know that a1=0modp and b1co=0modp. Comparing this to your first equation, then clearly boc1=0modp.
However, I don't think this is what you're asking. Perhaps you should state the assumptions of the theorem!
[ tex ] a^x_n [ /tex ] | 301 | 967 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2023-50 | latest | en | 0.936438 |
https://mathematics-monster.com/lessons/area_of_a_trapezoid.html | 1,558,530,507,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232256812.65/warc/CC-MAIN-20190522123236-20190522145236-00194.warc.gz | 563,026,604 | 5,682 | # How to Find the Area of a Trapezoid
## Finding the Area of a Trapezoid
The area of a trapezoid is found using the formula:
In this formula, b1 and b2 are the lengths of the bases of the trapezoid and h is the height of the trapezoid. The image below shows what we mean by the lengths of the bases and the height:
## Interactive Widget
Use this interactive widget to create a trapezoid and then calculate its area. Start by clicking in the shaded area.
Oops, it's broken!
Turn your phone on its side to use this widget.
## How to Find the Area of a Trapezoid
Finding the area of a trapezoid is easy.
### Question
What is the area of a trapezoid with bases of length 3 cm and 5 cm and a height of 2 cm, as shown below?
# 1
Area = ½(b1 + b2)h
Don't forget: ½(b1 + b2)h = ½ × (b1 + b2) × h
# 2
Substitute the length of the bases and the height into the formula. In our example, b1 = 3, b2 = 5 and h = 3.
Area = ½ × (3 + 5) × 2
Area = ½ × (8) × 2
Area = 8 cm2
Don't forget: Calculate what's in the brackets () first (using the order of operations). In our example, (3 + 5) = 8.
or: ½ × a number = 0.5 × a number = a number ÷ 2. | 365 | 1,145 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.25 | 4 | CC-MAIN-2019-22 | longest | en | 0.881519 |
https://groups.yahoo.com/neo/groups/primenumbers/conversations/topics/484?l=1 | 1,493,008,374,000,000,000 | text/html | crawl-data/CC-MAIN-2017-17/segments/1492917118963.4/warc/CC-MAIN-20170423031158-00041-ip-10-145-167-34.ec2.internal.warc.gz | 814,637,774 | 23,082 | ## Re: [PrimeNumbers] Re: A Record Prime Factor by Pollard's "p
Expand Messages
• RE : Factoring. Two of the sites of most use for this sort of stuff are ; Will Edgington s page : http://www.garlic.com/~wedgingt/mersenne.html and Paul
Message 1 of 5 , Mar 5, 2001
RE : Factoring.
Two of the sites of most use for this sort of stuff are ;
Will Edgington's page : http://www.garlic.com/~wedgingt/mersenne.html
and
Paul Zimmerman's page : http://www.loria.fr/~zimmerma/records/ecmnet.html
In terms of the list of methods below, you;ve missed out the Quadratic Sieve,
which was the mainstay for a long time before NFS came along and is still used
in various forms. I've implemented at various times everything except NFS and
ECM (the maths is too difficult for me). I'm amazed that UBASIC provided such a
result - it can't possibly be as fast as specifically designed code. However, by
it's nature, there is always a chance at finding reaonably big factors.
With finding factors of Fermat numbers, there are even more methods available,
since the form of divisors is so restrictive : Leonid Durman and Tony Forbes
both have freely available software that hunts specifically for Fermat factors.
Joe McLean.
Subject: [PrimeNumbers] Re: A Record Prime Factor by Pollard's "p-1"
Author: Phil Carmody <fatphil@...> at Internet
Date: 05/03/01 01:45
On Sun, 04 March 2001, Andy Steward wrote:
[On the NMBRTHRY mailing list]
> Early on 3rd March 2001, my own Ubasic "p-1" code found the following
> factor of 922^47-1:
>
> p39=188879386195169498836498369376071664143
> p-1=2.3.13.47.101.813613.1174951.1766201.3026227.99836987
>
> I therefore claim the world record factor found by this method (unless
> any of you know better).
Well done, Andy.
Is UBasic as fast for this kind of job as C would be with one of the standard
bignum packages. Or a dedicated number-theoretic package, such as pari or LiDIA?
I was looking on the prime pages, for some info on the p-1 factoring method, and
I noticed that there is no "Factoring into primes" section on the prime pages.
My favourite search engine quickly pointed me in the direction of Wolfram/Eric
Weisstein's Mathworld, several dead links, and some source code in an unknown
language... So perhaps we could put our heads together to create the kernel of a
new prime pages page of factoring methods? As far as my memory serves me
(traditionally very poorly), it would be nice to get information on the
following.
- Trial division
- Pollard p-1
- Pollard p+1
- Continued Fractions
- Pollard Rho
- NFS
- ECM
Note - I haven't got a clue what any of the above are (apart from one), I'm just
parroting.
I do have Knuth, so I am prepared to put my reading spec's on to learn a few of
the above, but that still leaves half of them for others.
Any takers?
Phil
Mathematics should not have to involve martyrdom;
Support Eric Weisstein, see http://mathworld.wolfram.com
Find the best deals on the web at AltaVista Shopping!
http://www.shopping.altavista.com
Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
The Prime Pages : http://www.primepages.org
Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
• ... Sieve, ... used Indeed, it s still doing sterling service and is probably the method of choice for general integers in the range 50-110 digits. I have
Message 2 of 5 , Mar 5, 2001
> In terms of the list of methods below, you;ve missed out the Quadratic
Sieve,
> which was the mainstay for a long time before NFS came along and is still
used
Indeed, it's still doing sterling service and is probably the method of
choice for general integers in the range 50-110 digits. I have several QS
jobs running right now.
At the moment, GNFS takes over for about 100-160 digits whereas SNFS will
handle special integers up to around 240 digits.
Finding small factors (8 - 30 digits say) of large integers is best done
with ECM; smaller factors are best found by trial division (<5 digits or so)
and Pollard rho for the intervening range.
Another good algorithm for factoring small integers, up to 25 digits say, is
SQFOF. Such a tool comes in handy at times.
> - Continued Fractions
Personally, I'd drop this one. It has long been superseded by QS.
Paul
• ... Yuppers. Amusingly, as I _thought_ QS, I typed _NFS_! I d counciously forgotten about NFS. ... In the words of my Finnish teacher you can t even pronounce
Message 3 of 5 , Mar 5, 2001
On Mon, 05 March 2001, Paul Leyland wrote:
> > In terms of the list of methods below, you;ve missed out the Quadratic
> Sieve,
> > which was the mainstay for a long time before NFS came along and is still
> used
Yuppers. Amusingly, as I _thought_ QS, I typed _NFS_! I'd counciously forgotten about NFS.
> Another good algorithm for factoring small integers, up to 25 digits say, is
> SQFOF. Such a tool comes in handy at times.
In the words of my Finnish teacher "you can't even pronounce it".
(actually it sounds like an insult!)
I've never heard of it, care to elaborate?
> > - Continued Fractions
>
> Personally, I'd drop this one. It has long been superseded by QS.
My source is Knuth, ACP vol 2 - SNA, alone. Knuth waxes lyrical about it. He makes it sound of great historical importance.
I'm not qualified to even sneeze in the presence of QS. Do I see a volunteer? (quick - everyone else step back 1 pace :-) )
Does anyone have _any_ information on "Valles two-thirds algorithm"? I know it _was_ on mathworld, but that was when mathworld was still available.
Phil
Mathematics should not have to involve martyrdom;
Support Eric Weisstein, see http://mathworld.wolfram.com
Find the best deals on the web at AltaVista Shopping!
http://www.shopping.altavista.com
• ... Implementing NFS from scratch is somewhat non-trivial. I d estimate about a year s work to get something worthwhile. QS, on the other hand, is rather
Message 4 of 5 , Mar 5, 2001
> Yuppers. Amusingly, as I _thought_ QS, I typed _NFS_! I'd
Implementing NFS from scratch is somewhat non-trivial. I'd estimate about a
year's work to get something worthwhile. QS, on the other hand, is rather
easier --- especially as several implementations are available as source
code.
> > Another good algorithm for factoring small integers, up to 25 digits
say, is
> > SQFOF. Such a tool comes in handy at times.
>
> In the words of my Finnish teacher "you can't even pronounce it".
> (actually it sounds like an insult!)
> I've never heard of it, care to elaborate?
Perhaps you've never heard of it, at least in part, because I didn't spell
it properly! Try SQUFOF, aka
Daniel Shanks' "Square Forms Factorization" algorithm. It finds factors of N
in about (logN)^0.25 and so is much faster than trial division which is
(logN)^0.5. Like CFRAC, it calculates the continued fraction expansion of
sqrt(N), but doesn't use a factor base but, rather looks for a square
directly. It's this last difference than makes SQUFOF an exponential
algorithm and CFRAC subexponential. OTOH, it's very fast for relatively
small integers.
If memory serves, and I haven't looked for a *long* time, there's an
implementation of SQUFOF in the good old RSA-129 version of the MPQS siever
still to be found at ftp://ftp.ox.ac.uk/pub/math/ The double large prime
version of MPQS requires the factorization of large numbers of relatively
small integers and SQUFOF ruled supreme in this role for several years.
These days, we tend to use ECM with small parameters but SQUFOF is still
competitive.
> > > - Continued Fractions
> >
> > Personally, I'd drop this one. It has long been superseded by QS.
>
> My source is Knuth, ACP vol 2 - SNA, alone. Knuth waxes
> lyrical about it. He makes it sound of great historical importance.
It is indeed of great historical importance, as is Fermat's method.
Nonetheless, both are quite outdated now. Certainly implement it if you
want to do so, but don't expect it to be competitive.
Paul
• This has turned into a quite fun day (sad sad man that I am - sue me!) http://www.math.niu.edu/~rusin/known-math/index/11Y05.html Seems to be a fairly good
Message 5 of 5 , Mar 5, 2001
This has turned into a quite fun day (sad sad man that I am - sue me!)
http://www.math.niu.edu/~rusin/known-math/index/11Y05.html
Seems to be a fairly good resource after the non-existant one.
(Hmmm, does anyone have a copy of Weisstein? I'm thinking nefarious thougts here...)
About 6 months ago I found some ultra-lucid p-1, p+1, and rho source code on the internet, I even mailed the author a few hints on how he could optimise a couple of them. So - can I find that link again? Not for the life of me.
Phil
Mathematics should not have to involve martyrdom;
Support Eric Weisstein, see http://mathworld.wolfram.com
Find the best deals on the web at AltaVista Shopping!
http://www.shopping.altavista.com
Your message has been successfully submitted and would be delivered to recipients shortly. | 2,362 | 8,891 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2017-17 | latest | en | 0.915601 |
https://writemycustomessay.com/independent-samples-t-test-with-spss-2/ | 1,660,214,827,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882571284.54/warc/CC-MAIN-20220811103305-20220811133305-00518.warc.gz | 555,512,400 | 17,284 | # Independent-Samples t-Test with SPSS
## Introduction
This paper contains the solutions to week 8’s application assignment, namely,
Independent-Samples t Test with SPSS. In that assignment there are nine tasks to be done. The first task is to state the statistical assumptions that underlie an independent-Samples t Test test. The second task is to select a dataset and from it select the independent and dependent variables. The third task is to formulate a null and an alternative hypothesis. The fourth task is to use SPSS to calculate an independent-samples t test. The fifth task is to report on the p value and the confidence interval obtained from the test. The sixth task is to interpret the confidence interval. The seventh task is to make a decision on whether or not to accept the null hypothesis. The eighth task is to generate the SPSS syntax and output files. The final task is to report the results of the SPSS analysis using correct APA format.
An Independent-Samples t test is a statistical inference procedure that compares the sample means of two samples derived from two different populations for a given variable. In statistical inference, the descriptive statistic sample mean is taken as the best estimator for the true mean.
## Assumptions underlying test
A number of assumptions underlie an independent-samples t test. One of the assumptions is that each piece of data or observation in the datasets being compared in a t test is independent of the other. Furthermore, it is also assumed that the datasets being compared in the t test are themselves independent of each other. Another assumption is that the datasets are assumed to be from two different populations that follow a normal distribution (Frankfort-Nachmias and Nachmias, 2008). In special cases of the t test it is also assumed that these populations have equal variances.
## Dependent and independent variables
In this assignment, the goal of the independent-samples t test will be to determine whether the number of people living with HIV/AIDS in two distinct African regions is significantly different. The regions to be compared in the test are eastern and western Africa. Therefore, the datasets used for this assignment comprise of data on HIV/AIDS in Eastern and Western Africa. The datasets are obtained from the article “Sub-Saharan Africa HIV/AIDS statistics”, which gives statistics on HIV/AIDS in Sub-Saharan Africa for the year 2009 (Avert, 2011). The dataset for eastern Africa comprises of data from six countries that make up the Eastern Africa region and is captured in the table shown in Appendix A. The dataset for western Africa comprises of data from fifteen countries that make up the Western Africa region and is captured in the table shown in Appendix B. For these two datasets, the dependent variable is Region and the independent variable is People living with HIV/AIDS.
## Null and alternative hypotheses
Given that, the goal of the independent-samples t test in this case is to determine whether the number of people living with HIV/AIDS in eastern and western African is significantly different we formulate the following hypotheses. H0 : σ1 = σ2 and (or versus) H1: σ1 ≠ σ2. H0 is the null hypothesis and when translated it means that there is no significant difference between the number of people living with HIV/AIDS in eastern and western Africa. H1 is the alternative hypothesis and when translated it means that there is a significant difference between the number of people living with HIV/AIDS in eastern and western Africa. From the above hypotheses, it should be noted that the independent-samples t test to be carried out is going to be 2-tailed.
When the null hypothesis is denied when in actuality it should be accepted a first kind of error occurs and when the reverse of this happens a second kind occurs. The first kind of error described is a type I error while the second is a type II error. In hypothesis testing we accept the null hypothesis with a certain level of confidence that there is no a type I error or we accept it on the basis that there is strong or very strong evidence indicating that there is no a type II error. To accept the null hypothesis using the first basis the sample statistic, which is computed from the samples (datasets) being used in the test must lie inside an appropriate confidence interval. To accept the null hypothesis using the second basis the probability of committing a type II (known as p value) error must be less than the probability of committing a type I error (Mason et al, 1999).
## The test
To test the above hypotheses using SPSS’s Independent-samples t Test, a SPSS file is created with two variables. The first variable is of a numeric data-type and is called Number_of_people_with_HIVAIDS. The second variable is also of a numeric data-type and is called Region. Region is the grouping variable and is of scale measure whereas Number_of_people_with_HIVAIDS is the test variable and is of nominal measure. The data has been grouped using SPSS’s use specified values option whereby the eastern Africa region is assigned the value 1 and the eastern Africa region is assigned the value 2. Having defined these variables appropriately the SPSS syntax shown in Appendix C is run. The output of this syntax is two tables, which are also shown in Appendix C. By default SPSS uses a 95% level of confidence.
## Test results
The first table, which is titled group statistics, shows some descriptive statistics for the two datasets. The sample mean and sample standard deviation obtained from the dataset for eastern Africa are 718833.33 and 718264.691 respectively. The standard error that is used in calculating the test statistic for the test is found to be 293230.332. The sample mean and sample standard deviation obtained from the dataset for western Africa are 314333.33 and 833757.045 respectively. The standard error from this dataset is 215275.143.
The second table shows the results of the independent samples test and is known as independent sample test table. The independent samples test procedure in SPSS constitutes of two tests. In the first test, the variance of the first sample and the variance of the second sample are taken to be equal. The Lavene’s test for equality is used to test if this assumption should be adopted.. In the second test, the variance of the first sample and the variance of the second sample are taken not to be equal. To choose between the results of these two tests which should be adopted we consider the significance value from the result of the Lavene’s test. For these datasets the Lavene’s significance value is 0.445. Since it is greater than 0.10, it is safe to assume that the datasets have equal variances and thus the results of the first test are taken. Therefore, the observed t statistic is 1.040, degrees of freedom are 19, sample statistic is 404500, p value is 0.311and the confidence interval lies between – 409348.613 and 1218348.613.
## Interpreting confidence interval and conclusion
Confidence intervals are interpreted as the frequency that they contain the parameter of interest, which for this case is the true mean. The region outside a confidence interval is referred to as the rejection region. If the parameter of interest, which is estimated using a sample statistic, does not fall inside the rejection region, the null hypothesis is taken to be true.
In the test above the sample statistic has a value of 404500, which lies inside the confidence interval. Furthermore, the p value of 0.311 is greater than 0.5, which is the level of significance used in the test. Thus, based on these two facts, the null hypothesis is accepted and the conclusion made that there is no significant difference between the number of people living with HIV/AIDS in eastern and western Africa.
## References
Avert. (2011). Sub-Saharan Africa HIV/AIDS statistics. Web.
Frankfort-Nachmias, C., and Nachmias, D., (2008). Research Methods in Social Sciences (7th ed.). Worth publishers; New York. 434-453.
Mason, R. D., Lind, D, A. and Marchal, W. G.. (1999). Statistical techniques in business and economics. (10th ed.) Irwin/McGraw-Hill; USA. 316. | 1,703 | 8,201 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2022-33 | latest | en | 0.901685 |
http://www.cse.chalmers.se/edu/year/2013/course/TDA382_Concurrent_Programming_HT2013/exam/ | 1,531,945,441,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676590329.25/warc/CC-MAIN-20180718193656-20180718213656-00509.warc.gz | 436,414,808 | 3,315 | # Course Requirement and Examinations
To pass the course, you must have passed the assignments (labs) and have passed the exam. See exam schedule (including re-exam/omtenta) for TDA381/TDA380/TDA211/INN390 Concurrent Programming. The only permitted material is a dictionary.
# Past exams
Some previous exams are available below for reference.
• Exam-1303(with JR)
• Hints to solve the exercises:
• Q1 : (a) The scenario q1, q2, p1, q1 gives n=0 at the end. The scenario p1, q1, q2, p2, p1, q1 gives n=1 at the end.
• Q2 : The abbreviated version leaves out p1, q1, p4 and q4. Each state now shows where the program counters of p and q are, and the values of wantp and wantq. Draw the state diagram and show there is no state with p5 and q5.
• Q3 : (a) The state diagram is on p108 of the text-book (b) the definitions of wait and signal are on p109 and p110. (c) See algorithm 6.8 on p119.
• Q4 : (a) See p 161. (b). eating(i) becomes true only by executing takeForks(i) completely, or by by being unblocked in releaseForks(i+1) or releaseForks(i-1). In both cases, we have fork[i]≈2.
• Q5 & Q6 : These are programming problems, not involving formal reasoning.
• Q7: (a) the processes can livelock, looping p- to p3 and q- to q3. The invariant is that exactly one of C, Lp and Lq is true, (b) We did this in class, in my 3rd lecture. If p does not progress, Lp must be false. So q must progress, and will then set C to true. Assuming fairness, p must then progress.
• Exam-1203(with JR)
• Exam-1103(with JR)
• Exam-1008(with JR)
• Exam-1010(with JR)
• Exam-0910(with JR)
• Exam-0710(with JR)
• Exam-0310(with MPD)
• Exam-0210(with MPD) | 503 | 1,641 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2018-30 | latest | en | 0.908577 |
https://www.easyelimu.com/qa/2420/pointer-attached-spring-balance-points-weight-suspended-hanged | 1,718,516,153,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861643.92/warc/CC-MAIN-20240616043719-20240616073719-00341.warc.gz | 673,567,854 | 7,386 | # A pointer attached to a spring balance points to 21.5 cm on a scale when no weight is suspended from it. When a mass of 165g is hanged from it, it points 38.0 cm.
361 views
A pointer attached to a spring balance points to 21.5 cm on a scale when no weight is suspended from it. When a mass of 165g is hanged from it, it points 38.0 cm.
1. Determine its spring constant.
2. What would be the extension if two similar springs A and B in (a) above are used as shown below.
1. e=38.0−21.5=16.5m=0.165m
k= 1.65N =10Nm−1
0.165m
2. K2= 10×10 5N/M
10+10
∴et= F = 1.65N
KT 5N/M
=0.33m
OR
e=2×0.165m
=0.33m | 234 | 616 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.71875 | 4 | CC-MAIN-2024-26 | latest | en | 0.87388 |
https://historyofprivacy.net/what-were-knights/ | 1,680,268,909,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296949642.35/warc/CC-MAIN-20230331113819-20230331143819-00479.warc.gz | 350,976,553 | 14,491 | # What Were Knights
Posted on
If you are looking for the answer of what were knights, you’ve got the right page. We have approximately 10 FAQ regarding what were knights. Read it below.
## what is the knight's answer to the question the queen
Ask: what is the knight’s answer to the question the queen posed? how does the knight discover the answer?
When the two appear at court, the young knight provides the answer to the riddle: Women want power over their husbands. The queen acknowledges that this is the correct answer. Staying true to his promise, the young knight does what is next asked of him by the old woman: he marries her.
Explanation:
hope its help
## Understanding the Text 1. Why is Arthur’s knight sentenced to
Ask: Understanding the Text
1. Why is Arthur’s knight sentenced to death?
2. What is the condition for the knight to live?
3. What are the various answers given to the knight during his quest?
4. What answer does the old hag give the knight on condition that he marries her if it’s right?
5. At the end of the story, what two choices does the old hag offer to the knight? 6. What does the knight choose?
can i ask may story poba yung question?
## The knight whom King Arthur grew up with when they
Ask: The knight whom King Arthur grew up with when they were still young
sir Ector and Sir Kay, his son.
Explanation:
#CarryOnLearning
## there were more dragons than knights in the battle.in fact,
Ask: there were more dragons than knights in the battle.in fact, the ratio of dragons to knights was 5 to 4. if there were 60 knights, how many dragons were there?
75 dragons
Solution:
Let x be the number of dragons.
5:4 = x:60
[tex]frac{5}{4}= frac{x}{60}[/tex]
x = 300/4
x = 75
## what is the meaning of knight
Ask: what is the meaning of knight
a brave man who saves someone from a dangerous situation..
## What is a knight to you?How do you see a
Ask: What is a knight to you?How do you see a knight? (3 sentences)/ with affix and compounds
A knight is a person granted an honorary title of knighthood by a head of state or representative for service to the monarch, the church or the country, especially in a military capacity. Knighthood finds origins in the Greek hippeis and hoplite and Roman eques and centurion of classical antiquity.
Explanation:
not sure tho
knight sentence: Ex
After he saved her life, Alex was her knight and shinning armor.
Explanation:
The knight showedreverenceto the king by Bowing before him.
Hope it’s helpful po
## what are the characteristics of knights of the altar
Ask: what are the characteristics of knights of the altar
Medieval Knighthood, in the service of manor lords, calls forth such ideals as honor, loyalty, justice, chivalry, and respect for all. In the use of the term knight, the Altar Server is reminded of his duty to serve the Lord of lords with fidelity and honor, to treat others with respect and justice, and to live an upright personal life, defending always the rights of God and His Holy Church. In the names page and squire, the server is reminded again of the years of practice and study that went into the training of a knight and should consider with what devotion and perseverance he should attend to his own training in the service of the Altar. The chevalier was a traveling knight, which should remind the server that he should be ever traveling toward his heavenly goal.
Explanation:
## The story "The training of a knight" describes the chivalry
Ask: The story “The training of a knight” describes the chivalry of a page,esquire and a knight .As a youth of today,what relevant experience can you relate to chivalry? What qualities of a page, esquire and a knight that are essential to the youth?
Explanation:
[tex]yellow{ rule{10pt}{999999pt}}[/tex]
## one third of 2 dozen knight were on horseback how
Ask: one third of 2 dozen knight were on horseback how many knights were not on horse back
16
Step-by-step explanation:
If 1/3 of 2 dozen knights were on horseback, then it means 2/3 were not. Calculate what is 2/3 of 2 dozens.
1 dozen = 12, so 2 dozens are 24
Get the 2/3 of 24
## 4. What military rank would a knight have today? Point
Ask: 4. What military rank would a knight have today? Point out the admirable qualities of the
knight?
Today, a number of orders of knighthood continue to exist in Christian Churches, as well as in several historically Christian countries and their former territories, such as the Roman Catholic Order of the Holy Sepulchre, the Protestant Order of Saint John, as well as the English Order of the Garter, the Swedish Royal Order of the Seraphim, and the Order of St. Olav. Each of these orders has its own criteria for eligibility, but knighthood is generally granted by a head of state, monarch, or prelate to selected persons to recognise some meritorious achievement, as in the British honours system, often for service to the Church or country.
### -Aphmyn<3
try lng, pa follow ako.
Not only you can get the answer of what were knights, you could also find the answers of one third of, there were more, What is a, The knight whom, and 4. What military. | 1,235 | 5,155 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2023-14 | longest | en | 0.976917 |
https://blog.finxter.com/how-to-check-if-items-in-a-python-list-are-also-in-another/ | 1,716,824,742,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059044.17/warc/CC-MAIN-20240527144335-20240527174335-00586.warc.gz | 110,355,513 | 25,660 | # How to Check if Items in a Python List Are Also in Another
5/5 - (1 vote)
There comes a time in all our coding lives when we need to compare lists to understand whether items in one list appear in a second list.
In this article we’ll start where we all started, using for-loops, before moving to a more classic Python list comprehension. We’ll then move beyond that to use Python built-in functions any() and all() before looking at a clever use of methods contained within the set() data type.
By the end of this article, you’ll be creating code that not only meets your need but also retains readability while being concise, fast, and elegantly pythonic.
We will also go a bit beyond the remit of checking whether items in one list appear in another, we’ll also find out an easy method to return those duplicates (or as a bonus, return the non-duplicated items of the list) for review or use.
## Method 1: List Comprehension
So if we were starting in Python coding we would use a for loop to iterate through the two lists and return a result. In the examples following we first return a True/False on whether a number in List C is also in List A. We then return the actual numbers from List C that are in List A.
# Main list
lst_a = [24, 17, 37, 16, 27, 13, 46, 40, 46, 51,
44, 29, 54, 77, 78, 73, 40, 58, 32, 48,
45, 55, 51, 59, 68, 34, 83, 65, 57, 50,
57, 93, 62, 37, 70, 62
]
# SOME items are in lst_a
lst_c = [93, 108, 15, 42, 27, 83]
# Empty list
lst_result = []
# Check True or False items are in both lists
for i in lst_c:
for j in lst_a:
if i == j:
print(True, end=' ')
else:
continue
print('\n')
# Return values that are in both lists
for i in lst_c:
for j in lst_a:
if i == j:
lst_result.append(i)
else:
continue
print(lst_result)
# Result #
True True True
[93, 27, 83]
So that’s fine as far as it goes; we’ve answered the question. Yet, it took six lines of code for the True/False query and another six lines of code plus the creation of a new list to identify the actual numbers that were common to both lists. Using list comprehension we can improve on that.
## Method 2: Lambda
If you read my blog post on the use of lambda expressions and you’ve fallen in love with them as I once did, we could use a lambda for this purpose.
# Main list
lst_a = [24, 17, 37, 16, 27, 13, 46, 40, 46, 51,
44, 29, 54, 77, 78, 73, 40, 58, 32, 48,
45, 55, 51, 59, 68, 34, 83, 65, 57, 50,
57, 93, 62, 37, 70, 62
]
# SOME items are in lst_a
lst_c = [93, 108, 15, 42, 27, 83]
print(list(filter(lambda i: i in lst_a, lst_c)))
# Result #
[93, 27, 83]
Yet, in my post on lambda expressions I did say that they can sometimes make code difficult to read and when looking at the above code, as much as it pains me to admit it, I’m not convinced a lambda expression is necessary in this case. The syntax doesn’t exactly roll off your tongue while reading so let’s look at a couple of easy one-liner list comprehensions that return the same information as the previous methods but they’re more concise and more readable. Here they are.
# Main list
lst_a = [24, 17, 37, 16, 27, 13, 46, 40, 46, 51,
44, 29, 54, 77, 78, 73, 40, 58, 32, 48,
45, 55, 51, 59, 68, 34, 83, 65, 57, 50,
57, 93, 62, 37, 70, 62
]
# SOME items are in lst_a
lst_c = [93, 108, 15, 42, 27, 83]
print([True for i in lst_a if i in lst_c], '\n')
print([i for i in lst_a if i in lst_c])
# Result #
[True, True, True]
[27, 83, 93]
So we have reached a tidy landing place for list comprehensions with short, readable code but now we should inject another variable into our thinking, that being the speed of execution. On small lists such as the ones we’ve used here any speed penalties of different function choices are minor, however, be careful that on a large list this method of list comprehension doesn’t come with a speed penalty. It would pay to check with a timer during a test.
## Method 3: Python’s any() and all() Built-in Functions
To avoid writing lengthy code, Python has a range of built-in functions that meet our need to understand whether items in one list are present in another.
The function any() checks if any of the items in a list are True and returns a corresponding True. Here’s a simple example of how it works:
a = [True, False, True, False, True]
print(any(a))
# Result #
True
That’s straightforward, so let’s apply it to our list of examples. I’ve screenshot all the lists again to save you from scrolling. So if we want a simple True/False response to our question as to whether any items in one list are in another, any() suits our need admirably.
print(any(x in lst_a for x in lst_b))
print(any(x in lst_a for x in lst_c))
print(any(x in lst_a for x in lst_d))
# Result #
True
True
False
Remember that lst_b items are all in lst_a; lst_c has some of its items in lst_a, and lst_d does not have any items in lst_a. Therefore the return of True, True, False makes sense as only the third list, lst_d, does not have any items duplicated in lst_a.
The issue with this method is that it doesn’t tell you if all the items in one list are in another, only that some are. If you need that degree of precision, the built-in function all() can do this for you.
# Main list
lst_a = [24, 17, 37, 16, 27, 13, 46, 40, 46, 51,
44, 29, 54, 77, 78, 73, 40, 58, 32, 48,
45, 55, 51, 59, 68, 34, 83, 65, 57, 50,
57, 93, 62, 37, 70, 62
]
# ALL items are in lst_a
lst_b = [59, 37, 32, 40]
# SOME items are in lst_a
lst_c = [93, 108, 15, 42, 27, 83]
# NO items are in lst_a
lst_d = [23, 101, 63, 35]
print(all(x in lst_a for x in lst_b))
print(all(x in lst_a for x in lst_c))
print(all(x in lst_a for x in lst_d))
# Result #
True
False
False
So in this case the only list that has all of its items contained in lst_a is lst_b, hence the True.
These two functions any() and all() are useful, provide readable code, and are concise, but in the basic list comprehension done previously, we were also able to list out the actual duplicate items. While you could do that using any() and all() the extra code to make it work begs the question of why you’d bother so let’s leave those two to return just True or False and turn our attention to some different approaches.
## Method 4: Introducing The set() Data Type And Methods
Now it may seem strange and a bit arbitrary to introduce a new data type when we’re working with lists but the method I’m about to show is an elegant way to answer our question on whether items in one list are in another, and we’ll even return the answer as a list to remain coherent with our code.
For those who don’t do much with sets, they are one of the four Python built-in data types. They are an unordered and unindexed collection of data, and they come with some very clever methods that we can use. There are 17 methods for use on sets and I’ll first introduce you to two of those that I feel suit best this application. The first one gives us much the same as we’ve done using any() and all() , while the second provides an elegant way of returning the items common to two lists.
• issubset() – returns whether another set contains this set or not
• intersection() – returns a set, that is the intersection of two other sets
And here is the code using both methods on each of our three list comparisons.
# Main list
lst_a = [24, 17, 37, 16, 27, 13, 46, 40, 46, 51,
44, 29, 54, 77, 78, 73, 40, 58, 32, 48,
45, 55, 51, 59, 68, 34, 83, 65, 57, 50,
57, 93, 62, 37, 70, 62
]
# ALL items are in lst_a
lst_b = [59, 37, 32, 40]
# SOME items are in lst_a
lst_c = [93, 108, 15, 42, 27, 83]
# NO items are in lst_a
lst_d = [23, 101, 63, 35]
print(set(lst_b).issubset(lst_a))
print(set(lst_c).issubset(lst_a))
print(set(lst_d).issubset(lst_a), '\n')
print(list(set(lst_a).intersection(set(lst_b))))
print(list(set(lst_a).intersection(set(lst_c))))
print(list(set(lst_a).intersection(set(lst_d))))
# Result #
True
False
False
[32, 40, 59, 37]
[27, 83, 93]
[]
Note that in both cases we needed to convert the lists into sets using the set(lst_a), set(lst_b) syntax shown, before allowing the intersection method to do its work. If you want the response returned as a list then you’ll need to convert the response using the list() command as shown. If that’s not important to you, you’ll save a bit of code and return a set.
## Methods 5-7: Three Bonus Methods
While moving slightly away from our original question of whether items in one list are in another, there are three other methods in set( ) that may suit your needs in making list comparisons although the answers they return approach the problem from another angle. These are:
• difference() – returns a set containing the difference between two or more sets
• isdisjoint() – returns whether two sets have an intersection or not
• issuperset() – returns whether one set contains another set or not
As you can tell from the descriptions they’re effectively the inverse of what we have done previously with intersection() and issubset().
Using our code examples, difference() will return the numbers in lst_a that are not in lst_b, c or d while isdisjoint() will return False if there is an intersection and a True if there is not (which seems a little counterintuitive until you reflect on the name of the method), and issuperset() will check whether our large lst_a contains the smaller lst_b, c or d in its entirety.
Here’s an example of the three methods in use on our lists.
# Main List
lst_a = [24, 17, 37, 16, 27, 13, 46, 40, 46, 51,
44, 29, 54, 77, 78, 73, 40, 58, 32, 48,
45, 55, 51, 59, 68, 34, 83, 65, 57, 50,
57, 93, 62, 37, 70, 62
]
# ALL items are in lst_a
lst_b = [59, 37, 32, 40]
# SOME items are in lst_a
lst_c = [93, 108, 15, 42, 27, 83]
# NO items are in lst_a
lst_d = [23, 101, 63, 35]
print(set(lst_a).isdisjoint(lst_b))
print(set(lst_a).isdisjoint(lst_c))
print(set(lst_a).isdisjoint(lst_d), '\n')
print(list(set(lst_a).difference(set(lst_b))))
print(list(set(lst_a).difference(set(lst_c))))
print(list(set(lst_a).difference(set(lst_d))), '\n')
print(set(lst_a).issuperset(set(lst_b)))
print(set(lst_a).issuperset(set(lst_c)))
print(set(lst_a).issuperset(set(lst_d)))
# Result #
False
False
True
[65, 68, 70, 73, 13, 77, 78, 16, 17, 83, 24, 27, 29, 93, 34, 44, 45, 46, 48, 50, 51, 54, 55, 57, 58, 62]
[65, 68, 70, 73, 13, 77, 78, 16, 17, 24, 29, 32, 34, 37, 40, 44, 45, 46, 48, 50, 51, 54, 55, 57, 58, 59, 62]
[65, 68, 70, 73, 13, 77, 78, 16, 17, 83, 24, 27, 29, 93, 32, 34, 37, 40, 44, 45, 46, 48, 50, 51, 54, 55, 57, 58, 59, 62]
True
False
False
At the risk of labouring a point, remember that isdisjoint() will return False if any items in one list appear in the other. It will only return True when the two lists are entirely separate without any duplication.
🌍 Recommended Tutorial: How to Find Common Elements of Two Lists
## In Summary
To summarise what we’ve covered today, we looked at an oft posed question of how best to check if items in one list are also in another.
• We started with basic list comprehension using for-loops, before checking to see if a lambda expression was more suitable. We finally concluded with a one-line list comprehension that returned True or False on whether each list item was in another list. We also used a one-line list comprehension to return the actual values of the duplicates as a list.
• We then explored Python built-in functions any() and all() which return true or false depending on whether any or all items in one list repeat in another.
• Finally, we introduced some of the methods used within the set() datatype. Using issubset() we were able to return true or false on whether list items duplicated in another list and using intersection() we returned the values of the duplicated items.
• As a bonus, we also introduced some other set() methods which allowed us to further manipulate the lists to return useful data.
I hope the examples used in this article have been useful. They are not the only way to solve the original problem but there are enough examples here to get you started on your journey with lists. I highly recommend the following articles for those who wish to go deeper into some of the methods shown today.
To explore Python’s built-in functions I suggest starting here; https://blog.finxter.com/python-built-in-functions/ | 3,597 | 12,240 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2024-22 | latest | en | 0.879339 |
https://www.coursehero.com/file/6794262/lecture-9/ | 1,527,164,348,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794866276.61/warc/CC-MAIN-20180524112244-20180524132244-00309.warc.gz | 729,260,147 | 71,351 | {[ promptMessage ]}
Bookmark it
{[ promptMessage ]}
# lecture 9 - Math 482(Lecture 9 Duality II This lecture...
This preview shows pages 1–2. Sign up to view the full content.
Math 482 (Lecture 9): Duality II This lecture completes discussion of Section 3.1 of the textbook. Recall that given a primal problem: max c'x s.t Ax≤ b; x≥ 0 there is a dual problem min y'b s.t. y'A≥ c'; y≥ 0 We introduced the notion of shadow price . Mathematically: These are the coefficients of the slack variables y i in the final tableau at optimum. Economics interpretation: This is the additional profit for have one additional unit of resource i. (This being a math class, we can concern ourselves only with the mathematical definition, but I hope this economics interpretation is useful.) As yet unproved observation/claim: The shadow prices are the optimal solution to the dual problem. The following are some first results about duality. But first: Meta-exercise: Explain all the exercises below in terms of the UIUCbucks example. (WARNING: we have not mathematically justified the shadow price story, but it might be helpful to use it anyway as a sanity check.) In class exercise 1: (Weak Duality) Prove that if x is feasible for the primal problem, and y is feasible for the dual problem, then c'x≤ y'b. Hint: c'x≤ y'Ax (why?) ≤ ? Solution: c'x ≤ y'Ax ≤ y'b. The first inequality holds because x≥0 and c'≤ y'A. The second holds because?
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up to access the rest of the document.
{[ snackBarMessage ]}
### What students are saying
• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.
Kiran Temple University Fox School of Business ‘17, Course Hero Intern
• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.
Dana University of Pennsylvania ‘17, Course Hero Intern
• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.
Jill Tulane University ‘16, Course Hero Intern | 597 | 2,572 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.796875 | 4 | CC-MAIN-2018-22 | latest | en | 0.874377 |
https://www.convert-measurement-units.com/convert+Terawatt+hour+to+Therm.php | 1,669,663,603,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710534.53/warc/CC-MAIN-20221128171516-20221128201516-00035.warc.gz | 764,956,738 | 13,204 | Convert TWh to thm (Terawatt hour to Therm)
## Terawatt hour into Therm
numbers in scientific notation
https://www.convert-measurement-units.com/convert+Terawatt+hour+to+Therm.php
## How many Therm make 1 Terawatt hour?
1 Terawatt hour [TWh] = 34 121 416.331 279 Therm [thm] - Measurement calculator that can be used to convert Terawatt hour to Therm, among others.
# Convert Terawatt hour to Therm (TWh to thm):
1. Choose the right category from the selection list, in this case 'Energy'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), square root (√), brackets and π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Terawatt hour [TWh]'.
4. Finally choose the unit you want the value to be converted to, in this case 'Therm [thm]'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '895 Terawatt hour'. In so doing, either the full name of the unit or its abbreviation can be usedas an example, either 'Terawatt hour' or 'TWh'. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case 'Energy'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought. Alternatively, the value to be converted can be entered as follows: '40 TWh to thm' or '89 TWh into thm' or '85 Terawatt hour -> Therm' or '41 TWh = thm' or '70 Terawatt hour to thm' or '9 TWh to Therm' or '74 Terawatt hour into Therm'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(76 * 30) TWh'. But different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '895 Terawatt hour + 2685 Therm' or '62mm x 77cm x 38dm = ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
The mathematical functions sin, cos, tan and sqrt can also be used. Example: sin(π/2), cos(pi/2), tan(90°), sin(90) or sqrt(4).
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 9.999 999 909 ×1025. For this form of presentation, the number will be segmented into an exponent, here 25, and the actual number, here 9.999 999 909. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket calculators, one also finds the way of writing numbers as 9.999 999 909 E+25. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 99 999 999 090 000 000 000 000 000. Independent of the presentation of the results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications. | 944 | 3,905 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2022-49 | latest | en | 0.844825 |
http://www.algebra.com/algebra/homework/Percentage-and-ratio-word-problems/Percentage-and-ratio-word-problems.faq?hide_answers=1&beginning=630 | 1,369,257,473,000,000,000 | text/html | crawl-data/CC-MAIN-2013-20/segments/1368702447607/warc/CC-MAIN-20130516110727-00081-ip-10-60-113-184.ec2.internal.warc.gz | 303,944,413 | 11,996 | # Questions on Word Problems: Problems on percentages, ratios, and fractions answered by real tutors!
Algebra -> Algebra -> Percentage-and-ratio-word-problems -> Questions on Word Problems: Problems on percentages, ratios, and fractions answered by real tutors! Log On
Ad: Algebrator™ solves your algebra problems and provides step-by-step explanations! Ad: Algebra Solved!™: algebra software solves algebra homework problems with step-by-step help!
Word Problems: Problems on percentages, ratios, and fractions Solvers Lessons Answers archive Quiz In Depth
Question 103139: How many quarts of pure sulphuric acid should be added to 5 quarts of water to obtain a 40% solution? Click here to see answer by HyperBrain(694)
Question 103173: Dear Sir/Madam: This is from a worksheet that I received from the teacher. Origination unknown. The denominator of a fraction is one more than the numerator. When 4 is added to the numerator and 6 is added to the denominator the result equals 3/4. Find the fraction. this is what I have: x/x+1 + 4/6 = (x + 4)/(1x + 6) = 4 / 2x + 7 I feel stuck. What did I not capture correctly out of the word problem? Thank you so very much. Click here to see answer by ankor@dixie-net.com(15649)
Question 103508: A pair of jeans costs \$35.98 before taxes. If the tax rate is 8.5%, what is the price of the jeans after taxes? Click here to see answer by checkley75(3666)
Question 103793: Julio works as a quality control expert in a beverage factory. The assembly line that he monitors produces about 20,000 bottles in 24-hour period. Julio samples 120 bottles an hour and rejects the line if 1/50 of the sample is defective. About how many defective bottles should Julio allow before rejecting the entire line? I tried to mutiply 120 by 1/50. I am not sure where to start on this one. Click here to see answer by stanbon(57337)
Question 103872: Army regulations require that their pickle relish use 10 pickles per pound of relish. In one bowl they have relish that only used 8 pickles per pound. In another bowl they have relish that used 16 pickles per pound. They need to make 200 pounds of relish. How many pounds of each bowl should they use? Click here to see answer by stanbon(57337)
Question 104241: A college increased it's student population by 6.72% over last fall. They now have 270 students. what is the original number of students? Use the formula changePC= [original-new]/original X 100 6.72 = {x-270}/x X 100(please note I'm using brackets to denote absolute value Multiply each side by 100 0.0672= 100x-2700/x Multiply each side by x, .0672x = 100x squared -2700x Help! TX Click here to see answer by Fombitz(13828)
Question 104348: here is the question you invested \$8000 in two funds paying 2% and 5% annual interest respectively. At the end of the year, the interest from the 5% investment exceeded the the interest from the 2% investment by \$85. How much money was invested at each rate? Click here to see answer by ankor@dixie-net.com(15649)
Question 104686: How many ounces of water must be added to 100 ounces of 40% antifreeze solution to obtain a 16% solution Click here to see answer by elima(1433)
Question 104821: A man weighs 230 pounds. How many pounds must he gain in order to weigh 1/8 of a ton? Click here to see answer by pwac(253)
Question 104872: NUts to You sells trail mix in 16-ounce packages. Half the weight is peanuts. There are also 2oz of almonds, 1oz of cashews, and 3oz of raisins. The rest is chocolate chips. What fraction of the mix is chocolate chips? Click here to see answer by oberobic(2304)
Question 104869: Joe Ravioli went running 3 days this week. He ran 21/2mi on Monda, 23/10mi on Wednesday, and 32/5mi on Friday. How far did he run altogether this week? Click here to see answer by edjones(7569)
Question 104922: Doc O had 10,000 invested at 5%. How much would he have to invest at 8% so that his total interest per year would equal 7% of the two investments? Click here to see answer by edjones(7569)
Question 104921: Mr. Rich invested 50,000. Part of it he put in a gold mine stock from which he hoped to receive 20% per year. The rest he invested in bank stock which was paying 6%. If he received 400 more the first year from the bank stock than from the mining stock, how much did he invest in each stock? Click here to see answer by Fombitz(13828)
Question 105121: Could you help me? I usually good in solving word problems in statistics. But I am confused, and when and if I get an answer I'm not confident it's right, or if it was necessary to do some of the steps. The problem is: A local country club has a membership of 600 and operates facilities that include an 18-hole championship golf course and 12 tennis courts. Before deciding whether to accept new members, the club president would like to know how many members regularly use each facility. A survey of the membership indicates that 63% regularly use the golf course, 48% regularly use the tennis courts, and 7% use neither of these facilities regularly. What percentage of the 600 use at least one of the golf or tennis facilities? I thought first I should convert the percents into decimals So then .63, .48, and .07 I'm stuck from there! I needed help asap, and I'm not sure where to go from here. Click here to see answer by scott8148(6628)
Question 105082: Genetics : Approximately one-seventh of the people in the world are left-handed. Write down and solve an equation to estimate how many people in the US are left-handed if the population of the US is about 300 million. Click here to see answer by checkley75(3666)
Question 105385: Helen has 1/5 of her portfolio in U.S. stocks, 1/8 of her portfolio in European stock and, 1/10 of her portfolio in Japanese stock. The remainder is invested in municipal bonds. What fraction of her portfolio is invested in municiple bonds? What percent is invested in municiple bonds? Click here to see answer by Earlsdon(6287)
Question 105472: If there are 500 people in a town. 22% have dogs and cats, 44% only have dogs, 28% only have cats. 6% have neither. How many people doesn't have a cat or a dog. Click here to see answer by checkley75(3666)
Question 105475: determine which two equations represent parallel lines. a. y= 5/4x+3 b. y= 4/5x+5 c. y= 4/5x-5 d. y= -5/4x+5 Click here to see answer by checkley75(3666)
Question 105531: In a family there are two cars. In a given week, the first car gets an average of 15 miles per gallon, and the second car gets 40 miles per gallon. The two cars combined drive a total of 65 miles in that week, for a total gas consumption of 1600 gallons. How many gallons were consumed by each of the two cars that week? Click here to see answer by ptaylor(2048)
Question 105771: The sum of two numbers is 133. The second is 7 more than 5 times the first. What are the two numbers? Click here to see answer by kmcruz09(38)
Question 105820: I could use some help please.I am taking an on line couase and this did not come out of my text book. It came off my on line test. The question is For the following function,C computes the cost in millions of dollars of implementing a city recycling project when x percent of the citizens particapte. C(x)=1.5x/100-x. Using this model approximately what is the cost if 60% ofthe citizens participate? This is what I tried. C(60%)=1.5(60%)/100-60%, .9/66.66=.014 C=.014 Thank You for your help. I found that I am totally lost Click here to see answer by checkley75(3666)
Question 106195: Standing next to each other, a woman casts a 46.9 inch shadow and her 50 inch tall daughter casts a 35 inch shadow. What is the height of the woman to the nearest inch? I tried forming a ratio between the woman and the daughter x:46.9= 50: 35 is this the right way to do it? if not, how can I proceed with the question? Click here to see answer by bucky(2189)
Question 106228: In a Mathematics seminar 1/4 of the people attending were married men. If the number of married women was 2/7 greater than the number of married men, what fraction of people attending the seminar were not married? Click here to see answer by edjones(7569)
Question 106240: In the high school parking lot 16% of the vehicles are trucks and 8% of the vehicles are painted yellow. If these characteristics are mutually exclusive, what is the probability that a vehicle in the high-school parking lot will be a yellow truck? Click here to see answer by edjones(7569)
Question 106135: Problem solving using chart:Question: Brian O'Reilly earns twice as much each week as a tutor than he does pumping gas. His total weekly wages are \$150 more than that of his younger sister. She earns one quarter as much as Brian does as a tutor. How much does Brian earn as a tutor? Click here to see answer by ptaylor(2048)
Question 106417: 3. P can do a piece of work in 9 days. Q is 50% more efficient than P. The number of days it takes Q to do the same piece of work is? Click here to see answer by MathLover1(6628)
Question 106847: Last year Ray Watson took over as manager of the production department. Under Ray's leadership, production increased from 42,000 to 57,000 units a month. By what percentage did output increase? If output were to increase by the same percentage next month, how many units would the department produce? Click here to see answer by checkley75(3666)
Question 106846: American products plans to lay off 10% of its employees in its aerospace divison and 15 % of its employees in its argicultural division. If altogether 12 % of the 3000 employees in these two divison will be laid off, then how many employees are in each division? Click here to see answer by checkley75(3666)
Question 106857: A sales price for a car is \$12,590 which is 20% off the original price, what is the original price? Click here to see answer by MathLover1(6628)
Question 107259: The first angle of a triangle is 15 degrees more than the second triangle. The third angle of a triangle is three times the second angle. Use x to represent the unknown Value. ? The first angle ? the second angle ? the third angle Click here to see answer by Annabelle1(69)
Question 107259: The first angle of a triangle is 15 degrees more than the second triangle. The third angle of a triangle is three times the second angle. Use x to represent the unknown Value. ? The first angle ? the second angle ? the third angle Click here to see answer by scott8148(6628)
Question 107398: susan invested a total of \$14,000. Part of it was at 6% interest and the other was at 8% interest. The total interest earned in one year was \$1,060.00. how mush interest was warned at each rate? what is the formula used? Thank you Melinda Click here to see answer by Annabelle1(69)
Question 107385: If a discount of 20% off the retail price of a desk saves Mark \$45, how much did he pay for the desk? Click here to see answer by checkley75(3666)
Question 107360: Find the original price of a pair of shoes if the sale price is \$78.00 after a 25% discount. Click here to see answer by josmiceli(9674)
Question 107414: What is the perimeter of a rectangle with a width that is 7 feet shorter than its length? Click here to see answer by Annabelle1(69)
Question 107497: If a patient is 5 ft 8in tall . What should his or her weight be to have a BMI of 25? Round to nearest whole pound. I know that BMI is B=705w/h2 h-height so in this case, 68 inches w-weight so it would just remain w right? and b-bmi so 25? Click here to see answer by stanbon(57337)
Question 107672: Sally took four tests in science class. On each successive test, her score improved by 3 points. If her mean score was 69.5%, what did she score on the first test? Click here to see answer by scott8148(6628)
Question 107672: Sally took four tests in science class. On each successive test, her score improved by 3 points. If her mean score was 69.5%, what did she score on the first test? Click here to see answer by bucky(2189)
Question 107678: Eva can paint three rooms in seven hours. Eva and Anna working together can paint four rooms in nine hours. How long would it take Anna to paint two rooms on her own? I used the equation 3/7 + 2/X = 4/9. Rooms/ Hours. Then got a common Denominator of 63X. Came up with 2 rooms in 126 Hours. Doesn't seem right. I have asked every one I know and they all come up with a different answer. Am I setting up the equation right? X=126 makes this equation work but seems un logistic. Click here to see answer by solver91311(16877)
Question 107737: The U.S. population in 1990 was approximately 250 million, and the average growth rate for the past 30 years gives a doubling time of 66 years. The above formula for the United States then becomes P (in millions) 250 2( y1990)66 82. Social science. What will the population of the United States be in 2025 if this growth rate continues? Click here to see answer by stanbon(57337)
Question 107797: One invention saves 30 % on fuel; a second saves 45 percent; a third saves 25%. If you use all three inventions at once, how much fuel can you save Click here to see answer by lauramg012(4)
Question 108133: A lever 12 m long balances weights of 48kg one end and 24 on the other end. Find the lengths of the two arms of the lever. Click here to see answer by checkley71(8403)
Older solutions: 1..45, 46..90, 91..135, 136..180, 181..225, 226..270, 271..315, 316..360, 361..405, 406..450, 451..495, 496..540, 541..585, 586..630, 631..675, 676..720, 721..765, 766..810, 811..855, 856..900, 901..945, 946..990, 991..1035, 1036..1080, 1081..1125, 1126..1170, 1171..1215, 1216..1260, 1261..1305, 1306..1350, 1351..1395, 1396..1440, 1441..1485, 1486..1530, 1531..1575, 1576..1620, 1621..1665, 1666..1710, 1711..1755, 1756..1800, 1801..1845, 1846..1890, 1891..1935, 1936..1980, 1981..2025, 2026..2070, 2071..2115, 2116..2160, 2161..2205, 2206..2250, 2251..2295, 2296..2340, 2341..2385, 2386..2430, 2431..2475, 2476..2520, 2521..2565, 2566..2610, 2611..2655, 2656..2700, 2701..2745, 2746..2790, 2791..2835, 2836..2880, 2881..2925, 2926..2970, 2971..3015, 3016..3060, 3061..3105, 3106..3150, 3151..3195, 3196..3240, 3241..3285, 3286..3330, 3331..3375, 3376..3420, 3421..3465, 3466..3510, 3511..3555, 3556..3600, 3601..3645, 3646..3690, 3691..3735, 3736..3780, 3781..3825, 3826..3870, 3871..3915, 3916..3960, 3961..4005, 4006..4050, 4051..4095, 4096..4140, 4141..4185, 4186..4230, 4231..4275, 4276..4320, 4321..4365, 4366..4410, 4411..4455, 4456..4500, 4501..4545, 4546..4590, 4591..4635, 4636..4680, 4681..4725, 4726..4770, 4771..4815, 4816..4860, 4861..4905, 4906..4950, 4951..4995, 4996..5040, 5041..5085, 5086..5130, 5131..5175, 5176..5220 | 4,195 | 14,683 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.4375 | 3 | CC-MAIN-2013-20 | latest | en | 0.922782 |
https://www.excelforum.com/excel-formulas-and-functions/1265023-sum-based-on-dates-with-specific-cells.html | 1,660,045,828,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882570921.9/warc/CC-MAIN-20220809094531-20220809124531-00337.warc.gz | 687,497,538 | 18,823 | # sum based on dates with specific cells
1. ## sum based on dates with specific cells
hello
In F40 – I need a formula to work with the exiting formula to look at a date in f38 and then sum highlighted cells… sums between the dates in f38 & h38
I’ve entered it manually at the moment to show expected results
thanks
2. ## Re: sum based on dates with specific cells
=IFERROR(INDEX('mfg weekly'!\$A\$1:\$N\$1240,MATCH(A40,'mfg weekly'!\$A\$1:\$A\$1240,),7)-SUM(\$J\$6:\$S\$28*(\$J\$3:\$S\$3>=F\$38)*(\$J\$3:\$S\$3<H\$38)*(\$A\$6:\$A\$28=\$A40)),"0")
3. ## Re: sum based on dates with specific cells
Translated:
=IFERROR(INDEX('mfg weekly'!\$A\$1:\$N\$1240,MATCH(\$A40,'mfg weekly'!\$A\$1:\$A\$1240,),7)-SUM(\$J\$6:\$S\$28*(\$J\$3:\$S\$3>=F\$38)*(\$J\$3:\$S\$3<H\$38)*(\$A\$6:\$A\$28=\$A40)),"0")
Courtesy of https://en.excel-translator.de/translator/
4. ## Re: sum based on dates with specific cells
Not sure I understand everything, but try something like this.
F37 formula
Cell F40 formula and copy down
Greetings.
5. ## Re: sum based on dates with specific cells
Translated:
=IFERROR(INDEX('mfg weekly'!\$A\$1:\$N\$1240,MATCH(\$A40,'mfg weekly'!\$A\$1:\$A\$1240,),7)-SUM(\$J\$6:\$S\$28*(\$J\$3:\$S\$3>=F\$38)*(\$J\$3:\$S\$3<H\$38)*(\$A\$6:\$A\$28=\$A40)),"0")
6. ## Re: sum based on dates with specific cells
Please post your formulae in English in future, Tim, then there won’t be any need for translations. This is an English-speaking forum, after all. Of course, you could have just said thanks!
7. ## Re: sum based on dates with specific cells
Spasibo!________
8. ## Re: sum based on dates with specific cells
this also worked very well
There are currently 1 users browsing this thread. (0 members and 1 guests)
#### Posting Permissions
• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts
Search Engine Friendly URLs by vBSEO 3.6.0 RC 1 | 633 | 1,932 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.59375 | 4 | CC-MAIN-2022-33 | latest | en | 0.835869 |
http://reference.wolfram.com/legacy/v6/tutorial/SearchingAndReadingStrings.html | 1,511,348,824,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934806569.66/warc/CC-MAIN-20171122103526-20171122123526-00322.warc.gz | 254,346,955 | 10,202 | This is documentation for Mathematica 6, which was
based on an earlier version of the Wolfram Language.
Mathematica Tutorial
Functions like Read and Find are most often used for processing text and data from external files. In some cases, however, you may find it convenient to use these same functions to process strings within Mathematica. You can do this by using the function StringToStream, which opens an input stream that takes characters not from an external file, but instead from a Mathematica string.
StringToStream["string"] open an input stream for reading from a string Close[stream] close an input stream
Treating strings as input streams.
This opens an input stream for reading from the string.
Out[1]=
This reads the first "word" from the string.
Out[2]=
This reads the remaining words from the string.
Out[3]=
This closes the input stream.
Out[4]=
Input streams associated with strings work just like those with files. At any given time, there is a current position in the stream, which advances when you use functions like Read. The current position is given as the number of characters from the beginning of the string by the function StreamPosition[stream]. You can explicitly set the current position using SetStreamPosition[stream, n].
Here is an input stream associated with a string.
Out[5]=
The current position is initially 0 characters from the beginning of the string.
Out[6]=
This reads a number from the stream.
Out[7]=
The current position is now 3 characters from the beginning of the string.
Out[8]=
This sets the current position to be 1 character from the beginning of the string.
Out[9]=
If you now read a number from the string, you get the 23 part of 123.
Out[10]=
This sets the current position to the end of the string.
Out[11]=
If you now try to read from the stream, you will always get EndOfFile.
Out[12]=
This closes the stream.
Out[13]=
Particularly when you are processing large volumes of textual data, it is common to read fairly long strings into Mathematica, then to use StringToStream to allow further processing of these strings within Mathematica. Once you have created an input stream using StringToStream, you can read and search the string using any of the functions discussed for files above.
This puts the whole contents of textfile into a string.
Out[14]=
This opens an input stream for the string.
Out[15]=
This gives the lines of text in the string that contain is.
Out[16]=
This resets the current position back to the beginning of the string.
Out[17]=
This finds the first occurrence of the in the string, and leaves the current point just after it.
Out[18]=
This reads the "word" which appears immediately after the.
Out[19]=
This closes the input stream.
Out[20]= | 619 | 2,754 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2017-47 | latest | en | 0.886393 |
http://poj.org/problem?id=2565 | 1,708,634,166,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947473824.45/warc/CC-MAIN-20240222193722-20240222223722-00281.warc.gz | 26,748,261 | 3,294 | Online JudgeProblem SetAuthorsOnline ContestsUser
Web Board
F.A.Qs
Statistical Charts
Problems
Submit Problem
Online Status
Prob.ID:
Register
Authors ranklist
Current Contest
Past Contests
Scheduled Contests
Award Contest
Register
Language:
Average is not Fast Enough!
Time Limit: 1000MS Memory Limit: 65536K Total Submissions: 2361 Accepted: 1017
Description
A relay is a race for two or more teams of runners. Each member of a team runs one section of the race. Your task is to help to evaluate the results of a relay race.
You have to process several teams. For each team you are given a list with the running times for every section of the race. You are to compute the average time per kilometer over the whole distance. That's easy, isn't it?
So if you like the fun and challenge competing at this contest, perhaps you like a relay race, too. Students from Ulm participated e.g. at the "SOLA" relay in Zurich, Switzerland. For more information visit http://www.sola.asvz.ethz.ch/ after the contest is over.
Input
The first line of the input specifies the number of sections n followed by the total distance of the relay d in kilometers. You may safely assume that 1<=n<=20 and 0.0 < d < 200.0. Every following line gives information about one team: the team number t (an integer, right-justified in a field of width 3) is followed by the n results for each section, separated by a single space. These running times are given in the format "h:mm:ss" with integer numbers for the hours, minutes and seconds, respectively. In the special case of a runner being disqualified, the running time will be denoted by "-:--:--". Finally, the data on every line is terminated by a newline character. Input is terminated by EOF.
Output
For each team output exactly one line giving the team's number t right aligned in a field of width 3, and the average time for this team rounded to whole seconds in the format "m:ss". If at least one of the team's runners has been disqualified, output "-" instead. Adhere to the sample output for the exact format of presentation.
Sample Input
```2 12.5
5 0:23:21 0:25:01
42 0:23:32 -:--:--
7 0:33:20 0:41:35
```
Sample Output
``` 5: 3:52 min/km
42: -
7: 6:00 min/km
```
Source
[Submit] [Go Back] [Status] [Discuss] | 582 | 2,265 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2024-10 | latest | en | 0.913108 |
https://www.physicsforums.com/search/4201226/ | 1,628,109,125,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046154897.82/warc/CC-MAIN-20210804174229-20210804204229-00341.warc.gz | 915,059,565 | 8,570 | # Search results
1. ### Uncertainty Again
Homework Statement 4 glasses of liquid must be transfferred into a beaker that has a cylinder shape without any markings. The teacher says that the glass will hold 250mL, but the uncertainty was 10mL in which this was determined. The radius and the height of the beaker was measured within an...
2. ### Uncertainty physics problem
Homework Statement Drivers that come to a stop leave different amount of gaps between their car and the car in front. It was found that the average gap was 1.45m, but as the values varied, the uncertainty was 25cm. It was also reported that the car is 5.1 ± 0.5m in average. What is the range... | 163 | 671 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2021-31 | latest | en | 0.984838 |
https://calculat.io/en/length/feet-to-cm/6-feet--0-inches | 1,713,332,807,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817144.49/warc/CC-MAIN-20240417044411-20240417074411-00099.warc.gz | 137,068,094 | 25,017 | # Convert 6 Feet to Centimeters
## How many centimeters in 6 Feet?
6 Feet is equal to 182.88 Centimeters
## Explanation of 6ft to Centimeters Conversion
Feet to Centimeters Conversion Formula: cm = ft × 30.48
According to 'feet to cm' conversion formula if you want to convert 6 Feet to Centimeters you have to multiply 6 by 30.48.
Here is the complete solution:
6′ × 30.48
=
182.88 cm
(one hundred eighty-two point eight eight centimeters)
## About "Feet to Centimeters" Calculator
This converter will help you to convert Feet to Centimeters (ft to cm). For example, it can help you find out how many centimeters in 6 Feet? (The answer is: 182.88). Or how tall is 6 Feet in cm? Enter the number of Feet (e.g. '6'), number of Inches (e.g. '0') and hit the 'Convert' button.
6' = 182.88 cm | 226 | 801 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.296875 | 3 | CC-MAIN-2024-18 | latest | en | 0.777461 |
https://whomadewhat.org/how-many-city-blocks-is-1000-feet/ | 1,695,982,493,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510501.83/warc/CC-MAIN-20230929090526-20230929120526-00470.warc.gz | 676,440,085 | 12,722 | 3.0303 city block
Moreover, How many blocks is 1000 feet?
Feet to Blocks (table conversion)
——- —————————-
1000 ft = 3.7878787878788 bl
2000 ft = 7.5757575757576 bl
4000 ft = 15.151515151515 bl
5000 ft = 18.939393939394 bl
In respect to this, How many city blocks is 1000 feet?
3.0303 city block
How long is a mile in blocks?
Furthermore, How long is 1 mile block?
But how many NYC blocks are in a mile? The average length of a north-south block in Manhattan runs approximately 264 feet, which means there are about 20 blocks per mile.
## How long is a city block in feet?
Oblong blocks range considerably in width and length. The standard block in Manhattan is about 264 by 900 feet (80 m × 274 m). In Chicago, a typical city block is 330 by 660 feet (100 m × 200 m), meaning that 16 east-west blocks or 8 north-south blocks measure one mile, which has been adopted by other US cities.
200 steps
## How many steps is four blocks?
Two steps take four blocks. Five steps take twenty-five blocks. “A” steps will take “A”-squared blocks (A x A). So the rule is any number of steps times itself equals the number of blocks you use.
0.60606
## How many blocks is 3 miles?
Blocks to Miles (table conversion)
—— —————————
30 bl = 1.5 mi
40 bl = 2 mi
50 bl = 2.5 mi
60 bl = 3 mi
## How many feet is a half block?
A block is not really defined by distance, but rather is defined by the distance between cross streets, which could be 50 feet or 200 feet, depending on the place. Most blocks in cities tend to be between 200 – 300 feet apart, so the distance between 2 blocks would be roughly 400 – 600 feet.
## Does 4 blocks equal a mile?
It depends on how large a block is.. According to Wikipedia, the typical city block in Chicago or Minneapolis is 660 feet (ft) by 330 ft. 1320 ft / quarter mile X 1 block / 330 ft = 4 block per quarter mile on the short blocks. Typically 12 blocks per mile is a typical general estimate.
20
## How many blocks is a 1000 feet?
Feet to Blocks (table conversion)
——- —————————-
1000 ft = 3.7878787878788 bl
2000 ft = 7.5757575757576 bl
4000 ft = 15.151515151515 bl
5000 ft = 18.939393939394 bl
## How many steps are in 6 city blocks?
How many city block in 1 steps? The answer is 0.009469696969697.
## How many miles are in 6 blocks?
Blocks to Miles (table conversion)
—— —————————
6 bl = 0.3 mi
7 bl = 0.35 mi
8 bl = 0.4 mi
9 bl = 0.45 mi
## How many blocks are in a foot?
0.0037878787878788
## How many miles is 20 city blocks?
North-south is easy: about 20 blocks to a mile. The annual Fifth Avenue Mile, for example, is a race from 80th to 60th Street. The distance between avenues is more complicated. In general, one long block between the avenues equals three short blocks, but the distance varies, with some avenues as far apart as 920 feet.
## What is a block in distance?
A block is not really defined by distance, but rather is defined by the distance between cross streets, which could be 50 feet or 200 feet, depending on the place. Most blocks in cities tend to be between 200 – 300 feet apart, so the distance between 2 blocks would be roughly 400 – 600 feet.
## How many blocks equals 2 miles?
Each block is typically square and each side of the square is 1/8 of a mile. So an 8 x 8 group of blocks is 1 square mile. You will have to walk 4 blocks in any direction (and this will be north, south, east or west as the city is laid out in a regular grid) to go 1/2 mile. | 968 | 3,445 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.5625 | 4 | CC-MAIN-2023-40 | latest | en | 0.925112 |
https://communitycorrespondent.com/how-many-slices-in-10-inch-pizza/ | 1,695,890,300,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510368.33/warc/CC-MAIN-20230928063033-20230928093033-00864.warc.gz | 196,093,207 | 49,558 | # How Many Slices in 10 Inch Pizza
There are between 8 and 12 slices in a 10 inch pizza, depending on how thin or thick the slices are.
## Girlfriend Can't Understand Pizza Problem
If you’re ever wondering how many slices are in a 10 inch pizza, the answer is eight. This is based on the industry standard for pizza slice size, which is one ounce per slice. So, if you have a 10 inch pizza that weighs 28 ounces total, then it will have eight slices.
Of course, this can vary depending on the toppings and crust thickness, but generally speaking, you can expect eight slices from a 10 inch pizza.
## 10 Inch Pizza Serves How Many
A standard 10 inch pizza is cut into 8 slices. However, the size and number of slices can vary depending on the type of pizza and how it is cut. For example, a deep dish pizza or one with thicker crust will have fewer pieces than a thin crust pizza.
A pie cut into wedges will also have more pieces than traditional slices. So, if you’re wondering how many people a 10 inch pizza will serve, it really depends on the circumstances! In general, though, you can expect a 10 inch pie to feed 3-4 people comfortably.
## 12 Inch Pizza How Many Slices
Pizza is one of America’s favorite foods. In fact, about 93% of us have eaten pizza in the last month. That’s a lot of pizza!
When it comes to pizza, there are all sorts of different sizes and shapes. But one thing that always remains the same is how many slices are in a pizza. So, how many slices are in a 12 inch pizza?
The answer may surprise you. A 12 inch pizza actually has 8 slices in it. Of course, this can vary depending on how thick or thin the crust is and how much topping is added.
But generally speaking, a 12 inch pizza will have 8 slices. So next time you’re wondering how to divide up that large pie, now you know!
## Is 10-Inch Pizza Enough for 2
If you’re like most people, you probably think that 10-inch pizza is more than enough for two people. After all, it’s a lot of food! But the truth is, 10-inch pizza is actually quite small when you compare it to other types of food.
For example, a 12-inch pizza has 50% more food than a 10-inch pizza. And if you’re really hungry, you might even want to get a 16-inch pizza! So next time you’re ordering pizza for two people, make sure to get at least a 12-inch pie.
## Is a 10 Inch Pizza Enough for One Person
A 10 inch pizza is definitely enough for one person… as long as that person isn’t a teenage boy! For the average person, a 10 inch pizza is more than sufficient. It’s especially perfect if you’re pairing it with other dishes or side items.
If you have a hearty appetite, you might be able to eat an entire 12 inch pizza by yourself, but most people would probably feel pretty full after just 10 inches. So if you’re looking for a single-serving size, go with the 10 incher.
## 10 Inch Pizza Calories
A 10-inch pizza is a medium-sized pizza that is usually cut into 6 slices. A slice of 10-inch pizza has about 250 calories. The calorie content of a 10-inch pizza depends on the toppings and the crust.
If you are eating a plain cheese pizza, it will have less calories than a pizza with meat and vegetables toppings. The type of crust also affects the calorie content of a 10-inch pizza. A thin crust pizza has fewer calories than a thick crust or stuffed crust pizza.
Credit: brickschicago.com
## How Many Does a 10 Inch Pizza Feed?
Assuming you’re talking about a standard 10″ pizza from a pizzeria: A standard 10″ pizza has 8 slices in it, so it would feed 2-3 people depending on how hungry they are.
## Is a 10 Inch Pizza Good for One Person?
A 10-inch pizza is a good size for one person. It’s big enough to fill you up, but not so big that you’ll feel stuffed. Plus, it’s a manageable size to eat by yourself.
## Is 10 Inches a Large Pizza?
No, 10 inches is not a large pizza. A large pizza is typically around 16 inches in diameter.
## Conclusion
Assuming you would like a summary of the blog post titled “How Many Slices in a 10 Inch Pizza”, the answer is eight. The blog post goes on to say that this is based on the standard size for pizza slices, which are two inches wide and four inches long. | 1,008 | 4,190 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.09375 | 3 | CC-MAIN-2023-40 | latest | en | 0.950037 |
http://forums.codeguru.com/printthread.php?t=516254&pp=15&page=1 | 1,527,478,408,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794870771.86/warc/CC-MAIN-20180528024807-20180528044807-00285.warc.gz | 109,494,777 | 4,552 | # Matlab functions need conversion
• September 12th, 2011, 08:34 PM
stunner08
Matlab functions need conversion
Hi,
I need help in converting a few functions from matlab to VC++. Here's the code that I have done partially in C++ and some are still in MATLAB.
#include "mbed.h"
#include <stddef.h>
#include <stdlib.h>
#include <stdio.h>
#include "iostream"
//#include "math.h"
Norma::Norma(){}//constructor
Norma::~Norma(){}//destructor
void Norma::norma(float Dn)
{
float D;
int ni;
float Dn;
[~,ni] = sizeof(D);
if (ni == 1)
Dn = (D - min(D))./(max(D)-min(D)); //convert this equation in C++
else
vmaxD = max(D); vminD = min(D);
for (int i = 1; i<= ni; i++)
{
Dn(:,i) = (D(:,i) - vminD(i))./(vmaxD(i)-vminD(i));//convert this equation in C++
}
}
• September 12th, 2011, 09:41 PM
Paul McKenzie
Re: Matlab functions need conversion
Quote:
Originally Posted by stunner08
Hi,
I need help in converting a few functions from matlab to VC++. Here's the code that I have done partially in C++ and some are still in MATLAB.
Instead of hoping that someone knows MATLAB in a C++ forum, why not tell us the name of the formula you're trying to implement?
The reason why is that it may not be as easy as translating line-by-line a math formula that works in MATLAB to C++. There are things to consider such as round-off error, possible overflow/underflow, etc.
If we knew what you were trying to do, then we won't need MATLAB syntax to figure it out.
Regards,
Paul McKenzie
• September 12th, 2011, 11:15 PM
stunner08
Re: Matlab functions need conversion
Hi,
I'm developing an API which that will be embedded to a ARM micro controller. I was have the simulation version which is in .m or Matlab file. But it seems that I have to convert it into C++ since the micro controller compiler doesn't support calling other third party libraries.
The function that I need to convert is from the equation itself which are the min(D) and max(D).
• September 13th, 2011, 08:58 AM
Lindley
Re: Matlab functions need conversion
From what I can tell, D and Dn are actually arrays or vectors even though you've defined them as a single float here. You'll need to use a loop to sequentially do an operation on each element. However, the min() and max() functions can be replaced almost directly by std::min_element() and std::max_element(). The only difference is, those return an iterator which you will need to dereference rather than the element itself.
Precompute the min and max values outside the loop, of course, for efficiency.
• September 13th, 2011, 10:01 AM
superbonzo
Re: Matlab functions need conversion
Quote:
Originally Posted by Lindley
From what I can tell, D and Dn are actually arrays or vectors even though you've defined them as a single float here. You'll need to use a loop to sequentially do an operation on each element. However, the min() and max() functions can be replaced almost directly by std::min_element() and std::max_element(). The only difference is, those return an iterator which you will need to dereference rather than the element itself.
I don't use matlab, but look at the "vminD(i)" expression; it seems D is a matrix and min(D) is a vector of min values ... perhaps, the OP should look at a ublas C++ library instead ...
BTW, note that there's also the new minmax_element to get the min and max of a sequence simultaneously ...
• September 13th, 2011, 11:26 AM
Paul McKenzie
Re: Matlab functions need conversion
Quote:
Originally Posted by stunner08
Hi,
I'm developing an API which that will be embedded to a ARM micro controller. I was have the simulation version which is in .m or Matlab file. But it seems that I have to convert it into C++ since the micro controller compiler doesn't support calling other third party libraries.
What I'm asking is for you help us out here -- tell us what all those symbols mean, or at the very least, using mathematical nomenclature, what that code is doing.
For example, what is this?
Code:
`float D;`
This is perfectly legal syntax in C++. It declares a single float variable called D. Is this what you want, or is it something else (maybe an array)?
Then this:
Code:
`[~,ni] = sizeof(D);`
What is that tilde supposed to mean? And sizeof() is legal syntax in C++.
In other words, something like "I want to have an array of floating point values called D, vMin(D) means get the minimum value in an array called D,...". etc..
Regards,
Paul McKenzie
• September 13th, 2011, 12:00 PM
Lindley
Re: Matlab functions need conversion
Quote:
Originally Posted by Paul McKenzie
Then this:
Code:
`[~,ni] = sizeof(D);`
What is that tilde supposed to mean? And sizeof() is legal syntax in C++.
The size() function in MATLAB returns the dimensions of a matrix. The tilde indicates that a particular returned value should be ignored rather than stored, for functions with multiple return values. So the above code (if it were using size() rather than sizeof()) would store the number of columns in the matrix D while ignoring the number or rows.
Of course, sizeof() does not do anything even remotely similar in C++, so it is almost certainly not what you want there. | 1,274 | 5,138 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.875 | 3 | CC-MAIN-2018-22 | latest | en | 0.906519 |
http://mathhelpforum.com/algebra/182181-quick-question-about-cubing-polynomials-print.html | 1,516,576,257,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084890893.58/warc/CC-MAIN-20180121214857-20180121234857-00576.warc.gz | 241,078,942 | 3,612 | # quick question about cubing polynomials.
• Jun 1st 2011, 10:20 AM
sara213
I just want to understand how you get the three when you have to cube three binomials together.
$(a+b)^3=a^3+b^3+3a^2b+3ab^2$
my tutor gave me this formula to follow when i come across anything I have to cube. However I'm not sure how she got the 3 there.
• Jun 1st 2011, 10:26 AM
Plato
Quote:
Originally Posted by sara213
I just want to understand how you get the three when you have to cube three binomials together.
$(a+b)^3=a^3+b^3+3a^2b+3ab^2$
have to cube. However I'm not sure how she got the 3 there.
$(a+b)(a+b)(a+b)=(a)(a)(a)+(a)(a)(b)+(a)(b)(a)+(a)( b)(b)+\cdots$
• Jun 1st 2011, 10:42 AM
sara213
Quote:
Originally Posted by Plato
$(a+b)(a+b)(a+b)=(a)(a)(a)+(a)(a)(b)+(a)(b)(a)+(a)( b)(b)+\cdots$
Thanks but that doesn't answer my question, i know that part already.
I'm talking about was the actually about the 3 in front of $a^2b$
• Jun 1st 2011, 10:47 AM
Plato
Quote:
Originally Posted by sara213
Thanks but that doesn't answer my question, i know that part already. I'm talking about was the actually about the 3 in front of $a^2b$
If you follow how to multiply then you know that $a^2b$ comes from $(a)(a)(b)+(a)(b)(a)+(b)(a)(a)=3a^2b.$
• Jun 1st 2011, 10:50 AM
topsquark
Quote:
Originally Posted by sara213
Thanks but that doesn't answer my question, i know that part already.
I'm talking about was the actually about the 3 in front of $a^2b$
His point is that there are terms (a)(a)(b), (a)(b)(a), and (b)(a)(a) that come up. Since there are three of them, you get a 3 in front of the $a^2b$.
However let's look at this differently. $(a + b)^2 = a(a + b) + b(a + b)$, right? Then you expand and simplify to get $(a + b)^2 = a^2 + 2ab + b^2$. The 2 in front of the ab appears because we have a term ab and ba.
Similarly:
$(a + b)^3 = (a + b)(a + b)^2 = (a + b)(a^2 + 2ab + b^2)$
$= a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2)$
Expand this and simplify. Here you will find, for instance, that you have a $2a^2b$ and a $ba^2$ which gives you $3a^2b$.
-Dan
• Jun 1st 2011, 10:51 AM
e^(i*pi)
The order doesn't matter in multiplication - 5*3 is the same as 3*5. This extends to letters too so $(a)(a)(b) = (a)(b)(a) = (b)(a)(a)$ and since you have three lots in your expansion it gives $3a^2b$
If "proving" it isn't all that important look up Pascal's triangle - that gives you the coefficients necessary. Or, better yet, the binomial theorem
• Jun 1st 2011, 11:04 AM
sara213
Okay got it...sorry plato, I did not understand what you are trying to say in the first post. | 904 | 2,572 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2018-05 | longest | en | 0.950495 |
http://brainly.in/question/114664 | 1,481,061,048,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698542002.53/warc/CC-MAIN-20161202170902-00187-ip-10-31-129-80.ec2.internal.warc.gz | 34,802,209 | 10,194 | # The fisherwomen are shouting out their prices to the buyers. Mini — ‘‘Come here! Come here! Take sardines at Rs 40 a kg’’. Gracy — ‘‘Never so cheap! Get sword-fish for Rs 60 a kg’’. Floramma sells prawns for Rs 150 a kg. Karuthamma sells squid for Rs 50 a kg. 1) At what price per kg did Fazila sell the kingfish? 2) Floramma has sold 10 kg prawns today. How much money did she get for that? 3) Gracy sold 6 kg sword fish. Mini has earned as much money as Gracy. How many kg of sardines did Mini sell? 4) Basheer has Rs 100. He spends one-fourth of the money on squid and another three-fourthonprawns. a. How many kilograms of squid did he buy? b. How many kilograms of prawns did he buy?
2
by kalidasCheema357 | 215 | 713 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2016-50 | latest | en | 0.964474 |
https://www.coursehero.com/file/5626035/ECE313Lecture18/ | 1,490,763,113,000,000,000 | text/html | crawl-data/CC-MAIN-2017-13/segments/1490218190181.34/warc/CC-MAIN-20170322212950-00400-ip-10-233-31-227.ec2.internal.warc.gz | 878,436,281 | 299,084 | ECE313.Lecture18
# ECE313.Lecture18 - ECE 313 Probability with Engineering Applications Independent Events Professor Dilip V Sarwate Department of Electrical and
This preview shows pages 1–8. Sign up to view the full content.
Independent Events Professor Dilip V. Sarwate Department of Electrical and Computer Engineering © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign. All Rights Reserved ECE 313 Probability with Engineering Applications
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
ECE 313 - Lecture 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 2 of 45 What is independence? Repeated independent trials The outcome of any trial of the experiment does not influence or affect the outcome of any other trial The trials are said to be physically independent Physical independence is a belief It cannot be proved that the trials are independent; we can only believe
ECE 313 - Lecture 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 3 of 45 Simple vs Compound Experiments Consider a simple experiment with sample space = {a 1 , a 2 , … } The result of repeated independent trials of this experiment is a sequence or vector of outcomes, say, (a 5 , a 2 , a 7 , a 9 , a 1 , … ) This vector is regarded as the outcome of a compound experiment with sample space × × × Simple experiments are sub experiments
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
ECE 313 - Lecture 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 4 of 45 Events on Compound Experiments The outcome of a compound experiment is a sequence or vector of outcomes of the form (a 5 , a 2 , a 7 , a 9 , a 1 , … ) The simple event A occurred on i-th sub experiment if the i-th outcome in this sequence is a member of the event A ⊂ Ω The compound event (A, B, C, A c , …) occurred if a 5 A, a 2 B, a 7 C, a 9 A c , …
ECE 313 - Lecture 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 5 of 45 Declaration of Independence The belief in independence is reflected in the assignment of probabilities to the events of the compound experiment If the trials are (believed to be) independent, then we set P(A, B, C, A c , …) = P(A)P(B)P(C)P(A c )… Both A and A c cannot occur on the same trial of the simple experiment: here they are occurring on different sub experiments
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
ECE 313 - Lecture 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 6 of 45 What is the event A? We defined the event A on the sample space of the simple experiment The occurrence of A on the i-th trial can be viewed as an event A i defined on the compound experiment Which outcomes of the compound experiment comprise A i ? All outcomes of the form ( , , … , x, , , … ) where x A and means “don’t care”
ECE 313 - Lecture 18 © 2000 Dilip V. Sarwate, University of Illinois at Urbana-Champaign, All Rights Reserved Slide 7 of 45 Independence of the A i Consider arbitrary events A i and B j defined on the compound experiment; i ≠ j B = A and even B = A c are acceptable choices as long as i ≠ j Because of the physical independence of the subexperiments, we have that P(A i B j ) = P(A i )P(B j ) More generally, P(A 1 B 2 C 3 …) = P(A 1 )P(B 2 )P(C 3 ) …
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up to access the rest of the document.
## This note was uploaded on 09/29/2009 for the course ECE 123 taught by Professor Mr.pil during the Spring '09 term at University of Iowa.
### Page1 / 47
ECE313.Lecture18 - ECE 313 Probability with Engineering Applications Independent Events Professor Dilip V Sarwate Department of Electrical and
This preview shows document pages 1 - 8. Sign up to view the full document.
View Full Document
Ask a homework question - tutors are online | 1,052 | 4,193 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.578125 | 4 | CC-MAIN-2017-13 | longest | en | 0.883947 |
https://isabelle.in.tum.de/repos/isabelle/comparison/477c05586286/src/HOL/MiniML/W.thy | 1,627,419,162,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046153491.18/warc/CC-MAIN-20210727202227-20210727232227-00578.warc.gz | 347,649,799 | 3,208 | src/HOL/MiniML/W.thy
changeset 2525 477c05586286 parent 1900 c7a869229091 child 4502 337c073de95e
equal inserted replaced
2524:dd0f298b024c 2525:477c05586286
` 1 (* Title: HOL/MiniML/W.thy`
` 1 (* Title: HOL/MiniML/W.thy`
` 2 ID: \$Id\$`
` 2 ID: \$Id\$`
` 3 Author: Dieter Nazareth and Tobias Nipkow`
` 3 Author: Wolfgang Naraschewski and Tobias Nipkow`
` 4 Copyright 1995 TU Muenchen`
` 4 Copyright 1996 TU Muenchen`
` 5 `
` 5 `
` 6 Type inference algorithm W`
` 6 Type inference algorithm W`
` 7 *)`
` 7 *)`
` 8 `
` 8 `
` `
` 9 `
` 9 W = MiniML + `
` 10 W = MiniML + `
` 10 `
` 11 `
` 11 types `
` 12 types `
` 12 result_W = "(subst * typ * nat)maybe"`
` 13 result_W = "(subst * typ * nat)option"`
` 13 `
` 14 `
` 14 (* type inference algorithm W *)`
` 15 (* type inference algorithm W *)`
` `
` 16 `
` 15 consts`
` 17 consts`
` 16 W :: [expr, typ list, nat] => result_W`
` 18 W :: [expr, ctxt, nat] => result_W`
` 17 `
` 19 `
` 18 primrec W expr`
` 20 primrec W expr`
` 19 "W (Var i) a n = (if i < length a then Ok(id_subst, nth i a, n)`
` 21 "W (Var i) A n = `
` 20 else Fail)"`
` 22 (if i < length A then Some( id_subst, `
` 21 "W (Abs e) a n = ( (s,t,m) := W e ((TVar n)#a) (Suc n);`
` 23 bound_typ_inst (%b. TVar(b+n)) (nth i A), `
` 22 Ok(s, (s n) -> t, m) )"`
` 24 n + (min_new_bound_tv (nth i A)) ) `
` 23 "W (App e1 e2) a n =`
` 25 else None)"`
` 24 ( (s1,t1,m1) := W e1 a n;`
` 26 `
` 25 (s2,t2,m2) := W e2 (\$s1 a) m1;`
` 27 "W (Abs e) A n = ( (S,t,m) := W e ((FVar n)#A) (Suc n);`
` 26 u := mgu (\$s2 t1) (t2 -> (TVar m2));`
` 28 Some( S, (S n) -> t, m) )"`
` 27 Ok( \$u o \$s2 o s1, \$u (TVar m2), Suc m2) )"`
` 29 `
` `
` 30 "W (App e1 e2) A n = ( (S1,t1,m1) := W e1 A n;`
` `
` 31 (S2,t2,m2) := W e2 (\$S1 A) m1;`
` `
` 32 U := mgu (\$S2 t1) (t2 -> (TVar m2));`
` `
` 33 Some( \$U o \$S2 o S1, U m2, Suc m2) )"`
` `
` 34 `
` `
` 35 "W (LET e1 e2) A n = ( (S1,t1,m1) := W e1 A n;`
` `
` 36 (S2,t2,m2) := W e2 ((gen (\$S1 A) t1)#(\$S1 A)) m1;`
` `
` 37 Some( \$S2 o S1, t2, m2) )"`
` 28 end`
` 38 end` | 1,073 | 2,744 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2021-31 | latest | en | 0.406139 |
https://www.teachoo.com/4812/1313/Example-31---Evaluate-definite-integral-sin2-x-dx/category/Definate-Integration-by-properties---P7/ | 1,656,869,424,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656104248623.69/warc/CC-MAIN-20220703164826-20220703194826-00361.warc.gz | 1,070,080,003 | 31,260 | Definite Integration by properties - P7
Chapter 7 Class 12 Integrals
Concept wise
Introducing your new favourite teacher - Teachoo Black, at only ₹83 per month
Transcript
Example 31 Evaluate −𝜋4𝜋4sin2𝑥 𝑑𝑥 Let f(x) = 𝑠𝑖𝑛2𝑥 f(-x) = 𝑠𝑖𝑛2−𝑥=−sin𝑥2=𝑠𝑖𝑛2𝑥 Since f(x) = f(-x) Hence, 𝑠𝑖𝑛2𝑥 is an even function −𝜋4𝜋4sin2𝑥 𝑑𝑥=0𝜋4sin2𝑥 𝑑𝑥 = 0𝜋41 − cos2 𝑥2 𝑑𝑥 = 20𝜋412−cos2𝑥2 𝑑𝑥 = 2 𝑥2−sin2𝑥2×20𝜋4 = 2 𝑥2−sin2𝑥40𝜋4 Putting Limits = 2𝜋412−sin2𝜋44 – 2 02−sin204 = 2𝜋8−sin𝜋24−0 = 2 𝜋8−14 = 𝝅𝟒−𝟏𝟐 | 951 | 687 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.84375 | 4 | CC-MAIN-2022-27 | latest | en | 0.458424 |
https://www.essayby.com/applying-analytic-techniques-to-business/ | 1,670,143,013,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710968.29/warc/CC-MAIN-20221204072040-20221204102040-00657.warc.gz | 782,748,564 | 17,995 | Posted: August 23rd, 2021
# Applying Analytic Techniques to Business
Must Review attached example, instructions, scoring guide, and previous assignments to successfully complete this assignment
In the last assessment, you were asked to prepare the first part of your analytics report by creating graphs and calculating some descriptive statistics. In this assessment, you will write your 6-8 page analytics report by interpreting those graphs and statistics, and explicitly connecting those interpretations to implications in the practical business context.
The first step in creating meaningful information from raw data is to represent the data effectively in graphical format and to calculate any required statistics. The second step is interpreting and explaining those graphs and statistics in order to apply them in the business context.
In the previous assessment, you were asked to create the first part of your analytics report by preparing graphs and calculating some descriptive statistics. In this assessment, you will complete your analytics report by interpreting those graphs and statistics, and connecting those interpretations explicitly to implications in the business context.
In business and applied analytics, oftentimes you are interested in drawing conclusions about a population of interest. However, it may not be feasible or practical to gather data on the entire population. In those cases, data is gathered from a sample or subset of the population. Analyses done on the sample are then used to draw inferences regarding the overall population; this mathematic process is referred to as inferential statistics. In this assessment, we begin discussing the topics of sampling and drawing inferences.
All the inferential statistical techniques and methods covered in this course are considered parametric techniques and require certain assumptions to be used and for the results to be reliable, many of which are assumptions about an underlying distribution. Nonparametric techniques require no assumption about underlying distributions and are often used when the assumptions of parametric techniques are not met. Although these are beyond the scope of this introductory course, they are a great option for additional reading and research.
Analytics projects often result in two distinct types of reports or summaries: one tailored to the executive level, which takes the form of a presentation, and the other, a detailed analytics report, which documents an analysis so thoroughly that another analyst can reproduce the analysis exactly. Many times, the latter type is referred to by other departments or analysts wishing to conduct a similar analysis on similar data or by the same analyst who wants to repeat the analysis on a new or revised set of data. In this assessment, you will learn the essential elements that should be included in a report at this level of detail and you will create your own analytics report addressing the business problem you have been working on.
Must Review attached example, instructions, scoring guide, and previous assignments to successfully complete this assignment
### Expert paper writers are just a few clicks away
Place an order in 3 easy steps. Takes less than 5 mins.
## Calculate the price of your order
You will get a personal manager and a discount.
We'll send you the first draft for approval by at
Total price:
\$0.00 | 607 | 3,406 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2022-49 | latest | en | 0.938928 |
https://www.answers.com/search?q=athenion-1 | 1,561,072,692,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627999291.1/warc/CC-MAIN-20190620230326-20190621012326-00339.warc.gz | 665,335,735 | 16,570 | # Results for: athenion-1
### Was harriet Tubman the first slave to escape slavery?
Yes in recent times but there were three servile wars during the Roman Empire that were lead by Eunus and Cleon [the first servile war], by Athenion and Tryphon [the second servile war] and Spartacus [the third servile war]. Full Answer
### Who led a slave revolt in ancient Rome?
Notably , the Thracian slave turned gladiator named Spartacus . [Also , 135 BC -- 132 BC in Sicily, led by Eunus, a former slave claiming to be a prophet, and Cleon and 104 BC -- 100 BC in Sicily… Full Answer
### How many ones are in 7000?
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1… Full Answer
### What is 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1?
590295810358705651711 is the number you are looking for if you are looking for a binary to decimal conversion. Full Answer
### What multiply to get 12 add to get negative 8?
12, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1. Full Answer
### What square numbers add up to23?
I assume by "square numbers" you mean perfect squares. You didn't say how many of each were allowed: 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² + 1² +… Full Answer
### What adds up to -2 and multiplies to -27?
The elements of a set consisting of -27 and twentyfive 1s. ie -27*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1*1 = -27 and -27+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 = -2 Full Answer
### What is 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1-1 plus 1 plus 1 plus 10?
1+1+1+1+1+1+1+1+1+-1+1+1+10 1+1+1+1+1+1+1+1+1=9 1+1+1+1+1+1+1+1+1+-1=8 1+1+1+1+1+1+1+1+1+-1+1+1+10=20 Full Answer
### How many ways can you make change for a quarter?
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1… Full Answer
### What is 1 1 1 equal 1?
1+1-1=1, 1x1x1=1, 1-1+1=1, 1/1/1=1, 1/1x1=1... Full Answer
### What can you add up to get 46?
1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and 1 and… Full Answer
### How many ways are there to make 16 cents?
10+5+1 10+1+1+1+1+1+1 5+5+5+1 5+5+1+1+1+1+1+1 5+1+1+1+1+1+1+1+1+1+1+1 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 In normal circulating US coins there are 6 ways Full Answer
### What numbers add up to 21 that have 1 digit?
1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 is one example. Full Answer
### How many different ways can you make 25 cents from dimes nickels and pennies?
[Legend: Each number represents the denomination (amount) of the coin.] 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 = 25 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+5 = 25 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+5+5 = 25 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+10 = 25 1+1+1+1+1+1+1+1+1+1+5+5+5 = 25 1+1+1+1+1+5+5+10 = 25 1+1+1+1+1+5+5+5+5 = 25 1+1+1+1+1+10+10 = 25 5+5+5+5+5 = 25 5+5+5+10 = 25… Full Answer
### What is 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11?
A string of ones and an eleven. There is no operator defined. Full Answer
### What does 11111x11111 equal?
11111 x 11111 = 123454321 1 1 1 1 1 x 1 1 1 1 1 _______________ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 + 1 1… Full Answer
### What is the answer of 1 1?
1+1 is 2 and 1-1 is 0 and 1*1 is 1 and 1/1 is 1 Full Answer
### How can exactly 50 coins be worth exactly 1 if the coins include at least one quarter one dime one nickel and one penny.?
1 quarter, 2 dimes, 2 nickels, and 45 pennies 1+2+2+45=50 25+10+10+5+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=100 Full Answer
### How can exactly 50 coins be worth exactly 1 if the coins include at least one quarter one dime one nickel and one penny?
1 quarter, 2 dimes, 2 nickels, and 45 pennies. 1+2+2+45=50 25+10+10+5+5+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 Full Answer
### How many swings does a pendulum does in one second?
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1… Full Answer
### What 25 numbers equal 36?
There are infinitely many possible solutions. One of them is 1 + 1 + 1 + 1 + 1 + 1 +1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1… Full Answer
### What is an Example of freshwater herbivore?
I hAvE aBsOlUtLy pOsItIvLy nO iDeA!1!1!1!1!1!1!1!1!1!1!1!1!1!1!!!!!!!!!!!!!!!!!!!!!!!!!!! Full Answer
### What is the exponent of 1 1 1 1 1 1 1 1 1?
1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 = 19 Full Answer
### What is the answer to -1 -1?
Note that -1 - 1 is expressed as -1 + (-1) or -(1 + 1). Then, -(1 + 1) = -2. Full Answer
### What is 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 11 1 1 3 4 5 0 9 8 7 61 1 1?
A series if digits Full Answer
### What equals 32 in adding?
There are infinitely many possible answers. One such is 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+16 Full Answer
### What is 1 plus 1-1?
1 + 1 - 1 = 1 Full Answer
### How do you make 15 with 1 1 1 1 1 1?
you take 1 of the 1s and put it in the tenths place. Then you add the 1+1+1+1+1 and put the answer(5) in the ones place! that is how you get 15 out of 1 1 1 1 1 1. Full Answer
### What is 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 11-9999999?
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 11 - 9,999,999 = - 9,999,978 Full Answer
### What is the truth-table of IC 74147?
x1 x2 x3 x4 x5 x6 x7 x8 x9 y3 y2 y1 y0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 x… Full Answer
### What is a 2 step equation that equals 1?
# 1x+-1=0 # +1 | +1 # 1x = 1 # 1/1=1 # 1x+-1=0 # +1 | +1 # 1x = 1 # 1/1=1 Full Answer
### What multiplies to be -20 and adds to be -3?
(-20, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) multiply to -20 and their sum is -3. Full Answer | 3,028 | 5,819 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2019-26 | latest | en | 0.654638 |
https://hackerranksolution.in/shortestbridgegoogle/ | 1,723,216,778,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640767846.53/warc/CC-MAIN-20240809142005-20240809172005-00456.warc.gz | 228,147,082 | 13,623 | # Shortest Bridge - Google Top Interview Questions
### Problem Statement :
```Given a two-dimensional list of integers matrix containing 0s and 1s, 0 represents water and 1 represents land.
An island is a group of connecting 1s in 4 directions that are either surrounded by 0s or by the edges.
Find the shortest bridge that connects two islands.
It is guaranteed that there are two and only two islands.
Constraints
n, m ≤ 250 where n and m are the number of rows and columns in matrix
Example 1
Input
matrix = [
[0, 1],
[1, 0]
]
Output
1
Explanation
Either of the two water elements can be used as the bridge.
Example 2
Input
matrix = [
[1, 0, 0],
[0, 0, 0],
[0, 0, 1]
]
Output
3
Explanation
There's six shortest bridges such as [(0, 1), (0, 2), (1, 2)] or [(1, 0), (2, 0), (2, 1)]```
### Solution :
``` ```Solution in C++ :
void dfs(vector<vector<int>> &matrix, int x, int y, queue<pair<int, int>> &q) {
// Invalid position or cell-state
if (x < 0 || y < 0 || x >= matrix.size() || y >= matrix[0].size() || matrix[x][y] == 2 ||
matrix[x][y] < 0) {
return;
}
// If the cell is a water cell adjacent to the island
// mark it and place it into the bfs queue and return
if (matrix[x][y] == 0) {
matrix[x][y] = -1;
q.emplace(make_pair(x, y));
return;
}
// Mark island cell with different color
if (matrix[x][y] == 1) {
matrix[x][y] = 2;
}
// run dfs to mark neighbors
vector<vector<int>> dirs{{-1, 0}, {0, -1}, {1, 0}, {0, 1}};
for (auto &dir : dirs) {
int x1 = x + dir[0];
int y1 = y + dir[1];
dfs(matrix, x1, y1, q);
}
}
int solve(vector<vector<int>> &matrix) {
int m = matrix.size(), n = m == 0 ? 0 : matrix[0].size();
// Find the first land cell (can be any of the two islands)
// and run dfs to mark the island with a different color
bool landFound = false;
queue<pair<int, int>> q;
for (int x = 0; x < m && !landFound; ++x) {
for (int y = 0; y < n && !landFound; ++y) {
if (matrix[x][y] == 1) {
landFound = true;
dfs(matrix, x, y, q);
}
}
}
// Means that there's no island at all
// so return error
if (q.empty()) {
return -1;
}
// Run bfs from the edges of the marked island
vector<vector<int>> dirs{{-1, 0}, {0, -1}, {1, 0}, {0, 1}};
while (!q.empty()) {
pair<int, int> curr = q.front();
q.pop();
for (auto &dir : dirs) {
int x = curr.first + dir[0];
int y = curr.second + dir[1];
if (x >= 0 && y >= 0 && x < m && y < n) {
// If we have found the second island
// return the distance
// distance is -ve of the current cell
// since we've marked water cells with
// increasing negative values
if (matrix[x][y] == 1) {
return 0 - matrix[curr.first][curr.second];
}
// If it is a water cell, update its distance
// and then add it to the queue
else if (matrix[x][y] == 0) {
matrix[x][y] = matrix[curr.first][curr.second] - 1;
q.emplace(make_pair(x, y));
}
}
}
}
// We didn't find the second island so return error
return -1;
}```
```
``` ```Solution in Java :
import java.util.*;
class Solution {
private static final int[] DIRS = {-1, 0, 1, 0, -1};
private static final int NEWCOLOR = 2;
public int solve(int[][] grid) {
final int R = grid.length, C = grid[0].length;
Queue<int[]> q = new ArrayDeque<>();
findIsland(grid, R, C, q);
for (int step = 0; q.isEmpty() == false; step++) {
for (int b = q.size(); b != 0; b--) {
int[] now = q.poll();
for (int i = 0; i != 4; i++) {
final int nr = now[0] + DIRS[i], nc = now[1] + DIRS[i + 1];
if (nr == -1 || nr == R || nc == -1 || nc == C || grid[nr][nc] == NEWCOLOR)
continue;
if (grid[nr][nc] == 1)
return step;
grid[nr][nc] = NEWCOLOR;
q.offer(new int[] {nr, nc});
}
}
}
return -1;
}
private void findIsland(int[][] grid, final int R, final int C, Queue<int[]> q) {
for (int r = 0; r != R; r++)
for (int c = 0; c != C; c++)
if (grid[r][c] == 1) {
drown(grid, R, C, r, c, NEWCOLOR, q);
return;
}
}
private void drown(int[][] grid, int R, int C, int r, int c, int color, Queue<int[]> q) {
if (r == R || c == C || r == -1 || c == -1 || grid[r][c] != 1)
return;
grid[r][c] = color;
q.offer(new int[] {r, c});
for (int i = 0; i != 4; i++) drown(grid, R, C, r + DIRS[i], c + DIRS[i + 1], color, q);
}
}```
```
``` ```Solution in Python :
class Solution:
def solve(self, matrix):
cnt = 0
rows = len(matrix)
columns = len(matrix[0])
visited = {}
for i in range(2):
flag = True
while flag:
column = cnt % columns
row = cnt // columns
if (row, column) not in visited:
visited[(row, column)] = 0
if matrix[row][column] == 1:
flag = False
matches = {(row, column): 0}
self.find_touching(matrix, visited, matches, row, column, rows, columns)
cnt += 1
if i == 0:
island_one = self.find_outline(matches)
print(island_one)
else:
island_two = self.find_outline(matches)
print(island_two)
best = -1
for pos1 in island_one:
for pos2 in island_two:
needed = abs(pos1[0] - pos2[0]) + abs(pos1[1] - pos2[1]) - 1
if best == -1 or needed < best:
best = needed
return best
def find_touching(self, matrix, visited, matches, row, column, rows, columns):
moves = [(-1, 0), (0, -1), (0, 1), (1, 0)]
for move in moves:
new_row = row + move[0]
new_column = column + move[1]
if (new_row, new_column) not in visited:
if new_row >= 0 and new_row < rows:
if new_column >= 0 and new_column < columns:
visited[(new_row, new_column)] = 0
if matrix[new_row][new_column] == 1:
matches[(new_row, new_column)] = 0
self.find_touching(
matrix, visited, matches, new_row, new_column, rows, columns
)
return matrix
def find_outline(self, matches):
keys = list(matches.keys())
for item in keys:
flag = True
for direction in [(-1, 0), (1, 0), (0, -1), (0, 1)]:
if (item[0] + direction[0], item[1] + direction[1]) not in matches:
flag = False
break
if flag:
matches[item] = -1
return [x for x in matches if matches[x] == 0]```
```
## Direct Connections
Enter-View ( EV ) is a linear, street-like country. By linear, we mean all the cities of the country are placed on a single straight line - the x -axis. Thus every city's position can be defined by a single coordinate, xi, the distance from the left borderline of the country. You can treat all cities as single points. Unfortunately, the dictator of telecommunication of EV (Mr. S. Treat Jr.) do
## Subsequence Weighting
A subsequence of a sequence is a sequence which is obtained by deleting zero or more elements from the sequence. You are given a sequence A in which every element is a pair of integers i.e A = [(a1, w1), (a2, w2),..., (aN, wN)]. For a subseqence B = [(b1, v1), (b2, v2), ...., (bM, vM)] of the given sequence : We call it increasing if for every i (1 <= i < M ) , bi < bi+1. Weight(B) =
Meera teaches a class of n students, and every day in her classroom is an adventure. Today is drawing day! The students are sitting around a round table, and they are numbered from 1 to n in the clockwise direction. This means that the students are numbered 1, 2, 3, . . . , n-1, n, and students 1 and n are sitting next to each other. After letting the students draw for a certain period of ti
## Mr. X and His Shots
A cricket match is going to be held. The field is represented by a 1D plane. A cricketer, Mr. X has N favorite shots. Each shot has a particular range. The range of the ith shot is from Ai to Bi. That means his favorite shot can be anywhere in this range. Each player on the opposite team can field only in a particular range. Player i can field from Ci to Di. You are given the N favorite shots of M
## Jim and the Skyscrapers
Jim has invented a new flying object called HZ42. HZ42 is like a broom and can only fly horizontally, independent of the environment. One day, Jim started his flight from Dubai's highest skyscraper, traveled some distance and landed on another skyscraper of same height! So much fun! But unfortunately, new skyscrapers have been built recently. Let us describe the problem in one dimensional space
## Palindromic Subsets
Consider a lowercase English alphabetic letter character denoted by c. A shift operation on some c turns it into the next letter in the alphabet. For example, and ,shift(a) = b , shift(e) = f, shift(z) = a . Given a zero-indexed string, s, of n lowercase letters, perform q queries on s where each query takes one of the following two forms: 1 i j t: All letters in the inclusive range from i t | 2,517 | 8,272 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.65625 | 4 | CC-MAIN-2024-33 | latest | en | 0.665032 |
https://www.theprintableprincess.com/product/build-the-number-using-mini-erasers-kindergarten-math-numbers-to-10/ | 1,713,601,511,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296817491.77/warc/CC-MAIN-20240420060257-20240420090257-00775.warc.gz | 967,779,578 | 47,614 | # Build the Number Using Mini Erasers {Kindergarten Math: Numbers to 10}
\$4.50
Total Pages: 40
File Size: 34 MB
Also available in a BUNDLE:
Original price was: \$49.00.Current price is: \$29.00.
## Description
Building a solid number sense foundation is critical for a students' long-term math success – and it starts at an early age. It is important for our students to develop a deep understanding of individual numbers, their quantities, and how they relate to other numbers. Number sense is developed through lots of practice and exposure, which is why it's important for us to have lots of activities to help our students strength their skills.
Build the Number {Using Mini Erasers} is an activity that will make teaching {and learning} numbers fun, engaging, and hands-on! Plus it's pretty easy to prep! Just print, laminate, cut along the straight lines, then add some mini erasers. If you don't have mini erasers, plastic cubes or other small classroom manipulatives would work as well.
There are four different types of activities included in this resource. Students will use mini erasers to form numbers. This is a fun way to practice counting and number recognition while strengthening fine motor skills.
Here's what's included:
•Build the Number 1: This will provide students will guided practice to form the numbers using mini erasers. There are two types of mats included. On one mat students will make the number on top of guided directional arrows. At the bottom of the mat are pictures showing the number quantity. The second type of mat has the same numbers with directional arrows, but at the bottom students can practice writing the number using a dry erase marker.
•Build the Number 2: Students will select a picture card and identify the number represented. Students will form that number independently on the activity mat. A number formation reference pages are included.
•Build the Number 3 {What's the Number?}: Students will determine the secret number from a series of three pictures. Students will form the number that matches the pictures. A number formation reference page is included.
•Build the Number 4: This activity is perfect for introducing numbers in small group guided math. Small mini erasers work best with this activity. Students will use the mini erasers to form the number on the mat.
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
For more Mini Eraser Activities check out:
• Build a Letter Using Mini Eraser {Alphabet Activities}
## Reviews
There are no reviews yet.
Only logged in customers who have purchased this product may leave a review. | 617 | 2,642 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.40625 | 3 | CC-MAIN-2024-18 | longest | en | 0.904128 |
https://www.smarthelping.com/2022/09/how-to-calculate-cac-for-saas-and.html | 1,721,712,514,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763518014.29/warc/CC-MAIN-20240723041947-20240723071947-00669.warc.gz | 861,940,803 | 17,590 | ## Places Of Interest
### How to Calculate CaC for SaaS and Considerations
This is actually one of the more straight forward performance metrics that is calculated in the SaaS industry. However, it is really important to understand if you are scaling up and is involved in a few other important metrics. So, it is at the heart of SaaS business analysis.
Check out full SaaS financial models here (and recurring revenue business models).
In general CaC means customer acquisition costs. That pertains to all the direct costs related to acquiring new customers. This could include sales teams, marketing activities, ad spend, SEO service spend, and any other costs that are specifically to acquire more customers directly.
Your entire organization's goal is likely to acquire more customers if you are ramping up a SaaS business (except maybe your legal team and accounting), but we are limiting costs to essentially anything that can be classified as Sales and Marketing.
So, to do the calculation, you simply divide the total Sales and Marketing costs in a period by the total customers added in a period. It is very likely that some of your advertising and marketing activities may not yield new customers for months or even years. This is why it is just an average calculation because you can't tie every single new customer directly to the source cost of the acquisition.
Because of the above reality, sometimes the context of your CaC calculation is really important. What was going on during the period measured, how long is the period measured, and so it may be hard to quantify until you have some data behind the number and years of operations.
Calculation: (Total Sales and Marketing Costs in Period) / (Total New Customers Acquired in Period)
It is really important to understand how much value you are getting from customers over the time they are active and compare that to the average cost to acquire. If the lifetime value of a customer is less than the cost to acquire a customer, the ramping is not sustainable and your SaaS business will lose money.
You can also look at the number of months it takes a customer to pay back the cost to acquire. If that number of months is longer than the average customer retention, you have a problem.
Tracking CaC over time is healthy and you can do it on a monthly, quarterly, or annual basis. Comparative analysis is a good way to see how your efforts of customer retention are going and if the unit economics make sense. | 491 | 2,487 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2024-30 | latest | en | 0.970908 |
https://www.slideshare.net/SiteriCR2/2-q09-presentation-13314462 | 1,493,175,389,000,000,000 | text/html | crawl-data/CC-MAIN-2017-17/segments/1492917121121.5/warc/CC-MAIN-20170423031201-00466-ip-10-145-167-34.ec2.internal.warc.gz | 985,380,106 | 37,874 | Upcoming SlideShare
×
# 2 q09 presentation
163 views
Published on
0 Likes
Statistics
Notes
• Full Name
Comment goes here.
Are you sure you want to Yes No
• Be the first to comment
• Be the first to like this
Views
Total views
163
On SlideShare
0
From Embeds
0
Number of Embeds
6
Actions
Shares
0
1
0
Likes
0
Embeds 0
No embeds
No notes for slide
### 2 q09 presentation
1. 1. 1 % 3 "4 % % , 0 # % + , " , \$ % # # \$ ! & ( )!* - , + , 1 , 1 , % 2 , % . !/ " , " # # \$ # \$ +, \$ , 8 \$ , # \$ % & 5 - , , 3 2 , , 5 + , 2 9:9 -+ , / ; , , & # 6 )9/ 3 2 , ):* - ; , > # ) & ? , % " , %,7 < , " + # + , \$ % , % - , = " 2
2. 2. %< , !!*7!/ , \$ ,> # &? , @ \$7 % !! 3 , 4# \$ &? , *! \$ 4 4 % A : ) @ B % ! , 8 " 5 % + " C &? % # % % 1 , -1 , 2 D< + % " 5 \$ # , -+ , % %E + , ! \$ 3
3. 3. ,& + ; ,& + A B F 9 9:F ) F*9: )!F*! 9F9) G!/ )G!/ :G!/ G! G! 4
4. 4. < + < % !" #\$ % & \$ / H \$ H \$ :*H \$> ) H \$ ) H \$I ) H \$ 9H \$. H \$& !H \$@ !H \$J H \$< H \$1 H \$E )H \$( H !! ! % () * \$ H 4 H \$G 4 !H \$ 4 !H 5
5. 5. 5 \$3 < , * 0! - 1 , 3- 1 * !+ ( ,% & ( -%) . / (% 4 , .2 2 .2 2 @ & K && & & E & L A @B % / F /) /*-! /!- )H 3 < 2 + A@ B M A B *: ! ))-! /-: / H3 < 2 % A N O B F@F A B / ) ) -! *-9 / H 1 P , A B / : ! ):-! - / H 2 8 && @ A @B 9 ! 9-! !-! /!H F O K1 & E & L A @B % 9! F ) /!-! :-! /!H @ A @B )* 9!* !-! /:-! *!H F O K1 && E & L % A @B 9! F /!-! :-! /!H 8F Q% && = L A @B F9 /*-! : -) /!H 8F Q% && & = L A @B * : /:-! *- /!H 8F Q% &2 = L A @B / : /:-! *- /!H R , @ A @B 9 )/ )!-! -! *!H E & L A @B % * )!F /) F /!-! F /!-! !!H 0! 5 5 655 56 7 56 7 54 6
6. 6. G! . %, %, , :* AS 9H G! B T , )9 AS/ HG! B #\$ , \$ % E * AS) H G! B- )- AS * H G! B- < !-!/ A !-!) G! B % #\$ %, ,!( ) " : -* - :-! :- 9-9 ):- )-! - - :G!/ G! G! 7
7. 7. & \$ 1 " 2 U 8 (!& (* ,!( ) " ,!( ) " 8 ( ! &3 (*- %!" ; () , 3 8 (!& (* 9 % ( !: ((% , 3 ! : ! ; () %( ,!( ) " . #(" !:- %!" - %!" - %!" !! !-! ))-: - -9 )9-*H - !-! G!* -9 -: ) -9 9 -: )*- H )- *- G!* 9 -: !-: /-9 9:-) 9-:H /-/ -* )G!* 9:-) )- ) -: /9-! *- H !-: -! :G!* /9-! *!-) 9 - - )- H ) -) - G!/ - :*- *- -/ - H 9- -) G!/ -/ ) -) ::-: !)-* : -9H )-* 9!- )G!/ !)-* )- *- 9!-! -)H ::- )-: :G!/ 9!-! !-! - )*-9 9-!H !-! - G! )*-9 !-! - /-: /-!H !-! - G! /-: !-! ):- /)-/ 9-/H !-! ):-( A ; , BV W , 7A , # % % , S , 2 , ,\$ , % BX < = 4 !: !" >#\$ % & 8
8. 8. & \$ 1 " 2 8 (!& (* 2 ? 1 ,!( )% !( 8 (!& (* 2 ? 1 * -% ) )H *H :H Y Z )!" ) H ) "4 )9!" )9 " 4 9!!" H /)H 4 !:% ( ! & (* %% ! :% ()% ( ( 9
9. 9. 5 > \$4 , % & ) " * / (% ! " (" 8 ( ! & (* F 9!! F :* F !!! F * * F 9!! F !!! //) /H 9 *)H 9!! / H :H * 4 H )G! :G! . ! . ! . & \$+ !@ ) ! )!() " " % ( A% ! - 1 !: .5 5 (" , > . = 4 !: 7 (% ! " % * "%( "& ! " 10
10. 10. I . %, %, G! B+ % ( * ( . C ! - !:% . D C ! - !:% 2 %(/!F !!! * F/)* F !!! /-/H -:H !-!H*!F !!! :F !!! )F /-!H -!H!F !!! F !!! 9 F/ :-!H9!F !!! F*) !F !!! -!H:!F !!! /F !!! !-!H)!F !!! F !!! /-!H -!H!F !!! :F !!! :-!H!F !!! F !!! -!H 4 4 !-!H G! G! G! G! #98 0 . D #98 0 2 % ( - !:% . D - !:% 2 % ( F !!! !-/H -!H :F !!! -!H )F99/F !!! )F 9!! !-!H 9-!H*F !!! *F*/ !!! )F :- H F !!! /-!H :-!H F 9!!9F !!! -!H F !!! -9H )-!H:F !!! :-:H F 9!! F/)F !!! :-!H -!H F / F !!! F !!! -!H -!H F !!! 9!! 4 !-!H 4 !-!H G!/ G! G!/ G! 11
11. 11. # % G! # % /9 # % A) - H % B ( # , # 9 * G! - +, , 9 , 4* ( (" ! )! (% " E . = = # % 9)F ) /:F *: ) F !*: 9F / Y # % * F: )F ! ! F ! ) ) F ) # % * F * F *)! !:F : * 99F 9 1 % # % )!-)H ) - H ))-*H /- H 12
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
13. 13. & Investors Relations Rogério Furtado CFO and IR Officer Daniel Grozdea Finance and IRManager www.cr2.com.br/ir phone: (21) 3095-4600 (21) 3031-4600 14
14. 14. DisclaimerThis presentation contains certain statements that are neither reported financial results or other historical information. Theyare forward-looking statements.Because these forward-looking statements are subject to risks and uncertainties, actual future results may differ materiallyfrom those expressed in or implied by the statements. Many of these risks and uncertainties relate to factors that arebeyond CR2’s ability to control or estimate precisely, such as future market conditions, currency fluctuations, the behavior ofother market participants, the actions of governmental regulators, the Company ability to continue to obtain sufficient sfinancing to meet its liquidity needs; and changes in the political, social and regulatory framework in which the Companyoperates or in economic or technological trends or conditions, inflation and consumer confidence, on a global, regional ornational basis.Readers are cautioned not to place undue reliance on these forward-looking statements, which speak only as of the date ofthis document. CR2 does not undertake any obligation to publicly release any revisions to these forward looking statementsto reflect events or circumstances after the date of this presentation. 15 | 2,832 | 5,739 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.390625 | 3 | CC-MAIN-2017-17 | latest | en | 0.122412 |
https://www.coursehero.com/file/6295762/Act-11/ | 1,493,271,056,000,000,000 | text/html | crawl-data/CC-MAIN-2017-17/segments/1492917121869.65/warc/CC-MAIN-20170423031201-00124-ip-10-145-167-34.ec2.internal.warc.gz | 899,921,822 | 27,296 | # Act_11 - Name _ Section _ Period 11 Activity Sheet:...
This preview shows pages 1–3. Sign up to view the full content.
41 Name ___________________________ Section ____________________ Period 11 Activity Sheet: Electric Current Activity 11.1: How Can Electric Charge Do Work? Your instructor will demonstrate a Wimshurst machine, which separates electric charge. a) Describe what happens to the hanging soda cans as electric charge from the Wimshurst machine flows onto the cans. Explain how the separated charge does work on the cans. b) Why do you see sparks between the cans or between the balls of the Wimshurst machine? Activity 11.2: What is an Electric Circuit? a) Lighting a bulb Arrange one battery, one connecting wire, and one small light bulb (not in a tray), so that the bulb lights. You may need to try several different arrangements. 1) Draw a diagram showing your arrangement of the battery, wire and bulb that worked. 2) Explain why did this arrangement worked and other arrangements you tried did not. b) Circuit in a flashlight In part (a), you found how to light a bulb with a battery and one wire. A flashlight uses the same principle. Examine a flashlight to find the path that the current takes. 1) Draw a diagram of the flashlight showing the path the current follows. 2) If a flashlight does not work, it must have an open circuit. List problems that could cause an open circuit in a flashlight. ( Note : please put the flashlight back together.) c) Plumbing Analogies Your instructor will demonstrate plumbing analogies for circuits. Fill in the electrical concepts represented by the plumbing display. 1) Water _______________ 4) Plastic tubes __________________ 2) Flowing water _______________ 5) Narrow plastic tubes _______________ 3) Water pressure _______________ 6) Pump __________________ d) Group Discussion Question: Are charges “used up” to make a bulb light? If not, what happens to make it light?
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
42 Activity 11.3: What is Electric Current? a)
This is the end of the preview. Sign up to access the rest of the document.
## This note was uploaded on 06/11/2011 for the course PHYSICS 103 taught by Professor Staff during the Spring '10 term at Ohio State.
### Page1 / 4
Act_11 - Name _ Section _ Period 11 Activity Sheet:...
This preview shows document pages 1 - 3. Sign up to view the full document.
View Full Document
Ask a homework question - tutors are online | 560 | 2,514 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.609375 | 4 | CC-MAIN-2017-17 | longest | en | 0.861753 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.