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The quality of water and nutrient solution used in controlled environment agriculture (CEA) production systems, such as greenhouses and vertical farms, is one of the most important factors that affect plant health and yield. Growers monitor water and nutrient solution quality by sending samples for analysis to determine the levels of nutrients and salts. They also use sensors to monitor pH and electrical conductivity (EC) regularly to determine necessary adjustments for the nutrient solution. Growers may also analyze the microbiome, the genetic material of all the bacteria, fungi, and viruses that live in their water and nutrient solution, to evaluate levels of harmful pathogens, such as Pythium and Phytophthora species that cause root rot.
Most greenhouses and vertical farms use recirculated nutrient solution in order to reduce their water use and wastewater. This practice is not only friendly to the environment, but it reduces operating costs as well. Challenges with using recirculated nutrient solution include build-up of salts, pathogens, and biofilm in the circulation system and the root zone of the plants. Without proper management, these conditions can lead to unhealthy plants that are more prone to disease. To reduce salt accumulation in recirculated nutrient solution, its EC and pH need to be monitored and adjusted regularly. Treatments to reduce pathogen and biofilm levels include ozone, ultraviolet (UV) light, pasteurization, or a combination of these systems. However, these treatments kill not only pathogenic microbes but might kill beneficial microbes as well. Some growers prefer not to sterilize their nutrient solutions to promote beneficial microbes, though this may come at a cost of higher levels of pathogens and subsequent disease development. | https://www.verticalfarmdaily.com/article/9248563/can-on-nutrient-solution-analysis-projects-for-greenhouses-and-vertical-farms/ |
Music Theory with Richard Bleil (and, no, I’m not an expert)
Just a moment ago (as of writing this blog so roughly a week ago today) I put a challenge on my social media page for anybody who has never played piano or had given up on it. I told them to forget any specific song, forget the sheet music and forget the black keys. Just sit at a keyboard and play from the heart using only the white keys, and I’ll bet that anything they play will sound good. Then I told them that if they want to know why, they should text me after the experiment. If YOU want to play along, stop reading for now, and try it. The blog will be here afterwards if you want.
When I learn something, I need to know why I’m doing what I’m doing. I tried to learn sheet music (teaching myself), but never learned music theory, without which piano (or any instrument) simply becomes an exercise in memorization. I’m not good at memorization.
So the first thing that we have to realize about music theory is that keys, a set of notes that are played (like the C Major key, which is what you are playing with only the white keys) are simply notes that sound pleasing to the ear. Originally, they were constructed by trial and error. People just started playing, but a pattern eventually emerged. To understand the pattern, we have to understand the concept of “steps”.
Looking at the keyboard, a full step is always two keys (including the black keys) ahead of the previous. A half-step is the adjacent key. We’ll call a whole step (two keys) “W” and a half step (adjacent key) “H”. Pretty clever, eh? Looking at the keyboard, you’ll see sets of black keys, set in two, then three, then two, then three and so forth. Looking at the two black keys, the white key immediately to the left of the first in that pair is called the “C” note. The white notes then proceed alphabetically up (to the right) of the keyboard; D, E, F, G, A, B. The black keys are the sharps (to the right of a note) and flats (to the left of the note). We’ll not really get into that today other than to acknowledge that they exist and for use in understanding steps.
So, once you’ve identified the C note, you’ll notice that the D note is a full step (two keys, the black and then the D) ahead of C. Similarly, the E note is a full step ahead of D, but the F note is only a half-step ahead of E. G is a full note ahead of E, then A is a full step, and B is a full note. One more half step brings us back to C, but higher on the scale (we call that the next octave).
Now comes the music theory. Starting at one of the C’s, playing only the white note (and not counting that C), you’ll notice that a pattern emerges for the steps. If we look at the C Major, the pattern of steps that emerges is WWHWWWH. Music theory tells us that if we play this step pattern, it will sound pleasing to our ears. C Major (we’ll talk about major and minor momentarily) is the easiest key to play because it simply means playing only the white keys, but understanding this pattern, notice that we can start on any key, white or black, and play in a key provided we remember WWHWWWH. This, by the way, is a “C” key because it starts with the note of C; C, D, E, F, G, A, B. Starting on any other key, say the E key, means you start on E, but don’t forget that pattern. Looking at a keyboard, you’ll see that E Major is E, F#, G#, A, B, C#, D#, and back to E. It’s a more challenging chord to play because you have to remember which black keys to play, and which white keys to avoid, but the theory is the same. That’s why I suggest starting simply with the C major key.
Now, major keys are often described as “happy” keys. Christmas songs are probably mostly in major keys, but the Eagles wrote a very sad holiday song called “Please Come Home for Christmas.” Playing a bluesy song in a major chord is like sending a “Dear John” letter with a “heart” stamp. It just doesn’t fit. So, instead, you’d want to play in a minor key, which are usually described as mournful and sad. But major and minor keys are very closely related with only one minor difference in the pattern. Here, we switch one, just one, whole step with a half step. A minor key, then, would be WWWHWWWH. This means that C minor would be C, D, E, F#, G, A, B and back to C. Notice that the only difference in the C key is that F# that was an F in the major cord.
So there you go. It’s a matter of steps, stumbled upon by trial and error. What sounded good (the steps) and what sounded happy versus sad (major and minor) becomes the foundation of music theory. Happy playing!!! | https://bleilbanter.blog/2022/02/08/piano-2-8-22/ |
Supplies:
Directions: In a small sauce pot melt butter over medium heat. Then, add water, sugar, and several drops of food coloring. Mix well and cook until mixture comes to a boil. Simmer for three minutes. In a large bowl pour the mixture over the popcorn and stir until popcorn is coated. Cool popcorn for 10 minutes. Eat and enjoy!
St. Patrick’s Day Surprise Box
3/12/2014
Fruit Loop Rainbow
3/10/2014
Supplies needed; Cotton batting, fruit loops, and glue
Directions;
Assist children in making a fruit loop rainbow. Use small amounts of glue to hold fruit loops in place, if too much glue is used the fruit loops will begin to melt. For younger children, consider drawing colored lines to show them where to place their fruit loops. Next glue cotton batting at the ends of the rainbow to finish the ends.
Rainbow Paper Chain
3/10/2014
Supplies Needed; Rainbow colored construction paper, scissors and a stapler
Directions;
Cut strips of paper about 2" wide and 6" long. We used 5 of each color. Create a loop, overlap the ends, and staple over the ends. Thread each new paper through the previous loop, until finished.
Rainbow and Pot of Gold Painting
3/10/2014
Supplies Needed; Orange construction paper, rainbow colored paints, cotton balls, glue and scissors
Directions;
Using your construction paper, cut out a pot shape, and paste it to the bottom corner of your paper. In the opposite corner of your paper paste cotton balls. Assist the children in using their fingers to paint rainbow colored lines from the pot to the cloud. We chose to add a sun to shine on the rainbow.
Pot of Gold Hand Print
3/10/2014
Supplies Needed; Rainbow colors and brown paint, white paper, colored pens, and a paint brush.
Directions;
Paint the child's palm brown, and the fingers colors of the rainbow. Instruct the child to put their fingers close together. Stamp the child's hand on the paper and pull the hand directly upwards to make a clean and print. Allow the paint to dry, then add details. | https://www.123playandlearn.com/saint-patricks-day |
Combined Effect of Nitrogen Fertilization and Seeding Rate on Regional Wheat Yield and Yield Components in Bangladesh
Asian Research Journal of Agriculture,
Page 77-85
DOI: 10.9734/arja/2022/v15i430169
Abstract
Seed rate and nitrogen fertilization are two main critical factors that affect wheat (Triticum aestivum L.) growth and yield. But a little knowledge on the interaction effects of these two factors in wheat cultivation under course, strong acidic nutrient deficit soil condition aimed us to conduct this research. We grew wheat at the field laboratory of the department of Agronomy and Haor Agriculture, Sylhet Agricultural University, Sylhet-3100, Bangladesh, during crop growing period (November-March) of wheat 2021-2022 with split plot design assigning seed rate (100, 120, 140 & 160 kg ha-1) in main plot and nitrogen fertilizer (0, 120, 140, 160 & 180 kg ha-1) in sub plot. The maximum grain yield (3.0 t ha-1) was obtained at 140 kg ha-1 and 160 kg ha-1 seeding rate and nitrogen fertilization respectively. Whereas, the minimum grain yield (1.5 t ha-1) was recorded in control treatment (100 kg ha-1 seed with zero nitrogen). The highest value of the entire yield contributing parameters i.e. effective tillers plant-1 (2.9), spike length (14.7 cm), spikelet spike-1 (17.0), florets spike-1 (52.3), grain spike-1 (51.0) and 1000 seed weight (42.6 g) were recorded at 140 kg ha-1 and 160 kg ha-1 seeding rate and nitrogen fertilization respectively and the values were increased with the increase of seeding rate and nitrogen fertilization rate upto 140 kg ha-1 and 160 kg ha-1 respectively then declined. The growth parameter i.e. plant height (114.1 cm) increased with the increase of seeding rate 160 kg ha-1 and nitrogen fertilization upto 160 kg ha-1 and the minimum value (54.1 cm) was recorded at control treatment.
How to Cite
References
Available: ttps://doi.org/10.4060/cb9427en
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OEM Off-Highway magazine has been the resource for engineers and product development team members at mobile heavy-duty on- and off-road equipment manufacturers for over 30 years. Our in-depth reporting on trends, technology developments, engineering innovations and new product releases keep our readers informed of the latest information in a dynamic and rapidly changing global industry.
Improved Data Collection Benefits Autonomous Vehicle Development
In this week’s episode of OEM Industry Update, editor Sara Jensen speaks with Jared Aho, Director of Marketing, Transportation at NI, about the company’s new collaboration with Seagate which aims to improve testing and development of autonomous vehicles. NI’s high performance in-vehicle data logging systems will be brought together with Seagate’s data transfer and edge storage services to help original equipment manufacturers (OEMs) and suppliers efficiently leverage data to ensure the safety and reliability of autonomous vehicles.
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OEM Off-Highway magazine has been the resource for engineers and product development team members at mobile heavy-duty on- and off-road equipment manufacturers for over 30 years. Our in-depth reporting on trends, technology developments, engineering innovations and new product releases keep our readers informed of the latest information in a dynamic and rapidly changing global industry.
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9 and 8 being the given differences, we have only to multiply 9 by 8, and 8 by 9 to obtain equal products.
This process constitutes the basis of the rule in what is termed Alligation Alternate, which rule, where the proportions of two ingredients only are concerned, may be stated briefly as follows :--Take the difference between the rate of each ingredient and the mean rate, and place it opposite to the other rate. The operation is usually conducted as follows:Mean Rates of
Proportion of
Ingredients.
1 7d. .................... 8
...
9
Proof:
9 at 24d. = 216d. 8 at 7d. = 56d. 17
32722.(16d. mean rate. Having obtained in this manner, one set of proportional quantities, which are shown to fulfil the conditions of the question, we may deduce from them an infinite variety of other proportions, either whole or fractional, by simply multiplying or dividing both quantities by any number whatever : e. g.,
9 multiplied by 2, 3, 4, 5, 1, }, o, &c. = 18, 27, 36, 45, 41, 14, L. &c.
8 , by 2, 3, 4, 5, 1, 1, I, &c. = 16, 24, 32, 40, 4, 11, 4. &c. It will be seen on trial that any two of these proportions will produce a compound of the required value. The rule of Alligation, therefore, furnishes one very ready solution of the question, from which other solutions may be had at pleasure.
When more than two ingredients are proposed, an extension of the same method is still applicable. For if we select any two of the rates, one greater and one less than the mean, and proceed as above indicated with these alone, we shall obtain proportions giving the mean value required, apart from the influence of the other ingredients. And if we repeat the operation on two other rates respectively greater and less than the mean, until all are exhausted, we shall thus form, as it were, so many independent additional compounds each of the necessary value, the sum or admixture of which together cannot affect the average of any single pair of ingredients. Should an odd number of rates be given, or should those which are greater than the mean exceed in number those which are less, or the reverse, we may unite the game rate several times with others of the group, using the sum of the alternate differences thus set out opposite to each rate. The reason of this will best appear from an example.
Example (2.) Form a spirituous mixture of the strength of 24 per cent., U. P., from ingredients respectively 18, 22, 28, 33, and 54 per cent., U. P.
Diffs.
Diffs. , 18. . 30 +4. . 34
4 + 9 Mean
22 . . . . . 9 30 + 4 = 34 4 + 30 = 34 Strength. 28
| 33 . . . . . 2
57
9
= 13
6 +
2
* When the given strengths are all below or all above Proof, it is unnecessary to take the differences between the rates per cent, and 100, as in other cases.
The three sets of proportional quantities here presented are the results of different modes of coupling the given rates, but all will produce a mixture of the desired strength, as may readily be verified on taking the average value in each case. It is evident that several other sets of differences can be obtained by vary. ing the manner of pairing the rates. An account of the process of deriving the first set of differences will sufficiently explain this part of the subject.
The five given rates are formed into the following pairs:-18 and 54; 22 and 33; 18 (again) and 28; where each pair consists of a rate greater and a rate less than the mean. The difference between 18 and the mean is placed opposite to 54, and that between 54 and the mean opposite to 18. A similar operation is gone through with the remaining pairs, and the two differences standing opposite to 18 are made into a total, as 34 at 18 combined with 6 at 54, and 6 at 28, must have the same collective value as 30 at 18, and 4 at 18, combined separately with 6 at 54, and 6 at 28.
In working questions under the rule of Alligation, it has, until recently, been the practice to connect the several rates together in pairs by means of curves or rectangular figures, for the purpose, no doubt, of indicating more clearly where the alternate differences should be placed. Hence the name, Alligation, as has already been remarked. But the use of such devices is now almost discarded, as having an awkward appearance, and serving rather to confuse than assist the eye.
When the compound is limited to a certain quantity, that is, when the question requires that the ingredients shall amount to a stated sum, it is necessary after finding the alternate differences in the manner just exemplified, to apply the rule of division into proportional parts. (See page 118.)
Example (3.) It is required to compound 865 gallons at 7.5 per cent. U. P., from spirits at 34.6 per cent. O. P.; Proof ; 10.5 per cent., U. P., and water. What should be the quantity of each ingredient ?
( 134.6 . . . . . . 92-5 99.5 100. . . . . . . . . 3.0 89-5 . . . . . .
.
7.5
145.1
When the quantity of one of the ingredients is given, we have, as before, to take the difference between each rate and the mean, and then say by the Rule of Three,-As the difference standing against the rate of the ingredient, whose quantity is specified, is to each of the other differences, so is the given quantity to each of the required quantities.
The reason of the latter part of this rule needs no explanation.
Example (4.) How much spirits at 13s. 6d., 128. 9d., and 10s. 6d. per gallon, respectively, must be added to 38 gallons at 9s. 5d. per gallon, so that the mixture shall be worth 11s. per gallon ?
* For an easier method of performing the work of division into proportional parts, sec page 119, example 3.
( 162d. . . . 19 132d. / 153d.... 6
(126d. . . . 21
113d. . .. 30 30 : 19 :: 38 : 24.07
24.07 galls. at 138. 6d. 30 : 6 :: 38 : 760 Answer. 7.60 „ at 12s. 9d. 30 : 21 :: 38 : 26.60
26 60 „ at 10s. 6d. As the first and the third terms are the same in the three statements, the shorter way would be to form at once the quotient of 38 • 30, and employ it as a common multiplier. Thus, 88 = 1.267. And 1.267 x 19 = 24.07 )
1.267 x 6 = 7.60
1.267 x 21 = 26.60 In most questions of this class, as only one of the component quantities is limited, various answers may be obtained, according to the number of the ingredients, and the mode in which the rates are coupled together. A different treatment of the last example gives, for instance, as a second solution,
10 86 galls. at 13s. Od.)
54:28 galls. at 10s. 6d. It is left to the officer to verify the forgoing results by compounding the several rates and quantities, and finding the average value of the mixture.
Example (4.) In a distillery vessel there are 5650 gallons of worts at 24° gravity. How many gallons of worts at 70° must be added to raise the former gravity to 500 ?
mean 24o..... 20
50, 70° ..... 26 20 : 26 :: 5650 7345. Answer 7345 galls. If by the terms of the question, more than one of the ingredients should be limited as to quantity, the obvious course will be, first to compute the average value of all the ingredients so limited, and then substitute the total quantity at this value for its several component parts.
Example (5.) Required the quantities of worts at 18°, 26°, and 32°, gravity respectively, which will form with 856 gallons at 44°, and 972 gallons at 55°, a mixture having the gravity of 40°.
856 x 44 = 37664
limited ingredients. 18 ................ 9.85
9.85 32 ................ 9.85
( 49.85. . 22 + 14 + 8 = 44:
44 : 9.85 :: 1828 : 409.22 Answer, 409.22 galls. at each of the gravities 18°, 26°, and 32°, respectively.
EXERCISES IN AVERAGES. (1.) The dry malt obtained from 1200 quarters of barley, exceeded that quantity by 5.1 per cent. In another instance, the increase on 1800 quarters amounted to 81 per cent. What was the total or average increase per cent. ?
Answer, 7.15. (2.) Compute the average gravity of 262,580 gallons of worts at 55° ; 359,845 gallons at 51°; 76,283 gallons at 47°, and 56,781 gallons at 35° gravity.*
Answer, 50.7o. (3.) What is the strength of a mixture of 322 gallons of spirits at 22 per cent. 0.P. ; 150 gallons at 80 per cent. U.P., and 14 gallons of water ?
Answer, 13 per cent. U.P. (4.) In what proportions should spirits at the strengths respectively of 50 per cent. O.P., and 11 per cent. O.P. be mixed, so as to produce a compound at the strength of 25 per cent. O.P. ? Answer. In the proportion of 14 measures at 50 O.P. to 25 equal measures at 11 O.P.
(5.) From spirits worth respectively 11s. 9d., 10s. 3d., 8s. 6d., and 6s. 4d. a gallon, prepare a compound worth 9s. a gallon.
Answer. Take 32 galls. at 11s. 9d.
6 galls. at 10s, 3d. 15 galls. at 8s. 6d.
33 galls. at 6s. 4d. (6.) A vessel is found to contain 16,800 gallons of wash at the gravity of 45°. A subsequent account shows an increase of 5 per cent. in the quantity and 2 degrees in the gravity. What was the gravity of the wort that must have been added ?
Answer. 87° (7.) What quantity of spirits worth 14s. per gallon should be blended with 41 gallons of other spirits at 9s. 6d., and 59 gallons at 10s. 8d., so that the mixture shall be worth 11s. 6d. per gallon ?
Answer. 52 15 galls. (8.) A cask of the content of 120 gallons is filled with spirits at 22 per cent. O.P. But this being too weak, what quantity must be taken out and replaced with spirits at 60 per cent. O.P., so that the strength of the whole shall be 25 per cent. O.P.
Answer. 9.47 gallons or 9 gallons 3 pints.
(10.) EVOLUTION.-In treating of this subject it is necessary first to explain the meaning and notation of the powers of numbers. Evolution or the extracting of roots, is the reverse of Involution or the raising of powers. By a power of a number is signified the product formed when that number is multiplied once or several times successively into itself. Thus, 5 x 5 = 25. Here two equal factors are multiplied together, and their product, 25, is termed accordingly the second power of 5. Again, 125 is the third power of 5, since 5 x 5 x 5 = 125, whero three equal factors are concerned ; 625 is the fourth power of 5, for 5 x 5 x 5 x 5 (four factors) = 625, and so on of all other numbers, whole and fractional, the different powers in each case taking their names from the number of equal factors required to produce them respectively.
* A device by wbich the labour of finding the average of large quantities at numerous rates may be greatly abridged, is described in the Inland Revenue Almanack for 1859, page 15. According to the inexact method now deemed sufficient for the purpose of the aunual distillery returns, the mean gravity in Example (2) above, would be shown 47°, instead of 50.7.
It should be observed that any power of a fraction proper, * is necessarily less than the fraction itself. For instance, .4 x 4= .16, and .16 is not so great as •4, since 4-10ths, of 4-10ths can be only a fraction of .4. Similarly, XX X gives a product, which is only ths of . (See page 72.)
Powers are briefly denoted by placing an appropriate small figure called the index or exponent, close to the right hand upper corner of the number to be raised to the given power. Thus, 52 stands for the second power of 5; 59 for the third power of 5, &c. The index shows how often the number to which it is annexed, is to be repeated as factor.
It is customary to employ the word square instead of second power, and cube instead of third power, from the connection which exists between the process of finding the area of a square figure and the second power of a number, and from the similar relation between the content or volume of a solid body and the third power of a number. The fourth power is sometimes called also the biquadratic power, but there is no short term in general use for powers higher than the square and cube.
To complete the scale of powers, a number is said to be its own first power. 5° = 5, that is, 5 raised to the 1st power is 5.
There is nothing in the method of Involution that differs from ordinary multiplication. The object is always to find the actual product of a number multiplied continually by itself, until the number of factors equals the number of units in the index of the given power. If it were required to involve 7 to the fifth power, as indicated by the expression 79 the process would be to multiply 7 four times into itself, that is, to multiply five sevens together. In computing powers beyond the cube, certain abbreviations are possible, which will readily occur to any person from the following considerations. 5 x 5 x 5 x 5 = (5 x 5) x 5 x 5) = 5°x 5 = 5, that is, the 4th
power is the 2nd power squared, or multiplied by itself. 5 x 5 x 5 x 5 x 5 = (5 x 5 x 5) x (5 x 5) = 5°x 5° = 5, or the
5th power is the cube x the square. 5 x 5 x 5 x 5 x 5 = (5 x 5) x (5 x 5) x 5= 5 x 5 x 5 = 5,
or the 5th power is also the square x the square x the given number. It is evident that the same principle may be extended to any number of factors.
By a root of a number or power, is meant such a number as when multiplied a stated number of times into itself will produce the given number or power. Thus, as 25 is the square of 5, so 5 is the square root of 25, As .000512 is the cube of .08, 80 .08 is the cube root of .000512, and similarly with regard to all other powers and their corresponding roots.
Roots are usually denoted in arithmetic by the sign / called the radical sign, prefixed to the number of which a certain root is to be extracted. The square
* That is a fraction, the numerator of which is less than the denominator, and not a whole number or a whole number and fraction merely written in the form of a fraction. See page 66. | https://books.google.com.bd/books?id=SEUIAAAAQAAJ&pg=PA144&vq=%22direct+or+indirect.+A.+direct+tax+is+one+which+is+demanded+from+the+very%22&dq=editions:UOM39015075032303&output=html_text |
Kids won’t be able to resist this banana sushi! It’s simple – roll, cut and add sliced strawberries and kiwi for added flair.
Print Recipe
Pin Recipe
Prep Time
5
mins
Total Time
5
mins
Course
Lunchbox Ideas
Servings
2
servings
Calories
301
kcal
Ingredients
1x
2x
3x
1
large
6-inch whole wheat tortilla
3
tablespoons
natural peanut butter
divided
1
banana
peeled
2
strawberries
sliced
1/2
kiwifruit
sliced
1
tablespoon
unsweetened shredded coconut
Instructions
Lay tortilla on flat surface. Spread with half of the peanut butter. Place banana on one end of tortilla and roll up. Cut into even slices and place strawberries and kiwi sliced on top.
Microwave remaining peanut butter on HIGH 30 seconds, or until melted, and drizzle over banana sushi.
Top with shredded coconut.
Nutrition
Calories:
301
kcal
Carbohydrates:
36
g
Protein:
9
g
Fat:
15
g
Sodium:
249
mg
Fiber:
6
g
Sugar:
10
g
Tried this recipe?
Let us know
how it was! | https://healthyfamilyproject.com/wprm_print/recipe/18924 |
T20 Cricket or twenty20 cricket or 20-20 cricket is the shortest form of cricket sport which is played all over the world. Two teams face-off each other in a complete 40 overs t20 match and each team is permitted to play a maximum of 20 overs.
England and Wales Cricket Board initiated such kind of format back in 2003 in domestic tournament inter-county. First twenty20 international cricket match was played between Australia and New Zealand in 2005. The shortest format of the game gets popular quickly and cricket’s governing council International Cricket Council (ICC) introduced T20 championship.
ICC introduced ICC World Twenty20 championship now known as t20 world cup whose first edition was hosted by South Africa in 2007. India won the inaugural edition of ICC World T20 by defeating arch rivals Pakistan in the final by 5 runs. Since then five editions of prestigious event have been organized successfully and sixth T20 world cup is scheduled to be hosted in India in 2016.
T20 Match Format
- A typical twenty20 match completes in 3 hours. The time duration for the completion of one inning in the match is 75 to 90 minutes.
- In an uninterrupted T20 match a bowler can bowl a maximum of 4 overs as one team can play max to max 20 overs. If the match is interrupted and delayed due to any reason, each bowler will be able to bowl only one-fifth of the total overs in the innings.
- The first six-overs in a 20 over inning are reserved for powerplay. Powerplay is a fielding restriction in limited overs game and in Twenty20s, only two players can be put outside the 20-yeard circle by fielding team.
- After completion of six overs, maximum five fielders can be put outside the 30-yard circle.
- A maximum of 5 players can field on the leg-side at any time in match after powerplay.
- To make the game more interesting and exciting, ICC introduced one more rule to this format as if a bowler bowls a No-Ball then batting team will be awarded one run and the next ball will be free-hit. On the free-hit ball, batsman can’t be dismissed except he gets run-out, obstructing the field, hitting the ball twice or handling the ball.
- If a match ends at draw or tie, Super Over comes into play to decide the winner of match. In Super-Over, each team nominates three batsmen and one bowler to play one-over ‘mini match’. The team who bats second in the match, does bat first in super-over.
- Initially the tied matches were decided by the Bowl-Out.
International debut of Teams in T20
The first Twenty20 international match was played on 17 February in 2005 between Australia and New Zealand. Hence New Zealand and Australia are the first two teams to make debut in T20 internationals. Since then 21 teams have made debut and played at least one 20-20 match. Let’s have a look at the countries who have appeared in a T20 game. | https://twenty20wiki.com/twenty20-cricket/ |
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H
Haas, Michael R.
Hagiwara, Yoshiaki
Hahn, Joseph M.
Hall, Patrick B.
Hameury, Jean-Marie
Hanawa, Tomoyuki
Handley, Thomas H.
Hanisch, R. J.
Hanner, Martha S.
Hao, L,
Hao, L.
Hardee, Philip E.
Harnden, F. R., Jr.
Harper, Eamon
Harra, Louise K.
Harris, Andrew I.
Harrison, Sandra
Hartkopf, W. I.
Hartmann, D. H.
Harutyunian, H. A.
Harvey, Jack
Harvey, Karen L.
Haschick, Aubrey D.
Hatzidimitriou, Despina
Haverkorn, M.
Hawkins, Isabel
Hayes, J. J. E.
Heap, Sally
Hearnshaw, J. B.
Hearnshaw, John
Heber, U.
Heber, Ulrich
Heck, Andre
Heiles, C.
Heinzel, Petr
Hellier, Coel
Hemenway, M. K.
Hendry, M. A.
Henning, Patricia A.
Henning, Th.
Henning, Thomas
Henrichs, Huib F.
Henry, Gregory W.
Hensberge, Herman
Hibbard, John E.
Hidayat, Bambang
Higdon, James
Higdon, Sarah
Hilditch, Ronald W.
Hill, F.
Hill, Frank
Hirayama, Tomohiro
Hiromoto Shibahashi
Ho, L.
Ho, Luis, C.
Ho, Paul T. P.
Hoeksema, J. Todd
Hofmann, A.
Holl, A.
Holt, Stephen S.
Hook, R. N.
Hook, Richard N.
Horne, K.
Howarth, Ian
Howe, R.
Howell, Debbie
Howell, S.
Howell, Steve B.
Hron, J.
Hu Q.
Hubeny, Ivan
Huchra, J.
Hughes, Joanne D.
Humphreys, Roberta M.
Hunt, Gareth
Hunt, Gareth C.
Huttemeister, Susanne
Hwang, Chorng-Yuan
Astronomical Society of the Pacific
Conference Series
© Copyright 1988 - 2021 - Astronomical Society of the Pacific
No part of the material protected by this copyright may be reproduced or used in any form other than for personal use without written permission from the ASP. | http://aspbooks.org/a/volumes/editor_index?alpha=H |
North Atlantic right whale calf denoted as Catalog #3970 in the Bay of Fundy in Canada on Aug. 9, 2016.New England Aquarium
Using genetic testing, scientists have discovered new information about North Atlantic right whale calves, according to a study published Thursday.
“The results of this study have changed what we know about the separation time between a mother and calf as well as calves’ physical development, all crucial information for a critically endangered species that numbers less than 350 individuals,” Philip Hamilton, lead author of the study and senior scientist at the New England Aquarium’s Anderson Cabot Center for Ocean Life, said in a statement.
The North Atlantic right whale is a critically endangered species, with a total population of 336 as of 2020, according to the aquarium statement. The animals typically travel close to shore along the US and Canadian coastlines, spanning from Florida to Canada’s Gulf of St. Lawrence.
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The study, which was published in the journal Mammalian Biology, has been in the works for over 40 years, Hamilton said in a telephone interview Thursday. He said North Atlantic right whales have been tracked via photo identification since 1980 and tracked genetically, through skin and blubber biopsies, since 1988. Data for this study was collected until 2018, he said.
“We regularly compare the two databases because you can obtain identifications from either but using very different metrics,” he said. He said that in a number of cases, researchers were able to use genetic testing to identify whales that they had not been able to identify using photographs.
Braces and her calf in Florida on Feb. 2, 2009.New England Aquarium, taken under permit #655-1652-01
Researchers said it was previously assumed that if mothers were always seen alone on the feeding ground in the calf’s birth year, then their calves were dead. But the study found, with the help of genetic testing, that four calves missing and presumed to be dead had survived. Two of the four possibly had weaned earlier than expected, the researchers said.
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One of the 13 case studies, for example, involved an unnamed calf (denoted as Catalog #3970) born in 2009 and genetically sampled on the calving grounds in January 2009, with his mother, according to the statement.
The calf and his mother, Braces, were last seen together in mid-February 2009, according to the statement. But four months later, in mid-June, a young unidentified whale was spotted alone on a feeding ground 1,000 miles north.
After the whale was genetically sampled in September, it was identified as Braces’ calf, who had separated from his mother at only 7- to 8-months-old. This discovery, the statement said, helped researchers conclude that whales can wean from their mothers earlier than the typical 10 to 12 months.
“I don’t think it will have a big impact on the actual survival estimates because it’s just a few animals,” he said. “But everything helps to make our estimates more precise. And all of those estimates are built into assessments of, you know, what do we need in the way of protections for this species in order for them to survive?”
Right now, he said, the species is in “bad shape” with the population dwindling rapidly due to a decrease in reproduction and increases in mortality caused by vessel strikes and entanglement in ropes. And as someone who has studied whales for 35 years, he said he hopes that with this study, people recognize their importance.
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“I think one thing about this study is that it shows, yet again, the power of knowing the individual,” he said. “By knowing an individual whale, we can track their behavior and their survival, and you link genetics in there, and it just makes it more refined. ... The stories in this paper are about individuals, and I hope that makes the information more interesting and accessible to the reader.” | |
This application is the U.S. national phase of International Application No. PCT/IB2009/050179, filed 19 Jan. 2009, which designated the U.S. and claims the benefit of FR Application No. 08/00275, filed 18 Jan. 2008, the entire contents of each of which are hereby incorporated by reference.
The invention relates to tetrahydrocyclopenta[c]acridine derivatives as kinase inhibitors and is directed toward the use thereof as pharmacological tools and as medicaments.
It also relates to those of these derivatives which constitute new products.
The invention also relates to a process for the production thereof.
The inventors have a great deal of expertise regarding acridine derivatives which have led them to develop a particularly advantageous synthesis pathway, with a low number of stages starting, most generally, from commercially available products.
The development of their studies has resulted in a broadening of the family of these derivatives by synthesizing new tetrahydrocyclopenta[c]acridines.
The study of all these derivatives has made it possible to demonstrate, unexpectedly, inhibitory properties with respect to kinases which control cell division, for instance cyclin-dependent kinases (CDKs) and Aurora kinases, but also glycogen synthase kinase-3 (GSK-3).
By virtue of these inhibitory activities, these derivatives are particularly useful as active ingredients of medicaments for treating serious pathological conditions associated with dysregulation of these kinases.
The invention is therefore directed toward tetrahydrocyclopenta[c]acridine derivatives, as kinase inhibitors.
It also relates to these inhibitors for use as medicaments.
The invention also relates, as products, to those of these derivatives which are novel.
It is also directed toward a process for preparing these derivatives.
According to a first aspect, the invention is directed toward, as kinase inhibitors, tetrahydrocyclopenta[c]acridine derivatives corresponding to formula (I)
1
4
n
1
12
2
9
10
2
3
—Rto R, which may be identical or different, represent H; an ether or polyether radical —(OR′)—OR, R and R′, which may be identical or different, representing an optionally substituted, linear or branched C-Calkyl radical; an amino group NHor N(R, R); NO; NH-carbamate of —NH—CO-OM type, with M representing R (or R′), as defined above or a salt; NH—CO—R, with R as defined above; Nand derivatives thereof of 1,2,3-triazole type;
5
2
9
10
Rrepresents an —OH group; halogen; —OR with R as defined above; OH-carbamate of —O—CO—NHM type, with M representing R (or R′), as defined above; OH-carbonate of —O—CO-OM type, with M representing R (or R′), as defined above; NH, NH-carbamate of —NH—CO-OM type, with M representing R (or R′), as defined above or a salt; NH—CO—R, with R as defined above; N3 and derivatives thereof of 1,2,3-triazole type; N(R, R), M and R being as defined above;
5
1
12
R′ represents H or a C-Calkyl radical as defined above,
5
5
or R/R′ together represents an ═O group;
6
3
Rrepresents H; the R radical; an (R or R′)—Si group, R being as defined above; an aryl radical, where appropriate substituted, a heteroaryl radical; a halogen (iodine); or an alkynyl radical —C≡C—R, with R as defined above;
7
8
1
12
Rand R, which may be identical or different, represent an H or a C-Calkyl radical as defined above;
9
10
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8
5
5
5
5
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3
6
5
1
4
1
4
7
8
5
3
5
6
4
Rand R, which may be identical or different, represent H or the R (or R′) radical as defined above, with the exception of the compounds in which R-R, Rand R═H; Rand R′ form a —C═O group, or R═OH and R′═H (or vice versa); R=—(CH)—Si, —CH, or Cor Calkyl; and of the compound in which R-R, Rand R═H, R=—OCHand R′═H (or vice versa), and R═Calkyl.
In the description and the claims,
“alkyl” relates to a linear or branched, where appropriate substituted, hydrocarbon-based chain containing from 1 to 12 carbon atoms, preferably from 1 to 5 carbon atoms;
3
“halogen” represents F, Cl, Br, I and also the CFgroup;
“aryl” represents one or more aromatic rings, where appropriate substituted, preferably a phenyl radical;
“heteroaryl” represents a heterocycle with N, O or S as heteroatom, which is, where appropriate, substituted, preferably a pyridyl or pyridinyl radical.
in which:
The invention is also directed toward the racemic forms of the above derivatives and also the enantiomeric forms thereof taken individually, more particularly the position-5, -7 and/or -8 isomers.
Advantageously, these derivatives are capable of blocking the ATP site of target kinases which are abnormally activated and therefore dysregulated, thus preventing their phosphorylation activity. Furthermore, these derivatives exhibit a selectivity with respect to these kinases in tests carried out on a panel of 70 kinases.
In this application as kinase inhibitors, the derivatives defined above make it possible to study the functions of the kinases in cell models and the effects resulting from the dysregulation of said kinases (overexpression or abnormal activation) in pathological conditions such as cancers, neurodegenerative diseases, diabetes, in particular type II diabetes, inflammatory diseases, depression and bipolar disorders or viral infections.
50
50
Derivatives which are preferred for use as kinase inhibitors correspond to inhibitors which are CDK-selective and which exhibit ICvalues of less than 20 μM with respect to CDK1 and CDK5, in particular less than 10 μM, particularly advantageous derivatives having ICvalues of less than 2 μM.
5-hydroxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-8-methoxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-8,9-dimethoxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-9-methoxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-1-tert-butyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-8-methoxy-1-tert-butyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxyl-1-trimethylsilanyl-3-methyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-9-methoxy-1-trimethylsilanyl-3-methyl 3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-chloro-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-keto-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-1-butanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-keto-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one.
Derivatives corresponding to these characteristics are chosen from the group comprising:
50
FIG. 1
FIG. 1
5-Hydroxyl-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one constitutes a particularly preferred kinase inhibitor, with ICvalues of 0.56 to 0.74 μM with respect to CDK1 and 1.6 to 2.3 μM with respect to CDK5. This derivative was co-crystallized in the ATP site of CDK2-cyclin A (see ). This co-crystal constitutes a new product and, in this respect, is part of the field of the invention. The representation given in was performed with the group at R6 of tert-butyl type in place of the trimethylsilanyl group actually present, given that the silicon atom (Si) is not available in the processing software used.
50
Derivatives of this group which are also advantageous exhibit an ICwith respect to GSK-3 of less than 10 μM.
The inhibitory activity of the derivatives defined above confers on them a great advantage for treating pathological conditions related to kinase dysregulation.
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According to a second aspect, the invention is therefore directed toward the derivatives of formula (I) above, for use as medicaments, including those in which R-R, Rand R═H; Rand R′ form a —C═O group, or R═OH and R′═H (or vice versa); R=—(CH)—Si, —CH, or a Cor Calkyl; and the compound in which R-R, Rand R═H, R=—OCHand R′═H (or vice versa), and R═Calkyl.
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6
4
The invention is thus more particularly directed toward pharmaceutical compositions characterized in that they contain a therapeutically effective amount of at least one tetrahydrocyclopenta[c]acridine derivative as defined above, and also the compounds in which R-R, Rand R═H; Rand R′ form a —C═O group, or R═OH and R′═H (or vice versa); R=—(CH)—Si, —CH, or Cor Calkyl; and of the compound in which R-R, Rand R═H, R=—OCHand R′═H (or vice versa), and R═Calkyl, in combination with a pharmaceutically acceptable carrier.
These pharmaceutical compositions are advantageously in a form suitable for a given treatment according to the state of the patient and the pathological condition to be treated. Mention will more particularly be made of galenic forms for oral, parenteral or injectable administration.
In order to prepare these galenic forms, the active ingredients, used in therapeutically effective amounts, are mixed with the carriers that are pharmaceutically acceptable for the chosen method of administration.
For oral administration, the pharmaceutical compositions are more particularly in the form of tablets, gel capsules, capsules, pills, sugar-coated tablets, drops and the like.
Such compositions can contain from 1 to 100 mg of active ingredient per unit to be taken, in particular from 40 to 60 mg.
For intravenous, subcutaneous or intramuscular administration by injection, the pharmaceutical compositions are advantageously in the form of sterile or sterilizable solutions.
They contain from 10 to 50 mg of active ingredient, in particular from 20 to 30 mg.
These compositions are particularly effective for blocking the ATP site of CDKs and can thus in particular stop the anarchic cell division of cancer cells.
In addition to the treatment of cancers, these pharmaceutical compositions are also effective for treating neurodegenerative diseases, diabetes, in particular type II diabetes, inflammatory diseases, depression and bipolar disorders.
1
9
According to a third aspect, the invention is directed toward the derivatives of formula (I) above corresponding to new products. They are derivatives in which Rto Rare as defined above, with the exception of 5-hydroxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one, 5-hydroxy-1-butanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one and 5-keto-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one.
5-hydroxy-8-methoxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-8,9-dimethoxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-9-methoxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-1-tert-butyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-8-methoxy-1-tertbutyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-1-trimethylsilanyl-3-methyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-9-methoxy-1-trimethylsilanyl-3-methyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-chloro-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one,
5-hydroxy-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one.
Preferred derivatives comprise:
FIG. 2
The derivatives of the invention are advantageously obtained according to the methodology described by Patin and Belmont (1) and illustrated by the scheme given in .
The principle of this process is also applied for obtaining the novel derivatives of the invention.
the reaction of a derivative of formula (II)
According to a fourth aspect, the invention is thus directed toward a synthesis process comprising:
2
8
R1 to R7 are as defined above, and R8, defined as above, can be derivatized by means of a cross-metathesis reaction from the allyl or R8 represents H, in the presence of a catalyst such as Co(CO)(or a rhodium or molybdenum complex), according to the Pauson-Khand reaction (1) (abbreviated to PKR), under conditions which make it possible to give a derivative of formula (I)
in which:
The derivatives in which R5 represents an OM group can be subjected to an oxidation step so as to obtain a derivative of formula (I) in which R5/R5′ represent a ketone function.
3
The derivatives in which one of the substituents R1 to R5 represents an Nderivative of 1,2,3-triazole type are advantageously obtained by means of 1,3-dipolar reactions of “click chemistry” type (3).
1
12
The compound of formula (II) is advantageously obtained by means of a Sonogashira or Negishi reaction, using a 2-chloro-3-quinolinecarboxaldehyde derivative (R5′ represents H or a C-Calkyl radical as defined above) of formula (III)
6
with an alkyne of formula (IV) R—C≡CH, followed by a Grignard reaction with the addition of allylmagnesium bromide or of another Grignard reagent substituted on the allyl function (R8).
The derivative (III) is itself preferably obtained from a derivative of formula (V)
3
3
where Ac═CHCO—, by carrying out the process in an organic solvent such as DMF in the presence of POClunder the conditions described by Meth-Cohn et al. (2).
The synthesis intermediate quinolinecarbaldehyde derivatives of formula (II) are new products and are therefore, as such, also covered by the invention.
Intermediate derivatives comprise 2-(trimethylsilanylethynyl)quinoline-3-carbaldehyde, 6-methoxy-2-(trimethylsilanylethynyl)quinoline-3-carbaldehyde, 6,7-dimethoxy-2-(trimethylsilanylethynyl) quinoline-3-carbaldehyde, and 7-methoxy-2-(trimethylsilanylethynyl)quinaline-3-carbaldehyde. Preferably, they are 1-(2-(trimethylsilanylethynyl) quinolin-3-yl)but-3-en-1-ol, 1-(6-methoxy-2-(trimethylsilanylethynyl)quinolin-3-yl)but-3-en-1-ol, 1-(6,7-dimethoxy-2-(trimethylsilanylethynyl) quinolin-3-yl)but-3-en-1-ol and 1-(7-methoxy-2-(trimethylsilanylethynyl) quinolin-3-yl)but-3-en-1-ol.
Sonogashira Reaction:
2
3
2
The halogenated quinoline-type derivative of formula (III) (1.00 mmol), PdCl(PPh)(35 mg, 0.05 mmol) and CuI (9 mg, 0.05 mmol) are mixed under an argon atmosphere. Once the system has been degassed, DMF (1 ml) and TEA (0.6 ml) are added to the reaction medium. The alkyne (1.10 mmol) is then added dropwise. The reaction medium is stirred at ambient temperature for 12 hours. The reaction medium is then filtered through silica and then evaporated. The residue obtained is purified by flash chromatography.
Mp 125° C.
−1
IR: 2954, 2850, 2359, 2338, 1694, 1579, 1369, 1149, 1096 cm.
1
3
H NMR (300 MHz, CDCl): δ=10.70 (s, 1H), 8.72 (s, 1H), 8.16 (dd, 1H, J=8.5, 1.0 Hz), 7.95 (dd, 1H, J=8.1, 1.4 Hz), 7.85 (ddd, 1H, J=8.5, 7.0, 1.4 Hz), 7.63 (ddd, 1H, J=8.1, 7.0, 1.0 Hz), 0.34 (s, 9H);
13
3
3
C NMR (75 MHz, CDCl): δ=191.0 (CH), 150.0 (C), 143.6 (C), 136.8 (CH), 133.0 (CH), 129.7 (CH), 129.4 (CH), 128.8 (C), 128.4 (CH), 126.5 (C), 102.5 (C), 100.1 (C), −0.3 (CH);
+
+
+
MS: m/z (%)=286 (81) [MNa], 254 (100) [MH], 180 (17) [MH-TMS].
+
15
15
MS-HR: m/z [MH] calculated for CHNOSi: 254.1001; found: 254.0997.
Mp 155-156° C.
−1
IR: 3051, 3001, 2964, 2840, 2158, 1694, 1243, 1226, 837 cm.
1
3
H NMR (300 MHz, CDCl): δ=10.69 (s, 1H), 8.59 (s, 1H), 8.05 (d, 1H, J=9.3 Hz), 7.49 (dd, 1H, J=9.3, 2.8 Hz), 7.16 (d, 1H, J=2.8 Hz), 3.96 (s, 3H), 0.33 (s, 9H);
13
3
3
3
C NMR (75 MHz, CDCl): δ=191.3 (CH), 159.1 (C), 146.4 (C), 141.2 (C), 135.0 (CH), 130.8 (CH), 129.1 (C), 127.9 (C), 126.3 (CH), 106.2 (CH), 101.4 (C), 100.2 (C), 55.8 (CH), −0.2 (CH);
+
+
3
MS: m/z (%)=284 (28) [MH], 316 (100) [M+CHOH+H].
+
16
17
2
MSHR: m/z [MH] calculated for CHNOSi: 284.1107; found: 284.1112.
Mp 188° C.
−1
IR: 3015, 2957, 2931, 2860, 2830, 2163, 1688, 1257, 1215, 1113, 1008, 841 cm.
1
3
H NMR (300 MHz, CDCl): δ=10.65 (s, 1H), 8.54 (s, 1H), 7.47 (s, 1H), 7.12 (s, 1H), 4.05 (s, 3H), 4.04 (s, 3H), 0.33 (s, 9H);
13
3
3
3
3
C NMR (75 MHz, CDCl): δ=191.2 (CH), 155.6 (C), 151.3 (C), 148.0 (C), 142.1 (C), 134.1 (CH), 127.9 (C), 122.8 (C), 107.9 (CH), 106.2 (CH), 101.4 (C), 100.4 (C), 56.6 (CH), 56.4 (CH), 0.2 (CH);
+
+
3
MS: m/z (%)=314 (100) [MH], 346 (85) [M+CHOH+H].
+
17
19
3
MSHR: m/z [MH] calculated for CHNOSi: 314.1212; found: 314.1207.
Mp 142° C.
−1
IR: 3008, 2959, 2896, 2856, 2830, 1687, 1495, 1210, 1131, 1016, 841 cm.
1
3
H NMR (300 MHz, CDCl): δ=10.60 (s, 1H), 8.56 (s, 1H), 7.75 (d, 1H, J=9.0 Hz), 7.40 (d, 1H, J=2.3 Hz), 7.20 (dd, 1H, J=9., 2.3 Hz), 3.92 (s, 3H), 0.31 (s, 9H);
13
3
3
3
C NMR (75 MHz, CDCl): δ=190.8 (CH), 163.7 (C), 152.2 (C), 144.3 (C), 136.0 (CH), 130.8 (CH), 127.4 (C), 122.1 (C), 122.0 (CH), 107.2 (CH), 102.1 (C), 100.2 (C), 55.9 (CH), 0.2 (CH);
+
+
3
MS: m/z (%)=284 (58) [MH], 316 (100) [M+CHOH+H].
+
16
17
2
MSHR: m/z [MH] calculated for CHNOSi: 284.1107; found: 284.1111.
Grignard Reaction:
2
4
The derivative of 2-ethynylquinoline-3-carbaldehyde type (1.00 mmol) is dissolved in 10 ml of freshly distilled THF under an argon atmosphere. The reaction medium is cooled to 78° C. The commercially available 1M solution of allyl magnesium bromide in EtO (1.50 ml, 1.50 mmol) is then added dropwise. The reaction medium is stirred for 4 hours at −78° C. The reaction medium is then run into a saturated aqueous solution of NHCl, the aqueous phase is extracted with ethyl acetate and the resulting organic phase is rinsed with a saturated aqueous solution of NaCl, dried over NaaSCa, filtered and evaporated. The residue obtained is purified by flash chromatography,
Mp 111° C.
−1
IR: 3232, 3074, 2958, 2899, 2161, 1247, 1060 cm.
1
3
H NMR (300 MHz, CDCl): δ=8.29 (s, 1H), 8.09 (dd, 1H, J=8.4, 1.1 Hz), 7.79 (d, 1H, J=8.0, 1.4 Hz), 7.69 (ddd, 1H, J=8.5, 7.0, 1.4 Hz), 7.53 (ddd, 1H, J=8.0, 7.0, 1.1 Hz), 5.97-5.83 (m, 1H), 5.36-5.33 (m, 1H), 5.24 (dd, 1H, J=7.0, 1.1 Hz), 5.20 (s, 1H), 2.85 (m, 1H), 2.44 (m, 2H), 0.31 (s, 9H);
13
3
2
2
3
C NMR (75 MHz, CDCl): δ=147.3 (C), 141.2 (C), 138.8 (C), 134.4 (CH), 132.7 (CH), 129.9 (CH), 129.3 (CH), 129.2 (C), 127.8 (CH), 127.6 (CH), 119-1 (CH), 102.1 (C), 77.5 (C), 70.2 (CH), 42.9 (CH), 0.1 (CH);
+
MS: m/z (%)=296 (100) [MH].
+
18
21
MSHR: m/z [MH] calculated for CHNOSi: 296.1474; found: 296.1474.
Mp 149° C.
−1
IR: 3252, 3075, 3012, 2961, 2937, 2901, 2830, 2161, 1621, 1492, 1239, 1027, 827 cm.
1
3
H NMR (300 MHz, CDCl): δ=8.19 (s, 1H), 8.00 (d, 1H, J=8.8 Hz), 7.32 (dd, 1H, J=8.8, 2.7 Hz), 7.05 (d, 1H, J=2.7 Hz), 5.97-5.83 (m, 1H), 5.33-5.30 (m, 1H), 5.24 (d, 1H, J=6.4 Hz), 5.20 (s, 1H), 3.93 (s, 3H), 2.85 (m, 1H), 2.44 (m, 2H), 0.31 (s, 9H);
13
3
2
3
2
3
C NMR (75 MHz, CDCl): δ=147.3 (C), 141.2 (C), 138.8 (C), 134.5 (CH), 132.7 (CH), 129.9 (CH), 129.3 (CH), 129.2 (C), 127.8 (CH), 127.6 (CH), 119.2 (CH), 105.2 (CH), 102.1 (C), 77.5 (C), 70.3 (CH), 55.8 (CH), 43.0 (CH), 0.1 (CH);
+
MS: m/z (%)=326 (100) [MH].
+
19
23
2
MSHR m/z [MH] calculated for CHNOSi: 326.1576; found: 326.1571.
Mp 65-67° C.
−1
IR: 3367, 3077, 3003, 2959, 2929, 2851, 2159, 1621, 1497, 1244, 1213, 1008, 840 cm.
1
3
H NMR (300 MHz, CDCl): δ=8.10 (s, 1H), 7.40 (s, 1H), 7.00 (s, 1H), 5.97-5.82 (m, 1H), 5.33-5.27 (m, 1H), 5.24 (dd, 1H, J=6.4, 1.5 Hz), 5.19 (s, 1H), 4.00 (s, 3H), 3.99 (s, 3H), 2.85-2.79 (m, 1H), 2.50-2.40 (m, 1H), 2.36 (s, 1H), 0.30 (s, 9H);
13
3
2
3
3
2
3
C NMR (75 MHz, CDCl): δ=152.4 (C), 150.2 (C), 143.8 (C), 138.1 (C), 137.6 (C), 134.6 (CH), 130.7 (CH), 123.2 (C), 117.9 (CH), 107.1 (CH), 104.6 (CH), 102.1 (C), 99.2 (C), 69.9 (CH), 55.9 (CH), 55.8 (CH), 42.7 (CH), −0.3 (CH);
+
MS: m/z (%)=356 (100) [MH].
+
20
25
3
MSHR m/z [MH] calculated for CHNOSi: 356.1682; found: 356.1677.
Mp 176-177° C.
−1
IR: 3196, 3078, 3013, 2958, 2901, 2840, 2160, 1622, 1497, 1234, 1215, 1026, 839, 816 cm.
1
3
H NMR (300 NHz, CDCl): δ=8.21 (s, 1H), 7.68 (d, 1H, J=9.0 Hz), 7.32 (d, 1H, J=2.5 Hz), 7.19 (dd, 1H, J=9.0, 2.5 Hz), 5.97-5.83 (m, 1H), 5.34-5.29 (m, 1H), 5.24 (d, 1H, J=6.0 Hz), 5.19 (s, 1H), 3.92 (s, 3H), 2.86-2.77 (m, 1H), 2.50-2.39 (m, 1H), 2.35 (d, 1H, J=3.6 Hz), 0.31 (s, 9H);
13
3
2
3
2
3
C NMR (75 MHz, CDCl): δ=161.0 (C), 149.0 (C), 141.1 (C), 136.7 (C), 134.4 (CH), 132.4 (CH), 128.7 (CH), 122.9 (C), 121.0 (CH), 119.0 (CH), 106.9 (CH), 102.2 (C), 100.3 (C), 70.2 (CH), 55.6 (CH), 43.0 (CH), −0.1 (CH);
+
MS: m/z (%)=326 (100) [MH].
+
19
23
2
MSHR m/z [MH] calculated for CHNOSi: 326.1576; found: 326.1582.
Pauson-Khand Reaction:
2
8
The quinoline enyne derivative of formula (II) (1.00 mmol) is dissolved in 10 ml of freshly distilled DCM under an argon atmosphere. Co(CO)(420 mg, 1.20 mmol) is then added. The reaction medium is stirred for 2 hours at ambient temperature and the complexation of the metal on the alkyne is monitored by TLC. NMO (1171 mg, 10.00 mmol) is then added portionwise and the reaction medium is stirred for 12 hours at ambient temperature. The reaction medium is subsequently filtered through silica and then evaporated. The residue obtained is purified by flash chromatography.
Mp 167-168° C.
−1
IR: 2968, 2950, 2894, 1686, 1273, 1157, 856 cm.
1
3
H NMR (300 MHz, CDCl): δ=8.22 (s, 1H), 8.12 (dd, 1H, J=8.4, 0.9 Hz), 7.85 (dd, 1H, J=8.1, 0.9 Hz), 7.70 (ddd, 1H, J=8.4, 6.9, 0.9 Hz), 7.59 (ddd, 1H, J=8.1, 6.9, 0.9 Hz), 5.21-5.18 (m, 1H), 3.72-3.64 (m, 1H), 2.84 (dd, 1H, J=11.4, 6.6 Hz), 2.55-2.48 (m, 1H), 2.27 (dd, 1H, J=18.0, 3.9 Hz), 1.95 (ddd, 1H, J=13.5, 13.5, 3.3 Hz), 1.68 (m, 1H), 0.35 (s, 9H);
13
3
2
2
3
C NMR (75 MHz, CDCl): δ=212.1 (C), 179.3 (C), 149.9 (C), 147.6 (C), 142.7 (C), 137.4 (CH), 132.7 (C), 130.6 (CH), 129.5 (CH), 128.4 (C), 128.0 (CH), 127.8 (CH), 67.7 (CH), 43.7 (CH), 37.9 (CH), 35.4 (CH), 0.9 (CH);
+
+
2
MS: m/z (%)==324 (68) [MH], 306 (100) [MH—HO].
+
19
21
2
MSHR m/z [MH] calculated for CHNOSi: 324.1420; found: 324.1422.
Elemental analysis: found (calculated) C, 70.02 (70.55); H, 6.42 (6.54); N, 4.12 (4.33);
Mp 186° C.
−1
IR: 3357, 3001, 2955, 2888, 2825, 1659, 1490, 1216, 851; 840, 827 cm.
3
NMR (300 MHz, CDCl): δ=8.05 (s, 1H), 7.95 (d, 1H, J=9.3 Hz), 7.35 (dd, 1H, J=9.3, 2.6 Hz), 7.01 (d, 1H, J=2.6 Hz), 5.10-5.06 (m, 1H), 3.88 (s, 3H), 3.69-3.60 (m, 1H), 2.72 (dd, 1H, J=17.8, 6.8 Hz), 2.47-2.42 (m, 1H), 2.17 (dd, 1H, J=17.9, 4.1 Hz), 1.85 (ddd, 1H, J=13.5, 13.5, 3.2 Hz), 1.25 (m, 1H), 0.35 (s, 9H);
13
3
3
2
2
3
C NMR (75 MHz, CDCl): δ=212.5 (C), 180.5 (C), 159.0 (C), 147.3 (C), 143.8 (C), 141.0 (C), 136.0 (CH), 133.3 (C), 130.8 (CH), 129.7 (C), 123.7 (CH), 104.9 (CH), 67.4 (CH), 55.7 (CH), 43.6 (CH), 37.9 (CH), 35.4 (CH), 0.9 (CH);
4
+
+
+
MS: m/z (%)=338 (84) [MH-CH], 354 (100) [MH], 729 (33) [2MNa].
+
20
23
3
MSHR m/z [MH] calculated for CHNOSi: 354.1525; found: 354.1519.
Elemental analysis: found (calculated) C, 68.16 (67.96); H, 6.58 (6.56); N, 3.92 (3.96);
Mp 221-222° C.
−2
IR: 3388, 2962, 2936, 2891, 2825, 1691, 1497, 1240, 846, 830 cm.
1
3
H NMR (300 MHz, CDCl): δ=7.99 (s, 1H), 7.28 (s, 1H), 6.98 (s, 1H), 5.11-5.07 (m, 1H), 4.03 (s, 3H), 3.97 (s, 3H), 3.64-3.59 (m, 1H), 2.77 (dd, 1H, J=17.9, 6.8 Hz), 2.58-2.44 (m, 1H), 2.21 (dd, 1H, J=17.9, 4.1 Hz), 1.88 (ddd, 1H, J=13.5, 13.5, 3.2 Hz), 1.24 (m, 1H), 0.35 (s, 9H);
13
3
3
3
2
2
3
C NMR (75 MHz, CDCl): δ=212.2 (C), 180.4 (C), 153.5 (C), 151.2 (C), 147.5 (C), 144.8 (C), 140.8 (C), 135.3 (CH), 131.3 (C), 124.6 (C), 107.2 (CH), 104.9 (CH), 67.7 (CH), 56.3 (CH), 56.2 (CH), 43.7 (CH), 38.1 (CH), 35.4 (CH), 1.0 (CH);
4
+
+
+
MS: m/z (%)=368 (79) [MH-CH], 384 (100) [MH], 789 (29) [2MNa].
+
21
25
3
MSHR m/z [MH] calculated for CHNOSi: 384.1631; found: 384.1636.
2
Elemental analysis: found (calculated +0.5HO) C, 63.82 (64.26); H, 6.36 (6.68); N, 3.57 (3.57);
Mp 187° C.
−1
IR: 3440, 2962, 2947, 2903, 2851, 1693, 1621, 1228, 1140, 1019, 848, 835, 819 cm.
1
3
H NMR (300 MHz, CDCl): δ=8.09 (s, 1H), 7.66 (d, 1H, J=9.0 Hz), 7.31 (d, 1H, J=2.2 Hz), 7.19 (dd, 1H, J=9.0, 2.2 Hz), 5.10-5.08 (m, 1H), 3.95 (s, 3H), 3.66-3.56 (m, 1H), 2.75 (dd, 1H, J=18.0, 6.7 Hz), 2.48-2.42 (m, 1H), 2.17 (dd, 1H, J=18.0, 4.0 Hz), 1.85 (ddd, 1H, J=13.5, 13.5, 3.3 Hz), 0.35 (s, 9H);
13
3
3
2
2
3
C NMR (75 MHz, CDCl): δ=212.3 (C), 180.3 (C), 161.4 (C), 149.9 (C), 149.3 (C), 142.0 (C), 137.1 (CH), 130.8 (C), 128.8 (CH), 123.8 (C), 121.3 (CH), 106.8 (CH), 67.5 (CH), 55.6 (CH), 43.7 (CH), 38.1 (CH), 35.4 (CH), 0.9 (CH);
4
+
+
+
MS: m/z (%)=338 (66) [MH-CH], 354 (100) [MH], 729 (17) [2MNa].
+
20
23
3
MSHR m/z [MH] calculated for CHNOSi: 354.1525; found: 354.1531.
2
3
2
4
The 5-hydroxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one (323 mg, 1.00 mmol) is dissolved in 10 ml of freshly distilled DCM under an argon atmosphere at 0° C. SOCl(182 μl, 2.5 mmol) is then added dropwise to the reaction medium, which is stirred at 0° C. for 15 min. The reaction medium is then run into a saturated aqueous solution of NaHCO, the aqueous phase is extracted with DCM and the resulting organic phase is rinsed with a saturated aqueous solution of NaCl, dried over NaSO, filtered and evaporated. The residue obtained is purified by flash chromatography.
Mp 169-170° C.
−1
IR: 3038, 2952, 2897, 1687, 1491, 1219, 1195, 1157, 841, 770 cm.
1
3
H NMR (300 MHz, CDCl): δ=8.24 (s, 1H), 8.11 (dd, 1H, J=8.4, 1.1 Hz), 7.85 (dd, 1H, J=8.1, 1.4 Hz), 7.78 (ddd, 1H, J=8.4, 7.0, 1.4 Hz), 7.61 (ddd, 1H, J=8.1, 7.0, 1.1 Hz), 5.64 (dd, 1H, J=3.5, 2.2 Hz), 3.87-3.77 (m, 1H), 2.90 (dd, 1H, J=17.9, 6.9 Hz), 2.71 (ddd, 1H, J=14.2, 3.9, 2.2 Hz), 2.34-2.24 (m, 2H), 0.37 (s, 9H);
13
3
2
2
3
C NMR (75 MHz, CDCl): δ=211.2 (C), 178.1 (C), 148.9 (C), 147.7 (C), 143.4 (C), 138.0 (CH), 131.3 (C), 131.0 (CH), 129.5 (CH), 128.3 (C), 128.2 (CH), 128.0 (CH), 57.1 (CH), 43.3 (CH), 38.7 (CH), 35.8 (CH), 0.9 (CH);
4
+
+
MS: m/z (%)=326 (92) [MH-CH], 342 (100) [MH].
+
19
20
MSHR m/z [MH] calculated for CHClNOSi: 342.1081; found: 342.1079.
The enantiomer forms are obtained according to scheme 1 below:
Variant Synthesis of Derivatives According to the Invention
This variant is illustrated by scheme 2 below, relating to the synthesis of 5-hydroxy-7-amino-8-methoxy-1-trimethylsilyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one:
50
The tests are carried out as follows: the enzyme to be assayed was purified, for example by affinity chromatography on agarose beads. The catalytic activity was measured using radiolabeled ATP, at a standard final concentration. The test compounds were added at various concentrations making it possible to establish dose-response curves (activity of the enzyme as a function of the concentration). The ICvalues were calculated from these curves and are given in μM. They represent the value at which a 50% inhibition of the enzyme is observed.
The procedure for the tests, attesting to the selectivity of the compounds of type (I) for the target kinases (versus 70 other kinases), was recently reported (4).
50
The value of the ICs measured with compounds of the invention, with respect to CDK1 and CDK5, are reported in the following table 1:
TABLE 1
Compound
CDK1
CDK5
5-hydroxy-1-trimethylsilanyl-
0.56 to
1.6 to
3,3a,4,5-tetrahydro-2H-
0.74
2.3
cyclopenta[c]acridin-2-one (racemic)
5-hydroxy-1-trimethylsilanyl-
0.62
3
3,3a,4,5-tetrahydro-2H-
cyclopenta[c]acridin-2-one
(+ enantiomer)
5-hydroxy-1-trimethylsilanyl-
7.8
26
3,3a,4,5-tetrahydro-2H-
cyclopenta[c]acridin-2-one
(− enantiomer)
5-chloro-1-trimethylsilanyl-
3.6
63
3,3a,4,5-tetrahydro-2H-
cyclopenta[c]acridin-2-one
5-hydroxy-8-methoxy-1-
1.7
4
trimethylsilanyl-3,3a,4,5-
tetrahydro-2H-cyclopenta[c]acridin-
2-one
5-hydroxy-8,9-methoxy-1-
1.7
3.3
trimethylsilanyl-3,3a,4,5-
tetrahydro-2H-cyclopenta[c]acridin-
2-one
5-hydroxy-9-methoxy-1-
1.6
4.8
trimethylsilanyl-3,3a,4,5-
tetrahydro-2H-cyclopenta[c]acridin-
2-one
50
50
The 5-hydroxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro 2H-cyclopenta[c]acridin-2-one has an ICwith ICvalues of 0.54 μM with respect to CDK1 and of 1.6 μM with respect to CDK5.
The tests are carried out on HT29 cells (human colon adenocarcinoma, deposit ATCC HTB 38) with the procedure as follows:
4
The HT29 cells are cultured in Dulbecco's MEM medium supplemented with 10% FCS. The cells originating from a log-phase culture are seeded into 24-well microplates (1 ml-5×10cells/well) and incubated for 2 days. The compounds tested, in solution in DMSO (dimethyl sulfoxide), are added in a minimum volume (5 μl) at increasing concentration. The control cells receive only 5 μl of DMSO alone. The plates are incubated for 24 h, then the medium is removed and the cells are washed twice with PBS (phosphate buffered saline solution) before medicament-free fresh medium is added. The plates are re-incubated for 3 days before evaluation of the cell survival using the MTT test (5) which comprises Incubating 3-[4,5-dimethylthiazol-2-yl]-2,5-diphenyl tetrazolium bromide (MTT, Sigma) for 30 min in wells, in a proportion of 100 μg/well. After removal of the medium, the formazan crystals are recovered with 100 μl of DMSO and the absorbance is measured at 540 nm with a microplate reader (model 450, Bio-Rad). The cell survival is expressed as % of the controls treated with DMSO.
The results are given in the following Table 2:
TABLE 2
Compound tested
IC<sub>50 </sub>(HT 29-24 h)
5-Hydroxy-1-trimethylsilanyl-3,3a,4,5-
26
tetrahydro-2H-cyclopenta[c]acridin-2-one
5-Hydroxy-8-methoxy-1-trimethylsilanyl-
21
3,3a,4,5-tetrahydro-2H-
cyclopenta[c]acridin-2-one
5-Hydroxy-8,9-dimethoxy-1-
41
trimethylsilanyl-3,3a,4,5-tetrahydro-2H-
cyclopenta[c]acridin-2-one
5-Hydroxy-9-methoxy-1-trimethylsilanyl-
6.5
3,3a,4,5-tetrahydro-2H-
cyclopenta[c]acridin-2-one
The viability of SHSY cells is determined by measuring the MTS reduction as described in (6).
The results obtained are given in the following Table 3:
TABLE 3
% survival
of SHSY
cells at
IC<sub>50 </sub>at
Compound
10 μM
48 h
1-(2-(Trimethylsilanylethynyl)quinolin-
0.4
6.1
3-yl)propan-2-en-1-ol
1-(2-[3-(Tetrahydropyran-2-yloxy)prop-
2
5.1
1-ynyl]quinolin-3-yl)but-3-en-1-ol
1-(2-(Diethoxyethynyl)quinolin-3-
2
5.2
yl)but-3-en-1-ol
1-(2-Trimethylsilanylethynyl)quinolin-
47
15
3-yl)propan-3-nitro-1-one
5-Keto-1-butyl-3,3a,4,5-tetrahydro-2H-
38
13
cyclopenta[c]acridin-2-one
1-(2-(Pyridin-2-ylethynyl)quinolin-3-
4
10
yl)ethanone
5-Hydroxy-9-methoxy-1-trimethylsilanyl-
44
13
3,3a,4,5-tetrahydro-2H-
cyclopenta[c]acridin-2-one
5-Hydroxy-1-trimethylsilanyl-3,3a,4,5-
85
>10
tetrahydro-2H-cyclopenta[c]acridin-2-
one (racemic)
1. Patin A. and Belmont P., Synthesis, 2005, 2400-2406
2. Meth-Cohn O., Narine B., Tarnowski B., J. Chem. Soc, Perkin Trans. 1, 1981, 1520 and 1531.
3. Kolb H. C, Finn M. G. and Sharpless K. B., 2001, Angew. Chem. Int. Ed. 40, 2004-2021.
4. Bain J., Plater L., Elliott M., Shpiro N., Hastie C. J., Mclauchlan H., Klevernic I., Arthur J. S. C, Alessi D. R. and Cohen P., Biochem. J., 2007, 408, 297-315.
5. Mossmann T., J. Immunol. Meth., 1983, 65, 55-63.
6. Ribas J. and Boix J., 2004, Exp. Cell Res., 295, 9-24.
Other characteristics and advantages of the invention are given in the examples which follow.
FIGS. 1 and 2
represent, respectively, the structure of the co-crystal of 5-hydroxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one with the ATP site of CDK2-cyclin A, and a scheme for the synthesis of tetrahydrocyclopenta[c]acridine derivatives.
EXAMPLE 1
Synthesis of Tetrahydrocyclopenta[c]Acridine Derivatives According to the Invention
2-(Trimethylsilanylethynyl)quinoline-3-carbaldehyde
6-Methoxy-2-(trimethylsilanylethynyl)quinoline-3-carbaldehyde
6,7-Dimethoxy-2-(trimethylsilanylethynyl)quinoline-3-carbaldehyde
7-Methoxy-2-(trimethylsilanylethynyl)quinoline-3-carbaldehyde
1-(2-(Trimethylsilanylethynyl)quinolin-3-yl)but-3-en-1-ol
1-(6-Methoxy-2-(trimethylsilanylethynyl)quinolin-3-yl)but-3-en-1-ol
1-(6,7-Dimethoxy-2-(trimethylsilanylethynyl)quinolin-3-yl)but-3-en-1-ol
1-(7-Methoxy-2-(trimethylsilanylethynyl)quinolin-3-yl)but-3-en-1-ol
5-Hydroxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one
5-Hydroxy-8-methoxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one
5-Hydroxy-8,9-dimethoxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one
5-Hydroxy-9-methoxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one
5-Chloro-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one
Obtaining the enantiomers of 5-hydroxy-1-trimethylsilanyl-3,3a,4,5-tetrahydro-2H-cyclopenta[c]acridin-2-one
EXAMPLE 2
Enzyme Inhibition Tests
EXAMPLE 3
Cytotoxic Activity Tests
EXAMPLE 4
MTS Tests
REFERENCES | |
Summary:
Kids can make unique works of art by dipping string into paint and making pulled string art.
Objectives:
- aprons or smocks, white or colored construction paper
- string cut in 2 feet pieces
- tempera paint in different colors
- paper plates for paint
- a covered surface
What You Need:
- Lay out a piece of paper and have the child fold it in half, then open it back up laying it flat on a covered surface.
- Next have the child take a piece of string and submerge 3/4 of it in the paint leaving an end free of paint to hold on to.
- Next lay the paint covered string on one side of the folded paper in any type pattern, leaving the clean end of the string out so you can hold on to it.
- Then fold the paper back together and put your hand on the paper so you can feel the string through the paper and begin pulling the string out moving it around the edges of the paper.
- It’s really fun to have more than one string with different colors pull the strings out and open up the paper and waaaa laaaa a beautiful creation! | https://kinderart.com/art-lessons/painting/pulled-string-paintings/ |
By: Elise Gowen
For the third year in a row, the EMS Library will be giving graduate students a chance to make the most of the quiet on campus during Spring Break by the EMS Writers’ Retreat in the library! Aimed primarily at graduate students enrolled in the College of Earth and Mineral Sciences, but open to all colleges, the retreat offers a quiet, convivial atmosphere where participants can retreat to and focus on finishing their writing projects. Originally conceived as a thesis and dissertation bootcamp, the event took on a more relaxed mood, where the goal is for students to move at a pace that works them while staying focused and surrounding themselves with other writers in a productive atmosphere.
The Writers’ Retreat offers attendees a free ceramic mug, unlimited coffee, light breakfast, snacks throughout the day, and lunch. Books on writing and time management are also available for check-out, and librarians are also on-hand to provide citation management support.
Attendance for Penn State students, faculty, and staff is free.
The Writers’ Retreat takes place in the EMS Library during spring break, Monday – Friday, Mar. 9–13. 9 a.m.- 4 p.m. | https://sites.psu.edu/librarynews/2020/03/09/ |
OPPO A74 6+128GB 6.43" Prism black DS
OPPO A74. Screen size: 16.3 cm (6.43 "), display resolution: 2400 x 1080 pixels, display type: AMOLED. Processor family: Qualcomm Snapdragon, processor model: 662. RAM capacity: 6 GB, RAM type: LPDDR4X, internal storage capacity: 128 GB.
Free shipping throughout Europe by express courier. | https://www.esdorado.com/gb/smartphone/3412-oppo-a74-6128gb-643-prism-black-ds-6944284685888.html |
Q:
How to automatically determine the minimum number of filling dots with the following constraint?
I want to display a fraction
but with an overbrace saying number 7 appears n times and an underbrace saying number 8 appears n times.
I want the minimum number of dots to be automatically determined such that
the most right character of the upper label, 7, 5, and the right end of the horizontal line are all aligned with the same vertical line.
the most left character of the lower label, 8, 1, and the left end of the horizontal line are all aligned with the same vertical line.
MWE
\documentclass[border=12pt,12pt]{standalone}
\usepackage{amsmath}
\begin{document}
$\displaystyle\frac{1\overbrace{7\dots 7}^{\text{number 7 appears $n$ times}}}{\underbrace{8\dots 8}_{\text{number 8 appears $n$ times}}5}$
\end{document}
How to automatically determine the minimum number of filling dots with the constraint given above?
Edit
By a trial and error, I got the following but I want a smarter way.
\documentclass[border=12pt,12pt]{standalone}
\usepackage{amsmath}
\begin{document}
$\displaystyle\frac{1\overbrace{7..........................7}^{\text{number 7 appears $n$ times}}}{\underbrace{8..........................8}_{\text{number 8 appears $n$ times}}5}$
\end{document}
A:
I'm not sure I understood the constraints but perhaps
\documentclass[border=12pt,12pt]{standalone}
\usepackage{amsmath}
\def\zz#1#2#3#4{%
\setbox0\hbox{$\scriptstyle#4$}%
#1{\hbox to \wd0{$#2$}}#3{#4}%
}
\begin{document}
$\displaystyle
\frac
{1\zz\overbrace{7\dotfill7}^{\text{number 7 appears $n$ times}}}
{\zz\underbrace{8\dotfill 8}_{\text{number 8 appears $n$ times}}5}
$
\end{document}
or as requested by jfbu with aligned dots (needs luatex)
\RequirePackage{luatex85}
#1{\hbox to \wd0{$\let\cleaders\gleaders#2$}}#3{#4}%
}
$
not sure it looks better
A:
Here are some options to play with:
\documentclass{article}
\usepackage{mathtools}
\begin{document}
\[
\frac
{1\overbrace{\makebox[5em]{7\dotfill 7}}^{\clap{\scriptsize\begin{tabular}{c} number 7 appears \\ $n$ times \end{tabular}}}}
{\underbrace{\makebox[5em]{8\dotfill 8}}_{\clap{\scriptsize\begin{tabular}{c} number 8 appears \\ $n$ times \end{tabular}}}5}
\]
\[
\frac
{1\overbrace{\makebox[5em]{7\dotfill 7}}^{\mathclap{\text{number 7 appears $n$ times}}}}
{\underbrace{\makebox[5em]{8\dotfill 8}}_{\mathclap{\text{number 8 appears $n$ times}}}5}
\]
\[
\frac
{1\makebox[5em]{7\dotfill 7}\mathllap{\overbrace{\rule{5em}{0pt}\vphantom{7}}^{\mathclap{\text{number 7 appears $n$ times}}}}}
{\mathrlap{\underbrace{\rule{5em}{0pt}\vphantom{8}}_{\mathclap{\text{number 8 appears $n$ times}}}}\makebox[5em]{8\dotfill 8}5}
\]
\end{document}
You can adjust the width 5em I've chosen to space out the content more. The difference between options 2 and 3 is in the spacing beside the non-braced numbers.
A:
I believe.
\documentclass{article}
\usepackage{amsmath}
\newlength{\ntimeslen}
\newcommand{\ntimes}[2]{%
\settowidth{\ntimeslen}{$\scriptstyle\text{number $#2$ appears $n$ times}$}%
#1{\makebox[\ntimeslen][s]{$#2\dotfill#2$}}%
\ifx#1\overbrace^\else_\fi
{\text{number $#2$ appears $n$ times}}%
}
\begin{document}
\begin{equation*}
\frac{1\!\ntimes{\overbrace}{7}}{\ntimes{\underbrace}{8}\!5}=
\frac{10^n+7\dfrac{10^n-1}{9\mathstrut}}{80\dfrac{10^n-1\mathstrut}{9}+5}=
\frac{16\cdot 10^n-7}{80\cdot 10^n-35}=\frac{1}{5}
\end{equation*}
\end{document}
| |
Freshly grated zucchini helps make this rich cake moist. When you're testing the cake for doneness, insert the wooden pick in several different places. You may hit a melted chocolate chip the first time, which might make you think the cake isn't done even if it is.
How to Make It
Preheat oven to 350°.
To prepare cake, coat a 12-cup Bundt pan with cooking spray; dust pan with 1 tablespoon flour.
Place granulated sugar and next 3 ingredients in a large bowl, and beat with a mixer at medium speed until well blended (about 5 minutes). Add eggs and egg whites, 1 at a time, beating well after each addition. Beat in 1 teaspoon vanilla.
Weigh or lightly spoon 25 ounces flour (about 2 1/2 cups) into dry measuring cups; level with a knife. Combine flour and next 5 ingredients in a medium bowl, stirring with a whisk.
Add flour mixture and buttermilk alternately to sugar mixture, beginning and ending with flour mixture. Stir in zucchini and chocolate chips. Pour batter into prepared pan. Bake at 350° for 1 hour or until a long wooden pick inserted in cake comes out clean. Cool pan on a wire rack 10 minutes. Remove cake from pan; cool completely on wire rack.
To prepare glaze, combine 3/4 cup powdered sugar and 3 tablespoons cocoa in a small bowl; stir with a whisk. Combine milk, 2 tablespoons chocolate chips, and 1/2 teaspoon vanilla in a 1-cup glass measure. Microwave at HIGH 30 seconds or until chocolate melts. Combine powdered sugar mixture with chocolate mixture, stirring with a whisk. Drizzle glaze over cake.
Young Chefs can:
Measure shredded zucchini and chocolate chips
Add zucchini and chocolate chips to batter
Older Chefs can: | https://www.myrecipes.com/recipe/chocolate-zucchini-cake-1 |
Despite mounting scientific evidence that viruses can cause changes in learning and memory, the reasons have remained elusive.
In a new study, scientists reported finding that viruses affect the immune system in a way that results in loss of connections between nerve cells in mice brains and those mice performed worse on tests of learning ability.
"This study in animals resonates with what we see in the clinic, where patients with acute or chronic infectious diseases often have weaker performance on motor skills and experience memory decline," study co-author Guang Yang said in a press release.
The research, conducted by scientists at the New York University School of Medicine was published online May 15th in Nature Medicine.
How a Virus Uses the Immune System to Change the Brain
One route of entry for viruses into the body is through the blood. Even if a virus initially enters through the lungs (like the flu virus) or the genital tract (like HIV), the virus often ends up in the blood.
Experiments in mice conducted by the investigators found that the effects of a virus-like infection — created by injecting the mice with a non-pathogenic compound that the body reacts to as if it were an infection — on the brain started in the bloodstream.
They found that when a virus enters the bloodstream, it triggers the immune system to respond. The first cells that respond to fight the foreign invader are immune cells that go by the name CX3CR1highLY6Clow monocytes in mice — a very unique subset of white cells. In humans, they go by the name CX3CR1highCD14dimCD16+, but, they both are the patrolling cells of the immune system.
Dendrites are projections from the brain's neuron cells that carry information from another neuron into the dendrite cell's body. Some dendrites have small projections called dendritic spines that process information received through the senses into memories. The responding monocytes in the mice released tumor necrosis factor alpha (TNFa) — a protein involved in modulating immune responses — that then traveled to the brain. Once in the brain, the study team found that TNFa blocked the formation of dendritic spines.
To test effects of infection on the animals' ability to learn, mice were trained to run on an accelerating rotating rod, while the investigators took images of the dendrites in their brain. Two days after the researchers infected the mice with the viral mimic, they did significantly worse on the rotating rod manipulation they had learned just days before.
Brain images showed the mice with the virus-like infection had lost than half the percentage of dendritic spines as uninfected mice did and led the scientist to conclude that immune disruption of synaptic networks was the cause of the impaired learning process they observed.
The study authors said that the unique subset of monocytes, as well as TNFa, may present potential therapeutic targets for preventing infection-induced cognitive dysfunction. | https://invisiverse.wonderhowto.com/news/viruses-might-cause-brain-changes-learning-problems-0177641/ |
Q:
How can I find why my merge sorting algorithm crash when sorting an array of 1 million element?
I'm a French student and trying to calculate the execution time of the Merge Sort algorithm for different size of array.
I also want to write the different execution time in a .csv file. But when my program tries to sort an array with 1 million elements the process returns -1073741571 (0xC00000FD) in Code::Blocks. So if you could point me to a way to find a solution I would be very grateful!
Here is my code:
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <time.h>
void genTab(int *tab, int n) {
int i;
for (i = 0; i < n; i++) {
tab[i] = rand() % 100;
}
}
void fusion(int *tab, int deb, int mid, int fin) {
int i = deb;
int j = mid + 1;
int k = deb;
int temp[fin + 1];
while ((i <= mid) && (j <= fin)) {
if (tab[i] <= tab[j]) {
temp[k] = tab[i];
i++;
} else {
temp[k] = tab[j];
j++;
}
k++;
}
while (i <= mid) {
temp[k] = tab[i];
i++;
k++;
}
while (j <= fin) {
temp[k] = tab[j];
k++;
j++;
}
for (i = deb; i <= fin; i++) {
tab[i] = temp[i];
}
}
void triFusion(int *tab, int i, int j) {
if (i < j) {
triFusion(tab, i, (int)((i + j) / 2));
triFusion(tab, (int)((i + j) / 2 + 1), j);
fusion(tab, i, (int)((i + j) / 2), j);
}
}
void reset(int *tab1, int *tab2, int n) {
for (int i = 0; i < n; i++) {
tab2[i] = tab1[i];
}
}
int main() {
srand(time(NULL));
clock_t start, end;
int nbrTest[15] = {
1000, 5000, 10000, 50000, 80000, 100000, 120000, 140000,
150000, 180000, 200000, 250000, 300000, 450000, 1000000
};
FILE *fp;
char *tpsExecution = "exeTime.csv";
fp = fopen(tpsExecution, "w");
fprintf(fp, "Array Size; Merge Time");
for (int i = 0; i < 15; i++) {
int n = nbrTest[i];
printf("Calculating time for an array of %d \n", n);
int *tab = malloc(sizeof(int) * n);
genTab(tab, n);
int *copie = malloc(sizeof(int) * n);
reset(tab, copie, n);
start = clock();
triFusion(tab, 0, n - 1);
end = clock();
float tpsFusion = (float)(end - start) / CLOCKS_PER_SEC;
reset(tab, copie, n);
printf("writing in the file\n");
fprintf(fp, "\n%d;%f", n, tpsFusion);
free(tab);
free(copie);
}
fclose(fp);
return 0;
}
A:
(Note: posted after the answer from @Eric Postpischil).
The function
void fusion(int * tab, int deb, int mid, int fin)
Has the line
int temp[fin+1];
and the value of fin comes through another function from the number of elements n to be sorted
triFusion(tab, 0, n-1);
and as an automatic variable, breaks the stack when n is large.
I suggest replacing the line with
int *temp = malloc((fin+1) * sizeof *temp);
if(temp == NULL) {
puts("malloc");
exit(1);
}
// ...
free(temp);
| |
These flavors are then accented by warming spices such as cinnamon, ginger and nutmeg to create a treat that’s as comforting as it is healthy.
Light and fluffy, these fall-inspired bars boast a complex flavor profile that combines the earthy taste of pumpkin with the deeper taste of dark chocolate.
Preheat your oven to 350°F (177°C) and prepare an 8 x 8-inch (20.3 x 20.3-cm) baking pan by lining it with a sheet of aluminum foil or parchment paper, leaving a few inches of overhang on the sides to allow for easy removal.
Lightly grease the foil and set aside.
Add the pumpkin puree, eggs, almond butter, sugar, almond milk, cinnamon, ginger and nutmeg to a high-speed blender, and process on high for about 10 seconds or until all of the ingredients are combined and the mixture is smooth. Add the almond flour, coconut flour and baking powder, and continue processing for about 30–40 seconds until the batter becomes smooth and creamy. Finally, fold in the chocolate chips by hand, reserving a couple tablespoons to sprinkle on the top. The batter will be a little thick and paste-like, which is normal.
Transfer the batter into your prepared pan and use a spatula to distribute it evenly before topping with the remaining chocolate chips. Bake for 25–27 minutes, until the bars begin to turn golden brown around the edges and a toothpick inserted into the center comes out clean. Remove them from the oven and let them cool in the pan for about 15 minutes before transferring them to a wire rack to cool completely. Use a sharp knife to cut them into individual bars and store them in an airtight container at room temperature for up to 5 days. | http://origin-www.besthealthmag.ca/recipes/pumpkin-spice-bars/ |
Occasionally on this blog I will write about my beliefs and the things I learn about the world based on those beliefs. To learn more about what I believe, please visit mormon.org.
Bayes' Theorem has also helped me increase my faith. Before I explain that, though, here is a quick intro to how Bayes' Theorem works.
This formula gives you the probability of X (the unknown) given Y (the observation). A few examples should help clear things up a bit. The example on Wikipedia is great. We'll work through another example here.
Now, I sit up and look out the window and I see sagebrush. I have a little bit more information. Does my probability of being in Nevada change? Is it higher or lower?
There is a lot of sagebrush in Nevada.
When I woke up, the probability that I was in Nevada was just 0.5. But, when I sat up and saw sagebrush the probability increased to 0.635. A single observation, and my probability of being in Nevada increased by 27% ( (0.635 - 0.5)/0.5 = 0.27). If I also see a casino or an Elvis impersonator, the probability will keep going up. If I see a tree or rain, the probability will go down.
In the Book of Mormon, a prophet named Alma teaches about faith (Alma 32), and his explanation fits very well with Bayes' Theorem. Thinking of Bayes' Theorem in terms of faith helped me understand Bayes' Theorem. Then, thinking of faith in terms of Bayes' Theorem gave me a new perspective on faith.
The prior belief must be greater than 0 and less than 1, else the posterior belief will always be the same as the prior belief.
The unknown (Nevada) and the observation (sagebrush) must be in some way connected. If not, the observation will have no effect on the posterior belief.
Our prior belief has to be greater than 0 (we must at least have a desire to believe), but less than 1 (if we already know something, there is no need for faith).
We have to be able to observe something that is correlated to the unknown, else our faith can't grow (this correlation and observation are promised by heavenly law).
With Bayes' Theorem P(X|Y) will never reach 1 (or 0). You cannot have a perfect knowledge of something that can be only indirectly observed. However, after each observation, the posterior belief becomes the prior belief of the next observation, and as we continue to make observations we can eventually have very high confidence that our belief is the truth.
But, the posterior belief continues to approach the truth only if we are actively experimenting and making observations. In the context of faith, that means that we must act according to our faith, and consciously observe the result. The first time I fast and pray for something, and that thing happens, I may credit coincidence. After years of fasting and praying every month, if I carefully observe the result each time, I can see that coincidence has little to do with it. Faith comes from experimenting on the Word again and again, and always observing the result. Count your blessings, for it is mathematically proven to increase your faith! | http://learningtoprogrambook.com/blog/Faith-and-Bayes-Theorem |
A tourist traveled 10 km away from the city by bus, and then continued his journey by foot in the same direction at the speed of 5 km/h. At which distance y was he from the city in x hours of walking?
-
college algebra
Kiran drove from City A to City B, a distance of 242 mi. She increased her speed by 10 mi/h for the 351-mi trip from City B to City C. If the total trip took 12 h, what was her speed from City A to City B?
-
Social studies
In which area did Sparta differ most from Athens? A. the role of the city's assembly B. the city's economic basis C. the city's overall military strength D. the exclusion of women from politics
-
algebra
An airplane flies from City A in a straight line to City B, which is 70 kilometers north and 140 kilometers west of City A. How far does the plane fly?
-
physics
An airplane flies 200 km due west from city A to city B and then 280 km in the direction of 30.5° north of west from city B to city C. (a) In straight-line distance, how far is city C from city A? (b) Relative to city A, in what
You can view more similar questions or ask a new question. | https://www.jiskha.com/questions/1816696/a-boat-on-a-river-traveled-from-city-a-to-city-c-with-a-stop-at-city-b-on-the-first-part |
A symbol of freedom and democracy towering above the New York Harbor she has welcomed the tired the poor and the huddled masses into America for more than 100 years. Authoritative and richly detailed THE STATUE OF LIBERTY explores the remarkable steps leading to it's creation from the initial struggles to find adequate funding to it's celebratory centennial restoration. Rare footage and archival photos document the radical methods the French devised for shaping the massive sculpture and for transporting their incredible creation to America. Discover the engineering marvel of it's internal framework designed by Gustave Eiffel and the dimensions of it's American-built pedestal. Finally after measuring the length of her index finger (8 feet) take a historical tour of the island on which it stands. Originally a friendly gesture between nations the "Mother of Exiles" stands today and into a new millennium as a testament to human ingenuity freedom and hope. DVD Features: Save Our History: Ellis Island; Statue of Liberty Facts; Interactive Menus; Scene Selection. | https://www.wowhd.co.nz/statue-of-liberty/733961712537 |
GW842166X shows similar potency and efficacy for rat and human recombinant CB2 receptors with EC50 of 91 nM and 63nM, respectively. GW842166X exhibits full agonist potency with an EC50 of 133 nM and Emax of 101% in cyclase assays. GW842166X exhibits weak agonist potency with an EC50 of 7.780 μM and Emax of 84% in FLIPR assays.GW842166X has an oral bioavailability of 58% and a half-life of 3 h when dosed orally in the rat. GW842166X has extremely high potency with an oral ED50 of 0.1 mg/kg and shows full reversal of hyperalgesia at 0.3 mg/kg in the FCAa model of inflammatory pain. GW842166X orally administrated at a dose of 15 mg/kg for 8 days produced a significant reversal of the CCI induced decrease in paw withdrawal threshold in a rat model of neuropathic pain.
|Cell Experiment|
|Cell lines|
|Preparation method|
|Concentrations|
|Incubation time|
|Animal Experiment|
|Animal models||rat model of neuropathic pain|
|Formulation||saline|
|Dosages||15 mg/kg|
|Administration||Orally administrated once daily for 8 days|
|Species||Mouse||Rat||Rabbit||Guinea pig||Hamster||Dog|
|Weight (kg)||0.02||0.15||1.8||0.4||0.08||10|
|Body Surface Area (m2)||0.007||0.025||0.15||0.05||0.02||0.5|
|Km factor||3||6||12||8||5||20|
|Animal A (mg/kg) = Animal B (mg/kg) multiplied by||Animal B Km|
|Animal A Km|
For example, to modify the dose of resveratrol used for a mouse (22.4 mg/kg) to a dose based on the BSA for a rat, multiply 22.4 mg/kg by the Km factor for a mouse and then divide by the Km factor for a rat. This calculation results in a rat equivalent dose for resveratrol of 11.2 mg/kg.
|Molecular Weight||449.25|
|Formula||C18H17Cl2F3N4O2|
|CAS Number||666260-75-9|
|Purity||>98%|
|Solubility||DMSO 10 mg/mL|
|Storage||at -20°C|
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An automobile tire contains a certain volume of air at 30 psig and 70 °F. The barometric pressure is 29.50 Hg. If due to running conditions, the temperature of air in the tire rises to 160 °F. What will be the gauge pressure?
-
Science
1. The air temperature is 70 degrees, and the relative humidity is 90%. Which conclusion can be made? a. The dew point temperature is 90% of the air temperature. b. The air holds little water vapor and is relatively dry. c. The
-
Physics
A hot-air balloon is accelerating upward under the influence of two forces, its weight and the buoyant force. For simplicity, consider the weight to be only that of the hot air within the balloon, thus ignoring the balloon fabric
-
physics
A 1.6 air bubble is released from the sandy bottom of a warm, shallow sea, where the gauge pressure is 1.6 . The bubble rises slowly enough that the air inside remains at the same constant temperature as the water. What is the
-
Physics
A sound wave is traveling in warm air when it hits a layer of cold, dense air. If the sound wave hits the cold air interface at an angle of 23 degree angle, what is the angle of refraction? Assume that the cold air temperature is
-
ecology
An air conditioner lowers the temperature of the surrounding air. water condensed from the air is most likely to run out of the bottom water hose of a automobile air condition when the air entering the air conditioner. a- contains
-
science
Which of the following would most likely cause rising air? A warm air and low pressure B cold air and low pressure C warm air and high pressure D cold air and high pressure Please give us the answer.
-
Chemistry
The total mass that can be lifted by a balloon is given by the difference between the mass of air displaced by the balloon and the mass of the gas inside the balloon. Consider a hot air balloon that approximates a sphere 5.00 m in
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sci
a radiator is attached to a wall on one side of a room. which is mostlikely way its heat will warm the air on the other side of the room? a. by conduction b. by convection c. the warm air will sink and condense d. the warm air
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The authors argue that the global refugee regime, distinct from its component organizations, lacks a clearly defined system of governance due to: diffuse governance arrangements; conflation of governance of the regime with governance of UNHCR; and lack of effective coordination, dialogue and political engagement necessary for international cooperation and the realization of the regime’s core objectives of protection and solutions for refugees. Individual states are responsible for implementing the regime’s norms within their jurisdictions, with control over the quantity and quality of asylum they grant to refugees on their territory, while outcomes for refugees are increasingly shaped by decisions taken in other fields (e.g. development, humanitarianism, human rights, labor migration, travel, security). Additionally, there are no binding obligations on states to cooperate to ensure the functioning of the regime or to share the burden or responsibility for refugee protection.
In response to these gaps, the authors propose enhanced governance arrangements for the global refugee regime that would contribute to enhanced protection and solutions for refugees and more predictability for states and the international system. They identify four functions needed to facilitate collective action—dialogue, facilitation, expertise and oversight—and propose: (a) a forum for dialogue between refugee-hosting and donor states and other stakeholders, including the private sector, NGOs and refugees themselves; (b) the capacity for political facilitation between actors, i.e. to identify principled yet practical bargains that can meet states’ interests while advancing refugee protection and solutions; (c) enhanced capacity for analysis and evidence-based planning; and (d) oversight and accountability to ensure compliance with international norms.
Specific recommendations are as follows:
- New governance mechanisms: The Global Refugee Forum (GRF) and Support Platform, detailed in the Global Compact on Refugees (GCR), should be supported as new governance mechanisms that, if combined, could provide a mechanism for dialogue, facilitation, expertise, delivery and oversight. Working groups should be authorized to develop responses to specific refugee situations and make proposals that require political and material support. If the GRF proves inadequate, the ten largest host and ten largest donor countries should establish a ‘R20’ mechanism. New governance mechanisms should be supported by a secretariat that can provide political analysis and research.
- Ensuring coherence: A special representative of the UN Secretary-General for displaced persons should be tasked with ensuring sustained engagement and complementarity across the UN system and with regional organizations and other actors, and more predictable efforts to address root causes of displacement and to respond to displacement when it occurs.
- Strengthening accountability: Mechanisms are needed to ensure more consistent state compliance, including through authoritative and legitimate monitoring, enforcement and accountability mechanisms to address causes of displacement and provision of protection and solutions.
- Addressing gaps: Notwithstanding the potential benefits of the GCR, the reliability of the refugee regime would benefit from additional instruments and mechanisms to ensure that burden and responsibility sharing for refugees is ultimately predictable, equitable and sufficient in both scope and scale. | https://www.jointdatacenter.org/literature_review/governance-of-the-global-refugee-regime/ |
---
abstract: |
In this article, a sensitivity analysis of long-term cash flows with respect to perturbations in the underlying process is presented. For this purpose, we employ the [*martingale extraction*]{} through which a pricing operator is transformed into what is easier to address. The method of Fournie will be combined with the martingale extraction. We prove that the sensitivity of long-term cash flows can be represented in a simple form.
Key Words: Long-term cash flows, Martingale extraction, Malliavin calculus
author:
- |
Hyungbin Park[^1]\
\
\
\
\
\
title: 'Sensitivity Analysis of Long-Term Cash Flows[^2]'
---
Introduction
============
In finance, we often encounter the quantity of the form: $$p_T:=\mathbb{E}^{\mathbb{Q}}[e^{-\int_0^T r(X_t)\,dt} f(X_T)]\;.$$ For example, if $Q$ is a risk-neutral measure and $r(X_t)$ is a short interest rate, then the quantity is the current price of the option with payoff $f(X_T)$ at time $T.$ If $\mathbb{Q}$ is an objective measure, $f$ is a utility function of an agent and $r(X_t)$ is a discount rate of the agent, then the quantity is the discounted expected utility of the agent. This article examines a sensitivity analysis of the quantity $p_T$ for large $T$ with respect to perturbations in the underlying process $X_t.$
The underlying process $X_t$ in this article is a conservative diffusion process in a Brownian environment. Let $W_t=(W_1(t),W_2(t),\cdots,W_d(t))^{\top}$ be a standard $d$-dimensional Brownian motion.
\[assume:Markov\_X\] The underlying process $X_t$ is a $d$-dimensional time-homogeneous Markov diffusion process. Assume that $X_t$ satisfies the following stochastic differential equation: $$\begin{aligned}
&dX_{t}=b(X_{t})\, dt+\sigma(X_{t})\, dW_{t}\;,\quad X_{0}=\xi\;.
\end{aligned}$$ Here, $b$ is a $d$-dimensional column vector and $\sigma$ is a $d\times d$ matrix. $b(\cdot)$ and $\sigma(\cdot)$ are continuously differentiable and $\sum_{i,j=1}^{d}\sigma_{ij}(x)v_iv_j>0$ for all $v\in\mathbb{R}^d-\{0\}.$ In addition, we assume that the range of $X_t$ is $\mathbb{R}^d,$ that is, the process does not explode in finite time $t.$
$r(\cdot)$ is a continuously differentiable function on $\mathbb{R}^d.$
We explore a sensitivity analysis for the quantity $p_T$ with respect to the perturbation in the underlying process $X_{t}.$ Let $X_{t}^{\epsilon}$ be a perturbed process of $X_{t}$ (with the same initial value $\xi=X_{0}=X_{0}^{\epsilon}$) of the form: $$%\label{eqn:form_perturb}
dX_t^{\epsilon}=b_\epsilon(X_{t}^{\epsilon})\,dt+\sigma_\epsilon(X_{t}^{\epsilon})\,dW_{t}$$ with $b_0(\cdot)=b(\cdot)$ and $\sigma_0(\cdot)=\sigma(\cdot).$ The perturbed quantity is given by $$\label{eqn:perturbed_quant_intro}
p_T^\epsilon:=\mathbb{E}^{\mathbb{Q}}[e^{-\int_{0}^{T}r(X_{s}^{\epsilon})\,ds}f(X_{T}^{\epsilon})\,]\;.$$ For the sensitivity analysis, we compute $$\left.\frac{\partial }{\partial\epsilon}\right|_{\epsilon=0}
p_{T}^{\epsilon}$$ and investigate the behavior of this quantity for large $T.$ The sensitivity with respect to the perturbation of the drift term $b_\epsilon(X_{t})$ is called the [*rho*]{}, and the sensitivity with respect to the diffusion term $\sigma_\epsilon(X_{t})$ is called the [*vega.*]{} The sensitivity with respect to the initial value $X_0=\xi$ is given by $$\frac{\partial p_{T}}{\partial\xi}\quad\left(=
\frac{\partial }{\partial \xi}\,\mathbb{E}^{\mathbb{Q}}[e^{-\int_{0}^{T}r(X_{s})\,ds}\,f(X_{T})|X_0=\xi\,]
\right)$$ and is called the [*delta.*]{}
The main contribution of this article is the use of the martingale extraction method to the sensitivity analysis. Assume that $(X_t^{\epsilon},r)$ admits the martingale extraction that stabilizes $f$ (Definition \[def:martingale\_extract\] and \[def:stabilizing\_martingale\_extract\]), then it can be easily shown that $$p_{T}^\epsilon\simeq e^{-\lambda(\epsilon) T}l_\epsilon(\xi)$$ for some number $\lambda(\epsilon)$ and function $l_\epsilon(\xi).$ Here, for two nonzero functions $p_T$ and $q_T$ of $T,$ the notation $p_{T}\simeq q_{T}$ means that $\lim_{T\rightarrow\infty}\frac{p_{T}}{q_{T}}=1.$ When $T$ is large, because $e^{-\lambda(\epsilon) T}$ dominates the perturbed quantity $p_{T}^\epsilon,$ we can anticipate that the long-term behavior of $p_T^\epsilon$ is mainly determined by $e^{-\lambda(\epsilon) T}.$ We may then expect $$\left.\frac{\partial }{\partial\epsilon}\right|_{\epsilon=0}p_{T}^{\epsilon}\,\simeq\, -\lambda'(0)\,T \cdot e^{-\lambda T}l(\xi)+e^{-\lambda T}\left.\frac{\partial }{\partial\epsilon}\right|_{\epsilon=0}l_{\epsilon}(\xi)$$ and we thus obtain the following simple equation: $$\frac{\left.\frac{\partial }{\partial\epsilon}\right|_{\epsilon=0}p_{T}^{\epsilon}}{T\cdot p_{T}}
\,\simeq\, -\lambda'(0)\;.$$ For the delta, because $\lambda$ is independent of the initial value of $X_{t}$ - as we will see soon - we have $$\frac{\partial p_{T}}{\partial\xi}\,\simeq\, e^{-\lambda T}\,l'(\xi)\;,$$ thus we obtain $$\frac{\ \frac{\partial p_{T}}{\partial x} \ }{p_{T}}\,\simeq\,\frac{\ l'(\xi)\ }{\ l(\xi)\ }\;.$$
To justify these arguments, we employ the method of Fournie [@Fournie], in which there is a remarkable technique for sensitivity analysis. See [@Alos], [@Nualart] and [@Nunno] as references for the method. Unfortunately, this method cannot be applied to functionals of the following form: $$\mathbb{E}^{\mathbb{Q}}[e^{-\int_{0}^{T}r(X_{s})\,ds}f(X_{t})]\;,$$ and this is the form that interests us. This method (for calculating the delta and vega) is valid only for [*discretely monitored functionals*]{} of the following form: $$\mathbb{E}^{\mathbb{Q}}[f(X_{t_{1}},X_{t_{2}},\cdots,X_{t_{m}})\,]$$ such that the process $X_{t}$ is detected only for finite times up to maturity $T.$ In our case, however, the expectation contains the term $$e^{-\int_{0}^{T}r(X_{s})\,ds}$$ which depends on the entire path of $X_{t}$ up to time $T.$ The martingale extraction is useful in overcoming this problem. It is largely because the martingale extraction transforms the functionals depending on the entire path of $X_{t}$ up to time $T$ to the discretely monitored functionals. Thus, while applying the martingale extraction, the Fournie method is able to be successfully applied to our cases.
Another contribution of this article is a generalization of the result of Fournie for the rho. In the paper of Fournie, the perturbation is linear of the form $b_\epsilon=b+\epsilon\overline{b}$ and the function $\overline{b}$ is bounded. In addition, the diffusion matrix $\sigma$ satisfies the uniform ellipticity condition and the payoff function satisfies the $L^2$-condition, that is, $\mathbb{E}^{\mathbb{Q}}[f^2(\cdot)]<\infty.$ We slightly generalize these conditions in Proposition \[prop:rho\] in Appendix. Many financial models including the examples in this paper satisfy the generalized conditions.
Many authors employed the martingale extraction to investigate financial and economic problems. Hansen and Scheinkman explored long-term risk in [@Hansen],[@Hansen2] and [@Hansen3], in which the martingale extraction was used to show that a pricing operator consists with three components: an exponential term, a martingale and a transient term. They offered financial and economic meanings of the terms.
Borovicka, Hansen, Hendricks and Scheinkman [@Borovicka11] exploit the martingale extraction for a sensitivity analysis. They investigate shock exposure in terms of shock elasticity, which measures the impact of a current shock. Let $G_t$ be cash flow at time $t.$ It is assumed that $G_t$ is a multiplicative functional. They consider the following perturbation form, which is somewhat different from the perturbation form in this paper. Set $$H_t^\epsilon:=e^{\int_0^T\kappa_\epsilon(X_s)\,ds+\epsilon\int_0^T\alpha(X_s)\,dW_s}\;.$$ Here, $\kappa_\epsilon(\cdot)$ and $\alpha(\cdot)$ are given functions and define the direction of perturbation. Put the perturbed cash flow by $q_T^\epsilon:=\mathbb{E}^{\mathbb{Q}}\left[G_TH_T^\epsilon\,\right].$ The quantity $\left.\frac{\partial }{\partial\epsilon}\right|_{\epsilon=0}
q_{T}^{\epsilon}$ is called the shock elasticity. The shock elasticity for large $T$ was analyzed in their work. The shock elasticity is not the same, but is somewhat similar with the notion of delta. Their result coincides with Theorem \[thm:delta\] in this article
We now review the risk elasticity, which is similar to the rho and vega. The perturbed expected return is defined by $$R_T^\epsilon:=\frac{\mathbb{E}[G_TH_T^\epsilon]}{\mathbb{E}[e^{-\int_0^Tr_s\,ds}\,G_TH_T^\epsilon]}$$ and the quantity $\left.\frac{\partial }{\partial\epsilon}\right|_{\epsilon=0}
R_{T}^{\epsilon}$ is of interest to us and is called the risk elasticity. In their paper, a more general form of discount factor than $e^{-\int_0^Tr_s\,ds}$ is considered. They do not provide a long-term analysis for risk elasticity. Borovicka, Hansen and Scheinkman [@Borovicka14b] present more direct way of computing the shock elasticities.
The martingale extraction method is linked to several financial and economic topics. The connection to spectral theory can be found in [@Davydov], [@Gorovoi], [@Lewis], [@Li13], [@Lim12], [@Linetsky04], [@Linetsky08] and [@Linetsky]. Ross recovery is also closely related to the martingale extraction. Refer to [@Borovicka14], [@Han], [@Park], [@Qin14] and [@Ross13].
The following provides an overview of this article. We present the martingale extraction method in Section \[sec:martingale\_extraction\]. In Section \[sec:sen\_drift\_vol\] and \[sec:delta\], the sensitivity analysis for long-term cash flows is investigated. Sections \[sec:ex\_option\_prices\] and \[sec:ex\_utility\] present examples, and the last section summarizes the paper. The proofs of main results and the details of examples are in Appendices.
Martingale extraction {#sec:martingale_extraction}
=====================
In this section, we explore the notion of the martingale extraction. Let $\mathcal{L}$ be the infinitesimal generator corresponding to the operator $$f\mapsto p_T=\mathbb{E}^{\mathbb{Q}}[e^{-\int_0^T r(X_t)\,dt} f(X_T)]\;.$$ Then, $$\mathcal{L}:=\frac{1}{2}\sum_{i,j=1}^da_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j}+\sum_{i=1}^db_i(x)\frac{\partial}{\partial x_i}-r(x)$$ where $a=\sigma\sigma^{\top}.$ We are interested in an eigenpair $(\lambda,\phi)$ of $\mathcal{L}\phi=-\lambda\phi$ with positive function $\phi.$ There are two possibilities.
- there is no positive solution $\phi$ for any $\lambda\in\mathbb{R}$, or
- there exists a number $\overline{\lambda}$ such that it has positive solutions for $\lambda\leq\overline{\lambda}$ and has no positive solution for $\lambda>\overline{\lambda}.$
Refer to [@Pinsky] for proof. In this article, we assume the second case.
Let $(\lambda,\phi)$ be an eigenpair of $\mathcal{L}\phi=-\lambda\phi$ with positive function $\phi.$ It is easily checked that $$M_{t}:=e^{\lambda t-\int_{0}^{t} r(X_{s})ds}\,\phi(X_{t})\,\phi^{-1}(\xi)$$ is a local martingale.
\[def:martingale\_extract\] When the local martingale $M_{t}$ is a martingale, we say that $(X_t,r)$ admits the [*martingale extraction*]{} with respect to $(\lambda,\phi).$
When $M_t$ is a martingale, we can define a new measure $\mathbb{P}$ by $$\mathbb{P}(A):=\int_{A}M_{t}\;d\mathbb{Q}
=\mathbb{E}^{\mathbb{Q}}\left[\mathbb{I}_{A}
M_{t}\right]\quad\text { for }\; A\in \mathcal{F}_{t}\,.$$ The measure $\mathbb{P}$ is called [*the transformed measure*]{} from $\mathbb{Q}$ with respect to $(\lambda,\phi).$ The definition is well defined: If $A\in\mathcal{F}_{t}\,,$ then for $0<t<s,$ we have $\mathbb{E}^{\mathbb{Q}}\left[\mathbb{I}_{A}
M_{t}\right]=
\mathbb{E}^{\mathbb{Q}}\left[\mathbb{I}_{A}
M_{s}\right].$ Using this transformed measure $\mathbb{P},$ $p_T$ can be expressed by $$\label{eqn:operator_decomposition}
\begin{aligned}
p_{T}
=\mathbb{E}^{\mathbb{Q}}
[e^{-\int_{0}^{T} r(X_{s})ds}f(X_{T})]=\phi(\xi)\,e^{-\lambda T}\cdot
\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}f)(X_{T})]
\,.
\end{aligned}$$ This relationship implies that the quantity $p_T$ can be expressed in a relatively more manageable manner. The term $\mathbb{E}^{\mathbb{P}}[(\phi^{-1}f)(X_{T})]$ depends on the final value of $X_T,$ whereas $\mathbb{E}^{\mathbb{Q}}[e^{-\int_{0}^{T} r(X_{s})ds}f(X_{T})]$ depends on the whole path of $X_t$ at $0\leq t\leq T.$ This advantage makes it easier to analyze the sensitivity of long-term cash flows. As a special case, if the density function of $X_t$ under $\mathbb{P}$ is known, one can directly analyze the term $\mathbb{E}^{\mathbb{P}}[(\phi^{-1}f)(X_{T})].$
We now observe how the dynamic of $X_{t}$ is changed when the underlying measure is changed from the measure $\mathbb{Q}$ to the transformed measure $\mathbb{P}.$ We know that the Radon-Nikodym derivative of $\mathbb{Q}$ with respect to $\mathbb{P}$ on $\mathcal{F}_{t}$ is $$M_{t}=e^{\lambda t-\int_{0}^{t} r(X_{s})ds}\;\phi(X_{t})\,\phi^{-1}(\xi)\;.$$ For convenience, let $$\varphi:=\sigma^\top\cdot\frac{\nabla \phi}{\phi}\;,$$ where $\nabla\phi$ is the $d\times 1$ gradient vector of $\phi.$ We say $\varphi$ is the [*martingale exponent*]{} of $M_t.$ According to the Girsanov theorem, we know that a process $B_{t}$ defined by $$B_{t}:=W_{t}-\int_{0}^{t}\varphi(X_{s})\,ds$$ is a Brownian motion under $\mathbb{P}.$ Therefore, $X_{t}$ follows $$\begin{aligned}
dX_{t}
&=b(X_{t})\, dt+\sigma(X_{t})\, dW_{t} \\
&=(b(X_{t})+\sigma(X_{t})\varphi(X_{t}))\, dt+\sigma(X_{t})\, dB_{t} \;.
\end{aligned}$$ This equation gives us the dynamic of $X_{t}$ under $\mathbb{P}.$
Among all possible martingale extractions, we choose a special one, which will be useful for sensitivity analysis of long-term cash flows. The choice depends on the function $f$ and is not unique in general.
\[def:stabilizing\_martingale\_extract\] Let $(\lambda,\phi)$ be an eigenpair of $\mathcal{L}\phi=-\lambda\phi$ with positive $\phi.$ Assume that $(X_t,r)$ admits the martingale extraction with respect to $(\lambda,\phi).$ We say the martingale extraction of $(\lambda,\phi)$ [*stabilizes*]{} $f$ if $$\mathbb{E}^{\mathbb{P}}\left[(\phi^{-1}f)(X_{T})\right]$$ converges to a nonzero constant as $T\rightarrow\infty,$ where $\mathbb{P}$ is the transformed measure with respect to $(\lambda,\phi).$
The definition of the term ‘stabilize’ is somewhat different from the meaning used in [@Hansen2]. It is noteworthy that if $(\lambda,\phi)$ and $(\beta,\pi)$ are two eigenpairs that induce the martingale extractions stabilizing the common $f,$ then $\lambda=\beta.$ The stabilizing martingale extraction characterizes the exponential decay (or growth) rate of the quantity $p_T$ as $T\rightarrow\infty.$ If the martingale extraction of $(\lambda,\phi)$ stabilizes $f,$ then $$\lim_{T\rightarrow\infty}\ln p_T=-\lambda\;.$$ For more about the stabilizing martingale extraction, refer to Appendix \[app:stabiling\_mart\_extrac\], in which there are sufficient conditions for martingale extractions to stabilize the function $f.$
Sensitivity on drift and volatility {#sec:sen_drift_vol}
===================================
We now investigate how the martingale extraction is used for the sensitivity analysis. For the rho and the vega, consider the perturbed process $X_{t}^{\epsilon}$ expressed by $$dX_{t}^{\epsilon}=b_\epsilon(X_{t}^{\epsilon})\,dt+\sigma_\epsilon (X_{t}^{\epsilon})\,dW_{t}$$ where $b_0(\cdot)=b(\cdot)$ and $\sigma_0(\cdot)=\sigma(\cdot).$ Assume that the perturbed process $X_t^\epsilon$ satisfies the conditions in Assumption \[assume:Markov\_X\]. We slightly generalize the form of perturbed quantity in equation to $$p_{T}^{\epsilon}:=\mathbb{E}^{\mathbb{Q}}
[ e^{-\int_{0}^{T}r_\epsilon(X_{s}^{\epsilon})ds}
f_\epsilon(X_{T}^{\epsilon})]$$ with $r_0(\cdot)=r(\cdot)$ and $f_0(\cdot)=f(\cdot).$ Then $$\left.\frac{\partial }{\partial \epsilon}\right|_{\epsilon=0}p_{T}^{\epsilon}$$ for large $T$ is of interest to us.
The perturbed quantity $p_T^\epsilon$ can be expressed in a relatively more manageable manner by using the martingale extraction. We assume that $(X_{t}^{\epsilon},\, r_\epsilon)$ admits a martingale extraction stabilizing $f_\epsilon.$ Denote the corresponding eigenpair, the martingale exponent and the transformed measure by $(\lambda(\epsilon),\phi_{\epsilon}),$ $\varphi_{\epsilon}$ and $\mathbb{P}_{\epsilon},$ respectively. Then, $$\label{eqn:perturbed_quant}
\begin{aligned}
p_{T}^{\epsilon}
&=\phi_{\epsilon}(\xi)\,e^{-\lambda(\epsilon) T} \cdot\mathbb{E}^{\mathbb{P_{\epsilon}}}
[(\phi^{-1}_{\epsilon}f_\epsilon)(X_{T}^{\epsilon})]\;. \\
\end{aligned}$$
We will explore $\left.\frac{\partial }{\partial \epsilon}\right|_{\epsilon=0}p_{T}^{\epsilon}$ by analyzing the components $\phi_{\epsilon}(\xi),$ $e^{-\lambda(\epsilon) T}$ and $ \mathbb{E}^{\mathbb{P_{\epsilon}}}
[(\phi^{-1}_{\epsilon}f_\epsilon)(X_{T}^{\epsilon})].$ Differentiate with respect to $\epsilon$ and evaluate at $\epsilon=0,$ then $$\label{eqn:differ_rho}
\begin{aligned}
\frac{\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}p_{T}^{\epsilon}}
{T\cdot p_{T}}
&=-\lambda'(0) +\frac{\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\phi_{\epsilon}(\xi)}{T\cdot\phi(\xi)} \\
&+\frac{\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}_{\epsilon}
f_\epsilon)(X_{T})]}
{T\cdot\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}
f)(X_{T})]}
+\frac{\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}
^{\mathbb{P_{\epsilon}}}
[(\phi^{-1}
f)(X_{T}^{\epsilon})]}
{T\cdot\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}
f)(X_{T})]}
\;\; .
\end{aligned}$$ Since this is a stabilizing martingale extraction, we know that $\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}
f)(X_{T})]$ in the denominator in the last two terms converge to a nonzero constant as $T\rightarrow\infty.$ When the perturbations are small in some sense, the last three terms converges to zero as $T\rightarrow\infty,$ thus we can anticipate the following simple relationship: $$\label{eqn:final_eqn}
\lim_{T\rightarrow\infty}\frac{1}{T}\left.\frac{\partial}{\partial\epsilon}\right|_{\epsilon=0}\ln p_T^\epsilon
=\lim_{T\rightarrow\infty}\frac{\left.\frac{\partial }{\partial\epsilon}\right|_{\epsilon=0}p_{T}^{\epsilon}}{T\cdot p_{T}}= -\lambda'(0)\;.$$
We now shift our attention to the four terms in equation . Only the last term is involved with the perturbation in the underlying process. The main contribution of this article is to control the last term. In the first term, $\lambda(\epsilon)$ is differentiable at $\epsilon=0$ for many financially meaningful cases. In the second term, $\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\phi_{\epsilon}(\xi)$ is independent of $T.$ In the third term, $\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}_{\epsilon}
f_\epsilon)(X_{T})]$ is just of an ordinary problem of differentiation and integration. Those conditions can be checked case-by-case, thus we do not go further details of the first three terms here. Assume the following conditions.
$\lambda(\epsilon)$ and $\phi_{\epsilon}(\xi)$ are differentiable at $\epsilon=0.$
$\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}_{\epsilon}
f_\epsilon)(X_{T})]$ is differentiable at $\epsilon=0$ and $\frac{1}{T}\cdot\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}_{\epsilon}
f_\epsilon)(X_{T})]\rightarrow 0$ as $T\rightarrow\infty.$
These conditions are satisfied for many financially meaningful perturbations as we will see soon. It is noteworthy that we can occasionally interchange the differentiation and the integration: $$\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}
\left[(\phi^{-1}_{\epsilon}
f_\epsilon)(X_{T})\right]=\mathbb{E}^{\mathbb{P}}
\left[\left(\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\phi^{-1}_{\epsilon}
f_\epsilon\right)(X_{T})\right]\;.$$ This holds, for example, if $h_\epsilon:=\phi^{-1}_{\epsilon}f_\epsilon$ satisfies the hypothesis of Theorem \[thm:payoff\].
To achieve the relationship in equation , we have to show that the last part satisfies $$\frac{1}{T}\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}
^{\mathbb{P_{\epsilon}}}
[(\phi^{-1}
f)(X_{T}^{\epsilon})]\rightarrow 0\;.$$ The differentiability and the convergence to zero do not look clear. We will find sufficient conditions when this holds. The conditions for the perturbation on the drift $b_\epsilon(\cdot)$ and the volatility $\sigma_\epsilon(\cdot)$ are demonstrated in Section \[sec:rho\] and \[sec:vega\], respectively.
Rho {#sec:rho}
---
In this section, the [*rho*]{} of the quantity $p_T$ is investigated for large $T.$ Consider the perturbed process $X_{t}^{\epsilon}$ expressed by $$dX_{t}^{\epsilon}=b_\epsilon(X_{t}^{\epsilon})\,dt+\sigma (X_{t}^{\epsilon})\,dW_{t}\;,$$ where $b_0(\cdot)=b(\cdot).$ Define $k_\epsilon(x):=(\sigma^{-1}b_\epsilon+\varphi_{\epsilon})(x)$ and $k(x):=k_0(x).$ Assume that $k_\epsilon(x)$ is continuously differentiable at $\epsilon=0$ for each $x.$ Denote the derivative at $\epsilon=0$ by $\overline{k}(x),$ that is, $$\overline{k}(x):=\left.\frac{\partial}{\partial\epsilon}\right|_{\epsilon=0}k_\epsilon(x)\;.$$ We write the usual $d$-dimensional Euclidean norm by $|\cdot|.$ Assume that there exists a function $g:\mathbb{R}^d\rightarrow\mathbb{R}$ such that $$\left|\frac{\partial k_\epsilon(x)}{\partial\epsilon}\right|\leq g(x)$$ on $(\epsilon,x)\in I\times\mathbb{R}^d$ for an open interval $I$ containing $0.$ Refer to Appendix \[app:pf\_rho\_expo\_condi\], \[app:pf\_ext\_rho\] and \[app:pf\_rho\_coro\] for the proofs of the following theorems and the corollary.
\[thm:rho\_expo\_condi\] Suppose that the following conditions hold.
- there exists a positive number $\epsilon_0$ such that $$\mathbb{E}^{\mathbb{P}}\left[\exp\left(\epsilon_0\int_0^Tg^2(X_t)\,dt\right)\right]
=c(T)\,e^{aT}$$ for some constants $a$ and $c=c(T)$ with $c(T)$ bounded on $0<T<\infty.$
- for each $T>0,$ there is a positive number $\epsilon_1$ such that $\mathbb{E}^{\mathbb{P}}\int_0^Tg^{2+\epsilon_1}(X_t)\,dt$ is finite.
- $\frac{1}{T}\cdot\mathbb{E}^{\mathbb{P}}[(\phi^{-1}f^{})^2(X_T)]\rightarrow0$ as $T\rightarrow\infty.$
Then, $\mathbb{E}
^{\mathbb{P_{\epsilon}}}
[(\phi^{-1}
f)(X_{T}^{\epsilon})]$ is differentiable at $\epsilon=0$ and $$\frac{1}{T}\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}
^{\mathbb{P_{\epsilon}}}
[(\phi^{-1}
f)(X_{T}^{\epsilon})]\rightarrow 0\;.$$ In conclusion, $$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{1}{T}\left.\frac{\partial}{\partial\epsilon}\right|_{\epsilon=0}\ln p_T^\epsilon=-\lambda'(0)\;.
\end{aligned}$$
\[thm:variation\_rho\_expo\_condi\] The $L^2$-condition [(iii)]{.nodecor} on $\phi^{-1}f$ in the above theorem can be relaxed if $g$ satisfies a stronger condition. Condition [(ii)]{.nodecor} and [(iii)]{.nodecor} can be replaced by the following way.
- for each $T>0$ and $n\in\mathbb{N},$ $\mathbb{E}^{\mathbb{P}}\int_0^Tg^n(X_t)\,dt$ is finite,
- $\frac{1}{\sqrt{T}}\cdot\mathbb{E}^{\mathbb{P}}[|\phi^{-1}f|^{1+\epsilon_2}(X_T)]\rightarrow0$ as $T\rightarrow\infty$ for some positive $\epsilon_2.$
\[cor:rho\_expo\_condi\] We have that $\mathbb{E}
f)(X_{T}^{\epsilon})]\rightarrow 0$$ if there exists a positive number $\epsilon_0$ such that $\mathbb{E}^{\mathbb{P}}[\exp(\epsilon_0\,g^2(X_T)) ]$ is finite on $0<T<\infty$ and if [(iii)]{.nodecor}$'$ is satisfied.
Vega {#sec:vega}
----
### The Lamperti transform for univariate processes {#sec:Lamperti_trans}
In this section, assume that the underlying process $X_t$ is a one-dimensional process. Let $X_{t}^{\epsilon}$ be a perturbed process expressed by $$\label{eqn:general_perturb}
dX_{t}^{\epsilon}=b_\epsilon(X_{t}^{\epsilon})\,dt+\sigma_\epsilon(X_{t}^{\epsilon})\,dW_{t}\,,\quad X_0^{\epsilon}=X_0=\xi\;,$$ with $b_0=b$ and $\sigma_0=\sigma.$ This form of perturbation covers the vega. The initial value is not perturbed. We are interested in the perturbed quantity $p_T^\epsilon$ given by equation . Because it is difficult to analyze the volatility term, we use the Lamperti transform to convert the perturbation of volatility into perturbations of drift. Define a function $$\label{eqn:Lamperti_trans}
u_\epsilon(x):=\int_{\xi}^{x}\sigma_\epsilon^{-1}(y)\,dy\;,$$ then we have $$du_\epsilon(X_t^\epsilon)=(\sigma_\epsilon^{-1}b_\epsilon-\frac{1}{2}\sigma_\epsilon')(X_t^\epsilon)\,dt
+dW_t\;,\;u_\epsilon(X_0^\epsilon)=u_\epsilon(\xi)=0\;.$$ Here, $\sigma_\epsilon(x)$ is assumed to be a continuously differentiable function of $x.$ We denote the inverse function of $u_\epsilon(\cdot)$ by $v_\epsilon(\cdot).$ Set $U_t^\epsilon:=u_\epsilon(X_t^\epsilon),$ then $$\label{eqn:Lamperti_transformed_SDE}
dU_t^\epsilon=\delta_\epsilon(U_t^\epsilon)\,dt
+dW_t\;,\;U_0^\epsilon=0\;,$$ where $\delta_\epsilon(\cdot):=\left(\left(\sigma_\epsilon^{-1}b_\epsilon-\frac{1}{2}\sigma_\epsilon'\right)\circ v_\epsilon\right)(\cdot).$ The perturbation of form is transformed into a perturbation in drift. $$\begin{aligned}
p_{T}^{\epsilon}
=&\mathbb{E}^{\mathbb{Q}}[ e^{-\int_{0}^{T}r_\epsilon(X_{s}^{\epsilon})ds}\,f_\epsilon(X_{T}^{\epsilon})]
=\mathbb{E}^{\mathbb{Q}}[ e^{-\int_{0}^{T}(r_\epsilon\circ v_\epsilon)(U_{s}^{\epsilon})ds}(f_\epsilon\circ v_\epsilon)(U_{T}^{\epsilon})]\\
=&\mathbb{E}^{\mathbb{Q}}[e^{-\int_{0}^{T}R_\epsilon(U_{s}^{\epsilon})ds}\,F_\epsilon(U_{T}^{\epsilon})]
\end{aligned}$$ where $R_\epsilon:=r\circ v_\epsilon$ and $F_\epsilon:=f\circ v_\epsilon.$ In conclusion, the behavior of the long-term vega is obtained by applying Theorem \[thm:rho\_expo\_condi\], \[thm:variation\_rho\_expo\_condi\] or Corollary \[cor:rho\_expo\_condi\] to $U_t^\epsilon,\,R_\epsilon$ and $F_\epsilon.$
There is an invariant property between $(X_t^\epsilon,r_\epsilon,f_\epsilon)$ and $(U_t^\epsilon,R_\epsilon,F_\epsilon).$ Suppose $(X_t^\epsilon,r_\epsilon)$ admits the martingale extraction stabilizing $f_\epsilon$ with the eigenpair $(\lambda(\epsilon),\phi_{\epsilon})$ and the martingale exponent $\varphi_{\epsilon}.$ Then $(U_t^\epsilon,R_\epsilon)$ admits the martingale extraction stabilizing $F_\epsilon$ with the eigenpair $(\lambda(\epsilon),\phi_{\epsilon}\circ v_\epsilon)$ and the martingale exponent $\varphi_{\epsilon}\circ v_\epsilon.$
For the remainder of this section, we introduce a slight variation of the Lamperti transform . For a real number $c,$ define $$u_\epsilon(x):=\int_{c}^{x}\sigma_\epsilon^{-1}(y)\,dy\;.$$ Denote the inverse function of $u_\epsilon(\cdot)$ by $v_\epsilon(\cdot).$ Set $U_t^\epsilon:
=u_\epsilon(X_t^\epsilon)$ and $q(\epsilon):=U_0^\epsilon=\int_{c}^{\xi}\sigma_\epsilon^{-1}(y)\,dy,$ then $$\label{eqn:Lamperti_transformed_SDE}
dU_t^\epsilon=\delta_\epsilon(U_t^\epsilon)\,dt
+dW_t\;,\;U_0^\epsilon=q(\epsilon)\;,$$ where $\delta_\epsilon(\cdot):=\left(\left(\sigma_\epsilon^{-1}b_\epsilon-\frac{1}{2}\sigma_\epsilon'\right)\circ v_\epsilon\right)(\cdot).$ By choosing suitable $c,$ one can find a simple form of $\delta_\epsilon,$ which is useful for the sensitivity analysis. However, different from the previous transform , the initial value is perturbed. Thus, the sensitivity analysis of the initial value is required. Let $R_\epsilon:=r_\epsilon\circ v_\epsilon,$ $F_\epsilon:=f_\epsilon\circ v_\epsilon$ as before and let $\Phi_\epsilon:=\phi_\epsilon\circ v_\epsilon.$ Then $$\begin{aligned}
p_{T}^{\epsilon}
=&\mathbb{E}_\xi^{\mathbb{Q}}[ e^{-\int_{0}^{T}r(X_{s}^{\epsilon})ds}\,f(X_{T}^{\epsilon})]\\
=&\mathbb{E}^{\mathbb{Q}}_{q(\epsilon)}[e^{-\int_{0}^{T}R_\epsilon(U_{s}^{\epsilon})ds}\,F_\epsilon(U_{T}^{\epsilon})]\\
=&\phi_{\epsilon}(\xi)\,e^{-\lambda(\epsilon) T} \cdot\mathbb{E}^{\mathbb{P_{\epsilon}}}_{q(\epsilon)}
[(\Phi^{-1}_{\epsilon}F_\epsilon)(U_{T}^{\epsilon})]\;.
\end{aligned}$$ By applying the chain rule, we have $$\begin{aligned}
\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}p_{T}^{\epsilon}
=&\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\phi_{\epsilon}(\xi)\,e^{-\lambda(\epsilon) T} \cdot\mathbb{E}_{q(0)}^{\mathbb{P_{\epsilon}}}
[(\Phi^{-1}_{\epsilon}F_\epsilon)(U_{T}^{\epsilon})]\\
&+\phi(\xi)\,e^{-\lambda T} \left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}_{q(\epsilon)}
[(\Phi^{-1}F)(U_{T})]\;.
\end{aligned}$$ The first term can be analyzed by the method in Section \[sec:rho\] because the initial value and the volatility are not perturbed. The second term is involved with the perturbation of initial value and can be analyzed, for example, by Theorem \[thm:delta\] in Section \[sec:delta\] below.
### The Fournie method with bounded-derivative coefficients {#sec:vega_fournie}
We present how the Fournie method can be applied to the sensitivity analysis with respect to the perturbation in volatility. In this section, we consider the following perturbed process $X_t^\epsilon:$ $$dX_{t}^{\epsilon}=b(X_{t}^{\epsilon})\,dt+(\sigma +\epsilon\overline{\sigma})(X_{t}^{\epsilon})\,dW_{t}$$ and assume the hypothesis of the paper of Fournie [@Fournie]. The coefficients $b,$ $\sigma$ and $\overline{\sigma}$ are continuously differentiable with bounded derivatives. The diffusion matrix $\sigma+\epsilon\overline{\sigma}$ satisfies the uniform ellipticity condition for small $\epsilon\geq0.$
Consider the martingale extraction. Under the corresponding transformed measure $\mathbb{P}_\epsilon,$ the dynamics of $X_t^\epsilon$ satisfies $$\begin{aligned}
dX_{t}^\epsilon
=(b+(\sigma+\epsilon\overline{\sigma})\varphi_\epsilon)(X_{t}^\epsilon)\, dt+(\sigma+\epsilon\overline{\sigma})(X_{t}^\epsilon)\, dB_{t}^\epsilon
\end{aligned}$$ with a Brownian motion $B_t^\epsilon$ on $\mathbb{P}_\epsilon.$ Thus, the perturbation is induced by two part: the drift term and the volatility term. We take apart two perturbations by the chain rule. Let $X_t^\rho$ and $X_t^\nu$ be the processes corresponding to the perturbations in the drift and in the volatility, respectively: $$%\label{eqn:vega_separable}
\begin{aligned}
dX_{t}^\rho
&=(b+(\sigma+\rho\overline{\sigma})\varphi_\rho)(X_{t}^\rho)\, dt+\sigma(X_{t}^\rho)\, dB_{t}^\rho\;, \\
dX_{t}^\nu
&=(b+\sigma\varphi)(X_{t}^\nu)\, dt+(\sigma+\nu\overline{\sigma})(X_{t}^\nu)\, dB_{t}^\nu \;.
\end{aligned}$$ Then we have $$\label{eqn:vega_chain_rule}
\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P_{\epsilon}}}[(\phi^{-1}f)(X_{T}^{\epsilon})]
=\left.\frac{\partial}{\partial \rho}\;\right|_{\rho =0}\mathbb{E}^{\mathbb{P_{\rho}}}[(\phi^{-1}f)(X_{T}^{\rho})]
+\left.\frac{\partial}{\partial \nu}\;\right|_{\nu =0}\mathbb{E}^{\mathbb{P_{\nu}}}[(\phi^{-1}f)(X_{T}^{\nu})]\;.$$ The perturbation in the drift term can be analyzed by the the method in Section \[sec:rho\].
We now shift our attention to the perturbation in the volatility term. The main purpose of this section is to use the result of Fournie to investigate when the second term $$\frac{1}{T}\left.\frac{\partial}{\partial \nu}\;\right|_{\nu =0}\mathbb{E}^{\mathbb{P_{\nu}}}[(\phi^{-1}f)(X_{T}^{\nu})]$$ goes to zero as $T\rightarrow\infty.$ Suppose that $b+\sigma\varphi$ and $\phi^{-1}f$ are continuously differentiable with bounded derivatives. Then $$\left.\frac{\partial}{\partial \epsilon}\;\right|_{\nu =0}\mathbb{E} ^{\mathbb{P_{\nu}}} [(\phi^{-1}
f)(X_{T}^{\nu})]
=\mathbb{E}^{\mathbb{P}}[\nabla(\phi^{-1}f)(X_{T}^{\nu})\, Z_T]\;.$$ Here, $Z_t$ is the variation process given by $$\begin{aligned}
dZ_t=(b+\sigma\varphi)'(X_t)Z_t\,dt+\overline{\sigma}(X_t)dB_t+\sum_{i=1}^{d}\sigma_i'(X_t)Z_t\,dB_{i,t}\;,\;Z_0=0_d
\end{aligned}$$ where $\sigma_i$ is the $i$-th column vector of $\sigma$ and $0_d$ is the $d$-dimensional zero column vector. From this observation, we have the following theorem.
\[thm:vega\_Fournie\_condi\] Suppose that $b+\sigma\varphi$ and $\phi^{-1}f$ are continuously differentiable with bounded derivatives. If $\frac{1}{T}\cdot\mathbb{E}^{\mathbb{P}}[|Z_T|]\rightarrow0$ as $T\rightarrow\infty,$ then $$\lim_{T\rightarrow\infty}\frac{1}{T}\left.\frac{\partial}{\partial \nu}\;\right|_{\nu =0}\mathbb{E}^{\mathbb{P_{\nu}}}[(\phi^{-1}f)(X_{T}^{\nu})]=0\;.$$
Sensitivity on initial value {#sec:delta}
============================
The sensitivity analysis with respect to the initial perturbation is presented. Set $p_{T}:=\mathbb{E}_\xi^{\mathbb{Q}}
[ e^{-\int_{0}^{T}r(X_{s})ds}
f(X_{T})],$ then the quantity of interest is $$\nabla_\xi\,p_T$$ for large $T.$ Applying the martingale extraction, by equation , it follows that $$\begin{aligned}
\frac{\,\nabla_\xi\,p_T\,}{p_T}=\frac{\nabla_\xi\,\phi}{\phi(\xi)}+\frac{\nabla_\xi\,\mathbb{E}_\xi^{\mathbb{P }}[(\phi^{-1}f)(X_{T})]}{\mathbb{E}_\xi^{\mathbb{P }}[(\phi^{-1}f)(X_{T})]}\;.
\end{aligned}$$
\[thm:delta\] Suppose that $\phi(\xi)$ and $\mathbb{E}_\xi^{\mathbb{P }}[(\phi^{-1}f)(X_{T})]$ are differentiable functions of $\xi.$ If $|\nabla_\xi\,\mathbb{E}_\xi^{\mathbb{P }}[(\phi^{-1}f)(X_{T})]|\rightarrow0$ as $T\rightarrow\infty,$ then $$\lim_{T\rightarrow\infty}\frac{\,\nabla_\xi\, p_T\,}{p_T}=\frac{\nabla_\xi\,\phi}{\phi(\xi)}\;.$$
\[cor:delta\] Assume that the functions $b+\sigma\varphi$ and $\sigma$ are continuously differentiable with bounded derivatives. If $\mathbb{E}_\xi^\mathbb{P}(\phi^{-1}f)^2(X_T)$ and $\mathbb{E}_\xi^\mathbb{P}|\!|\sigma^{-1}(X_T)Y_T|\!|^2$ are bounded on $0<T<\infty,$ then $\mathbb{E}_\xi^{\mathbb{P }}[(\phi^{-1}f)(X_{T})]$ is differentiable by $\xi$ and $|\nabla_\xi\,\mathbb{E}_\xi^{\mathbb{P }}(\phi^{-1}f)(X_{T})|\rightarrow0$ as $T\rightarrow\infty.$ Here, $|\!|\cdot|\!|$ is the matrix 2-norm and $Y_t$ is the first variation process defined by $$\begin{aligned}
dY_t=(b+\sigma\varphi)'(X_t)Y_t\,dt+\sum_{i=1}^{d}\sigma_i'(X_t)Y_t\,dB_{i,t}\;,\;Y_0=I_d
\end{aligned}$$ where $\sigma_i$ is the $i$-th column vector of $\sigma$ and $I_d$ is the $d\times d$ identity matrix.
This theorem is obtained from the result of Fournie [@Fournie]. Refer to Appendix \[app:pf\_delta\] for proof. The $L^2$-condition on $\mathbb{E}_\xi^\mathbb{P}(\phi^{-1}f)^2(X_T)$ can be relaxed when $\mathbb{E}_\xi^\mathbb{P}|\!|(\sigma^{-1}(X_T)Y_T)|\!|^n$ is bounded on $0<T<\infty$ for a larger number $n.$
Examples of option prices {#sec:ex_option_prices}
=========================
The geometric Brownian motion {#sec:ABM}
-----------------------------
Consider the classical Black-Scholes model. The short interest is constant $r$ and the stock price, denoted by $S_{t},$ follows a geometric Brownian motion: $$\begin{aligned}
dS_{t}=\mu S_{t}\, dt+\sigma S_{t}\, dW_{t}
\end{aligned}$$ with $\mu-\frac{1}{2}\sigma^{2}>0.$ In this section, we assume that the payoff function $f_\alpha:[0,\infty)\rightarrow\mathbb{R}$ for $\alpha>0$ is a continuous function with growth rate $s^\alpha$ as $s\rightarrow\infty,$ that is, $\lim_{s\rightarrow\infty}f_\alpha(s)/s^\alpha$ exists and is nonzero constant. For example, $f_\alpha(s)=s^{\alpha},$ $f_\alpha(s)=(s^{\alpha}-K)_{+}$ or $f_\alpha(s)=(s-K)_{+}^\alpha$ for a positive $K.$ Set $$p_{T}
=\mathbb{E}^{\mathbb{Q}}
[e^{-rT}f_\alpha(S_{T})]\;.$$
We analyze the sensitivity analysis of the long-term option prices with payoff function $f_\alpha.$ Consider the corresponding infinitesimal generator $$(\mathcal{L}\phi)(s)=\frac{1}{2}\sigma^{2}s^{2}\phi''(s)+\mu s\phi'(s)-r\phi(s)\;.$$ It can be shown that the martingale extraction with respect to $$(\lambda,\phi(s)):=(r-\mu\alpha-\frac{1}{2}\sigma^{2}\alpha(\alpha-1),s^\alpha)$$ stabilizes $f_\alpha.$ With this $(\lambda,\phi),$ we conclude that $$\begin{aligned}
&\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\mu}\ln p_T
=-\frac{\partial\lambda}{\partial\mu}= \alpha\;,\\
&\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\sigma}\ln p_T=-\frac{\partial\lambda}{\partial\sigma}
=\sigma \alpha(\alpha-1)\;,\\
&\lim_{T\rightarrow\infty}\frac{\partial}{\partial S_{0}}\ln p_{T}=\frac{\phi'(S_{0})}{\phi(S_{0})}=\frac{\alpha}{S_{0}}\;.
\end{aligned}$$
The analysis also can be applied to the expected utility of an investor. Suppose that $\mathbb{Q}$ is an objective measure, $f_\alpha(s)=s^\alpha$ with $0<\alpha<1$ is the utility function of the investor and $r$ is the discount rate of the investor. Then $p_T$ is the discounted expected utility. Thus we can obtain the sensitivity of the expected utility of the long-term investor.
The CIR model {#sec:CIR}
-------------
We explore the sensitivity analysis of option prices whose underlying process is the Cox–Ingersoll–Ross (CIR) model. Under a risk-neutral measure $\mathbb{Q},$ the interest rate $r_{t}$ follows $$dr_{t}=(\theta-ar_{t})\,dt + \sigma \sqrt{r_{t}} \,dW_{t}$$ with $\theta,\sigma>0$ and $a\in\mathbb{R}.$ We assume $2\theta>\sigma^{2}$ so that the original interest rate process and the perturbation process stay strictly positive for small perturbation.
The quantity $$p_T:=\mathbb{E}^{\mathbb{Q}}[e^{-\int_0^T r_t\,dt} f(r_T)]$$ for large $T$ is of interest to us. Here, $f$ is a payoff function and we assume that $f(r)$ is a nonnegative continuous function on $r\in[0,\infty),$ which is not identically zero, and that the growth rate at infinity is equal to or less than $e^{mr}$ with $m<\frac{a}{\sigma^2}.$ The associated second-order equation is $$\mathcal{L}\phi(r)=\frac{1}{2}\sigma^{2}r\phi''(r)+(\theta-ar)\phi'(r)-r\phi(r)=-\lambda\phi(r)\;.$$ Set $\kappa:=\frac{\sqrt{a^{2}+2\sigma^{2}}-a}{\sigma^{2}}.$ It can be shown that the martingale extraction with respect to $$(\lambda,\phi(r)):=(\theta \kappa,e^{-\kappa r})$$ stabilizes $f.$
By using this $(\lambda,\phi),$ the sensitivities of the quantity can be analyzed. The sensitivities of the long-term option prices with respect to $\theta,$ $a,$ $\sigma$ and $r_0$ are given by $$\begin{aligned}
&\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\theta}\ln p_T=-\frac{\partial\lambda}{\partial\theta}= -\frac{\sqrt{a^{2}+2\sigma^{2}}-a}{\sigma^{2}}\;,\\
&\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial a}\ln p_T=-\frac{\partial\lambda}{\partial a}
=\frac{\theta(\sqrt{a^{2}+2\sigma^{2}}-a)}{\sigma^{2}\sqrt{a^{2}
+2\sigma^{2}}}\;,\\
&\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\sigma}\ln p_T=-\frac{\partial\lambda}{\partial\sigma}
=\frac{\theta(\sqrt{a^{2}+2\sigma^{2}}-a)^2}{\sigma^{3}\sqrt{a^{2}+2\sigma^{2}}}\;,\\
&\lim_{T\rightarrow\infty}\frac{\partial}{\partial r_0}\ln p_T=\frac{\phi'(r_0)}{\phi(r_0)}
=-\frac{\sqrt{a^{2}+2\sigma^{2}}-a}{\sigma^{2}}\;.
\end{aligned}$$ For more details about the sensitivity analysis of the CIR model, refer to Appendix \[app:CIR\].
Quadratic models
----------------
We present the sensitivity analysis of short-interest option prices whose underlying process is a quadratic term structure model. This section is indebted to [@Qin14]. Suppose $X_t$ is a $d$-dimensional OU process satisfying the SDE under a risk-neutral measure $\mathbb{Q}:$ $$dX_t=(b+BX_t)\,dt+\sigma\,dW_t$$ where $b$ is a $d$-dimensional column vector, $B$ is a $d\times d$ matrix, and $\sigma$ is a non-singular $d\times d$ matrix, so that $a=\sigma\sigma^{\top}$ is strictly positive definite. The short interest rate is given by $$r(x)=\beta+\langle\alpha,x\rangle+\langle \Gamma x,x\rangle$$ where the constant $\beta,$ vector $\alpha$ and symmetric positive definite $\Gamma$ are taken to be such that the short interest rate is non-negative for all $x\in\mathbb{R}^d.$
The quantity $$p_T=\mathbb{E}^\mathbb{Q}[e^{-\int_0^Tr(X_t)\,dt}f(X_T)]$$ is the option price with payoff function $f$ and we explore the sensitivity analysis of this quantity with respect to a perturbation of the underlying process $X_t.$ Assume that $f$ is bounded and has a bounded support on $\mathbb{R}^d.$ Let $V$ be the [*stabilizing solution*]{} of $$2VaV-B^\top V-VB-\Gamma=0\;,$$ then it is well-known that $B-2aV$ is non-singular and the eigenvalues of $B-2aV$ have negative real parts. Define a vector $u$ by $$u=(2Va-B^\top)^{-1}(2Vb+\alpha)\;.$$ It can be shown that the martingale extraction with respect to $$(\lambda,\phi(x))=(\beta-\frac{1}{2}u^\top au+tr(aV)+u^\top b\,,\,e^{-\langle u,x\rangle-\langle Vx,x\rangle})$$ stabilizes $f.$ The sensitivity of the quantity $p_T$ for large $T$ is given by $$\begin{aligned}
&\lim_{T\rightarrow\infty}\frac{1}{T} \frac{\partial }{\partial b_i}\ln p_T=\frac{\partial \lambda}{\partial b_i}\,,\;
\lim_{T\rightarrow\infty} \frac{1}{T} \frac{\partial }{\partial B_{ij}}\ln p_T=\frac{\partial \lambda}{\partial B_{ij}}\,,\;
\lim_{T\rightarrow\infty}\frac{\,\nabla_\xi\,p_T\,}{p_T}=-u-2V\xi
\end{aligned}$$ for $1\leq i,j\leq d.$ If $f$ is continuously differentiable with compact support, then we have $$\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial }{\partial \sigma_i}\ln p_T=\frac{\,\partial \lambda\,}{\partial \sigma_i}\;.$$ Refer to Appendix \[app:QTSM\] for more details.
Examples of expected utilities {#sec:ex_utility}
==============================
The Heston model
----------------
The sensitivity analysis of the expected utility with respect to the parameters of the Heston model is presented. Under the objective measure $\mathbb{Q},$ suppose that an asset $X_t$ follows $$\begin{aligned}
&dX_t=\mu X_t\, dt+ \sqrt{v_t}X_t\,dZ_t\;,\\
&dv_t=(\gamma-\beta v_t)\,dt+\delta\sqrt{v_t}\,dW_t\;,
\end{aligned}$$ where $Z_t$ and $W_t$ are two standard Brownian motions with $\langle Z,W\rangle_t=\rho t$ for the correlation $-1\leq \rho\leq 1.$ Assume that $\mu,\gamma,\beta,\delta>0$ and $2\gamma>\delta^2.$ We consider a power utility function of the form $$u(c)={c}^{\alpha}\,,\;0<\alpha< 1\;.$$ The sensitivity of the quantity $$p_T:=\mathbb{E}^\mathbb{Q}[u(X_T)]=\mathbb{E}^\mathbb{Q}[X_T^\alpha]=\mathbb{E}^\mathbb{Q}[e^{\alpha\int_0^T\sqrt{v_t}dZ_t-\frac{\alpha}{2}\int_0^Tv_t\,dt}]\,e^{\alpha\mu T}X_0^\alpha$$ for large $T$ is of interest to us. We have that $$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\mu}\ln p_T
&=\alpha\\
\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\gamma}\ln p_T
&=-\frac{1}{2}\alpha(1-\alpha)\cdot\frac{\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\beta+\rho\alpha\delta}{\delta^2}\\
\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\beta}\ln p_T
&=\frac{\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\beta+\rho\alpha\delta}{\delta^2\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}}\\
\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\delta}\ln p_T
&=-\rho\alpha\cdot\frac{\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\beta+\rho\alpha\delta}{\delta^2\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}}\\
&+\frac{(\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\beta+\rho\alpha\delta)^2}{\delta^3\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}}\\
\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\rho}\ln p_T
&=-\frac{\alpha\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\alpha\beta+\rho\alpha^2\delta}{\delta\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}}\\
\lim_{T\rightarrow\infty}\frac{\partial}{\partial X_0}\ln p_T\,
&=\frac{\alpha}{X_0}\\
\lim_{T\rightarrow\infty}\frac{\partial}{\partial v_0}\ln p_T\;\;
&=-\frac{1}{2}\alpha(1-\alpha)\cdot\frac{\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\beta+\rho\alpha\delta}{\delta^2}\;.
\end{aligned}$$ Refer to Appendix \[app:Heston\] for the details.
The $3/2$ LEFT model {#sec:3/2_model}
--------------------
The sensitivity analysis of the expected utility and the return of an exchange-traded fund (ETF) is explored. We investigate the leveraged ETF (LETF), which promises a fixed leverage ratio with respect to a given underlying asset or index process $X_t.$ Assume that $X_t$ stays positive. A long-leveraged ETF $L_t$ on $X_t$ with leverage ratio $\beta\geq 1$ is constructed by the following way. At time $t,$ the cash amount of $\beta L_t$ ($\beta$ times the fund value) is invested in $X_t,$ and the amount $(\beta-1)L_t$ is borrowed at the risk-free rate $r.$ For a short-leveraged ETF $L_t$ with ratio $\beta\leq -1,$ the cash amount of $|\beta|L_t$ is shorted on $X_t,$ and $(1-\beta)L_t$ is kept in the money market account with the risk-free rate $r.$ In practice, most typical leverage ratios are $\beta=1,2,3$ (long) and $\beta=-1,-2,-3$ (short), thus we assume $|\beta|\leq3.$ The LEFT $L_t$ satisfies $$\begin{aligned}
\frac{dL_t}{L_t}
&=\beta\left(\frac{dX_t}{X_t}\right)-((\beta-1)r)\,dt\\
&=\left(\beta\left(\frac{\mu(X_t)}{X_t}\right)-(\beta-1)r\right)\,dt+\beta\left(\frac{\sigma(X_t)}{X_t}\right)\,dB_t
\end{aligned}$$ and can be written by $$\label{eqn:LETF_undelying} \frac{L_t}{L_0}=\left(\frac{X_t}{X_0}\right)^\beta e^{-r(\beta-1)t-\frac{\beta(\beta-1)}{2}\int_0^t\sigma^2(X_u)/X_u^2\,du}\;.$$ In this section, we assume $X_0=L_0=1.$
The underlying process $X_t$ and the utility function are as follows. The dynamics of $X_t$ is given by the $3/2$ model $$dX_t=(\theta-aX_t)X_t\,dt+\sigma X_t^{3/2}\,dW_t$$ with $\theta,a,\sigma>0$ under the objective measure $\mathbb{Q}.$ This process stays positive and is recurrent. As a practical example, one can consider the leveraged volatility-index EFT. We consider a power function of the form $$u(c)={c}^{\alpha}\,,\;0<\alpha\leq 1\;.$$
The sensitivity analysis of the quantity $$p_T:=\mathbb{E}^\mathbb{Q}[u(L_T)]$$ is of interest to us. This quantity is the expected utility of $L_T$ if $0<\alpha< 1$ and is the expected return of $L_T$ if $\alpha=1.$ For the sensitivity on $\theta,$ $$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{1}{T} \frac{\,\partial }{\partial \theta}\ln p_T=-\sqrt{\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)^2+\alpha\beta(\beta-1)}-\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)\;.
\end{aligned}$$ For sensitivities on $a$ and $\sigma,$ we have $$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{1}{T} \frac{\,\partial }{\partial a}\ln p_T=\frac{\theta\left(\sqrt{\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)^2+\alpha\beta(\beta-1)}-\left(\frac{\sigma}{2}+a\right)\right)}{\sigma^2\sqrt{\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)^2+\alpha\beta(\beta-1)}}
\end{aligned}$$ $$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{1}{T} \frac{\,\partial }{\partial \sigma}\ln p_T=\frac{{2a\theta}\left(\sqrt{\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)^2+\alpha\beta(\beta-1)}-\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)\right)}{{\sigma^3}\sqrt{\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)^2+\alpha\beta(\beta-1)}}
\end{aligned}$$ when $\frac{a}{\sigma^2}+1-\alpha\beta>0.$ Refer to Appendix \[app:3/2\_model\] for more details.
Conclusion
==========
In this article, the sensitivity analysis of long-term cash flows was investigated. We explored the sensitivity of $p_T=\mathbb{E}^{\mathbb{Q}}[e^{-\int_0^T r(X_t)\,dt} f(X_T)]$ with respect to the perturbation on the process $X_t.$ Essentially, two types of perturbation were presented. First, we discussed the sensitivities with respect to the perturbation on the drift and the volatility. Under the assumption that the perturbed process $X_t^\epsilon$ and the function $r$ admits the martingale extraction stabilizing $f,$ the perturbed quantity $p_T^\epsilon$ was transformed into what is easier to address $$\begin{aligned}
p_{T}^{\epsilon}
&=\phi_{\epsilon}(\xi)\,e^{-\lambda(\epsilon) T} \cdot\mathbb{E}^{\mathbb{P_{\epsilon}}}
[(\phi^{-1}_{\epsilon}f )(X_{T}^{\epsilon})]
\end{aligned}$$ with an eigenpair $(\lambda(\epsilon),\phi_\epsilon)$ and the transformed measure $\mathbb{P}_{\epsilon}.$ The method of Fournie was useful to analyze the last component in the above expression of $p_{T}^{\epsilon}.$ We proved that the sensitivity of $p_T^\epsilon$ on $\epsilon$ is expressed in the a simple form for large $T$ under some conditions: $$\lim_{T\rightarrow\infty}\frac{1}{T}\left.\frac{\partial}{\partial \epsilon}\right|_{\epsilon=0}\ln p_T^\epsilon=-\lambda'(0)\;.$$
Second, the sensitivity to the initial value $X_0$ was investigated. Assuming that the process $X_t$ and the function $r$ admits the martingale extraction stabilizing $f,$ the quantity $p_T$ was expressed by $$\begin{aligned}
p_{T}
=\phi(\xi)\,e^{-\lambda T}\cdot
\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}f)(X_{T})]
\end{aligned}$$ with an eigenpair $(\lambda,\phi)$ and the transformed measure $\mathbb{P}.$ It was shown that the sensitivity of $p_T$ is expressed in the following simple form for large $T$ under appropriate conditions: $$\lim_{T\rightarrow\infty}\frac{\,\nabla_\xi\, p_T\,}{p_T}=\frac{\nabla_\xi\,\phi}{\phi(\xi)}\;.$$
We suggest the following extension for further research. It would be interesting to find the sensitivities of $p_T$ with path-dependent functionals $f(\cdot)$ instead of $f(X_T),$ which depends only on the final time $T.$ It is straightforward to extend the results in the paper to discretely monitored functionals $f(X_{t_{1}},X_{t_{2}},\cdots,X_{t_{m}}).$ However, it will be challenging to find the sensitivities for a general form of functionals including the payoff form of barrier and American options.
Stabilizing martingale extractions {#app:stabiling_mart_extrac}
==================================
We investigate sufficient conditions on $f$ such that the martingale extraction stabilizes $f$ when a martingale extraction is given.
Suppose that $(X_t,r)$ admits the martingale extraction of $(\lambda,\phi).$ If $X_t$ is [*positive*]{} recurrent under the transformed measure $\mathbb{P}$ with respect to $(\lambda,\phi)$ and if $\phi^{-1}f$ is nonzero and bounded, then the martingale extraction of $(\lambda,\phi)$ stabilizes $f.$ In this case, $$\lim_{T\rightarrow\infty}\mathbb{E}^{\mathbb{P}}\left[(\phi^{-1}f)(X_{T})\right]=\int\phi^{-1}f\,d\nu$$ where $\nu$ is the invariant probability of $X_t$ under $\mathbb{P}.$
The condition that $\phi^{-1}f$ is bounded can be relaxed by using the $L^2$-ergodic property or the Lyapunov criteria. For convenience, put $h:=\phi^{-1}f.$
\[prop:L2\_ergodicity\][($L^{2}$-ergodicity)]{.nodecor} Assume that $X_{t}$ has an invariant probability $\nu$ under $\mathbb{P}.$ For $h\in L^{2}(\nu),$ we have $$\lim_{T\rightarrow\infty}\mathbb{E}^{\mathbb{P}}\left[h(X_{T})\right]=\int h\,d\nu$$ $\xi$-pointwise and in $L^{2}(\nu).$
\[thm:Lyapunov\] [(Lyapunov criteria)]{.nodecor} Assume that $X_{t}$ has an invariant measure $\nu$ (not necessarily a probability) under $\mathbb{P}.$ Let $h\geq 0.$ If there are constants, $a>0$ and $b<\infty,$ such that $$\mathcal{L}^{\mathbb{P}}h(x) \leq -ah(x) +b\mathbb{I}_{K}(x)\;,$$ where $\mathcal{L}^{\mathbb{P}}$ is the infinitesimal generator of $X_t$ under $\mathbb{P}$, and $K$ is a compact set, then $$\lim_{T\rightarrow\infty}\mathbb{E}^{\mathbb{P}}\left[h(X_{T})\right]=\int h\,d\nu\;.$$
For more details, refer to [@Meyn].
We can apply spectral theory to explore another condition that possesses a martingale extraction that stabilizes $f$ when $X_t$ is a one-dimensional process. Consider the [*speed measure*]{} $\mu$ of $X_t$ under $\mathbb{Q}$ defined by $d\mu:=w(x)dx$, where $$w(x)= \frac{1}{\sigma^2(x)} e^{\int_{\xi}^{x} \frac{2b(z)}{\sigma^2(z)} dz}\; .$$ It is well known that the infinitesimal generator $\mathcal{L}$ is a [*densely defined symmetric nonpositive*]{} operator from $L^{2}(\mu)$ to itself. Let $A$ be a self-adjoint extension of $-\mathcal{L}.$ Denote the domain of $A$ by $\text{Dom}(A),$ which is in $L^2(\mu).$
\[thm:spectral\_theory\] Suppose that the operator $A$ has at least one eigenvalue. Assume that the spectral gap is positive when $A$ has a continuum spectrum. Let $\beta$ be the minimum eigenvalue and denote its eigenfunction by $\phi.$ Assume that $M_t$, induced by $(\beta,\phi)$, is a martingale. If $f\in \text{Dom}(A)$ and $f\geq 0,\,f\neq 0\,,$ then the martingale extraction with respect to $(\beta,\phi)$ stabilizes $f.$ In this case, $$\lim_{T\rightarrow\infty}\mathbb{E}^{\mathbb{P}}\left[h(X_{T})\right]=\langle f,\phi\rangle/\langle \phi,\phi\rangle$$ where $$\langle f, g\rangle:=\int fg\,d\mu\;.$$
Refer to [@Bakry], [@Fulton], [@Lax] and [@Zettl] for more details.
Perturbation of payoff function {#sec:payoff}
===============================
In this section, we are interested in a sufficient condition that the differentiation and expectation are interchangeable: $$\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}\left[\,h_\epsilon(X)\,\right]=
\mathbb{E}^{\mathbb{P}}\left[\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}h_\epsilon(X)\right]\,.$$ The following theorem is a well-known fact and it is noteworthy because we will use this theorem frequently.
\[thm:payoff\] Let $X$ be a random variable and let $h_\epsilon(x)$ be a continuously differentiable function at $\epsilon=0$ for each $x.$ Suppose that there exists a random variable $G$ such that $$\left|\frac{\partial}{\partial\epsilon}\,h_\epsilon(X)\right|\leq G\;\;\textnormal{on}\;\;
(\epsilon,x)\in I\times\mathbb{R}^d\;\;\textnormal{for an open interval } I \textnormal{ containing } 0$$ and $$\mathbb{E}^{\mathbb{P}}\left[\,G\,\right]<\infty\;.$$ Then, $\mathbb{E}^{\mathbb{P}} [h_\epsilon(X)]$ is differentiable at $\epsilon=0$ and $$\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}} [h_\epsilon(X)]=
\mathbb{E}^{\mathbb{P}}\left[\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}h_\epsilon(X)\right]\,.$$
Proof of Theorem \[thm:rho\_expo\_condi\] {#app:pf_rho_expo_condi}
=========================================
We first prove the following proposition. This proposition is a generalization of the result of Fournie and gives an implication how to control the last term in equation in Section \[sec:sen\_drift\_vol\].
\[prop:rho\] Suppose that $\mathbb{E}^{\mathbb{P}}[\exp(\epsilon_1\int_0^Tg^2(X_t)\,dt)]$ and $\mathbb{E}^{\mathbb{P}}\int_0^Tg^{2+\epsilon_1}(X_t)\,dt$ are finite for some positive $\epsilon_1.$ Then for any given function $f(\cdot)$ with $\mathbb{E}^{\mathbb{P}}[(\phi^{-1}f^{})^2(X_T)]<\infty,$ we have $$\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P_{\epsilon}}}
\left[(\phi^{-1}
f)(X_{T}^{\epsilon})\right]=\mathbb{E}^{\mathbb{P}}
\left[(\phi^{-1}
f)(X_{T})\int_{0}^{T}\overline{k}(X_{s})\; dB_{s}\right]\;.$$
We slightly modify the proof in [@Fournie]. A process $B_{t}^{\epsilon}$ defined by $dB_{t}^{\epsilon}=dW_{t}-\varphi_{\epsilon}(X_{t}^{\epsilon})dt$ is a $d$-dimensional Brownian motion under $\mathbb{P}_{\epsilon}\,.$ $$\begin{aligned}
dX_{t}^{\epsilon}
&=b_\epsilon(X_{t}^{\epsilon})\,dt+\sigma (X_{t}^{\epsilon})\,dW_{t}\\
&=(b_\epsilon+\sigma\varphi_{\epsilon})(X_{t}^{\epsilon})\,dt+\sigma (X_{t}^{\epsilon})\,dB_{t}^{\epsilon}\\
&=(\sigma k_\epsilon)(X_{t}^{\epsilon})\,dt+\sigma (X_{t}^{\epsilon})\,dB_{t}^{\epsilon} \;.
\end{aligned}$$ A process $\tilde{X}_{t}^{\epsilon}$ defined by $$\begin{aligned}
d\tilde{X}_{t}^{\epsilon}
&=(\sigma k_{\epsilon})(\tilde{X}_{t}^{\epsilon})\,dt+\sigma (\tilde{X}_{t}^{\epsilon})\,dB_{t}
\end{aligned}$$ under $\mathbb{P}$ has the same distribution with $X_{t}^{\epsilon}$ under $\mathbb{P}_{\epsilon}.$ Thus, $$\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P_{\epsilon}}}
[(\phi^{-1}
f)(X_{T}^{\epsilon})]
=\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}
f)(\tilde{X}_{T}^{\epsilon})]\,.$$ Because $k_\epsilon$ is continuously differentiable at $\epsilon=0,$ by using the Taylor expansions, we write $k_{\epsilon}=k+\epsilon\eta_\epsilon$ for some $d\times 1$ vector $\eta_\epsilon.$ $$\begin{aligned}
d\tilde{X}_{t}^{\epsilon}
&=(\sigma k+\epsilon\sigma\eta_\epsilon)(\tilde{X}_{t}^{\epsilon})\,dt+\sigma (\tilde{X}_{t}^{\epsilon})\,dB_{t}\;.
\end{aligned}$$
We now show that $\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}
f)(\tilde{X}_{T}^{\epsilon})]$ is differentiable at $\epsilon=0$ and $$\label{eqn:pf_rho_derivative}
\begin{aligned}
\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}
f)(\tilde{X}_{T}^{\epsilon})]
&=\mathbb{E}^{\mathbb{P}}
\left[(\phi^{-1}
f)(X_{T})\int_{0}^{T}\overline{k}(X_{t})\; dB_{t}\right]\;.
\end{aligned}$$ By the assumption that $\mathbb{E}^{\mathbb{P}}[\exp(\epsilon_1\int_0^Tg^2(X_t)\,dt)]$ is finite, we know that $$Z^\epsilon(T)
:=\exp\left(-\epsilon\int_0^T\eta_\epsilon(X_t)\,dB_t
-\frac{\epsilon^2}{2}\int_0^T|\eta_\epsilon|^2(X_t)\,dt\right)$$ is martingale for small $\epsilon$ because the Novikov condition is satisfied. By the Girsanov theorem, we know $\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}
f)(\tilde{X}_{T}^{\epsilon})]=\mathbb{E}^{\mathbb{P}}[Z^\epsilon(T)\,(\phi^{-1}f)(X_{T})].$ Thus, $$\begin{aligned}
\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}
f)(\tilde{X}_{T}^{\epsilon})]
&=\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}[Z^\epsilon(T)\,(\phi^{-1}f)(X_{T})]\\
&=\lim_{\epsilon\rightarrow0}\,\mathbb{E}^{\mathbb{P}} [(\epsilon^{-1}(Z^\epsilon(T)-1)(\phi^{-1}f)(X_{T})]\\
&=\lim_{\epsilon\rightarrow0}\,\mathbb{E}^{\mathbb{P}} \left[(\phi^{-1}f)(X_{T})\int_0^TZ^\epsilon(t)\,\eta_\epsilon(X_t)\,dB_t\right]\;.
\end{aligned}$$ Here, for the last equality, we used ${\epsilon}^{-1}(Z^\epsilon(T)-1)=\int_0^TZ^\epsilon(t)\,\eta_\epsilon(X_t)\,dB_t.$
To prove equation , it will be shown that $$\lim_{\epsilon\rightarrow0}\,\mathbb{E}^{\mathbb{P}} \left[(\phi^{-1}f)(X_{T})\int_0^TZ^\epsilon(t)\eta_\epsilon(X_t) -\overline{k}(X_{t})\;dB_t\right]=0\;,$$ which is obtained from the above equation subtracted by $\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}
f)(X_{T})\int_{0}^{T}\overline{k}(X_{t})\; dB_{t}].$ From the condition $\mathbb{E}^{\mathbb{P}}[(\phi^{-1}f^{})^2(X_T)]<\infty,$ by the Cauchy-Schwarz inequality, it is enough to show that $\int_0^TZ^\epsilon(t)\eta_\epsilon(X_t) -\overline{k}(X_{t})\;dB_t$ converges to zero in $L^2$ as $\epsilon\rightarrow0.$ Since we know $$\int_0^TZ^\epsilon(t)\eta_\epsilon(X_t) -\overline{k}(X_{t})\;dB_t
=\int_0^T(Z^\epsilon(t)-1)\eta_\epsilon(X_t)\,dB_t+\int_0^T(\eta_\epsilon-\overline{k})(X_{t})\;dB_t\;,$$ it will be shown that each term on the right hand side converges to zero in $L^2.$ For the second term, we use the Lebesgue dominated convergence theorem. Because $|\eta_\epsilon-\overline{k}|^2\leq2(|\eta_\epsilon|^2+|\overline{k}|^2)\leq 4g^2$ and $\mathbb{E}^{\mathbb{P}}\int_0^Tg^2(X_{t})\;dt$ is finite, we have that as $\epsilon\rightarrow0,$ $$\mathbb{E}^{\mathbb{P}}\left(\int_0^T(\eta_\epsilon-\overline{k})(X_{t})\;dB_t\right)^2
=\mathbb{E}^{\mathbb{P}}\int_0^T|\eta_\epsilon-\overline{k}|^2(X_{t})\;dt\rightarrow0\;.$$
We now prove that $$\lim_{\epsilon\rightarrow0}\,\mathbb{E}^{\mathbb{P}}\left(\int_0^T(Z^\epsilon(t)-1)\,\eta_\epsilon(X_t)\,dB_t\right)^2=0\;.$$ Choose a positive integer $p$ and a positive number $q$ such that $\frac{1}{p}+\frac{1}{q}=1$ and $1<q<1+\frac{\epsilon_1}{2}.$ Then $$\begin{aligned}
&\mathbb{E}^{\mathbb{P}}\left(\int_0^T(Z^\epsilon(t)-1)\,\eta_\epsilon(X_t)\,dB_t\right)^2\\
=&\,\mathbb{E}^{\mathbb{P}}\int_0^T(Z^\epsilon(t)-1)^2\,|\eta_\epsilon|^2(X_t)\,dt \\
\leq\,&\left(\int_0^T\mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)-1)^{2p}\,dt\right)^{\frac{1}{p}}\cdot
\left(\int_0^T\mathbb{E}^{\mathbb{P}}|\eta_\epsilon|^{2q}(X_t)\,dt\right)^{\frac{1}{q}}\\
\leq\,&\left(\int_0^T\mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)-1)^{2p}\,dt\right)^{\frac{1}{p}}\cdot
\left(\int_0^T\mathbb{E}^{\mathbb{P}}g^{2q}(X_t)\,dt\right)^{\frac{1}{q}}\,.
\end{aligned}$$ The second term is finite by the assumption because $2q<2+\epsilon_1.$ We now prove that the first term converges to zero. Consider $(Z^\epsilon(t)-1)^{2p}=\sum_{l=0}^{2p}{2p \choose l} (-1)^{l}Z^\epsilon(t)^l.$ It is enough to show that $\int_0^T\mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)^l)\,dt$ converges to $T$ as $\epsilon\rightarrow0$ for $l=1,2,\cdots,2p,$ because $$\int_0^T\mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)-1)^{2p}\,dt\!=\!\sum_{l=0}^{2p}{2p \choose l} (-1)^{l}\int_0^T \mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)^l)\,dt\rightarrow T\sum_{l=0}^{2p}{2p \choose l} (-1)^{l}=0\;.$$ To show this, we use the Lebesgue dominated convergent theorem: prove that $\mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)^l)$ is uniformly bounded for small $\epsilon$ and $0\leq t\leq T$ and that $\mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)^l)$ converges to $1$ as $\epsilon$ goes to zero for fixed $t.$ $$\begin{aligned}
&\;\quad\mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)^l)\\
&=\mathbb{E}^{\mathbb{P}}\exp\left(-l\epsilon\int_0^t\eta_\epsilon(X_s)\,dB_s
-\frac{l\epsilon^2}{2}\int_0^t|\eta_\epsilon|^2(X_s)\,ds\right) \\
&=\mathbb{E}^{\mathbb{P}}\exp\left(-l\epsilon\int_0^t\eta_\epsilon(X_s)\,dB_s
-l^2\epsilon^2\int_0^t|\eta_\epsilon|^2(X_s)\,ds\right)
\cdot\exp\left(l(l-1/2)\epsilon^2\int_0^t|\eta_\epsilon|^2(X_s)\,ds\right) \\
&\leq\left(\mathbb{E}^{\mathbb{P}}\exp\left(-2l\epsilon\int_0^t\eta_\epsilon(X_s)\,dB_s
-2l^2\epsilon^2\int_0^t|\eta_\epsilon|^2(X_s)\,ds\right)\right)^{\frac{1}{2}} \\
&\quad\cdot\left(\mathbb{E}^{\mathbb{P}}\exp\left(l(2l-1)\epsilon^2\int_0^t|\eta_\epsilon|^2(X_s)\,ds\right)\right)^{\frac{1}{2}}\\
&=\left(\mathbb{E}^{\mathbb{P}}\exp\left(l(2l-1)\epsilon^2\int_0^t|\eta_\epsilon|^2(X_s)\,ds\right)\right)^{\frac{1}{2}} \;\because\textnormal{the former term is a martingale for small } \epsilon \\
&\leq\left(\mathbb{E}^{\mathbb{P}}\exp\left(l(2l-1)\epsilon^2\int_0^tg^2(X_s)\,ds\right)\right)^{\frac{1}{2}}\\
&\leq \left(\mathbb{E}^{\mathbb{P}}\exp\left(\epsilon_1\int_0^Tg^2(X_s)\,ds\right)\right)^{\frac{1}{2}} \quad\textnormal{ for $t<T$ and small } \epsilon \\
&<\infty \quad\textnormal{ by assumption}\,.
\end{aligned}$$ Thus, $\mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)^l)$ is uniformly bounded for small $\epsilon$ and $0\leq t\leq T.$
We now show that $\mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)^l)$ converges to $1$ as $\epsilon$ goes to zero for fixed $t.$ We use the Lebesgue dominated convergent theorem to $$\exp\left(l(2l-1)\epsilon^2\int_0^tg^2(X_s)\,ds\right)$$ as $\epsilon$ goes to zero. Because this is dominated pathwise by $$\exp\left(\epsilon_1\int_0^tg^2(X_s)\,ds\right)$$ whose expectation is finite, we know that $$\mathbb{E}^{\mathbb{P}}\exp\left(l(2l-1)\epsilon^2\int_0^tg^2(X_s)\,ds\right)$$ converges to $1$ as $\epsilon$ goes to zero. $$\begin{aligned}
1&=\mathbb{E}^{\mathbb{P}}\left(\liminf_{\epsilon\rightarrow0}Z^\epsilon(t)^l\right)
\leq \liminf_{\epsilon\rightarrow0}\mathbb{E}^{\mathbb{P}}\left(Z^\epsilon(t)^l\right)
\leq
\limsup_{\epsilon\rightarrow0}\mathbb{E}^{\mathbb{P}}\left(Z^\epsilon(t)^l\right) \\
&\leq\lim_{\epsilon\rightarrow0}\mathbb{E}^{\mathbb{P}}\exp\left(l(2l-1)\epsilon^2\int_0^tg^2(X_s)\,ds\right) =1\,.
\end{aligned}$$ Thus, we obtained the desired result. This completes the proof.
Now we prove Theorem \[thm:rho\_expo\_condi\].
It suffices to show that $\frac{1}{T}\cdot\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}f)(X_{T})\int_{0}^{T}\overline{k}(X_{s})\; dB_{s}]$ converges to zero as $T$ goes to infinity. From the assumption, $$\begin{aligned}
c(T)\,e^{aT}
&=\mathbb{E}^{\mathbb{P}}\left[\exp\left(\epsilon_0\int_0^Tg^2(X_t)\,dt\right)\right]\\
&\geq \exp\left(\epsilon_0\,\mathbb{E}^{\mathbb{P}}\left[\int_0^Tg^2(X_t)\,dt\right]\right)\;,
\end{aligned}$$ thus we have that $$\mathbb{E}^{\mathbb{P}}\left[\int_{0}^{T}|\overline{k}|^2(X_{s})\; ds\right]\leq\mathbb{E}^{\mathbb{P}}\left[\int_0^Tg^2(X_t)\,dt\right]
\leq \frac{1}{\epsilon_0}(\log c(T)+aT)\leq a_1\,T$$ for some positive number $a_1$ when $T$ is large. $$\begin{aligned}
&\frac{1}{T}\cdot\mathbb{E}^{\mathbb{P}}
\left[(\phi^{-1}f)(X_{T})\int_{0}^{T}\overline{k}(X_{s})\; dB_{s}\right]\\
\leq\;&\frac{1}{T}\cdot\mathbb{E}^{\mathbb{P}}
\left[(\phi^{-1}f)^2(X_{T})\right]^{1/2}\cdot\mathbb{E}^{\mathbb{P}}\left[\int_{0}^{T}|\overline{k}|^2(X_{s})\; ds\right]^{1/2}\\
\leq\;&\sqrt{\frac{a_1}{T}}\cdot\mathbb{E}^{\mathbb{P}}
\left[(\phi^{-1}f)^2(X_{T})\right]^{1/2}\;.
\end{aligned}$$ This completes the proof.
Proof of the Theorem \[thm:variation\_rho\_expo\_condi\] {#app:pf_ext_rho}
========================================================
We first show the following fact, which is a variation of proposition \[prop:rho\].
\[prop:rho\_vari\] Let $g:\mathbb{R}^d\rightarrow\mathbb{R}$ be a function such that $$\left|\frac{\partial k_\epsilon(x)}{\partial\epsilon}\right|\leq g(x)$$ on $(\epsilon,x)\in I\times\mathbb{R}^d$ for an open interval $I$ containing $0.$ Suppose that $\mathbb{E}^{\mathbb{P}}[\exp(\epsilon_1\int_0^Tg^2(X_t)\,dt)]$ is finite for some positive $\epsilon_1$ and $\mathbb{E}^{\mathbb{P}}\int_0^Tg^{n}(X_t)\,dt$ is finite for all $n>0.$ Then for any given function $f(\cdot)$ with $\mathbb{E}^{\mathbb{P}}[|\phi^{-1}f|^{1+\epsilon_2}(X_T)]<\infty$ for some positive $\epsilon_2,$ we have $$\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P_{\epsilon}}}
The following theorem will be essentially used for the proof of this proposition.
\[thm:McKean\] Let $q$ be a positive even integer and let $Y_t$ be a stochastic process with $\mathbb{E}\int_0^\infty Y_t^2\, dt<\infty.$ Then $$\mathbb{E}\left(\int_0^\infty Y_t\, dB_t\right)^q\leq c_q\cdot\mathbb{E}\left(\int_0^\infty Y_t^2\, dt\right)^{\frac{q}{2}}$$ for some positive constant $c_q.$
For proof, see page 40 in [@McKean].
We now prove proposition \[prop:rho\_vari\]. Using the same argument in the proof of Proposition \[prop:rho\], it will be shown that $$\lim_{\epsilon\rightarrow0}\,\mathbb{E}^{\mathbb{P}} \left[(\phi^{-1}f)(X_{T})\int_0^TZ^\epsilon(t)\eta_\epsilon(X_t) -\overline{k}(X_{t})\;dB_t\right]=0\;,$$ where $Z^\epsilon(t)$ and $\eta_\epsilon$ are as defined in the proof of Proposition \[prop:rho\]. Let $q'$ be a positive number such taht $1/q'+1/(1+\epsilon_2)=1.$ From the condition $\mathbb{E}^{\mathbb{P}}[|\phi^{-1}f|^{1+\epsilon_2}(X_T)]<\infty,$ by the Cauchy-Schwarz inequality, it is enough to show that $\int_0^TZ^\epsilon(t)\eta_\epsilon(X_t) -\overline{k}(X_{t})\;dB_t$ converges to zero in $L^q$ as $\epsilon\rightarrow0$ for any positive even integer $q$ with $q\geq q'.$ Since we know $$\int_0^TZ^\epsilon(t)\eta_\epsilon(X_t) -\overline{k}(X_{t})\;dB_t
=\int_0^T(Z^\epsilon(t)-1)\eta_\epsilon(X_t)\,dB_t+\int_0^T(\eta_\epsilon-\overline{k})(X_{t})\;dB_t\;,$$ it will be shown that each term on the right hand side converges to zero in $L^q.$ For the second term, we use the Lebesgue dominated convergence theorem. Because $|\eta_\epsilon-\overline{k}|^q\leq c\cdot(|\eta_\epsilon|^q+|\overline{k}|^q)\leq 2cg^q$ for some positive constant $c$ and $\mathbb{E}^{\mathbb{P}}\int_0^Tg^q(X_{t})\;dt$ is finite, we have that $$\begin{aligned}
\mathbb{E}^{\mathbb{P}}\left(\int_0^T(\eta_\epsilon-\overline{k})(X_{t})\;dB_t\right)^q
&\leq c_q\cdot\mathbb{E}^{\mathbb{P}}\left(\int_0^T|\eta_\epsilon-\overline{k}|^2(X_{t})\;dt\right)^{\frac{q}{2}}\\
&\leq c_q T^{\frac{q}{2}-1}\cdot \mathbb{E}^{\mathbb{P}}\int_0^T|\eta_\epsilon-\overline{k}|^q(X_{t})\;dt\rightarrow0
\end{aligned}$$ as $\epsilon\rightarrow0$ for some constant $c_q,$ which is independent of $\epsilon.$
We now prove that $$\lim_{\epsilon\rightarrow0}\,\mathbb{E}^{\mathbb{P}}\left(\int_0^T(Z^\epsilon(t)-1)\,\eta_\epsilon(X_t)\,dB_t\right)^q=0\;.$$ It follows that $$\begin{aligned}
&\mathbb{E}^{\mathbb{P}}\left(\int_0^T(Z^\epsilon(t)-1)\,\eta_\epsilon(X_t)\,dB_t\right)^q\\
\leq&\,c_qT^{\frac{q}{2}-1}\cdot\mathbb{E}^{\mathbb{P}}\int_0^T(Z^\epsilon(t)-1)^q\,|\eta_\epsilon|^q(X_t)\,dt \\
\leq\,&\,c_qT^{\frac{q}{2}-1}\left(\int_0^T\mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)-1)^{2q}\,dt\right)^{\frac{1}{2}}\cdot
\left(\int_0^T\mathbb{E}^{\mathbb{P}}|\eta_\epsilon|^{2q}(X_t)\,dt\right)^{\frac{1}{2}}\\
\leq\,&\,c_qT^{\frac{q}{2}-1}\left(\int_0^T\mathbb{E}^{\mathbb{P}}(Z^\epsilon(t)-1)^{2q}\,dt\right)^{\frac{1}{2}}\cdot
\left(\int_0^T\mathbb{E}^{\mathbb{P}}g^{2q}(X_t)\,dt\right)^{\frac{1}{2}}\,.
\end{aligned}$$ The second term is finite from the assumption. By the same argument in the proof of Proposition \[prop:rho\], it can be shown that the first term goes to zero as $\epsilon$ goes to zero. This completes the proof.
We need the following proposition to prove Theorem \[thm:variation\_rho\_expo\_condi\].
\[prop:expo\_p\_th\_power\_inequ\] Let $p$ be a positive integer. Then for any positive random variable $Y,$ we have $$\mathbb{E}[Y^p]^{\frac{1}{p}}\leq\ln\mathbb{E}[e^{Y}]\;.$$
By direct calculation, we obtain $$\begin{aligned}
\mathbb{E}[e^{Y}]
&=\sum_{n=0}^\infty\frac{\mathbb{E}[Y^n]}{n!}\\
&=\sum_{n=0}^\infty\frac{\mathbb{E}[Y^{pn}]}{(pn)!}+\sum_{n=0}^\infty\frac{\mathbb{E}[Y^{pn+1}]}{(pn+1)!}+\cdots+\sum_{n=0}^\infty\frac{\mathbb{E}[Y^{pn+p-1}]}{(pn+p-1)!}\\
&\geq\sum_{n=0}^\infty\frac{\mathbb{E}[Y^{p}]^n}{(pn)!}+\sum_{n=0}^\infty\frac{\mathbb{E}[Y^{p}]^{n+\frac{1}{p}}}{(pn+1)!}+\cdots+\sum_{n=0}^\infty\frac{\mathbb{E}[Y^p]^{n+\frac{p-1}{p}}}{(pn+p-1)!}\\
&=\sum_{n=0}^\infty\frac{(\mathbb{E}[Y^{p}]^{\frac{1}{p}})^{pn}}{(pn)!}+\sum_{n=0}^\infty\frac{(\mathbb{E}[Y^{p}]^{\frac{1}{p}})^{pn+1}}{(pn+1)!}+\cdots+\sum_{n=0}^\infty\frac{(\mathbb{E}[Y^p]^{\frac{1}{p}})^{pn+p-1}}{(pn+p-1)!}\\
&=e^{\mathbb{E}[Y^{p}]^{\frac{1}{p}}}
\end{aligned}$$ This completes the proof.
We now prove the Theorem \[thm:variation\_rho\_expo\_condi\]. It suffices to show that $$\frac{1}{T}\cdot\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}f)(X_{T})\int_{0}^{T}\overline{k}(X_{s})\; dB_{s}]$$ converges to zero as $T$ goes to infinity. Let $q'$ be the positive number such that $1/q'+1/(1+\epsilon_2)=1$ and let $q$ be any positive even integer $q$ with $q\geq q'.$ From the assumption and the proposition above, $$\begin{aligned}
\ln c(T) +aT &=\ln\mathbb{E}^{\mathbb{P}}\left[\exp\left(\epsilon_0\int_0^Tg^2(X_t)\,dt\right)\right]\\
&\geq \,\left(\mathbb{E}^{\mathbb{P}}\left(\epsilon_0\int_0^Tg^2(X_t)\,dt\right)^{\frac{q}{2}}\right)^{\frac{2}{q}} \;,
\end{aligned}$$ it is obtained that $$\mathbb{E}^{\mathbb{P}}\left(\int_{0}^{T}|\overline{k}|^2(X_{s})\; ds\right)^{\frac{q}{2}}
\leq\mathbb{E}^{\mathbb{P}}\left(\int_{0}^{T}g^2(X_{s})\; ds\right)^{\frac{q}{2}}
\leq \left(\frac{\ln c(T)+aT}{\epsilon_0}\right)^{\frac{q}{2}}
\leq a_1\,T^{\frac{q}{2}}$$ for some positive number $a_1$ when $T$ is large.
$$\begin{aligned}
&\frac{1}{T}\cdot\mathbb{E}^{\mathbb{P}}
\left[(\phi^{-1}f)(X_{T})\int_{0}^{T}\overline{k}(X_{s})\; dB_{s}\right]\\
\leq\;&\frac{1}{T}\cdot\mathbb{E}^{\mathbb{P}}
\left[|\phi^{-1}f|^{1+\epsilon_2}(X_{T})\right]^{\frac{1}{1+\epsilon_2}}\cdot\mathbb{E}^{\mathbb{P}}\left[\left(\int_{0}^{T}\overline{k}(X_{s})\; dB_s\right)^{q}\right]^{\frac{1}{q}}\\
\leq\;&\frac{c_q}{T}\cdot\mathbb{E}^{\mathbb{P}}
\left[|\phi^{-1}f|^{1+\epsilon_2}(X_{T})\right]^{\frac{1}{1+\epsilon_2}}\cdot\mathbb{E}^{\mathbb{P}}\left[\left(\int_{0}^{T}|\overline{k}|^2(X_{s})\; ds\right)^{\frac{q}{2}}\right]^{1/q}\\
\leq\;&\frac{c_qa_1^{\frac{1}{q}}}{\sqrt{T}}\cdot\mathbb{E}^{\mathbb{P}}
\left[|\phi^{-1}f|^{1+\epsilon_2}(X_{T})\right]^{\frac{1}{1+\epsilon_2}}\\
\leq\;&\frac{c_qa_1^{\frac{1}{q}}}{\sqrt{T}}\cdot\mathbb{E}^{\mathbb{P}}
\left[|\phi^{-1}f|^{1+\epsilon_2}(X_{T})\right]\;.
\end{aligned}$$
This completes the proof.
Proof of Corollay \[cor:rho\_expo\_condi\] {#app:pf_rho_coro}
==========================================
It suffices to show the condition (i) and (ii)$'$. First show that $$\mathbb{E}^{\mathbb{P}}\left[\exp\left(\frac{1}{T}\int_0^T\epsilon_0\,g^2(X_t)\,dt\right)\right]$$ is bounded on $0<T<\infty.$ Let $c_1$ be such that $\mathbb{E}^{\mathbb{P}}[\exp(\epsilon_0\,g^2(X_t))]\leq c_1$ for all $t>0.$ Then, $$\begin{aligned}
\mathbb{E}^{\mathbb{P}}\left[\exp\left(\frac{1}{T}\int_0^T\epsilon_0\,g^2(X_t)\,dt\right)\right]
&\leq \mathbb{E}^{\mathbb{P}}\left[\frac{1}{T}\int_0^T\exp(\epsilon_0\,g^2(X_t))\,dt\right]\\
&=\frac{1}{T}\int_0^T\mathbb{E}^{\mathbb{P}}[\exp(\epsilon_0\,g^2(X_t))]\,dt\\
&\leq c_1\;,
\end{aligned}$$ which is the desired result. Now we prove that $$\mathbb{E}^{\mathbb{P}}\left[\exp\left(\epsilon_0\int_0^Tg^2(X_t)\,dt\right)\right]
=c(T)\,e^{aT}$$ for some constants $a$ and $c=c(T)$ with $c(T)$ bounded on $T>0.$ For any positive integer $n,$ by using Proposition \[prop:expo\_p\_th\_power\_inequ\], we have that $$\begin{aligned}
\left(\mathbb{E}^{\mathbb{P}}\left(\frac{1}{T}\int_0^T\epsilon_0\,g^2(X_t)\,dt\right)^n\right)^{\frac{1}{n}}
&\leq\ln \mathbb{E}^{\mathbb{P}}\left[\exp\left(\frac{1}{T}\int_0^T\epsilon_0\,g^2(X_t)\,dt\right)\right]
\leq\ln c_1\;.\\
\end{aligned}$$ Thus, $$\begin{aligned}
\mathbb{E}^{\mathbb{P}}\left(\int_0^T\epsilon_0\,g^2(X_t)\,dt\right)^n
\leq(T\ln c_1)^n\;.\\
\end{aligned}$$ We obtain $$\begin{aligned}
\mathbb{E}^{\mathbb{P}}\left[\exp\left(\epsilon_0\int_0^Tg^2(X_t)\,dt\right)\right]
&\leq \sum_{n=0}^\infty\frac{1}{n!}\,\mathbb{E}^{\mathbb{P}}\left(\int_0^T\epsilon_0\,g^2(X_t)\,dt\right)^n\\
&\leq \sum_{n=0}^\infty\frac{(T\ln c_1)^n}{n!}=e^{T\ln c_1}\;.\\
\end{aligned}$$ Thus, condition (i) is proved and condition (ii)$'$ is trivial. This completes the proof.
Proof of Corollary \[cor:delta\] {#app:pf_delta}
================================
From Proposition 3.2 in Fournie [@Fournie], we have that $$\nabla_\xi\,\mathbb{E}_\xi^{\mathbb{P }}(\phi^{-1}f)(X_{T})=\frac{1}{T}\,\mathbb{E}^\mathbb{P}\left[(\phi^{-1}f)(X_T)\int_0^T(\sigma^{-1}(X_t)Y_t)^\top dB_t\right]\;.$$ By the Cauchy-Schwarz inequality, it follows that $$|\nabla_\xi\,\mathbb{E}_\xi^{\mathbb{P }}(\phi^{-1}f)(X_{T})|\leq \frac{1}{T}\,\left(\mathbb{E}^\mathbb{P}(\phi^{-1}f)^2(X_T)\right)^{\frac{1}{2}}\cdot\left(\mathbb{E}^\mathbb{P}
\int_0^T|\!|\sigma^{-1}(X_t)Y_t |\!|^2dt
\right)^{\frac{1}{2}}\;,$$ which gives the desired result.
The CIR model {#app:CIR}
=============
The martingale extraction {#app:the_mart_extrac}
-------------------------
We explore the sensitivity analysis of option prices whose underlying process is the Cox–Ingersoll–Ross (CIR) model. Under a risk-neutral measure $\mathbb{Q},$ the interest rate $r_{t}$ follows $$dr_{t}=(\theta-ar_{t})\,dt + \sigma \sqrt{r_{t}} \,dW_{t}$$ with $\theta,a,\sigma>0.$ We assume $2\theta>\sigma^{2}$ so that the original interest rate process and the perturbation process stay strictly positive for small perturbation. The associated second-order equation is $$\mathcal{L}\phi(r)=\frac{1}{2}\sigma^{2}r\phi''(r)+(\theta-ar)\phi'(r)-r\phi(r)=-\lambda\phi(r)\;.$$ Set $\kappa:=\frac{\sqrt{a^{2}+2\sigma^{2}}-a}{\sigma^{2}}.$
We explore the martingale extraction with respect to $$(\lambda,\phi(r)):=(\theta\kappa,e^{-\kappa r})\;.$$ First, by direct calculation, it can be shown that this is an eigenpair. The dynamics of the diffusion process induced by this pair satisfies equation below and is recurrent, thus this pair admits a martingale extraction. Let $\mathbb{P}$ be the transformed measure with respect to $(\theta \kappa,\,e^{-\kappa r}).$ The corresponding martingale exponent is $\varphi(r):=-\sigma \kappa\sqrt{r}.$ We know that a process $B_{t}$ defined by $$dB_{t}=dW_{t}+\sigma \kappa\sqrt{r_{t}} \,dt$$ is a Brownian motion under $\mathbb{P}.$ The interest rate $r_{t}$ follows $$\label{eqn:r_under_P}
dr_{t}=\left(
\theta-\sqrt{a^{2}+2\sigma^{2}}\,r_{t} \right)dt + \sigma \sqrt{r_{t}}\, dB_{t} \; .$$
We see that this martingale extraction stabilizes $f.$ Here, $f(r)$ is a nonnegative continuous function on $r\in[0,\infty),$ which is not identically zero, and whose growth rate at infinity is equal to or less than $e^{mr}$ with $m<\frac{a}{\sigma^2}.$ Even more, it can be shown that $\mathbb{E}^{\mathbb{P}}[(\phi^{-1}f)^2(r_t)]$ is convergent as $t$ approaches to infinity. To achieve this, it is enough to prove that $\mathbb{E}^{\mathbb{P}}[
e^{cr_{t}}]$ is convergent as $t$ goes to infinity for $c<\frac{2\sqrt{a^{2}+2\sigma^{2}}}{\sigma^{2}}.$ Set $b:=\sqrt{a^2+2\sigma^2}.$ We consider the density function of $r_t$ under $\mathbb{P}.$ The CIR process has an explicit formula of the density function: $$g(r;t)=h_t\,e^{-u-v}\left(\frac{v}{u}\right)^{q/2}I_q(2\sqrt{uv})$$ where $I_q$ is the modified Bessel function of the first kind of order $q$ and $$h_t=\frac{2b}{\sigma^2(1-e^{-bt})}\,,\;q=\frac{2\theta}{\sigma^2}-1\,,\;
u=h_tr_0e^{-bt}\,,\;v=h_t\,r\;.$$ After rewriting slightly, we find $$g(r;t)=k_t\,h_t\,e^{-h_tr}r^{q/2}I_q(2h_te^{-bt/2}\sqrt{r_0r})\;.$$ Here, $k_t=e^{-h_tr_0e^{-bt}}(r_0e^{-bt})^{-q/2}$ and $$I_q(z)=\frac{(z/2)^q}{\pi^{1/2}\,\Gamma(q+1/2)}\int_0^\pi (e^{z\cos u}\sin^{2q}u)\,du\leq
\frac{\pi^{1/2}(z/2)^qe^z}{\Gamma(q+1/2)}
\;.$$ For large $t,$ we have $$\label{eqn:CIR_density}
g(r;t)\leq B\,e^{-h_tr} r^{q}e^{2h_t\sqrt{r_0r}}$$ for some constant $B.$ Because $c<\frac{2b}{\sigma^2}<h_t,$ we know that $e^{cr}g(r;t)$ is dominated by $$B\,e^{\left(c-\frac{2b}{\sigma^2}\right)r} r^{q}e^{2h_1\sqrt{r_0r}},$$ whose integration over $(0,\infty)$ is finite. By the Lebesgue dominated convergent theorem, we have that $\mathbb{E}^{\mathbb{P}}\left[
e^{cr_{t}}\right]$ is convergent and the limit is $$\int_0^\infty e^{cr}g(r;\infty)\,dr$$ where $g(r;\infty)=\lim_{t\rightarrow\infty} g(r;t),$ which is equal to the invariant density function of $r_t$ under $\mathbb{P}.$ For more details of the density of the CIR model, refer to [@Benth].
Sensitivity on $\theta$ {#sec:sen_theta_CIR}
-----------------------
Now, we see the sensitivity analysis with respect to $\theta$ of long-term option prices. Consider the perturbed process $r_{t}^{\epsilon}$ with respect to $\theta:$ $$dr_{t}^{\epsilon}=((\theta+\epsilon)-ar_{t}^{\epsilon})\,dt+
\sigma\sqrt{r_{t}^{\epsilon}}\,dW_{t}\,.$$ We already know $$\begin{aligned}
(\lambda(\epsilon),\phi_{\epsilon}(r)):=((\theta+\epsilon)\kappa,e^{-\kappa r})
\end{aligned}$$ stabilizes $f$ described above. The dynamics of $r_t^\epsilon$ follows $$dr_{t}^\epsilon=\left(
(\theta+\epsilon)-\sqrt{a^{2}+2\sigma^{2}}\,r_{t}^\epsilon \right)dt + \sigma \sqrt{r_{t}^\epsilon}\, dB_{t}^\epsilon$$ where $B_t^\epsilon$ is a Brownian motion under the corresponding measure $\mathbb{P}_\epsilon.$
We apply Theorem \[thm:rho\_expo\_condi\] to conclude that $$\begin{aligned}
&\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\theta}\ln p_T=-\kappa=-\frac{\sqrt{a^{2}+2\sigma^{2}}-a}{\sigma^{2}}\;.
\end{aligned}$$ First, Condition 1 and 2 in Section \[sec:sen\_drift\_vol\] are clearly satisfied. Now it will be shown in the following proposition that one of the conditions of the theorem is satisfied. Using $$k_\epsilon(r)=\frac{\theta+\epsilon}{\sigma \sqrt{r}}-\frac{\sqrt{a^{2}+2\sigma^{2}}}{\sigma}\,\sqrt{r}\;,$$ we know $$\frac{\partial}{\partial \epsilon}k_\epsilon(r)=\frac{1}{\sigma\sqrt{r}}\;,$$ and thus $$\overline{k}(r)=\frac{1}{\sigma\sqrt{r}}\;.$$ Since $\frac{\partial}{\partial \epsilon}k_\epsilon(r)$ is independent of $\epsilon,$ we set $g(r):=\overline{k}(r)$ in the theorem. The following proposition is enough to confirm one of the conditions of the theorem.
For $\epsilon_0$ with $\epsilon_0\leq \frac{1}{2}\left(\frac{\sigma}{2}-\frac{\theta}{\sigma}\right)^2,$ we have $$\mathbb{E}^{\mathbb{P}}\left[\exp\left(\epsilon_0\int_0^Tr^{-1}_t\,dt\right)\right]
\leq c(T)\,e^{aT}$$ for some constants $a$ and $c(T)$ with $c(T)$ bounded on $T.$
The main idea of the proof is from [@Ahn]. We know that $r_t$ satisfies $dr_{t}=\left(
\theta-br_{t}\right)dt + \sigma \sqrt{r_{t}}\, dB_{t}$ where $b=\sqrt{a^{2}+2\sigma^{2}}.$ Define $X_t:=r_t^{-1}.$ By direct calculation, we have $$dX_t=((\sigma^2-\theta)X_t+b)X_t\,dt-\sigma X^{3/2}\,dB_t\;.$$
We find a positive function $V(x,t)$ on $(x,t)\in \mathbb{R}^+\times [0,T] $ such that $V(X_t,t)\exp\left(\epsilon_0\int_0^tX_t\,dt\right)$ is a local martingale and $V(x,T)$ is a constant function of $x.$ It follows that $$V_t+\frac{1}{2}\sigma^2x^3V_{xx}+((\sigma^2-\theta)x+b)xV_x+\epsilon_0xV=0\;.$$ Try this form: $V(x,t)=f(y)y^\gamma$ where $y=a(t)/x.$ $$\begin{aligned}
&V_x=-\frac{1}{a(t)}f'(y)y^{\gamma+2}-\frac{\gamma}{a(t)}f(y)y^{\gamma+1}\\
&V_{xx}=\frac{1}{a^2(t)}f''(y)y^{\gamma+4}+\frac{2(\gamma+1)}{a^2(t)}f'(y)y^{\gamma+3}+\frac{\gamma(\gamma+1)}{a^2(t)}f(y)y^{\gamma+2}\\
&V_t=\frac{a'(t)}{a(t)}f'(y)y^{\gamma+1}+\frac{a'(t)}{a(t)}\gamma f(y)y^{\gamma}\;.
\end{aligned}$$ Then we have $$\begin{aligned}
&\quad\frac{1}{2}\sigma^2a(t)y^{\gamma+1}f''(y)
+\left(\frac{a'(t)}{a(t)}y^{\gamma+1}-by^{\gamma+1}-(\sigma^2-\theta)a(t)y^{\gamma}
+\sigma^2(\gamma+1)a(t)y^\gamma\right)f'(y)\\
&+\left(\frac{a'(t)}{a(t)}\gamma y^{\gamma}-b\gamma y^{\gamma}
+\frac{1}{2}\sigma^2\gamma(\gamma+1)a(t)y^{\gamma-1}-(\sigma^2-\theta)\gamma a(t)y^{\gamma-1}+\epsilon_0a(t)y^{\gamma-1}\right)f(y)=0
\end{aligned}$$ Let $$\left\{\quad
\begin{aligned}
&\frac{a'(t)}{a(t)}-b=a(t) \\
&\frac{1}{2}\sigma^2\gamma(\gamma+1)-(\sigma^2-\theta)\gamma+\epsilon_0=0\;.
\end{aligned}\right.$$ It follows that $$\frac{1}{2}\sigma^2yf''(y)+(y+\sigma^2\gamma+\theta)f'(y)+\gamma f(y)=0\;.$$ Define a new variable $z$ by $y=-\frac{1}{2}\sigma^2z$ and set $g(z):=f(y).$ We have $$zg''(z)+(\kappa-z)g'(z)-\gamma g(z)=0$$ where $\kappa=2\left(\gamma+\frac{\theta}{\sigma^2}\right).$ A solution of this equation is the standard confluent hypergeometric function: $$f(y)=g(z)=M(\gamma,\kappa;z)\;.$$
We now find an explicit expression for $V(x,t).$ From $\frac{a'(t)}{a(t)}-b=a(t),$ we obtain $$a(t)=\frac{b}{e^{b(T-t)}-1}$$ for $t<T.$ Solving $\frac{1}{2}\sigma^2\gamma(\gamma+1)-(\sigma^2-\theta)\gamma+\epsilon_0=0$ yields $$\gamma=\frac{1}{2}-\frac{\theta}{\sigma^2}+\sqrt{\left(\frac{1}{2}-\frac{\theta}{\sigma^2}\right)^2-\frac{2\epsilon_0}{\sigma^2}}\;,$$ which is a real number by the assumption on $\epsilon_0.$ We also have that $\gamma<0$ and $\kappa=2\left(\gamma+\frac{\theta}{\sigma^2}\right)>0.$ The solution $V(x,t)$ is given by $$\begin{aligned}
V(x,t)&=f(y)y^\gamma=g\left(z\right)\left(-\frac{1}{2}\sigma^2z\right)^\gamma\\
&=\left(\frac{1}{2}\sigma^2\right)^\gamma\cdot
M\left(\gamma,\kappa;z\right)(-z)^\gamma \\
&=\left(\frac{1}{2}\sigma^2\right)^\gamma\cdot M(\kappa-\gamma,\kappa;-z)(-z)^\gamma e^z
\end{aligned}$$ with $$z=-\frac{2y}{\sigma^2}=-\frac{2a(t)}{\sigma^2x}=-\frac{2b}{\sigma^2(e^{b(T-t)}-1)x}\;.$$ Here, we used $M(\gamma,\kappa;z)=M(\kappa-\gamma,\kappa;-z)e^z.$
We show that $$\begin{aligned} V(x,T):=
\lim_{t\rightarrow T}V(x,t)=\left(\frac{1}{2}\sigma^2\right)^\gamma\,\frac{\Gamma(\kappa)}{\Gamma(\kappa-\gamma)}\;.
\end{aligned}$$ It is obtained by $$\begin{aligned}
\lim_{t\rightarrow T}V(x,t)
&=\left(\frac{1}{2}\sigma^2\right)^\gamma\,\lim_{z\rightarrow -\infty}
M(\kappa-\gamma,\kappa;-z)(-z)^\gamma e^z\\
&=\left(\frac{1}{2}\sigma^2\right)^\gamma\,\lim_{u\rightarrow \infty}
M(\kappa-\gamma,\kappa;u)u^\gamma e^{-u} \\
&=\left(\frac{1}{2}\sigma^2\right)^\gamma\,\frac{\Gamma(\kappa)}{\Gamma(\kappa-\gamma)\Gamma(\gamma)}\,\lim_{u\rightarrow \infty}u^\gamma e^{-u}
\int_0^1 e^{us}s^{\kappa-\gamma-1}(1-s)^{\gamma-1}\,ds \\
&=\left(\frac{1}{2}\sigma^2\right)^\gamma\,\frac{\Gamma(\kappa)}{\Gamma(\kappa-\gamma)\Gamma(\gamma)}\,\lim_{u\rightarrow \infty}u^\gamma
\int_0^1 e^{-us}(1-s)^{\kappa-\gamma-1}s^{\gamma-1}\,ds \\
&=\left(\frac{1}{2}\sigma^2\right)^\gamma\,\frac{\Gamma(\kappa)}{\Gamma(\kappa-\gamma)\Gamma(\gamma)}\,\lim_{u\rightarrow \infty}
\int_0^u e^{-t}(1-t/u)^{\kappa-\gamma-1}t^{\gamma-1}\,dt \\
&=\left(\frac{1}{2}\sigma^2\right)^\gamma\,\frac{\Gamma(\kappa)}{\Gamma(\kappa-\gamma)\Gamma(\gamma)}\,\int_0^\infty e^{-t}t^{\gamma-1}\,dt \\
&=\left(\frac{1}{2}\sigma^2\right)^\gamma\,\frac{\Gamma(\kappa)}{\Gamma(\kappa-\gamma)}\;.
\end{aligned}$$ On the other hand, we have $$V(x,0)=c_1(T;x)\cdot e^{-\gamma b T}$$ with $c_1(T;x)$ bounded for large $T.$ It is because $$V(x,0)=c_2(T;x)\cdot\left(\frac{\sigma^2(1-e^{-bT})x}{2b}\right)^{-\gamma}\cdot e^{-\gamma bT}$$ where $$c_2(T;x)=
\left(\frac{1}{2}\sigma^2\right)^\gamma\cdot M\left(\kappa-\gamma,\kappa;\frac{2b}{\sigma^2(e^{bT}-1)x}\right) \exp\left(-\frac{2b}{\sigma^2(e^{bT}-1)x}\right)$$ and $c_2(T;x)$ is bounded for large $T.$ It is known that $\lim_{u\rightarrow 0}M(\kappa-\gamma,\kappa,u)=1.$
Because $V(X_t,t)\exp\left(\epsilon_0\int_0^tX_t\,dt\right)$ is a positive local martingale, it is a supermartingale. Thus, we have $$\begin{aligned}
&\left(\frac{1}{2}\sigma^2\right)^\gamma\,\frac{\Gamma(\kappa)}{\Gamma(\kappa-\gamma)}
\cdot\mathbb{E}\left[\exp\left(\epsilon_0\int_0^TX_t\,dt\right)\right]\\
=\;&\mathbb{E}\left[V(X_T,T)\exp\left(\epsilon_0\int_0^TX_t\,dt\right)\right]\\
\leq\; & V(X_0,0)\\
=\; & c_1(T;X_0)\cdot e^{-\gamma b T}\;.
\end{aligned}$$ By setting $c(T):=\left(\frac{1}{2}\sigma^2\right)^{-\gamma}\,\frac{\Gamma(\kappa-\gamma)}{\Gamma(\kappa)}\,c_1(T;r_0^{-1})$ and $a=-\gamma b,$ we obtain the desired result.
We now prove that the other conditions in Theorem \[thm:rho\_expo\_condi\] are satisfied. Let $\epsilon_1$ be a positive number with $\frac{\epsilon_1}{2}<\frac{2\theta}{\sigma^2}-1.$ We show that $\mathbb{E}^{\mathbb{P}}[(1/\sqrt{r_t})^{2+\epsilon_1}]$ is convergent to a constant as $t$ approaches to infinity. From equation , we know that $$\mathbb{E}^{\mathbb{P}}[(1/\sqrt{r_t})^{2+\epsilon_1}]=\int_0^{\infty} r^{-1-\frac{\epsilon_1}{2}} g(r;t)\,dr
\leq B\int_0^{\infty}e^{-h_tr} r^{\frac{2\theta}{\sigma^2}-2-\frac{\epsilon_1}{2}}e^{2h_t\sqrt{r_0r}}\,dr\;.$$ For large $t,$ the integrand is dominated by $e^{-\frac{2b}{\sigma^2}r}r^{\frac{2\theta}{\sigma^2}-2-\frac{\epsilon_1}{2}}e^{2h_1\sqrt{r_0r}},$ whose integration over $(0,\infty)$ is finite because $\frac{2\theta}{\sigma^2}-2-\frac{\epsilon_1}{2}>-1.$ By the Lebesgue dominated convergent theorem, we obtain the desired result. We already showed that the condition $\frac{1}{T}\cdot\mathbb{E}^{\mathbb{P}}[(\phi^{-1}f^{})^2(X_T)]\rightarrow0$ as $T\rightarrow\infty$ is satisfied with $f$ described above.
Sensitivity on $a$ {#sec:sen_a}
------------------
Now, we explore the sensitivity of variable $a$ in the drift coefficient. The perturbed process $r_{t}^{\epsilon}$ with respect to $a$ is $$dr_{t}^{\epsilon}=(\theta-(a+\epsilon)r_{t}^{\epsilon})\,dt+
\sigma\sqrt{r_{t}^{\epsilon}}\,dW_{t}\,.$$ We know that $$\begin{aligned}
(\lambda(\epsilon),\phi_{\epsilon}(r)):=(\theta \kappa(\epsilon),e^{-\kappa(\epsilon)r})
\end{aligned}$$ stabilizes $f$ describe above, where $$\kappa(\epsilon)=\frac{\sqrt{(a+\epsilon)^{2}+2\sigma^{2}}-(a+\epsilon)}{\sigma^{2}}\,.$$ The dynamics of $r_t^\epsilon$ follows $$dr_{t}^\epsilon=\left(
\theta-\sqrt{(a+\epsilon)^{2}+2\sigma^{2}}\,r_{t}^\epsilon \right)dt + \sigma \sqrt{r_{t}^\epsilon}\, dB_{t}$$ under the corresponding measure $\mathbb{P}_\epsilon.$
First, we check Condition 1 and 2 in Section \[sec:sen\_drift\_vol\]. The first condition is clear, thus the second condition will be proved. We will use Theorem \[thm:payoff\], so the function $G$ in the theorem is constructed. By direct calculation, $$\frac{\partial}{\partial\epsilon}(\phi_\epsilon^{-1}f)(r)=\kappa'(\epsilon)re^{\kappa(\epsilon)r}f(r)\;.$$ Since $$\kappa(0)=\kappa=\frac{\sqrt{a^{2}+2\sigma^{2}}-a}{\sigma^{2}}<
\frac{2\sqrt{a^2+\sigma^2}-a}{\sigma^2}\;,$$ there exits $\eta>0$ such that for $|\epsilon|<\eta,$ $$\kappa(\epsilon)<\frac{2\sqrt{a^2+\sigma^2}-a}{\sigma^2}\;.$$ Set $$G=G(r_T)=C\,r_T\,e^{\frac{2\sqrt{a^2+\sigma^2}-a}{\sigma^2}r_T}\,f(r_T)$$ where $C=\sup_{|\epsilon|<\eta}|\kappa'(\epsilon)|.$ Clearly, $$\sup_{|\epsilon|<\eta}\left|\frac{\partial}{\partial\epsilon}\,(\phi_\epsilon^{-1}f)(r_T)\right|\leq G\;.$$ We now see that $$\mathbb{E}^{\mathbb{P}}[G]<\infty\,.$$ It was assumed that the growth rate of $f(r)$ is less than $e^{mr}$ with $m<\frac{a}{\sigma^2},$ so the growth rate of $G(r)$ is less than $re^{\frac{2\sqrt{a^2+\sigma^2}}{\sigma^2}r},$ which is less than $e^{\frac{2\sqrt{a^2+2\sigma^2}}{\sigma^2}r}.$ We showed in Appendix \[app:the\_mart\_extrac\] that $\mathbb{E}^{\mathbb{P}}[e^{cr_{T}}]$ is convergent as $T$ goes to infinity for $c<\frac{2\sqrt{a^{2}+2\sigma^{2}}}{\sigma^{2}}.$ In particular, we have $\mathbb{E}^{\mathbb{P}}[G]<\infty.$ Therefore, we obtain that $\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}_{\epsilon}
f)(r_{T})]$ is differentiable at $\epsilon=0$ and $$\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}_{\epsilon}
f)(r_{T})]=\mathbb{E}^{\mathbb{P}}
\left[\left(\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\phi^{-1}_{\epsilon}
f\right)(r_{T})\right]=\kappa'(0)\cdot\mathbb{E}^{\mathbb{P}}
[ r_Te^{\kappa r_T}f(r_T)]\;.$$ By the same argument above, $\mathbb{E}^{\mathbb{P}}[r_Te^{\kappa r_T}f(r_{T})]$ converges to a constant as $T$ approaches to infinity because the growth rate of $re^{\kappa r}f(r)$ is less than $e^{cr}$ with $c<\frac{2\sqrt{a^{2}+2\sigma^{2}}}{\sigma^{2}}.$ In conclusion, we have that $$\frac{1}{T}\cdot\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}_{\epsilon}
f)(r_{T})]=\frac{k'(0)}{T}\cdot\mathbb{E}^{\mathbb{P}}[r_Te^{kr_T}
f(r_{T})]\rightarrow 0$$ as $T$ approaches to infinity.
We apply Theorem \[thm:rho\_expo\_condi\] to conclude that $$\begin{aligned}
&\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial a}\ln p_T=-\theta \kappa'(0)
=\frac{\theta(\sqrt{a^{2}+2\sigma^{2}}-a)}{\sigma^{2}\sqrt{a^{2}
+2\sigma^{2}}}\;.
\end{aligned}$$ Using that $$k_\epsilon(r)=\frac{\theta}{\sigma \sqrt{r}}-\frac{\sqrt{(a+\epsilon)^{2}+2\sigma^{2}}}{\sigma}\,\sqrt{r}\;,$$ we know $$\frac{\partial}{\partial \epsilon}k_\epsilon(r)=-\frac{(a+\epsilon)\sigma}{\sqrt{(a+\epsilon)^{2}+2\sigma^{2}}}{\sqrt{r}}\;,$$ and thus $$\overline{k}(r)=-\frac{a\sigma}{\sqrt{a^{2}+2\sigma^{2}}}{\sqrt{r}}\;.$$ Define $g(r)=\sigma\sqrt{r}.$ We show that $g$ satisfies the hypothesis of the theorem. First, it is trivial that $$\left|\frac{\partial}{\partial \epsilon}k_\epsilon(r)\right|\leq \sigma\sqrt{r}=g(r)$$ because $\frac{|(a+\epsilon)|\sigma}{\sqrt{(a+\epsilon)^{2}+2\sigma^{2}}}\leq
\sigma.$ Now it suffices to prove that $$\mathbb{E}^{\mathbb{P}}\left[\exp\left(\int_0^Tr_t\,dt\right)\right]
\leq c(T)\,e^{dT}$$ for some constants $d$ and $c(T)$ with $c(T)$ bounded on $T.$ For proof, refer to Lemma 3.1 on page 6 in [@Wong]. For another condition, let $\epsilon_1=2.$ It can be easily shown that $\mathbb{E}^{\mathbb{P}}[r_t^2]$ is convergent to a constant as $t$ approaches to infinity. We already showed that the condition $\frac{1}{T}\cdot\mathbb{E}^{\mathbb{P}}[(\phi^{-1}f^{})^2(X_T)]\rightarrow0$ as $T\rightarrow\infty$ is satisfied with $f$ described above.
Sensitivity on $\sigma$ {#app:sens_sigma_CIR}
-----------------------
The sensitivity analysis of variable $\sigma$ in the diffusion coefficient is explored. The perturbed process $r_{t}^{\epsilon}$ follows $$dr_{t}^{\epsilon}=(\theta-ar_{t}^{\epsilon})\,dt+(\sigma+\epsilon)\sqrt{r_{t}^{\epsilon}}\,dW_{t}
\,.$$ The initial value is not perturbed, that is, $r_0^\epsilon=r_0.$ We know that $$\begin{aligned}
(\lambda(\epsilon),\phi_{\epsilon}(r)):=(\theta \ell(\epsilon),e^{-\ell(\epsilon)r})
\end{aligned}$$ stabilizes $f$ described above, where $$\ell(\epsilon):=\frac{\sqrt{a^{2}+2(\sigma+\epsilon)^{2}}-a}{(\sigma+\epsilon)^{2}}\;.$$ Motivated by the discussion in Section \[sec:Lamperti\_trans\], define $u_\epsilon(r)=\frac{2}{\sigma+\epsilon}\sqrt{r}$ and $U_t^\epsilon=u_\epsilon(r_t^\epsilon),$ then we have $$\begin{aligned}
dU_t^\epsilon
&=\left(\left(\frac{2\theta}{(\sigma+\epsilon)^2}-\frac{1}{2}\right)\frac{1}{U_t^\epsilon}-\frac{a}{2}U_t^\epsilon\right)dt+dW_t \\
&=\left(\left(\frac{2\theta}{(\sigma+\epsilon)^2}-\frac{1}{2}\right)\frac{1}{U_t^\epsilon}-\frac{b(\epsilon)}{2}U_t^\epsilon\right)dt+dB_t^\epsilon \;,\\
U_0^\epsilon&=\frac{2}{\sigma+\epsilon}\sqrt{r_0}
\end{aligned}$$ where $b(\epsilon)=\sqrt{a^2+2(\sigma+\epsilon)^2}.$ Here, $B_t^\epsilon$ is a Brownian motion under the corresponding transformed measure $\mathbb{P}_\epsilon.$ The quantity $p_T^\epsilon$ can be expressed by $$\begin{aligned}
p_T^\epsilon:=\mathbb{E}^{\mathbb{Q}}[e^{-\int_0^T r_t^\epsilon\,dt}
f(r_T^\epsilon)]
&=\mathbb{E}^{\mathbb{Q}}[e^{-\int_{0}^{T}R_\epsilon(U_{s}^{\epsilon})ds}\,F_\epsilon(U_{T}^{\epsilon})]\\
&=e^{-\ell(\epsilon)r_0}\,e^{-\theta\ell(\epsilon) T}\cdot
\mathbb{E}^{\mathbb{P}_\epsilon}_{q(\epsilon)}
[(\Phi^{-1}_\epsilon F_\epsilon)(U_{T}^\epsilon)]
\end{aligned}$$ where $R_\epsilon(u):=(\sigma+\epsilon)^2u^2/4,$ $F_\epsilon(u):=f((\sigma+\epsilon)^2u^2/4),$ $\Phi_\epsilon(u)=\phi_\epsilon((\sigma+\epsilon)^2u^2/4)$ and $q(\epsilon)=\frac{2}{\sigma+\epsilon}\sqrt{r_0}.$ Differentiate with respect to $\epsilon$ and evaluate at $\epsilon=0,$ then $$\label{eqn:differ_vega_CIR}
\begin{aligned}
\frac{\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}p_{T}^{\epsilon}}
{T\cdot p_{T}}
&=-\theta\ell'(0)-\frac{\ell'(0)r_0}{T} \\
&+\frac{\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}_{q(\epsilon)}
[(\Phi^{-1}_{\epsilon}
F_\epsilon)(U_{T})]}
{T\cdot \mathbb{E}^{\mathbb{P}}
[(\Phi^{-1}
F)(U_{T})]}
+\frac{\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}
^{\mathbb{P_{\epsilon}}}
[(\Phi^{-1}
F)(U_{T}^{\epsilon})]}
{T\cdot\mathbb{E}^{\mathbb{P}}
[(\Phi^{-1}
F)(U_{T})]}
\;\; .
\end{aligned}$$
We now prove that $$\begin{aligned}
&\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial \sigma}\ln p_T=-\theta \ell'(0)
=\frac{\theta(\sqrt{a^{2}+2\sigma^{2}}-a)^2}{\sigma^{3}\sqrt{a^{2}+2\sigma^{2}}}
\end{aligned}$$ by showing that the third and the last terms go to zero as $T$ goes to infinity. For the last term, it is enough to show that the conditions in Theorem \[thm:rho\_expo\_condi\] are satisfied. To check the condition of the theorem, define $$k_\epsilon(u)=\left(\frac{2\theta}{(\sigma+\epsilon)^2}-\frac{1}{2}\right)\frac{1}{u}-\frac{b(\epsilon)}{2}u\;.$$ By direct calculation of $\frac{\partial}{\partial \epsilon}k_\epsilon(u),$ it can be shown that there exists a number $C>0$ such that $$\left|\frac{\partial}{\partial \epsilon}k_\epsilon(u)\right|\leq C\left(\frac{1}{u}+u\right)$$ for $\epsilon$ near $0$ and for all $u>0.$ Set $g(u):=C\left(\frac{1}{u}+u\right).$ Because $$g^2(U_t)=C^2\left(\frac{1}{U_t}+U_t\right)^2\leq 2C^2\left(\frac{1}{U_t^2}+U_t^2\right)=
2C^2\left(\frac{\sigma^2}{4r_t}+\frac{4r_t}{\sigma^2}\right)\leq C_1\left(\frac{1}{r_t}+{r_t}\right)$$ for sufficiently large $C_1>0,$ to confirm the condition of the theorem, it suffices to show that for a small positive number $\epsilon_0,$ $$\mathbb{E}^{\mathbb{P}}\left[\exp\left(\epsilon_0\int_0^T(r_t+r^{-1}_t)\,dt\right)\right]
\leq c(T)\,e^{aT}$$ for some constants $a$ and $c(T)$ with $c(T)$ bounded on $T.$ This was proven in section \[sec:sen\_theta\_CIR\] and \[sec:sen\_a\]. We already showed that the other conditions in the theorem are satisfied.
Now it will be proven that the third term of eqation goes to zero as $T$ goes to infinity. We will use that $\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}=\frac{\partial}{\partial \sigma}.$ The parameter $\sigma$ is involved with both $\Phi^{-1}
F$ and the dynamics of $U_{T}.$ However, in the third term, the differentiation is involved only with the parameter $\sigma$ in $\Phi^{-1}
F.$ To distinguish the parameter $\sigma$ in $\Phi^{-1}
F$ with that in the dynamics of $U_{T},$ we will use parameter $s.$ Define $$\begin{aligned}
\eta(s)&=\frac{\sqrt{a^2+2s^2}-a}{s^2}\;,\\
\pi_s(r)&=e^{-\eta(s)r}\;,\\
\Pi_s(u)&=\pi_s(s^2u^2/4)\;,\\
G_s(u)&=f(s^2u^2/4)\;,\\
\zeta(s)&=\frac{2\sqrt{r_0}}{s}\;.\\
\end{aligned}$$ Then $$\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}_{q(\epsilon)}
[(\Phi^{-1}_{\epsilon}
F_\epsilon)(U_{T})]=\left.\frac{\partial}{\partial s}\right|_{s=\sigma}\mathbb{E}^{\mathbb{P}}_{\zeta(s)}
[(\Pi^{-1}_sG_s)(U_{T})]\;.$$ It suffices to show that $\frac{1}{T}\cdot\left.\frac{\partial}{\partial s}\right|_{s=\sigma}\mathbb{E}^{\mathbb{P}}_{\zeta(s)}
[(\Pi^{-1}_sG_s)(U_{T})]\rightarrow0$ as $T\rightarrow\infty.$
\[prop:sens\_sigma\_CIR\] $$\left.\frac{\partial}{\partial s}\right|_{s=\sigma}\mathbb{E}^{\mathbb{P}}_{\zeta(s)}
[(\Pi^{-1}_sG_s)(U_{T})]$$ is bounded on $0<T<\infty.$
It is clear that $(\Pi^{-1}_sG_s)(U_T)=(\pi^{-1}_sf)(s^2U^2_T/4)=(\pi^{-1}_sf)(Z_t)$ where $Z_t=Z_t(s)=s^2U_t^2/4.$ By direct calculation, we have $$dZ_t=\left(\frac{\theta s^2}{\sigma^2}-bZ_t\right)dt+s\sqrt{Z_t}\,dB_t\,,\,Z_0=r_0\;.$$ It is noteworthy that two parameters $\sigma$ and $s$ are involved in the dynamics. One of nice properties of this process is that the initial value is not perturbed. We know that $Z_t$ is a CIR process and the density function is given by $$g(z;t)=g_s(z;t)=h_t\,e^{-u-v}\left(\frac{v}{u}\right)^{q/2}I_q(2\sqrt{uv})$$ where $I_q$ is the modified Bessel function of the first kind of order $q$ and $$h_t=\frac{2b}{s^2(1-e^{-bt})}\,,\;q=\frac{2\theta}{\sigma^2}-1\,,\;
u=h_tz_0e^{-bt}\,,\;v=h_t\,z\;.$$ After rewriting slightly, we find $$g(z;t)=e^{-h_tz_0e^{-bt}}(z_0e^{-bt})^{-q/2}\,h_t\,e^{-h_tz}z^{q/2}I_q(2h_te^{-bt/2}\sqrt{z_0z})\;.$$
We prove that $$\label{eqn:interchange_int_diff_cir_sigma}
\left.\frac{\partial}{\partial s}\right|_{s=\sigma}\int_0^\infty (\pi_s^{-1}f)(z)\,g(z;t)\,dz
=\int_0^\infty f(z)\,\left.\frac{\partial}{\partial s}\right|_{s=\sigma} \,\pi_s^{-1}(z)g(z;t)\,dz$$ by using theorem \[thm:payoff\]. It is enough to show that there exists a function $G(z)$ such that $$\int_0^\infty G(z)\,dz<\infty\;,$$ and for $s$ near $\sigma$ and for all $z>0,$ $$\left|f(z)\,\frac{\partial}{\partial s} \,\pi_s^{-1}(z)g(z;t)\right|\leq G(z)\;.$$ Using $\frac{\partial}{\partial s}h_t=-2h_t/s,$ we have $$\begin{aligned}
\frac{\partial}{\partial s}\,g(z;t)&=\frac{\,2\,}{s}h_tz_0\,e^{-bt}\,g(z;t)-\frac{\,2\,}{s}g(z;t)+\frac{\,2\,}{s}z\,h_t\,g(z;t)\\
&-\frac{\,2\,}{s}e^{-h_tz_0e^{-bt}}z_0^{(-q+1)/2}e^{(q-1)bt/2}\,h_t^2\,e^{-h_tz}z^{(q+1)/2}(I_{q-1}+I_{q+1})\;.\\
\end{aligned}$$ Here, we used $I_q'(\cdot)=\frac{1}{2}(I_{q-1}(\cdot)+I_{q+1}(\cdot)).$
Now we can find the decay rate of $|f(z)\frac{\partial}{\partial s} \,\pi_s^{-1}(z)g(z;t)|.$ For large $t$ and for $s$ near $\sigma,$ each term of $\frac{\partial}{\partial s} \,g(z;t)$ above is dominated by, up to constant multiples, one of $$g(z;t)\,,\;zg(z;t)\,,\;e^{-h_tz}z^{(q+1)/2}(I_{q-1}+I_{q+1})\;.$$ The decay rate of each term is $e^{-h_tz}$ up to polynomial decay or growth rate. Thus, the decay rate of $|\frac{\partial}{\partial s} \,\pi_s^{-1}(z)g(z;t)|$ is less than or equal to $e^{(\eta(s)-h_t)z}$ up to polynomial decay or growth rate. It was assumed that the growth rate of $f(z)$ is less than $e^{mz}$ with $m<\frac{a}{\sigma^2}.$ We obtain that the decay rate of $|f(z)\frac{\partial}{\partial s} \,\pi_s^{-1}(z)g(z;t)|$ is less than $e^{(a/\sigma^2+\eta(s)-h_t)z},$ whose exponent satisfies $$\begin{aligned}
\frac{a}{\sigma^2}+\eta(s)-h_t
&=\frac{a}{\sigma^2}+\frac{\sqrt{a^2+2s^2}-a}{s^2}-\frac{2b}{s^2(1-e^{-bt})}\\
&<\frac{a}{\sigma^2}+\frac{\sqrt{a^2+2s^2}-a}{s^2}-\frac{2b}{s^2}\\
&=\frac{a}{\sigma^2}+\frac{\sqrt{a^2+2s^2}-a}{s^2}-\frac{2\sqrt{a^2+2\sigma^2}}{s^2}\;.
\end{aligned}$$ When $s=\sigma,$ the last term is $-\frac{\sqrt{a^2+2\sigma^2}}{\sigma^2},$ thus, less than half of which the last term is for $s$ near $\sigma$ by the continuity argument. We have that for $s$ near $\sigma,$ $$\frac{a}{\sigma^2}+\eta(s)-h_t<-\frac{\sqrt{a^2+2\sigma^2}}{2\sigma^2}\;.$$ Thus by setting $G(z):=Ce^{-\frac{\sqrt{a^2+2\sigma^2}}{2\sigma^2}z}$ for sufficiently large $C,$ we obtain the equation . Moreover, from this observation, we know that $$\int_0^\infty f(z)\,\left.\frac{\partial}{\partial s}\right|_{s=\sigma} \,\pi_s^{-1}(z)g(z;T)\,dz$$ is finite and converges to a constant as $T$ goes to infinity. Since $$\begin{aligned}
\left.\frac{\partial}{\partial s}\right|_{s=\sigma}\mathbb{E}^{\mathbb{P}}_{\zeta(s)}
[(\Pi^{-1}_sG_s)(U_{T})]
=&\left.\frac{\partial}{\partial s}\right|_{s=\sigma}\mathbb{E}^{\mathbb{P}}
[(\pi^{-1}_sf)(Z_T)]\\
=&\left.\frac{\partial}{\partial s}\right|_{s=\sigma}\int_0^\infty (\pi_s^{-1}f)(z)\,g(z;T)\,dz\\
=&\int_0^\infty f(z)\,\left.\frac{\partial}{\partial s}\right|_{s=\sigma} \,\pi_s^{-1}(z)g(z;T)\,dz\;,
\end{aligned}$$ we conclude that $ \left.\frac{\partial}{\partial s}\right|_{s=\sigma}\mathbb{E}^{\mathbb{P}}
[(\pi^{-1}_sf)(s^2U_T^2/4)]$ is bounded on $T,$ which is the desired result.
Sensitivity on $r_0$
--------------------
The sensitivity on the initial value $r_0$ is presented in this section. From the discussion in Section \[sec:delta\], we can write the quantity $\frac{\partial}{\partial r_0}p_T$ by $$\begin{aligned}
\frac{\frac{\partial}{\partial r_0}p_T}{p_T}
=-\kappa
+\frac{\frac{\partial}{\partial r_0}\mathbb{E}_{r_0}^{\mathbb{P }}[(\phi^{-1}f)(r_{T})]}{\mathbb{E}_{r_0}^{\mathbb{P }}[(\phi^{-1}f)(r_{T})]}\;.
\end{aligned}$$ It is enough to show that $\frac{\partial}{\partial r_0}\mathbb{E}_{r_0}^{\mathbb{P }}[(\phi^{-1}f)(r_{T})]\rightarrow0$ as $T\rightarrow\infty.$ Corollary \[cor:delta\] cannot be used because the drift and the volatility do not satisfy the conditions of the corollary. However, the expectation depends only on the final value $r_T,$ which is the beauty of the martingale extraction, we can easily calculate the expectation. Recall the density function of $r_t$ from Appendix \[app:the\_mart\_extrac\]: $$g(r;t)=e^{-h_tr_0e^{-bt}}(r_0e^{-bt})^{-q/2}\,h_t\,e^{-h_tr}r^{q/2}I_q(2h_te^{-bt/2}\sqrt{r_0r})\;.$$ It is well-known that the density $g(r;t)$ converges to the invariant density function, denoted by $g(r;\infty),$ as $t\rightarrow\infty.$ Since $g(r;\infty)$ is independent of $r_0,$ one can expect that $$\lim_{t\rightarrow\infty}\frac{\partial g(r;t)}{\partial r_0}=0$$ and the proof is as follows.
By direct calculation, we have $$\begin{aligned}
\frac{\partial g}{\partial r_0}
=\left(-h_te^{-bt}-\frac{q}{2r_0}+\frac{1}{2}\sqrt{\frac{r}{r_0}}\,h_te^{-bt/2}\frac{I_{q-1}(z)+I_{q+1}(z)}{I_q(z)}\right)g
\end{aligned}$$ where $z=2h_te^{-bt/2}\sqrt{r_0r}.$ Here, we used $I_q'(\cdot)=\frac{1}{2}(I_{q-1}(\cdot)+I_{q+1}(\cdot)).$ Observe that $z\rightarrow0$ when $t\rightarrow\infty.$ It is well-known that the modified Bessel function $I_q$ of order $q$ satisfies $$\lim_{z\rightarrow0}\frac{\;\;I_q(z)\;\;}{\frac{(z/2)^q}{\Gamma(q+1)}}=1\;.$$ We have that $$\lim_{t\rightarrow\infty}h_te^{-bt/2}\frac{I_{q-1}(z)+I_{q+1}(z)}{I_q(z)}
=\lim_{t\rightarrow\infty}h_te^{-bt/2}\frac{\frac{(h_te^{-bt/2}\sqrt{r_0r})^{q-1}}{\Gamma(q)}+\frac{(h_te^{-bt/2}\sqrt{r_0r})^{q+1}}{\Gamma(q+2)}}{\frac{(h_te^{-bt/2}\sqrt{r_0r})^q}{\Gamma(q+1)}}=\frac{q}{\sqrt{r_0r}}\;.$$ Thus, $\lim_{t\rightarrow\infty}\frac{\partial g}{\partial r_0}=0.$
Now we prove that $\frac{\partial}{\partial r_0}\mathbb{E}_{r_0}^{\mathbb{P }}[(\phi^{-1}f)(r_{T})]\rightarrow0$ as $T\rightarrow\infty.$ It follows that $$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{\partial}{\partial r_0}\mathbb{E}_{r_0}^{\mathbb{P }}[(\phi^{-1}f)(r_{T})]
&=\lim_{T\rightarrow\infty}\frac{\partial}{\partial r_0}\int_0^\infty (\phi^{-1}f)(r) \,g(r;T)\,dr\\
&=\lim_{T\rightarrow\infty}\int_0^\infty (\phi^{-1}f)(r)\,\frac{\partial g(r;T)}{\partial r_0}\,dr\\
&=\int_0^\infty (\phi^{-1}f)(r)\,\lim_{T\rightarrow\infty}\frac{\partial g(r;T)}{\partial r_0}\,dr\\
&=0
\end{aligned}$$ which is the desired result. The interchangeability of the differentiation with the integration and the limit with the integration can be easily justified.
The quadratic term structure model {#app:QTSM}
==================================
The martingale extraction with respect to $$(\lambda,\phi(x))=(\beta-\frac{1}{2}u^\top au+tr(aV)+u^\top b\,,\,e^{-\langle u,x\rangle-\langle Vx,x\rangle})$$ stabilizes $f.$ The dynamics of $X_t$ follows $$\label{eqn:SDE_QTSM_under_P}
dX_t=(b-au+(B-2aV)X_t)\,dt+\sigma\,dB_t$$ where $B_t$ is a Brownian motion under the corresponding transformed measure.
Sensitivity on $b$ {#app:sen_b_QTSM}
------------------
We present the sensitivity analysis of the quantity $p_T$ with respect to $b=(b_1,b_2,\cdots,b_d)^\top.$ Consider the following perturbed process $X_{t}^{\epsilon}:$ $$dX_t^\epsilon=(b+\epsilon\overline{b}+BX_t^\epsilon)\,dt+\sigma\,dW_t$$ for some $d$-dimensional column vector $\overline{b}.$ By the chain rule, we may assume that $\overline{b}=(1,0,0,\cdots,0)^\top.$ We already know the martingale extraction with respect to $$(\lambda(\epsilon),\phi_\epsilon(x))=(\beta-\frac{1}{2}u_\epsilon^\top au_\epsilon+tr(aV)+u_\epsilon^\top b_\epsilon\,,\,e^{-\langle u_\epsilon,x\rangle-\langle Vx,x\rangle})$$ stabilizes $f,$ where $b_\epsilon:=b+\epsilon\overline{b}$ and $u_\epsilon=(2Va-B^\top)^{-1}(2Vb_\epsilon+\alpha)=u+\epsilon(2Va-B^\top)^{-1}(2V\overline{b}+\alpha).$ The dynamics of $X_t^\epsilon$ follows $$dX_t^\epsilon=(b_\epsilon-au_\epsilon+(B-2aV)X_t^\epsilon)\,dt+\sigma\,dB_t^\epsilon$$ where $B_t^\epsilon$ is a Brownian motion under the corresponding transformed measure $\mathbb{P}_\epsilon.$
We apply Theorem \[thm:rho\_expo\_condi\] to conclude that $$\begin{aligned}
&\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial b_1}\ln p_T=-\lambda'(0)\;.
\end{aligned}$$ First, we check Condition 1 and 2 in Section \[sec:sen\_drift\_vol\]. The first condition is clear by the above observation. For the second condition, we use Theorem \[thm:payoff\]. By direct calculation, $$\frac{\partial}{\partial\epsilon} (\phi_\epsilon^{-1}f)=\left\langle \frac{\partial}{\partial\epsilon} u_\epsilon,x\right\rangle\cdot e^{\langle u_\epsilon,x\rangle+\langle Vx,x\rangle}f(x)
=\langle (2Va-B^\top)^{-1}(2V\overline{b}+\alpha),x\rangle\cdot e^{\langle u_\epsilon,x\rangle+\langle Vx,x\rangle}f(x)\;.$$ Since $f$ is bounded and has a bounded support, we know that $|\frac{\partial}{\partial\epsilon} (\phi_\epsilon^{-1}f)|$ is bounded by a constant for $\epsilon$ near $0$ and all $x.$ It follows that $\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}_{\epsilon}
f)(X_{T})]$ is differentiable at $\epsilon=0$ and $\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P}}
[(\phi^{-1}_{\epsilon}
f)(X_{T})] $ is bounded on $T.$ This gives the desired result. We now check the conditions of the theorem. Let $$k_\epsilon(x)=\sigma^{-1}b_\epsilon-\sigma^{\top}u_\epsilon+(\sigma^{-1}B-2\sigma^{\top}V)x\;.$$ Then, $\frac{\partial}{\partial \epsilon}k_\epsilon(x)$ is a constant vector independent of $\epsilon.$ Define $g(x)=c_1$ for sufficiently large constant $c_1$ so that $|\frac{\partial}{\partial \epsilon}k_\epsilon(x)|\leq c_1.$ Since $g$ is a constant function, clearly $g$ satisfies the conditions of the theorem. It is clear that $\mathbb{E}^\mathbb{P}[(\phi^{-1}f)^{2}(X_T)]$ is bounded on $0<T<\infty$ by considering the Gaussian density of $X_T$ because $f$ is bounded and has bounded support.
Sensitivity on $B$ {#app:sen_B_QTSM}
------------------
We investigate the sensitivity analysis of the quantity $p_T$ with respect to the matrix $B.$ Consider the following perturbed process $X_{t}^{\epsilon}:$ $$dX_t^\epsilon=(b+(B+\epsilon\overline{B})X_t^\epsilon)\,dt+\sigma\,dW_t$$ for some $d\times d$ matrix $\overline{B}.$ For convenience, set $B_\epsilon=B+\epsilon\overline{B}.$ Let $V_\epsilon$ be the [*stabilizing solution*]{} of $$2V_\epsilon aV_\epsilon-B_\epsilon^\top V_\epsilon-V_\epsilon B_\epsilon-\Gamma=0\;,$$ and let $u_\epsilon:=(2V_\epsilon a-B_\epsilon^\top)^{-1}(2V_\epsilon b+\alpha).$ We know the martingale extraction with respect to $$(\lambda(\epsilon),\phi_\epsilon(x))=(\beta-\frac{1}{2}u_\epsilon^\top au_\epsilon+tr(aV_\epsilon)+u_\epsilon^\top b\,,\,e^{-\langle u_\epsilon,x\rangle-\langle V_\epsilon x,x\rangle})$$ stabilizes $f.$ The dynamics of $X_t^\epsilon$ follows $$\label{eqn:SDE_perturbed_QTSM_under_P}
dX_t^\epsilon=(b-au_\epsilon+(B_\epsilon-2aV_\epsilon)X_t^\epsilon)\,dt+\sigma\,dB_t^\epsilon$$ where $B_t^\epsilon$ is a Brownian motion under the corresponding transformed measure $\mathbb{P}_\epsilon.$
We apply Corollary \[cor:rho\_expo\_condi\] to conclude that $$\begin{aligned}
&\lim_{T\rightarrow\infty}\frac{1}{T}\left.\frac{\partial \,}{\partial\epsilon}\right|_{\epsilon=0}\ln p_T^\epsilon=-\lambda'(0)\;.
\end{aligned}$$ First, we check Condition 1 and 2 in Section \[sec:sen\_drift\_vol\]. For the first condition, it is enough to show that $V_\epsilon$ and $u_\epsilon$ are differentiable at $\epsilon=0.$ Here, the differentiability of a matrix means that all components are differentiable.
This proof is indebted to Appendix D in [@Goodwin]. Consider the stabilizing solution $V_\epsilon$ of $$2V_\epsilon aV_\epsilon-B_\epsilon^\top V_\epsilon-V_\epsilon B_\epsilon-\Gamma=0\;.$$ The solution $V_\epsilon$ can be expressed by the following way. Define $$H_\epsilon=
\begin{pmatrix}
B_\epsilon & -2a \\
-\Gamma & -B_\epsilon^\top
\end{pmatrix}\;.$$ Since a similarity transformation preserves the eigenvalues, the eigenvalues of $H_\epsilon$ are the same as those of $-H_\epsilon^\top.$ On the other hand, the eigenvalues of $H_\epsilon$ and $H_\epsilon^\top$ must be same. Hence the spectral set of $H_\epsilon$ is the union of two sets $\Lambda_\epsilon^a$ and $\Lambda_\epsilon^b$ such that if $\beta\in\Lambda_\epsilon^a,$ then $-\beta\in\Lambda_\epsilon^b.$ According to the continuous-time algebraic Riccati equation theory, $H$ does not contain any eigenvalue on the imaginary axis when $\Gamma$ is positive definite. We can form $\Lambda_\epsilon^a$ such that it contains only the eigenvalues of $H_\epsilon$ that lie in the open left-half plane. Then there always exists a nonsingular matrix $P_\epsilon$ such that $$P_\epsilon^{-1}H_\epsilon P_\epsilon
=\begin{pmatrix}
H_\epsilon^a & {\bf 0} \\
{\bf 0} & H_\epsilon^b
\end{pmatrix}$$ where $H_\epsilon^a$ and $H_\epsilon^b$ are diagonal matrices with eigenvalues sets $\Lambda_\epsilon^a$ and $\Lambda_\epsilon^b,$ respectively. Write $$P_\epsilon=\begin{pmatrix}
P_{\epsilon,11} & P_{\epsilon,12} \\
P_{\epsilon,21} & P_{\epsilon,22}
\end{pmatrix}\;,$$ then $V_\epsilon=P_{\epsilon,21}P_{\epsilon,11}^{-1}$ is the stabilizing solution.
Form this observation, we can prove that $V_\epsilon$ is differentiable at $\epsilon=0.$ Since the eigenvalues of a matrix are continuously differentiable by the linear-perturbation in the components (see [@Lancaster]), we know that $H_\epsilon^a$ and $H_\epsilon^b$ are differentiable, so $P_\epsilon$ is also differentiable. Hence $V_\epsilon$ is differentiable, which induces that $u_\epsilon$ is also differentiable. This gives the desired result.
The second condition can be proven by the same way in the previous section. We now check the conditions of the theorem. Let $$k_\epsilon(x)=\sigma^{-1}b-\sigma^{\top}u_\epsilon+(\sigma^{-1}B_\epsilon-2\sigma^{\top}V_\epsilon)x\;.$$ Since $V_\epsilon$ and $u_\epsilon$ are continuously differentiable at $\epsilon=0,$ there exist sufficiently large constants $c_1$ and $c_2$ such that $|\frac{\partial}{\partial \epsilon}k_\epsilon(x)|\leq c_1+c_2|x|$ for $\epsilon$ near $0$ and for all $x\in\mathbb{R}^d.$ Define $g(x)=c_1+c_2|x|.$ To check the hypothesis of the corollary with $g,$ it suffices to show that there exists a positive $\epsilon_0$ such that $\mathbb{E}^{\mathbb{P}}[\exp(\epsilon_0\,|X_T|^2) ]$ is finite on $0<T<\infty.$ Consider the density function of $X_T,$ which is a multivariate normal random variable. $$\mathbb{E}^{\mathbb{P}}[\exp(\epsilon_0\,|X_T|^2)]
=\frac{1}{\sqrt{(2\pi)^d\det\Sigma_T}}\int_{\mathbb{R}^d} e^{\epsilon_0 |z|^2-\frac{1}{2}(z-\mu_T)^\top\Sigma_T^{-1}(z-\mu_T)}\,dz$$ where $\mu_T$ and $\Sigma_T$ are the mean vector and the covariance matrix of $X_T,$ respectively. Under $\mathbb{P},$ the coefficient of $X_t$ in the drift term of equation is $B-2aV,$ all of whose eigenvalues have negative real parts. Thus, the distribution of $X_T$ is convergent to an invariant distribution, which is a non-degenerate multivariate normal random variable. Let $\Sigma_\infty$ be the covariance matrix of the invariant distribution. Choose $\epsilon_0$ less than the smallest eigenvalue of $\Sigma_\infty^{-1},$ then the above integral converges to a constant as $T\rightarrow\infty.$ Lastly, it is clear that $\mathbb{E}^\mathbb{P}[(\phi^{-1}f)^{2}(X_T)]$ is bounded on $0<T<\infty$ by considering the Gaussian density of $X_T$ because $f$ is bounded and has bounded support.
Sensitivity on $\sigma$ {#app:sen_sigma_QTSM}
-----------------------
We investigate the sensitivity analysis of the quantity $p_T$ with respect to the volatility matrix $\sigma.$ Consider the following perturbed process $X_{t}^{\epsilon}:$ $$dX_t^\epsilon=(b+BX_t^\epsilon)\,dt+(\sigma+\epsilon\overline{\sigma})\,dW_t$$ Set $a_\epsilon=(\sigma+\epsilon\overline{\sigma})(\sigma+\epsilon\overline{\sigma})^\top.$ Let $V_\epsilon$ be the [*stabilizing solution*]{} of $$2V_\epsilon a_\epsilon V_\epsilon-B^\top V_\epsilon-V_\epsilon B-\Gamma=0\;,$$ and let $u_\epsilon:=(2V_\epsilon a_\epsilon-B^\top)^{-1}(2V_\epsilon b+\alpha).$ We know the martingale extraction with respect to $$(\lambda(\epsilon),\phi_\epsilon(x))=(\beta-\frac{1}{2}u_\epsilon^\top a_\epsilon u_\epsilon+tr(a_\epsilon V_\epsilon)+u_\epsilon^\top b\,,\,e^{-\langle u_\epsilon,x\rangle-\langle V_\epsilon x,x\rangle})$$ stabilizes $f.$ The dynamics of $X_t^\epsilon$ follows $$dX_t^\epsilon=(b-a_\epsilon u_\epsilon+(B-2a_\epsilon V_\epsilon)X_t^\epsilon)\,dt+(\sigma+\epsilon\overline{\sigma})\,dB_t^\epsilon$$ where $B_t^\epsilon$ is a Brownian motion under the corresponding transformed measure $\mathbb{P}_\epsilon.$
We apply Theorem \[thm:vega\_Fournie\_condi\] to conclude that $$\begin{aligned}
&\lim_{T\rightarrow\infty}\frac{1}{T}\left.\frac{\partial \,}{\partial\epsilon}\right|_{\epsilon=0}\ln p_T^\epsilon=-\lambda'(0)\;.
\end{aligned}$$ Condition 1 and 2 in Section \[sec:sen\_drift\_vol\] can be proven by the same way in the previous section. Define $X_t^\rho$ and $X_t^{\nu}$ as in Section \[sec:vega\_fournie\] and recall the equation : $$\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}\mathbb{E}^{\mathbb{P_{\epsilon}}}[(\phi^{-1}f)(X_{T}^{\epsilon})]
=\left.\frac{\partial}{\partial \rho}\;\right|_{\rho =0}\mathbb{E}^{\mathbb{P_{\rho}}}[(\phi^{-1}f)(X_{T}^{\rho})]
+\left.\frac{\partial}{\partial \nu}\;\right|_{\nu =0}\mathbb{E}^{\mathbb{P_{\nu}}}[(\phi^{-1}f)(X_{T}^{\nu})]\;.$$ We want to show $\lim_{T\rightarrow\infty}\frac{\partial}{\partial \epsilon}\left|_{\epsilon =0}\mathbb{E}^{\mathbb{P_{\epsilon}}}[(\phi^{-1}f)(X_{T}^{\epsilon})]\right.=0.$ For the first term, it can be shown that $\lim_{T\rightarrow\infty}\frac{\partial}{\partial \rho}\left|_{\rho =0}\mathbb{E}^{\mathbb{P_{\rho}}}[(\phi^{-1}f)(X_{T}^{\rho})]\right.
=0$ by the same argument in Section \[app:sen\_b\_QTSM\] and \[app:sen\_B\_QTSM\] because $X_t^\rho$ is involved with the perturbation in the drift. For the second term, it will be proven that $$\lim_{T\rightarrow\infty}\left.\frac{\partial}{\partial \nu}\;\right|_{\nu =0}\mathbb{E}^{\mathbb{P_{\nu}}}[(\phi^{-1}f)(X_{T}^{\nu})]=0\;.$$ Since $X_t^\nu$ satisfies $$dX_{t}^\nu=(b-au+(B-2aV )X_{t}^\nu)\, dt+(\sigma+\nu\overline{\sigma})\, dB_{t}^\nu \;,$$ the corresponding process $Z_t$ is given by $$dZ_t=(B-2aV)Z_t\,dt+\overline{\sigma}\,dB_t\;.$$ The process $Z_t$ is an OU process and we know that all eigenvalues of $B-2aV$ have negative real parts, thus $\mathbb{E}^\mathbb{P}[|Z_T|^2]$ is convergent as $T$ goes to infinity. Thus the conditions of Theorem \[thm:vega\_Fournie\_condi\] are satisfied for $f$ continuously differentiable with compact support.
Sensitivity on $\xi$ {#app:sen_xi_QTSM}
--------------------
We apply Corollary \[cor:delta\] to show that $$\lim_{T\rightarrow\infty}\frac{\,\nabla_\xi\, p_T\,}{p_T}=\frac{\nabla_\xi\,\phi(\xi)}{\phi(\xi)}=-u-2V\xi\;.$$ It is enough to show that $\mathbb{E}_\xi^\mathbb{P}|\!|Y_T|\!|^2$ is bounded on $0<T<\infty.$ The first variation process $Y_t$ is given by $dY_t=(B-2aV)Y_t\,dt$ with $Y_0=I_d,$ where $I_d$ is the $d\times d$ identity matrix. Thus, $$\mathbb{E}_\xi^\mathbb{P}|\!|Y_T|\!|^2=|\!|Y_T|\!|^2=|\!|e^{(B-2aV)T}|\!|^2$$ and since all eigenvalues of $B-2aV$ have negative real parts, we obtain the desired result.
The Heston model {#app:Heston}
================
The sensitivity of the quantity $$p_T:=\mathbb{E}^\mathbb{Q}[u(X_T)]=\mathbb{E}^\mathbb{Q}[X_T^\alpha]=\mathbb{E}^\mathbb{Q}[e^{\alpha\int_0^T\sqrt{v_t}dZ_t-\frac{\alpha}{2}\int_0^Tv_t\,dt}]\,e^{\alpha\mu T}X_0^\alpha$$ for large $T$ is of interest to us. Let $\mathbb{L}$ be a measure defined by $$\left.\frac{d\mathbb{L}}{d\mathbb{Q}}\right|_{\mathcal{F}_T}=e^{\alpha\int_0^T\sqrt{v_t}dZ_t-\frac{\alpha^2}{2}\int_0^Tv_t\,dt}\;,$$ then, using the Girsanov theorem, a process $U_t$ given by $dU_t=-\alpha\sqrt{v_t}\,dt+dZ_t$ with $U_0=1$ is a Brownian motion under $\mathbb{L}.$ It follows that $$\begin{aligned}
p_T=\mathbb{E}^\mathbb{L}[e^{-\frac{1}{2}\alpha(1-\alpha)\int_0^Tv_t\,dt}]\,e^{\alpha\mu T}X_0^\alpha\;.
\end{aligned}$$ For convenience, put $q_T:=\mathbb{E}^\mathbb{L}[e^{-\frac{1}{2}\alpha(1-\alpha)\int_0^Tv_t\,dt}].$ For some Brownian motion $\overline{Z}_t$ independent of $Z_t,$ we have $$\begin{aligned}
dv_t
&=(\gamma-\beta v_t)\,dt+\delta\sqrt{v_t}\,dW_t\\
&=(\gamma-\beta v_t)\,dt+\rho\delta\sqrt{v_t}\,dZ_t+\sqrt{1-\rho^2}\delta\sqrt{v_t}\,d\overline{Z}_t\;.
\end{aligned}$$ It follows that $$\begin{aligned}
dv_t
&=(\gamma-(\beta-\alpha\rho\delta) v_t)\,dt+\rho\delta\sqrt{v_t}\,dU_t+\sqrt{1-\rho^2}\delta\sqrt{v_t}\,d\overline{Z}_t\\
&=(\gamma-(\beta-\alpha\rho\delta) v_t)\,dt+\delta\sqrt{v_t}\,dB_t
\end{aligned}$$ for a Brownian motion $B_t$ under $\mathbb{L}.$ Define $r_t=\frac{1}{2}\alpha(1-\alpha)v_t,$ then $r_t$ is a CIR model expressed by $$\begin{aligned}
dr_t
&=\left(\frac{1}{2}\alpha(1-\alpha)\gamma-(\beta-\alpha\rho\delta)r_t\right)\,dt+\frac{\delta\sqrt{2\alpha(1-\alpha)}}{2}\sqrt{r_t}\,dB_t
\end{aligned}$$ and $q_T$ is written by $q_T=\mathbb{E}^\mathbb{L}[e^{-\int_0^Tr_t\,dt}].$ We already analyzed the sensitivity of $q_T$ for large $T$ in Section \[sec:CIR\] by using the martingale extraction. To apply the chain rule, put $\theta=\frac{1}{2}\alpha(1-\alpha)\gamma,$ $a=\beta-\rho\alpha\delta,$ $\sigma=\frac{\delta\sqrt{2\alpha(1-\alpha)}}{2}$ and $r_0=\frac{1}{2}\alpha(1-\alpha)v_0,$ then $r_t$ is expressed by $$dr_t=(\theta-ar_t)\,dt+\sigma\sqrt{r_t}\,dB_t\;.$$ In conclusion, $$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\mu}\ln p_T
&=\alpha\\
\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\gamma}\ln p_T
&=\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\gamma}\ln q_T
=\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\theta}\ln q_T\cdot\frac{\partial\theta}{\partial\gamma}\\
&=\frac{1}{2}\alpha(1-\alpha)\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\theta}\ln q_T\\
&=-\frac{1}{2}\alpha(1-\alpha)\cdot\frac{\sqrt{a^{2}+2\sigma^{2}}-a}{\sigma^{2}}\\
&=-\frac{1}{2}\alpha(1-\alpha)\cdot\frac{\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\beta+\rho\alpha\delta}{\delta^2}
\end{aligned}$$
$$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\beta}\ln p_T
&=\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\beta}\ln q_T
=\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial a}\ln q_T\cdot\frac{\partial a}{\partial\beta}\\
&=\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial a}\ln q_T\\
&=\frac{\theta(\sqrt{a^{2}+2\sigma^{2}}-a)}{\sigma^{2}\sqrt{a^{2}+2\sigma^{2}}}\\
&=\frac{\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\beta+\rho\alpha\delta}{\delta^2\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}}
\end{aligned}$$
$$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\delta}\ln p_T
&=\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\delta}\ln q_T\\
&=\lim_{T\rightarrow\infty}\frac{1}{T}\left(\frac{\partial}{\partial a}\ln q_T\cdot\frac{\partial a}{\partial\delta}+\frac{\partial}{\partial\sigma}\ln q_T\cdot\frac{\partial \sigma}{\partial\delta}\right)\\
&=-\rho\alpha\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial a}\ln q_T
+\frac{\sqrt{2\alpha(1-\alpha)}}{2}\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial \sigma}\ln q_T\\
&=-\rho\alpha\cdot\frac{\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\beta+\rho\alpha\delta}{\delta^2\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}}\\
&+\frac{(\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\beta+\rho\alpha\delta)^2}{\delta^3\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}}\\
\end{aligned}$$
$$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\rho}\ln p_T
&=\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial\rho}\ln q_T\\
&=\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial a}\ln q_T\cdot\frac{\partial a}{\partial\rho}\\
&=-\alpha\delta\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial}{\partial a}\ln q_T\\
&=-\frac{\alpha\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\alpha\beta+\rho\alpha^2\delta}{\delta\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}}
\end{aligned}$$
$$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{\partial}{\partial X_0}\ln p_T
=\lim_{T\rightarrow\infty}\frac{\partial}{\partial X_0}\ln(q_T\,e^{\alpha\mu T}X_0^\alpha)=\frac{\alpha}{X_0}
\end{aligned}$$
$$\begin{aligned}
\lim_{T\rightarrow\infty}\frac{\partial}{\partial v_0}\ln p_T
&=\lim_{T\rightarrow\infty}\frac{\partial}{\partial v_0}\ln q_T
=\lim_{T\rightarrow\infty}\frac{\partial}{\partial r_0}\ln q_T\cdot\frac{\partial r_0}{\partial v_0}\\
&=\frac{1}{2}\alpha(1-\alpha)\lim_{T\rightarrow\infty}\frac{\partial}{\partial r_0}\ln q_T\\
&=-\frac{1}{2}\alpha(1-\alpha)\cdot\frac{\sqrt{a^{2}+2\sigma^{2}}-a}{\sigma^{2}}\\
&=-\frac{1}{2}\alpha(1-\alpha)\cdot\frac{\sqrt{(\beta-\rho\alpha\delta)^{2}+\delta^2\alpha(1-\alpha)}-\beta+\rho\alpha\delta}{\delta^2}
\end{aligned}$$
The $3/2$ model {#app:3/2_model}
===============
Denote a perturbed process of $X_t$ and the induced perturbed process of $L_t$ by $X_t^\epsilon$ and $L_t^\epsilon,$ respectively. The quantity $$\left.\frac{\partial}{\partial\epsilon}\right|_{\epsilon=0}\mathbb{E}^\mathbb{Q}[u(L_T^\epsilon)]$$ will be investigated for large $T.$ We now find the corresponding martingale extraction. Using equation , the quantity $p_T$ can be expressed by $$p_T:=\mathbb{E}^\mathbb{Q}[u(L_T)]
=\mathbb{E}^\mathbb{Q}[e^{-\frac{\alpha\beta(\beta-1)\sigma^2}{2}\int_0^TX_u\,du}\,X_T^{\alpha\beta}]\cdot e^{-r\alpha(\beta-1)T}\;.$$ Consider the operator $$f\mapsto \mathbb{E}^\mathbb{Q}[e^{-\frac{\alpha\beta(\beta-1)\sigma^2}{2}\int_0^tX_u\,du}\,f(X_t)]\;.$$ The corresponding infinitesimal generator is $$\frac{1}{2}\sigma^2x^3\phi''(x)+(\theta-ax)x\phi'(x)-\frac{1}{2}\alpha\beta(\beta-1)\sigma^2x\phi(x)=-\lambda\phi(x)\;.$$ Set $$\ell:=\sqrt{\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)^2+\alpha\beta(\beta-1)}-\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)\;.$$ It can be shown that the martingale extraction with respect to $$(\lambda,\phi(x)):=\left(\theta\ell\,,\,x^{-\ell}\right)$$ stabilizes $f(x):=x^{\alpha\beta}.$ Then, $X_t$ follows $$dX_t=(\theta-(a+\sigma^2\ell)X_t)X_t\,dt+\sigma X_t^{3/2}\,dB_t$$ where $B_t$ is a Brownian motion under the corresponding transformed measure $\mathbb{P}.$
Sensitivity on $\theta$ {#sensitivity-on-theta}
-----------------------
Now, we see the sensitivity analysis of the expected utility and the return of $L_t$ with respect to $\theta.$ Consider the following perturbed process $X_{t}^{\epsilon}:$ $$dX_t^\epsilon=((\theta+\epsilon)-aX_t^\epsilon)X_t^\epsilon\,dt+\sigma X_t^{\epsilon\,{3/2}}\,dW_t\;.$$ We already know $$\begin{aligned}
(\lambda(\epsilon),\phi_{\epsilon}(r)):=((\theta+\epsilon)\ell\,,\,x^{-\ell})
\end{aligned}$$ stabilizes $f(x)=x^{\alpha\beta}.$ The dynamics of $X_t^\epsilon$ follows $$dX_t^\epsilon=((\theta+\epsilon)-(a+\sigma^2\ell)X_t^\epsilon)X_t^\epsilon\,dt+\sigma X_t^{\epsilon\,{3/2}}\,dB_t^\epsilon$$ where $B_t^\epsilon$ is a Brownian motion under the corresponding transformed measure $\mathbb{P}_\epsilon.$
Theorem \[thm:variation\_rho\_expo\_condi\] will be applied to conclude that $$\begin{aligned}
&\lim_{T\rightarrow\infty}\frac{1}{T}\frac{\partial }{\partial \theta}\ln p_T=-\lambda'(0)=-\ell\;.
\end{aligned}$$ Condition 1 and 2 in Section \[sec:sen\_drift\_vol\] are clearly satisfied. For the second condition, $\phi_\epsilon(x)=x^{-\ell}$ is independent of $\epsilon.$ We now show that the conditions of the theorem are satisfied. Using that $$k_\epsilon(x)=\left(\frac{\theta+\epsilon}{\sigma}\right)\frac{1}{\sqrt{x}}-\left(\frac{\,a\,}{\sigma}+\sigma\ell\right)\sqrt{x}\;,$$ we know $\frac{\partial}{\partial \epsilon}k_\epsilon(x)=\frac{1}{\sigma \sqrt{x}},$ and thus $\overline{k}(x)=\frac{1}{\sigma \sqrt{x}}.$ Define $g(x)=\frac{1}{\sigma \sqrt{x}}.$ It suffices to prove that $$\mathbb{E}^{\mathbb{P}}\left[\exp\left(\int_0^T\frac{1}{X_t}\,dt\right)\right]
\leq c(T)\,e^{aT}$$ for some constants $a$ and $c(T)$ with $c(T)$ bounded on $T.$ Define $r_t=1/X_t,$ then $r_t$ is the CIR model and we already proved this condition is satisfied. For the CIR process $r_t,$ it is well-known that $\mathbb{E}^{\mathbb{P}}[r_T^n]$ is convergent to a constant as $T\rightarrow\infty$ for any $n\in\mathbb{N}.$ By considering the density function of $r_t=1/X_t,$ it can be easily checked that for small $\epsilon_2>0,$ $$\mathbb{E}^\mathbb{P}[(\phi^{-1}f)^{1+\epsilon_2}(X_T)]=\mathbb{E}^\mathbb{P}[X_T^{(1+\epsilon_2)(\alpha\beta+\ell)}]$$ is convergent as $T\rightarrow\infty.$
Sensitivity on $a$ {#app:sen_a_3/2}
------------------
In this section, the sensitivity analysis of the expected utility and the return of $L_t$ with respect to $a$ is explored. Consider the following perturbed process $X_{t}^{\epsilon}:$ $$dX_t^\epsilon=(\theta-(a+\epsilon)X_t^\epsilon)X_t^\epsilon\,dt+\sigma X_t^{\epsilon\,{3/2}}\,dW_t\;.$$ We already know $$\begin{aligned}
(\lambda(\epsilon),\phi_{\epsilon}(r)):=(\theta\ell(\epsilon)\,,\,x^{-\ell(\epsilon)})
\end{aligned}$$ stabilizes $f(x)=x^{\alpha\beta},$ where $$\ell(\epsilon):=\sqrt{\left(\frac{1}{2}+\frac{a+\epsilon}{\sigma^2}\right)^2+\alpha\beta(\beta-1)}-\left(\frac{1}{2}+\frac{a+\epsilon}{\sigma^2}\right)\;.$$ The dynamics of $X_t^\epsilon$ follows $$dX_t^\epsilon=(\theta-(a+\epsilon+\sigma^2\ell(\epsilon))X_t^\epsilon)X_t^\epsilon\,dt+\sigma X_t^{\epsilon\,3/2}\,dB_t^\epsilon$$ where $B_t^\epsilon$ is a Brownian motion under the corresponding transformed measure $\mathbb{P}_\epsilon.$
Theorem \[thm:rho\_expo\_condi\] will be applied to conclude that $$\begin{aligned}
&\lim_{T\rightarrow\infty} \frac{\frac{\,\partial p_T\,}{\partial a}}{T\cdot p_{T}}=-\theta\ell'(0)=\frac{\left(\sqrt{\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)^2+\alpha\beta(\beta-1)}-\left(\frac{\sigma}{2}+a\right)\right)\theta}{\sigma^2\sqrt{\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)^2+\alpha\beta(\beta-1)}}\;.
\end{aligned}$$ We first check Condition 1 and 2 in Section \[sec:sen\_drift\_vol\]. The first condition is trivial. For the second condition, Theorem \[thm:payoff\] is used. By direct calculation, $$\left|\frac{\partial}{\partial\epsilon}(\phi_\epsilon^{-1}f)(X_t)\right|=\left|\ell'(\epsilon)X_t^{\alpha\beta+\ell(\epsilon)}\ln X_t\right|\leq c_2X_t^{\alpha\beta+\ell+1}$$ near $\epsilon=0$ for some positive constant $c_2.$ Since $\mathbb{E}^{\mathbb{P}}[X_t^{\alpha\beta+\ell+1}] $ is finite by considering the density of $r_t:=1/X_t,$ we obtain the desired result. We now show that the conditions of the theorem are satisfied. Using that $$k_\epsilon(x)=\frac{\theta}{\sigma\sqrt{x}}-\left(\frac{a+\epsilon}{\sigma}+\sigma\ell(\epsilon)\right)\sqrt{x}\;,$$ we know $\frac{\partial}{\partial \epsilon}k_\epsilon(x)=-\left(\frac{1}{\sigma}+\sigma\ell'(\epsilon)\right)\sqrt{x},$ and thus $\overline{k}(x)=-\left(\frac{1}{\sigma}+\sigma\ell'(0)\right)\sqrt{x}.$ For sufficiently large $c_1>0,$ we have that $$\left|\frac{\partial}{\partial \epsilon}k_\epsilon(x)\right|\leq c_1\sqrt{x}$$ near $\epsilon=0$ and for all $x>0.$ Define $g(x)=c_1\sqrt{x}.$ It suffices to prove that there exists a positive $\epsilon_0$ such that $$\mathbb{E}^{\mathbb{P}}\left[\exp\left(\epsilon_0\int_0^TX_t\,dt\right)\right]
\leq c(T)\,e^{aT}$$ for some constants $a$ and $c(T)$ with $c(T)$ bounded on $T.$ Define $r_t=1/X_t,$ then $r_t$ is the CIR model and we already proved this condition in Appendix \[sec:sen\_theta\_CIR\]. For another condition, it is enough to show that $\mathbb{E}^{\mathbb{P}}[(1/\sqrt{r_T})^{2+\epsilon_1}]$ is convergent as $T\rightarrow\infty$ and it was shown in Appendix \[sec:sen\_theta\_CIR\]. For the last condition, by considering the density function of $r_t,$ we know that for $m$ with $0<m<\frac{2a}{\sigma^2}+2\ell+2,$ $\mathbb{E}^{\mathbb{P}}[X_t^{m}]$ converges to a constant. Because $(\phi^{-1}f)^2(X_t)=X_t^{2\alpha\beta+2\ell},$ it follows that the condition is satisfied when $\frac{a}{\sigma^2}+1-\alpha\beta>0.$
Sensitivity on $\sigma$ {#sensitivity-on-sigma}
-----------------------
The sensitivity analysis of variable $\sigma$ in the diffusion coefficient is explored. The perturbed process $X_{t}^{\epsilon}$ follows $$dX_t^\epsilon=(\theta-aX_t^\epsilon)X_t^\epsilon\,dt+(\sigma +\epsilon)X_t^{\epsilon\,{3/2}}\,dW_t\;,\;X_0^\epsilon=\xi\;.$$ We know that $$\begin{aligned}
(\lambda(\epsilon),\phi_{\epsilon}(r)):=(\theta\ell(\epsilon)\,,\,x^{-\ell(\epsilon)})
\end{aligned}$$ stabilizes $f(x)=x^{\alpha\beta},$ where $$\ell(\epsilon):=\sqrt{\left(\frac{1}{2}+\frac{a}{(\sigma+\epsilon)^2}\right)^2+\alpha\beta(\beta-1)}-\left(\frac{1}{2}+\frac{a}{(\sigma+\epsilon)^2}\right)\;.$$ The dynamics of $X_t^\epsilon$ follows $$dX_t^\epsilon=(\theta-(a+(\sigma+\epsilon)^2\ell(\epsilon))X_t^\epsilon)X_t^\epsilon\,dt+(\sigma+\epsilon) X_t^{\epsilon\,3/2}\,dB_t^\epsilon$$ where $B_t^\epsilon$ is a Brownian motion under the corresponding transformed measure $\mathbb{P}_\epsilon.$
Motivated by the discussion in section \[sec:Lamperti\_trans\], define $u_\epsilon(x)=\frac{2}{(\sigma+\epsilon)\sqrt{x}}$ and $U_t^\epsilon=u_\epsilon(X_t^\epsilon),$ then we have $$\begin{aligned}
dU_t^\epsilon
&=\left(\left(\frac{2a}{(\sigma+\epsilon)^2}+2\ell(\epsilon)+\frac{3}{2}\right)\frac{1}{U_t^\epsilon}-\frac{\theta}{2}U_t^\epsilon\right)dt-dB_t^\epsilon \;,\\
U_0^\epsilon&=\frac{2}{(\sigma+\epsilon)\sqrt{\xi}}\;.
\end{aligned}$$ Here, $B_t^\epsilon$ is a Brownian motion under the corresponding transformed measure $\mathbb{P}_\epsilon.$ The quantity $p_T^\epsilon$ can be expressed by $$\begin{aligned}
p_T^\epsilon:
&=e^{-r\alpha(\beta-1)T}\cdot\mathbb{E}^\mathbb{Q}[e^{-\frac{\alpha\beta(\beta-1)(\sigma+\epsilon)^2}{2}\int_0^TX_u^\epsilon\,du}\,X_T^{\epsilon\,\alpha\beta}]\\
&=\xi^{-\ell(\epsilon)}e^{-(r\alpha(\beta-1)+\theta\ell(\epsilon))T}\cdot\mathbb{E}^{\mathbb{P}_\epsilon}_{q(\epsilon)}[((\sigma +\epsilon)U_t^\epsilon/2)^{-2\alpha\beta-2\ell(\epsilon)}]\\
&=\xi^{-\ell(\epsilon)}e^{-(r\alpha(\beta-1)+\theta\ell(\epsilon))T}\cdot
\mathbb{E}^{\mathbb{P}_\epsilon}_{q(\epsilon)}
[(\Phi^{-1}_\epsilon F_\epsilon)(U_{T}^\epsilon)]
\end{aligned}$$ where $F_\epsilon(u):=((\sigma +\epsilon)u^\epsilon/2)^{-2\alpha\beta},$ $\Phi_\epsilon(u)=\phi_\epsilon(((\sigma +\epsilon)u^\epsilon/2)^{-2})=((\sigma +\epsilon)u^\epsilon/2)^{-2\ell(\epsilon)}$ and $q(\epsilon)=\frac{2}{(\sigma+\epsilon)\sqrt{\xi}}.$ Differentiate the above equation with respect to $\epsilon$ and evaluate at $\epsilon=0,$ then $$\begin{aligned}
\frac{\left.\frac{\partial}{\partial \epsilon}\;\right|_{\epsilon =0}p_{T}^{\epsilon}}
{T\cdot p_{T}}
&=-\theta\ell'(0) -\frac{-\ell'(0)\ln\xi}{T}\\
\;\; .
&\lim_{T\rightarrow\infty} \frac{\frac{\,\partial p_T\,}{\partial\sigma}}{T\cdot p_{T}}
=-\theta \ell'(0)
=\frac{{2a\theta}\left(\sqrt{\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)^2+\alpha\beta(\beta-1)}-\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)\right)}{{\sigma^3}\sqrt{\left(\frac{1}{2}+\frac{a}{\sigma^2}\right)^2+\alpha\beta(\beta-1)}}
\end{aligned}$$ by showing that the third and the last terms go to zero as $T\rightarrow\infty.$ Using the same method in Proposition \[prop:sens\_sigma\_CIR\], it can be proven that the third term goes to zero as $T$ goes to infinity. For the last term, Theorem \[thm:rho\_expo\_condi\] is applied. Define $$k_\epsilon(u)=\left(\frac{2a}{(\sigma+\epsilon)^2}+2\ell(\epsilon)+\frac{3}{2}\right)\frac{1}{u}-\frac{\theta}{2}u\;.$$ By direct calculation of $\frac{\partial}{\partial \epsilon}k_\epsilon(u),$ we have that there exists a number $c_1>0$ such that $$\left|\frac{\partial}{\partial \epsilon}k_\epsilon(u)\right|\leq\frac{c_1}{u}$$ for $\epsilon$ near $0$ and for all $u>0.$ Set $g(u):=\frac{c_1}{u}.$ Using $g^2(U_t)=\frac{c_1^2}{U_t^2}=c_2X_t$ for sufficiently large $c_2>0,$ it can be shown that the conditions of the theorem are satisfied by the same method in Appendix \[app:sen\_a\_3/2\] when $\frac{a}{\sigma^2}+1-\alpha\beta>0.$
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[^1]: [email protected], [email protected]
[^2]: Most of work in the present article was done when the author was affiliated to Courant Institute of Mathematical Sciences, New York University, NY, USA. The author thanks to Jonathan Goodman and Srinivasa Varadhan for helpful comments.
| |
With October just around the corner, we need your help to change Columbus Day to Indigenous Peoples Day in recognition that Indigenous Peoples are still here. Community organizers are coming together to get an Indigenous Peoples Day resolution passed in Boston to recognize the true history of the city and honor the continued resiliency of Indigenous communities in Boston today. Moonanum James, of the Wampanoag tribe of Aquinnah, explains: "The Massachusett people had villages here.
February 21, 2012 is International Mother Language Day, or Mother Tongue Day, first observed by the international community in 2000 expressly to promote linguistic diversity and multilingualism—this year’s theme is “Mother tongue instruction and inclusive education.”
VISIT OURMOTHERTOUNGES.ORG
Cultural Survival's Endangered Languages Program invites you to explore American Indian language revitalization efforts nationwide in preparation for the November 17 national broadcast of the triumphant story of the reawakening and return home of the Wampanoag language. We Still Live Here - ?s Nutayune?n, starring the W?pan?ak Language Reclamation Project, airing nationally on PBS's Independent Lens series,
This year the Jennifer Easton Community Spirit Award was awarded to Cultural Survival board member, Ramona Peters (Mashpee Wampanoag) for her commitment to sustaining the cultural values of her people by the First Peoples Fund. Ramona works with clay and other natural materials making ceramic vessels.
From my apartment door, I see a blue sky hovering over the tin roof of the neighboring house. I see the blended tops of coconut trees standing so close, their palms so intertwined, that it’s hard to tell where one ends and the next begins. I see lined telephone poles and a busy road down to my right, the bus stop and grocery store across the street, and wild chickens running around below. They almost always seem to be in a hurry, but I never know where they’re going.
Long before the arrival of the settlers, the land which we call Turtle Island was bountiful of rich foods, clean water, and a vast amount of biodiversity. Cornfields wrapped around the coastline for miles, schools of fish swam so thick, and trees were so healthy they produced many nuts and fruits. Our ancestors celebrated thanksgiving about 13 times a year. In the Northeast, the first thanksgiving is the Strawberry Thanksgiving as it is the first berry of the season.
With Native American Heritage Month well underway and Thanksgiving/National Day of Mourning occurring tomorrow, it is an excellent time to celebrate Indigenous Peoples’ brilliance, honor and acknowledge truth in history, recognize whose land we are on, and work towards true allyship. We call upon our Cultural Survival community to learn from Indigenous Peoples and their true account of this federal holiday, confront settler mythologies of this country's history, understand how American colonialism and imperialism continue to impact Indigenous communities today, and to take steps towards tru
I am a Na Ñuu Savi (Person of the Place of Rain, Mixtec) born in Santa Maria, California, United States, to Nivi Ñuu Savi (People of the Place of Rain) who migrated there to work as farmers in the California agricultural economy. Ñuu Savi (the Place of Rain) is in Oaxaca, Puebla, and Guerrero, Mexico, where many pueblos are known by names that describe our history.
In the Northeastern Coastal Algoquin language, our word for dugout canoe is “mishoon.” Our coastal Tribes have utilized the waterways as ancient highways for thousands of years traveling in mishoon which are considered carbon neutral water vessels. As the original population of the American northeastern region, we have faced European assimilation.
For 50 years, Cultural Survival has partnered with Indigenous communities to advance Indigenous Peoples' rights and cultures worldwide. We envision a future that respects and honors Indigenous Peoples' inherent rights and dynamic cultures, deeply and richly interwoven in lands, languages, spiritual traditions, and artistic expression and rooted in self-determination and self-governance. | https://www.culturalsurvival.org/country/united-states |
In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.
The concept of the Nash equilibrium (NE) is not exactly original to Nash (e.g., Antoine Augustin Cournot showed how to find what we now call the Nash equilibrium of the Cournot duopoly game). Consequently, some authors refer to it as a Nash-Cournot equilibrium. However, Nash showed for the first time in his dissertation, Non-cooperative games (1950), that Nash equilibria must exist for all finite games with any number of players. Until Nash, this had only been proven for 2-player zero-sum games by John von Neumann and Oskar Morgenstern (1947).
Formal definition
Let (S, f) be a game, where S is the set of strategy profiles and f is the set of payoff profiles. When each player chooses strategy xi resulting in strategy profile x = (x1,...,xn) then player i obtains payoff fi(x). Note that the payoff depends on the strategy profile chosen, i.e. on the strategy chosen by player i as well as the strategies chosen by all the other players. A strategy profile is a Nash equilibrium (NE) if no deviation in strategy by any single player is profitable, that is, if for all i
A game can have a pure strategy NE or an NE in its mixed extension (that of choosing a pure strategy stochastically with a fixed frequency). Nash proved that, if we allow mixed strategies (players choose strategies randomly according to pre-assigned probabilities), then every n-player game in which every player can choose from finitely many strategies admits at least one Nash equilibrium.
Proof sketch
Let σ − i be a mixed strategy profile of all players except for player i. We can define a best response correspondence for player i, bi. bi is a relation from the set of all probability distributions over opponent player profiles to a set of player i's strategies, such that each element of
- bi(σ − i)
is a best response to σ − i. Define
- .
One can use the Kakutani fixed point theorem to prove that b has a fixed point. That is, there is a σ * such that . Since b(σ * ) represents the best response for all players to σ * , the existence of the fixed point proves that there is some strategy set which is a best response to itself. No player could do any better by deviating, and it is therefore a Nash equilibrium.
Examples
Competition game
|Player 2 chooses '0'||Player 2 chooses '1'||Player 2 chooses '2'||Player 2 chooses '3'|
|Player 1 chooses '0'||0, 0||2, -2||2, -2||2, -2|
|Player 1 chooses '1'||-2, 2||1, 1||3, -1||3, -1|
|Player 1 chooses '2'||-2, 2||-1, 3||2, 2||4, 0|
|Player 1 chooses '3'||-2, 2||-1, 3||0, 4||3, 3|
Consider the following two-player game: both players simultaneously choose a whole number from 0 to 3. Both players then win the minimum of the two numbers in points. In addition, if one player chooses a larger number than the other, then s/he has to give up two points to the other. This game has a unique Nash equilibrium: both players choosing 0. Any other choice of strategies can be improved if one of the players lowers his number to one less than the other player's number. In the table to the left, for example, when starting at the green square it is in player 1's interest to move to the purple square by choosing a smaller number, and it is in player 2's interest to move to the blue square by choosing a smaller number. If the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 3 Nash equilibria.
Coordination game
|Player 2 adopts strategy 1||Player 2 adopts strategy 2|
|Player 1 adopts strategy 1||A, A||B, C|
|Player 1 adopts strategy 2||C, B||D, D|
The coordination game is a classic ( symmetric) two player, two strategy game, with the payoff matrix shown to the right, where the payoffs are according to A>C and D>B. The players should thus cooperate on either of the two strategies to receive a high payoff. Players in the game have to agree on one of the two strategies in order to receive a high payoff. If the players do not agree, a lower payoff is rewarded. An example of a coordination game is the setting where two technologies are available to two firms with compatible products, and they have to elect a strategy to become the market standard. If both firms agree on the chosen technology, high sales are expected for both firms. If the firms do not agree on the standard technology, few sales result. Both strategies are Nash equilibria of the game.
Driving on a road, and having to choose either to drive on the left or to drive on the right of the road, is also a coordination game. For example, with payoffs 100 meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix:
|Drive on the Left||Drive on the Right|
|Drive on the Left||100, 100||0, 0|
|Drive on the Right||0, 0||100, 100|
In this case there are two pure strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit mixed strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%,100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where the probabilities for each player is (50%, 50%).
Prisoner's dilemma
- (but watch out for differences in the orientation of the payoff matrix)
The Prisoner's Dilemma has the same payoff matrix as depicted for the Coordination Game, but now C > A > D > B. Because C > A and D > B, each player improves his situation by switching from strategy #1 to strategy #2, no matter what the other player decides. The Prisoner's Dilemma thus has a single Nash Equilibrium: both players choosing strategy #2 ("betraying"). What has long made this an interesting case to study is the fact that D < A ("both betray" is globally inferior to "both remain loyal"). The globally optimal strategy is unstable; it is not an equilibrium.
As Ian Stewart put it, "sometimes rational decisions aren't sensible!"
Nash equilibria in a payoff matrix
There is an easy numerical way to identify Nash Equilibria on a Payoff Matrix. It is especially helpful in two person games where players have more than two strategies. In this case formal analysis may become too long. This rule does not apply to the case where mixed (stochastic) strategies are of interest. The rule goes as follows: if the first payoff number, in the duplet of the cell, is the maximum of the column of the cell and if the second number is the maximum of the row of the cell - then the cell represents a Nash equilibrium.
We can apply this rule to a 3x3 matrix:
|Option A||Option B||Option C|
|Option A||0, 0||25, 40||5, 10|
|Option B||40, 25||0, 0||5, 15|
|Option C||10, 5||15, 5||10, 10|
Using the rule, we can very quickly (much faster than with formal analysis) see that the Nash Equlibria cells are (B,A), (A,B), and (C,C). Indeed, for cell (B,A) 40 is the maximum of the first column and 25 is the maximum of the second row. For (A,B) 25 is the maximum of the second column and 40 is the maximum of the first row. Same for cell (C,C). For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns.
This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and check if the second member of the tuple has maximum of the row. If yes - you've got a Nash Equilibrium. Check all columns this way to find all NE cells. An NxN matrix may have between 0 and N pure strategy Nash equilibria.
Stability
The concept of stability, useful in the analysis of many kinds of equilibrium, can also be applied to Nash equilibria.
A Nash equilibrium for a mixed strategy game is stable if a small change (specifically, an infinitesimal change) in probabilities for one player leads to a situation where two conditions hold:
- the player who did not change has no better strategy in the new circumstance
- the player who did change is now playing with a strictly worse strategy
If these cases are both met, then a player with the small change in his mixed-strategy will return immediately to the Nash equilibrium. The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an infinite number of optimal strategies for the player who changed. John Nash showed that the latter situation could not arise in a range of well-defined games.
In the "driving game" example above there are both stable and unstable equilibria. The equilibria involving mixed-strategies with 100% probabilities are stable. If either player changes his probabilities slightly, they will be both at a disadvantage, and his opponent will have no reason to change his strategy in turn. The (50%,50%) equilibrium is instability. If either player changes his probabilities, then the other player immediately has a better strategy at either (0%, 100%) or (100%, 0%).
Stability is crucial in practical applications of Nash equilibria, since the mixed-strategy of each player is not perfectly known, but has to be inferred from statistical distribution of his actions in the game. In this case unstable equilibria are very unlikely to arise in practice, since any minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium.
Note that stability of the equilibrium is related to, but distinct from, stability of a strategy.
Occurrence
If a game has a unique Nash equilibrium and is played among players with certain characteristics, then it is true (by definition of these characteristics) that the NE strategy set will be adopted. Sufficient conditions to be met by the players are:
- The players all will do their utmost to maximize their expected payoff as described by the game.
- The players are flawless in execution.
- The players have sufficient intelligence to deduce the solution.
- There is common knowledge that all players meet these conditions, including this one. So, not only must each player know the other players meet the conditions, but also they must know that they all know that they meet them, and know that they know that they know that they meet them, and so on.
Where the conditions are not met
Examples of game theory problems in which these conditions are not met:
- The first condition is not met if the game does not correctly describe the quantities a player wishes to maximize. In this case there is no particular reason for that player to adopt an equilibrium strategy. For instance, the prisoner’s dilemma is not a dilemma if either player is happy to be jailed indefinitely.
- Pong has an equilibrium which can be played perfectly by a computer, but to make human vs. computer games interesting the programmers add small errors in execution, violating the second condition.
- In many cases, the third condition is not met because, even though the equilibrium must exist, it is unknown due to the complexity of the game, for instance in Chinese chess. Or, if known, it may not be known to all players, as when playing tic-tac-toe with a small child who desperately wants to win (meeting the other criteria).
- The fourth criterion of common knowledge may not be met even if all players do, in fact, meet all the other criteria. Players wrongly distrusting each other's rationality may adopt counter-strategies to expected irrational play on their opponents’ behalf. This is a major consideration in “ Chicken” or an arms race, for example.
Where the conditions are met
Due to the limited conditions in which NE can actually be observed, they are rarely treated as a guide to day-to-day behaviour, or observed in practice in human negotiations. However, as a theoretical concept in economics, and evolutionary biology the NE has explanatory power. The payoff in economics is money, and in evolutionary biology gene transmission, both are the fundamental bottom line of survival. Agents failing to maximize these for whatever reason will be competed out of the market or environment, which are ascribed the ability to test all strategies. This conclusion is drawn from the " stability" theory above. In these situations the assumption that the strategy observed is actually a NE has often been borne out by research. | https://www.cs.mcgill.ca/~rwest/wikispeedia/wpcd/wp/n/Nash_equilibrium.htm |
---
abstract: 'In this paper, we reduce the computational complexities of partial and dual partial cyclotomic FFTs (CFFTs), which are discrete Fourier transforms where spectral and temporal components are constrained, based on their properties as well as a common subexpression elimination algorithm. Our partial CFFTs achieve smaller computational complexities than previously proposed partial CFFTs. Utilizing our CFFTs in both transform- and time-domain Reed–Solomon decoders, we achieve significant complexity reductions.'
author:
- 'Ning Chen, and Zhiyuan Yan, [^1] [^2]'
bibliography:
- 'IEEEabrv.bib'
- 'rs.bib'
title: 'Reduced-Complexity Reed–Solomon Decoders Based on Cyclotomic FFTs'
---
Common subexpression elimination (CSE), Decoding, Discrete Fourier transforms, Galois fields, Reed–Solomon codes.
Introduction
============
Due to the widespread applications of Reed–Solomon (RS) codes [@Blahut83] in various digital communication and storage systems, efficient RS decoding has been an important research topic (see, for example, [@Jeng99; @Costa04; @Truong06a; @Lin07; @Fedorenko06; @Fedorenko02; @Truong01]). Since all syndrome-based hard-decision decoding methods for RS codes involve discrete Fourier transforms (DFTs) over finite fields [@Blahut83], fast Fourier transform (FFT) algorithms can be used to reduce the complexity of RS decoders (see, for example, [@Costa04; @Truong06a; @Fedorenko06]).
Using an approach similar to those in previous works (see, for example, [@Zakharova92]), cyclotomic FFTs (CFFTs) were recently proposed [@Trifonov03] and two variants were subsequently considered [@Costa04; @Fedorenko06]. To avoid confusion, in this paper we refer to the CFFTs proposed in [@Trifonov03] as direct CFFTs (DCFFTs) and those in [@Costa04] and [@Fedorenko06] as inverse CFFTs (ICFFTs) and symmetric CFFTs (SCFFTs), respectively. Given a primitive element $\alpha \in \mathrm{GF}(2^m)$, the DFT of a vector $\boldsymbol{f} = (f_0,f_1,\dots,f_{n-1})^T$ is defined as $\boldsymbol{F}\triangleq\bigl(f(\alpha^0), f(\alpha^1), \dots,
f(\alpha^{n-1})\bigr)^T$, where $f(x) \triangleq \sum_{i=0}^{n-1}f_i x^i
\in \mathrm{GF}(2^m)[x]$. A DCFFT is given by $\boldsymbol{F}=\boldsymbol{ALf'}
=\boldsymbol{AQ}(\boldsymbol{c}\cdot \boldsymbol{Pf}')$, where $\boldsymbol{A}$ is an $n\times n$ binary matrix, $\boldsymbol{L}$ is a block diagonal matrix, $\boldsymbol{f}'$ is a permutation of the input vector $\boldsymbol{f}$, $\boldsymbol{c}$ is a pre-computed vector, $\cdot$ stands for pointwise multiplications, and $\boldsymbol{Q}$ and $\boldsymbol{P}$ are both sparse binary matrices. Similarly, an SCFFT is given by $\boldsymbol{F}'=\boldsymbol{L}^T\boldsymbol{A}'^{T}
\boldsymbol{f}'=\boldsymbol{P}^T\bigl(\boldsymbol{c}\cdot(\boldsymbol{A}'\boldsymbol{Q})^T
\boldsymbol{f}'\bigr)$, where both $\boldsymbol{F}'$ and $\boldsymbol{f}'$ are permuted by the same permutation matrix, and $\boldsymbol{L}^T\boldsymbol{A}'^{T}$ is symmetric. Finally, based on inverse DFTs, an ICFFT is given by $\boldsymbol{F}''=\boldsymbol{L}^{-1}\boldsymbol{A}^{-1}\boldsymbol{f}
=\boldsymbol{P}^T(\boldsymbol{c^*}\cdot
\boldsymbol{Q}^T\boldsymbol{A}^{-1}\boldsymbol{f})$, where $\boldsymbol{F}''$ is also a permutation of $\boldsymbol{F}$ and $\boldsymbol{c^*}$ is a pre-computed vector. Since all CFFTs are in bilinear forms [@Blahut83], we refer to $\boldsymbol{P}$, $(\boldsymbol{A}'\boldsymbol{Q})^T$, and $\boldsymbol{Q}^T\boldsymbol{A}^{-1}$ as *pre-addition* matrices and $\boldsymbol{AQ}$, $\boldsymbol{P}^T$, and $\boldsymbol{P}^T$ as *post-addition* matrices for DCFFTs, SCFFTs, and ICFFTs, respectively. The numbers of non-one elements in $\boldsymbol{c}$ or $\boldsymbol{c^*}$ are the number of multiplications required, and the pre- and post-addition matrices determine the additive complexities of CFFTs. Though CFFTs in [@Trifonov03; @Costa04; @Fedorenko06] achieve low multiplicative complexities, their additive complexities (numbers of additions required) are very high, with or without the various methods used in [@Trifonov03; @Costa04; @Fedorenko06] to reduce the additive complexities. In our previous work [@Chen08a], we proposed a novel common subexpression elimination (CSE) algorithm, and then used it to reduce the additive complexities of **full** CFFTs.
This paper has three main contributions. First, we reduce both multiplicative and additive complexities of *partial CFFTs*, which compute only **part** of the spectral components [@Costa04; @Fedorenko06], based on their properties as well as our CSE algorithm in [@Chen08a]. Our partial CFFTs have smaller complexities than those in [@Costa04]. Second, we propose *dual partial CFFTs*, where only a subset of temporal components are nonzero, and reduce their complexities. Finally, applying our partial and dual partial CFFTs, we reduce the complexities of time- and transform-domain RS decoders significantly.
Partial and Dual Partial CFFTs {#sec:cfft}
==============================
We now consider CFFTs in two special cases. One special case is when only a subset of frequency components are needed, and we refer to such CFFTs as partial CFFTs following the convention in [@Costa04; @Fedorenko06]. The other special case is when a subset of temporal components are all zeros. The two special cases can be viewed as dual to each other; Thus, for the lack of a better term, we refer to CFFTs in the second special case as dual partial CFFTs.
In a partial CFFT, some frequency components are not needed. Thus, we first eliminate the rows corresponding to the unnecessary frequency components from the post-addition matrices, possibly resulting in all-zero columns. We then remove the all-zero columns from the reduced post-addition matrices, as well as the corresponding entries in $\boldsymbol{c}$ or $\boldsymbol{c}^\ast$ and the corresponding rows from the pre-addition matrices. For dual partial CFFTs, some temporal components are zeros. Thus, we first remove the corresponding columns in the pre-addition matrices, leading to all-zero rows. We then remove the all-zero rows from the reduced pre-addition matrices and the corresponding entries in $\boldsymbol{c}$ or $\boldsymbol{c}^\ast$ as well as the corresponding columns from the post-addition matrices.
It was shown [@Chen08a] full SCFFTs and ICFFTs are equivalent in terms of complexities; Using a similar argument we can show that SCFFTs and ICFFTs are also equivalent in partial and dual partial DFTs. In both special cases of CFFTs, removing rows or columns from pre- and post-addition matrices leads to reduced additive complexities, and eliminating entries in $\boldsymbol{c}$ or $\boldsymbol{c}^\ast$ results in reduced multiplicative complexities. Both multiplicative and additive complexity reductions depend on the type of CFFTs. Note that $\boldsymbol{P}^T$ and $\boldsymbol{P}$ are sparse, while $\boldsymbol{AQ}$, $(\boldsymbol{A}'\boldsymbol{Q})^T$, and $\boldsymbol{Q}^T\boldsymbol{A}^{-1}$ are not. Thus, removing a certain number of rows or columns from $\boldsymbol{P}^T$ ($\boldsymbol{P}$, respectively) leads to less significant reductions in additive complexities than from $\boldsymbol{AQ}$ ($(\boldsymbol{A}'\boldsymbol{Q})^T$ and $\boldsymbol{Q}^T\boldsymbol{A}^{-1}$, respectively). On the other hand, after removing some rows (columns, respectively), a reduced $\boldsymbol{P}^T$ ($\boldsymbol{P}$, respectively) is more likely to have all-zero columns (rows, respectively) that eliminate entries in $\boldsymbol{c}$ or $\boldsymbol{c}^*$ than $\boldsymbol{AQ}$ ($(\boldsymbol{A}'\boldsymbol{Q})^T$ and $\boldsymbol{Q}^T\boldsymbol{A}^{-1}$, respectively). Thus, partial DCFFTs have higher multiplicative complexities but lower additive complexities than partial SCFFTs/ICFFTs; similarly, dual partial DCFFTs lead to lower multiplicative complexities but higher additive complexities than dual partial SCFFTs/ICFFTs.
The savings in multiplicative complexities by partial SCFFTs/ICFFTs (dual partial DCFFTs, respectively) are improved by considering different permutations of $\boldsymbol{F}'$ and $\boldsymbol{F}''$ ($\boldsymbol{f}'$, respectively) while preserving all cyclotomic cosets. These permutations do not impact $\boldsymbol{P}^T$ ($\boldsymbol{P}$, respectively) in a full CFFT, but by permuting $\boldsymbol{F}'$ and $\boldsymbol{F}''$ ($\boldsymbol{f}'$, respectively) the removed rows (columns, respectively) in $\boldsymbol{P}^T$ ($\boldsymbol{P}$, respectively) result in more all-zero columns (rows, respectively) and thus achieve greater savings in multiplicative complexities. This technique is equivalent to the rotation of normal bases in [@Costa04].
In addition to the complexity reduction techniques discussed above, which utilizes only the properties of the DFTs, we also apply our CSE algorithm [@Chen08a] to further reduce the additive complexities of both partial and dual partial CFFTs.
Partial CFFTs and their applications in syndrome computation were considered in [@Costa04; @Fedorenko06], while dual partial CFFTs have not been considered in the literature to the best of our knowledge. In Section \[sec:synd\], we compare the complexities of syndrome computation based on a variety of approaches, including our partial CFFTs and those in [@Costa04]. We do not compare to [@Fedorenko06] because no details were provided.
Reduced-Complexity RS Decoders {#sec:rs}
==============================
Using full CFFTs [@Chen08a] as well as partial and dual partial CFFTs described above, we propose both time- and transform-domain RS decoders with reduced complexities.
Syndrome Computation {#sec:synd}
--------------------
We implement syndrome computation, which is used in both time- and transform-domain decoders, with partial CFFTs. For $(255, 223), (511, 447), (1023, 895)$ RS codes, which are selected due to their widespread applications [@Truong06a], we compare the complexities of syndrome computation based on our partial SCFFTs/ICFFTs with the complexities of syndrome computation based on partial CFFTs in [@Costa04] and other approaches such as Horner’s rule, Zakharova’s algorithm [@Zakharova92], and the prime-factor FFT [@Truong06a] in Table \[tab:pfft\]. The results for the length-255 RS code using Horner’s rule as well as the algorithms in [@Zakharova92] and [@Costa04] are obtained from [@Costa04]; The results for RS codes of lengths 511 and 1023 using Horner’s rule and the algorithm in [@Lin07] are reproduced from [@Lin07]; the numbers of multiplications and additions for the prime-factor FFT [@Truong06a] are reproduced from [@Truong06a]. In comparison to these approaches except the prime-factor FFT [@Truong06a], our partial SCFFTs/ICFFTs apparently require smaller complexities for syndrome computation. To compare to the prime-factor FFT [@Truong06a], we use the metric for the total complexities $N_{total} = (2m-1)N_{mult} + N_{add}$ as in [@Chen08a]. Syndrome computation based on our partial SCFFTs/ICFFTs requires smaller total complexities than those based on the prime-factor FFT [@Truong06a]. We provide the details of the syndrome computation for the $(255, 223)$ RS code based on our partial SCFFT in the appendix.
Chien Search and Forney’s Formula {#sec:chien}
---------------------------------
For errors-only (errors-and-erasures, respectively) decoders, the Chien search evaluates the error locator polynomial of degree at most $t$ (errata locator polynomial of degree at most $2t$, respectively) over all the elements of the underlying field; each root leads to one error (errata) location. This evaluation is essentially a DFT of a vector for which only first $t+1$ ($2t+1$, respectively) temporal components are not zeros. Note that the Chien search in errors-and-erasures decoders needs to evaluate only the error locator polynomial if it is available. For errors-only (errors-and-erasures, respectively) decoders, Forney’s formula evaluates two polynomials: one is the error (errata, respectively) evaluator polynomial $A(x)$ and the other polynomial $x\tau'(x)$ is based on the derivative of error (errata, respectively) locator polynomial $\tau(x)$. The degree of the error (errata, respectively) evaluator polynomial is less than $t$ ($2t$, respectively), while the degree of $x\tau'(x)$ is no more than $t$ ($2t$, respectively). Roughly half of the coefficients in $x\tau'(x)$ are zero. Given these information, the techniques explained above for dual partial CFFTs are again applicable. For simplicity, we assume errors-and-erasures decoders henceforth, and our results can be easily extended to errors-only decoders.
The errata locator polynomial $\tau(x)$ satisfies $\tau(x)=\hat{\tau}_e(x^2)+x\hat{\tau}_o(x^2)$, where $\hat{\tau}_e(x^2)$ and $x\hat{\tau}_o(x^2)$ consist of terms with even and odd degrees, respectively. Note that $\hat{\tau}_e(x)$ and $\hat{\tau}_o(x)$ have degrees at most $t$ and $t-1$, respectively. It is easily verified that $\hat{\tau}_o(x^2)=\tau'(x)$ for characteristic-$2$ fields.
While the Chien search evaluates $\tau(x)$ at all $n=2^m-1$ points, Forney’s formula evaluates $A(x)$ and $\tau'(x)$ at up to $2t$ errata locations. Thus, **given** the errata locations, the evaluations of $A(x)$ and $\tau'(x)$ are DFTs, in which not only part of temporal components are zeros but also only part of frequency components are needed. Thus, the complexity reduction techniques for both partial and dual partial CFFTs are applicable, and our CSE algorithm can be applied. However, since the errata locations vary, it is infeasible to minimize the computational complexities “on the fly.” Thus, we also evaluate $A(x)$ and $\tau'(x)$ over all $n=2^m-1$ points. Since $\tau'(x)$ is evaluated over all the points, its evaluation is useful for both the Chien search and Forney’s formula. Thus, the Chien search and Forney’s formula are carried out jointly.
The evaluation of the $A(x)$ is directly implemented as a dual partial CFFT. For any $\alpha \neq 0$ in GF$(2^m)$, we can either obtain $\hat{\tau}_e(x^2)|_{x=\alpha}$ by dual partial CFFTs, or first evaluate $\hat{\tau}_e(x)|_{x=\alpha}$ by dual partial CFFTs and then obtain $\hat{\tau}_e(x^2)|_{x=\alpha}$ by properly permuting the frequency components. Although $\hat{\tau}_e(x^2)$ and $\hat{\tau}_e(x)$ have the same number of non-zero terms, the non-zero terms of $\hat{\tau}_e(x)$ fall into fewer cosets than those of $\tau_o(x)$, so its evaluation based on dual partial CFFTs has smaller multiplicative and additive complexities. Similar to our approach for $\hat{\tau}_e(x^2)$, we have two options to obtain $x\hat{\tau}_o(x^2)|_{x=\alpha}$. The first option is treat $x\hat{\tau}_o(x^2)$ as a polynomial of degree at most $2t-1$ and obtain $x\hat{\tau}_o(x^2)|_{x=\alpha}$ using dual partial CFFTs. The other option is to first compute $\hat{\tau}_o(x)|_{x=\alpha}$ using dual partial CFFTs, then obtain $\hat{\tau}_o(x^2)|_{x=\alpha}$ by permutation, and finally compute $x \hat{\tau}_o(x^2)|_{x=\alpha}$. For the latter option, similar to the reason given above, the evaluation of $\hat{\tau}_o(x)$ based on dual partial CFFTs requires fewer multiplications and additions than that of $\hat{\tau}_o(x^2)$, although they have the same number of non-zero terms. However, the latter option also requires $n$ extra multiplications. Thus, the latter option has higher multiplicative complexities but lower additive complexities as opposed to the former option.
We present the computational complexities of combined Chien search and Forney’s formula for errors-and-erasures decoders based on our dual partial CFFTs in Table \[tab:forney\]. Note that to evaluate $\tau(x)$ at all $2t$ errata locations, $2t$ additions are needed; Also, $2t$ divisions are needed to compute the errata values in Forney’s formula; Both are accounted for in the rows marked by “Misc.” The rows marked by “Sum” sum up the numbers of field operations required to evaluate $A(x)$, $\hat{\tau}_e(x^2)$, and $x\hat{\tau}_o(x^2)$, as well as $2t$ additions and $2t$ divisions mentioned above. As in Section \[sec:synd\], the total complexities of each individual step and the sum are measured by the metric in [@Chen08a], and we assume division has the same complexity as multiplication. They are presented in the columns marked by “Total.” The complexities of the two options for evaluating $x\hat{\tau}_o(x^2)$ are both given; the $n$ extra multiplications in the second option are shown in the second terms of the summations. Due to the $n$ extra multiplications, for lengths 255 and 511 the first option has a smaller total complexity; for length 1023, the second option has a smaller total complexity. For evaluating $x\hat{\tau}_o(x^2)$, the option with smaller **total** complexity is used.
The computational complexities based on our dual partial DCFFTs are compared to the complexities based on Horner’s rule in Table \[tab:forney\]. We also reproduce the complexities of the Chien search and Forney’s formula in [@Jeng99] from [@Truong06a]. The combined Chien search and Forney’s formula based on our partial dual CFFTs achieves significantly smaller computational complexities than those based on Horner’s rule and in [@Jeng99]. We do not compare with the approaches in [@Fedorenko02; @Lin07; @Truong01] because their computational complexities are not available.
Example
-------
We provide a simple example to illustrate syndrome computation and Chien search based on our CFFTs. For simplicity, let us consider errors-and-erasures decoding of a $(31, 25)$ cyclic RS code over ${\mathrm{GF}}(2^5)$ defined by the primitive polynomial $p(x) = x^5 + x^2 + 1$. The generator polynomial for the RS code is given by $g(x) = (x - 1) (x - \alpha) \cdots (x - \alpha^{5})$, where $\alpha$ is a root of $p(x)$. There are seven cyclotomic cosets over this field.
Suppose the received vector is $\boldsymbol{r} = (r_0, r_1, \dots,
r_{n - 1})$. To compute the syndromes $S_i = \sum_{j = 0}^{n - 1}
r_j \alpha^{ij}$ for $0 \le i \le 5$, it involves only four cyclotomic cosets: $\{0\}, \{2, 4, 8, 16, 1\}$, $\{6, 12, 24, 17,
3\}, \{10, 20, 9, 18, 5\}$. As explained in Section \[sec:cfft\], we have rotated the cosets in this order to reduce multiplicative complexity. We do not specify the other cosets since they are irrelevant to our purpose. Using the normal basis $(\alpha^3, \alpha^6, \alpha^{12}, \alpha^{24}, \alpha^{17})$ and the length-5 convolution algorithm in [@Blahut83], we first construct a full SCFFT $\boldsymbol{S}' =
\boldsymbol{P}^T (\boldsymbol{c} \cdot
(\boldsymbol{A}'\boldsymbol{Q})^T\boldsymbol{r}')$, in which $\boldsymbol{S}'$ and $\boldsymbol{r}'$ are permuted versions of $\boldsymbol{S} = (S_0, S_1, \dotsc, S_{30})^T$ and $\boldsymbol{r} = (r_0, r_1, \dotsc, r_{30})^T$, both ordered in the chosen cosets. For these four cosets, their $\boldsymbol{P}_i^T$ and $\boldsymbol{c}_i$ are given by $\boldsymbol{P}_0^T = [1]$, $\boldsymbol{c}_0 = (1)$, $\boldsymbol{c}_1 = \boldsymbol{c}_2 = \boldsymbol{c}_3=
(1, \alpha, \alpha^{25}, \alpha^7, \alpha^2, \alpha^{16},
\alpha^4, \alpha^{28}, \alpha^{14}, \alpha^{27})^T$, and $$\boldsymbol{P}_1^T = \boldsymbol{P}_2^T = \boldsymbol{P}_3^T =
\begin{bmatrix}
1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1\\
1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1\\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0\\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 1\\
1 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1
\end{bmatrix}.$$ Let $\boldsymbol{A}' = [\boldsymbol{A}'_0 \mid \boldsymbol{A}'_1
\mid \dots \mid \boldsymbol{A}'_6]$, $\boldsymbol{A}'\boldsymbol{Q} = [\boldsymbol{A}'_0\boldsymbol{Q}_0
\mid \boldsymbol{A}'_1\boldsymbol{Q}_1 \mid \dots \mid
\boldsymbol{A}'_6 \boldsymbol{Q}_6]$.
In the coset $\{2, 4, 8, 16, 1\}$, we need only $\{2, 4, 1\}$, and thus we remove the third and fourth rows in $\boldsymbol{P}_1^T$, resulting in the eighth column being all-zero. So we strike out the column and save one more multiplication. Let $\boldsymbol{P}'^T_1$ denote the reduced matrix. Note that other orders of the coset cannot produce all-zero columns. This can also be achieved by rotating the normal basis. Since we only need $\{3\}$ and $\{5\}$ in the cosets $\{6,12,24,17,3\}$ and $\{10,20,9,18,5\}$, respectively, we obtain $\boldsymbol{P}'^T_2 = \boldsymbol{P}'^T_3$ by keeping only the last row in $\boldsymbol{P}_2^T$ and striking out the all-zero fourth, seventh, eighth and ninth columns. Correspondingly, we remove the eighth row from $\boldsymbol{Q}_1^T$ and $\boldsymbol{c}_1$. $\boldsymbol{Q}'^T_2 = \boldsymbol{Q}'^T_3$ and $\boldsymbol{c}_2' = \boldsymbol{c}_3'$ are given by removing the fourth, seventh, eighth, and ninth rows from $\boldsymbol{Q}^T_2$ and $\boldsymbol{c}_2$, respectively. Hence the syndromes can be computed by a partial SCFFT as $$\begin{bmatrix} S_0\\ S_2\\ S_4\\ S_1\\ S_3\\ S_5 \end{bmatrix} =
\begin{bmatrix}
\boldsymbol{P}_0^T & & &\\
& \boldsymbol{P}'^T_1 & &\\
& & \boldsymbol{P}'^T_2 &\\
& & & \boldsymbol{P}'^T_3&
\end{bmatrix}
\left(
\begin{bmatrix} \boldsymbol{c}_0\\ \boldsymbol{c}_1'\\ \boldsymbol{c}_2'\\ \boldsymbol{c}_3'\end{bmatrix} \boldsymbol{\cdot}
\begin{bmatrix}
(\boldsymbol{A}'_0\boldsymbol{Q}_0)^T\\
(\boldsymbol{A}'_1\boldsymbol{Q}'_1)^T\\
(\boldsymbol{A}'_2\boldsymbol{Q}'_2)^T\\
(\boldsymbol{A}'_3\boldsymbol{Q}'_3)^T\\
\end{bmatrix}
\boldsymbol{r}'\right).$$ If there are all-zero columns in $\boldsymbol{Q}_1'^T, \boldsymbol{Q}'^T_2, \boldsymbol{Q}'^T_3$, we can strike out those columns and further remove corresponding rows from $\boldsymbol{A}'^T_i$’s.
In the Chien search, the errata locator polynomial $\tau(x) = \sum_{i=0}^6\tau_i x^i$ has degree up to six. So we need to use $\{6, 3\}$ for the third coset. Thus the Chien search can by done by a dual partial DCFFT $$\begin{bmatrix}
\boldsymbol{A}'_0\boldsymbol{Q}_0 \mid \boldsymbol{A}'_1\boldsymbol{Q}'_1 \mid
\boldsymbol{A}'_2\boldsymbol{Q}''_2 \mid \boldsymbol{A}'_3\boldsymbol{Q}'_3
\boldsymbol{P}_0 & & &\\
& \boldsymbol{P}'_1 & &\\
& & \boldsymbol{P}''_2 &\\
& & & \boldsymbol{P}'_3\\
\end{bmatrix}
\begin{bmatrix} \tau_0\\ \tau_2\\ \tau_4\\ \tau_1\\ \tau_6\\ \tau_3\\ \tau_5\end{bmatrix}\right),$$ where $\boldsymbol{P}''_2$ is obtained by keeping only the first and last columns of $\boldsymbol{P}_2$, $\boldsymbol{c}''_2$ and $\boldsymbol{Q}''_2$ are obtained by removing the eighth and ninth rows from $\boldsymbol{c}_2$ and the corresponding columns from $\boldsymbol{Q}_2$.
The Chien search can be split into evaluating $\tau_e(x)=\hat\tau_e(x^2)$ and $\tau_o(x)=x\hat\tau_o(x^2)$ to accommodate Forney’s formula. The direct evaluation of $\tau_o(x)$ can be carried out by $$\begin{bmatrix}
\boldsymbol{A}'_1\boldsymbol{Q}''_1 \mid \boldsymbol{A}'_2\boldsymbol{Q}'_2 \mid
\boldsymbol{A}'_3\boldsymbol{Q}'_3 \end{bmatrix}
\left(
\begin{bmatrix} \boldsymbol{c}_1''\\ \boldsymbol{c}_2'\\ \boldsymbol{c}_3'\end{bmatrix} \boldsymbol{\cdot}
\begin{bmatrix}
\begin{bmatrix} \tau_1\\ \tau_3\\ \tau_5 \end{bmatrix}\right),$$ where $\boldsymbol{Q}''_1 = \boldsymbol{Q}'_2, \boldsymbol{c}''_1 = \boldsymbol{c}'_2$, and $\boldsymbol{P}''_1 = \boldsymbol{P}'_2$. Alternatively, the evaluation of $\hat{\tau}_o(x^2)$ can be carried out by $$\begin{bmatrix}
\boldsymbol{A}'_0\boldsymbol{Q}_0 \mid \boldsymbol{A}'_1\boldsymbol{Q}'''_1
\end{bmatrix}
\left(
\begin{bmatrix} \boldsymbol{c}_0\\ \boldsymbol{c}_1'''\end{bmatrix} \boldsymbol{\cdot}
\begin{bmatrix}
\boldsymbol{P}_0 &\\
& \boldsymbol{P}'''_1
\end{bmatrix}
\begin{bmatrix} \hat{\tau}_{o,0}\\ \hat{\tau}_{o,2}\\ \hat{\tau}_{o,1}\end{bmatrix}\right),$$ where $\boldsymbol{Q}'''_1 = \boldsymbol{Q}''_2, \boldsymbol{c}'''_1 = \boldsymbol{c}''_2 $, and $\boldsymbol{P}'''_1 = \boldsymbol{P}''_2$. Similarly, the evaluation of $\hat{\tau}_e(x^2)$ can be carried out by $$\begin{bmatrix}
\boldsymbol{A}'_0\boldsymbol{Q}_0 \mid \boldsymbol{A}'_1\boldsymbol{Q}'''_1 \mid \boldsymbol{A}'_2\boldsymbol{Q}'_2
\end{bmatrix}
\left(
\begin{bmatrix} \boldsymbol{c}_0\\ \boldsymbol{c}_1'''\\ \boldsymbol{c}_2'\end{bmatrix} \boldsymbol{\cdot}
\end{bmatrix}
\begin{bmatrix} \hat{\tau}_{e,0}\\ \hat{\tau}_{e,2}\\ \hat{\tau}_{e,1}\\ \hat{\tau}_{e,3} \end{bmatrix}\right).$$
Then we can apply our CSE algorithm to these reduced pre- and post-addition matrices and furthur reduce the numbers of additions, but such details are omitted. It is easy to see that the Chien search based on our partial dual CFFTs achieves significantly smaller computational complexities.
Transform-Domain and Time-Domain RS Decoders {#sec:rrs}
--------------------------------------------
Replacing the prime-factor FFT [@Truong06a] by our CFFTs proposed above, we propose a transform-domain RS decoder with the following steps: (T.1) Compute the syndromes by our partial SCFFT; (T.2) Use the inverse-free BMA [@Jeng99] to obtain the errata locator polynomial $\tau(x)$; (T.3) Compute the remaining syndromes by recursive extension using $\tau(x)$; (T.4) Compute the error vector by full CFFT of the syndrome vector. Finally, the corrected codeword is obtained by adding the received vector and the error vector. Similarly, we propose a time-domain RS decoder with the following steps: t.1 and t.2 are the same as T.1 and T.2; (t.3) Compute the errata evaluator polynomial $A(x)$; (t.4) Find the error locations and error values by applying our combined Chien search and Forney’s formula based on dual partial DCFFTs to $\tau(x)$ and $A(x)$.
We compare the complexities of our time- and transform-domain RS decoders for $(255, 223)$, $(511, 447)$, and $(1023, 895)$ RS codes with those in [@Jeng99] and [@Truong06a] respectively in Table \[tab:ttrs\]. We are aware of the vast literature on RS decoding, and [@Jeng99] and [@Truong06a] are compared here since their data are directly comparable. The computational complexities of the time-domain decoder in [@Jeng99] and the transform-domain decoder in [@Truong06a] are all reproduced from [@Truong06a]. The complexities of T.4 are reproduced from [@Chen08a Table I]. Note that all the computational complexities are for errors-and-erasures decoders. The complexities of T.1/t.1 and t.4 are already presented in Tables \[tab:pfft\] and \[tab:forney\].
We first compare the overall complexities of our transform-domain RS decoder with those in [@Truong06a], presented in the row marked by “T.1+T.2+T.3+T.4.” Clearly our transform-domain decoder achieves smaller complexities. However, this comparison is somewhat misleading since our decoder differs from that in [@Truong06a] only in T.1 and T.4. We further compare the combined complexities of T.1 and T.4 of our transform-domain decoder and that in [@Truong06a], presented in the row marked by “T.1+T.4.” Here, the transform portion of our decoder achieves complexity reductions of 39%, 72%, and 29%.
For time-domain RS decoders, in comparison to the decoder considered in [@Jeng99], the overall complexities of our RS decoder are 90%, 91%, and 92% smaller. Again, since the focus of this paper is on transformation, it is more meaningful to compare only the steps using DFTs: t.1 and t.4. The sums of the total complexities of t.1 and t.4 are presented separately in the row marked by “t.1+t.4.” It can be seen that the transformation portion of our decoder achieves 95%, 96%, and 97% complexity savings over that in [@Jeng99] for the three RS codes, respectively.
Finally, based on our results, time-domain decoders have smaller complexities than transform-domain decoders. This conclusion is different from that in [@Truong06a]. However, the conclusion in [@Truong06a] is based on the comparison of transform-domain decoder using FFT and time-domain decoder without FFT. In our comparison, both decoders use CFFTs.
In this paper, we assume that RS decoders are implemented by integrated circuits, and each CFFT consists of combinational logic and requires no memory. Hence, we consider only the total complexity due to finite field operations above since they directly correspond to combinational logic. The total complexities in Tables \[tab:pfft\], \[tab:forney\], and \[tab:ttrs\] also assume that the maximum of received symbols are processed concurrently so as to increase throughput. Thus, reduced total complexities by CFFTs translate into smaller areas. However, decoders based on CFFTs have fixed and irregular adder trees for pre- and post-additions, and thus are less conducive to transformations that trade time (throughput) for area than decoders based on other approaches (for example, Horner’s rule).
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors are grateful to Prof. P. Trifonov for providing details of CFFTs and Prof. P. D. Chen for valuable discussions.
We provide the details of the syndrome computation for the (255, 223) RS code based on our partial SCFFT.
First, we reorder the received vector $\boldsymbol{r}$ based on cosets:
$\boldsymbol{r}' = (r_{0}$, $r_{1}$, $r_{2}$, $r_{4}$, $r_{8}$, $r_{16}$, $r_{32}$, $r_{64}$, $r_{128}$, $r_{3}$, $r_{6}$, $r_{12}$, $r_{24}$, $r_{48}$, $r_{96}$, $r_{192}$, $r_{129}$, $r_{5}$, $r_{10}$, $r_{20}$, $r_{40}$, $r_{80}$, $r_{160}$, $r_{65}$, $r_{130}$, $r_{131}$, $r_{7}$, $r_{14}$, $r_{28}$, $r_{56}$, $r_{112}$, $r_{224}$, $r_{193}$, $r_{66}$, $r_{132}$, $r_{9}$, $r_{18}$, $r_{36}$, $r_{72}$, $r_{144}$, $r_{33}$, $r_{11}$, $r_{22}$, $r_{44}$, $r_{88}$, $r_{176}$, $r_{97}$, $r_{194}$, $r_{133}$, $r_{67}$, $r_{134}$, $r_{13}$, $r_{26}$, $r_{52}$, $r_{104}$, $r_{208}$, $r_{161}$, $r_{195}$, $r_{135}$, $r_{15}$, $r_{30}$, $r_{60}$, $r_{120}$, $r_{240}$, $r_{225}$, $r_{34}$, $r_{68}$, $r_{136}$, $r_{17}$, $r_{98}$, $r_{196}$, $r_{137}$, $r_{19}$, $r_{38}$, $r_{76}$, $r_{152}$, $r_{49}$, $r_{138}$, $r_{21}$, $r_{42}$, $r_{84}$, $r_{168}$, $r_{81}$, $r_{162}$, $r_{69}$, $r_{226}$, $r_{197}$, $r_{139}$, $r_{23}$, $r_{46}$, $r_{92}$, $r_{184}$, $r_{113}$, $r_{70}$, $r_{140}$, $r_{25}$, $r_{50}$, $r_{100}$, $r_{200}$, $r_{145}$, $r_{35}$, $r_{141}$, $r_{27}$, $r_{54}$, $r_{108}$, $r_{216}$, $r_{177}$, $r_{99}$, $r_{198}$, $r_{71}$, $r_{142}$, $r_{29}$, $r_{58}$, $r_{116}$, $r_{232}$, $r_{209}$, $r_{163}$, $r_{31}$, $r_{62}$, $r_{124}$, $r_{248}$, $r_{241}$, $r_{227}$, $r_{199}$, $r_{143}$, $r_{37}$, $r_{74}$, $r_{148}$, $r_{41}$, $r_{82}$, $r_{164}$, $r_{73}$, $r_{146}$, $r_{39}$, $r_{78}$, $r_{156}$, $r_{57}$, $r_{114}$, $r_{228}$, $r_{201}$, $r_{147}$, $r_{43}$, $r_{86}$, $r_{172}$, $r_{89}$, $r_{178}$, $r_{101}$, $r_{202}$, $r_{149}$, $r_{45}$, $r_{90}$, $r_{180}$, $r_{105}$, $r_{210}$, $r_{165}$, $r_{75}$, $r_{150}$, $r_{47}$, $r_{94}$, $r_{188}$, $r_{121}$, $r_{242}$, $r_{229}$, $r_{203}$, $r_{151}$, $r_{51}$, $r_{102}$, $r_{204}$, $r_{153}$, $r_{53}$, $r_{106}$, $r_{212}$, $r_{169}$, $r_{83}$, $r_{166}$, $r_{77}$, $r_{154}$, $r_{55}$, $r_{110}$, $r_{220}$, $r_{185}$, $r_{115}$, $r_{230}$, $r_{205}$, $r_{155}$, $r_{59}$, $r_{118}$, $r_{236}$, $r_{217}$, $r_{179}$, $r_{103}$, $r_{206}$, $r_{157}$, $r_{61}$, $r_{122}$, $r_{244}$, $r_{233}$, $r_{211}$, $r_{167}$, $r_{79}$, $r_{158}$, $r_{63}$, $r_{126}$, $r_{252}$, $r_{249}$, $r_{243}$, $r_{231}$, $r_{207}$, $r_{159}$, $r_{85}$, $r_{170}$, $r_{87}$, $r_{174}$, $r_{93}$, $r_{186}$, $r_{117}$, $r_{234}$, $r_{213}$, $r_{171}$, $r_{91}$, $r_{182}$, $r_{109}$, $r_{218}$, $r_{181}$, $r_{107}$, $r_{214}$, $r_{173}$, $r_{95}$, $r_{190}$, $r_{125}$, $r_{250}$, $r_{245}$, $r_{235}$, $r_{215}$, $r_{175}$, $r_{111}$, $r_{222}$, $r_{189}$, $r_{123}$, $r_{246}$, $r_{237}$, $r_{219}$, $r_{183}$, $r_{119}$, $r_{238}$, $r_{221}$, $r_{187}$, $r_{127}$, $r_{254}$, $r_{253}$, $r_{251}$, $r_{247}$, $r_{239}$, $r_{223}$, $r_{191}).$
Pre-additions (3793 additions): $\boldsymbol{p} = (\boldsymbol{A}\boldsymbol{Q})^T \boldsymbol{r'}.$
$ t_{2399} = r'_{150} + r'_{197},
t_{2404} = r'_{228} + t_{2399},
t_{2247} = r'_{189} + r'_{190},
t_{2085} = r'_{92} + r'_{234},
t_{2056} = r'_{78} + r'_{153},
t_{1917} = r'_{33} + r'_{122},
t_{1853} = r'_{20} + r'_{219},
t_{1788} = r'_{34} + r'_{70},
t_{1707} = r'_{83} + r'_{203},
t_{2397} = r'_{152} + t_{1707},
t_{1662} = r'_{35} + r'_{92},
t_{1572} = r'_{8} + r'_{233},
t_{1571} = r'_{1} + r'_{193},
t_{1561} = r'_{19} + r'_{41},
t_{1553} = r'_{227} + r'_{245},
t_{1516} = r'_{82} + r'_{113},
t_{1455} = r'_{54} + r'_{121},
t_{1617} = r'_{153} + t_{1455},
t_{1425} = r'_{50} + r'_{223},
t_{1371} = r'_{222} + r'_{242},
t_{1326} = r'_{29} + r'_{114},
t_{1313} = r'_{152} + r'_{199},
t_{1308} = r'_{122} + r'_{220},
t_{1277} = r'_{25} + r'_{35},
t_{1244} = r'_{232} + r'_{246},
t_{1240} = r'_{168} + r'_{191},
t_{1221} = r'_{18} + r'_{166},
t_{1218} = r'_{105} + r'_{133},
t_{1177} = r'_{21} + r'_{49},
t_{1150} = r'_{7} + r'_{176},
t_{1368} = r'_{151} + t_{1150},
t_{1134} = r'_{9} + r'_{179},
t_{1127} = r'_{23} + r'_{219},
t_{1118} = r'_{3} + r'_{85},
t_{1101} = r'_{60} + r'_{63},
t_{1092} = r'_{172} + r'_{239},
t_{1085} = r'_{183} + r'_{214},
t_{1082} = r'_{37} + r'_{89},
t_{1066} = r'_{2} + r'_{184},
t_{1060} = r'_{83} + r'_{119},
t_{1059} = r'_{210} + r'_{226},
t_{1054} = r'_{169} + r'_{182},
t_{1420} = r'_{122} + t_{1054},
t_{1043} = r'_{141} + r'_{195},
t_{1024} = r'_{97} + r'_{178},
t_{1022} = r'_{22} + r'_{194},
t_{1676} = r'_{249} + t_{1022},
t_{1008} = r'_{17} + r'_{162},
t_{1532} = r'_{145} + t_{1008},
t_{1365} = r'_{160} + t_{1008},
t_{1547} = r'_{243} + t_{1365},
t_{999} = r'_{177} + r'_{181},
t_{993} = r'_{188} + r'_{236},
t_{983} = r'_{125} + r'_{131},
t_{970} = r'_{47} + r'_{139},
t_{957} = r'_{33} + r'_{34},
t_{1310} = r'_{205} + t_{957},
t_{1438} = r'_{129} + t_{1310},
t_{950} = r'_{158} + r'_{217},
t_{1041} = r'_{241} + t_{950},
t_{948} = r'_{81} + r'_{248},
t_{946} = r'_{48} + r'_{160},
t_{940} = r'_{199} + r'_{254},
t_{931} = r'_{197} + r'_{221},
t_{1457} = r'_{178} + t_{931},
t_{913} = r'_{39} + r'_{216},
t_{2122} = r'_{199} + t_{913},
t_{1263} = r'_{112} + t_{913},
t_{908} = r'_{127} + r'_{149},
t_{1020} = r'_{59} + t_{908},
t_{907} = r'_{4} + r'_{171},
t_{905} = r'_{55} + r'_{56},
t_{1052} = r'_{118} + t_{905},
t_{903} = r'_{150} + r'_{252},
t_{987} = r'_{140} + t_{903},
t_{901} = r'_{109} + r'_{121},
t_{1179} = r'_{76} + t_{901},
t_{892} = r'_{62} + r'_{84},
t_{864} = r'_{32} + r'_{123},
t_{862} = r'_{51} + r'_{180},
t_{2150} = r'_{83} + t_{862},
t_{1343} = t_{862} + t_{1024},
t_{1039} = r'_{87} + t_{862},
t_{858} = r'_{25} + r'_{89},
t_{817} = r'_{157} + r'_{208},
t_{1387} = r'_{14} + t_{817},
t_{812} = r'_{129} + r'_{152},
t_{811} = r'_{13} + r'_{222},
t_{795} = r'_{91} + r'_{192},
t_{1446} = r'_{224} + t_{795},
t_{1276} = r'_{7} + t_{795},
t_{934} = r'_{66} + t_{795},
t_{793} = r'_{0} + r'_{155},
t_{1193} = r'_{99} + t_{793},
t_{788} = r'_{16} + r'_{101},
t_{1230} = r'_{77} + t_{788},
t_{785} = r'_{15} + r'_{217},
t_{874} = r'_{77} + t_{785},
t_{1189} = r'_{21} + t_{874},
t_{783} = r'_{74} + r'_{115},
t_{1204} = r'_{234} + t_{783},
t_{779} = r'_{168} + r'_{210},
t_{1056} = r'_{165} + t_{779},
t_{774} = r'_{161} + r'_{196},
t_{754} = r'_{46} + r'_{220},
t_{1622} = r'_{164} + t_{754},
t_{936} = r'_{162} + t_{754},
t_{2082} = r'_{178} + t_{936},
t_{748} = r'_{52} + r'_{141},
t_{1564} = r'_{66} + t_{748},
t_{977} = r'_{128} + t_{748},
t_{742} = r'_{86} + r'_{178},
t_{1133} = r'_{43} + t_{742},
t_{1470} = r'_{174} + t_{1133},
t_{737} = r'_{115} + r'_{119},
t_{1996} = r'_{32} + t_{737},
t_{1376} = r'_{135} + t_{737},
t_{1149} = r'_{161} + t_{737},
t_{735} = r'_{65} + r'_{243},
t_{1488} = r'_{88} + t_{735},
t_{846} = r'_{246} + t_{735},
t_{732} = r'_{215} + r'_{223},
t_{731} = r'_{49} + r'_{240},
t_{965} = r'_{15} + t_{731},
t_{1158} = r'_{198} + t_{965},
t_{1669} = r'_{99} + t_{1158},
t_{728} = r'_{185} + r'_{189},
t_{720} = r'_{14} + r'_{108},
t_{973} = r'_{204} + t_{720},
t_{718} = r'_{92} + r'_{118},
t_{1131} = r'_{145} + t_{718},
t_{1690} = t_{1131} + t_{1572},
t_{769} = r'_{130} + t_{718},
t_{1620} = t_{769} + t_{858},
t_{716} = r'_{146} + r'_{236},
t_{2370} = r'_{160} + t_{716},
t_{1050} = r'_{96} + t_{716},
t_{2406} = r'_{55} + t_{1050},
t_{713} = r'_{101} + r'_{212},
t_{797} = r'_{150} + t_{713},
t_{709} = r'_{3} + r'_{144},
t_{841} = t_{709} + t_{774},
t_{1272} = r'_{169} + t_{841},
t_{707} = r'_{28} + r'_{114},
t_{1412} = r'_{215} + t_{707},
t_{1110} = r'_{225} + t_{707},
t_{706} = r'_{165} + r'_{245},
t_{1069} = t_{706} + t_{735},
t_{2149} = r'_{61} + t_{1069},
t_{705} = r'_{17} + r'_{234},
t_{704} = r'_{25} + r'_{201},
t_{825} = r'_{175} + t_{704},
t_{1028} = t_{825} + t_{864},
t_{1145} = t_{1028} + t_{1092},
t_{701} = r'_{142} + r'_{198},
t_{954} = r'_{44} + t_{701},
t_{976} = r'_{172} + t_{954},
t_{696} = r'_{203} + r'_{207},
t_{1288} = r'_{251} + t_{696},
t_{695} = r'_{24} + r'_{233},
t_{1459} = r'_{252} + t_{695},
t_{693} = r'_{69} + r'_{169},
t_{889} = r'_{253} + t_{693},
t_{688} = r'_{29} + r'_{85},
t_{1036} = t_{688} + t_{705},
t_{1114} = t_{709} + t_{1036},
t_{2170} = r'_{48} + t_{1114},
t_{1226} = t_{797} + t_{1114},
t_{679} = r'_{120} + r'_{134},
t_{678} = r'_{19} + r'_{81},
t_{1713} = r'_{33} + t_{678},
t_{2398} = t_{1713} + t_{2397},
t_{1383} = t_{678} + t_{1376},
t_{1528} = r'_{214} + t_{1383},
t_{677} = r'_{228} + r'_{237},
t_{1520} = r'_{213} + t_{677},
t_{861} = r'_{100} + t_{677},
t_{674} = r'_{90} + r'_{124},
t_{842} = r'_{26} + t_{674},
t_{664} = r'_{80} + r'_{172},
t_{663} = r'_{157} + r'_{160},
t_{1078} = r'_{144} + t_{663},
t_{1672} = r'_{18} + t_{1078},
t_{1299} = r'_{93} + t_{1078},
t_{1402} = t_{1299} + t_{1326},
t_{660} = r'_{10} + r'_{104},
t_{1574} = t_{660} + t_{905},
t_{1038} = t_{660} + t_{993},
t_{1403} = r'_{12} + t_{1038},
t_{759} = r'_{221} + t_{660},
t_{659} = r'_{74} + r'_{239},
t_{1344} = r'_{4} + t_{659},
t_{1526} = r'_{236} + t_{1344},
t_{658} = r'_{8} + r'_{175},
t_{1021} = t_{658} + t_{679},
t_{651} = r'_{84} + r'_{176},
t_{647} = r'_{5} + r'_{214},
t_{865} = t_{647} + t_{679},
t_{1422} = r'_{146} + t_{865},
t_{646} = r'_{9} + r'_{44},
t_{1874} = t_{646} + t_{1272},
t_{656} = r'_{159} + t_{646},
t_{1530} = t_{656} + t_{1221},
t_{644} = r'_{91} + r'_{117},
t_{2363} = t_{644} + t_{1343},
t_{1130} = r'_{107} + t_{644},
t_{1926} = r'_{168} + t_{1130},
t_{755} = r'_{102} + t_{644},
t_{643} = r'_{38} + r'_{238},
t_{828} = r'_{133} + t_{643},
t_{1197} = r'_{14} + t_{828},
t_{638} = r'_{11} + r'_{211},
t_{2280} = t_{638} + t_{1056},
t_{634} = r'_{158} + r'_{162},
t_{632} = r'_{113} + r'_{235},
t_{981} = r'_{251} + t_{632},
t_{866} = r'_{68} + t_{632},
t_{1664} = t_{866} + t_{1060},
t_{627} = r'_{173} + r'_{249},
t_{623} = r'_{30} + r'_{193},
t_{804} = t_{623} + t_{674},
t_{1517} = t_{804} + t_{1101},
t_{2019} = t_{1517} + t_{1564},
t_{1492} = r'_{147} + t_{804},
t_{1503} = t_{977} + t_{1492},
t_{1860} = r'_{11} + t_{1503},
t_{622} = r'_{56} + r'_{73},
t_{909} = r'_{170} + t_{622},
t_{1760} = t_{731} + t_{909},
t_{1049} = r'_{57} + t_{909},
t_{1784} = r'_{6} + t_{1049},
t_{770} = t_{622} + t_{705},
t_{1037} = r'_{83} + t_{770},
t_{1102} = r'_{141} + t_{1037},
t_{1300} = t_{647} + t_{1102},
t_{621} = r'_{40} + r'_{186},
t_{1167} = t_{621} + t_{861},
t_{914} = t_{621} + t_{664},
t_{1529} = t_{647} + t_{914},
t_{1769} = t_{1529} + t_{1574},
t_{1227} = r'_{56} + t_{914},
t_{619} = r'_{51} + r'_{148},
t_{1400} = t_{619} + t_{720},
t_{1007} = r'_{142} + t_{619},
t_{618} = r'_{174} + r'_{250},
t_{751} = r'_{137} + t_{618},
t_{1033} = r'_{231} + t_{751},
t_{615} = r'_{67} + r'_{167},
t_{1274} = r'_{28} + t_{615},
t_{614} = r'_{88} + r'_{126},
t_{953} = r'_{31} + t_{614},
t_{1494} = r'_{177} + t_{953},
t_{745} = r'_{242} + t_{614},
t_{1095} = r'_{77} + t_{745},
t_{613} = r'_{54} + r'_{191},
t_{951} = t_{613} + t_{658},
t_{1443} = r'_{75} + t_{951},
t_{768} = r'_{72} + t_{613},
t_{1120} = t_{768} + t_{828},
t_{607} = r'_{151} + r'_{155},
t_{604} = r'_{39} + r'_{57},
t_{2361} = r'_{242} + t_{604},
t_{1144} = r'_{72} + t_{604},
t_{1534} = t_{1144} + t_{1455},
t_{2003} = t_{618} + t_{1534},
t_{2010} = t_{1024} + t_{2003},
t_{727} = r'_{79} + t_{604},
t_{986} = t_{659} + t_{727},
t_{603} = r'_{145} + r'_{213},
t_{2083} = r'_{156} + t_{603},
t_{654} = t_{603} + t_{627},
t_{1139} = t_{654} + t_{948},
t_{602} = r'_{105} + r'_{154},
t_{2291} = t_{602} + t_{643},
t_{1014} = r'_{208} + t_{602},
t_{781} = r'_{209} + t_{602},
t_{1486} = t_{732} + t_{781},
t_{1569} = r'_{38} + t_{1486},
t_{1654} = r'_{100} + t_{1569},
t_{601} = r'_{70} + r'_{248},
t_{897} = r'_{40} + t_{601},
t_{1576} = r'_{55} + t_{897},
t_{683} = r'_{96} + t_{601},
t_{1186} = r'_{48} + t_{683},
t_{600} = r'_{103} + r'_{107},
t_{598} = r'_{170} + r'_{189},
t_{853} = r'_{235} + t_{598},
t_{1053} = r'_{42} + t_{853},
t_{595} = r'_{122} + r'_{136},
t_{1548} = t_{595} + t_{812},
t_{762} = r'_{200} + t_{595},
t_{1235} = r'_{73} + t_{762},
t_{1137} = r'_{35} + t_{762},
t_{1626} = t_{660} + t_{1137},
t_{933} = r'_{137} + t_{762},
t_{594} = r'_{132} + r'_{206},
t_{1169} = r'_{244} + t_{594},
t_{593} = r'_{109} + r'_{226},
t_{1661} = t_{593} + t_{601},
t_{1718} = r'_{247} + t_{1661},
t_{612} = r'_{19} + t_{593},
t_{1416} = t_{612} + t_{1134},
t_{1790} = r'_{27} + t_{1416},
t_{721} = r'_{171} + t_{612},
t_{927} = r'_{143} + t_{721},
t_{1575} = t_{927} + t_{1470},
t_{1382} = r'_{107} + t_{927},
t_{592} = r'_{22} + r'_{231},
t_{670} = r'_{47} + t_{592},
t_{1076} = t_{670} + t_{769},
t_{1605} = t_{934} + t_{1076},
t_{836} = t_{595} + t_{670},
t_{1621} = r'_{198} + t_{836},
t_{949} = r'_{237} + t_{836},
t_{2052} = t_{949} + t_{1457},
t_{591} = r'_{45} + r'_{219},
t_{1969} = t_{591} + t_{1095},
t_{815} = r'_{10} + t_{591},
t_{1081} = r'_{153} + t_{815},
t_{1493} = r'_{228} + t_{1081},
t_{2027} = r'_{217} + t_{1493},
t_{590} = r'_{49} + r'_{53},
t_{588} = r'_{149} + r'_{195},
t_{1121} = t_{588} + t_{817},
t_{1225} = t_{618} + t_{1121},
t_{1974} = t_{1225} + t_{1969},
t_{673} = r'_{140} + t_{588},
t_{587} = r'_{43} + r'_{225},
t_{1862} = t_{587} + t_{1860},
t_{1863} = t_{1066} + t_{1862},
t_{1100} = r'_{78} + t_{587},
t_{616} = r'_{59} + t_{587},
t_{1338} = r'_{226} + t_{616},
t_{810} = t_{616} + t_{688},
t_{1433} = r'_{28} + t_{810},
t_{583} = r'_{135} + r'_{139},
t_{1129} = r'_{64} + t_{583},
t_{582} = r'_{111} + r'_{123},
t_{2156} = t_{582} + t_{751},
t_{2158} = r'_{15} + t_{2156},
t_{2161} = r'_{175} + t_{2158},
t_{1108} = r'_{95} + t_{582},
t_{1454} = t_{1014} + t_{1108},
t_{712} = r'_{50} + t_{582},
t_{2344} = r'_{34} + t_{712},
t_{764} = r'_{76} + t_{712},
t_{920} = r'_{6} + t_{764},
t_{1533} = r'_{47} + t_{920},
t_{581} = r'_{62} + r'_{183},
t_{2276} = t_{581} + t_{1707},
t_{682} = r'_{100} + t_{581},
t_{888} = t_{656} + t_{682},
t_{1501} = t_{788} + t_{888},
t_{580} = r'_{94} + r'_{98},
t_{1906} = t_{580} + t_{897},
t_{576} = r'_{20} + r'_{229},
t_{1685} = r'_{106} + t_{576},
t_{633} = r'_{8} + t_{576},
t_{1212} = t_{633} + t_{1053},
t_{574} = r'_{1} + r'_{218},
t_{1340} = t_{574} + t_{701},
t_{1414} = t_{682} + t_{1340},
t_{1198} = t_{574} + t_{1007},
t_{1307} = t_{1198} + t_{1276},
t_{1638} = t_{1149} + t_{1307},
t_{572} = r'_{7} + r'_{197},
t_{599} = r'_{188} + t_{572},
t_{702} = r'_{216} + t_{599},
t_{884} = r'_{34} + t_{702},
t_{1023} = t_{663} + t_{884},
t_{571} = r'_{93} + r'_{184},
t_{890} = t_{571} + t_{768},
t_{1164} = r'_{57} + t_{890},
t_{681} = r'_{209} + t_{571},
t_{947} = t_{681} + t_{940},
t_{570} = r'_{58} + r'_{179},
t_{1580} = t_{570} + t_{858},
t_{859} = r'_{230} + t_{570},
t_{1909} = t_{859} + t_{1561},
t_{1248} = t_{859} + t_{1145},
t_{569} = r'_{71} + r'_{75},
t_{1877} = r'_{149} + t_{569},
t_{1879} = t_{1874} + t_{1877},
t_{1881} = t_{664} + t_{1879},
t_{1105} = t_{569} + t_{634},
t_{2339} = r'_{39} + t_{1105},
t_{567} = r'_{36} + r'_{190},
t_{2152} = t_{567} + t_{901},
t_{730} = t_{567} + t_{651},
t_{2086} = t_{730} + t_{755},
t_{1124} = r'_{237} + t_{730},
t_{1994} = r'_{59} + t_{1124},
t_{564} = r'_{112} + r'_{138},
t_{609} = r'_{164} + t_{564},
t_{1637} = t_{609} + t_{949},
t_{772} = r'_{131} + t_{609},
t_{561} = r'_{12} + r'_{16},
t_{845} = t_{561} + t_{600},
t_{559} = r'_{78} + r'_{82},
t_{558} = r'_{2} + r'_{143},
t_{665} = r'_{194} + t_{558},
t_{1087} = r'_{45} + t_{665},
t_{1432} = t_{947} + t_{1087},
t_{2215} = t_{1212} + t_{1432},
t_{2218} = t_{764} + t_{2215},
t_{1354} = r'_{60} + t_{1087},
t_{1682} = r'_{65} + t_{1354},
t_{2337} = t_{1020} + t_{1682},
t_{771} = t_{665} + t_{678},
t_{556} = r'_{95} + r'_{99},
t_{553} = r'_{27} + r'_{31},
t_{837} = t_{553} + t_{556},
t_{552} = r'_{106} + r'_{128},
t_{1040} = r'_{156} + t_{552},
t_{1295} = r'_{244} + t_{1040},
t_{630} = r'_{202} + t_{552},
t_{1765} = r'_{21} + t_{630},
t_{1772} = t_{908} + t_{1765},
t_{1775} = r'_{47} + t_{1772},
t_{767} = t_{619} + t_{630},
t_{1972} = t_{558} + t_{767},
t_{550} = r'_{60} + r'_{64},
t_{2023} = t_{550} + t_{1414},
t_{547} = r'_{32} + r'_{110},
t_{1489} = t_{547} + t_{889},
t_{1968} = t_{842} + t_{1489},
t_{650} = r'_{205} + t_{547},
t_{1849} = r'_{195} + t_{650},
t_{860} = t_{650} + t_{673},
t_{1170} = t_{860} + t_{1082},
t_{519} = r'_{60} + r'_{82},
t_{672} = r'_{37} + t_{519},
t_{799} = r'_{63} + t_{672},
t_{1332} = t_{664} + t_{799},
t_{1560} = r'_{11} + t_{1332},
t_{1607} = t_{1164} + t_{1560},
t_{2062} = t_{1607} + t_{1622},
t_{1080} = t_{759} + t_{799},
t_{1581} = t_{594} + t_{1080},
t_{1521} = t_{884} + t_{1080},
t_{515} = r'_{34} + r'_{38},
t_{514} = r'_{99} + r'_{251},
t_{1257} = t_{514} + t_{934},
t_{715} = r'_{163} + t_{514},
t_{814} = t_{715} + t_{716},
t_{510} = r'_{49} + r'_{75},
t_{967} = t_{510} + t_{772},
t_{1334} = t_{651} + t_{967},
t_{1409} = t_{1167} + t_{1334},
t_{1796} = t_{1409} + t_{1784},
t_{509} = r'_{158} + r'_{189},
t_{1667} = r'_{238} + t_{509},
t_{1088} = r'_{116} + t_{509},
t_{1419} = t_{591} + t_{1088},
t_{508} = r'_{27} + r'_{107},
t_{585} = r'_{224} + t_{508},
t_{675} = r'_{129} + t_{585},
t_{906} = t_{638} + t_{675},
t_{740} = t_{598} + t_{675},
t_{1562} = r'_{154} + t_{740},
t_{944} = t_{740} + t_{815},
t_{2173} = t_{944} + t_{1149},
t_{506} = r'_{133} + r'_{137},
t_{504} = r'_{236} + r'_{240},
t_{502} = r'_{12} + r'_{155},
t_{1044} = r'_{76} + t_{502},
t_{1405} = r'_{87} + t_{1044},
t_{499} = r'_{167} + r'_{168},
t_{2091} = t_{499} + t_{613},
t_{2093} = t_{1085} + t_{2091},
t_{893} = r'_{26} + t_{499},
t_{2270} = t_{893} + t_{1204},
t_{1266} = t_{677} + t_{893},
t_{498} = r'_{58} + r'_{62},
t_{497} = r'_{149} + r'_{153},
t_{494} = r'_{128} + r'_{132},
t_{493} = r'_{136} + r'_{140},
t_{492} = r'_{162} + r'_{185},
t_{2318} = t_{492} + t_{1718},
t_{2265} = t_{492} + t_{600},
t_{807} = r'_{41} + t_{492},
t_{1084} = t_{634} + t_{807},
t_{549} = t_{492} + t_{509},
t_{491} = r'_{59} + r'_{63},
t_{982} = t_{491} + t_{846},
t_{2125} = r'_{132} + t_{982},
t_{490} = r'_{150} + r'_{154},
t_{1384} = r'_{104} + t_{490},
t_{1254} = t_{490} + t_{561},
t_{577} = t_{490} + t_{506},
t_{489} = r'_{204} + r'_{208},
t_{488} = r'_{102} + r'_{106},
t_{919} = r'_{205} + t_{488},
t_{1122} = t_{919} + t_{1050},
t_{1640} = t_{1122} + t_{1400},
t_{2124} = t_{553} + t_{1640},
t_{1062} = t_{643} + t_{919},
t_{1703} = t_{1062} + t_{1526},
t_{628} = t_{488} + t_{494},
t_{1807} = t_{628} + t_{728},
t_{487} = r'_{10} + r'_{14},
t_{998} = t_{487} + t_{556},
t_{773} = t_{487} + t_{489},
t_{486} = r'_{180} + r'_{184},
t_{1190} = r'_{24} + t_{486},
t_{485} = r'_{95} + r'_{247},
t_{1171} = r'_{52} + t_{485},
t_{1885} = r'_{180} + t_{1171},
t_{955} = t_{485} + t_{603},
t_{1140} = t_{955} + t_{986},
t_{1392} = r'_{223} + t_{1140},
t_{1406} = t_{707} + t_{1392},
t_{1551} = t_{502} + t_{1406},
t_{791} = r'_{186} + t_{485},
t_{1485} = t_{705} + t_{791},
t_{1297} = r'_{50} + t_{791},
t_{2135} = t_{489} + t_{1297},
t_{2137} = t_{2125} + t_{2135},
t_{2141} = t_{1085} + t_{2137},
t_{2142} = t_{2124} + t_{2141},
t_{596} = t_{485} + t_{514},
t_{484} = r'_{141} + r'_{145},
t_{482} = r'_{171} + r'_{175},
t_{1243} = t_{482} + t_{576},
t_{2095} = t_{1243} + t_{2093},
t_{1971} = t_{1243} + t_{1637},
t_{972} = t_{482} + t_{572},
t_{481} = r'_{52} + r'_{56},
t_{818} = t_{481} + t_{559},
t_{2244} = t_{818} + t_{1534},
t_{480} = r'_{227} + r'_{231},
t_{1265} = t_{480} + t_{970},
t_{938} = t_{480} + t_{683},
t_{1270} = t_{892} + t_{938},
t_{624} = t_{480} + t_{486},
t_{1293} = t_{624} + t_{1105},
t_{479} = r'_{109} + r'_{113},
t_{2268} = t_{479} + t_{1257},
t_{1011} = t_{479} + t_{588},
t_{1496} = t_{1011} + t_{1382},
t_{478} = r'_{36} + r'_{40},
t_{1236} = r'_{200} + t_{478},
t_{1395} = t_{1069} + t_{1236},
t_{548} = t_{478} + t_{498},
t_{1634} = t_{548} + t_{728},
t_{1214} = t_{548} + t_{845},
t_{1694} = t_{491} + t_{1214},
t_{477} = r'_{213} + r'_{217},
t_{575} = t_{477} + t_{484},
t_{1658} = t_{575} + t_{1419},
t_{476} = r'_{152} + r'_{156},
t_{1468} = t_{476} + t_{506},
t_{475} = r'_{67} + r'_{246},
t_{2269} = r'_{224} + t_{475},
t_{2274} = r'_{23} + t_{2269},
t_{2275} = r'_{136} + t_{2274},
t_{1602} = t_{475} + t_{781},
t_{1336} = t_{475} + t_{1124},
t_{803} = t_{475} + t_{754},
t_{474} = r'_{193} + r'_{197},
t_{473} = r'_{188} + r'_{192},
t_{472} = r'_{16} + r'_{151},
t_{962} = r'_{177} + t_{472},
t_{1478} = t_{731} + t_{962},
t_{645} = t_{472} + t_{594},
t_{2335} = r'_{86} + t_{645},
t_{2347} = t_{1520} + t_{2335},
t_{2153} = t_{645} + t_{1414},
t_{2162} = t_{2153} + t_{2161},
t_{733} = r'_{227} + t_{645},
t_{1715} = t_{733} + t_{1127},
t_{922} = t_{650} + t_{733},
t_{2073} = r'_{29} + t_{922},
t_{1341} = t_{477} + t_{922},
t_{565} = t_{472} + t_{502},
t_{824} = t_{480} + t_{565},
t_{471} = r'_{238} + r'_{242},
t_{1347} = t_{471} + t_{508},
t_{1566} = t_{903} + t_{1347},
t_{1232} = t_{471} + t_{515},
t_{2061} = t_{645} + t_{1232},
t_{2063} = t_{2061} + t_{2062},
t_{2065} = r'_{116} + t_{2063},
t_{470} = r'_{219} + r'_{223},
t_{469} = r'_{77} + r'_{81},
t_{1161} = t_{469} + t_{715},
t_{847} = t_{469} + t_{550},
t_{694} = t_{469} + t_{583},
t_{1219} = t_{638} + t_{694},
t_{2189} = t_{1219} + t_{1626},
t_{1222} = t_{672} + t_{1219},
t_{468} = r'_{93} + r'_{97},
t_{467} = r'_{17} + r'_{21},
t_{2241} = t_{467} + t_{774},
t_{1482} = r'_{5} + t_{467},
t_{2261} = t_{634} + t_{1482},
t_{466} = r'_{214} + r'_{218},
t_{1508} = t_{466} + t_{580},
t_{538} = t_{466} + t_{471},
t_{714} = t_{504} + t_{538},
t_{465} = r'_{110} + r'_{114},
t_{464} = r'_{9} + r'_{13},
t_{698} = r'_{181} + t_{464},
t_{1187} = r'_{173} + t_{698},
t_{1677} = t_{1187} + t_{1266},
t_{463} = r'_{65} + r'_{244},
t_{1851} = t_{463} + t_{1602},
t_{1614} = r'_{42} + t_{463},
t_{1911} = t_{1614} + t_{1906},
t_{1913} = t_{616} + t_{1911},
t_{462} = r'_{33} + r'_{37},
t_{1539} = t_{462} + t_{1226},
t_{1237} = t_{462} + t_{538},
t_{461} = r'_{51} + r'_{55},
t_{881} = r'_{194} + t_{461},
t_{1380} = t_{881} + t_{1218},
t_{1540} = t_{1300} + t_{1380},
t_{1280} = r'_{68} + t_{881},
t_{1910} = t_{1280} + t_{1402},
t_{1912} = r'_{215} + t_{1910},
t_{1920} = t_{1912} + t_{1913},
t_{460} = r'_{139} + r'_{203},
t_{1601} = r'_{13} + t_{460},
t_{1372} = t_{460} + t_{906},
t_{459} = r'_{98} + r'_{181},
t_{964} = r'_{232} + t_{459},
t_{1194} = r'_{249} + t_{964},
t_{458} = r'_{237} + r'_{241},
t_{1287} = t_{458} + t_{466},
t_{925} = t_{458} + t_{481},
t_{544} = t_{458} + t_{462},
t_{2176} = t_{544} + t_{561},
t_{1785} = t_{538} + t_{544},
t_{952} = t_{471} + t_{544},
t_{1635} = t_{952} + t_{1197},
t_{457} = r'_{31} + r'_{103},
t_{1479} = t_{457} + t_{1110},
t_{830} = t_{457} + t_{755},
t_{1570} = t_{751} + t_{830},
t_{1720} = t_{815} + t_{1570},
t_{912} = t_{812} + t_{830},
t_{1498} = t_{460} + t_{912},
t_{1373} = t_{912} + t_{993},
t_{1702} = r'_{202} + t_{1373},
t_{739} = t_{457} + t_{508},
t_{1051} = t_{482} + t_{739},
t_{589} = r'_{97} + t_{457},
t_{456} = r'_{4} + r'_{8},
t_{1148} = t_{456} + t_{706},
t_{760} = t_{456} + t_{575},
t_{1424} = r'_{39} + t_{760},
t_{2271} = t_{1424} + t_{2265},
t_{2273} = t_{1517} + t_{2271},
t_{455} = r'_{64} + r'_{78},
t_{1599} = r'_{110} + t_{455},
t_{736} = t_{455} + t_{519},
t_{597} = r'_{133} + t_{455},
t_{1286} = t_{597} + t_{797},
t_{988} = t_{597} + t_{811},
t_{1999} = t_{614} + t_{988},
t_{2000} = r'_{104} + t_{1999},
t_{667} = r'_{254} + t_{597},
t_{819} = r'_{71} + t_{667},
t_{1172} = t_{819} + t_{1041},
t_{1697} = t_{1172} + t_{1248},
t_{935} = t_{623} + t_{819},
t_{2130} = t_{732} + t_{935},
t_{2131} = t_{772} + t_{2130},
t_{1491} = r'_{36} + t_{935},
t_{454} = r'_{70} + r'_{74},
t_{843} = r'_{177} + t_{454},
t_{1260} = r'_{27} + t_{843},
t_{1116} = t_{698} + t_{843},
t_{537} = t_{454} + t_{481},
t_{766} = t_{537} + t_{596},
t_{1656} = t_{766} + t_{1494},
t_{1290} = t_{760} + t_{766},
t_{453} = r'_{170} + r'_{174},
t_{1659} = r'_{3} + t_{453},
t_{586} = t_{453} + t_{473},
t_{452} = r'_{80} + r'_{84},
t_{610} = t_{452} + t_{487},
t_{2031} = t_{610} + t_{952},
t_{963} = t_{490} + t_{610},
t_{451} = r'_{211} + r'_{215},
t_{1027} = t_{451} + t_{714},
t_{1947} = t_{963} + t_{1027},
t_{566} = t_{451} + t_{491},
t_{871} = t_{506} + t_{566},
t_{1241} = t_{498} + t_{871},
t_{746} = t_{544} + t_{566},
t_{450} = r'_{50} + r'_{54},
t_{1609} = t_{450} + t_{553},
t_{857} = t_{450} + t_{783},
t_{1604} = r'_{23} + t_{857},
t_{1436} = r'_{53} + t_{857},
t_{449} = r'_{26} + r'_{30},
t_{1401} = t_{449} + t_{739},
t_{711} = r'_{251} + t_{449},
t_{1797} = t_{711} + t_{785},
t_{448} = r'_{18} + r'_{22},
t_{975} = t_{448} + t_{537},
t_{1418} = r'_{200} + t_{975},
t_{573} = t_{448} + t_{474},
t_{838} = t_{573} + t_{624},
t_{447} = r'_{144} + r'_{148},
t_{1176} = t_{447} + t_{519},
t_{446} = r'_{79} + r'_{83},
t_{1946} = t_{446} + t_{550},
t_{1958} = t_{1241} + t_{1946},
t_{1321} = r'_{188} + t_{446},
t_{2243} = t_{1321} + t_{1540},
t_{445} = r'_{43} + r'_{47},
t_{1348} = r'_{125} + t_{445},
t_{1359} = t_{590} + t_{1348},
t_{1636} = t_{1359} + t_{1528},
t_{910} = t_{445} + t_{586},
t_{1477} = r'_{51} + t_{910},
t_{1259} = t_{480} + t_{910},
t_{2123} = t_{1259} + t_{2122},
t_{2128} = r'_{2} + t_{2123},
t_{444} = r'_{57} + r'_{61},
t_{959} = t_{444} + t_{847},
t_{1421} = r'_{0} + t_{959},
t_{443} = r'_{117} + r'_{121},
t_{994} = t_{443} + t_{506},
t_{1047} = r'_{60} + t_{994},
t_{1070} = r'_{103} + t_{1047},
t_{1378} = r'_{182} + t_{1070},
t_{539} = t_{443} + t_{479},
t_{442} = r'_{94} + r'_{177},
t_{1546} = t_{442} + t_{497},
t_{1495} = t_{442} + t_{1120},
t_{1652} = t_{1129} + t_{1495},
t_{848} = t_{442} + t_{793},
t_{1519} = t_{848} + t_{1169},
t_{1063} = t_{602} + t_{848},
t_{1870} = r'_{94} + t_{1063},
t_{655} = t_{442} + t_{589},
t_{1030} = t_{655} + t_{771},
t_{551} = t_{442} + t_{459},
t_{1973} = t_{551} + t_{1546},
t_{1675} = t_{551} + t_{695},
t_{441} = r'_{87} + r'_{91},
t_{2421} = t_{441} + t_{468},
t_{776} = t_{441} + t_{633},
t_{2340} = r'_{154} + t_{776},
t_{2345} = r'_{94} + t_{2340},
t_{1565} = t_{681} + t_{776},
t_{1042} = r'_{171} + t_{776},
t_{440} = r'_{72} + r'_{76},
t_{1251} = t_{440} + t_{897},
t_{2353} = r'_{254} + t_{1251},
t_{1183} = t_{440} + t_{575},
t_{560} = t_{440} + t_{468},
t_{1178} = t_{560} + t_{998},
t_{439} = r'_{186} + r'_{190},
t_{438} = r'_{228} + r'_{232},
t_{1721} = t_{438} + t_{1295},
t_{1871} = t_{1297} + t_{1721},
t_{437} = r'_{53} + r'_{71},
t_{1168} = t_{437} + t_{612},
t_{1147} = t_{437} + t_{767},
t_{1732} = r'_{67} + t_{1147},
t_{642} = t_{437} + t_{574},
t_{2172} = t_{642} + t_{1037},
t_{1094} = t_{630} + t_{642},
t_{1317} = t_{842} + t_{1094},
t_{1000} = t_{638} + t_{642},
t_{541} = t_{437} + t_{510},
t_{436} = r'_{104} + r'_{108},
t_{1031} = t_{436} + t_{444},
t_{1731} = t_{1031} + t_{1056},
t_{780} = t_{436} + t_{497},
t_{1717} = t_{549} + t_{780},
t_{535} = t_{436} + t_{450},
t_{1837} = t_{535} + t_{714},
t_{939} = t_{497} + t_{535},
t_{763} = t_{445} + t_{535},
t_{1349} = t_{650} + t_{763},
t_{1585} = t_{1230} + t_{1349},
t_{435} = r'_{248} + r'_{252},
t_{1071} = r'_{126} + t_{435},
t_{2002} = t_{1071} + t_{1996},
t_{2004} = t_{2000} + t_{2002},
t_{2007} = t_{1994} + t_{2004},
t_{652} = t_{435} + t_{452},
t_{1573} = t_{652} + t_{983},
t_{2320} = t_{538} + t_{1573},
t_{434} = r'_{19} + r'_{23},
t_{2105} = t_{434} + t_{1237},
t_{527} = t_{434} + t_{438},
t_{877} = t_{494} + t_{527},
t_{1233} = t_{581} + t_{877},
t_{700} = t_{527} + t_{539},
t_{433} = r'_{202} + r'_{206},
t_{1255} = t_{433} + t_{474},
t_{2171} = t_{1095} + t_{1255},
t_{2175} = t_{1601} + t_{2171},
t_{2177} = t_{1043} + t_{2175},
t_{657} = r'_{156} + t_{433},
t_{1117} = t_{656} + t_{657},
t_{832} = t_{510} + t_{657},
t_{1115} = r'_{218} + t_{832},
t_{584} = t_{433} + t_{493},
t_{790} = t_{548} + t_{584},
t_{1397} = t_{790} + t_{999},
t_{432} = r'_{212} + r'_{216},
t_{1017} = t_{432} + t_{535},
t_{729} = t_{432} + t_{451},
t_{2178} = r'_{250} + t_{729},
t_{2179} = t_{2173} + t_{2178},
t_{1502} = t_{552} + t_{729},
t_{1098} = t_{466} + t_{729},
t_{666} = t_{432} + t_{434},
t_{1579} = t_{666} + t_{773},
t_{606} = t_{432} + t_{515},
t_{1026} = t_{606} + t_{736},
t_{974} = t_{498} + t_{606},
t_{431} = r'_{159} + r'_{163},
t_{1234} = r'_{250} + t_{431},
t_{2090} = t_{1234} + t_{2085},
t_{1032} = r'_{61} + t_{431},
t_{1350} = t_{1032} + t_{1038},
t_{1445} = t_{1115} + t_{1350},
t_{1559} = t_{1063} + t_{1445},
t_{1716} = t_{982} + t_{1559},
t_{896} = t_{431} + t_{489},
t_{531} = t_{431} + t_{439},
t_{1915} = r'_{235} + t_{531},
t_{1919} = t_{1915} + t_{1917},
t_{1922} = t_{455} + t_{1919},
t_{747} = t_{531} + t_{541},
t_{1535} = t_{739} + t_{747},
t_{430} = r'_{111} + r'_{115},
t_{995} = t_{430} + t_{700},
t_{1610} = t_{476} + t_{995},
t_{2364} = t_{1610} + t_{2152},
t_{1984} = t_{1255} + t_{1610},
t_{429} = r'_{165} + r'_{166},
t_{2134} = t_{429} + t_{848},
t_{937} = r'_{66} + t_{429},
t_{1374} = t_{462} + t_{937},
t_{511} = r'_{0} + t_{429},
t_{1123} = r'_{138} + t_{511},
t_{1588} = r'_{192} + t_{1123},
t_{1758} = t_{858} + t_{1588},
t_{1388} = t_{458} + t_{1123},
t_{1886} = r'_{96} + t_{1388},
t_{794} = r'_{116} + t_{511},
t_{2220} = t_{794} + t_{962},
t_{2228} = t_{1000} + t_{2220},
t_{1089} = r'_{159} + t_{794},
t_{1304} = t_{972} + t_{1089},
t_{428} = r'_{160} + r'_{164},
t_{1215} = t_{428} + t_{539},
t_{1029} = t_{428} + t_{435},
t_{427} = r'_{229} + r'_{233},
t_{426} = r'_{3} + r'_{7},
t_{1597} = t_{426} + t_{694},
t_{2346} = t_{1597} + t_{2337},
t_{2348} = t_{2345} + t_{2346},
t_{2349} = t_{2347} + t_{2348},
t_{2355} = t_{1571} + t_{2349},
t_{787} = t_{426} + t_{476},
t_{1009} = t_{494} + t_{787},
t_{516} = t_{426} + t_{447},
t_{2430} = t_{516} + t_{714},
t_{1487} = t_{516} + t_{1017},
t_{826} = t_{516} + t_{803},
t_{425} = r'_{222} + r'_{226},
t_{800} = t_{425} + t_{477},
t_{1464} = t_{800} + t_{1387},
t_{1426} = t_{800} + t_{1116},
t_{424} = r'_{119} + r'_{123},
t_{1698} = r'_{180} + t_{424},
t_{1447} = t_{424} + t_{539},
t_{1057} = t_{424} + t_{810},
t_{806} = t_{424} + t_{440},
t_{423} = r'_{1} + r'_{5},
t_{2107} = t_{423} + t_{2105},
t_{1281} = t_{423} + t_{1084},
t_{1318} = r'_{31} + t_{1281},
t_{1469} = r'_{164} + t_{1318},
t_{738} = t_{423} + t_{590},
t_{822} = t_{439} + t_{738},
t_{422} = r'_{143} + r'_{147},
t_{421} = r'_{221} + r'_{225},
t_{2342} = t_{421} + t_{2339},
t_{2350} = t_{1676} + t_{2342},
t_{2351} = t_{2344} + t_{2350},
t_{1386} = t_{421} + t_{874},
t_{758} = t_{421} + t_{573},
t_{1608} = t_{454} + t_{758},
t_{1192} = r'_{9} + t_{758},
t_{1497} = r'_{207} + t_{1192},
t_{2216} = t_{1497} + t_{1547},
t_{2224} = t_{1130} + t_{2216},
t_{2225} = t_{727} + t_{2224},
t_{529} = t_{421} + t_{445},
t_{1954} = t_{489} + t_{529},
t_{1512} = t_{529} + t_{700},
t_{926} = t_{516} + t_{529},
t_{1964} = t_{458} + t_{926},
t_{420} = r'_{96} + r'_{100},
t_{1003} = t_{420} + t_{654},
t_{1199} = t_{1003} + t_{1060},
t_{1206} = t_{1089} + t_{1199},
t_{662} = t_{420} + t_{431},
t_{902} = t_{504} + t_{662},
t_{2192} = t_{425} + t_{902},
t_{419} = r'_{86} + r'_{90},
t_{2016} = t_{419} + t_{1519},
t_{2018} = t_{592} + t_{2016},
t_{2021} = t_{779} + t_{2018},
t_{2022} = t_{1498} + t_{2021},
t_{2024} = t_{2022} + t_{2023},
t_{2028} = t_{2024} + t_{2027},
t_{856} = r'_{139} + t_{419},
t_{536} = t_{419} + t_{449},
t_{1841} = t_{467} + t_{536},
t_{418} = r'_{20} + r'_{24},
t_{1333} = t_{418} + t_{936},
t_{1356} = t_{674} + t_{1333},
t_{1568} = r'_{231} + t_{1356},
t_{1763} = t_{1551} + t_{1568},
t_{1766} = t_{1546} + t_{1763},
t_{699} = t_{418} + t_{443},
t_{854} = t_{699} + t_{780},
t_{2072} = t_{854} + t_{1421},
t_{2078} = t_{806} + t_{2072},
t_{671} = r'_{247} + t_{418},
t_{1335} = t_{671} + t_{711},
t_{534} = t_{418} + t_{427},
t_{417} = r'_{134} + r'_{138},
t_{1724} = r'_{26} + t_{417},
t_{697} = t_{417} + t_{565},
t_{1239} = t_{612} + t_{697},
t_{1471} = t_{1089} + t_{1239},
t_{530} = t_{417} + t_{476},
t_{1945} = t_{530} + t_{696},
t_{1951} = t_{577} + t_{1945},
t_{611} = t_{464} + t_{530},
t_{1513} = t_{577} + t_{611},
t_{416} = r'_{172} + r'_{176},
t_{1525} = t_{416} + t_{1485},
t_{543} = t_{416} + t_{435},
t_{882} = t_{452} + t_{543},
t_{415} = r'_{194} + r'_{198},
t_{1413} = t_{415} + t_{549},
t_{1004} = t_{415} + t_{441},
t_{2217} = t_{1004} + t_{1635},
t_{1352} = r'_{98} + t_{1004},
t_{414} = r'_{69} + r'_{73},
t_{2264} = t_{414} + t_{1118},
t_{2272} = t_{1352} + t_{2264},
t_{2281} = t_{2272} + t_{2275},
t_{2286} = t_{567} + t_{2281},
t_{761} = t_{414} + t_{447},
t_{1411} = t_{596} + t_{761},
t_{918} = t_{580} + t_{761},
t_{533} = t_{414} + t_{461},
t_{2309} = t_{480} + t_{533},
t_{1107} = r'_{110} + t_{533},
t_{1342} = r'_{130} + t_{1107},
t_{687} = t_{533} + t_{551},
t_{1714} = t_{687} + t_{747},
t_{929} = t_{465} + t_{687},
t_{413} = r'_{157} + r'_{161},
t_{1282} = t_{413} + t_{634},
t_{1162} = t_{413} + t_{584},
t_{775} = t_{413} + t_{470},
t_{412} = r'_{66} + r'_{243},
t_{1441} = t_{412} + t_{589},
t_{1655} = r'_{186} + t_{1441},
t_{1002} = r'_{65} + t_{412},
t_{1249} = r'_{230} + t_{1002},
t_{1630} = t_{1249} + t_{1446},
t_{1789} = r'_{135} + t_{1630},
t_{802} = t_{412} + t_{732},
t_{1309} = r'_{33} + t_{802},
t_{1005} = t_{567} + t_{802},
t_{1370} = t_{572} + t_{1005},
t_{503} = t_{412} + t_{463},
t_{1699} = t_{503} + t_{826},
t_{1223} = r'_{185} + t_{503},
t_{528} = t_{499} + t_{503},
t_{891} = t_{459} + t_{528},
t_{1247} = t_{771} + t_{891},
t_{1484} = t_{882} + t_{1247},
t_{2379} = r'_{95} + t_{1484},
t_{411} = r'_{35} + r'_{39},
t_{1631} = t_{411} + t_{444},
t_{629} = t_{411} + t_{497},
t_{542} = t_{411} + t_{446},
t_{1692} = t_{452} + t_{542},
t_{1558} = t_{443} + t_{542},
t_{1673} = r'_{12} + t_{1558},
t_{1997} = t_{736} + t_{1673},
t_{2001} = t_{1186} + t_{1997},
t_{2009} = t_{2001} + t_{2007},
t_{2011} = t_{2009} + t_{2010},
t_{1557} = t_{542} + t_{1031},
t_{2316} = t_{457} + t_{1557},
t_{2324} = t_{425} + t_{2316},
t_{2326} = t_{2241} + t_{2324},
t_{2328} = t_{1131} + t_{2326},
t_{835} = t_{542} + t_{549},
t_{1977} = t_{835} + t_{1290},
t_{410} = r'_{42} + r'_{46},
t_{869} = r'_{147} + t_{410},
t_{1680} = r'_{218} + t_{869},
t_{1859} = t_{1443} + t_{1680},
t_{1323} = t_{869} + t_{1161},
t_{2242} = t_{1323} + t_{1519},
t_{2252} = t_{907} + t_{2242},
t_{1275} = t_{470} + t_{869},
t_{409} = r'_{187} + r'_{191},
t_{2266} = t_{409} + t_{1110},
t_{2279} = t_{2266} + t_{2268},
t_{2283} = t_{432} + t_{2279},
t_{1181} = t_{409} + t_{533},
t_{2168} = t_{761} + t_{1181},
t_{668} = t_{409} + t_{416},
t_{2336} = r'_{108} + t_{668},
t_{1632} = t_{543} + t_{668},
t_{792} = t_{542} + t_{668},
t_{900} = t_{488} + t_{792},
t_{1267} = t_{763} + t_{900},
t_{540} = t_{409} + t_{428},
t_{1213} = r'_{70} + t_{540},
t_{1283} = t_{678} + t_{1213},
t_{2020} = t_{1283} + t_{2019},
t_{778} = t_{540} + t_{560},
t_{408} = r'_{44} + r'_{48},
t_{1143} = t_{408} + t_{671},
t_{2352} = t_{460} + t_{1143},
t_{2356} = t_{2351} + t_{2352},
t_{1128} = t_{408} + t_{599},
t_{2222} = r'_{169} + t_{1128},
t_{1360} = t_{1021} + t_{1128},
t_{1481} = r'_{73} + t_{1360},
t_{1908} = r'_{237} + t_{1481},
t_{684} = t_{408} + t_{438},
t_{1152} = t_{684} + t_{715},
t_{1061} = t_{611} + t_{684},
t_{899} = t_{684} + t_{775},
t_{1366} = t_{854} + t_{899},
t_{532} = t_{408} + t_{425},
t_{1523} = t_{532} + t_{582},
t_{839} = t_{532} + t_{611},
t_{407} = r'_{249} + r'_{253},
t_{1678} = t_{407} + t_{468},
t_{1914} = r'_{196} + t_{1678},
t_{710} = t_{407} + t_{615},
t_{2239} = r'_{227} + t_{710},
t_{523} = t_{407} + t_{422},
t_{978} = t_{461} + t_{523},
t_{1933} = t_{530} + t_{978},
t_{1428} = t_{978} + t_{1259},
t_{945} = t_{450} + t_{523},
t_{1653} = t_{945} + t_{1183},
t_{637} = t_{469} + t_{523},
t_{2079} = t_{490} + t_{637},
t_{1200} = r'_{241} + t_{637},
t_{984} = r'_{159} + t_{637},
t_{1995} = t_{970} + t_{984},
t_{2005} = r'_{221} + t_{1995},
t_{1550} = t_{450} + t_{984},
t_{2180} = t_{1386} + t_{1550},
t_{406} = r'_{101} + r'_{105},
t_{1068} = t_{406} + t_{695},
t_{405} = r'_{127} + r'_{131},
t_{1385} = r'_{69} + t_{405},
t_{546} = t_{405} + t_{430},
t_{404} = r'_{118} + r'_{122},
t_{1627} = t_{404} + t_{1051},
t_{1112} = t_{404} + t_{701},
t_{1725} = t_{412} + t_{1112},
t_{1556} = t_{679} + t_{1112},
t_{725} = t_{404} + t_{465},
t_{1567} = t_{725} + t_{939},
t_{1111} = t_{405} + t_{725},
t_{1591} = r'_{90} + t_{1111},
t_{2136} = t_{1562} + t_{1591},
t_{2139} = t_{2134} + t_{2136},
t_{2144} = t_{2139} + t_{2142},
t_{2145} = r'_{72} + t_{2144},
t_{403} = r'_{41} + r'_{45},
t_{1268} = t_{403} + t_{1040},
t_{2219} = r'_{223} + t_{1268},
t_{921} = t_{403} + t_{430},
t_{1650} = t_{921} + t_{963},
t_{2103} = t_{445} + t_{1650},
t_{518} = t_{403} + t_{404},
t_{887} = t_{470} + t_{518},
t_{2385} = t_{486} + t_{887},
t_{915} = t_{471} + t_{887},
t_{741} = t_{427} + t_{518},
t_{1767} = r'_{136} + t_{741},
t_{1774} = t_{1766} + t_{1767},
t_{402} = r'_{169} + r'_{173},
t_{956} = t_{402} + t_{421},
t_{813} = t_{402} + t_{417},
t_{1612} = r'_{166} + t_{813},
t_{886} = t_{619} + t_{813},
t_{1125} = t_{616} + t_{886},
t_{1301} = r'_{58} + t_{1125},
t_{401} = r'_{135} + r'_{207},
t_{1577} = t_{401} + t_{1068},
t_{863} = t_{401} + t_{463},
t_{1693} = t_{803} + t_{863},
t_{1434} = r'_{245} + t_{863},
t_{703} = r'_{187} + t_{401},
t_{1582} = t_{621} + t_{703},
t_{1660} = r'_{163} + t_{1582},
t_{2392} = t_{1317} + t_{1660},
t_{2395} = r'_{88} + t_{2392},
t_{2410} = t_{2395} + t_{2406},
t_{1119} = t_{703} + t_{1042},
t_{1216} = r'_{108} + t_{1119},
t_{1444} = t_{946} + t_{1216},
t_{2278} = t_{1444} + t_{2273},
t_{2284} = t_{2278} + t_{2280},
t_{507} = t_{401} + t_{460},
t_{966} = t_{456} + t_{507},
t_{1104} = t_{409} + t_{966},
t_{2195} = t_{1104} + t_{2192},
t_{2201} = t_{1374} + t_{2195},
t_{2054} = t_{1104} + t_{1139},
t_{2055} = t_{1107} + t_{2054},
t_{2060} = t_{1422} + t_{2055},
t_{400} = r'_{230} + r'_{234},
t_{620} = t_{400} + t_{467},
t_{840} = t_{417} + t_{620},
t_{517} = t_{400} + t_{424},
t_{852} = t_{439} + t_{517},
t_{1155} = t_{407} + t_{852},
t_{649} = t_{467} + t_{517},
t_{1642} = t_{649} + t_{760},
t_{1987} = t_{1487} + t_{1642},
t_{1229} = t_{404} + t_{649},
t_{1511} = t_{546} + t_{1229},
t_{399} = r'_{125} + r'_{129},
t_{1339} = t_{399} + t_{806},
t_{1930} = t_{852} + t_{1339},
t_{1013} = t_{399} + t_{704},
t_{1475} = t_{623} + t_{1013},
t_{1315} = t_{1013} + t_{1095},
t_{1362} = t_{933} + t_{1315},
t_{717} = t_{399} + t_{478},
t_{1555} = t_{696} + t_{717},
t_{1935} = t_{780} + t_{1555},
t_{1034} = t_{694} + t_{717},
t_{2412} = t_{772} + t_{1034},
t_{1515} = t_{532} + t_{1034},
t_{680} = t_{399} + t_{415},
t_{1381} = r'_{63} + t_{680},
t_{1035} = t_{539} + t_{680},
t_{2053} = t_{474} + t_{1035},
t_{875} = t_{680} + t_{711},
t_{557} = t_{399} + t_{465},
t_{1246} = t_{557} + t_{1176},
t_{1647} = t_{420} + t_{1246},
t_{398} = r'_{28} + r'_{32},
t_{1331} = r'_{218} + t_{398},
t_{397} = r'_{68} + r'_{245},
t_{923} = t_{397} + t_{429},
t_{1404} = r'_{113} + t_{923},
t_{1704} = r'_{143} + t_{1404},
t_{1305} = t_{923} + t_{1052},
t_{1648} = t_{1117} + t_{1305},
t_{483} = r'_{210} + t_{397},
t_{1430} = t_{483} + t_{484},
t_{831} = r'_{182} + t_{483},
t_{1584} = t_{569} + t_{831},
t_{1067} = t_{831} + t_{864},
t_{1273} = t_{693} + t_{1067},
t_{2126} = t_{1273} + t_{1530},
t_{2129} = t_{1516} + t_{2126},
t_{2133} = t_{2129} + t_{2131},
t_{2138} = t_{1119} + t_{2133},
t_{2143} = t_{698} + t_{2138},
t_{524} = r'_{209} + t_{483},
t_{1872} = t_{524} + t_{720},
t_{545} = t_{475} + t_{524},
t_{1316} = t_{545} + t_{1194},
t_{1611} = t_{1316} + t_{1418},
t_{579} = t_{511} + t_{545},
t_{608} = t_{528} + t_{579},
t_{396} = r'_{195} + r'_{199},
t_{1449} = t_{396} + t_{499},
t_{941} = t_{396} + t_{414},
t_{1554} = t_{557} + t_{941},
t_{1086} = t_{482} + t_{941},
t_{1681} = t_{704} + t_{1086},
t_{2338} = t_{1681} + t_{2336},
t_{2354} = t_{1342} + t_{2338},
t_{2359} = t_{2354} + t_{2355},
t_{2226} = t_{1653} + t_{1681},
t_{2227} = t_{2218} + t_{2226},
t_{1361} = t_{1086} + t_{1098},
t_{1473} = t_{800} + t_{1361},
t_{395} = r'_{250} + r'_{254},
t_{2240} = t_{395} + t_{745},
t_{2251} = t_{1632} + t_{2240},
t_{2253} = t_{2251} + t_{2252},
t_{2256} = t_{1664} + t_{2253},
t_{1184} = t_{395} + t_{406},
t_{1453} = t_{537} + t_{1184},
t_{1156} = t_{395} + t_{416},
t_{648} = t_{395} + t_{453},
t_{1510} = t_{474} + t_{648},
t_{1146} = t_{410} + t_{648},
t_{1351} = r'_{188} + t_{1146},
t_{2322} = t_{1351} + t_{2320},
t_{2323} = r'_{51} + t_{2322},
t_{928} = t_{540} + t_{648},
t_{520} = t_{395} + t_{413},
t_{1423} = r'_{75} + t_{520},
t_{821} = t_{520} + t_{586},
t_{1467} = t_{821} + t_{1026},
t_{1458} = r'_{137} + t_{821},
t_{2058} = t_{1458} + t_{2053},
t_{2066} = t_{2058} + t_{2065},
t_{2067} = t_{2052} + t_{2066},
t_{1850} = t_{1270} + t_{1458},
t_{1865} = t_{1850} + t_{1863},
t_{394} = r'_{178} + r'_{182},
t_{1979} = t_{394} + t_{1977},
t_{782} = t_{394} + t_{532},
t_{992} = t_{484} + t_{782},
t_{1875} = r'_{27} + t_{992},
t_{1882} = t_{1871} + t_{1875},
t_{1883} = t_{1631} + t_{1882},
t_{724} = t_{394} + t_{550},
t_{1328} = t_{724} + t_{761},
t_{961} = t_{672} + t_{724},
t_{1587} = t_{961} + t_{987},
t_{555} = t_{394} + t_{441},
t_{979} = t_{555} + t_{837},
t_{1624} = t_{450} + t_{979},
t_{809} = t_{474} + t_{555},
t_{1522} = t_{809} + t_{835},
t_{2191} = t_{666} + t_{1522},
t_{2199} = t_{548} + t_{2191},
t_{1019} = t_{454} + t_{809},
t_{393} = r'_{235} + r'_{239},
t_{1729} = t_{393} + t_{456},
t_{1103} = t_{393} + t_{856},
t_{1298} = t_{558} + t_{1103},
t_{878} = t_{393} + t_{629},
t_{689} = t_{393} + t_{468},
t_{796} = t_{529} + t_{689},
t_{2197} = t_{524} + t_{796},
t_{2202} = t_{2197} + t_{2201},
t_{2203} = r'_{244} + t_{2202},
t_{1595} = t_{488} + t_{796},
t_{1898} = t_{845} + t_{1595},
t_{1835} = t_{544} + t_{1595},
t_{521} = t_{393} + t_{444},
t_{911} = t_{428} + t_{521},
t_{1719} = t_{911} + t_{1521},
t_{1970} = r'_{211} + t_{1719},
t_{1975} = t_{502} + t_{1970},
t_{1976} = t_{1972} + t_{1975},
t_{1978} = t_{1971} + t_{1976},
t_{1980} = t_{1974} + t_{1978},
t_{1981} = t_{1968} + t_{1980},
t_{1982} = t_{1979} + t_{1981},
t_{1983} = t_{1973} + t_{1982},
t_{60} = t_{890} + t_{1983},
t_{808} = t_{60} + t_{570},
t_{930} = t_{462} + t_{911},
t_{1065} = t_{652} + t_{930},
t_{669} = t_{507} + t_{521},
t_{2325} = t_{669} + t_{1699},
t_{2331} = t_{1130} + t_{2325},
t_{1641} = t_{669} + t_{1039},
t_{1079} = t_{479} + t_{669},
t_{883} = t_{491} + t_{669},
t_{392} = r'_{179} + r'_{183},
t_{1330} = t_{392} + t_{552},
t_{851} = t_{392} + t_{504},
t_{1895} = t_{851} + t_{1513},
t_{1596} = t_{560} + t_{851},
t_{1957} = t_{1426} + t_{1596},
t_{1083} = t_{668} + t_{851},
t_{1880} = t_{1083} + t_{1331},
t_{1888} = t_{1605} + t_{1880},
t_{1889} = t_{1883} + t_{1888},
t_{526} = t_{392} + t_{420},
t_{868} = t_{449} + t_{526},
t_{1452} = t_{443} + t_{868},
t_{719} = t_{526} + t_{543},
t_{1753} = t_{719} + t_{1065},
t_{1537} = t_{548} + t_{719},
t_{1182} = t_{418} + t_{719},
t_{391} = r'_{201} + r'_{205},
t_{1174} = t_{391} + t_{847},
t_{872} = t_{391} + t_{464},
t_{2394} = t_{872} + t_{901},
t_{2405} = t_{2394} + t_{2398},
t_{1531} = t_{714} + t_{872},
t_{1153} = r'_{216} + t_{872},
t_{1099} = t_{493} + t_{872},
t_{743} = t_{391} + t_{448},
t_{1450} = t_{609} + t_{743},
t_{2254} = t_{1450} + t_{2247},
t_{2258} = t_{1153} + t_{2254},
t_{1357} = t_{666} + t_{743},
t_{1686} = t_{775} + t_{1357},
t_{1097} = t_{743} + t_{814},
t_{1852} = t_{1097} + t_{1849},
t_{1857} = t_{918} + t_{1852},
t_{1514} = t_{1097} + t_{1298},
t_{522} = t_{391} + t_{406},
t_{1695} = t_{522} + t_{1222},
t_{2157} = t_{1695} + t_{2150},
t_{2159} = t_{2149} + t_{2157},
t_{2160} = t_{655} + t_{2159},
t_{2166} = t_{710} + t_{2160},
t_{867} = t_{522} + t_{559},
t_{980} = t_{583} + t_{867},
t_{2116} = t_{436} + t_{980},
t_{2117} = t_{606} + t_{2116},
t_{1988} = t_{839} + t_{980},
t_{636} = t_{522} + t_{577},
t_{1986} = t_{494} + t_{636},
t_{1989} = t_{882} + t_{1986},
t_{1990} = t_{928} + t_{1989},
t_{1663} = t_{425} + t_{636},
t_{855} = t_{493} + t_{636},
t_{1992} = t_{855} + t_{1017},
t_{1618} = t_{773} + t_{855},
t_{390} = r'_{220} + r'_{224},
t_{1073} = r'_{196} + t_{390},
t_{1598} = t_{1073} + t_{1244},
t_{2233} = t_{1193} + t_{1598},
t_{2234} = t_{2228} + t_{2233},
t_{2089} = t_{1598} + t_{2083},
t_{639} = t_{390} + t_{410},
t_{2368} = t_{639} + t_{1168},
t_{2371} = t_{2363} + t_{2368},
t_{1839} = t_{639} + t_{1835},
t_{1843} = t_{1837} + t_{1839},
t_{1399} = t_{639} + t_{747},
t_{1683} = t_{945} + t_{1399},
t_{960} = t_{541} + t_{639},
t_{2306} = t_{596} + t_{960},
t_{1625} = t_{877} + t_{960},
t_{389} = r'_{126} + r'_{130},
t_{1759} = t_{389} + t_{859},
t_{1761} = r'_{208} + t_{1759},
t_{1762} = t_{1760} + t_{1761},
t_{1768} = t_{1758} + t_{1762},
t_{1771} = t_{1768} + t_{1769},
t_{1773} = t_{503} + t_{1771},
t_{1776} = t_{1773} + t_{1775},
t_{1778} = t_{582} + t_{1776},
t_{1010} = t_{389} + t_{656},
t_{2071} = t_{792} + t_{1010},
t_{2074} = t_{709} + t_{2071},
t_{2081} = t_{2074} + t_{2079},
t_{2084} = t_{2081} + t_{2082},
t_{2088} = r'_{94} + t_{2084},
t_{990} = t_{389} + t_{517},
t_{2388} = t_{990} + t_{2385},
t_{500} = t_{389} + t_{396},
t_{723} = t_{489} + t_{500},
t_{2296} = t_{723} + t_{867},
t_{1398} = t_{549} + t_{723},
t_{968} = r'_{238} + t_{723},
t_{1242} = t_{659} + t_{968},
t_{2403} = t_{627} + t_{1242},
t_{640} = t_{500} + t_{534},
t_{1733} = t_{640} + t_{1608},
t_{1465} = t_{620} + t_{640},
t_{388} = r'_{142} + r'_{146},
t_{2401} = r'_{39} + t_{388},
t_{1780} = t_{388} + t_{1508},
t_{997} = t_{388} + t_{396},
t_{1320} = t_{861} + t_{997},
t_{1583} = r'_{126} + t_{1320},
t_{756} = t_{388} + t_{569},
t_{1480} = t_{756} + t_{1341},
t_{1303} = t_{540} + t_{756},
t_{1649} = t_{566} + t_{1303},
t_{1090} = t_{662} + t_{756},
t_{1619} = t_{918} + t_{1090},
t_{722} = t_{388} + t_{553},
t_{2075} = r'_{166} + t_{722},
t_{2080} = t_{1654} + t_{2075},
t_{2087} = t_{2078} + t_{2080},
t_{2094} = t_{2087} + t_{2088},
t_{2097} = t_{2094} + t_{2095},
t_{1329} = t_{425} + t_{722},
t_{1563} = t_{564} + t_{1329},
t_{924} = t_{584} + t_{722},
t_{1670} = t_{561} + t_{924},
t_{505} = t_{388} + t_{423},
t_{2427} = t_{505} + t_{1467},
t_{2428} = t_{1714} + t_{2427},
t_{1045} = t_{505} + t_{628},
t_{631} = t_{422} + t_{505},
t_{387} = r'_{25} + r'_{29},
t_{1838} = t_{387} + t_{481},
t_{1840} = t_{479} + t_{1838},
t_{1141} = t_{387} + t_{610},
t_{1460} = t_{473} + t_{1141},
t_{820} = t_{387} + t_{703},
t_{1324} = r'_{229} + t_{820},
t_{1456} = t_{1324} + t_{1362},
t_{1231} = r'_{102} + t_{820},
t_{1346} = t_{471} + t_{1231},
t_{708} = t_{387} + t_{520},
t_{1730} = t_{708} + t_{1663},
t_{827} = t_{708} + t_{728},
t_{1644} = t_{724} + t_{827},
t_{2104} = t_{425} + t_{1644},
t_{2106} = t_{2103} + t_{2104},
t_{2112} = t_{416} + t_{2106},
t_{386} = r'_{112} + r'_{116},
t_{870} = t_{386} + t_{546},
t_{1592} = t_{736} + t_{870},
t_{1252} = r'_{22} + t_{870},
t_{1220} = t_{402} + t_{870},
t_{1687} = t_{1220} + t_{1267},
t_{753} = t_{386} + t_{398},
t_{2194} = t_{738} + t_{753},
t_{1018} = t_{666} + t_{753},
t_{385} = r'_{196} + r'_{200},
t_{1736} = t_{385} + t_{553},
t_{1427} = t_{385} + t_{778},
t_{1209} = t_{385} + t_{824},
t_{895} = t_{385} + t_{639},
t_{1302} = t_{700} + t_{895},
t_{2420} = t_{687} + t_{1302},
t_{1210} = t_{814} + t_{895},
t_{786} = t_{385} + t_{655},
t_{1278} = t_{742} + t_{786},
t_{1394} = t_{427} + t_{1278},
t_{525} = t_{385} + t_{415},
t_{1524} = t_{525} + t_{606},
t_{2210} = t_{413} + t_{1524},
t_{1196} = t_{525} + t_{929},
t_{932} = t_{525} + t_{689},
t_{1689} = t_{932} + t_{990},
t_{2034} = t_{531} + t_{1689},
t_{2035} = t_{902} + t_{2034},
t_{686} = t_{406} + t_{525},
t_{1782} = t_{686} + t_{1475},
t_{1783} = t_{477} + t_{1782},
t_{1258} = t_{586} + t_{686},
t_{1269} = t_{422} + t_{1258},
t_{2127} = t_{454} + t_{1269},
t_{2132} = t_{1071} + t_{2127},
t_{2140} = t_{2128} + t_{2132},
t_{2146} = t_{1189} + t_{2140},
t_{2147} = t_{2145} + t_{2146},
t_{2148} = t_{2143} + t_{2147},
t_{384} = r'_{120} + r'_{124},
t_{1750} = t_{384} + t_{1683},
t_{1752} = t_{1045} + t_{1750},
t_{844} = t_{384} + t_{634},
t_{1684} = t_{844} + t_{1608},
t_{752} = t_{384} + t_{580},
t_{1735} = t_{752} + t_{1733},
t_{1737} = t_{728} + t_{1735},
t_{1743} = t_{575} + t_{1737},
t_{1188} = t_{453} + t_{752},
t_{512} = t_{384} + t_{386},
t_{1600} = t_{512} + t_{787},
t_{1195} = t_{512} + t_{681},
t_{798} = t_{441} + t_{512},
t_{1390} = t_{662} + t_{798},
t_{1379} = t_{391} + t_{798},
t_{2208} = t_{1379} + t_{1555},
t_{2151} = t_{838} + t_{1379},
t_{2155} = r'_{155} + t_{2151},
t_{2163} = t_{2155} + t_{2162},
t_{2165} = t_{1125} + t_{2163},
t_{2167} = t_{2165} + t_{2166},
t_{784} = t_{512} + t_{556},
t_{1389} = t_{784} + t_{1170},
t_{2248} = t_{1389} + t_{2239},
t_{1157} = t_{546} + t_{784},
t_{1505} = t_{915} + t_{1157},
t_{625} = t_{512} + t_{536},
t_{2169} = t_{625} + t_{818},
t_{2174} = t_{2169} + t_{2172},
t_{2181} = t_{2170} + t_{2174},
t_{2182} = t_{2177} + t_{2181},
t_{2183} = t_{2180} + t_{2182},
t_{2184} = t_{2168} + t_{2183},
t_{1628} = t_{525} + t_{625},
t_{1312} = t_{625} + t_{628},
t_{1606} = t_{680} + t_{1312},
t_{61} = t_{573} + t_{669} + t_{824} + t_{839} + t_{1051} + t_{1255} + t_{1290} + t_{1512} + t_{1522} + t_{1606},
t_{1415} = t_{61} + t_{1098},
t_{2193} = t_{407} + t_{1415},
t_{1055} = t_{515} + t_{625},
t_{2187} = t_{998} + t_{1055},
t_{1337} = t_{397} + t_{1055},
t_{2321} = t_{1337} + t_{2318},
t_{2327} = t_{2261} + t_{2321},
t_{2329} = t_{2323} + t_{2327},
t_{2332} = t_{2329} + t_{2331},
t_{383} = r'_{88} + r'_{92},
t_{2389} = t_{383} + t_{409},
t_{1764} = t_{383} + t_{579},
t_{1770} = t_{940} + t_{1764},
t_{1777} = t_{1770} + t_{1774},
t_{1779} = t_{1777} + t_{1778},
t_{1159} = t_{383} + t_{875},
t_{1825} = t_{1159} + t_{1287},
t_{1827} = t_{1729} + t_{1825},
t_{635} = t_{383} + t_{392},
t_{1369} = t_{627} + t_{635},
t_{1711} = r'_{16} + t_{1369},
t_{849} = t_{467} + t_{635},
t_{1046} = t_{752} + t_{849},
t_{801} = t_{389} + t_{635},
t_{916} = t_{419} + t_{801},
t_{496} = t_{383} + t_{398},
t_{1410} = t_{496} + t_{798},
t_{1916} = t_{546} + t_{1410},
t_{1918} = t_{1908} + t_{1916},
t_{1925} = t_{1914} + t_{1918},
t_{1928} = t_{1925} + t_{1926},
t_{1738} = t_{1410} + t_{1736},
t_{1264} = t_{496} + t_{560},
t_{1666} = t_{537} + t_{1264},
t_{894} = t_{496} + t_{652},
t_{563} = t_{470} + t_{496},
t_{1064} = t_{534} + t_{563},
t_{1781} = t_{779} + t_{1064},
t_{1787} = t_{951} + t_{1781},
t_{1791} = t_{1148} + t_{1787},
t_{1794} = t_{1789} + t_{1791},
t_{1798} = t_{775} + t_{1794},
t_{1801} = t_{1797} + t_{1798},
t_{1802} = t_{459} + t_{1801},
t_{1015} = t_{563} + t_{640},
t_{1472} = t_{518} + t_{1015},
t_{1474} = t_{637} + t_{1472},
t_{692} = t_{403} + t_{563},
t_{1507} = t_{637} + t_{692},
t_{2298} = t_{537} + t_{1507},
t_{1811} = t_{396} + t_{1507},
t_{991} = t_{692} + t_{926},
t_{382} = r'_{2} + r'_{6},
t_{1440} = r'_{167} + t_{382},
t_{1132} = t_{382} + t_{631},
t_{1205} = t_{629} + t_{1132},
t_{578} = t_{382} + t_{456},
t_{1705} = t_{578} + t_{1390},
t_{1096} = t_{578} + t_{826},
t_{942} = t_{533} + t_{578},
t_{1228} = t_{489} + t_{942},
t_{1959} = t_{1228} + t_{1958},
t_{1961} = t_{1554} + t_{1959},
t_{513} = t_{382} + t_{402},
t_{1377} = t_{513} + t_{753},
t_{1285} = t_{513} + t_{708},
t_{1201} = t_{513} + t_{710},
t_{1578} = t_{665} + t_{1201},
t_{958} = t_{486} + t_{513},
t_{58} = r'_{56} + t_{564} + t_{673} + t_{682} + t_{716} + t_{730} + t_{746} + t_{778} + t_{807} + t_{907} + t_{958} + t_{973} + t_{997} + t_{1076} + t_{1317} + t_{1457} + t_{1459} + t_{1550} + t_{1551},
t_{2008} = t_{58} + t_{451},
t_{2013} = t_{1580} + t_{2008},
t_{1072} = t_{58} + t_{871},
t_{1873} = t_{1072} + t_{1872},
t_{1876} = t_{437} + t_{1873},
t_{1878} = t_{1870} + t_{1876},
t_{1884} = t_{1697} + t_{1878},
t_{1887} = t_{1884} + t_{1886},
t_{1890} = t_{717} + t_{1887},
t_{1891} = t_{1889} + t_{1890},
t_{1892} = t_{1881} + t_{1891},
t_{1893} = t_{1707} + t_{1892},
t_{1674} = t_{1072} + t_{1662},
t_{1998} = t_{1674} + t_{1885},
t_{1500} = t_{697} + t_{958},
t_{2196} = t_{1401} + t_{1500},
t_{2198} = t_{2193} + t_{2196},
t_{2200} = t_{2198} + t_{2199},
t_{2204} = t_{2200} + t_{2203},
t_{676} = t_{482} + t_{513},
t_{2111} = t_{676} + t_{747},
t_{2113} = t_{611} + t_{2111},
t_{1822} = t_{441} + t_{676},
t_{1824} = t_{845} + t_{1822},
t_{1826} = t_{1141} + t_{1824},
t_{1828} = t_{1826} + t_{1827},
t_{1543} = t_{522} + t_{676},
t_{1327} = t_{573} + t_{676},
t_{917} = t_{505} + t_{676},
t_{1462} = t_{917} + t_{1079},
t_{1165} = t_{534} + t_{917},
t_{1590} = t_{891} + t_{1165},
t_{381} = r'_{11} + r'_{15},
t_{691} = t_{381} + t_{477},
t_{1355} = t_{622} + t_{691},
t_{1256} = t_{691} + t_{822},
t_{1461} = t_{636} + t_{1256},
t_{1250} = r'_{124} + t_{691},
t_{1311} = t_{1000} + t_{1250},
t_{1586} = t_{1311} + t_{1585},
t_{850} = t_{607} + t_{691},
t_{1948} = t_{850} + t_{1947},
t_{1950} = t_{462} + t_{1948},
t_{1952} = t_{1946} + t_{1950},
t_{1953} = t_{1951} + t_{1952},
t_{1955} = t_{974} + t_{1953},
t_{1001} = t_{421} + t_{850},
t_{2030} = t_{1001} + t_{1377},
t_{2036} = t_{1644} + t_{2030},
t_{2037} = t_{875} + t_{2036},
t_{2038} = t_{2035} + t_{2037},
t_{2039} = t_{924} + t_{2038},
t_{2040} = t_{1600} + t_{2039},
t_{1722} = t_{806} + t_{1001},
t_{495} = t_{381} + t_{390},
t_{1518} = t_{495} + t_{1155},
t_{1393} = t_{430} + t_{495},
t_{1808} = t_{629} + t_{1393},
t_{1814} = t_{418} + t_{1808},
t_{1818} = t_{1811} + t_{1814},
t_{1296} = t_{495} + t_{536},
t_{1668} = t_{686} + t_{1296},
t_{777} = t_{433} + t_{495},
t_{1358} = t_{624} + t_{777},
t_{1934} = t_{692} + t_{1358},
t_{1936} = t_{559} + t_{1934},
t_{1754} = t_{1358} + t_{1752},
t_{823} = t_{446} + t_{777},
t_{1949} = t_{823} + t_{1462},
t_{1956} = t_{590} + t_{1949},
t_{1962} = t_{1956} + t_{1957},
t_{1829} = t_{823} + t_{1828},
t_{1831} = t_{569} + t_{1829},
t_{1544} = t_{823} + t_{896},
t_{568} = t_{410} + t_{495},
t_{1006} = t_{490} + t_{568},
t_{1727} = t_{1006} + t_{1703},
t_{1253} = t_{620} + t_{1006},
t_{834} = t_{492} + t_{568},
t_{2393} = t_{579} + t_{834},
t_{2402} = t_{2393} + t_{2401},
t_{2408} = r'_{104} + t_{2402},
t_{2415} = t_{2408} + t_{2410},
t_{1077} = t_{629} + t_{834},
t_{1396} = t_{466} + t_{1077},
t_{1728} = t_{1396} + t_{1590},
t_{380} = r'_{85} + r'_{89},
t_{1012} = t_{380} + t_{925},
t_{1786} = t_{1012} + t_{1785},
t_{1792} = t_{1786} + t_{1790},
t_{1793} = t_{1100} + t_{1792},
t_{1795} = t_{1343} + t_{1793},
t_{1799} = t_{1788} + t_{1795},
t_{1800} = t_{1330} + t_{1799},
t_{1803} = t_{1783} + t_{1800},
t_{1804} = t_{781} + t_{1803},
t_{1805} = t_{1796} + t_{1804},
t_{1806} = t_{1802} + t_{1805},
t_{690} = t_{380} + t_{405},
t_{1657} = r'_{36} + t_{690},
t_{1854} = t_{1657} + t_{1851},
t_{1855} = t_{592} + t_{1854},
t_{1858} = t_{1054} + t_{1855},
t_{1861} = t_{1857} + t_{1858},
t_{1864} = t_{1853} + t_{1861},
t_{1866} = t_{1088} + t_{1864},
t_{1867} = t_{1859} + t_{1866},
t_{1407} = t_{586} + t_{690},
t_{1173} = t_{690} + t_{699},
t_{1126} = t_{690} + t_{840},
t_{829} = t_{513} + t_{690},
t_{1696} = t_{829} + t_{929},
t_{1207} = t_{778} + t_{829},
t_{1748} = t_{636} + t_{1207},
t_{1749} = t_{517} + t_{1748},
t_{501} = t_{380} + t_{387},
t_{1109} = t_{501} + t_{532},
t_{1671} = t_{507} + t_{1109},
t_{1536} = t_{473} + t_{1109},
t_{1932} = t_{1536} + t_{1780},
t_{734} = t_{486} + t_{501},
t_{1435} = t_{734} + t_{1023},
t_{2154} = t_{1088} + t_{1435},
t_{2367} = t_{2154} + t_{2361},
t_{2372} = t_{1413} + t_{2367},
t_{1142} = t_{568} + t_{734},
t_{63} = t_{61} + t_{513} + t_{746} + t_{1142} + t_{1427} + t_{1474} + t_{1511} + t_{1567} + t_{1618},
t_{2429} = t_{63} + t_{2428},
t_{2431} = t_{452} + t_{2429},
t_{2432} = t_{2430} + t_{2431},
t_{2433} = t_{1537} + t_{2432},
t_{23} = t_{596} + t_{600} + t_{668} + t_{741} + t_{760} + t_{778} + t_{844} + t_{916} + t_{966} + t_{1009} + t_{1116} + t_{1428} + t_{1461} + t_{2433},
t_{1751} = t_{23} + t_{1749},
t_{1755} = t_{697} + t_{1751},
t_{1756} = t_{1754} + t_{1755},
t_{1238} = t_{23} + t_{1019},
t_{122} = t_{404} + t_{422} + t_{436} + t_{447} + t_{635} + t_{666} + t_{689} + t_{844} + t_{878} + t_{895} + t_{910} + t_{930} + t_{1182} + t_{1238} + t_{1649} + t_{1670} + t_{1730} + t_{1895},
t_{0} = t_{608} + t_{2433},
t_{1106} = t_{0} + t_{63},
t_{1175} = t_{906} + t_{1106},
t_{626} = t_{0} + t_{23},
t_{744} = t_{608} + t_{626},
t_{54} = t_{381} + t_{504} + t_{624} + t_{744} + t_{761} + t_{849} + t_{899} + t_{1012} + t_{1029} + t_{1228} + t_{1335} + t_{1515} + t_{1592} + t_{1631} + t_{1670},
t_{2032} = t_{479} + t_{744},
t_{2033} = t_{2031} + t_{2032},
t_{2041} = t_{2033} + t_{2040},
t_{5} = t_{1143} + t_{2041},
t_{1594} = t_{5} + t_{946},
t_{1691} = t_{1436} + t_{1594},
t_{1138} = t_{400} + t_{744},
t_{53} = t_{449} + t_{468} + t_{487} + t_{518} + t_{530} + t_{607} + t_{668} + t_{699} + t_{763} + t_{902} + t_{921} + t_{959} + t_{1019} + t_{1138} + t_{1146} + t_{1258} + t_{1473} + t_{1631},
t_{1208} = t_{53} + t_{686},
t_{1665} = t_{1035} + t_{1208},
t_{8} = t_{390} + t_{408} + t_{415} + t_{427} + t_{484} + t_{555} + t_{583} + t_{690} + t_{818} + t_{941} + t_{1138} + t_{1397} + t_{1649} + t_{1686} + t_{1687},
t_{1603} = t_{63} + t_{1138},
t_{99} = t_{434} + t_{435} + t_{464} + t_{476} + t_{479} + t_{507} + t_{525} + t_{620} + t_{739} + t_{1393} + t_{1603} + t_{1619} + t_{1730},
t_{1960} = t_{99} + t_{743},
t_{1963} = t_{1960} + t_{1962},
t_{1965} = t_{1963} + t_{1964},
t_{1966} = t_{1961} + t_{1965},
t_{1967} = t_{1505} + t_{1966},
t_{55} = t_{1327} + t_{1967},
t_{1314} = t_{55} + t_{555},
t_{1504} = t_{1018} + t_{1314},
t_{91} = t_{517} + t_{530} + t_{541} + t_{600} + t_{628} + t_{838} + t_{883} + t_{1012} + t_{1196} + t_{1452} + t_{1465} + t_{1504} + t_{1992},
t_{1202} = r'_{247} + t_{55},
t_{1823} = t_{628} + t_{1202},
t_{1830} = t_{1531} + t_{1823},
t_{1832} = t_{1830} + t_{1831},
t_{1833} = t_{1473} + t_{1832},
t_{2310} = t_{394} + t_{1967},
t_{1483} = t_{99} + t_{445},
t_{30} = t_{53} + t_{380} + t_{426} + t_{474} + t_{652} + t_{915} + t_{963} + t_{1209} + t_{1366} + t_{1397} + t_{1483} + t_{1603} + t_{1988} + t_{2194},
t_{1810} = t_{30} + t_{1642},
t_{1812} = t_{1705} + t_{1810},
t_{1813} = t_{99} + t_{1812},
t_{1815} = t_{1238} + t_{1813},
t_{1816} = t_{1753} + t_{1815},
t_{1817} = t_{498} + t_{1816},
t_{1819} = t_{896} + t_{1817},
t_{1757} = t_{1483} + t_{1756},
t_{24} = t_{419} + t_{719} + t_{1757},
t_{1809} = t_{1757} + t_{1807},
t_{1820} = t_{1809} + t_{1819},
t_{1821} = t_{1818} + t_{1820},
t_{113} = t_{960} + t_{1821},
t_{32} = t_{575} + t_{662} + t_{790} + t_{840} + t_{1061} + t_{1064} + t_{1258} + t_{1696} + t_{1821} + t_{1949} + t_{2169} + t_{2187},
t_{605} = t_{61} + t_{63},
t_{1016} = t_{605} + t_{741},
t_{873} = t_{578} + t_{605},
t_{1367} = t_{873} + t_{916},
t_{2108} = t_{446} + t_{1367},
t_{1279} = t_{620} + t_{873},
t_{1261} = t_{394} + t_{1142},
t_{1896} = t_{531} + t_{1261},
t_{1902} = t_{556} + t_{1896},
t_{685} = t_{501} + t_{516},
t_{1180} = t_{551} + t_{685},
t_{1271} = r'_{30} + t_{1180},
t_{1151} = t_{526} + t_{685},
t_{816} = t_{631} + t_{685},
t_{1262} = t_{816} + t_{928},
t_{121} = t_{113} + t_{382} + t_{480} + t_{500} + t_{543} + t_{782} + t_{975} + t_{1111} + t_{1157} + t_{1262} + t_{1447} + t_{1714},
t_{653} = t_{113} + t_{121},
t_{1289} = t_{382} + t_{653},
t_{2044} = t_{877} + t_{1289},
t_{879} = t_{531} + t_{653},
t_{92} = t_{32} + t_{456} + t_{525} + t_{526} + t_{734} + t_{766} + t_{821} + t_{879} + t_{992} + t_{1086} + t_{1302} + t_{1554} + t_{2385},
t_{1552} = t_{92} + t_{121},
t_{66} = t_{738} + t_{1552} + t_{2204},
t_{554} = t_{32} + t_{92},
t_{108} = t_{489} + t_{554} + t_{1955},
t_{2400} = t_{108} + t_{680},
t_{1679} = t_{108} + t_{538},
t_{1615} = t_{554} + t_{1151},
t_{2382} = t_{1615} + t_{1696},
t_{2384} = t_{1004} + t_{2382},
t_{2386} = t_{1413} + t_{2384},
t_{2308} = t_{1615} + t_{2306},
t_{2312} = t_{2308} + t_{2310},
t_{2313} = t_{2309} + t_{2312},
t_{765} = t_{546} + t_{554},
t_{114} = t_{122} + t_{513} + t_{532} + t_{685} + t_{765} + t_{928} + t_{1339} + t_{1628} + t_{1683},
t_{617} = t_{114} + t_{122},
t_{2049} = t_{523} + t_{617},
t_{2307} = t_{692} + t_{2049},
t_{726} = t_{557} + t_{617},
t_{2422} = t_{765} + t_{2421},
t_{2423} = t_{553} + t_{2422},
t_{2424} = t_{2420} + t_{2423},
t_{1135} = t_{765} + t_{868},
t_{1203} = t_{521} + t_{1135},
t_{6} = t_{438} + t_{498} + t_{518} + t_{691} + t_{796} + t_{900} + t_{1126} + t_{1203} + t_{1269} + t_{1293} + t_{1328} + t_{1453} + t_{1554} + t_{1665} + t_{1679},
t_{1476} = t_{6} + t_{1030},
t_{72} = r'_{180} + t_{1476} + t_{1893},
t_{1745} = t_{1203} + t_{1743},
t_{1734} = t_{410} + t_{879},
t_{1739} = t_{1411} + t_{1734},
t_{1741} = t_{542} + t_{1739},
t_{1363} = t_{879} + t_{1016},
t_{2383} = t_{1363} + t_{1609},
t_{2387} = t_{2383} + t_{2386},
t_{2390} = t_{2387} + t_{2388},
t_{2391} = t_{2389} + t_{2390},
t_{97} = t_{390} + t_{2391},
t_{969} = t_{97} + t_{746},
t_{56} = t_{385} + t_{486} + t_{536} + t_{590} + t_{653} + t_{969} + t_{1046} + t_{1099} + t_{1141} + t_{1181} + t_{1279} + t_{1288} + t_{1678} + t_{1679} + t_{1686} + t_{2041},
t_{1509} = t_{56} + t_{649},
t_{1613} = t_{969} + t_{1106},
t_{22} = t_{465} + t_{520} + t_{560} + t_{608} + t_{1029} + t_{1474} + t_{1613} + t_{1619},
t_{1744} = t_{22} + t_{1741},
t_{1154} = t_{22} + t_{97},
t_{64} = t_{466} + t_{697} + t_{790} + t_{822} + t_{838} + t_{918} + t_{939} + t_{1083} + t_{1141} + t_{1154} + t_{1285} + t_{1312} + t_{1954},
t_{1466} = t_{64} + t_{1254},
t_{1448} = t_{64} + t_{1407},
t_{1740} = t_{467} + t_{1448},
t_{1742} = t_{1738} + t_{1740},
t_{1746} = t_{1742} + t_{1745},
t_{1747} = t_{1744} + t_{1746},
t_{100} = t_{605} + t_{1747},
t_{1442} = t_{451} + t_{1154},
t_{2212} = t_{1442} + t_{1543},
t_{2295} = t_{1262} + t_{1427},
t_{2297} = t_{1017} + t_{2295},
t_{1136} = t_{816} + t_{942},
t_{2206} = t_{1027} + t_{1136},
t_{2209} = t_{1535} + t_{2206},
t_{1431} = t_{484} + t_{1136},
t_{2418} = t_{838} + t_{1431},
t_{2419} = t_{1509} + t_{2418},
t_{2425} = t_{2419} + t_{2424},
t_{2426} = t_{537} + t_{2425},
t_{74} = t_{1625} + t_{2426},
t_{880} = t_{74} + t_{726},
t_{2046} = t_{880} + t_{1196},
t_{1091} = t_{554} + t_{880},
t_{98} = t_{64} + t_{389} + t_{415} + t_{426} + t_{482} + t_{752} + t_{917} + t_{921} + t_{975} + t_{1091} + t_{1264} + t_{1327} + t_{1335} + t_{1428} + t_{1509} + t_{2389},
t_{1931} = t_{98} + t_{1215},
t_{1939} = t_{1931} + t_{1936},
t_{1940} = t_{1935} + t_{1939},
t_{1937} = t_{1091} + t_{1930},
t_{1938} = t_{1933} + t_{1937},
t_{1941} = t_{1938} + t_{1940},
t_{1942} = t_{1932} + t_{1941},
t_{1943} = t_{1579} + t_{1942},
t_{1944} = t_{1466} + t_{1943},
t_{89} = t_{488} + t_{1944},
t_{1048} = t_{89} + t_{779},
t_{833} = t_{56} + t_{89},
t_{37} = t_{419} + t_{427} + t_{471} + t_{491} + t_{533} + t_{540} + t_{833} + t_{835} + t_{882} + t_{921} + t_{956} + t_{997} + t_{1009} + t_{1018} + t_{1453} + t_{1515} + t_{1518},
t_{29} = t_{747} + t_{833} + t_{991} + t_{1045} + t_{1465} + t_{1510} + t_{1606} + t_{2426},
t_{1499} = t_{29} + t_{833},
t_{76} = t_{54} + t_{518} + t_{531} + t_{551} + t_{928} + t_{979} + t_{1064} + t_{1091} + t_{1132} + t_{1413} + t_{1499},
t_{749} = t_{54} + t_{76},
t_{1834} = t_{607} + t_{749},
t_{1836} = t_{1135} + t_{1834},
t_{1842} = t_{91} + t_{1836},
t_{1844} = t_{1840} + t_{1842},
t_{1845} = t_{726} + t_{1844},
t_{1846} = t_{1841} + t_{1845},
t_{1847} = t_{1843} + t_{1846},
t_{1848} = t_{1504} + t_{1847},
t_{39} = t_{1205} + t_{1848},
t_{1589} = t_{640} + t_{749},
t_{90} = t_{30} + t_{526} + t_{680} + t_{806} + t_{995} + t_{1431} + t_{1589} + t_{1624},
t_{757} = t_{30} + t_{90},
t_{2249} = t_{717} + t_{757},
t_{106} = t_{477} + t_{629} + t_{729} + t_{736} + t_{790} + t_{1034} + t_{1254} + t_{1531} + t_{2249},
t_{1545} = r'_{247} + t_{106},
t_{1688} = r'_{79} + t_{1545},
t_{12} = r'_{37} + r'_{171} + r'_{226} + t_{529} + t_{705} + t_{834} + t_{842} + t_{888} + t_{913} + t_{923} + t_{1081} + t_{1209} + t_{1215} + t_{1336} + t_{1385} + t_{1450} + t_{1566} + t_{1638} + t_{1652} + t_{1685} + t_{1688},
t_{1593} = t_{12} + t_{537},
t_{102} = r'_{118} + r'_{140} + r'_{157} + r'_{187} + r'_{241} + t_{411} + t_{568} + t_{907} + t_{967} + t_{1103} + t_{1139} + t_{1274} + t_{1310} + t_{1332} + t_{1395} + t_{1398} + t_{1403} + t_{1412} + t_{1492} + t_{1516} + t_{1533} + t_{1535} + t_{1581} + t_{1593} + t_{1688} + t_{1720},
t_{1322} = t_{106} + t_{956},
t_{1723} = r'_{20} + t_{1322},
t_{2245} = t_{1723} + t_{2244},
t_{2246} = t_{2241} + t_{2245},
t_{2255} = t_{2243} + t_{2246},
t_{2250} = t_{2248} + t_{2249},
t_{2257} = t_{2250} + t_{2256},
t_{2259} = t_{2257} + t_{2258},
t_{2260} = r'_{148} + t_{2259},
t_{2262} = t_{2255} + t_{2260},
t_{2263} = t_{1233} + t_{2262},
t_{2311} = t_{1589} + t_{2307},
t_{2314} = t_{2311} + t_{2313},
t_{75} = t_{494} + t_{2314},
t_{641} = t_{55} + t_{75},
t_{86} = r'_{26} + r'_{28} + r'_{67} + r'_{71} + r'_{91} + r'_{116} + r'_{120} + r'_{132} + r'_{183} + r'_{229} + t_{102} + t_{434} + t_{641} + t_{668} + t_{768} + t_{791} + t_{806} + t_{1007} + t_{1043} + t_{1096} + t_{1111} + t_{1260} + t_{1282} + t_{1547} + t_{1575} + t_{1611} + t_{1620},
t_{1113} = t_{557} + t_{641},
t_{31} = t_{91} + t_{700} + t_{734} + t_{1113} + t_{1511} + t_{1624} + t_{1628} + t_{1666},
t_{661} = t_{31} + t_{91},
t_{107} = t_{471} + t_{497} + t_{504} + t_{565} + t_{610} + t_{661} + t_{746} + t_{883} + t_{1174} + t_{1241} + t_{1692},
t_{2369} = r'_{37} + t_{107},
t_{2373} = t_{2369} + t_{2370},
t_{2374} = t_{2372} + t_{2373},
t_{1894} = t_{107} + t_{540},
t_{1245} = t_{107} + t_{981},
t_{2006} = t_{1245} + t_{2005},
t_{2012} = r'_{105} + t_{2006},
t_{2014} = t_{2012} + t_{2013},
t_{2015} = t_{2011} + t_{2014},
t_{78} = t_{1728} + t_{1885} + t_{2015},
t_{1160} = r'_{42} + t_{78},
t_{1616} = t_{1160} + t_{1372},
t_{1706} = r'_{191} + t_{1616},
t_{1306} = r'_{34} + t_{1245},
t_{80} = r'_{30} + r'_{139} + t_{407} + t_{432} + t_{442} + t_{464} + t_{607} + t_{663} + t_{724} + t_{742} + t_{808} + t_{809} + t_{923} + t_{938} + t_{1096} + t_{1171} + t_{1200} + t_{1306} + t_{1321} + t_{1327} + t_{1523} + t_{1607} + t_{1653} + t_{1720},
t_{2186} = t_{1306} + t_{2184},
t_{2188} = t_{2179} + t_{2186},
t_{898} = t_{107} + t_{551},
t_{1364} = r'_{182} + t_{898},
t_{1643} = t_{514} + t_{1364},
t_{36} = t_{901} + t_{1435} + t_{1643} + t_{1706} + t_{2167},
t_{2164} = t_{1501} + t_{1643},
t_{2375} = t_{2164} + t_{2374},
t_{2377} = t_{1047} + t_{2375},
t_{750} = t_{107} + t_{473},
t_{2185} = t_{750} + t_{2176},
t_{2190} = t_{2185} + t_{2188},
t_{1325} = t_{414} + t_{750},
t_{123} = t_{99} + t_{439} + t_{446} + t_{522} + t_{640} + t_{809} + t_{930} + t_{952} + t_{1173} + t_{1174} + t_{1178} + t_{1232} + t_{1293} + t_{1325} + t_{1339} + t_{1694} + t_{2314},
t_{1093} = t_{64} + t_{750},
t_{2205} = t_{1093} + t_{1822},
t_{1463} = t_{486} + t_{1093},
t_{904} = t_{641} + t_{661},
t_{1549} = t_{904} + t_{1463},
t_{62} = r'_{247} + t_{649} + t_{711} + t_{725} + t_{829} + t_{904} + t_{992} + t_{1090} + t_{1632} + t_{1666} + t_{1684} + t_{1747} + t_{1841} + t_{2391},
t_{82} = r'_{108} + r'_{121} + r'_{135} + r'_{170} + r'_{239} + t_{62} + t_{571} + t_{667} + t_{764} + t_{1019} + t_{1032} + t_{1200} + t_{1227} + t_{1433} + t_{1522} + t_{1718} + t_{1998} + t_{2015} + t_{2164},
t_{562} = t_{62} + t_{82},
t_{71} = t_{562} + t_{2148},
t_{1897} = t_{562} + t_{1894},
t_{21} = t_{62} + t_{521} + t_{584} + t_{773} + t_{801} + t_{900} + t_{939} + t_{1016} + t_{1461} + t_{1500},
t_{1429} = t_{21} + t_{883},
t_{1712} = t_{62} + t_{1429},
t_{2109} = t_{1712} + t_{2108},
t_{2110} = t_{2107} + t_{2109},
t_{2114} = t_{2110} + t_{2112},
t_{2115} = t_{2113} + t_{2114},
t_{40} = t_{108} + t_{391} + t_{433} + t_{569} + t_{596} + t_{631} + t_{684} + t_{734} + t_{746} + t_{827} + t_{868} + t_{896} + t_{929} + t_{1104} + t_{1182} + t_{1557} + t_{1722} + t_{2115},
t_{876} = r'_{20} + t_{21},
t_{1639} = t_{841} + t_{876},
t_{96} = t_{1639} + t_{1779},
t_{1075} = t_{96} + t_{643},
t_{18} = r'_{102} + r'_{119} + r'_{121} + r'_{194} + r'_{195} + r'_{228} + r'_{234} + t_{21} + t_{466} + t_{591} + t_{721} + t_{727} + t_{776} + t_{785} + t_{795} + t_{902} + t_{954} + t_{1052} + t_{1053} + t_{1075} + t_{1118} + t_{1133} + t_{1286} + t_{1480} + t_{1494} + t_{1556} + t_{1573} + t_{1634} + t_{1729},
t_{1224} = t_{973} + t_{1075},
t_{971} = t_{860} + t_{876},
t_{1417} = t_{470} + t_{971},
t_{59} = r'_{162} + t_{501} + t_{649} + t_{794} + t_{808} + t_{893} + t_{895} + t_{931} + t_{965} + t_{1224} + t_{1252} + t_{1286} + t_{1392} + t_{1417} + t_{1529},
t_{20} = r'_{15} + r'_{98} + r'_{101} + r'_{225} + t_{59} + t_{441} + t_{520} + t_{606} + t_{657} + t_{665} + t_{699} + t_{722} + t_{889} + t_{892} + t_{899} + t_{1122} + t_{1331} + t_{1449} + t_{1471} + t_{1491} + t_{1566} + t_{1601} + t_{1647} + t_{1672} + t_{1983},
t_{1700} = t_{59} + t_{517},
t_{1058} = t_{59} + t_{808},
t_{1292} = t_{391} + t_{1058},
t_{2396} = t_{933} + t_{1292},
t_{2407} = t_{2396} + t_{2404},
t_{2409} = t_{1037} + t_{2407},
t_{2411} = t_{1523} + t_{2409},
t_{2413} = t_{2400} + t_{2411},
t_{2414} = t_{2405} + t_{2413},
t_{2416} = t_{2414} + t_{2415},
t_{2417} = t_{2403} + t_{2416},
t_{10} = t_{650} + t_{772} + t_{2417},
t_{2221} = r'_{32} + t_{10},
t_{2223} = t_{1587} + t_{2221},
t_{2232} = t_{1591} + t_{2223},
t_{1185} = t_{10} + t_{627},
t_{1708} = t_{564} + t_{1185},
t_{2017} = t_{632} + t_{1708},
t_{2025} = t_{1565} + t_{2017},
t_{2026} = t_{2020} + t_{2025},
t_{2029} = t_{2026} + t_{2028},
t_{77} = t_{1539} + t_{2029},
t_{2057} = t_{565} + t_{2029},
t_{2059} = t_{891} + t_{2057},
t_{2064} = t_{2056} + t_{2059},
t_{2068} = t_{2064} + t_{2067},
t_{2069} = t_{1108} + t_{2068},
t_{2070} = t_{2060} + t_{2069},
t_{103} = t_{633} + t_{2070},
t_{87} = r'_{1} + r'_{29} + r'_{112} + r'_{131} + t_{536} + t_{587} + t_{1150} + t_{1152} + t_{1159} + t_{1179} + t_{1229} + t_{1289} + t_{1351} + t_{1423} + t_{1432} + t_{1564} + t_{1568} + t_{1604} + t_{1611} + t_{1656} + t_{2070},
t_{1375} = t_{381} + t_{1292},
t_{120} = r'_{0} + r'_{10} + r'_{74} + r'_{109} + r'_{112} + r'_{149} + r'_{212} + r'_{236} + t_{86} + t_{543} + t_{577} + t_{614} + t_{625} + t_{688} + t_{807} + t_{846} + t_{1010} + t_{1045} + t_{1049} + t_{1126} + t_{1252} + t_{1308} + t_{1348} + t_{1375} + t_{1434} + t_{1454} + t_{1479} + t_{1491} + t_{1779},
t_{789} = t_{86} + t_{120},
t_{1710} = t_{658} + t_{789},
t_{2315} = t_{1355} + t_{1710},
t_{2317} = t_{1099} + t_{2315},
t_{2319} = t_{976} + t_{2317},
t_{2330} = t_{547} + t_{2319},
t_{2333} = t_{2328} + t_{2330},
t_{2334} = t_{2332} + t_{2333},
t_{85} = r'_{4} + r'_{44} + r'_{252} + t_{391} + t_{483} + t_{655} + t_{788} + t_{797} + t_{821} + t_{836} + t_{892} + t_{1144} + t_{1190} + t_{1257} + t_{1381} + t_{1385} + t_{1421} + t_{1438} + t_{1440} + t_{1477} + t_{1496} + t_{1532} + t_{1544} + t_{1583} + t_{1659} + t_{1693} + t_{1780} + t_{1806} + t_{2334} + t_{2412},
t_{101} = r'_{31} + r'_{68} + r'_{128} + r'_{210} + t_{85} + t_{380} + t_{556} + t_{591} + t_{593} + t_{598} + t_{709} + t_{742} + t_{749} + t_{757} + t_{920} + t_{1046} + t_{1186} + t_{1223} + t_{1235} + t_{1275} + t_{1359} + t_{1402} + t_{1447} + t_{1512} + t_{1525} + t_{1571} + t_{1621} + t_{1690},
t_{1163} = t_{651} + t_{789},
t_{1451} = r'_{190} + t_{1163},
t_{67} = r'_{205} + t_{1451} + t_{1626} + t_{2190},
t_{104} = r'_{38} + r'_{117} + r'_{133} + r'_{158} + t_{67} + t_{536} + t_{557} + t_{585} + t_{646} + t_{696} + t_{886} + t_{915} + t_{1029} + t_{1100} + t_{1251} + t_{1395} + t_{1464} + t_{1578} + t_{1586} + t_{1659} + t_{1708} + t_{1710} + t_{1986} + t_{1992},
t_{1541} = t_{104} + t_{1059},
t_{1651} = r'_{148} + t_{1541},
t_{2214} = r'_{125} + t_{1651},
t_{2229} = t_{2214} + t_{2227},
t_{2230} = t_{1657} + t_{2229},
t_{2231} = t_{2219} + t_{2230},
t_{2235} = t_{2231} + t_{2234},
t_{2236} = t_{2232} + t_{2235},
t_{2237} = t_{2225} + t_{2236},
t_{2238} = t_{2222} + t_{2237},
t_{2} = t_{92} + t_{1635} + t_{2238},
t_{1542} = t_{62} + t_{386},
t_{2267} = t_{555} + t_{1542},
t_{2277} = t_{1732} + t_{2267},
t_{2285} = t_{2277} + t_{2283},
t_{2288} = t_{1725} + t_{2285},
t_{2289} = t_{743} + t_{2288},
t_{1074} = t_{62} + t_{100},
t_{95} = r'_{146} + r'_{152} + r'_{196} + r'_{206} + t_{472} + t_{605} + t_{654} + t_{683} + t_{727} + t_{920} + t_{967} + t_{1022} + t_{1026} + t_{1036} + t_{1074} + t_{1092} + t_{1323} + t_{1346} + t_{1416} + t_{1430} + t_{1457} + t_{1613} + t_{1655} + t_{1698} + t_{1724} + t_{2082} + t_{2261},
t_{1527} = t_{95} + t_{1175},
t_{69} = t_{1527} + t_{2263},
t_{1856} = t_{1074} + t_{1259},
t_{1868} = t_{1856} + t_{1865},
t_{1869} = t_{1867} + t_{1868},
t_{17} = t_{1700} + t_{1869},
t_{3} = r'_{115} + r'_{128} + r'_{144} + r'_{146} + r'_{150} + r'_{173} + r'_{204} + r'_{211} + t_{491} + t_{504} + t_{745} + t_{759} + t_{786} + t_{827} + t_{894} + t_{972} + t_{1044} + t_{1048} + t_{1053} + t_{1085} + t_{1193} + t_{1261} + t_{1277} + t_{1279} + t_{1313} + t_{1409} + t_{1574} + t_{1577} + t_{1584} + t_{1599} + t_{1617} + t_{1682} + t_{1869},
t_{996} = r'_{212} + t_{62},
t_{57} = r'_{48} + r'_{113} + r'_{232} + t_{386} + t_{545} + t_{602} + t_{758} + t_{865} + t_{907} + t_{971} + t_{988} + t_{996} + t_{1133} + t_{1158} + t_{1223} + t_{1224} + t_{1227} + t_{1375} + t_{1401} + t_{1468} + t_{1498} + t_{1715},
t_{1633} = t_{667} + t_{996},
t_{1645} = t_{57} + t_{1633},
t_{119} = r'_{3} + r'_{76} + r'_{79} + r'_{91} + r'_{134} + r'_{173} + r'_{208} + r'_{248} + t_{555} + t_{739} + t_{935} + t_{1011} + t_{1077} + t_{1148} + t_{1152} + t_{1235} + t_{1324} + t_{1338} + t_{1384} + t_{1478} + t_{1602} + t_{1645} + t_{1655} + t_{1667} + t_{1697} + t_{1713} + t_{2217} + t_{2238},
t_{19} = r'_{9} + r'_{253} + t_{558} + t_{825} + t_{936} + t_{1313} + t_{1355} + t_{1366} + t_{1420} + t_{1475} + t_{1479} + t_{1495} + t_{1565} + t_{1618} + t_{1645} + t_{1656} + t_{1658} + t_{1672} + t_{1693},
t_{885} = t_{19} + t_{57},
t_{93} = r'_{6} + r'_{45} + r'_{174} + t_{485} + t_{488} + t_{560} + t_{857} + t_{885} + t_{906} + t_{945} + t_{1014} + t_{1022} + t_{1117} + t_{1456} + t_{1471} + t_{1532} + t_{1542} + t_{1609} + t_{1617} + t_{1641} + t_{1677} + t_{1719},
t_{1391} = r'_{1} + t_{885},
t_{1623} = t_{1066} + t_{1391},
t_{94} = r'_{52} + r'_{59} + r'_{138} + r'_{183} + r'_{190} + r'_{194} + r'_{241} + r'_{250} + t_{383} + t_{401} + t_{408} + t_{453} + t_{502} + t_{598} + t_{693} + t_{817} + t_{896} + t_{948} + t_{951} + t_{964} + t_{1010} + t_{1016} + t_{1150} + t_{1168} + t_{1309} + t_{1417} + t_{1584} + t_{1623} + t_{1640} + t_{1702},
t_{2282} = t_{1033} + t_{1623},
t_{2287} = t_{2282} + t_{2286},
t_{2290} = t_{2287} + t_{2289},
t_{2292} = t_{2284} + t_{2290},
t_{2293} = t_{2276} + t_{2292},
t_{2294} = t_{2291} + t_{2293},
t_{26} = r'_{32} + r'_{181} + r'_{233} + t_{392} + t_{428} + t_{449} + t_{512} + t_{549} + t_{560} + t_{657} + t_{670} + t_{786} + t_{825} + t_{874} + t_{974} + t_{1009} + t_{1101} + t_{1173} + t_{1231} + t_{1272} + t_{1371} + t_{1403} + t_{1404} + t_{1493} + t_{1540} + t_{1614} + t_{1691} + t_{2270} + t_{2294},
t_{112} = t_{26} + t_{2334},
t_{1353} = t_{976} + t_{1113},
t_{88} = r'_{96} + r'_{181} + t_{453} + t_{671} + t_{681} + t_{1177} + t_{1353} + t_{1452} + t_{1469} + t_{1489} + t_{1503} + t_{1553} + t_{1604} + t_{1620} + t_{1630} + t_{1651} + t_{1705},
t_{110} = r'_{58} + r'_{70} + r'_{85} + r'_{164} + r'_{253} + t_{88} + t_{519} + t_{520} + t_{773} + t_{812} + t_{878} + t_{968} + t_{1060} + t_{1070} + t_{1172} + t_{1377} + t_{1405} + t_{1481} + t_{1502} + t_{1704} + t_{1724} + t_{1725} + t_{1954} + t_{1955} + t_{2189} + t_{2190} + t_{2412} + t_{2417},
t_{25} = r'_{101} + r'_{106} + r'_{187} + r'_{231} + t_{382} + t_{506} + t_{597} + t_{689} + t_{693} + t_{777} + t_{789} + t_{898} + t_{925} + t_{987} + t_{1005} + t_{1020} + t_{1082} + t_{1190} + t_{1233} + t_{1288} + t_{1349} + t_{1353} + t_{1393} + t_{1394} + t_{1478} + t_{1563} + t_{1574} + t_{1575} + t_{1636} + t_{2148} + t_{2208} + t_{2270},
t_{1439} = t_{540} + t_{749},
t_{2042} = t_{422} + t_{1439},
t_{115} = t_{123} + t_{413} + t_{435} + t_{661} + t_{782} + t_{822} + t_{917} + t_{1074} + t_{1363} + t_{1367} + t_{1536} + t_{1684} + t_{2042} + t_{2049},
t_{2045} = t_{2042} + t_{2044},
t_{2047} = t_{2045} + t_{2046},
t_{2050} = t_{534} + t_{2047},
t_{1319} = t_{29} + t_{878},
t_{2118} = t_{1692} + t_{1944},
t_{2119} = t_{2117} + t_{2118},
t_{2120} = t_{1237} + t_{2119},
t_{2121} = t_{1694} + t_{2120},
t_{105} = t_{1319} + t_{2121},
t_{1025} = t_{105} + t_{562},
t_{16} = t_{385} + t_{394} + t_{470} + t_{531} + t_{840} + t_{959} + t_{1025} + t_{1157} + t_{1174} + t_{1537} + t_{1650},
t_{989} = t_{16} + t_{496},
t_{1217} = t_{544} + t_{989},
t_{1985} = t_{1217} + t_{1984},
t_{1991} = t_{1985} + t_{1987},
t_{1993} = t_{1990} + t_{1991},
t_{83} = t_{451} + t_{493} + t_{512} + t_{527} + t_{989} + t_{1253} + t_{1287} + t_{1398} + t_{1567} + t_{1848} + t_{1988} + t_{1993},
t_{1899} = t_{1099} + t_{1217},
t_{1900} = t_{904} + t_{1899},
t_{1901} = t_{1897} + t_{1900},
t_{1903} = t_{1901} + t_{1902},
t_{1904} = t_{507} + t_{1903},
t_{1905} = t_{1898} + t_{1904},
t_{14} = t_{478} + t_{1513} + t_{1905},
t_{1408} = t_{14} + t_{629},
t_{84} = t_{16} + t_{403} + t_{458} + t_{507} + t_{689} + t_{775} + t_{784} + t_{839} + t_{855} + t_{979} + t_{1408} + t_{1549} + t_{1717} + t_{2187},
t_{51} = r'_{132} + r'_{172} + r'_{220} + t_{84} + t_{456} + t_{470} + t_{580} + t_{600} + t_{601} + t_{671} + t_{674} + t_{860} + t_{937} + t_{957} + t_{964} + t_{1195} + t_{1206} + t_{1274} + t_{1346} + t_{1434} + t_{1448} + t_{1456} + t_{1580} + t_{1586} + t_{1621} + t_{1631} + t_{1648} + t_{1692} + t_{1704} + t_{2154},
t_{805} = t_{84} + t_{562},
t_{1490} = t_{805} + t_{1156},
t_{1166} = t_{405} + t_{805},
t_{2299} = t_{1166} + t_{2296},
t_{2300} = t_{561} + t_{2299},
t_{2301} = t_{2298} + t_{2300},
t_{2302} = t_{796} + t_{2301},
t_{2303} = t_{1287} + t_{2302},
t_{2304} = t_{402} + t_{2303},
t_{2305} = t_{2297} + t_{2304},
t_{38} = t_{487} + t_{2305},
t_{1506} = t_{38} + t_{1490},
t_{7} = t_{1506} + t_{1833},
t_{68} = t_{7} + t_{584} + t_{717} + t_{870} + t_{894} + t_{896} + t_{928} + t_{1002} + t_{1201} + t_{1202} + t_{1285} + t_{1288} + t_{1337} + t_{1398} + t_{1447} + t_{1449} + t_{1510} + t_{1612} + t_{1627} + t_{2296},
t_{2207} = t_{1166} + t_{2205},
t_{2211} = t_{2207} + t_{2209},
t_{2213} = t_{2210} + t_{2211},
t_{47} = t_{29} + t_{89} + t_{105} + t_{428} + t_{438} + t_{493} + t_{501} + t_{1015} + t_{1155} + t_{1460} + t_{1505} + t_{1513} + t_{1592} + t_{1722} + t_{2208} + t_{2212} + t_{2213},
t_{15} = t_{575} + t_{1543} + t_{1555} + t_{2213},
t_{81} = t_{15} + t_{824} + t_{878} + t_{898} + t_{975} + t_{991} + t_{1253} + t_{1467} + t_{1717} + t_{2305},
t_{11} = r'_{0} + r'_{86} + r'_{200} + t_{81} + t_{417} + t_{515} + t_{667} + t_{702} + t_{706} + t_{730} + t_{881} + t_{888} + t_{909} + t_{926} + t_{983} + t_{1059} + t_{1071} + t_{1101} + t_{1121} + t_{1139} + t_{1195} + t_{1271} + t_{1376} + t_{1466} + t_{1577} + t_{1727} + t_{1732},
t_{34} = r'_{193} + t_{15} + t_{405} + t_{498} + t_{567} + t_{622} + t_{655} + t_{908} + t_{922} + t_{1039} + t_{1068} + t_{1096} + t_{1129} + t_{1151} + t_{1242} + t_{1280} + t_{1338} + t_{1422} + t_{1488} + t_{1530} + t_{1578} + t_{1658} + t_{2189},
t_{52} = r'_{49} + r'_{98} + r'_{100} + t_{34} + t_{387} + t_{434} + t_{597} + t_{661} + t_{721} + t_{750} + t_{856} + t_{955} + t_{1047} + t_{1162} + t_{1212} + t_{1220} + t_{1273} + t_{1330} + t_{1368} + t_{1370} + t_{1405} + t_{1438} + t_{1464} + t_{1600} + t_{1637} + t_{1731} + t_{1893} + t_{1998},
t_{1907} = t_{52} + t_{1043},
t_{1921} = t_{916} + t_{1907},
t_{1923} = t_{1909} + t_{1921},
t_{1924} = t_{1920} + t_{1923},
t_{1927} = t_{1922} + t_{1924},
t_{1929} = t_{1927} + t_{1928},
t_{70} = r'_{8} + r'_{21} + r'_{40} + r'_{118} + r'_{127} + r'_{172} + t_{525} + t_{675} + t_{687} + t_{752} + t_{866} + t_{977} + t_{1118} + t_{1159} + t_{1240} + t_{1301} + t_{1309} + t_{1563} + t_{1583} + t_{1599} + t_{1698} + t_{1929},
t_{9} = t_{1716} + t_{1929},
t_{1191} = t_{9} + t_{1153},
t_{35} = r'_{16} + r'_{30} + r'_{41} + r'_{86} + r'_{102} + r'_{110} + t_{457} + t_{648} + t_{677} + t_{723} + t_{889} + t_{1100} + t_{1179} + t_{1191} + t_{1425} + t_{1484} + t_{1548} + t_{1622} + t_{1638} + t_{1660} + t_{1671} + t_{1695} + t_{2217},
t_{1538} = t_{35} + t_{1378},
t_{109} = r'_{7} + r'_{152} + r'_{159} + r'_{202} + t_{462} + t_{642} + t_{679} + t_{704} + t_{754} + t_{759} + t_{811} + t_{947} + t_{1083} + t_{1207} + t_{1221} + t_{1230} + t_{1240} + t_{1265} + t_{1275} + t_{1352} + t_{1412} + t_{1423} + t_{1488} + t_{1510} + t_{1538} + t_{1576} + t_{1579} + t_{1664} + t_{1667} + t_{1685},
t_{985} = r'_{69} + t_{35},
t_{44} = r'_{24} + r'_{27} + r'_{72} + r'_{98} + r'_{117} + r'_{176} + r'_{190} + t_{109} + t_{422} + t_{738} + t_{849} + t_{894} + t_{908} + t_{937} + t_{985} + t_{1032} + t_{1033} + t_{1041} + t_{1128} + t_{1137} + t_{1152} + t_{1278} + t_{1342} + t_{1411} + t_{1430} + t_{1544} + t_{1556} + t_{1581} + t_{1636} + t_{1727},
t_{2362} = t_{865} + t_{985},
t_{2365} = t_{2362} + t_{2364},
t_{2366} = t_{627} + t_{2365},
t_{2376} = t_{2167} + t_{2366},
t_{2378} = t_{2376} + t_{2377},
t_{2380} = t_{2378} + t_{2379},
t_{2381} = t_{2371} + t_{2380},
t_{79} = t_{1057} + t_{2381},
t_{1291} = t_{79} + t_{1210},
t_{1709} = t_{402} + t_{1291},
t_{117} = r'_{222} + t_{437} + t_{483} + t_{571} + t_{644} + t_{684} + t_{726} + t_{759} + t_{761} + t_{819} + t_{877} + t_{970} + t_{1023} + t_{1145} + t_{1162} + t_{1178} + t_{1250} + t_{1268} + t_{1305} + t_{1424} + t_{1494} + t_{1553} + t_{1675} + t_{1680} + t_{1699} + t_{1709} + t_{1711},
t_{43} = r'_{47} + r'_{90} + r'_{105} + r'_{123} + r'_{188} + r'_{253} + t_{440} + t_{453} + t_{499} + t_{533} + t_{623} + t_{901} + t_{1062} + t_{1115} + t_{1174} + t_{1225} + t_{1263} + t_{1336} + t_{1425} + t_{1480} + t_{1497} + t_{1538} + t_{1561} + t_{1647} + t_{1690} + t_{1709} + t_{1731} + t_{2263},
t_{42} = r'_{69} + r'_{135} + r'_{141} + r'_{147} + r'_{228} + r'_{243} + t_{384} + t_{492} + t_{559} + t_{595} + t_{740} + t_{757} + t_{832} + t_{866} + t_{894} + t_{924} + t_{938} + t_{974} + t_{1127} + t_{1134} + t_{1189} + t_{1234} + t_{1299} + t_{1304} + t_{1349} + t_{1440} + t_{1454} + t_{1502} + t_{1673} + t_{1675} + t_{1676} + t_{2381},
t_{2077} = t_{853} + t_{985},
t_{2076} = t_{1191} + t_{1669},
t_{2092} = t_{2076} + t_{2086},
t_{2096} = t_{2090} + t_{2092},
t_{2098} = t_{2073} + t_{2096},
t_{2099} = t_{1283} + t_{2098},
t_{2100} = t_{2077} + t_{2099},
t_{2101} = t_{2097} + t_{2100},
t_{2102} = t_{2089} + t_{2101},
t_{41} = r'_{113} + r'_{240} + t_{403} + t_{480} + t_{487} + t_{521} + t_{626} + t_{651} + t_{944} + t_{1011} + t_{1030} + t_{1061} + t_{1129} + t_{1160} + t_{1277} + t_{1304} + t_{1325} + t_{1371} + t_{1415} + t_{1420} + t_{1482} + t_{1520} + t_{1525} + t_{1553} + t_{1665} + t_{1732} + t_{2102},
t_{1} = r'_{111} + t_{2102},
t_{45} = t_{427} + t_{439} + t_{541} + t_{873} + t_{925} + t_{1045} + t_{1046} + t_{1156} + t_{1408} + t_{1468} + t_{1524} + t_{1627} + t_{1634} + t_{1712},
t_{118} = r'_{23} + r'_{68} + r'_{127} + r'_{167} + r'_{217} + r'_{229} + t_{45} + t_{94} + t_{381} + t_{400} + t_{713} + t_{811} + t_{932} + t_{1021} + t_{1037} + t_{1071} + t_{1094} + t_{1188} + t_{1232} + t_{1270} + t_{1370} + t_{1381} + t_{1384} + t_{1389} + t_{1469} + t_{1533} + t_{1546} + t_{1612} + t_{1641} + t_{1711} + t_{1721} + t_{1895} + t_{1905},
t_{1284} = t_{118} + t_{570},
t_{2341} = t_{1284} + t_{1588},
t_{2343} = t_{1648} + t_{2341},
t_{2357} = t_{2343} + t_{2356},
t_{2358} = t_{2353} + t_{2357},
t_{2360} = t_{2358} + t_{2359},
t_{33} = t_{615} + t_{2360},
t_{1294} = r'_{204} + t_{33},
t_{1646} = t_{1294} + t_{1514},
t_{50} = r'_{66} + r'_{82} + r'_{124} + r'_{186} + t_{37} + t_{82} + t_{390} + t_{493} + t_{588} + t_{596} + t_{710} + t_{713} + t_{885} + t_{988} + t_{1003} + t_{1049} + t_{1057} + t_{1067} + t_{1084} + t_{1095} + t_{1177} + t_{1260} + t_{1284} + t_{1368} + t_{1388} + t_{1444} + t_{1477} + t_{1548} + t_{1646},
t_{27} = r'_{46} + r'_{85} + r'_{115} + r'_{138} + r'_{209} + t_{702} + t_{741} + t_{761} + t_{782} + t_{979} + t_{1020} + t_{1032} + t_{1073} + t_{1102} + t_{1148} + t_{1274} + t_{1314} + t_{1374} + t_{1443} + t_{1487} + t_{1562} + t_{1576} + t_{1646} + t_{1652} + t_{1677} + t_{1715} + t_{1833},
t_{28} = r'_{5} + r'_{104} + r'_{110} + r'_{210} + r'_{244} + t_{461} + t_{464} + t_{494} + t_{565} + t_{626} + t_{748} + t_{856} + t_{921} + t_{961} + t_{962} + t_{1033} + t_{1263} + t_{1301} + t_{1308} + t_{1386} + t_{1459} + t_{1460} + t_{1496} + t_{1654} + t_{1689} + t_{1691} + t_{1702} + t_{2360},
t_{1437} = t_{108} + t_{1025},
t_{1701} = t_{440} + t_{1437},
t_{13} = t_{980} + t_{1701} + t_{1993},
t_{48} = t_{13} + t_{2115},
t_{73} = t_{48} + t_{411} + t_{473} + t_{500} + t_{541} + t_{766} + t_{959} + t_{991} + t_{999} + t_{1079} + t_{1282} + t_{1668} + t_{1753} + t_{2212},
t_{1726} = t_{73} + t_{500},
t_{4} = t_{1204} + t_{1726} + t_{2294},
t_{124} = t_{24} + t_{448} + t_{490} + t_{543} + t_{553} + t_{607} + t_{1027} + t_{1061} + t_{1285} + t_{1518} + t_{1596} + t_{1597} + t_{1687} + t_{2121},
t_{111} = r'_{1} + r'_{199} + t_{124} + t_{410} + t_{629} + t_{652} + t_{767} + t_{829} + t_{847} + t_{944} + t_{1048} + t_{1134} + t_{1188} + t_{1206} + t_{1265} + t_{1271} + t_{1394} + t_{1433} + t_{1455} + t_{1543} + t_{1587} + t_{1593} + t_{1605} + t_{1669} + t_{1723} + t_{2152},
t_{943} = t_{124} + t_{494},
t_{2043} = t_{943} + t_{1499},
t_{2048} = t_{1111} + t_{2043},
t_{2051} = t_{2048} + t_{2050},
t_{46} = t_{15} + t_{124} + t_{600} + t_{625} + t_{685} + t_{805} + t_{822} + t_{871} + t_{995} + t_{1208} + t_{1328} + t_{1426} + t_{1549} + t_{1753} + t_{2051},
t_{116} = t_{501} + t_{719} + t_{757} + t_{2051},
t_{65} = r'_{66} + t_{465} + t_{475} + t_{511} + t_{573} + t_{607} + t_{700} + t_{854} + t_{887} + t_{943} + t_{1048} + t_{1126} + t_{1668} + t_{1671} + t_{2194} + t_{2204},
t_{1211} = r'_{215} + t_{943},
t_{1345} = t_{116} + t_{1211},
t_{1629} = t_{27} + t_{1345},
t_{49} = t_{388} + t_{1629} + t_{1806},
p_{186} = t_{119} + t_{121},
p_{185} = t_{120},
p_{184} = t_{121} + t_{123},
p_{183} = t_{124},
p_{182} = t_{117} + t_{119},
p_{181} = t_{118} + t_{120},
p_{180} = t_{123},
p_{179} = t_{122} + t_{124},
p_{178} = t_{109} + t_{113},
p_{177} = t_{110},
p_{176} = t_{113},
p_{175} = t_{114},
p_{174} = t_{109} + t_{111},
p_{173} = t_{110} + t_{112},
p_{172} = t_{115},
p_{171} = t_{114} + t_{116},
p_{170} = t_{103} + t_{105},
p_{169} = t_{104},
p_{168} = t_{104},
p_{167} = t_{105} + t_{107} + t_{108},
p_{166} = t_{108},
p_{164} = t_{102} + t_{104},
p_{165} = t_{101} + t_{103} + p_{164},
p_{162} = t_{106} + t_{108},
p_{163} = t_{107} + p_{162},
p_{161} = t_{93} + t_{97},
p_{160} = t_{94},
p_{159} = t_{97},
p_{158} = t_{98},
p_{157} = t_{93} + t_{95},
p_{156} = t_{94} + t_{96},
p_{155} = t_{99},
p_{154} = t_{98} + t_{100},
p_{153} = t_{85} + t_{89},
p_{152} = t_{86},
p_{151} = t_{86},
p_{150} = t_{89} + t_{90},
p_{149} = t_{90},
p_{147} = t_{86} + t_{88},
p_{148} = t_{85} + t_{87} + p_{147},
p_{145} = t_{90} + t_{92},
p_{146} = t_{91} + p_{145},
p_{144} = t_{79} + t_{81},
p_{143} = t_{80},
p_{142} = t_{80},
p_{141} = t_{81} + t_{83} + t_{84},
p_{140} = t_{84},
p_{138} = t_{78} + t_{80},
p_{139} = t_{77} + t_{79} + p_{138},
p_{136} = t_{82} + t_{84},
p_{137} = t_{83} + p_{136},
p_{135} = t_{69} + t_{73},
p_{134} = t_{70},
p_{133} = t_{70},
p_{132} = t_{73} + t_{74},
p_{131} = t_{74},
p_{129} = t_{70} + t_{72},
p_{130} = t_{69} + t_{71} + p_{129},
p_{127} = t_{74} + t_{76},
p_{128} = t_{75} + p_{127},
p_{126} = t_{65},
p_{125} = t_{68},
p_{124} = t_{66},
p_{123} = t_{66},
p_{122} = t_{67},
p_{121} = t_{57} + t_{61},
p_{120} = t_{58},
p_{119} = t_{58},
p_{118} = t_{61},
p_{117} = t_{61} + t_{62},
p_{116} = t_{62},
p_{115} = t_{57} + t_{59},
p_{113} = t_{58} + t_{60},
p_{114} = p_{113} + p_{115},
p_{112} = t_{63},
p_{110} = t_{62} + t_{64},
p_{111} = t_{63} + p_{110},
p_{109} = t_{49} + t_{53},
p_{108} = t_{50},
p_{107} = t_{50},
p_{106} = t_{53},
p_{105} = t_{53} + t_{54},
p_{104} = t_{54},
p_{103} = t_{49} + t_{51},
p_{101} = t_{50} + t_{52},
p_{102} = p_{101} + p_{103},
p_{100} = t_{55},
p_{98} = t_{54} + t_{56},
p_{99} = t_{55} + p_{98},
p_{97} = t_{43} + t_{45},
p_{96} = t_{44},
p_{95} = t_{44},
p_{94} = t_{45} + t_{47},
p_{93} = t_{48} + p_{94},
p_{92} = t_{48},
p_{91} = t_{41} + t_{43},
p_{89} = t_{42} + t_{44},
p_{90} = p_{89} + p_{91},
p_{88} = t_{47},
p_{86} = t_{46} + t_{48},
p_{87} = t_{47} + p_{86},
p_{85} = t_{33} + t_{37},
p_{84} = t_{34},
p_{83} = t_{34},
p_{82} = t_{37},
p_{81} = t_{37} + t_{38},
p_{80} = t_{38},
p_{79} = t_{33} + t_{35},
p_{77} = t_{34} + t_{36},
p_{78} = p_{77} + p_{79},
p_{76} = t_{39},
p_{74} = t_{38} + t_{40},
p_{75} = t_{39} + p_{74},
p_{73} = t_{25} + t_{29},
p_{72} = t_{26},
p_{71} = t_{26},
p_{70} = t_{27} + t_{29},
p_{69} = t_{28},
p_{68} = t_{28},
p_{67} = t_{29} + t_{31} + t_{32},
p_{66} = t_{32},
p_{65} = t_{29},
p_{64} = t_{29} + t_{30},
p_{63} = t_{30},
p_{62} = t_{25} + t_{27},
p_{60} = t_{26} + t_{28},
p_{61} = p_{60} + p_{62},
p_{59} = t_{31},
p_{57} = t_{30} + t_{32},
p_{58} = t_{31} + p_{57},
p_{56} = t_{17} + t_{21},
p_{55} = t_{18},
p_{54} = t_{19} + t_{21},
p_{53} = t_{20},
p_{52} = t_{20},
p_{51} = t_{21} + t_{23},
p_{50} = t_{24} + p_{51},
p_{49} = t_{24},
p_{48} = t_{21},
p_{47} = t_{22},
p_{46} = t_{17} + t_{19},
p_{44} = t_{18} + t_{20},
p_{45} = p_{44} + p_{46},
p_{43} = t_{23},
p_{41} = t_{22} + t_{24},
p_{42} = t_{23} + p_{41},
p_{40} = t_{9} + t_{13},
p_{39} = t_{10},
p_{38} = t_{10},
p_{37} = t_{11} + t_{13},
p_{36} = t_{12},
p_{35} = t_{12},
p_{34} = t_{13} + t_{15},
p_{33} = t_{16} + p_{34},
p_{32} = t_{16},
p_{31} = t_{13},
p_{30} = t_{13} + t_{14},
p_{29} = t_{14},
p_{28} = t_{9} + t_{11},
p_{26} = t_{10} + t_{12},
p_{27} = p_{26} + p_{28},
p_{25} = t_{15},
p_{23} = t_{14} + t_{16},
p_{24} = t_{15} + p_{23},
p_{22} = t_{3} + t_{7},
p_{21} = t_{4} + t_{8},
p_{20} = t_{1} + t_{5},
p_{19} = t_{2},
p_{18} = t_{2},
p_{17} = t_{3} + t_{5},
p_{16} = t_{4},
p_{15} = t_{4},
p_{14} = t_{5} + t_{7},
p_{13} = t_{8} + p_{14},
p_{12} = t_{8},
p_{11} = t_{5},
p_{10} = t_{5} + t_{6},
p_{9} = t_{6},
p_{8} = t_{7},
p_{7} = t_{6} + t_{8},
p_{2} = t_{7} + p_{7},
p_{1} = p_{7},
p_{6} = t_{1} + t_{3},
p_{4} = t_{2} + t_{4},
p_{5} = p_{4} + p_{6},
p_{3} = t_{7},
p_{0} = t_{0}.$
Pointwise multiplication (149 multiplications): $\boldsymbol{g} = \boldsymbol{p} \cdot \boldsymbol{c}$, where $\boldsymbol{c} = (1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{199}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{68}$, $\alpha^{204}$, $\alpha^{68}$, $\alpha^{38}$, $\alpha^{222}$, $\alpha^{111}$, $\alpha^{137}$, $\alpha^{183}$, $\alpha^{219}$, $\alpha^{170}$, $1$, $1$, $\alpha^{170}$, $1$, $1$, $\alpha^{170}$, $1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{199}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{68}$, $\alpha^{38}$, $\alpha^{222}$, $\alpha^{111}$, $\alpha^{137}$, $\alpha^{183}$, $\alpha^{219}$, $\alpha^{170}$, $1$, $1$, $\alpha^{170}$, $1$, $1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{199}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{68}$, $\alpha^{38}$, $\alpha^{1}$, $\alpha^{11}$, $\alpha^{137}$, $\alpha^{183}$, $\alpha^{219}$, $\alpha^{170}$, $1$, $1$, $\alpha^{170}$, $1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{199}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{68}$, $\alpha^{38}$, $\alpha^{222}$, $\alpha^{111}$, $\alpha^{137}$, $\alpha^{183}$, $\alpha^{170}$, $1$, $1$, $\alpha^{170}$, $1$, $1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{199}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{68}$, $\alpha^{38}$, $\alpha^{222}$, $\alpha^{111}$, $\alpha^{170}$, $1$, $1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{199}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{68}$, $\alpha^{137}$, $\alpha^{183}$, $\alpha^{219}$, $\alpha^{17}$, $1$, $1$, $1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{199}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{68}$, $\alpha^{38}$, $\alpha^{222}$, $\alpha^{111}$, $\alpha^{170}$, $1$, $1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{199}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{68}$, $\alpha^{38}$, $\alpha^{222}$, $\alpha^{111}$, $\alpha^{17}$, $1$, $1$, $1$, $\alpha^{153}$, $\alpha^{170}$, $1$, $\alpha^{68}$, $1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{38}$, $\alpha^{222}$, $\alpha^{170}$, $1$, $1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{137}$, $\alpha^{183}$, $\alpha^{170}$, $1$, $1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{38}$, $\alpha^{222}$, $\alpha^{170}$, $1$, $1$, $\alpha^{5}$, $\alpha^{199}$, $\alpha^{204}$, $\alpha^{68}$, $\alpha^{38}$, $\alpha^{111}$, $\alpha^{170}$, $1$, $\alpha^{5}$, $\alpha^{143}$, $\alpha^{204}$, $\alpha^{136}$, $\alpha^{137}$, $\alpha^{183}$, $\alpha^{170}$, $1$, $1$, $\alpha^{5}$, $\alpha^{199}$, $\alpha^{204}$, $\alpha^{68}$, $\alpha^{38}$, $\alpha^{111}$, $\alpha^{170}$, $1$, $\alpha^{5}$, $\alpha^{199}$, $\alpha^{204}$, $\alpha^{68}$, $\alpha^{137}$, $\alpha^{219}$, $\alpha^{170}$, $1)$.
Post-additions (177 additions): $\boldsymbol{S}' = \boldsymbol{P}^T \boldsymbol{g}$. Note that in the following sequence we use $\boldsymbol{S}$ directly to avoid extra permutation.
$ t_{273} = g_{18} + g_{20},
t_{269} = g_{110} + g_{116},
t_{270} = g_{113} + t_{269},
t_{271} = g_{119} + t_{270},
t_{272} = g_{121} + t_{271},
S_{30} = g_{111} + g_{114} + g_{117} + g_{120} + t_{272},
t_{266} = g_{52} + g_{54},
t_{267} = g_{49} + t_{266},
t_{264} = g_{92} + g_{95},
t_{262} = g_{89} + g_{97},
t_{263} = g_{86} + t_{262},
t_{265} = t_{263} + t_{264},
S_{22} = g_{87} + g_{90} + g_{93} + g_{96} + t_{265},
t_{260} = g_{32} + g_{37},
t_{256} = g_{77} + g_{80},
t_{255} = g_{74} + g_{83},
t_{257} = g_{85} + t_{255},
t_{258} = t_{256} + t_{257},
S_{17} = g_{75} + g_{78} + g_{81} + g_{84} + t_{258},
t_{248} = g_{98} + g_{104},
t_{249} = g_{107} + t_{248},
t_{250} = g_{109} + t_{249},
t_{251} = g_{101} + t_{250},
S_{26} = g_{99} + g_{102} + g_{105} + g_{108} + t_{251},
t_{246} = g_{63} + g_{73},
t_{247} = g_{71} + t_{246},
t_{244} = g_{2} + g_{5},
t_{241} = g_{41} + g_{44},
t_{268} = t_{241} + t_{267},
S_{9} = g_{42} + g_{45} + g_{50} + g_{53} + t_{268},
t_{240} = g_{15} + g_{17},
t_{237} = g_{57} + g_{60},
t_{236} = g_{100} + g_{106},
t_{242} = g_{103} + t_{236},
S_{13} = t_{242} + t_{251},
t_{233} = g_{1} + g_{4},
t_{276} = g_{12} + t_{233},
t_{277} = t_{240} + t_{276},
S_{2} = g_{13} + g_{16} + t_{244} + t_{277},
t_{274} = g_{9} + t_{233},
t_{275} = t_{273} + t_{274},
S_{8} = g_{10} + g_{19} + t_{244} + t_{275},
t_{230} = g_{58} + g_{61},
t_{245} = t_{230} + t_{237},
S_{13} = g_{64} + g_{72} + t_{245} + t_{247},
S_{11} = g_{66} + g_{67} + g_{68} + g_{69} + g_{70} + t_{245},
t_{226} = g_{91} + g_{94},
t_{232} = g_{88} + t_{226},
S_{11} = t_{232} + t_{265},
t_{225} = g_{59} + g_{62},
t_{235} = g_{65} + t_{225},
S_{22} = t_{235} + t_{237} + t_{247},
t_{223} = g_{112} + g_{118},
t_{239} = g_{115} + t_{223},
S_{15} = t_{239} + t_{272},
t_{222} = g_{76} + g_{82},
t_{224} = g_{79} + t_{222},
S_{26} = t_{224} + t_{258},
t_{221} = g_{43} + g_{46},
S_{18} = g_{47} + g_{48} + g_{55} + g_{56} + t_{221} + t_{241},
t_{229} = g_{51} + t_{221},
S_{5} = t_{229} + t_{268},
t_{220} = g_{3} + g_{6},
t_{234} = g_{14} + t_{220},
S_{1} = t_{234} + t_{277},
S_{16} = g_{7} + g_{8} + g_{21} + g_{22} + S_{1} + t_{240},
t_{231} = g_{11} + t_{220},
S_{4} = t_{231} + t_{275},
t_{219} = g_{25} + g_{28},
t_{228} = g_{27} + t_{219},
t_{238} = g_{24} + t_{228},
t_{227} = g_{23} + t_{219},
t_{243} = g_{26} + t_{227},
t_{259} = g_{35} + t_{243},
t_{261} = t_{259} + t_{260},
S_{6} = g_{33} + g_{36} + t_{238} + t_{261},
S_{3} = g_{34} + t_{261},
t_{252} = g_{29} + t_{243},
t_{253} = g_{38} + t_{252},
t_{254} = g_{40} + t_{253},
S_{24} = g_{30} + g_{39} + t_{238} + t_{254},
S_{12} = g_{31} + t_{254},
S_{31} = g_{179} + g_{180} + g_{181} + g_{182} + g_{183} + g_{184} + g_{185} + g_{186},
S_{29} = g_{171} + g_{172} + g_{173} + g_{174} + g_{175} + g_{176} + g_{177} + g_{178},
S_{27} = g_{162} + g_{163} + g_{164} + g_{165} + g_{166} + g_{167} + g_{168} + g_{169} + g_{170},
S_{25} = g_{154} + g_{155} + g_{156} + g_{157} + g_{158} + g_{159} + g_{160} + g_{161},
S_{23} = g_{145} + g_{146} + g_{147} + g_{148} + g_{149} + g_{150} + g_{151} + g_{152} + g_{153},
S_{21} = g_{136} + g_{137} + g_{138} + g_{139} + g_{140} + g_{141} + g_{142} + g_{143} + g_{144},
S_{19} = g_{127} + g_{128} + g_{129} + g_{130} + g_{131} + g_{132} + g_{133} + g_{134} + g_{135},
S_{17} = g_{122} + g_{123} + g_{124} + g_{125} + g_{126},
S_{0} = g_{0}.$
Overall 149 multiplications and 3970 additions over ${\mathrm{GF}}(2^8)$ are needed.
[^1]: This work was supported in part by Thales Communications Inc. and in part by a grant from the Commonwealth of Pennsylvania, Department of Community and Economic Development, through the Pennsylvania Infrastructure Technology Alliance (PITA). The material in this paper was presented in part at the IEEE Workshop on Signal Processing Systems, Shanghai, China, October 2007.
[^2]: The authors are with the Department of ECE, Lehigh University, Bethlehem, PA 18015 USA (E-mails: {nic6, yan}@lehigh.edu).
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“I believe every human has a finite number of heartbeats. I don’t intend to waste any of mine running around doing exercises.”
– Neil Armstrong (5 August 1930 – 25 August 2012) (1)
There are 86,400 seconds/day, and 31.536 million seconds/year (365 days).
The normal resting adult human heart rate ranges from 60 to 100 beats-per-minute (bpm). Slow heartbeat rates of about 40-50 bpm during sleep are common and considered normal. Medically, heart rates of 50 to 60 bpm in apparently healthy people are taken as a good sign needing no further attention, while heart rates above 80 bpm may be due to some otherwise undetected unhealthy condition, if not caused by stimulants like caffeine, or bursts of exercise. The maximum heart rate a person can safely experience during bursts of strenuous activity decreases with age, being about 180-200 bpm for people in their 20s, 175-190 bpm for people in their 30s, 170-185 bpm for people in their 40s, 165-175 bpm for people in their 50s, 155-170 bpm for people in their 60s, and 145-160 bpm for people in their 70s. A human lifespan that is not prematurely interrupted may experience up to 3.5 billion heartbeats, or even more. (2)
Let us define a characteristic average heart rate, which we shall call the Armstrong Heart Rate (AHR) in honor of Neil Armstrong: test pilot, aeronautical engineer, university professor, and the astronaut who was the first human to step onto the surface of the Moon. Assume as typical an average heart rate of 66+2/3 bpm during three quarters of every day (18 hours), which includes periods of “calm” and periods of “activity” and “stress.” We assume that sleep occupies one quarter of every day (6 hours) with an average heart rate of 40 bpm. The daily average with these assumptions is
A human with a heart rate equal to 1 bps will experience 31.536 million heartbeats per year. Given this average heart rate, the total number of heartbeats over periods of time would be as follows.
Neil Armstrong’s lifetime of 82 years and 20 days experienced an estimated 2.58768 billion heartbeats.
The United States is listed 38th and ranked 34th among nations as regards average life expectancy. The overall life expectancy in the United States is 79 years. The U.S. is ranked 37th for male life expectancy, which averages 76 years, and it is ranked 36th for female life expectancy, which averages 81 years. (3)
By our AHR model of average heart rate, the average US male lifespan includes 2.396736 billion heartbeats, and the average US female lifespan includes 2.554416 billion heartbeats. The overall average (79 years) is 2.4913344 billion heartbeats.
So, the average US lifetime is one of about 2.5 billion heartbeats, assuming the typical heart rate is the AHR, which we defined as 1 bps.
Of course, heart rate can and will vary over the course of a lifetime, and human variability is wide, so in reality heart rates both above and below the AHR model will occur in the population. The AHR model helps us visualize the order of magnitude of total heartbeats experienced in a human lifetime.
The heartbeats per lifetime for a wide variety of non-human mammals ranges between 0.53-1.5 billion heartbeats; and is 2.17 billion for chickens that live 15 years, and 2.21 billion for humans that live 70 years. (4)
Since many animal species experience lifespans of about 1 billion heartbeats, we can think of them as “dying in our 30s.”
We can describe five stages of human life, based on the summation of heartbeats, as follows:
1 billion heartbeats to develop and grow into seasoned adults in three decades (to 31.71 years),
1 billion more heartbeats to experience three decades of productive adult life (to 63.42 years, 2 billion heartbeats),
1/2 billion more heartbeats over the course of 1.5 decades of retirement and denouement (to 79 years, 2.5 billion heartbeats),
a possible bonus of another 1/2 billion heartbeats and 1.5 decades of advanced old age (to 95.13 years, 3 billion heartbeats),
and a very few may experience another 1/2 billion heartbeats to live another 1.5 decades of extreme old age (to 111 years, 3.5 billion heartbeats).
For most of us who manage to avoid the fatal hazards of bad luck and disease, we can expect to experience lifespans of between 2 to 3 billion heartbeats, and most likely about 2.5 billion heartbeats.
The wise thing to do with your heartbeats is to spend the life they sustain on what you enjoy doing.
The only moral constraint (or aspiration) I would put on that enjoyment is: be kind.
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The 163-year old telegram service may have technically come to an end a week ago but for many who rushed to be a part of history by sending their last telegrams, the journey is not over yet.Even after a week of booking their cherished telegrams, many - including those who used 'Taar' for the first time in their lives - are still awaiting confirmation that the messages have indeed been delivered to the rightful recipients."I booked eight telegrams on Sunday but none of them has reached. I rushed to book telegram, the service that I never used in my life, as it was turning in to history," says Delhi-based businessman Sanjeev Yadav."With so much delay in delivery, government has killed the meaning of telegram," Yadav, who stood in the queue at Central Telegraph Office for around two hours, said.The telegram, once the fastest mode of communication, lost its sheen with advent of telephone and later with widespread of mobile phones.Still, hundreds crammed into 75 telegram offices in the country to send souvenir messages before the service was shutdown after running for 162 years at a stretch. As a result, over 20,000 telegrams booked on last day of its service compared to daily run of 5,000.Another individual, M S Seth expressed disappointment at not receiving telegrams till date which were booked for local addresses."I drove for 20 kilometres to book telegram, stood in the queue for around three hours and even in rain just because of emotions that this service will no longer exist. But after putting so much effort there has been no result. I just pray that my telegrams get delivered properly," Seth said.No comments were received from BSNL, which was in charge of telegram service operations.On July 15, BSNL claimed to have despatched 12,568 out of 20,000 telegrams that were booked on July 14.On July 16, a BSNL spokesperson said all booked telegrams have been despatched with the help of using company's own staff and India Post.Karuppiah, a 96-year old resident of Vadamalaipatti village near Trichy, received a telegram from his grandson Anand Sathiyaseelan after four days."This time I got telegram by post. It used to get delivered in around 2 hours even when I booked it from Ceylon (Sri Lanka) for my parents in this village. There were many telegram offices earlier but the number is now very less. The nearest telegram office to our village was closed , I think, around 5 to 6 years back," Karuppiah said on phone.Sathiyaseelan said his grandfather ran a business in Sri Lanka and had sent first telegram in 1934 to his parents.First 30 words in telegram cost Rs 29 and Re 1 thereafter for every word - umpteen times more expensive than short message service or e-mails used for communications at present.
BSNL decided to discontinue the services following huge gap between the average annual revenue of around Rs 75 lakh compared to cost of over Rs 100 crore.
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The Kazakh-Italian business forum was held online
Prospects for investment cooperation and opportunities for cooperation in the fields of green economy, agriculture and industrialization were discussed at the Kazakh-Italian business forum.
The event was organized by the foreign ministries of the two countries with the assistance of the General Confederation of Italian Industry «Confindustrie», the ICE Agency, the national company «KAZAKH INVEST» and the National Chamber of Entrepreneurs «Atameken».
The Forum was opened by Deputy Prime Minister-Minister of Foreign Affairs of the Republic of Kazakhstan Mukhtar Tileuberdi and Minister of Foreign Affairs and International Cooperation of the Italian Republic Luigi Di Maio.
In his speech, Mukhtar Tileuberdi noted that since 1992, Italian companies have invested about 9 billion US dollars in our country and today more than 270 enterprises with Italian capital are successfully operating in Kazakhstan.
The contribution to the Kazakh economy of the Italian multinational company «ENI», which has been present in our country since 1992 and has implemented a number of important projects in the oil and gas sector, was particularly noted. At the same time, in recent years, Kazakhstan has successfully implemented new projects in the field of renewable energy sources, such as the construction of a wind power plant in the Aktobe region and the start of construction of a solar power plant in the Turkestan region.
Also during the online meeting, the Chairman of the Management Board of «NC «KAZAKH INVEST» JSC Meirzhan Yussupov spoke in detail about the support measures provided by the national company «KAZAKH INVEST» to foreign investors in the implementation of investment projects.
«Our organization is a single national operator for supporting investment activities, providing a full range of services for investors. For potential investors, we provide comprehensive information about the country, as well as industry reviews and, importantly, specific investment projects prepared by our team. In addition, we organize targeted visits of investors to Kazakhstan. We arrange meetings with state authorities and local self-government bodies. We have representatives in all regions to help localize projects locally, as well as a network of foreign representatives. An important part of our work is to establish business contacts, we help to find a strong local partner», - said Meirzhan Yussupov.
The event was attended by about 600 Kazakh and Italian businessmen. Italian investors were presented with projects in such sectors of the Kazakh economy as the agribusiness, renewable energy sources, mechanical engineering and others.
In turn, foreign investors welcomed the planned launch in the near future by the Italian company «NEOS» of a direct flight between Milan and Almaty. This air service should make a significant contribution to the development of business and tourism relations between the two countries. | https://invest.gov.kz/media-center/press-releases/kazakhstansko-italyanskiy-biznes-forum-proshyel-v-onlayn-rezhime/ |
I have done a thorough IP review on my own project in 2007. You can find some info about this project in this slide deck: Startup Legal and IP. I encountered many situations that are similar to what you describe, although none of them were "criminal." These are some examples. Example 1: code that was proprietary to SUN, later open source by SUN under a ...
14
When you put code on GitHub, you retain all the copyright to your code. However, you do grant GitHub a license to host the code, and you also allow GitHub users a set of rights - namely the ability to look at, and fork your repository. These are terms that you have accepted when accepting their Terms of Service when creating a GitHub account. Even when you ...
12
In the 3D graphics world, there are free-licensed models, textures, and the like. When building a 3D graphical work, they're the rough equivalents of code libraries and the like. There's also the open-source hardware movement (OpenCores, RepRap, etc.), though particularly in computing hardware, the line between "software" and "not software" gets a bit ...
8
Han never had copyright on the code. He was unable to license it to anybody. He claimed he gave people a license anyway, but this wasn't true, since he couldn't. All derivative works who thought they had a license to do what they were doing, didn't have a license to do so at all. Han said he gave them a license to the work, but he was lying. The copyright ...
7
Intellectual property, while not the same thing as copyright, tends to follow similar patterns. In a single-owner project, the copyright is entirely theirs, and so is the IP. In a project with multiple contributors and no contributor agreement, the copyright is distributed: each contributor owns the copyright on any contributions he makes. The same applies ...
6
Say I'm the only one uploading code to someone's private repository, do they own the copyright to the code? Or do I own the copyright to the code since I wrote it? The author (or copyright holder) of the code "owns" the code, meaning you do. Where the code lives does not have much impact. Now, if this is work you did "for hire" as an employee or ...
6
Sometimes it is made clear in contributor agreements. When there is no contributor agreement, the author(s) of each part of the code own the intellectual property for the part they wrote.
6
When software is diffused under an open-source licence I imagine that a project becomes the "intellectual property" of the community. This is incorrect. Unless copyright has been reassigned, the content creator owns their contributions to a work. As a copyright holder, you can freely choose what rights you want to grant to other people, including different ...
5
There is legally no such thing as "industrial secret", but there are trade secrets. If Han Solo publishes code that contains trade secrets, then he is likely going to jail. Publishing trade secrets when you have no right to do so is criminal. However, most if the time the copying of source code is just plain old copyright violation, whether there are any ...
5
On the surface not, as the source is an integral part (every OSS license is demanding release of source code, not binaries). That is obviously not directly applicable to other copyrighted works, like for example stories. But the ideas of freedom, permission for everyone to copy, modify, redistribute and remix has arrived in other areas. Namely the Creative ...
4
Is the output of an ML algorithm a derived work, and if so from what? I'd argue that it is a derived work not of the ML software but only of the training data set as whole, so that this training set's database rights are relevant. Here, you seem to be dealing with an CC-BY-NC-SA 4.0 dataset, which is not suitable for your purposes. The first question is ...
4
Do Open Source/Free Software projects require submitters to submit proof of authorization to contribute?
Virtually all major open source projects require contributors to agree to a contributor licensing agreement that includes language asserting that their contributions are able to be legally contributed to the project. For example, Apache's Individual Contributor License Agreement says You represent that you are legally entitled to grant the above ...
4
Wikipedia notes that there are loads of uses, from transportation to robots to beer. Some fall under the umbrella of open source hardware. My favorite of these is the Hyperloop, first pushed forward by Elon Musk, the co-founder of PayPal and founder of SpaceX and Tesla Motors (some of the latter's ideas are open-source). Numerous groups have done studies ...
3
Simply, yes. Software is the industry in which the open source industry is strongest, but other industries or arts can also be licensed openly. A related industry is hardware. Some microcomputers are open-source licensed. DIY is one such industry: when you make a product, you can release the documents you used to help make it under an open source license. ...
3
As others have said, the Creative Commons licences are generally thought to be better for non-code forms of expression, because the distinction between source and executable form (which the GPL uses a lot) doesn't exist for simple written materials. Before the CC licences were created, the FSF made the GNU Free Documentation Licence specifically to cover ...
2
What widely available software license is appropriate if I want to freely distribute but keep my own right to sell?
No open source license would fit your requirements. The Open Source Definition lists things that open source licenses do/not do. OSD #6 is “No Discriminiation Against Fields of Endeavour”, which means that an open source license must not prevent commercial use. You want to allow only personal use; this is not compatible. You are not the only one who would ...
2
What widely available software license is appropriate if I want to freely distribute but keep my own right to sell?
There are various licenses which might achieve what you want. The very widely used Creative Commons CC-BY-NC-SA would seem to do everything you desire, although it is not the only one that would. You can read more about Creative Commons licenses in this Wikipedia article. Note that once a work is released under a creative commons license, people may ...
2
German copyright law distinguishes adaption/modification (Bearbeitung, §23 UrhG) and free use (Freie Benutzung, §24 UrhG) when using older works. A Bearbeitung means that you edit an existing work. The character of the original work is still plainly visible. You hold copyright to your changes, but you need permission from the original copyright holder to ...
2
LibreOffice is a fork of OpenOffice, which itself was based on StarOffice. StarOffice corp, before being acquired by Sun Microsystems, had reverse-engineered rough compatibility with MS Office 97 formats, something that several other companies (e.g. Corel) had done as well. When MS released OfficeXP, they published the full spec to their quasi-XML format ...
2
Does using LGPL libraries in my product prevent me from distributing it under my own license or extending it to cover all work, derived from it? The LGPL only requires that changes made to the LGPL code itself are published under the LGPL users of your product have the possibility (and right) to replace the LGPL code with a version of their own. The ...
1
If authors of open source software can be held liable for (accidentally) infringing (vaguely worded) software patents, why add one's name to it?
The last question is the easiest. You apply a licence to it because if you don't, the software is unusable. Nobody else will have the right to run, copy, modify, or redistribute it, so nobody will. As to why you'd put your name on it, I agree that in theory some patent troll could sue you. The thing is, patent trolls exist to make money via lawsuits. It'...
1
While there is a license from the code snippet authors to the code snippet hosting site, there is no license from anyone to you. In absence of an explicit license, all rights are reserved. You have the right to look at the code snippets, but nothing else.
1
Are there open source hardware licenses that require publishing modified sources when a physical product is distributed?
Modern electronic products are a combination of hardware and software. Firmware is software that is "burnt" into hardware, that may or may not be modifiable, which controls the functionality of the hardware. Whether the software is stored in a temporary medium such as hard drive or flash memory, or permanently etched into an ASIC, it is still software, ...
1
That's easy. Make sure you don't compile your GUI and your algorithms together statically as a single executable. If you do that, you don't have any GPL obligation for your algorithms. Run your algorithms as a command-line tool and use inter-process communication in your GUI. This is a very common implementation in computer chess engines. Run your ...
1
First: I am not a lawyer and this is not legal advice. The first link you gave is the license for the documentation, not the SDK itself. The second link has copyright information on the SDK iself and several components. The SDK itself does not have any particular license according to the document, just this paragraph: This product is developed and ...
1
The Apache license is "sublicenseable": Grant of Copyright License. Subject to the terms and conditions of this License, each Contributor hereby grants to You a perpetual, worldwide, non-exclusive, no-charge, royalty-free, irrevocable copyright license to reproduce, prepare Derivative Works of, publicly display, publicly perform, sublicense, and ...
1
The copyright holder for code (or other IP) can choose to apply one or more licenses to their work. That is, the person (or perhaps their employer, depending on their employment agreement) who actually wrote the code owns the copyright, and they can choose to provide it to you under a specific license. They can also provide it to someone else under a ... | https://opensource.stackexchange.com/tags/intellectual-property/hot |
Naval Aviation Warfighting Development Center (NAWDC) at Naval Air Station Fallon is the center of excellence for naval aviation training and tactics development. NAWDC provides service to aircrews, squadrons and air wings throughout the United States Navy through flight training, academic instructional classes, and direct operational and intelligence support. The command consists of more than 120 officers, 140 enlisted and 50 contract personnel. NAWDC flies and maintains F/A-18C/D Hornets, F/A-18E/F Super Hornets, E/A-18G Growlers, F-16 Fighting Falcons and MH-60S Seahawk helicopters.
History:
Prior to June 2015, NAWDC was known as Naval Strike and Air Warfare Center (NSAWC) which was the consolidation of three commands into a single command structure on July 11, 1996. NSAWC was comprised of the Naval Strike Warfare Center (STRIKE "U") based at NAS Fallon since 1984, and two schools from NAS Miramar, the Navy Fighter Weapons School (TOPGUN) and the Carrier Airborne Early Warning Weapons School (TOPDOME).
Mission:
NAWDC is the Navy's center of excellence for air combat training and tactics development. NAWDC trains naval aviation in advanced Tactics, Techniques and Procedures (TTP) across assigned combat mission areas at the individual, unit, integrated and joint levels, ensuring alignment of the training continuum; to set and enforce combat proficiency standards; to develop, validate, standardize, publish and revise TTPs. In addition, NAWDC provides subject matter expertise support to strike group commanders, numbered fleet commanders, Navy component commanders and combatant commanders; to lead training and warfighting effectiveness assessments and identify and mitigate gaps across all platforms and staffs for assigned mission areas as the supported WDC; collaborate with other WDCs to ensure cross-platform integration and alignment. NAWDC's individual mission requirements include:
N2: The Information Warfare Directorate at NAWDC is responsible for ensuring command leadership and personnel are provided the full capabilities of the Information Warfare Community (IWC) to support combat readiness and training of Carrier Air Wings and Strike Groups. The Directorate is comprised of four areas of focus: Air Wing Intelligence Training, the Maritime ISR (MISR) Cell, Targeting, and Command Information Services (CIS). The Air Wing Intelligence Training Division is responsible for training CVW Intelligence Officers and Enlisted Intelligence Specialists in strike support operations. The MISR Cell is tasked with providing ISR integration into Carrier Air Wing training as well as qualifying MISR Package Commanders and Coordinators. The Targeting Division trains and certifies all CVW Targeteer personnel and provides distributed reach-back support for deployed units worldwide regarding target development. CIS provides cyber security and computer network operations for the entire NAWDC enterprise.
N3: NAWDC Operations department (N3) is responsible for the coordination, planning, synchronization, and scheduling for the operations of the command, its assigned aircraft, and airspace and range systems within the Fallon Range Training Complex (FRTC).
N4: NAWDC's Maintenance Department is the heart of training for all the NAWDC schoolhouses. Maintenance's focus is providing mission-ready fleet and adversary aircraft configured with required weapons and systems for all training evolutions. We support day to day training missions with the F-16 Viper, F-18 Hornet and Super Hornet, EA-18G Growler, E-2C Hawkeye and the MH-60S Seahawk; conducting scheduled and un-scheduled maintenance on 39 individual aircraft. These aircraft and weapon systems are the foundation for all other NAWDC Department's training syllabi.
N5: Responsible for training Naval aviation in advanced Tactics, Techniques and Procedures (TTP) across assigned combat mission areas at the individual, unit, integrated and joint levels, ensuring alignment of the training continuum; to set and enforce combat proficiency standards; to develop, validate, standarize, publish and revise TTPs. Also provides subject matter expertise support to strike group commanders, numbered fleet commanders, Navy component commanders and combatant commanders; to lead training and warfighting effectiveness assessments and identify and mitigate gaps across all platforms and staffs for assigned mission areas as the supported WDC; and collaborates with other WDCs to ensure cross-platform intergration and alignment.
NAWDC's Joint Close-Air Support (JCAS) Division continues to answer the needs of current theater operations with increased production of Joint Terminal Attack Controllers Course (JTACC). NAWDC JCAS primarily trains Naval Special Warfare and Riverine Group personnel, but has this year also trained U.S. Army Special Operations, U.S. Marine Corps Air and Naval Gunfire Liaison Officers, international personnel, as well as U.S. Navy Fixed and Rotary Wing Forward-Air Controller (Airborne) personnel. NAWDC's JCAS branch is the U.S. Navy's designated representative to the Coalition JCAS Executive Steering Committee, and is a recognized authority on kinetic air support to information warfare (IW), tactical precision targeting, and digitally aided CAS.
N6: Carrier Airborne Early Warning Weapons School (CAEWWS), also referred to as TOP DOME, is the E-2 weapon school and responsible for Airborne Tactical Command and Control advanced individual training via the Hawkeye Weapons and Tactics Instructors (HEWTIs) class. CAEWWS is also responsible for development of community Tactics, Technique and Procedures and provides inputs to the acquisition process in the form of requirements and priorities for research and development (R&D), procurement, and training systems. CAEWWS works closely to support other Warfare Development Centers and Weapons Schools; such as the Surface and Mine Warfighting Development Center's Integrated Air Defense Course (IADC) and Integrated Air and Missile Defense WTI Integration Course (IWIC). Other functions include support to advanced integrated fleet training by way of WTI augmentation to the N5/STRIKE Department for CVW integrated training detachments; also known as Air Wing Fallon Detachment and support of squadron activities.
N7: In the early stages of the Vietnam War, the tactical performance of Navy fighter aircraft against seemingly technologically inferior adversaries, the North Vietnamese MiG-17, MiG-19, and MiG-21, fell far short of expectations and caused significant concern among national leadership. Based on an unacceptable ratio of combat losses, in 1967, ADM Tom Moorer, Chief of Naval Operations, commissioned an in-depth examination of the process by which air-to-air missile systems were acquired and employed. Among the multitude of findings within this report was the critical need for an advanced fighter weapons school, designed to train aircrew in all aspects of aerial combat including the capabilities and limitations of Navy aircraft and weapon systems, along with those of the expected threat.
In 1969, the United States Navy Fighter Weapons School (TOPGUN) was established to develop and implement a course of graduate-level instruction in aerial combat. Today, TOPGUN continues to provide advanced tactics training for FA-18A-F aircrew in the Navy and Marine Corps through the execution of the Strike Fighter Tactics Instructor (SFTI) Course. TOPGUN is the most demanding air combat syllabus found anywhere in the world. The SFTI Course ultimately produces graduate-level strike fighter tacticians, adversary instructors, and Air Intercept Controllers (AIC) who go on to fill the critical assignment of Training Officer in fleet units.
N8: Navy's Rotary Wing Weapons School is composed of a staff of 25 pilots and aircrewmen who instruct the Seahawk Weapons and Tactics Instructor program; provide tactics instructors to fleet squadrons; maintain and develop the Navy's helicopter tactics doctrine via the SEAWOLF Manual; instruct the Navy's Mountain Flying School; provide high-altitude, mountainous flight experience for sea-going squadrons; and provide academic, ground, flight, and opposing-forces instruction for visiting aircrew during Air Wing Fallon detachments.
N9: The NAWDC Safety Department (N9) serves as the principal advisor to the Commander on all matters pertaining to safe command operations and is responsible for administering the following safety programs: aviation, ground, ergonomics, motor vehicles (personal, commercial), recreation, and on- and off-duty. Our goal is to eliminate preventable mishaps while maximizing operational readiness. We accomplish this by preserving lives, preventing injury, and protecting equipment and material.
N10: The US Navy's Airborne Electronic Attack Weapons School, call sign "HAVOC", stood up in 2011 to execute the NAWDC mission as it pertains to Electronic Warfare and the EA-18G Growler. HAVOC is comprised of highly qualified Growler Tactics Instructors, or GTIs, that form the "tactical engine" of the EA-18G community, developing the tactics that get the most out of EA-18G sensors and weapons. HAVOC's mission is also to train Growler Aircrew and Intelligence Officers on those tactics during the Growler Tactics Instructor Course. The Growler Tactics Instructor Course is a rigorous 12 week syllabus of academic, simulator, and live fly events that earn graduates the Growler Tactics Instructor designation - the highest level of EA-18G tactical qualification that is recognized across Naval Aviation. The Growler brings the most advanced tactical Electronic Warfare capabilities to operational commanders creating a tactical advantage for friendly air, land, and maritime forces by delaying, degrading, denying, or deceiving enemy kill chains.
N20: The Tomahawk Land Attack Missile (TLAM) Department provides direct support to U.S. Fleet Forces Command (USFFC) in the development and standardization of tactics, techniques and procedures for the employment of the Tomahawk weapon system. In addition, TLAM provides training to the CVW, fleet, and joint commands on TLAM capabilities and strike integration
Command Address:
4755 Pasture Rd. Bldg 465
Fallon, NV 89496-5000
(775)426-3884
The "Fighting Saints" of VFC-13 can trace their origins back to 1946, when VF-753 was commissioned flying F6F-5 "Hellcats." Today's squadron was formed on Sept. 1, 1973, at N.A.S. New Orleans during the reorganization of the U.S. Naval Reserve.
Initially, the squadron operated the Chance Vought F-8H "Crusader" with a complement of 17 officers and 127 enlisted men, former members of VSF-76 and VSF-86. In April 1974, the "Saints" made the transition to the single-seat A-4L "Skyhawk."
As the demand for west coast adversary services and other fleet support missions increased, the squadron was permanently transferred to N.A.S. Miramar, arriving there in February 1976. That summer, a transition was made from the A-4L to the more reliable two-seat TA-4J. The year 1983 marked the return of the single seat aircraft to VFC-13 and the arrival of the A-4E.
October 1993 marked the end of an era when VFC-13 made the transition to the single-seat, two engine F/A-18 "Hornet." This change further enhanced the squadron's ability to perform its adversary mission by providing an even more capable and realistic threat aircraft. In April 1996, the command relocated to N.A.S. Fallon and made the transition to the F-5E/F Tiger II, supported by McDonnell Douglas contract maintenance.
The "Fighting Saints" provide a formidable dissimilar adversary threat. In recognition of that primary mission assignment, VC-13 was proudly redesignated as Fighter Squadron Composite Thirteen (VFC-13) April 22, 1988.
VFC-13 provides the highest quality adversary training for regular Navy fleet and replacement squadrons and air wings, reserve fighter and attack squadrons, U.S.A.F. and U.S.M.C. units, and Canadian forces. The Fighting Saints have received three consecutive CNO Safety Awards, three Wing Golden Wrench Maintenance Awards, and in 1994 and 2011, the Battle "E" Efficiency Award.
Fleet Readiness Center West Detachment Fallon (FRC West) is designated as a fullF/A-18, F-5 E/F and F-16 support site for detachments of Strike Fighter Wing Detachment, Fallon, VFC-13 and NSAWC. Additionally, FRC West provides transient maintenance for carrier air wings and for USMC aircraft. Non F/A-18 repair actions of common aircraft systems are rendered whenever possible. FRC West has evolved into a new complex that includes a modern production control and quality assurance division with state-of-the-art airframes, NDI and a welding facility over the past several years. There are 30 mobile maintenance vans that house test benches for more than 15 avionics systems included in the complex.
In addition to this, there is an extensive type IV calibration lab and a new state-of-the-art hydraulics work center that houses HYAC, T-10 and STS benches. An aircraft sound abatement building (hush house) is also available. The hush house gives tenant commands and transient air wings a completely enclosed area to perform high and low power aircraft engine runs. Additionally, the facility has been configured with an oil laboratory as well as a test cell for the J85-6E-21 engine (F-5E/F).
FRC West’s motto of professionalism and excellence is embodied in every maintenance evolution performed, and it will continue to represent that in quality intermediate maintenance support.
In December 1983, the first Strike Fighter maintenance personnel arrived in Fallon from Naval Air Station (NAS) Lemoore, Calif., to establish the VFA-125 detachment. The mission of the detachment was to maintain an operationally rich aircrew training environment by providing quality organizational level maintenance for Fleet Replacement Squadron (FRS) F/A-18 series aircraft, and limited support for transient F/A-18 series aircraft.
In March 1987, VFA-106 stood up a detachment in Fallon supporting FRS training from NAS Cecil Field, Fla. Both commands performed similar missions to support FRS training in Fallon. In 1994, the detachments combined under VFA-125 to form Strike Fighter Wing Detachment (SFWD). In late 1996, the detachment came under the control of Commander, Strike Fighter Wing Pacific (CSFWP), NAS Lemoore, Calif., and was renamed Strike Fighter Wing Pacific etachment (SFWPD) Fallon. This allowed better coordination of training for F/A-18 series Weet replacement pilots in strike and fighter weapons tactics for east and west coast Navy and Marine forces. SFWPD Fallon’s normal manning level consists of 110 enlisted personnel and two officers.
In November 2002, the detachment’s mission, functions and tasks became joint between CSFWP and Commander, Strike Fighter Wing Atlantic to support F/A-18 series Hornet and Super Hornet aircraft during FRS training detachments and Strike Fighter Tactics Instructor (SFTI) detachments in support of Naval Strike Air Warfare Center TOPGUN classes. A typical cycle at SFWPD Fallon usually consists of at least one two-week FRS strike training (Air-to-Ground) detachment and an eight week TOPGUN SFTI class per quarter. SFTI class includes two week Air-to-Air detachments to various Navy and Marine Corps Air Stations, and a two day detachment to Naval Air Warfare Center China Lake, Calif. This schedule continues year-round.
The Fallon Range Training Complex (FRTC), located in the high desert of northern Nevada approximately 65 miles east of Reno, NV, is a set of well defined geographic areas encompassing a land area and multiple air spaces. It is used primarily for training operations, with some capability to support research and development, and test and evaluation of military hardware, personnel, tactics, munitions, explosives, and electronic combat.
The geographic scope encompasses NAS Fallon and near-by range training areas, Bureau of Land Management (BLM) rights-of-way, and 13,000 square miles of Special Use Airspace (SUA). The SUA is comprised of the 11 Military Operations Areas (MOAs), nine Restricted areas, ten Air Traffic Control Assigned Areas (ATCAAs), and an Aerial Refueling Route (ARR). Additionally, 17 Instrument Flight Rules (IFR) Military Training Routes (MTRs), three helicopter MTRs, and 14 Low Level Visual Flight Rules (VFR) MTRs transit, terminate in, or are in close proximity to the FRTC. The FRTC encompasses over 234,124 acres of land area including the Bravo-16, Bravo-17, Bravo-19, Bravo-20, Dixie Valley, and Shoal Site training areas.
The Navy administers only 234,124 acres of the 6.5 million acres of land under the FRTC airspace, while the remainder consists largely of public land managed by the Bureau of Land Management (BLM).
Oakland and Salt Lake Air Route Traffic Control Center (ARTCC) control the airspace within the FRTC, which in turn delegate scheduling and coordination authority to the Naval Strike and Air Warfare Center (NSAWC).
The FRTC is particularly significant to the Department of Defense (DoD) because of its unique training and tactics development capabilities, extensive instrumentation and target sets, live ordnance impact areas, and its capability to provide Basic, Integration and Sustainment Phase training of Naval forces in the Fleet Readiness Training Plan (FRTP).
Mission
The mission of the FRTC is to support Navy and Marine Corps tactical training by providing the most realistic strike and integrated air warfare training available, maintaining and operating facilities, and providing services and equipment to support the U.S. Pacific Fleet, U.S. Atlantic Fleet, U.S. Marine Corps Forces Pacific, U.S. Marine Corps Forces Atlantic, and other operating forces. Research, Development, Test, and Evaluation (RDT&E) operations are supported on a not to interfere basis.
Target Ranges
Bravo-16. The B-16 area has typically saline soil characterized by extensive alkali flats and areas of patchy desert sand and sparsely vegetated by sagebrush. Located south of the Sheckler Reservoir and nine miles southwest of NAS Fallon at an elevation of 3,942 feet, B-16 includes two conventional bull’s-eye targets. The West Conventional Weapons Bull provides night lighting; the East Conventional Weapons Bull does not. At both targets, MK-76/BDU-33, MK-106/BDU-48, 2.75 FFAR (practice), and LUU-2 Paraflares are allowed.
Bravo-17. The most frequently used range at FRTC, B-17, is located west of Fairview Peak and south of U.S. Highway 50 and is contained within the Fairview NSAWC working area. The B-17 terrain is made up of the following: alkali flats in the northern section giving way to a rocky terrain along the west and east foothills, and patchy areas of desert sand sparsely vegetated by sagebrush along a gently sloping foothill at the southern extreme. The range is flanked on the west by the Sand Spring Mountains and State Highway 839 and on the east by Fairview Peak.
Located 25 miles east-southeast of NAS Fallon at an elevation of 4,153 feet, B-17 is split into an east (B-17E) and west (B-17W) component. These areas are further divided into a total of four surface areas. The B-17W target complex is comprised of No Drop Area (NDA) targets. The NDA targets include an Army compound target; Scud missile target, laser billboard; a bridge target; the West Petroleum, Oil, and Lubricant (POL) Facility target; and a motor pool target. Ordnance expenditure is forbidden in this area.
B-17E includes the Light Inert Impact Area, the Heavy Inert Impact Area, and the Live Impact Area. The Light Inert Impact Area includes a conventional weapons bull’s-eye target, a strafe target, an airfield complex, an air defense site, the East POL Facility, a Headquarters compound, the East Power Plant target, a helicopter tank target, a tank convoy and cave entrance target, a Scud missile launcher, a convoy target, a command and control center, a Close Air Support (CAS) target that simulates a below-ground POL, and another CAS target that represents a below-ground building. The targets in the Light Inert Impact Area collectively accommodate expenditure of the following ordnance types: MK-76/BDU-33, MK-106/BDU-48, Laser Guided Training Round (LGTR), BDU-45, LUU-2 Paraflares, and 2.75 FFAR (practice). Targets in the Light Inert Impact Area are Weapons Impact Scoring System (WISS) scored.
Forward Air Controller (FAC) platforms are designated areas from which approved artillery, small arms, and mortars are fired in support of CAS exercises. Each FAC position allows an unobstructed view of associated target areas. There is one FAC platform located within B-17 at the western edge of the Light Inert Area. There is also a helicopter Landing Zone (LZ) in addition to the DZ Bad Monkey within B-17 to support CAS training
The Heavy Inert Impact Area is in the northeastern corner of the B-17 complex. This area includes three targets: an Industrial Site target, a SAM site target, and a missile assembly target. All three of these targets accommodate expenditure of MK-76/BDU-33, MK-106/BDU-48, LGTR, MK-81 thru MK-84 practice ordnance, BDU-45, LUU-2 Paraflares, and 2.75 FFAR (practice). Targets in the Heavy Inert Impact Area are Weapons Impact Scoring System (WISS) scored.
The High Explosive impact (HEI) area is located in the southeastern section of the B-17 complex and allows expenditure of high explosive ordnance. The HEI area contains numerous tank vehicle targets and a camouflaged cave entrance. Targets in the HEI area are WISS scored.
Bravo-19. The B-19 area is comprised of alkali flats with areas of patchy desert sand sparsely vegetated by sagebrush. This target complex, which lies 16 nm south-southeast of NAS Fallon at an elevation of 3,882 feet, consists of a strafe target consisting of an acoustic transducer located behind an earthen berm, a HEI area with three distinct clusters of four M60 tank targets each, and a helicopter strafe area comprised of 14 different light armored vehicles. Night lighting is provided for the bull target. The HEI area is also designated as an alternate ordnance jettison area. There are two FAC Platforms in B-19 to support CAS training, one on the tower road and one at the east tower.
The targets within B-19 accommodate expenditure of MK-76/BDU-33, MK-106, BDU-48, LGTR, 2.75 FFAR (practice), LUU-2 Paraflares, BDU-45, .20mm TP, .25mm TP, 30mm TP, 7.62mm, 5.56mm, .50 cal (no HEI), 5.0 Zuni (practice), MK-80 series (live and practice Laser Guided Bombs [LGB]), 20mm HEI, and MK-77 (Napalm).
An open range for small arms (up to .50 cal) training is available at B-19 with firing lines located 50 feet north of the center tower area. The range includes a pistol/shotgun range, popup targets and a rifle/machine gun range.
Bravo-20. The B-20 target range is located in the northeastern section of the Carson Sink and lies within the Lone Rock NSAWC working area. Lone Rock, an igneous rock formation approximately 140 feet tall, is the center of this target area. The B-20 area is 31 nm north-northeast of NAS Fallon at an elevation of 4,040 feet at Lone Rock. The adjacent flats are at 3,890 feet above MSL. Drainage in the area surrounding this range is very poor, often leading to extensive areas of shallow surface water surrounding many of the target sites after heavy rains.
The Light Inert Impact Areas within B-20 include:
• Two conventional bull’s-eye targets with night lighting and WISS scoring
• Laser evaluation capabilities
• A laser-guided bomb target
• Two strafe targets
• A submarine target
• A broadcasting facility
• A radar van target
• Area 52, a simulated Weapons of Mass Destruction (WMD) facility
The Heavy Inert Impact Area within B-20 consists of an industrial site comprised of 22 large metal targets of various geometric designs. Adjacent to the Heavy Inert Impact Area is the Live Impact Area, which includes the Lone Rock target within an alkali flat, and the Hellfire target, a single, light-armored vehicle target. The primary ordnance jettison area at Fallon is the B-20 HE impact area.
The five Laser Target Areas (LTAs) aboard B-20 include the Live Impact Area, a submarine target, a laser-guided bull, and the North and South Conventional Bull targets. Delivery of inert Joint Direct Attack Munition (JDAM) is only authorized within the boundaries of the B-20 target range. The only authorized target for practice JDAM expenditure is the radar van target (B-20-12), which includes Sheridan Tank-1, Sheridan Tank-2, Sheridan Tank-3, and the Tactical Fuel Truck.
The targets within B-20 accommodate expenditure of MK-76/BDU-33, MK-106, BDU-48, LGTR, 2.75 FFAR (practice), LUU-2 Paraflares, BDU-45, .20mm TP, .25mm TP, 30mm TP, 7.62mm, .50 cal (no HEI), 5.0 Zuni (practice), MK-80 series (live and practice LGB), MK-77 (Napalm), JDAM, and AGM-114 (Hellfire).
Dixie Valley Training Area
The capability to provide forces with CSAR and non-ordnance CAS at FRTC is provided by approximately 80,000 acres of Navy-managed land within Dixie Valley. Four sub-areas make up the Dixie Valley training area:
• Leisy Ground Training Area
• Dixie Valley Settlement North
• Dixie Valley Settlement South
• Horse Creek
Dixie Valley Settlement South, situated on the valley floor, and the mountainous Horse Creek area, are the most frequently used areas within the Dixie Valley training area.
The 11 target clusters within the Dixie Valley training area are non-ordnance targets; lasing and ordnance drops are not authorized. The majority of these targets are found within Dixie Valley Settlement South, including Fort Apache–a 100,000 square foot complex of buildings, tracked and wheeled vehicles, tents, and a firing base for two 8-inch howitzers.
Within the Dixie North airspace working area, the Gabbs North MOA overlays the Dixie Valley Settlement and Horse Creek. The Gabbs North MOA extends from 100 feet AGL to FL180 but excludes restricted area R-4816N extending from 1,500 feet AGL to but not including FL180. Helicopter landings are permitted in the Navy-owned land within Horse Creek and Dixie Valley Settlement North and South at the aircrew’s discretion.
Shoal Site
NSW and CSAR training operations are conducted on the 2,560-acre Shoal Site training range. Located south of US Highway 50 and west of B-17, the Shoal Site is public land withdrawn by the Department of Energy (DOE). The Military Lands Withdrawal Act of 1999 authorized a secondary withdrawal by the Navy for military use on the surface position of the DOE site.
Small Arms Training Range
Target range B-19 contains a small arms training area. This area includes a pistol/shotgun range, a zero range, an automated-record fire range, and a rifle/machine gun range. The rifle/machine gun range accommodates M2, M60, Squad Automatic Weapon (SAW), and Sniper rifle firing. Munitions calibers authorized for use here include:
• 12 gauge Shotgun
• 9mm
• .22 cal
• .357 cal
• .38 cal
• .30 cal
• .44 cal
• .45 cal
• 5.56mm
• 7.62mm
• .50 cal
• 40mm PR/TR B546
Electronic Warfare Complex (EWC)
The Fallon EWC consists of a series of pre-approved fixed and mobile site locations spread through most of the FRTC as depicted in Figure 2-6. The fixed sites are centered in the Dixie Valley, 23 nm east of NAS Fallon at an elevation of 4,170 feet, which is characterized as high desert, moderately vegetated by sagebrush and a variety of high desert type flora. The EWC integrates with TACTS and R-4816 to provide a variety of EC training capabilities. EC services and strike/attack scenarios can be customized for specific mission training and include both fixed and mobile threat capabilities. The system supports specialized EC training, such as CSAR helicopter penetration and reconnaissance training, and provides real-time and post-engagement feedback. The EWC assets include SAM/AAA simulators; a command, control and communication network and emulator; search radar systems; and Electronic Support Measures/Electronic Countermeasures (ESM/ECM) systems.
NSAWC Working Areas
For safety and training efficiency, FRTC airspace is subdivided into NSAWC Working Areas. Though not strictly SUA, these areas can be scheduled only through NSAWC. The eight primary NSAWC working areas and their respective subdivisions are:
• Berlin East/West, High/Low
• Callaghan North/South
• Cortez North/South, High/Low
• Dixie North/South, High/Low
• Edwards North/South, High/Low
• Fairview
• Kingston
• Lone Rock
The ‘Low’ subdivisions encompass airspace below 10,000 feet MSL, while the ‘High’ sub-areas extend from 11,000 feet MSL to the top of the MOA boundary. One or more of these areas can be scheduled as required. The NSAWC working area boundaries are shown in Figure 2-5.
The eight primary working areas can be grouped into three major combined areas–NSAWC 1, NSAWC 2, and CAS 17/19. NSAWC 1 includes Lone Rock (including R-4813A), Dixie North, Edwards North, Cortez, Callaghan North, and Stillwater Corridor. NSAWC 2 consists of Fairview (including R-4804A), Dixie South, Berlin, Kingston, O’Toole, Shoshone, Middlegate, Edwards South, Callaghan South, and R-4812. NSAWC 1/2 are designed to provide support for training events involving 6 or more aircraft that require significant lateral dispersion. Scheduled together, the NSAWC 1/2 areas can be utilized jointly in COMMODORE events, which involve 12 or more aircraft. COMMODORE is an airspace and communications scheduling package that provides blanket airspace clearance for large-scale exercises. The area covered by the COMMODORE airspace clearance includes NSAWC 1 and NSAWC 2. The third major combined working area is CAS 17/19, which includes Dixie South, B-17 or B-19, R-4812, and Berlin West.
Four corridors, each 5 nm in width, have also been defined by NSAWC to facilitate safe and orderly transit within the FRTC airspace. The four corridors are named Middlegate, O’Toole, Shoshone, and Stillwater.
Explosive Ordnance Disposal (E O D) Detachment Fallon is a tenant command on N A S Fallon. With the large amount of ordnance dropped on N A S Fallon ranges, it's up to the unit to conduct "render safe" procedures on all types of military and unconventional ordnance. E O D conducts sweep operations for unexploded ordnance utilizing all-terrain vehicles and helicopters on Fallon's Range Training Complex. The unit also maintains the Navy's only high altitude dive locker.
Detachment Fallon personnel aid the local community in providing E O D service and training to local air law enforcement, including underwater search capabilities and demolition expertise.
Contact E O D at 775-426-3401
The Naval Criminal Investigative Service (N.C.I.S.) is responsible for the investigation of all major crimes occurring aboard N.A.S. Fallon properties, whether involving active duty, civilian, and/or contractor personnel. The N.C.I.S. also provides counterintelligence support to all local commands, and is responsible for all liaisons with federal, state, and local agencies with respect to criminal investigative, counterintelligence, and security matters. The N.C.I.S. maintains a fully staffed office, or "Resident Agency", located in Building 427. After working hours emergency requests or referrals should be forwarded through the N.A.S. Fallon Emergency Services (911) Dispatch Center, who will then contact the on-duty N.C.I.S. Special Agent. | https://cnrsw.cnic.navy.mil/Installations/NAS-Fallon/About/Tenant-Commands/ |
Discussion: Science Instructional Materials for Middle School: Informing Future Initiatives
posted by:
Barbara Fitzsimmons
on August 17, 1999 at 9:45AM
subject:
Reply from Barbara Fitzsimmons
>The questions we are addressing are:
>
>1. What instructional materials have you used for middle school science?
> What is your take on their strengths and weaknesses?
We use Science Kits (primarily STC) at the sixth grade. At grades 7 and 8 we've just selected the Prentice-Hall Science Explorer series, modular texts which allow us to address fewer topics in greater depth. These texts are intended to support the curriculum which is intended to be significantly hands-on. This was a compromise with the teachers who wanted a text based curriculum. It's difficult to determine strengths and weaknesses at this early date.
>
>2. In considering a new initiative for curriculum development, what do
>you think teachers and administrators want/need in instructional
>materials to provide high-quality science education to their students? At the middle school level, the greatest need is a consistent supply of laboratory materials such as is available at the high school. Laboratory sessions too often depend on the ingenuity of the teacher. Some teachers provide minimal lab experiences. I think teachers also need to have highter expectations for this population.
>
>3. Should new curricula materials for middle school be in earth, life,
>and physical science, or multidisciplinary, or interdisciplinary? Should
>they be all modular or year long? Should they be integrated across
>subject domains? Should they have texts that go along with the
>activities, as the high school programs have? Would you recommend a
>social/societal context, a historical context, or a traditional one?
We have chosen multidisciplinary. Teachers felt that addressed the needs and learning styles of this population. Modular units were deemed most appropriate. Our schools do integrate across subject domains as much as possible, probably less in science and math than in language arts and social studies. Texts were viewed as an absolute essential by the teachers but the shift is that the text supplements the curriculum rather than being the curriculum. Social/societal context may be the most likely to work for this group. They (students) have a need to see the reason for learning.
>4. What are the primary barriers to implementing such a curriculum
>(teacher certification/training, facitlities, materials)?
The greatest barrier, in my opinion, is teachers who are comfortable with a traditional, more lecture oriented approach to teaching. They consider themselves subject matter experts and responsible for "teaching the students what they need to know" rather than helping them to uncover through experimentation. On-going replenishment of materials is an issue - but local planning is improving. Knowledge of how to use technology most effectively is an issue - on-going training and support is helping. The schedule of the school day is an issue - too fragmented.
Barbara A. Fitzsimmons, Ed.D.
Director of Curriculum and Instruction
North Kingstown School District
Thread View
Welcome to the Forum: Science Instructional Materi...
- posted by Joni Falk on 08/10/99 - 09:49
Welcome, let's begin
- posted by Joni Falk on 08/12/99 - 07:14
Response from Gail Paulin-DESERT project
- posted by Gail Paulin on 08/12/99 - 11:32
"Non-official" perspectives welcome!
- posted by Joni Falk on 08/13/99 - 11:38
MS curriculum reply from Mary Kay Swanson
- posted by Joni Falk on 08/13/99 - 03:41
Gail Paulin's Comments
- posted by Scott Hays on 08/13/99 - 08:32
Middle School Science Curriculum
- posted by Janie West on 08/14/99 - 10:59
MS discussion
- posted by Gail Paulin on 08/16/99 - 11:19
Middle School Curriculum
- posted by Jerry Valadez on 08/16/99 - 11:23
Re: request for your input
- posted by Vicki Graber on 08/16/99 - 11:24
Brief reply
- posted by Peter Dow on 08/16/99 - 11:52
Middle School Science
- posted by Tom Archer on 08/16/99 - 03:43
Reply from Barbara Fitzsimmons
- posted by Barbara Fitzsimmons on 08/17/99 - 09:45
Middle School Science
- posted by Ron DeFronzo on 08/17/99 - 01:25
Questions for forum
- posted by Miriam Robin on 08/18/99 - 12:24
midddle school curriculum
- posted by [email protected] on 08/18/99 - 01:38
have you tried it yourself?
- posted by Barbara Sullivan on 08/20/99 - 11:00
Questions?
- posted by Scott Hays on 08/23/99 - 07:26
Role of Volunteer Scientists from the Private Sect...
- posted by [email protected] on 08/25/99 - 11:49
The volunteer aspect raised in the last post may b...
- posted by Richard Dinko on 08/27/99 - 02:30
Middle School Science Curriculum
- posted by Linda Gregg on 08/27/99 - 11:29
From: Rick Vanosdall,Mesa Public Schools
- posted by Joni Falk on 08/29/99 - 12:25
materials review, implementation barriers
- posted by Mack McCary on 08/30/99 - 09:52
Real World Earth Science, etc? | https://lsc-net.terc.edu/do/discussion_post/11505/show/use_set-discussions/sort-td.html |
Search results: Found 35
Objectives maintaining root canal anatomy through minimal canal transportation by usingRotary single file systems: One Shape, Wave One Gold and Reciproc systems used insimulated artificial curved root canals. Materials and Methods 30 simulated curved rootcanals in clear resin blocks were used in this study and divided into three groups containing10 samples each: group (1) represented canals instrumented with Rotary One Shape files,group (2) canals instrumented with reciprocating Wave One Gold files while Group (3)canals were prepared with Reciproc files. All canals were imaged pre- and postinstrumentationat three levels, 2,3 and 5 mm apically and compared using Adobe Photoshopsoftware program. Amount of transportation were assessed. The three groups were comparedwith ANOVA and LSD statistic tests Results the mean transportation at 3mm level of thecanal had no significant difference among all groups, all shown transportation. At 2mm, and5mm levels there were significant difference between groups, G2 shown the leasttransportation among all groups. Conclusions Within the limits of this study, the canalpreparation with Wave One Gold files showed lesser transportation than One Shape andReciproc files.
Root canal --- Rotary --- curve
Background: The solubility of the root canal sealers is undesirable, because the process of dissolution may result in gaps and voids along the sealer-dentine or sealer-gutta percha interface. The aim of this study was to assess the solubility of zinc oxide based sealer (Endofil) in normal saline solution at different time intervals.Materials and methods: Fifty standardized plastic ring moulds were constructed and filled with Endofil sealer. The specimens were allowed to dry for 24h then weighed to the nearest 0.0001g .The samples were divided randomly into 5 groups and immersed in normal saline solution for 1,7,14,28 and 56 days. The samples were removed from the solution after completing the specified immersion period and allowed to dry for 24h. Then they were weighed, the percentage weight loss was then determined.Results: The percentage weight loss was less that 3% for immersion periods not exceeding 14 days and increased up to 5.4% for the 56 days. The statistical analysis of the results revealed a non significant difference between 1 and 7 days immersion time groups, while the differences between other groups were statistically significant.Conclusion: Under the conditions of the present study, the Endofil sealer met the International Standard ISO requirement for at least 14 days. The solubility rate of the sealer increased gradually from the first day till the 56 th days, but it remained within the acceptable limits for only 14 days.
to determine the bactericidal efficiency of 0.75% Chlorhexidine in vitro . The time required for this irrigant to start its antimicrobial effect on the selected microorganisms isolated from the infected root canals and unidentified samples taken from root canals with necrotic pulps were evaluated. Materials and Methods: The substantive antimicrobial effect of 0.75% Chlorhexidine in vitro is also considered. Samples taken from 13 teeth with necrotic pulp from patients attended the Department of Conservative Dentistry, College of Dentistry at Mosul University. The turbidity method was applied to determine the antimicrobial effect of 0.75% Chlorhexidine and the combination of 0.5% Chlorhexidine and 0.5% sodium hypochlorite comparing with the antimicrobial effect of 2.5% sodium hypochlorite, the time required for these materials to start their antimicrobial effect on the selected microorganisms was determined using contact test. The antimicrobial effectiveness were evaluated at different time intervals, immediately, 5, 10, 15 minutes after the contact of the microorganisms with the irrigating solutions. Results: Both 0.75% Chlorhexidine and 2.5% sodium hypochlorite are effective on microorganisms collected from root canal. There is no significant difference between Chlorhexidine and sodium hypochlorite in their antimicrobial effect on the anaerobic microorganisms, but Chlorhexidine is more effective on the aerobic microorganisms. Conclusion: Chlorhexidine 0.75% and 2.5% sodium hypochlorite has an immediate effect on the selected microorganisms and unidentified samples from the teeth with necrotic pulps. The combination of Chlorhexidine 0.5% and 0.5% sodium hypochlorite has an effect started after 5 minutes
Aims: To evaluate the antimicrobial effect of the ethanolic extracts of Ruta graveolens (Rue) and Sal-via officinalis (Sage) in a concentration of 0.2% and compare the results with the same concentration of chlorhexidine 0.2% (CHX) and normal saline on root canal bacteria using the above plant extracts as an irrigating solutions clinically. Materials and Methods: Thirty five uniradicular teeth with necrotic pulps were chosen. The patients were divided randomly into four groups, 10 patients for groups I, II and III and 5 patients for group IV. Using 0.2% ethanolic extract of Sage, 0.2% ethanolic extract of Rue, 0.2% chlorhexidine gluconate (CHX) and normal saline, respectively. Samples were obtained from the canal at the beginning of the first and second appointments, at the end of the second appoint-ment and at the beginning of the third appointment using wet sterile paper points. Results: The results revealed that 0.2% of the ethanolic extract of both Sage and Rue have a significant antimicrobial effect when used clinically as an endodontic irrigant, and was significantly not different from 0.2% chlorhex-idine gluconate (CHX) and significantly different from normal saline. Conclusions: Rue and Sage demonstrated antimicrobial effects on the root canal bacteria (both aerobic and anaerobic) used as en-dodontic irrigants compared with CHX.
Antimicrobial effect --- Rue --- Sage --- root canal bacteria
The aim of this study was to evaluate the antimicrobialeffect of 10% water extraction of Salvadora persica (Miswak)when used clinically as an endodontic irrigant.Twenty four uniradicular teeth with necrotic pulps werechosen. The patients were divided randomly into 2 groups:Experimental group, in which water extract of Salvadorapersica (10%) was used as a root canal irrigant; and controlgroup, in which distilled water was used as a root canal irrigant.Bacteriological samples were obtained from the canal atthe step of working length determination (before the canalwas subjected to instrumentation and irrigation procedures),and at the end of the biomechanical instrumentation proceduresby using a sterile K–file. The file was separated from thehandle using a sterile wire cutter, and the severed portion wasplaced in a sterile screw–capped vial containing 5 ml of thioglycollatebroth as a transport media. A 0.1 ml of thioglycollatebroth was inoculated on each of two brain–heart infusionagar plates: One plate was incubated under aerobic conditions,and the other was incubated under anaerobic conditionsusing anaerobic jar and gas pack anaerobic system. Both plateswere incubated at 37 ºC for 24 hours; then, the number ofbacterial colonies was counted.The results revealed that 10% water extraction of Salvadorapersica is an effective antimicrobial agent when utilizedclinically as an irrigant in the endodontic treatment of teethwith necrotic pulps.
order to determine the number of root canals and the number of apicalforamens, 1528 endodontically treated. Teeth were examined. Out of 777 maxillary firstpremolars 9.1% had one canal, (8.6% of them had one apical foramen and 0.5% had twoforamens) and 89.7% had two canals (8.5% had one apical foramen and 81.2% had twoapical foramens). Only 1.2% of examined teeth had three canals and three foramens. Outof 751 maxillary second premolars 68.6% had one canal (65.1% of them had one apicalforamen and 3.5% had two apical foramens) and 31.4% of teeth had two canals (11.9% ofthem had one apical foramen and 19.6% had two apical forameris
Background: The aim of this study was to comparatively evaluate the push out bond strength (PBS) of root canalfillings using four different obturation techniques (single cone (SC), cold lateral compaction (CLC), continuous wave(CW), and carrier based gutta percha (CBG)).Materials and Methods: Forty mandibular premolar decoronated and instrumented with rotary ProTaper to F3 thenteeth were divided randomly into 4 groups of 10 teeth for each as follow: group (I) single- cone obturation withmatched-taper gutta-percha, group (II) cold lateral compaction technique, group (III) continuous wave ofobturation technique, and group( IV) carrier based gutta-percha technique. Zinc oxide eugenol (ZOE) sealer wasused as a root canal sealer for the four groups. After obturation of the root canals, all the roots were sectionedhorizontally at three levels in the apical, middle, and cervical thirds of each group. PBS test was performed usingdigital universal testing machine. Mode of failures was evaluated using digital stereomicroscope (40 X). Collecteddata were analyzed statistically using one way ANOVA and Tukey test.Results: PBS of CW and CBG significantly higher than SC and CLC, but significantly there were no differencesbetween CW and CBG, and between SC and CLC.Conclusion: Under the condition of this study it can be concluded that thermoplasticized techniques obtain superiorPBS of the filling materials in comparisons with cold gutta percha obturation techniques
The lower third molar is the most teeth failed to erupt in thealveolar process and the surgical extraction of lower third molar iswidely carried out in the dental clinic due to pathologic changeand prophylactically purpose, the damage of the inferior dentalcanal can be occurred when the lower third molar located deep andits root is closely to the inferior dental canal, the aim of this studywas to investigate the prevalence of the types of impacted lowerthird molar between male and female and to determine theradiographic relationship of the inferior dental canal to the rootapices of different types of the impacted lower third molar usingdental panoramic radiograph, eighty (80) panoramic radiographicimages ,40 male 40 female aged (18-41) were selected from thepool of data stored in the computer of the digital panoramicmachine, the teeth were divided according to the relation to thelower 7th molar (angulation), and the relation of the root apices tothe inferior dental canal according to the distance of root apicesfrom the inferior dental canal either far, close , superimposed. Theresult of this study showed that the mesioangular type ofimpaction is the most closely positioned to the inferior dentalcanal , and this may represent an independent risk factor forpostoperative paraesthesia, the majority of patients showed theposition of the inferior dental canal varies in relation to the rootapices of impacted mandibular third with the majority being inadjacent position , The superimposed relation of roots apices ofthe mesioangular impaction in male was 32% while in female was46% so the risk of the damage to the mandibular canal in femalemore than the male during the surgical extraction of lower thirdmolar, this variation should be appreciated by the oral surgeonwhen undertaking surgical extraction of the impacted mandibularthird molars.
Root canal --- impacted 3rd molar --- panoramic
Background Several root canal filling materials and techniques have been developed and studied, aiming to completely fill the root canals, their ramifications, and any anatomical variations, which are frequently observed. The objective is to investigate the push out bond strength of three different obturation materials GuttaFlow 2, Thermafil and GuttaCore at different levels. Materials and methods thirty extracted upper molars were collected and the palatal roots were sectioned at the CEJ of the tooth. The platal roots were instrumented with Hyflex CM rotary files to the size of 40/0.06. The instrumented samples were divided into three groups of ten samples each, the first group was obturated with GuttaFlow 2, the second group was obturated with Thermafil and the third group was obturated with GuttaCore obturating materials. After an incubation period of 7 days, each sample were sectioned into three sections of 2 mm thickness (apical, middle, coronal), each slice then introduced to the push out testing using a universal testing machine at a cross head speed of 0.5 mm/min.Results it showed Push-out bond strengths were significantly higher when canals were filled with GuttaCore than those filled with Thermafil and GuttaFlow 2. And Thermafil showed a higher significant difference than the GuttaFlow 2. It also showed that the bond strength values decreased from the coronal to the apical direction. Conclusion The thermoplasticized gutta-percha appears to achieve higher push out bond strength values than the cold flowable gutta-percha. With GuttaCore showed higher push out bond values than Thermafil. | https://iasj.net/iasj?func=search&query=kw:%22Root%20canal%22 |
In vivo models are employed in biomedical research to mimic human illness and create novel medications. However, they do not replicate the disease as it manifests in humans, and their application has not led to the discovery of innovative treatments that are efficient for a number of extremely common non-communicable diseases, including Alzheimer's disease. In fact, there is still a relatively high clinical failure rate in drug development, with an overall approval probability from Phase I of only 9.6%. On the other hand, human-based models, cutting-edge imaging methods, and human epidemiological research may enhance the development of secure and efficient treatments by improving our comprehension of illness aetiology and pathogenesis.
Animal models' usage in research to promote science and health has long been a contentious issue. Despite their contributions to several scientific advances, there are growing ethical and scientific issues regarding such research due to the development of new, cutting-edge methods that do not employ animals. According to the most recent data report on animal usage in EU Member States , basic and applied research accounts for the majority of animals used in science in the European Union (EU) and maybe elsewhere. However, these models sometimes produce conclusions that cannot be applied to the in vivo experience of a human. In light of this, the European Union Reference Laboratory for Alternatives to Animal Testing (EURL ECVAM) of the European Commission's Joint Research Centre conducted a number of studies on existing and new non-animal models in seven fields: respiratory tract diseases, neurodegenerative disorders, breast cancer, immune-oncology, autoimmunity, cardiovascular diseases, and immunogenicity of advanced pharmaceutical products. The regions were chosen after taking into account disease frequency and prevalence as well as the quantity of animal procedures carried out. This is mostly because preclinical animal testing on animals did not detect unanticipated toxicity or lack of efficacy in humans. Therefore, the development of animal-free, human-relevant techniques in various fields of the biological sciences is being driven by scientific concerns regarding the predictive value of animal models.
Each study was given a unique approach to generate a collection of pertinent non-animal models [2,3]. The approach used a predetermined set of search terms and was based on mutually accepted inclusion and exclusion criteria. Grey literature was also taken into consideration, including news, events, societies, etc.
Neurodegenerative diseases are chronic, incurable disorders that cause the progressive death or degeneration of nerve cells. They include Parkinson's disease (PD), Huntington's disease, motor neuron disease, Creutzfeldt-Jakob disease, multiple sclerosis, and Alzheimer's disease and other dementias. Of them, dementias have the highest disease burden , with Alzheimer's disease (AD) accounting for more than 60–70% of cases. Although AD does not presently have a cure [5,6], there are some symptomatic therapies available. Unfortunately, none of the animal models of Alzheimer's disease that are now available have construct or predictive validity, despite the fact that animal disease models are thought to be crucial in the development of new treatments. In the end, only human subjects can be used to test the hypothesis, and only the right equipment, including pharmacologically active intervention and clinical trials, may be used. | https://www.ijdrt.com/articles/advanced-nonanimal-models-for-respiratory-diseases-breast-cancer-and-neurodegenerative-disorders-eurl-ecvam-literature-review-seri-92840.html |
Antibodies continue to be developed as therapeutics for a variety of indications. Current methods of screening groups of antibodies typically focus on selecting for antibodies that bind to known proteins. Such selections can yield a large number of antibodies, few of which have therapeutically useful biological properties. Moreover, such antibodies are typically expressed as Fab or scFv fragments in prokaryotic or yeast systems. Most currently approved antibody therapeutics are, full-length antibodies, often human or humanized antibodies, that are usually expressed in mammalian cells. Therefore, development of therapeutic antibodies from libraries of antibodies typically involves a tedious, one-by-one conversion of selected antibody fragments to full-length antibodies. Subsequent testing of the full-length antibodies does not always yield results that correlate well with results obtained with the antibody fragments. The present invention presents a scheme for subjecting a moderately large group of multimeric, optionally Fc-containing, antibodies expressed by mammalian cells to a screen or a selection to directly identify antibodies that have a desired biological property.
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Governance and Anti-Fragility are Key to Robust Decentralized Systems, and Synthetix Intends to Stand Out
A prolonged debate in the space is that of whether decentralization is a spectrum or an absolute state. That is, can a system have different degrees of decentralization or is it simply just a binary function of ‘yes’ or ‘no’. Synthetix, one of the largest dApps on Ethereum, believes that the former holds, and the project intends to prove this by serving as a prime example of how control can move from a core team to the entire community, December 18, 2019.
Making Changes and Coordinating Decisions
At its core, decentralization of a protocol implies no entity or cartel can capture control and implement their will over the network at large. In addition to this, a network must be antifragile to attacks, meaning the attack should make the network stronger. In order to keep control out of the hands of the few, communities need to find a way to coordinate decision making in a robust manner, so as to keep everyone included and allow the best ideas to shine.
This is where the idea of a DAO comes in. Yes, the term ‘DAO’ has been taboo since the great hack of 2016, but this alluring concept has made a comeback by way of Moloch, MetaCartel, and others.
Synthetix aims to one day completely transition from the current state of governance towards a DAO. But before this can happen, the core team and community must be confident that the network can survive on its own.
How Synthetix Plans to Pivot to Decentralization
According to the co-founder of Synthetix, Kain Warwick, there are key aspects that need to be addressed in the context of this discussion: legal and operational structure, protocol changes, and product improvements.
Moving to a DAO solves most of the legal and operational issues. Given the sheer amount of progress in DAOs and smart contract coding since 2016, the quality and security of such organizations have drastically improved.
Protocol changes are already gradually finding their way towards a more community-driven approach. Synthetix recently changes its supply schedule as per the recommendations of a few active community members.
Finally, product improvement is supposedly being ushered through the partnership with ChainLink. As the most widely used oracle platform, the community can add new synths and request ChainLink to provide the necessary data to run said contracts.
Overall, the project seems to be on the right track as they have gained a stupendous amount of traction in a short period of time. Hopefully, Synthetix is able to prove that the shift from centralized control to decentralized management is very much possible. | https://btcmanager.com/governance-anti-fragility-decentralized-systems-synthetix/ |
Reviewing progress in mental health reform
The Commission undertook a mid-term review of Living Well: A Strategic Plan for Mental Health in NSW 2014-2024, the 10-year plan for mental health reform in NSW. As part of this review, the Commission consulted with a broad range of stakeholders around NSW to consider the progress that has been made against Living Well over the first five years and to identify priorities and opportunities for the remaining five years of the strategy.
Living Well in Focus 2020-2024 is the result of the mid-term review and identifies three whole-of-government strategic priorities that inform the direction of mental health reform over the next five years. These strategic priorities, which are underpinned by seven focus areas, will provide the best opportunity for good mental health and wellbeing of all people in NSW.
The strategic priorities will:
- strengthen community recovery and resilience
- strategically invest in community wellbeing and mental health
- ensure the right workforce for the future.
Our process
In partnership with people with lived experience of a mental health issue, their families’ carers and kinship groups, stakeholders, experts, clinicians and peer workers, the Commission commenced reviewing the progress made towards mental health reform in NSW in January 2019.
Over 18 months, the Commission held over 60 consultations to learn first-hand about the mental health and wellbeing issues facing communities and see examples of local reform. This information was analysed by independent researchers and supported by evidence, literature reviews, a community survey and tested with stakeholders to determine the priorities for reform over the next five years to develop Living Well in Focus.
Local examples of reform
Regional planning teams across NSW helped the Commission gather information on reform initiatives that demonstrate progress and align with one or more of the domains in Living Well. Read the definitions of the domains here.
Examples of local reform were showcased during the regional community consultations and 70 were selected to be published on the Commissions website. One case study from each region was voted by the regional planning team to be produced into a short video.
View the examples of reform by region below.
Resources developed throughout the review
In addition to regional consultations, the mid term review of Living Well is informed by evidence including:
- Community survey report
- Regional Consultation Report
- Literature reviews
- Evidence checks
Click here to view the resources.
Thank you for your participation!
It was a privilege to travel around NSW and learn from the voice and expertise of people with lived experience of mental health issues and caring, families and kinship groups, mental health and social support service providers and community members. We are grateful for the opportunity to hear from you about progress, challenges and future priorities for your region. Thank you for participating and sharing your experiences and expertise with us.
More information
To find out more you can:
- Watch this space for further updates.
- Subscribe to our newsletter for updates on the Living Well review project and other news. | https://nswmentalhealthcommission.com.au/living-well-agenda/living-well-mid-term-review |
New Student Orientation is seeking sophomores, juniors, and seniors to serve as crucial peer advisors, mentors, and leaders for the class of 2024 as they join the Dartmouth community. This role will require empathy, flexibility, creativity, patience, and the ability to find moments of connection and joy in the midst of an incredibly challenging time. The Class of 2024 will begin at Dartmouth at a moment and in a manner like no class in Dartmouth's history. This will demand a welcome and introduction like no class has received before.
Please consider applying to be an Orientation Peer Leader (OPL). The role of OPLs will combine elements of O-Team, First-Year Trips Leaders and Croos, and other upper-level student mentors plus new needs and responsibilities the moment demands.
Reflect on the ways you felt welcomed and supported when you arrived at Dartmouth. Draw on the values of community that inspire us to take care of each other, especially in challenging times. Join us in creating those feelings of connection in re-imagined ways for incoming students by highlighting what makes Dartmouth unique.
As part of New Student Orientation and welcoming the new class, Orientation Peer Leaders will:
Facilitate small-group, community-based experiences for a dedicated group of incoming students, for a period of approximately 7-10 days (which will include New Student Orientation) and potentially, in a more limited capacity, throughout their first term at Dartmouth.
Work both independently as well as collaboratively with other student leaders and staff to lead activities and conversations that will help foster a sense of connection, belonging, and responsibility in the Dartmouth community and the incoming class
- Create opportunities for relationship-building through both structured and informal activities and conversations during New Student Orientation.
Time commitment: As with everything at this time, this is a role and opportunity that is evolving. We anticipate this role will require both preparation and training before connecting with incoming students, as well as daily contact and conversations with those students during the period of New Student Orientation.
Compensation: OPLs who complete their commitments during training, New Student Orientation and in a more limited role throughout the fall term will receive a $500 stipend.
No residence requirement: All programming will happen virtually, so Orientation Peer Leaders do not need to be physically present in Hanover. However, internet access will be required; please let us know if you anticipate challenges regarding internet access.
Student standing: Orientation Peer Leaders must be in good academic and conduct standing at the College.
NOTE: OPLs will embody the best of Dartmouth and prioritize the programming intentions put forward by New Student Orientation in support of welcoming incoming students. However, that doesn't mean you have to love everything about Dartmouth. You do not even necessarily need previous leadership experience – the staff and students facilitating this program will do our best to help you thrive in the role. All you need is to be your authentic self and bring a sincere desire to welcome first-year students to Dartmouth and ease their transition in a thoughtful and supportive way. This program will be significantly stronger as students participate from all different corners of campus. Anyone can be an incredible peer leader; we want you to consider applying, because the '24s will value your perspective, care, and love.
Important accessibility notice:
We are committed to making this experience as accessible as possible for each leader. If you have any personal or ability-related concerns (regarding your own physical/mental/emotional health) that would be relevant to you serving in this role, please contact coordinators [email protected] or [email protected] to discuss possible accommodations. This information will not negatively impact review of your application. | https://students.dartmouth.edu/student-life/programs/orientation-peer-leaders |
Worker safety is crucial to responsible copper production. A critical safety parameter is the Occupational Exposure Limit (OEL), or the number of copper particles that can be in the air without physically or mentally affecting the worker. While copper is a naturally occurring element essential to the human body, continued exposure to copper particles in the air can have long-term adverse effects. Occupational exposure to copper particles may occur during the mining, smelting and refining processes of copper production. OELs ensure worker health is protected, and these limits must be tested and determined through scientific study. The International Copper Association (ICA) has commissioned research to examine the effects of exposure to airborne copper particles to help determine safe and optimal OELs.
Peer-Reviewed Studies on Copper Exposure in the Workplace
EXPOSURE TO COPPER-CONTAINING DUST PARTICLES: NO DETECTABLE ADVERSE EFFECT
A long-term medical surveillance study published in the Journal of Occupational and Environmental Medicine concluded there was no detectable effect of copper-containing dust particles on lung health or chronic inflammation among the 104 German copper smelter workers who participated in the study when compared to 70 of their colleagues who worked in the same facility with different metals. This study tracked worker health for a total of 22 years from 1972 to 2018.
BIOMARKER STUDY OF SMELTER WORKERS: NO ADVERSE EFFECTS FROM COPPER-CONTAINING DUST EXPOSURE
In 2019, follow-up research of the same German copper smelter workers examined the effects of copper particles in dust on biomarkers (via a blood and sputum analysis) connected to lung function and inflammation. The biomarker study also found there were no detectable effects from the copper-containing dust particles.
REPEATED DOSAGE OF COPPER COMPOUNDS FOUND TO BE NON-TOXIC IN RAT STUDY
A peer-reviewed study published in Toxicology examined the effects of repeated doses of two common copper compounds (e.g., Cu2O dicopper oxide and CuSO4.5 H2O copper sulphate pentahydrate) in rats over a period of 2 weeks followed by a 13-week recovery period. A longer study was conducted on just the Cu2O compound over a 28-day period. The studies aimed to evaluate whether respiratory toxicity resulted from continued exposure to copper in the air. The rats demonstrated some inflammation at the cellular level, but this was deemed to be a result of a localized and natural adaptive response, resolving itself fully during the recovery period. No adverse effect or toxicity was observed in the 14-or 28-day studies.
Copper is essential to the modern world.
Learn more about copper's uses in sustainable development. | https://copperalliance.org/policy-focus/health-safety/occupational-exposure/ |
MEDICAL STUDENTS AT NUI Galway are offering their services to the west’s teddy bears.
The 10th annual Teddy Bear Hospital at NUIG will take place next Thursday and Friday, 22 and 23 January. The event will see over 1,500 sick teddy bears admitted to the hospital, accompanied by their minders, 1,500 primary school children.
The event is organised by the Sláinte Society, the NUI Galway branch of the International Federation of Medical Students Associations, and up to 200 medical and science students will diagnose and treat the teddy bears. In the process, they hope to help children, ranging in age from 3-8 years, feel more comfortable around doctors and hospitals.
Over the years, children have come along with teddy bears suffering from an imaginative range of sore ears, sick tummies and all kinds of others weird and wonderful ailments.
Katie Lynam, a second year medical student at NUI Galway and co-auditor of Sláinte Society, said: “This year we are celebrating ten years of Teddy Bear Hospital and it is going to be our biggest ever with 1,500 children attending over the two mornings. We hope to create a fun, friendly atmosphere for both the children and our volunteers, and are looking forward to a busy couple of days!”
This year, 22 local primary schools are participating in the event. On arrival at the Teddy Bear Hospital on campus, the children will go to the ‘waiting room’, which contains jugglers and face painters.
Then the children and their teddy bears are seen by a team of Teddy Doctors and Teddy Nurses, who will examine them. The students will have specially designed X-ray and MRI machines on hand, should the teddy bears need them. | https://www.thejournal.ie/teddy-bear-hospital-in-galway-1881178-Jan2015/?switcher=touch |
Innergex Renewable Energy Inc (TSE:INE) – Stock analysts at Raymond James reduced their FY2020 earnings estimates for Innergex Renewable Energy in a research note issued to investors on Wednesday, May 15th. Raymond James analyst D. Quezada now expects that the company will earn $0.48 per share for the year, down from their prior estimate of $0.53. Raymond James currently has a “Outperform” rating and a $17.00 target price on the stock.
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INE has been the subject of several other reports. National Bank Financial upped their price target on shares of Innergex Renewable Energy from C$17.50 to C$18.50 and gave the company an “outperform” rating in a research report on Tuesday, February 19th. TD Securities lowered shares of Innergex Renewable Energy from a “buy” rating to a “hold” rating and increased their target price for the stock from C$15.00 to C$16.00 in a research note on Friday, March 1st. Desjardins reissued an “average” rating and set a C$15.50 target price on shares of Innergex Renewable Energy in a research note on Wednesday, April 24th. Finally, BMO Capital Markets raised their price objective on shares of Innergex Renewable Energy from C$13.50 to C$15.00 and gave the company a “market perform” rating in a research note on Wednesday, February 13th. Three research analysts have rated the stock with a hold rating and two have issued a buy rating to the company’s stock. The company currently has a consensus rating of “Hold” and a consensus target price of C$15.86.
INE stock opened at C$14.01 on Monday. The company has a market cap of $1.86 billion and a P/E ratio of 68.01. Innergex Renewable Energy has a 1 year low of C$11.66 and a 1 year high of C$14.75. The company has a quick ratio of 0.29, a current ratio of 0.36 and a debt-to-equity ratio of 490.74.
Innergex Renewable Energy (TSE:INE) last released its quarterly earnings data on Wednesday, February 27th. The company reported C$0.10 earnings per share (EPS) for the quarter, topping the Zacks’ consensus estimate of C$0.06 by C$0.04. The company had revenue of C$166.16 million for the quarter, compared to analyst estimates of C$162.30 million.
The firm also recently disclosed a quarterly dividend, which will be paid on Monday, July 15th. Shareholders of record on Friday, June 28th will be paid a $0.175 dividend. This represents a $0.70 dividend on an annualized basis and a dividend yield of 5.00%. The ex-dividend date of this dividend is Thursday, June 27th. Innergex Renewable Energy’s dividend payout ratio (DPR) is 330.10%.
In other Innergex Renewable Energy news, Director Daniel Lafrance acquired 2,000 shares of the company’s stock in a transaction on Thursday, May 16th. The shares were purchased at an average cost of C$13.82 per share, with a total value of C$27,640.00. Following the completion of the purchase, the director now owns 35,000 shares of the company’s stock, valued at C$483,700.
Innergex Renewable Energy Company Profile
Innergex Renewable Energy Inc operates as an independent renewable power producer. It develops, owns, and operates run-of-river hydroelectric facilities, wind farms, solar photovoltaic farms, and geothermal power facilities. The company operates through five segments: Hydroelectric Generation, Wind Power Generation, Solar Power Generation, Geothermal Generation, and Site Development. | |
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79
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MY LIST
21378
Humbodlt y Cordoba
City:
Buenos Aires
Neighborhood:
Palermo
Area:
Touristic
Code:
21378
People:
2
Bedrooms:
1
Category:
Fashion
M² / Sq.Feet:
45 / 484
Single beds:
0
Double beds:
1
Single sofa bed:
0
Double sofa bed:
0
Distance (in minutes)
Subway walking
10
Walking to downtown
45
Downtown by bus
30
Downtown in cab
25
Services
Amenities
Allowed
Internet
Air.
Balcony
Swim.
Kids
Pets
Heating
Cot
Garage
Gym
Smokers
Hand.
Yes-apartments.com
DETAILS
CONDITIONS AND AVAILABILITY
MAP AND NEIGHBORHOOD
Kitchen
Microwave
Oven
(Electric) Coffee maker
Refrigerator
Misc
Air conditioning
Heating
Towells
Sheets and towels
Home appliances
Hairdryer
Iron
Amenities
Internet Tv Music
Wifi
CD Stereo
Cable TV
Dvd
Phone
TV
External areas
Balcony
Brief description
SPECIAL CONSIDERATIONS: The use of the garage is not included in the price and we must confirm it's availability. Two room apartment (one bedroom) facing the street, balcony, kitchen and full bathroom. Very sunny and airy Wooden floors, heating, fully equipped, optional garage
Payment method
:
Payment in cash, on check-in
Security deposit (returned upon check out)
One week:
300
15 days:
300
More months:
800
Check in:
9 am/ Late check in 20 U$S
Check out:
7 pm /Late check out 20 U$S
Deliver keys in:
Restrictions
Not allowed
Not allowed
No facilities for handicapped
Maid service
: Semanal/weekly
Minimum stay (days)
: 3
Maximum stay
: 150
Availability
Available
Not available
Inquire
Barrio Palermo
Palermo is an area linked more directly to the Park on February 3 or Forests of Palermo, as it is popularly known. A beautiful place built on the attractiveness of the area green and recreational largest city. The villages of the widest Av. Libertador, the Japanese Garden, Rosedale, the golf course and the lake are just some of the walks daily and more picturesque.
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Date arrival: 05/04/2019 | Date departure: 05/11/2019
7 Night/s | Buenos Aires | Passengers
Price
U$D
58
daily
U$D
409
for 7 ds.
U$D
79
Res. (included)
BOOK
MY LIST
21378
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Contact
Argentina
Buenos Aires
(54) 11-5273 6361
España
Madrid
(34) 91- 151 6780
USA
New York
(1) 516- 693 0810
U.K. | http://www.yes-apartments.com/apartment/21378/Buenos_Aires/Palermo/Humbodlt_y_Cordoba |
Evaluation of historical info means that science, on many occasions has proved to be a curse for humanity somewhat than a blessing. Your membership promotes scientific literacy and gives hundreds of thousands of scholars opportunities to fall in love with science. Science is a discipline where conventional classroom teaching just isn’t sufficient and it turns into effective only when teachings are accompanied by creating experiments and deducing conclusions from the experiments.
E f 102 104 That is done partly through statement of natural phenomena, but also via experimentation that tries to simulate pure events underneath managed circumstances as acceptable to the discipline (within the observational sciences, resembling astronomy or geology, a predicted statement may take the place of a managed experiment).
26: Science has additionally brought medical equipments that help to save lots of human life. Science is the research of the world around us. Scientists learn about their topic by observing, describing, and experimenting. A lot of the science we all know at this time was found utilizing the Scientific Technique The Scientific Method is a method scientists use to get correct results from their experiments.
The content material requirements offered in this chapter define what college students should know, understand, and be capable to do in pure science. To find out whether or not students are influenced to turn out to be scientists (it’s human nature to ask questions and pay attention to what surrounds you; science workouts are good starting vehicles for academics to find out if their students are studying), it is important to allow them to perceive the strategies or processes of science by way of palms-on actions or laboratory work.
It’s potential we will make one of the biggest discoveries of all time throughout the subsequent 20 years if we proceed at the charge of technological development that we’re going at. We could possibly know the reply to if there’s other life out there, not from earth.
Instead, supernatural explanations needs to be left a matter of private belief outdoors the scope of science Methodological naturalism maintains that proper science requires strict adherence to empirical examine and impartial verification as a process for properly developing and evaluating explanations for observable phenomena.
For instance, Begbie explores the role that music plays in accommodating human experience to time, whereas Wolterstorff discovers a duty toward the visible aesthetics of public areas. In essence then, forensic science is the appliance of scientific knowledge in a legal context.
The government funding proportion in certain industries is larger, and it dominates analysis in social science and humanities Similarly, with some exceptions (e.g. biotechnology ) authorities provides the bulk of the funds for basic scientific analysis Many governments have dedicated businesses to assist scientific research.
Popcorn is one many peoples favorite snacks. Science coverage is an area of public coverage concerned with the policies that affect the conduct of the scientific enterprise, together with research funding , typically in pursuance of different nationwide coverage goals resembling technological innovation to promote business product growth, weapons growth, well being care and environmental monitoring.
A relational narrative for science that speaks to the need to reconcile the human with the material, and that pulls on historic wisdom, contributes to the construction of recent pathways to a healthier public discourse, and an interdisciplinary instructional challenge that’s trustworthy to the story of human engagement with the apparently chaotic, inhuman materiality of nature, yet one whose future have to be negotiated alongside our own.
In 1976, Congress established the White House Office of Science and Expertise Policy (OSTP) to supply the President and others inside the Government Workplace of the President with advice on the scientific, engineering, and technological features of the economy, national security, homeland safety, health, overseas relations, the atmosphere, and the technological recovery and use of assets, among other topics.
Pure sciences and social sciences are totally different only in what they study. Science coverage also refers back to the act of applying scientific data and consensus to the development of public insurance policies. Can be clarified through the experiments in a science camp.
Earth Science
Science and technology present many societal advantages, such because the enhancement of economic progress or quality of life. Within the wake of the latest developments and the new calls for that are being placed on the S&T system, it’s crucial for us to embark on some major science initiatives which have relevance to national wants and which will even be related for tomorrow’s technology.
Iron supplements, for example, can change foods’ style and coloration, making individuals much less possible to make use of them, says Ana Jaklenec, a biomedical engineer on the Massachusetts Institute of Expertise (MIT) in Cambridge. A science is the research of some aspect of human behaviour, for instance sociology or anthropology.the modern science of psychology.
Science Information
Natural sciences are typically referred to as the onerous sciences. Check out our science experiments for center faculty and highschool, they really are one in 1,000,000 yet straightforward.
Evaluation of historical info suggests that science, on many events has proved to be a curse for humanity slightly than a blessing. Your seek for a few of the most straightforward and fun science experiments gets over, right here. VoYS is a novel and dynamic community of early profession researchers committed to enjoying an energetic role in public discussions about science. I generally suppose persons are too fast to proclaim themselves “too dumb to grasp that science” when in truth the science is poor.
Though each theology and philosophy suffer frequent accusations of irrelevance, on this point of brokenness and confusion in the relationship of people to the world, present public debate on essential science and technology point out that each strands of thought are on the mark.
Science has brought sophistication to human life. Earth Science, which is the study of the Earth. In other cases, experiments are incomplete as a result of lack of materials or time. A science fair mission on testing consuming water can help them be taught what is within the water they use.
A superficial evaluation would possibly conclude that the charges of ‘intellectual imposture’ and ‘uncritical naivety’ levied from either aspect are merely the millennial manifestation of the earlier ‘two cultures’ conflict of F R Leavis and C P Snow, between the late-fashionable divided mental world of the sciences and the humanities.
fifth Grade Science Honest Projects For Profitable Science Experiments
In our fashionable world it appears as if pure science is completely incompatible with faith and (to a big lengthen) philosophy. Whereas, environmental science is multidisciplinary in nature, and consists of the examine of environmental techniques, integrating each its biological and physical points, with an interdisciplinary approach. Essentially the most conspicuous marker of this alteration was the alternative of “pure philosophy” by “pure science”.
It has many branches that embody, however usually are not limited to, anthropology , archaeology , communication studies , economics , historical past , human geography , jurisprudence , linguistics , political science , psychology , public well being , and sociology Social scientists could undertake numerous philosophical theories to review individuals and society.
4th Grade Science Fair Project Ideas
A brand new period of science has begun. The investigation of natural phenomena by means of statement, theoretical clarification, and experimentation, or the information produced by such investigation.♦ Science makes use of the scientific methodology, which includes the cautious observation of pure phenomena, the formulation of a speculation, the conducting of one or more experiments to check the speculation, and the drawing of a conclusion that confirms or modifies the hypothesis.
For example, positivist social scientists use methods resembling those of the natural sciences as tools for understanding society, and so outline science in its stricter fashionable sense Interpretivist social scientists, by contrast, might use social critique or symbolic interpretation somewhat than constructing empirically falsifiable theories, and thus deal with science in its broader sense.
science direct gratis, science adalah ilmu, medical science artinya
Science Undertaking Ideas are sometimes laborious to return by. When your baby comes home from faculty normally in January or February and says, hey Mom and Dad my trainer despatched dwelling this observe that we’ve got a science project due in three weeks. When science college students present true knowledge, it additionally gives legitimacy to their science initiatives. Natural sciences are generally called the onerous sciences. Their research areas embrace biodiversity, crop and meals sciences, environmental and food sciences, water and sanitation, and animal diet. | https://bernie2016events.org/constructive-psychology-is-a-booming-business-but-is-it-science-religion-or-one-thing-else.html |
Patient registries are organised systems that use observational methods to collect uniform data on a population defined by a particular disease, condition, or exposure, and that is followed over time. Patient registries can play an important role in monitoring the safety of medicines. The European Medicines Agency (EMA) has set up an initiative to make better use of existing registries and facilitate the establishment of high-quality new registries if none provide an adequate source of post-authorisation data for regulatory decision-making.
The initiative for patient registries, launched in September 2015, explores ways of expanding the use of patient registries by introducing and supporting a systematic and standardised approach to their contribution to the benefit-risk evaluation of medicines within the European Economic Area.
Regulators and pharmaceutical companies currently face a number of challenges in using existing registries or establishing new ones, including a lack of:
- coordination between ongoing initiatives at national and international levels;
- harmonised protocols, scientific methods and data structures;
- data sharing and transparency;
- sustainability.
These factors have led to inefficiency and a duplication of efforts. To address the problems, the EMA initiative seeks to create a European Union-wide framework on patient registries, facilitating collaboration between:
- registry coordinators, such as physicians' associations, patients' associations, academic institutions or national agencies responsible for overseeing healthcare services;
- potential users of registry data, such as medicines regulators and pharmaceutical companies.
To support the initiative, EMA has set up a cross-committee task force on registries, comprising representatives from EMA scientific committees and working parties and experts from national competent authorities. EMA has published the task force's strategy and mandate:
On 8 November 2018 the cross-committee task force published a discussion paper on methodological and operational considerations in the use of patient disease registries for regulatory purposes. The open consultation was closed on 30 June 2019. EMA will consider the comments and finalise the document in consultation with the relevant EMA committees by the end of 2019.
EMA has created an inventory of patient registries in the resources database of the European Network of Centres for Pharmacoepidemiology and Pharmacovigilance (ENCePP). The inventory aims to facilitate the interaction between stakeholders and existing patient registries.
EMA encourages patient registry owners whose registries are not listed in the inventory to add their registries to the database.
EMA also published a guidance document on how to search the ENCePP resources database for information about patient registries and on how to upload new registry details:
EMA has held a stakeholder workshop to better understand the barriers and facilitators to collaboration between stakeholders. The workshop report which provides recommendations on actiions to improve stakeholder collaboration and optimise the use of registries to support regulatory decision-making:
EMA has also held disease-specific workshops where participants provided recommendations on the use of registries in these disease areas, including on core data elements, consents, governance, data sharing and interoperability. EMA has published the workshop reports which may act as models for guiding use of patient registries in other disease areas:
- Haemophilia registries workshop (08/06/2018)
- Chimeric antigen receptor (CAR) T-cell therapy registries workshop (09/02/2018)
- Multiple sclerosis workshop - Registries initiative (07/07/2017)
- Cystic fibrosis workshop - Registries initiative (14/06/2017)
EMA has provided qualification opinions on two registries, the European Cystic Fibrosis Society (ECFS) patient registry and the Cellular Therapy module of the European Blood and Marrow Transplant (EBMT) registry, describing the contexts in which EMA considers the use of registry data suitable. They may also provide useful information for registry stakeholders on required data and quality standards:
- Qualification Opinion on The European Cystic Fibrosis Society Patient Registry (ECFSPR) and CF Pharmacoepidemiology Studies
- Qualification opinion on Cellular therapy module of the European Society for Blood & Marrow Transplantation (EBMT) Registry
The task force welcomes the opportunity to work with stakeholders in facilitating the development of implementation plans to support the delivery of the workshop recommendations.
It also welcomes feedback on these recommendations and interest from stakeholders in taking part in activities facilitating the use of registries for regulatory decision-making. | https://www.ema.europa.eu/en/human-regulatory/post-authorisation/patient-registries |
by S. H. Watkins, Sr.
The works of two Rhode Island-based photographers will be in the spotlight in January and February with a gallery exhibition called “Visual Jazz.” The presenters were kind enough to provide us with four samples of the fine photography for inclusion with this article. The month-long exhibition begins January 16, 2001 at the CapitolArts Gallery, Suite #49 in the historic Arcade building in downtown Providence and will focus on works documenting jazz and blues genres.
Photographers Ken Franckling and McDonald Wright specialize in capturing the spirit and “moments of truth” found in jazz performance. Sometimes they photograph side by side, yet come up with distinct interpretations and visions that complement each other’s work.
“In January, award-winning filmmaker Ken Burns will turn more of America’s eyes and ears onto the role jazz has played in shaping America’s cultural values throughout the 20th century through his JAZZ series on public television. In similar fashion, we want to spotlight the distinctive, yet complementary visions of two artists whose work has added to the legacy of great jazz photography in America and around the world,” said Bob Rizzo, director of CapitolArts Providence.
Franckling’s black & white and color works selected for Visual Jazz include timeless images of Miles Davis, Stan Getz, Dizzy Gillespie, Sonny Rollins, Wynton Marsalis, Dave Brubeck, Sarah Vaughan, Gerry Mulligan, Cassandra Wilson and blues legends Sippie Wallace and Eric Clapton.
Wright’s selected color photography has a painterly abstract quality resulting from multiple imaging in the camera. His subjects include Slide Hampton, Renee Rosnes, Geri Allen, Ahmad Jamal, John Scofield, Gato Barbieri, Wallace Roney, Jon Lucien Dominique Eade and Ben Allison.
Ken Franckling is a veteran arts writer and free-lance photographer. He covers the jazz scene throughout the Northeast with occasional journeys to other regions in pursuit of essential musical moments. His work since 1983 includes some of the most riveting photographs taken of Miles Davis during the final five years of his life. His images have been published extensively and are in many private collections.
McDonald Wright is a native of Henderson, N.C., and a 1996 graduate of the Rhode Island School of Design. He began a serious focus on the New England jazz scene in 1998. Over the past three summers his work intensified as he built a solid body of personal work, primarily at concerts in Boston, Newport and Providence.
Visual Jazz is open Tuesdays through Saturdays from January 16 to February 10 from 11 a.m. to 3 p.m. on Tuesdays through Fridays and 11 a.m. to 4 p.m. on Saturdays. An opening reception is scheduled from 5 to 8 p.m. on Thursday, January 18.
All photography on this page is ©2000 Ken Franckling and McDonald Wright.
Unauthorized use or reporduction without explicit written permission is strictly prohibited. | http://jazzusa.com/ken-franckling-and-mcdonald-wright-visual-jazz/ |
Work Inclusivity Research Centre (WIRC) sponsors Greater Birmingham Chambers of Commerce Growth Through People Campaign 2021 sharing expertise with local business for building back better workplaces.
WIRC brings together Birmingham Business School academics from a wide range of disciplinary perspectives and backgrounds to create one of the leading centres for inclusivity research in the UK. Working with key partners to coproduce research that is both academically rigorous and implementable in strategic decision-making at all levels of business practice; the Centre actively works to have a positive impact on policy making and wider society.
Regional business leaders can access the latest insights on managing flexible working, building trust in remote teams, and how to create family friendly workplace policies.
Join our WIRC experts along with prominent regional business advocates, Henrietta Brealey from Greater Birmingham Chambers of Commerce and Pam Sheemar from NatWest, for a discussion around building back more inclusive workplaces post pandemic. Chaired by Professor Joanne Duberley the webinar will cover how employers can create a more inclusive workplace, why this is needed and what the workplace of the future could be like.
Building Back Better Workplaces Webinar: Thursday 18th March from 8:00 – 9:30am
Book now to attend all the free Growth Through People expert-led webinars:
- Managing Flexible Working Post-Pandemic, 2nd March, 08:00-09:30
- Tricky Conversations, 3rd March, 12:30-14:00
- Re-Establishing Employee Engagement, 9th March, 14:30-16:00
- Navigating Employment Support to Access New Talent, 11th March, 08:00-09:30
- Making Progress in Diversity and Inclusivity – What Works? 16th March, 12:00-13:30
- Supporting Your Pipeline of Female Talent, 17th March, 14:30-16:00
- Harnessing Resilience for Future Challenges, 23rd March, 08:00-09:30
- Leading for Growth, 24th March, 12:00-13:30
The campaign will come to an end with a digital conference on Tuesday 30th March. | https://www.birmingham.ac.uk/news/2021/building-back-better-workplaces |
A fun art activity to explore organic shape.
Use this teaching resource when studying shape as one of the Elements of Visual Art. Students use their knowledge of organic shape to create a mobile. After searching for organic shapes in an artwork, they cut similar shapes from coloured card and hang them from a coat hanger. Pipe cleaners and beads are added for decoration.
Featured in
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National Curriculum Curriculum alignment
- Art and design
Pupils should be taught to develop their techniques, including their control and their use of materials, with creativity, experimentation and an increasing awareness of different kinds of art, craft and design. Pupils should be taught: to create sket...
- Key Stage 2 (KS2) - Lower
Key Stage 2 (KS2) - Lower covers students in Year 3 and Year 4.
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Are you having trouble downloading or viewing this resource? Please try the following steps: | https://www.teachstarter.com/gb/teaching-resource/organically-abstract-mobile-activity-2/ |
I had a request to do soap carving again, so I got a pack of Ivory soap bars and let the boys make a humongous mess with dull knives and various carving implements. Ian carved a couch and I got as far as a robot head before I realized I’d been abandoned with a mountain of soap shavings.
I rounded the boys back and up and we scooped the shavings and scraps onto a paper plate and popped them into the microwave to watch it grow into a giant Jabba the Soap monster (due to the moisture and air expansion).
We then floated the fluffy, leathery “island” in the bath, and all the boys got in with it and obliterated it. Instant bubble bath for the next 4 days. I pressed some more left overs into cookie cutters to make shaped soap for the bath. Not gorgeous, but passable.
Next, I got on a crayon kick. Inspired by a dinner “campfire,” I experimented with crayons and a candle to “draw with dots.” But, I got busted
using melting Elijah’s crayons without permission, so I put that idea on hold for later.
All over Pinterest, I’ve seen the projects where crayons are glued to a canvas or board and then melted down the page. We gave that a go. It was pretty cool. Next I want to try using a cluster of greens and then flip the picture over to look like a tree, or use warm colors to do a fire.
I’ve also seen some mixed-media projects that use a paint background with melted crayon shavings on top that I wanted to try. From left to right is mine, Ian’s, Isaac’s and Elijah’s. There were leftover shavings and paint, so I got some heavy paper and painters tape to play with blocking out a letter. | https://sahmhill.com/2013/10/08/soapy-waxy-stuff/ |
Programmes and initiatives also aimed at upskilling skills of IHL students and working professionals.
KUALA LUMPUR, 11 February 2015 – In the move to support Government efforts in transforming the national education agenda, the Multimedia Development Corporation (MDEC) scored another round of success in its ICT-focused talent enhancement programmes.
The completion of the programmes for 2014 was marked with the hosting of an awards ceremony, “Talent Partners Appreciation and Networking Event 2015”, which was held earlier this week. The event was organised by MDEC’s Talent Division, which oversees the Knowledge Workers Development Centre in Cyberjaya and undertakes various programmes and initiatives to enhance the skills, knowledge and experience of undergraduates, fresh graduates, newly employed graduates, working professionals besides generating interest among students in the fields of science, technology, engineering and mathematics.
Ms. Ng Wan Peng, Chief Operating Officer said, “MDEC has continuously nurtured knowledge workers via the collective efforts undertaken with private sectors in Malaysia, to ensure the consistent and sustainable supply of skilled workforce and also help address the talent skills gap in the local ICT sector. MDEC’s training initiatives continues to play a crucial part in the effort to boost the workforce skills in the ICT industry, as we head towards a high income knowledge-based digital economy,” added Wan Peng.
“Moving forward, we will focus more towards critical areas such as Big Data Analytics (BDA), catalyzing the growth of the creative content industry, expanding Global Business Services and Internet of Things, among others” she continued.
From 2011 to 2014, a total of 38,377 students (comprising undergraduates, graduates and unemployed graduates), as well as 12,363 working professionals have been trained and benefitted from the various skills development and enhancement programmes initiated and undertaken by MDEC’s Talent Division. These programmes have also included the participation of over 1,200 companies and 40 IHLs.
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Divination is an attempt to gain insight into a person’s present, past and future by way of an occult standardized process. The term collectively describes a variety of methods which are used around the world to foresee or foretell. Even though all these methods serve the same purpose which is to provide solutions to problems at hand, they are seen as different techniques in different cultures and religions.
The practice of divination connects individuals with divine wisdom or a higher mind. A systematic approach is followed to get answers to the questions that may be haunting a person. Whether it is something related to a job, career, love life, relationships, family or life in general, divination is useful in every respect. However, the practice is far more different than what the modern day practice of fortune-telling is.
As its name suggests, Divination is inspired by God. It has a formal or ritual and often social character within a religious context, which provides people with access to information that they aren’t able to perceive through ordinary means.
History
Divination has been around for decades. The idea of getting an insight into one’s past, present and future is as old as mankind itself. Despite religious persecution and scientific criticism, the tool of Divination has survived through the centuries and is being used by people as a spiritual resource to help them make well informed decisions in their life.
Categories of Divination
According to the American psychologist Julian Jaynes, divination is categorized into the following four types:
The question is asked while riffling the pages of a holy book, and the first paragraph that is laid upon by their eyes is considered as the answer to the question at hand. Other forms of spontaneous divination includes methods like Feng Shui, auras etc.
How Divination works?
Several techniques have been developed for Divination. However, the basic principle of each of these techniques remains the same - Concentrating on the target question and interpreting the signs and symbols that the diviner is encountered with. The major obstacle that beginners face in divination is difficulty in harnessing the subconscious mind. That is because, the human brain remains active all the day because of social influence, and is unable to do away with the clutter of regular life, and focus completely. However, once the subconscious mind is actuated, there are various techniques that can be adopted to foretell the future. Each diviner has a different method that works best for them, and the most important thing is to find out which method helps you have the quickest and accurate results.
Some common methods of Divination
Get the psychic reading you need here, by connecting with our gifted clairvoyant medium, Meryem. She is able to provide you with accurate and powerful readings, using her psychic clairvoyant gift, tarot cards, crystal ball and rune readings.
You may connect with her either using the phone, using web technologies such as Skype or through a face to face meeting. | http://www.meryem.com/psychic-article-details.aspx/divination--know-your-present-past-and-future/4551 |
. The figures given on this page are according to the data published by government of Punjab.
The details are divided into following sections:
Crops Grown in Punjab
Fruits Grown in Punjab
Vegetables Grown in Punjab
Rural Population and Agriculture Workers
Role of Agriculture in Economy
Punjab Agricultural University
Area under Agriculture
Forest Land
Irrigation - Rainfall, Canals and Tubewells
Agriculture Minister of Punjab
Agriculture based Industries
Crops Grown in Punjab
In Punjab, mainly the two cereal crops, wheat and rice, are grown in rotation during an year. Rice is the principal crop of Kharif season and wheat is the principal crop of Rabi season. Other than wheat and rice, some quantitiy of maize and barley is also grown. Other cereal crops like jowar, bajra etc. are either not sown or are produced in very small quantity.
Punjab, having only 1.54% area of India, is the largest contributor of wheat and rice in the central pool. During 1980-81, the state's share in the central pool was - rice 45% and wheat 73%. These figures for the year 2014-15 are Rice 24.2% and Wheat 41.5%. Being the largest contributor of main cereals to the central pool, Punjab has earned the title of
Granary of India
or
Food Basket of India
. Note that these figures are contribution to the central pool, not the percentage of total cereals produced in India. If we consider the data of last few years,
Punjab produces roughly 12% of the total cereals produced in India
.
Other main crops of Punjab are cotton and sugarcane. Gram and some other pulses are also grown in some areas of Punjab. In the oil seeds category, rapeseed and mustard are the major contributors. Groundnut, sesamum and sunflower are also cultivated, but only in small area. The production of vegetables and fruits is discussed in the next section.
The area under the major crops and their production quantity is given below (estimated figures for year 2016-17):
Crop
Area (Thousand Hectare)
Production (Thousand Metric Ton)
Year 2016-17
Year 2015-16
Year 2016-17
Year 2015-16
Wheat
3468
3506
17636
16068
Rice
3033
2970
12638
11803
Maize
118
127
445
474
Barley
9
12
32
44
Pulses
20
20
12
10
Rapeseed and Mustard
31
32
44
42
Sunflower
6
6
11
10
Total Oil Seeds
42
48
58
52
Sugarcane
87
92
591
558
Cotton
285
335
1267
1389
According to 2015-16 data, the total production of rice in India was 104408 thousand metric ton and that in Punjab was 11823 units. It means Punjab produced 11.32% of the total rice produced in India. The production of wheat in India was 92287 units and the same in Punjab was 16077 units, which is 17.4% of the wheat production in India. The total production of cereals (includes wheat, rice and other cereals like maize, jowar, bajra, barley etc.) in India was 251566 units and that in Punjab was 28400 units. According to these figures, Punjab produced almost 11.3% of the total cereals produced in the country.
Fruits Grown In Punjab
The state mainly produces the food grain crops and most of the fruits are imported from the other states of the country. The main fruit grown in Punjab is Kinnow. There are vast areas covered with Kinnow orchards, mostly in the Fazilka, Muktsar and Firozpur districts. The production is more than enough for consumption within the state and it is also sent to other states in large quantities. Other than Kinnow, Guava and Mango are the other main fruits of Punjab. Guava is mainly grown in Firozpur district and the production is enough for consumption within the state. Mango orchards are mainly located in the Gurdaspur and Hoshiarpur districts of Punjab, but the production is small as compared to the total consumption of the state. Pear, Peach, Lemon and Ber are also produced in good quantity and it is enough for the state's own consumption. For the commonly consumed fruits like apple, mango, banana, grapes, papaya and waterlemon, the state depends mostly on other parts of the country. The area and production of different fruits is given below:
Crop
Area (Hectare)
Production (Metric Ton)
Kinnow
50147
1162455
Guava
8103
182089
Mangoes
6748
113687
Pear
2910
66814
Peach
1782
31675
Litchi
2320
37637
Orange and Malta
2813
23250
Ber
1516
25420
Total Fruits
80079
1700462
Vegetables Grown in Punjab
The only major vegetable produce of the state is Potato. The other main vegetables produced in the state are cauliflower, lady finger, tomatoes, carrot and raddish. The total production of vegetables is small as compared to the overall consumption within the state. The area under vegetables is given below:
Crop
Area (Hectare)
Potato
91627
Onion
872
Other Vegetables (Winter)
22377
Other Vegetables (Summer)
18729
Total Vegetables
133885
Role of Agriculture in Economy
Agriculture and allied fields like dairy, fisheries, animal husbandry are a major source of employment in Punjab. These sectors play an important role in the economy of Punjab, although the percentage contribution of agriculture in the state income is decreasing every year. According to latest figures available (2016-17), the contribution of agriculture to the total state income or GDP of Punjab is 17.23%. The contribution of agriculture and allied industries to the gross state domestic product (GSDP) at current prices is 29.26%. It was 32.65% in 2004-2005 and is decreasing every year.
Contribution of Agriculture to GDP (2016-17)
Agriculture = 17.23%, Livestock = 9.34%
Forestry and Logging = 2.33%, Fisheries = 0.36%
Total = 29.26%, Total (2004-05) = 32.65%
Also check our page containing the detailed information about
GDP of Punjab
.
Rural Population and Agriculture Workers
In Punjab, majority of the population lives in villages where main occupation of the people is agriculture or related to agriculture like dairy or animal husbandry. According to 2011 census data, the total rural population of Punjab is 1,73,44,192 i.e. 1.73 crore, which is 62.5% of total population of the state.
Agriculture Workers
- According to 2001 figures, the total workers in Punjab is 91.27 lakh and number of agriculture workers is 35.55 lakh. The percentage of agriculture workers to total workers is around 39%.
Area under Agriculture
The total area of Punjab is 5036 thousand hectare (50362 square kilometers). The net area sown in 2013-14 was 4145 thousand hectare, which you can say is the total agricultural land in Punjab. It means that almost 82% land of Punjab comes under agriculture. Most of the agriculture land in Punjab is sown more than once and area sown more than once is 3703 thousand hectare. Hence the total cropped area in 2013-14 was 7848 thousand hectare. According to this, the crop intensity in Punjab is almost 189%.
Forest Land
The forest area in Punjab is very less as compared to the national average. The forest land is almost 6% of the total area of the state. The district Hoshiarpur has the largest forest cover and has almost 38% of the total forests of the state. Timber is the main forest produce in the state. Other than this, bamboos & canes are obtained from these forests.
Irrigation - Rainfall, Canals and Tubewells
Punjab receives good rainfall during the monsoon season from July to September which is very good for the Paddy (Rice) crop grown in that season. The state also has a good infrastructure for irrigation. Out of the total agriculture land of Punjab, almost 99% is irrigated through canals or tubewells. It is often said that Punjab has a extensive network of canals for irrigation, but the fact is that only about 27% of the total cultivated area is irrigated through canals and rest 73% is irrigated through tubewells. The state has surplus electricity production so power supply to the tubewells is not a problem. There are total 14.05 lakh tubewells, out of which 12.26 lakh are operated by electricity and 1.79 are diesel operated.
Annual Rainfall
- The
average rainfall in Punjab
was 619.7 mm in 2013 and it was 384.9 mm in 2014. The five year average for the period from 2010 to 2014 was 501.4 mm. The rainy season in Punjab is from July to September and 60-70 percent of annual rainfall occurs during these three months. Gurdaspur district receives the maximum rainfall, followed by Rupnagar (Ropar). The district with least rainfall is Mansa, followed by Firozpur district.
Punjab Agricultural University
Punjab Agriculture University (PAU) is one of the renowned agriculture universities in India, which has made a significant contribution to the development of agriculture in Punjab. It palyed an important role in bringing the green revolution in the country and to make the country self sufficient in the production of food grains. According to National Institutional Ranking Framework (NIRF) rankings for 2017, the PAU was ranked at 24th position in India amongst all universities in India and it was ranked 40th in the Overall category.
Kissan Mela
: PAU organizers Kisan Mela at the main campus of university at Ludhiana and some other cities of the state. Hign quality seeds produced at the university farms are distributed/sold during these melas. The Kisan Mela is held twice an year, once in March before the start of sowing for Kharif crops and then in September when the Rabi season is about to start.
Agriculture Based Industries
Agriculture plays a major role in the economy of Punjab and there are a number of industries which are directly related to or dependent upon agriculture. These industries range from an ironsmith making small agricultural implements to the farm machinery like tractor and combine manufacturing, making pickles at a shop to the large scale food processing units. There are hundreds of companies that manufacture farm equipments and machinery. Here we will give a list of large scale industrial units in Punjab that are directly related to agriculture.
Sonalika Tractors
- Sonalika Group, based in Hoshiarpur, is amongst the top tractor manufactures of India.
National Fertilizers Limited
- This public sector company has two urea plants in the state. One unit is located in Nangal and another is located in Bathinda.
Swaraj Tractors
- This company is owned by Mahindra Group and one of the largest tractor manufactures of the country. The compnay was established by the government of Punjab and later acquired by Mahindra & Mahindra. The company has main manufacturing unit in Mohali city under the name of Punjab Tractors Limited.
Standard Corporation
- Standard Corporation is a leading Harvester Combine and Tractor manufacturing company, based in Barnala.
Nestle
- Nestle has a factory in Moga where a huge quantity of milk is collected from surrounding areas and processed to form desi ghee, skimmed milk, butter and other products. Some of the household names like Maggi, Cerelac and EveryDay are manufactures in this factory.
Preet Tractors
- Preet tractors is another leading manufacturers of tractors, self propelled combine harvester and other such machinery and company has its manufacturing facility in Nabha, near Patiala.
ITC
: ITC, which manufacture the various food products under the brand names Sunfeast, Bingo, Aashirvad etc. has a large food processing unit in Kapurthala.
Markfed
: Markfed is a co-operative company of Punjab government and is one of country's largest agricultural products marketing company. The products are marketed under the brand name of Sohna and the portfolio includes various edible oils, basmati rice, jams and pickles, desi ghee and ready to serve food items.
Agriculture Minister of Punjab
Capt. Amrinder Singh, the current Chief Minister of Punjab, also heads the agriculture department. So if you are asked that who is the agriculture minister of Punjab, then you can say that Capt Amrinder Singh is the agriculture minister of Punjab. He is a senior leader of Congress party and is MLA from Patiala Urban assembly constituency. In the previous tenure of SAD government from March 2012 to March 2017, Tota Singh of Akali Dal was the head of this department.
To more details about the agriculture department and list of ministers of related departments, check our page related to
Agriculture Minister of Punjab
.
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Geography of Punjab
Rainfall In Punjab
Agriculture In Punjab
Agriculture Minister of Punjab
Groundwater Level in Punjab
Sangrand Dates 2019
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Purnima in July 2018
Rajya Sabha Members From Punjab
Cabinet Ministers of Punjab
Finance Minister of Punjab
Punjab Vidhan Sabha
Punjab GK
Chief Parliamentary Secretary Punjab
PCS Exam 2015
Population of Punjab
Punjab Assembly Constituency List
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Districts of Punjab
Literacy Rate In Punjab
Universities In Punjab
Punjabi Books
Railway Routes In Punjab
Railway Time Table of Ludhiana
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Who was the first chief minister of Punjab?
Which town of Punjab is known as 'Guru Ki Kashi'?
Which city was the capital of Punjab before India got independence?
Which city of Punjab is famous for manufacturing of sports goods?
To know the answers and more such questions, check this page -
Punjab GK
[Close]
ਕੀ ਤੁਸੀਂ ਜਾਣਦੇ ਹੋ?
ਤੁਹਾਡੇ ਜ਼ਿਲ੍ਹੇ ਵਿੱਚ ਕਿੰਨੀਆਂ ਤਹਿਸੀਲਾਂ ਹਨ?
ਜਵਾਬ ਜਾਣਨ ਲਈ ਕਲਿੱਕ ਕਰੋ -
Tehsils in Punjab
.
ਪੰਜਾਬ ਵਿੱਚ ਕੁੱਲ ਕਿੰਨੇ ਜ਼ਿਲ੍ਹੇ ਅਤੇ ਡਵੀਜ਼ਨਾਂ ਹਨ?
ਵਸੋਂ ਦੇ ਹਿਸਾਬ ਨਾਲ ਪੰਜਾਬ ਦਾ ਸਭ ਤੋਂ ਵੱਡਾ ਜ਼ਿਲ੍ਹਾ ਕਿਹੜਾ ਹੈ?
ਖੇਤਰਫ਼ਲ ਪੱਖੋਂ ਪੰਜਾਬ ਦਾ ਸਭ ਤੋਂ ਛੋਟਾ ਜ਼ਿਲ੍ਹਾ ਕਿਹੜਾ ਹੈ?
ਜਵਾਬ ਜਾਣਨ ਲਈ ਕਲਿੱਕ ਕਰੋ -
Districts of Punjab
. | http://punjabdata.com/Agriculture-In-Punjab.aspx |
Join Jill Butler for an in-depth discussion in this video 33 Choose serif or sans serif, based on aesthetics, part of The 33 Laws of Typography.
- Law 33: Choose Serif or Sans Serif Based on Aesthetics.…Let's define serif and sans serif typefaces…and look at the differences between them.…Serif typefaces have little lines, or what are called…serifs that appear at the end of each stroke.…In French, the word sans means without, and so…sans serif means without serifs and therefore,…a sans serif typeface does not have the little…serifs at the end of each stroke.…
Many designers love to discuss the legibility of…serif versus sans serif typefaces and you can…find a slew of articles stating that serif…typefaces are easier to read and then you can find…an equal number of articles stating that sans serif…typefaces are easier to read.…You can find articles that recommend sans serif…typefaces for screen reading and serif…for print reading and on and on and on.…In this debate, the Serif vs Sans Serif debate,…can make for some great party conversation if you're…hanging out with a bunch of people who love design.…
But, scientific research has shown that at sizes…larger than 10 points, there is no performance difference…
AuthorJill Butler
Released11/3/2014
- Maintaining a visual hierarchy
- Avoiding bad paragraph breaks and line-breaking hyphens
- Staying away from all caps and underlined text
- Using proper punctuation
- Choosing the right typeface
Skill Level Beginner
Duration
Views
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Graphic Design Foundations: Typographywith Ina Saltz2h 23m Beginner
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Typography: Hierarchy and Navigationwith Ina Saltz45m 12s Intermediate
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Typography: Color Contrast and Scalewith Ina Saltz45m 25s Intermediate
-
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Introduction
-
Introduction3m 31s
-
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1. How to Format a Document
-
2. How to Format Large Bodies of Text
-
3. How to Format Smaller Blocks of Text
-
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21 Avoid bad line breaks4m 29s
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4. How to Use Punctuation Properly
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5. How to Choose Typefaces
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Conclusion
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Goodbye3m 18s
-
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Notes are saved with you account but can also be exported as plain text, MS Word, PDF, Google Doc, or Evernote. | https://www.lynda.com/Design-Page-Layout-tutorials/33-Choose-serif-sans-serif-based-aesthetics/147012/360862-4.html |
Elementary particles like Electron, proton, neutron are discovered by Thomson, Golstine, and Chanweak which constitutes the atoms of all chemical elements in the periodic table. The idea about these elementary particles resulted from faradays famous experiment of electrolysis in 1837. Faraday made an intensive study of the decomposition of salt, acid, and base by the passage of electric current. He establishes the quantitative relations between the amount of electrolysis and the quantity of electricity. These relationships are known as Faraday’s law of electrolysis for learning chemistry. He suggested that current was carry in the solution by charged elementary particles or ions. Therefore, the idea of the atomicity of electricity was finely developed from the ionization of gases. Our present-day, for understanding the chemical bonding of the atom is based on the orbital structure and electronic configuration.
Discovery of Electron Particles
Thomson Cathode rays experiment shows that all the atoms possess negatively charged elementary particles like the photon. He suggested when gases at low pressures subjected to high potential form various luminous effects. When the pressure quite low (0.01 mm), the tube remains dark (Crooks dark space) but a streak of rays, named cathode rays, traveled from cathode to anode.
Cathode Rays and Electrons
- Production of fluorescence on the opposite wall where the rays impinge.
- The rays travel in straight lines confirmed by the shadows of an object placed on their path.
- The cathode rays defected from the path they travel by electric or magnetic field. The direction of deflection suggested that cathode rays are a negative charge electron.
- When these electromagnetic rays impinge on the crystalline solid metal targets placed on their path x-ray is produced.
Charge and Mass of Electron
An electron carrying negatively charged with the value of
= – 4.8 × 10-10 esu
= – 1.60 × 10-19 coulombs
Let the mass of an electron = m and charge = e
∴ e/m = 1.76 × 108 coulomb/gram.
Mass of an electron = (1.60 × 10-19)/(1.76 × 108) gram
= 9.11 × 10-28 gram
Determination of Charge of an Electron
Faraday’s law of electrolysis of silver nitrate used for the determination of the charge of an electron in chemical science. Metallic silver uses as reducing agents at the cathode by decreasing the oxidation number or gaining one electron.
Therefore, the Avogadro number of electrons produced 1 mole of silver at the electrode from this redox process. At the same time, 1 mole of electrons removed from the anode and 1 mole of nitrate ions discharged. But according to Faraday’s law, 96500 coulombs of electrical energy required for the production of 1 gm equivalent of the molecule at the electrode.
∴ The charge carried by each electron
e = (96500-coulomb mol-1)/(6.023 × 1023 mol-1)
= 1.60 × 10-19 coulombs
Discovery of Protons Particles by Goldstein
Electrons contribute negligibly to the total mass of an atom but an atom is electrically neutral. Thus nucleus of an atom must carry elementary particles protons which account both for the mass and positive charge. Therefore, Goldstein added a new feature to the discharge tubes by using holes in the cathode. With this modification, he observed that there appeared not only cathode rays but also a beam of positively charged ions traveling from anode to cathode. Thus some of the positively charged particles passed through the hole in the cathode and produce a spot on the far end of the discharge tube.
Positive Rays Protons Particles
The nature of these positive rays protons investigated by Thomson in physics or chemistry.
- On deflection by a magnetic and electric field, the positive ray beam produced a large diffuse spot on the tube.
- The e/m ratio and velocity of these elementary particles are not the same as electrons.
- Thomson further demonstrated that each different gas placed in the apparatus gave a different assortment of e/m.
Electrons Protons Neutrons in Hydrogen
The hydrogen is the simplest discovery with one electron and proton without any neutron. Therefore the nucleus of the hydrogen atom carry unit positive charge. When this electron of the hydrogen has removed the nucleus contains unit charge and mass. Therefore the particle represented by hydrogen ion called a proton considered as an elementary particle which accounts entire positive charge of the nucleus.
Charge and Mass of Proton
The proton carrying positively charged and the charge of the proton
= +4.8 × 10-10 esu
= +1.60 × 10-19 coulomb
Let the mass of proton = m and charge = +e
∴ e/m = 9.3 × 104 coulomb/gram
∴ Mass of proton = 1.6725 × 10-24 gm
Discovery of Neutron Particles by Chadwick
The entire mass of an atom concentrated in the nucleus and the weight of electrons being negligible. Atomic number and mass number of hydrogen = 1. Therefore protons alone account for the total mass of hydrogen atom.
But except hydrogen, the proton alone cannot account for the total mass of the nucleus. Helium atom 4 times heavy as an atom of hydrogen, hence helium nucleus must be 4 times heavier than a proton. Therefore, to resolve this anomaly Chanwick discovery a new elementary particle called the neutron. Let the mass number of an atom = A, nuclear charge, or the number of protons of the atom = Z. Therefore, (A – Z) shortfall of mass number due to other particles like a neuron.
Neutrons and Protons Particles in Atom
Rutherford suggested this shortfall must be made up by another elementary particle. These elementary particles have electrically neutral, and mass equal to that of the proton. Rutherford named this particle in advance as a neutron. But this glory of discovery neutron went to Chadwick, one of Rutherford students. Mass number and the atomic number of oxygen 16 and 8 respectively suggested that the atomic nucleus of oxygen composed of 8 protons and 8 neutrons and 8 electrons.
But some of the species with the same number of protons varying numbers of neutrons inside the nucleus are called isotopes. Such species must belong to the same element and must vary only in their mass numbers.
- Protium, deuterium, and tritium are three hydrogen isotopes with zero, one, and two protons in the nucleus of the hydrogen atom.
- Oxygen-16, oxygen-17, and oxygen-18 are three isotopes of oxygen with 8, 9, 10 number of protons on the nucleus. | https://www.priyamstudycentre.com/2019/02/elementary-particles.html |
Escherichia coli.
分子量
Approximately 13.1 kDa, a single non-glycosylated polypeptide chain containing 116 amino acids.
生物活性
Fully biologically active when compared to standard. The biologically active determined by a chemotaxis bioassay using human lymphocytes is in a concentration range of 5.0-50 ng/ml.
外观
Sterile Filtered White lyophilized (freeze-dried) powder.
配方
Lyophilized from a 0.2 um filtered concentrated solution in 20 mM PB, pH 7.4, 200 mM NaCl.
内毒素
Less than 1 EU/ug of rRtMEC/CCL28 as determined by LAL method.
溶解说明
We recommend that this vial be briefly centrifuged prior to opening to bring the contents to the bottom. Reconstitute in sterile distilled water or aqueous buffer containing 0.1 % BSA to a concentration of 0.1-1.0 mg/ml. Stock solutions should be apportioned into working aliquots and stored at ≤ -20 °C. Further dilutions should be made in appropriate buffered solutions.
储存条件
Use a manual defrost freezer and avoid repeated freeze-thaw cycles.- 12 months from date of receipt, -20 to -70 °C as supplied.- 1 month, 2 to 8 °C under sterile conditions after reconstitution.- 3 months, -20 to -70 °C under sterile conditions after reconstitution.
参考资料
1. Wang W, Soto H, Oldham ER, et al. 2000. J Biol Chem, 275: 22313-23.2. Hieshima K, Ohtani H, Shibano M, et al. 2003. J Immunol, 170: 1452-61.3. Eksteen B, Miles A, Curbishley SM, et al. 2006. J Immunol, 177: 593-603.4. Kagami S, Kakinuma T, Saeki H, et al. 2005. J Invest Dermatol, 124: 1088-90.
纯度
> 96 % by SDS-PAGE and HPLC analyses. | https://www.watson-bio.cn/recombinant-rat-mucosae-associated-epithelial-chemokine-ccl28-rrtmec-ccl28/ |
ETD cores have been designed to make optimum use of a given volume of ferrite material for maximum throughput power, specifically for forward converter transformers. The structure, which includes a round center post, approaches a nearly uniform cross-sectional area throughout the core and provides a winding area that minimizes winding losses. ETD cores are used mainly in switched-mode power supplies and permit off-line designs where IEC and VDE isolation requirements must be met.
ETD cores can be supplied with the center post gapped to a mechanical dimension or an ALvalue.
Weight indicated is per pair or set.
Weight: 94 (g)
|Dim||mm||mm tol||nominal inch||inch misc.|
|A||44||± 0.75||1.732||_|
|B||22.3||± 0.20||0.878||_|
|C||14.8||± 0.35||0.583||_|
|D||16.5||± 0.20||0.65||_|
|E||32.5||min||1.28||min|
|F||14.8||± 0.35||0.583||_|
Chart Legend
Σl/A : Core Constant, le : Effective Path Length, Ae : Effective Cross-Sectional Area, Ve : Effective Core Volume
AL : Inductance Factor
Explanation of Part Numbers: Digits 1 & 2 = product class and 3 & 4 = material grade.
|Electrical Properties|
|AL(nH)||5100 ±25%|
|Ae(cm2)||1.73|
|Σl/A(cm-1)||6|
|le(cm)||10.35|
|Ve(cm3)||17.94|
|Amin(cm2)||1.717|
AL value is measured at 1 kHz, B < 10 gauss
A low loss MnZn ferrite material for power applications up to 200 kHz with low temperature variation. New type of 95 material is a low loss power material which features less power loss variation over temperature (25-100°C) at moderate flux densities for operation below 200 kHz.
Available in 95 material:
Pot Cores
RM
PQ
EFD
EP
95 Material Characteristics
|Property||Unit||Symbol||Value|
|Initial Permeability @ B < 10 gauss||µi||3000|
|Flux Density @ Field Strength||Gauss
|
Oersted
|B
|
H
|5000
|
5
|Residual Flux Density||Gauss||Br||1200|
|Coercive Force||Oersted||Hc||0.18|
|Loss Factor @ Frequency||10 -6
|
MHz
|Tan δ/ µi||2.5
|
0.1
|Temperature Coefficient of Initial Permeability (20 -70°C)||%/°C||0.6|
|Curie Temperature||°C||Tc||>220|
|Resistivity||Ohm-cm||ρ||200|
**** Characteristic curves are measured on standard Toroids (18/10/6 mm) at 25°C and 10 kHz unless otherwise indicated. Impedance characteristics are measured on standard shield beads (3.5/1.3/6.0 mm) unless otherwise indicated. | https://www.fair-rite.com/product/etd-cores-9595444502/ |
St. Andrews Building Glasgow University.
As a celebration of the UN’s International Day of Democracy, Education Scotland is delighted to invite you to the second in a series of professional learning sessions on Learning for Democracy aimed at community learning and development (CLD) practitioners from across the public, voluntary and community sectors.
It follows on from the very positive first session held at the Scottish Parliament in February which shared perspectives on the role of CLD to encourage democratic renewal and showcased a range of practice examples.
This session will focus on the impact of poverty and inequalities on our democratic system. Public involvement in democracy was one of the most talked about issues in the recent national conversation on Creating a Fairer Scotland. Democracy is only as strong as the political participation of citizens but evidence suggests that there is a political poverty gap in the UK. Plainly put, the poorer you are the less likely you are to vote. We also know that some groups in society such as BME communities and people with disabilities are underrepresented in our political institutions. Despite the rhetoric – these facts affect the policies and decisions of political parties and of governments .
The event will include several speakers and will discuss the following key questions:
– What groups are currently underrepresented in our democratic system?
– Why are these groups underrepresented and what impact does this lack of representation have?
– What role can community learning and development practitioners play to help communities address this situation and what steps must we take to support this role?
Booking to attend –
There is no cost to attend and places can be booked online at: https://www.surveymonkey.co.uk/r/BVBCNZ2
Please register asap as places will be limited.
Learning for Democracy is a short series of professional learning opportunities focused on the role that community learning and development plays in supporting in democratic and civic participation. It is a partnership project between Learning Link Scotland, Education Scotland, The Workers Educational Association, the Scottish Community Development Network, Edinburgh and Glasgow Universities. | https://cldstandardscouncil.org.uk/professional-learning-opportunity-learning-for-democracy/ |
754 P.2d 599 (1988)
91 Or.App. 119
Eva Mae CHILDERS, Aka Eva Mae Janelli, Respondent,
v.
James J. SPINDOR, Appellant.
16-84-04654; CA A36636.
Court of Appeals of Oregon, In Banc.
Petition for Reconsideration April 22, 1987.
Decided May 11, 1988.
Thomas M. Christ, and Mitchell, Lang & Smith, Portland, for appellant.
No appearance contra.
On Appellant's Petition for Reconsideration April 22, 1987.
DEITS, Judge.
Defendant has filed a petition for review of our decision in this case. 84 Or. App. 407, 733 P.2d 1388 (1987). We treat the petition as a petition for reconsideration. ORAP 10.10. We are persuaded that we did not address the question actually presented and, had we done so, we would have reversed and remanded the judgment.[1] We therefore allow reconsideration, withdraw our former opinion in toto, reverse the judgment and remand the case for trial.
This malpractice case arose out of defendant's legal representation of plaintiff in the dissolution of her marriage to Janelli. After the dissolution trial, plaintiff filed this action, contending that, as a result of defendant's negligence in representing her in the dissolution trial, plaintiff failed to offer sufficient proof of various financial matters between the parties, which resulted in Janelli's receiving a disproportionately large share of the marital assets.[2] The *600 jury returned a general verdict for plaintiff. Defendant appeals, arguing that the court erred in failing to grant his motions for directed verdict on plaintiff's claim for relief and on each specification of negligence.[3]
Defendant asserts that his motions for directed verdict should have been granted, because plaintiff did not present adequate expert testimony to establish the relevant standard of care for an attorney in this type of case. In most negligence actions against professionals, expert testimony is necessary to inform the jury of the applicable standard of care. Getchell v. Mansfield, 260 Or. 174, 489 P.2d 953 (1971). A jury generally is not able to determine what is reasonable professional conduct without such testimony. There are some instances when the breach of a standard of care is within the ordinary knowledge and experience of lay persons; in such cases, a jury can determine the reasonableness of professional conduct without expert testimony. An example of a situation in which expert testimony is usually not required is when an attorney allows the Statute of Limitations to run. See Collins v. Greenstein, 61 Haw. 26, 595 P.2d 275 (1979). This case involves what an attorney is supposed to do in preparing for trial, presenting evidence and communicating with a client or the court. Generally, that is not within the knowledge or experience of a lay juror and, in most instances, at least some expert testimony concerning the expert's knowledge of the customary and proper method of handling such legal matters is required.
Plaintiff's expert evidence consisted of the testimony of one witness, a practicing attorney. His testimony includes a very general discussion of the practices of attorneys in preparing for trial, presenting evidence and communicating with clients and the court. The testimony on the parameters of acceptable conduct is quite limited. However, except for the fourth specification of negligence, the expert testimony taken together with the presumed general knowledge and experience of a lay juror, gave the jury sufficient knowledge to make an informed judgment on each specification of negligence.
Plaintiff contends in specification 4 that defendant was negligent in failing to depose Janelli. A lay juror likely would not have much knowledge concerning discovery and, thus, expert testimony explaining discovery and the standard of care in the use of discovery was necessary. Plaintiff's attorney asked the expert, "What obligation or what standard what duty does an attorney have in the discovery phase? What is discovery, first of all?" The expert explained generally the purpose of discovery *601 and the mechanics of a deposition, but he never discussed the parameters of acceptable conduct or explained a standard of care. Expert testimony was necessary and, therefore, the court erred in submitting that specification to the jury.
Because the jury returned a general verdict for plaintiff, it is not clear on which specification of negligence the jury based its verdict. It is conceivable that the jury based its verdict upon specification 4, which was improperly submitted. Therefore, there must be a new trial. Pavlik v. Albertson's, Inc., 253 Or. 370, 454 P.2d 852 (1969); Port of Portland v. Brady Hamilton, 62 Or. App. 92, 659 P.2d 995, rev. allowed and modified 63 Or. App. 146, 662 P.2d 790 (1983).
Reconsideration allowed; former opinion withdrawn; reversed and remanded.
NOTES
[1] Our earlier opinion addressed the issue of whether the standard of care for an attorney is specific to a local community or involves the statewide legal community. Because of our disposition of the case, it is not now necessary to address this issue.
[2] Her complaint alleged that defendant had acted negligently in one or more of the following particulars:
"1. In failing to present sufficient evidence at trial as to the 1976 value of the Janelli 60-acre ranch;
"2. In failing to present sufficient evidence at trial as to the 1982 value of the Janelli 60-acre ranch;
"3. In failing to inform Plaintiff that it would be necessary to establish the 1976 and 1982 valuation of the property in order to prove her valuation and to present evidence at trial to establish the increase in value of that property;
"4. In failing to take a deposition of Mr. Janelli, which resulted in the Defendant's not being prepared for trial in that he did not then know what Mr. Janelli's position would be regarding valuations of property;
"5. In failing to prove at trial a transfer of approximately $60,000 from Plaintiff's restaurant supply business to Mr. Janelli's holdings;
"6. In failing to inform the Court and argue to the Court that the valuation of the ranch equipment by the Court was incorrect in that it was less than both parties testified to at trial and, further that the court had apparently made an arithmetic error.
"7. In failing to prepare properly for trial, including development of proper evidence of valuation, the subpoenaing of witnesses necessary to support Plaintiff's position on valuation, and counselling of the Plaintiff; and
"8. In failing to prove to the Court the $71,096.04 indebtedness of Mr. Janelli at the time of the marriage, which was reduced and eliminated by Mrs. Janelli."
[3] Plaintiff argues that defendant's motion for directed verdict on the individual specifications of negligence was improper. Technically, plaintiff is correct, and defendant should have requested peremptory instructions. See NW Pac. Indem. v. Junction City Water Dist., 296 Or. 365, 677 P.2d 671 (1984). However, the effects of a peremptory instruction and a directed verdict are the same in that both remove issues from the jury's consideration. Hoekstre v. Golden B. Products, 77 Or. App. 104, 712 P.2d 149 (1985), rev. den. 300 Or. 563, 715 P.2d 94 (1986). We treat defendant's motion as sufficient for the purposes of review.
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1. Field of the Invention
The present invention relates to a power supply unit having a dimmer function for driving a semiconductor light emitting module so as to light the semiconductor light emitting module at suitably dimmed brightness, and a lighting unit having this power supply unit.
2. Description of the Related Art
Recently, from a viewpoint of energy saving, semiconductor light emitting modules such as light-emitting diodes are used as light sources for lighting units, and DC power supply units into which switching elements are incorporated are developed as power supplies which drive the semiconductor light emitting modules such as the light-emitting diodes. As these power supply units, it is known that they have a dimmer function for adjusting brightness of the light-emitting diodes according to a dimmer signal given from the outside.
Conventionally, the power supply unit having such a dimmer function is disclosed in, for example, JP-A 2003-157986 (KOKAI). The power supply unit disclosed in this publication has a voltage dimmer circuit which controls an applied voltage to light-emitting diodes, and a duty dimmer circuit which switching-controls an applied voltage to the light-emitting diodes. The voltage dimmer circuit and the duty dimmer circuit are changed over to be controlled according to a dimmer control signal.
In the power supply unit disclosed in JP-A 2003-157986 (KOKAI), a DC voltage given to the light-emitting diodes is adjusted according to a pulse width of the dimmer signal, and an applied voltage to the light-emitting diodes is switched so that the light-emitting diodes are controlled to be dimmed. Therefore, output light from the light-emitting diodes has a problem that flicker easily occurs. In addition to a current limiting function for controlling an output current according to the pulse width of the dimmer signal, a switching element which is in series or in parallel with the light-emitting diodes is necessary, and thus the number of parts increases and circuit efficiency is deteriorated. Since the pulse width is controlled, when a switching frequency for this control is in an audible area, a noise might be generated.
On the other hand, since the light-emitting diodes have an approximately constant voltage characteristic, a part or a device having the current limiting element is necessary for stable lighting. In order to control an electric power in a power supply unit using a switching element, current control is generally used. In the current control, an element temperature of the light-emitting diodes is determined by a value of an electric current flowing in the light-emitting diodes, and the element temperature influences an element life. Therefore, in the current control, the flowing electric current is the important control element due to design of the lighting unit.
The dimming of the light-emitting diodes can be realized comparatively more easily than a discharge lamp lighting unit. The light-emitting diodes as a load have stable electric characteristics, and fluctuation in the brightness of the light-emitting diodes due to an external factor such as temperature is small. For this reason, the dimming of the light-emitting diodes can be easily realized. In the application of deep dimmer control, namely, brightness control where a brightness control rate or a dimmer rate is set large and thus brightness of the light-emitting diodes is greatly reduced, the constant current control is adopted to the light-emitting diodes. In this constant current control system, the light-emitting diodes can be lighted stably in a control area where lighting current is high for full-emission lighting. In this system, however, the lighting current supplied to the light-emitting diodes is lowered in the deep dimmer control area, and a current detecting signal becomes minute according to the lowering of the lighting current, and a reference current for controlling the lighting current is a minute signal. Therefore, in a constant current control circuit, accuracy of a detecting circuit or a comparator requires high performance, and the control circuit is easily influenced by a noise, so that a stable operation becomes difficult. It is thus considered that a signal voltage for control is increased. However, the current detecting signal is generally detected by a resistor inserted in series into the light-emitting diodes, and resistance of the resistor should be increased in order to increase the detecting signal. As a result, in the control area where the electric current flowing in the light-emitting diodes is high, the electric power is greatly consumed by the detection resistor, or heat is generated from the detection resistor, and a countermeasure against this heat inhibits developments of products.
As a control system which solves these problems, a constant voltage control system which constantly controls an output voltage is also proposed. A voltage for turning on the light-emitting diodes is higher than that for a general. silicon diode. For example in a GaN type diode represented by blue one, an electric current starts to flow at about 2.5 V, and about 3.5 to 4.5 V in the full-emission lighting, and the brightness of the light-emitting diodes can be controlled comparatively stably without being influenced by the performance of the light-emitting diodes or a noise generated on the light-emitting diodes even in the deep dimmer control. However, a forward voltage of the light-emitting diodes has a negative temperature characteristic, and the forward voltage is decreased due to self heat generation at the time of applying an electric current to the light-emitting diodes, and the electric current increases. As a result, heat generation becomes large, and thus thermo-runaway might occur. The forward voltage of the light-emitting diodes greatly varies, and even if an output from the lighting unit is adjusted, output currents vary due to a individual difference of respective light-emitting diodes.
The above-mentioned problem arises not only in the semiconductor light emitting modules such as the light-emitting diodes but also in the power supply units which light a light source such as an organic EL light source or an inorganic EL light source developed in recent years, and this problem still remains unsolved.
The power supply unit and the lighting unit having the power supply unit which can realize the stable dimmer control are already proposed as a prior application in International Application No. PCT/JP2009/055871 filed on Mar. 24, 2009 by the same assignee. In the power supply unit of the International Application, first and second reference signals, which change according to a dimmer rate of a dimmer signal, namely, a dimmer level, are prepared. In the almost full-emission lighting control area where the dimmer rate is small, the first reference signal is selected, and light-emitting diodes are controlled with constant current with reference to the first signal. In a lighting control area where the dimmer rate is large and the brightness is reduced, the second reference signal is selected, and the light-emitting diodes are controlled with constant voltage with reference to the second reference signal. Since the first and second reference signals are selected so that the light emission from the light-emitting diodes is controlled, the stable dimmer control can be realized.
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For this assignment, you will create an e-newsletter for your corporation with a message from your CEO asking for support for an issue or partnership of your choosing. Earlier, we talked about issues that your company should care about (CSA/CSR lecture). Identify the issue and find a partnership (nonprofit or small business) that your company could collaborate with. (Think about the cookie company caring about blood donations and partnering with the Red Cross). This must be an original idea and not a collaboration that yor corporation is already involved in now or in the past.
Part A: Determine who your target audience is. Once you have determined your audience, craft a 200-word MAX message from your corporation’s actual CEO that announces the partnership with a call to action (donate/sign this petition/march/etc.). This must contain original copy written only by you (Do not use an existing message from your corporation.). In addition, include a subject line. The message that you write in Part A will be inserted into an e-newsletter that’s created and submitted through Mailchimp via email. (Open a free account to use at MailChimp.com). Use your corporate logo, brand colors, and tone in the email.
Do you have a: | https://americaukessays.com/index.php/2022/10/18/my-company-is-american-airlines-part-a-ceo-letter-word-doc-for-this-assignment-you-will-create-an-e-newsletter-2/ |
Position Overview:
The National Park Foundation (NPF) is entering an exciting phase in increasing impact, amplifying the stellar work of the National Park Service (NPS) and working toward a “collective” campaign that vastly increases the ability of NPF to support an extraordinary visitor experience for all people who visit any of the 420+ National Park sites. Our youth programs have reached well over one million children and young adults over the past few years and we are looking to engage another million in high-quality experiences that help foster feelings of stewardship and connection in hopes of developing a generation of park champions.
The work of NPF programs is split into two primary categories, Connect and Protect. The primary role of Manager, Youth Programs, is to meaningfully advance core projects that support the strategic theme of “Connecting Audiences” under the Connect pillars of NPF’s strategic plan. This work entails education, recreation, and visitor engagement programs that directly support the NPS and its partners. The programs Open OutDoors for Kids, Junior Ranger Angler and Field Science fall within the Youth Engagement and Education pillar and the Outdoor Exploration pillar which both work to connect youth, students, and families to meaningful, high-quality, safe and memorable experiences at parks across the country.
The Manager will be responsible for planning, guiding, administering and assessing a subset of NPF’s grant programs and projects focused on youth education and engagement in national parks. This involves managing and completing the operational aspects of the grant-making processes to deliver high-quality programs that connect youth to parks, ensuring high standards for grant management, extensive communication with parks and partners, and removing obstacles to program success.
Duties include:
- Implement and oversee an assigned portfolio of NPF grant programs including Open OutDoors for Kids, Junior Ranger Angler, and Field Science (formerly Citizen Science 2.0). Provide grant support, project management, and technical assistance to effectively deliver grant funds to NPS sites and a range of partners
- Manage the reconciliation of all grant budgets, agreements and payments in this portfolio, working closely with the Director, Grants and Administration and VP, Connecting Audiences. Serve as key point of contact for NPF grantees demonstrating superior customer service exhibited through timely, solution-oriented responses
- Manage diverse components of national programs by tracking the budget, timeline, deliverables, reporting, communications, and donor stewardship. Work closely with Grant Management and Finance teams to provide support that increases efficiency, equity and transparency of grant management and payment processes
- Budgeting tasks also include tracking both corporate and individual gifts, maintaining budgets for both granting and other relevant purposes and deliverables such as evaluation, site visit expenses, grantee travel, and resources and materials, and, ensuring that gifts intended for specific parks and projects are tracked and reported on
- Build and maintain strong communication and relationships with NPS and partner organizations to ensure timely response and awareness around high priority needs and internal processes.
- Support grant cohorts with capacity building opportunities and professional development to ensure implementation is happening at the highest level and responding to current trends and best practices
- Coordinate closely with all philanthropy teams to ensure donor stewardship is a hallmark of program management
- Understand youth education trends and best practices among leading organizations (i.e. nonprofit & corporate partners, other federal, state, or local municipalities) that drive equitable youth engagement in parks and other environmental, cultural and historic sites. Ensure that the programs NPF supports are accessible to all youth, including students at Title I schools, and that programs foster feelings of inclusivity, representation and respect
- Collaborate with Connecting Audiences team and external evaluation vendor(s) to ensure appropriate qualitative and quantitative metrics are captured
- Help ensure NPF and partners have current information on projects in the Connecting Audiences space to facilitate storytelling and help demonstrate success of investments through regular grants updates
- Develop communication materials including proposals, budgets, reports, blogs and infographics which demonstrate impact of youth programs, aid the expansion of philanthropy and reflect NPF mission
- Serve on cross-departmental teams to assist with strategic planning, donor proposal development, communications, fund disbursements, and stewardship
REQUIRED KNOWLEDGE, SKILLS, AND ABILITIES
Basic Qualifications
- Minimum 3 years (5 years for Senior Manager) of experience in program or grant management positions; park, historic or cultural resource programs for youth preferred
- Expertise in budgeting, managing complex spreadsheets and understanding of non-profit project management
- Experience in grants management processes, demonstrated experience managing approximately 50-100 grants per year preferred
- Youth program planning and management, management of statewide or national programs strongly preferred
- Undergraduate degree in related field required; master’s degree or equivalent experience preferred
- Experience with grant databases, proficiency in Fluxx preferred
- Proficient in Microsoft Office Suite, especially Excel
- Collaborative working style, responsive, considerate, flexible, and personable
- Strong written and verbal communication skills
- Exceptional organizational skills and attention to detail
- Experience in working with diverse stakeholder, partners, and vendors; demonstrated ability to engage diverse youth organizations preferred
- Some travel may be required
Preferred Experiences
- Experience working in partnership with a federal agency, park, museum, historic/cultural site, or youth education organization
- Understanding of philanthropic gift lifecycle from proposal development to final reporting and donor stewardship
- Ability to analyze and review grant requests for completeness, viability, inclusivity, and effectiveness
- Evaluation & monitoring experience; knowledge and experience in developing and demonstrating effectiveness of program outcomes
***For consideration, please include a resume plus cover letter. Review of applications will be based on submission date. The deadline for application submission is November 11, 2022. NPF is currently registered to allow remote work in the following states: | https://www.greenlatinos.org/jobs/manager%2C-youth-programs |
Data is everywhere in our daily lives, behind everything from the advertisements we see online to the predicted drive times our favourite mapping software supplies whenever we need directions.
Given the increasing importance of data analytics in many fields and the unprecedented amount of information we have access to in the internet age, a basic understanding of statistical concepts and the ability to manipulate data sets are highly valuable assets for the future job candidate and the general consumer of information alike.
It was in part because of the rapidly changing position data plays in our world that UBC’s first dedicated data science course — being offered for the first time in the winter 2018/19 term — was created.
“It used to be that data collection was a problem and that’s not a problem anymore,” said Dr. Tiffany Timbers, the course instructor.
“We have devices like our cell phones that are collecting all kinds of information, we’re wearing Fitbits, you can track people’s activity on the internet, sensors in your home are collecting information so information is being collected quite widely and the bottleneck now comes down to making sense of that data,” she said.
According to Timbers, what makes the class unique is its dual emphasis on both coding and statistical methods.
As can be seen from the course outline, publicly available on GitHub, the course material can be broken into two general sections. The first part of the material introduces students to the basics of using R — a program commonly used for data science — to organize and visualize data. The second part of the material moves on to cover some basic statistical methods for analyzing and understanding data.
The course, open to students from all faculties, is structured to provide ample opportunity for hands-on learning, with much of the twice-weekly lectures set aside for students to work through exercises using R and Jupyter Notebooks, another popular data science tool.
All of the class exercises will utilize real data from different fields, allowing students to see more concretely the potential real-world applications of what they are learning. In keeping with the emphasis on experiential learning and real-world applicability, the course will culminate in a final group project in addition to a final exam, allowing students to put some of the skills they have learned into practice on actual data sets.
According to the course learning objectives — also available on GitHub — by the end of the term, students will be able to collect data from a variety of online sources, shape it into a usable format and use some basic statistical tools to analyze and interpret it.
Timbers believes that this new course will offer students valuable preparation for both future coursework at UBC and their future careers in the work world.
“To keep up with the world, it’s important to develop these skills,” she said. | https://www.ubyssey.ca/science/new-ubc-data-science-course/ |
The writer is President of Queen’s College, Cambridge and Chancellor of Allianz and Gramercy
With so much going on in the global economy and financial markets, the dollar’s recent strong rally has attracted less attention than expected given historical experience.
Theoretically, the appreciation of the currency of the world’s most economically performing companies should help them adjust to the global economy. It helps boost exports from weaker countries while relieving inflationary pressures in the United States by lowering the cost of imports.
But in the current circumstances, there are risks of a rapid appreciation in the dollar both for the well-being of the already volatile global economy and unstable financial markets.
Since the start of the year, the dollar has appreciated about 10 percent as measured by DXY, a widely followed indicator of the currency’s global value. While it was a remarkably broad-based move involving the currencies of the vast majority of economies, the 12-month overall rise of 16 percent in the index brought the index to levels not seen in 20 years.
Three factors come into play: expectations that the Federal Reserve will raise interest rates more aggressively than other central banks in the developed world; superior economic performance of the United States attracting capital from the rest of the world; and the attractiveness of the relative haven of its financial markets.
So far, there has been little political rollback from the development that is undermining the competitiveness of the United States and contributing to its record trade deficit. In the past, such rises in the dollar have threatened trade wars. Now the strong US labor market faced potential tensions.
However, the lack of US political hostility over the dollar’s rise does not mean it is sailing smoothly for global economic and financial stability. The risks are particularly acute for developing countries that already face the clear and present dangers of crises to the economy, energy, food and debt.
For most, a rising dollar translates to higher import prices, an increase in the cost of servicing foreign debt, and an increased risk of financial instability. It is placing more pressure on countries that have already drained their resources and policy responses by combating the scourge of Covid.
The concern is particularly acute for low-income countries, which are also hampered by high food and energy price inflation. The cost of living crisis here also poses a threat of starvation to the most vulnerable.
If I were allowed to burn more, what I called “Little Fire Syndrome Everywhere” – that is, multiplying economic and financial instability in countries – could coalesce into a larger and more dangerous mix of damaged global growth, debt defaults, social, political and geopolitical instability.
The indirect repercussions for advanced economies are likely to be more problematic than any direct effect on them of the appreciation of the dollar. In addition to weakening these economies’ external growth drivers at a time of rising stagflation at home, a turbulent developing world can add volatility to financial markets that are already dealing with multiple risks.
Financial markets have already had to contend with a significant increase in interest rate risk due to the persistently high inflation that has caused the Federal Reserve to infiltrate significantly. In the process, turmoil in government bonds has spread to other market sectors as concerns grow about tightening financial conditions. Now the markets have to worry more about slowing global economic growth.
As alarming as this year’s wealth destruction has been, its impact on economic activity has been mild and the market performance risks have not yet begun. Having said that, for those with sharp noses, there is actually some smell from this due to the cipher massacrealong with frequent price gaps in global benchmarks for the US Treasury market.
Even if this develops into something larger due to the payments turmoil in the developing world, the Fed will find it difficult to return to its usual policy of flooding markets with liquidity given its ballooning balance sheet and inflationary concerns.
The way to reduce the risks associated with a very rapid appreciation of the dollar is for the rest of the world to advance faster with structural reforms that boost growth and productivity, improve capital returns, and increase economic resilience.
Without it, the theoretical promise of orderly global adjustment, including external reinforcements to underperforming countries, would become a difficult source of economic and financial instability. | https://mtilending.com/the-rapid-appreciation-of-the-dollar-increases-risks-to-the-global-economy/ |
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Tatyana Kholodkov
Psychologist, PhD, DYRT
Verified
Suffering presents itself in a number of ways- behaviors and relationships we desire to change, uncomfortable emotions, chronic health problems, and feeling out of touch with meaning in our lives. I aim to help clients not only feel better, but live a life according to their values. I work with adults across the lifespan, and address a range of concerns from adjustment to life changes, stress and health management, relationship concerns, and significant mental health conditions. I particularly enjoy helping others overcome trauma, phobias, panic disorder, and depression. I support diversity- all are welcome, including non traditional relationships;
sex positive
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(919) 351-8251
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Tess M. Kilwein, PhD, LP
Psychologist, PhD, LP
Verified
Looking to improve your sport/mental performance? Better understand your gender, sexuality, or relationship identity? Process a history of interpersonal trauma? Our stories are our greatest power, and yet, can be difficult to share with others. In therapy, I encourage clients to tell their stories as accurately and meaningfully as possible (including the "ugly" stuff!) to pave a path for moving forward. I equally recognize that not all mental health concerns are individually determined, and encourage clients to explore the impact of their social, intergenerational, and environmental histories and resources to promote healing/growth.
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Lindsay Wortham
Counselor, LPC, LAC
Verified
I believe in healing through therapeutic relationship. Establishing trust creates a safe container in which to experience change. I am LGBTQIA+-allied, BIPOC-allied, and I also work with cis-white folx who are working on reducing their contribution to and healing trauma associated with white supremacy and capitalist patriarchy. My graduate work was at Naropa University, where I deeply connected with the Buddhist roots of my education. I welcome people from all religions and backgrounds, and often incorporate mindfulness practices into sessions.
(307) 227-4160
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Laramie,
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Dr. Deb Smith, LPC
Licensed Professional Counselor, PhD, LPC
Verified
My theoretical perspective foundation is very person-centered and I strive for unconditional positive regard for all my clients. I utilize existential philosophies to help my clients dig deep to find meaning, purpose, and even JOY in their lives. I have extensive experience in working with adults who are struggling with depression, anxiety, relationships, sexuality, and gender identity. I work with adolescents with sexuality and gender identity questions and concerns. I am LGBTQIA affirming.
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A Better Way Counseling & Consulting, LLC
Clinical Social Work/Therapist, MSW, LCSW, ADS
Verified
Hi, I'm LaRae. I am passionate about helping people heal. I have worked with people who have experienced trauma and PTSD, substance-abuse, co-occurring disorders, serious and persistent mental illness, people who self-harm, people involved in the criminal justice system, and people with disabilities. I enjoy working with adults, and children 10 years and older. I understand Spanish and have counseled people whose first language is Spanish. I have personal and counseling experience with Hispanic populations as well as Native American populations. Additionally, I have counseled people involved in relationships with domestic/interpersonal abuse and/or violence.
(307) 316-4574
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Julio Brionez
Psychologist, PhD, LP
Verified
Thank you for visiting this page. As a psychologist and human, my passions are suicide prevention through action, connection, and therapy. While I can work with most people, I aim to provide services to people of color wrestling with thoughts of suicide and emotional pain. I come to this work with a compassionate lens and training to reduce suicide using two types of therapy (e.g., ACT, BCBT). Conversely, I am a suicide survivor, attempt survivor, and psychologist licensed to practice in Wyoming, Colorado, and Montana.
(307) 207-5023
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Ashlee Mickelson
Clinical Social Work/Therapist, LCSW
Verified
Hello! I am a directive and evidence based therapist. My practice focuses on people who struggle with self-harm, suicidal thoughts, and persistent personality disorders. I am also skilled and honored to work with individuals who struggle in difficult situations. I believe in the change process of therapy and believe in working with people on achieving their own goals and needs, and work on cultivating an environment and space to do that.
(307) 223-4753
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Ty Tedmon-Jones; Somatic Revelations, LLC
Licensed Professional Counselor, LPC, LCAT, BC-DMT
Verified
Are you serious about creating change in your life? Let me be a support to your goals! My integrative approach provides a client-centered and supportive environment where my clients learn skills, process trauma, and create change. My work draws from numerous psychological and counseling theories including: existential-humanistic theory, psychodynamic theories and practice, object-relations, systems theories, DBT and CBT. My approach to counseling and psychotherapy is a collaborative process. To make the changes you desire, we will work together as a team and I will encourage you to give me feedback about our work together.
(307) 522-1812
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Alicia Brock
Licensed Professional Counselor, MA, LPC
Verified
Most people have at least one time in their lives when they need some outside support. Sometimes all we need is supportive friends and family to listen, but sometimes we need more. That could be because the circumstances are too overwhelming, or the issues are too personal to share with family and friends. You might need specific information about an issue. This is where I come in. I believe therapy is most effective when it is collaborative. I provide an objective perspective, experience and knowledge. You provide expertise on yourself. Together we discover what will help you on your journey.
(307) 227-3781
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Laniece M Schleicher
Counselor, MS, LPC, NCC, RYT
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Hello, I am a Licensed Professional Counselor in Laramie, Wyoming. I see the therapeutic relationship as a collaborative brave space to move through obstacles from the past that hinder present experiences. I specialize in working with clients to relieve symptoms from complex and simple trauma, anxiety, depression, issues related to relationships, transitions, grief and loss.
(307) 271-8853
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Susan Weinstein & City Park Counseling Center,PLLC
Clinical Social Work/Therapist, MA, MSW, LCSW
Verified
I have a passion for working with adults in big or small life transitions and emerging adults staring their adult lives! I have a trauma focused practice. I work with those who seek treatment for life experiences such as: anxiety, depression, sexual, physical, or emotional abuse, social isolation, teen friendships/bullying/dating, teen parenting, teen pregnancy counseling and support, low energy, stress, panic attacks, a traumatic event or other stressors, self sufficiency development, divorce, low self esteem, ADD/ADHD, mood dysregulation, oppositional or angry behaviors, and relational and attachment issues. EMDR Certified and CIT
(720) 583-8711
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Dr. Julie Morris
Psychologist, PhD
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Life gets complicated -- an unexpected medical diagnosis, a loss, parenting concerns, work or school stress, anxiety, depression -- it happens. When these moments show up, we can feel as though we are removed from the here and now and have been taken away from the things and people that matter the most to us. In essence, we can end up stuck in a rut, doing fewer or none of the things that bring us joy, enrichment, or meaning. Perhaps one of these moments has shown up and you are looking for help getting out of your rut. Would you like to feel more grounded and present? Would you like to be living the life you imagined?
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Sex-Positive, Kink Allied Therapists
Sex-positive counseling in Laramie for sex-positive clients
If you are sex-positive or are looking for help with being sex-positive in Laramie or for a Laramie sex-positive therapist, these professionals provide kink allied therapy in Laramie and counseling for sex-positive clients. | https://www.psychologytoday.com/us/therapists/wy/laramie?category=sex-positive-kink-allied |
Anna-Maria Helsing is at the rostrum of North Iceland Symphony & Songbird Philharmonic, Choir of the North with soloists to conduct two concerts of Mozart's Requiem and Piano Concerto d Minor KV 466 in Akureyri on 18 April and Reykjavik on 19 April.
Grete Pedersen conducts the Choeur de l'Orchestre de Paris at Philharmonie Paris on 27 April with Rachmaninov's Vesper op. 37.
Paul Hillier conducts the Hungarian Radio Choir in Budapest on 30 April with the programme "500 years of British Sacred Music".
Les Vents Francais: New CD "Moderniste"
Warner Classics has released a further album of Les Vents Francais with music by Philippe Hersant and Thierry Escaich.
On 4 May Edin Karamazov gives a lute recital at Hulencourt Festival in Genappe, Belgiuum, performing music by J.S. Bach, Giovanni Zamboni and Alessandro Piccinini.
Tonu Kaljuste conducts Orchestra and Choir of the Norrlandsopera in Umea on 16 May featuring Schönberg: Chamber Symphony No.1 op. 9, Brahms: Five Songs op. 104, Alma Mahler/Staffan Storm: Five Songs, Brahms: Schicksalslied and Schönberg: Kol Nidre.
Edin Karamazov performs at Thüringer Bachwochen in Wechmar on 13 April with Bach's cello suites arranged for lute in interaction with vocal compositions.
Deutschlandradio Kultur sends the broadcast on 14 April, 20:03-22:00 (CET).
On 7 April Risto Joost conducts the Orchestre de Chambre de Lausanne with an entire Mozart programme: Symphony E flat major KV 385 "Haffner" and Violin Concerto G major KV 216.
On 4 April Risto Joost is at rostrum of Janacek Philharmonic in Ostrava to conduct Bartok The Miraculous Mandarin, Liszt Les Preludes, Ligeti Lontano among others.
Tonu Kaljuste conducts the Gulbenkian Orchestra and Chorus with soloists in Lisbon on 5 April featuring Arvo Pärt's Adam's Lament and Tigran Mansurian's Requiem. | https://www.hoertnagel.com/en/news.html?tx_news_pi1%5Byear%5D=2018&tx_news_pi1%5Bcontroller%5D=News&cHash=ab154e5ccc479bc82a6ba8c08da0b395 |
Successfully baking bread takes practice, but it is very doable. Some of the most common problems faced by new and not-so-new bakers have avoidable causes and easy solutions.
Don’t let the constant fear of bread-baking mishaps keep you out of the kitchen. Dust off the bread pans, pull out the big bag of flour, and get ready to knead some dough worry-free.
Follow Your Instinct
How many times have you moaned, “But I followed the recipe perfectly!” while looking at an inedible mess covering your kitchen counter? What went wrong? How could it have gotten so messed up when you meticulously followed the bread recipe, step by step?
That’s what went wrong. You followed the directions, not your hands or eyes.
In generations past, baking was an enjoyable activity that rarely required a cookbook. Remember how Grandma would just toss handfuls of flour and a glass of water in the bowl, mixing it until it looked “right?” It turned out just fine! Today’s bakers need to adopt that sort of trust in their culinary instincts.
This is not to say that people should toss cookbooks out the window and just grab a bit of this and a lot of that, expecting perfectly formed loaves of bread as the final results. Not at all. Instead, experts encourage bakers to learn what dough should look and feel like, and not be afraid to “tweak” the recipe here and there to get the desired result.
Bread varies depending upon the environment. A wet and rainy day can result in a different dough than a loaf baked on a dry and hot summer afternoon. The dough might need less liquid on that wet day. Novice bakers often don’t know this, and they instead add the ingredients as listed in the recipe without paying attention to the dough in front of them.
Top Bread-baking Troubles
Aside from practicing the art of baking and learning to trust what you see and feel, there are other things you can do to avoid and fix common bread-baking problems.
Gummy bread
Gummy bread can result from slicing into the bread loaf before it’s completely cooled. It might feel nearly impossible to resist taking just a nibble, but launching into fresh-from-the-oven bread can ruin the whole loaf.
Before cutting or storing bread, let it cool completely on the kitchen counter, not in the bread pan. Slide the loaf out of the pan, and rest it on its side on a wire cake rack. When it is time to cut off a slice (or 10), always use a sharp serrated knife.
Flour dilemma
All flours are not created equal. A recipe that calls for unbleached all-purpose white flour can’t be interchanged equally in a recipe with whole-wheat flour or bread flour. Results will not be the same or even desirable, leading to a failed loaf or one with too tough a crust.
Even unbleached and bleached flours differ in protein content, which results in different outcomes. When a recipe does not specify which flour to use, default to the all-purpose unbleached variety.
Kneading time
Kneading improperly stands out as a top cause of bread-baking snafus. Sure, you can let a stand mixer do the job, but kneading by hand will produce the best results, if done properly.
Too thick of a crust or large holes in the bread can result from too much kneading, while a sour flavor, flat bread top, soggy loaf or one that sags in the middle can stem from too little kneading. Remain patient, work the dough, and know when the dough is done. It should be soft, smooth and springy but not very sticky.
Eight to 10 minutes of kneading with lightly floured hands on a lightly floured surface is a crucial bread-baking step. Kneading remains a skill best learned through practice, but once mastered, it can result in amazing loaves.
Too much flour
Adding too much flour to the dough as you start to knead is counterproductive. Experts recommend waiting a bit to see if the dough seemingly too wet to work with begins to form a more workable texture before you dump handfuls of flour on the work surface. When needed, use pinches of flour—not handfuls—as you knead.
Dough doesn’t rise
Using old yeast, even if it has not yet expired, usually causes bread failure. Proof the yeast first to make sure it’s good, and use lukewarm water—not hot or cold water—when adding it to the recipe. Yeast can even be stored in the freezer to help it last longer.
Loaves that don’t rise can also develop from old yeast, improper flour choice, wrong baking temperatures, the addition of salt to the yeast instead of the dry ingredients, improper measurement of ingredients and incorrect kneading techniques.
Collapsed loaf
Bread that collapses in the oven results from letting the loaf over-rise for too long of a time or in too hot of temperatures. This also can happen if the oven is not hot enough during baking. Use an oven thermometer to test that the temperature gauge and the internal temperature of the oven match.
Despite many potential trouble spots, remember that there are just as many answers and solutions to smooth out the journey from mixing ingredients to eating a fresh slice of homemade bread.
Don’t feel afraid of baking bread. Instead, view it as a creative outlet. Anyone involved in the creative pursuits in life knows that mistakes make great stepping stones toward success, so don’t fear them. Experts compare baking to modern-day art, so feel free to let your artistic side take over in the kitchen and bake away. | https://www.hobbyfarms.com/bread-baking-problems/ |
Last modified on May 27th, 2019
What is DOM(Document object model)?
The HTML page is a collection of object called “Elements” and these elements are called “DOM Elements.
Document Object Model Is basically a map or model of a web page. Its job is to describe the structure of an HTML document and the relationship between different elements like tags, attributes, and texts on a page. If we doing addition, deletion or modification in existing element on our website, so basically we are developing the structure of DOM. As a W3C specification, one important objective for the Document Object Model is to provide a standard programming interface that can be used in a wide variety of environments and applications. The Document Object Model can be used with any programming language.
Every DOM obj. is different and It has its own “properties, events and functionality”. And every DOM obj. has its own Documentation, according to “DOM standard”. There are additional documented functions that can be used to manage an element. These are called the DOM API.
There are three types of API’s which are specified by DOM Standard:
- Properties
We can say that DOM has a property of style sheets that can be used to get the list of CSS files, it has a title property etc.
- Events
We call event method e.g “onreadystatechnge” this code is for the event handler.
- Methods
We also use methods e.g files.delete, files.info, files.list. These are the methods of the file.
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Example 1 of DOM:
If we have this code
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<span style="font-size: 14pt;"><html>
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<Table>
<tr>
<th>Firstname</th>
<th>Lastname</th>
<th>Gender</th>
</tr>
<tr>
<td>Sameed</td>
<td>Khan</td>
<td>Male</td>
</tr>
</table>
</body>
</html></span>
- Now you can see the DOM representation of this code
Example 2 of DOM:
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</body></html></span>
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Why is DOM used for?
The DOM standard is a set of API functions that specifies how to programmatically
- Add DOM elements to a tree
- Remove DOM elements
- Add, Remove or Modify attributes.
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DOM History
- There are two model of DOM
- Model 1 concentrates on HTML and XML DOM. Model 1 contains the functionality for having document navigation and management. level 1 became a 1st recommendation of W3C in October 1998.
- Model 2 became a recommendation of W3C on 13 November 2000. In this, we can add the style object to the DOM to manage the style information in the document. Level 2 also defines an event model and provide the support for XML namespaces.
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Advantages of DOM
The DOM provides us lot of advantages. These are:
- XML DOM provides us platform and independence in languages.
- XML DOM is traversable.
- XML DOM is editable and dynamic. It allows developers to do addition, updating, deletion or move nodes on the tree.
- Robust API for the DOM tree.
- Relatively simple to modify the data structure and extract data.
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Disadvantages of DOM
The DOM has some disadvantages also. These are:
- It consumes more memory when the XML structure becomes large.
- Its operational speed is slower due to the larger usage of memory.
- Stores the entire document in memory.
- As DOM was written for any language, method naming conventions don’t follow standard Java programming conventions.
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When to use DOM?
Suppose you are working with normal data structure & if XML Schema is not a part of your plans, then you can find the one or more object-oriented standards.
In early stages, Document Object Model was made to be neutral in languages because it was developed for using dom with many or different languages. DOM doesn’t take illegal advantage of Java’s object-oriented features. | https://t4tutorials.com/what-is-domdocument-object-model-in-website-development/ |
IntroductionAntelope are herbivorous even-toed animals of the Bovidae family with social structures ranging from the solitary Steenbok to the very social Wildebeest. Kruger Park has a strong antelope population with larger species such as Sable antelope found in mixed savannah woodlands around Pretoriuskop and also near Phalaborwaa and the Roan in the open grasslands in north-eastern Kruger.
What is the difference between pronking and stotting?Stotting, also known as pronking, is a gait used by gazelles where they jump high in the air with all feet off the ground. Pronking is used to describe this behaviour in Springbok of southern Africa and comes from the Afrikaans word 'pronk' which means to strut or show off.
Do Antelope mothers abandon their young often?Some antelope mothers leave their young hidden from predators when the mothers go out and feed. In some cases such as the Impala the mothers leave their young in nursery herds that are cared for by adults. These two examples may give the impression that antelope mothers abandon their young.
What is an Antelope and why does it differ from a Gazelle?A Gazelle is a genus of the Antelope group with only one species, Springbok, occurring in Southern Africa. The Springbok does not occur in the Kruger Park.
Do Antelope horns re-grow?Horn re-growth in antelope does not generally occur. If the horn breaks off completely there will be no chance of regrowth but if the animal is young and the horns breaks somewhere along the shaft there may be some form of growth, although the growth will more than likely be deformed. | http://www.krugerpark.co.za/faq-on-antelope-africa.html |
FORMER Zimbabwe national cricket team coach Heath Streak looks set to continue his sabbatical from international cricket after ruling himself out of contention for the vacant Scotland job following reports linking him to the post, which fell vacant in September.
BY DANIEL NHAKANISO
Streak was believed to have applied for the job following the departure of former New Zealand all-rounder Grant Bradburn.
Bradburn (52) spent more than four years in charge of Scotland before leaving to become Pakistan assistant coach.
Streak, who was roped in by Cricket Scotland as bowling consultant ahead of their Twenty20 International tri-series against the Netherlands and Ireland in June, had been touted as a possible replacement together with former England all-rounder Paul Collingwood.
The former Zimbabwe captain, however, told The Sports Hub in an interview that although he had considered the job, he had not submitted an application.
Streak cited his commitments in the Indian Premier League (IPL) and other Twenty20 tournaments around the world as the main reason behind his decision.
“I haven’t applied for the Scotland role,” Streak said. “It (the Scotland job) is something that I have thought of and am still thinking about, but it’s difficult to commit to it with some of the other (Twenty20) T20 tournaments that I’m involved in. So at this moment there’s nothing concrete,” he said.
Streak, who picked up 216 wickets from 65 Tests and 239 wickets from 189 one-day internationals for Zimbabwe, is set to feature in the IPL early next year following a successful stint with Kolkata Knight Riders.
The well-travelled coach has also enjoyed successful stints with the Bangladesh national side as bowling coach.
Streak is currently in the midst of a legal battle with his former employer, Zimbabwe Cricket (ZC), and recently engaged the services of top local lawyer Thabani Mpofu in a US$1 million defamation lawsuit against ZC chairman Tavengwa Mukuhlani over racism and corruption accusations.
Streak launched the lawsuit in April after the ZC boss accused him in statements of biased selection, based on race, following Zimbabwe’s failure to qualify for next year’s World Cup in England.
Addressing reporters in April, Mukuhlani seemed to insinuate that Zimbabwe’s shock defeat to minnows United Arab Emirates in the World Cup qualifiers, which dealt the team’s qualification bid a hammer blow, could have been engineered by underhand dealings under the oversight of Streak.
The affable former Zimbabwe captain was sacked by ZC in the aftermath of that tournament, which led a series of clashes between the ZC board and a spirited group of disgruntled fans, who instead laid blame on the board for the team’s failure.
Streak, who was replaced by Indian Lalchand Rajput, has also filed a court application seeking the liquidation of the local cricket governing body for non-payment of money owed to him and other creditors. | https://thestandard.newsday.co.zw/2018/12/02/heath-streak-speaks-scotland-coaching-job/ |
Electrical Characterization of Si and SiGe Semiconductor-core Optical Fibers.
Fried, Michael Aaron
Master thesis
URI
http://hdl.handle.net/11250/2624525
Date
2019
Metadata
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Institutt for fysikk
Abstract
The optoelectronic properties of semiconductors in a fiber waveguide form make semiconductor core fibers a promising platform for many applications including infrared transmission, fiber detectors and nonlinear optics. In this work, the four point probe method is used to characterize core from fibers produced using the molten core draw (MCD) method. Sample preparation techniques are introduced to perform four point probe and Hall measurements. Maskless lithography is used to pattern electrical contacts on fiber cores with high positioning accuracy and the use of a permanent epoxy based resist is shown to be effective at covering cracks in the fiber cladding. Resistivity measurements performed on Si fibers with low purity CaO based interface coatings showed that the fiber drawing process introduces impurities into the fiber core, causing a drop in resistivity. As-drawn SiGe fibers showed no change in resistivity with high purity CaO based and boron containing coatings. This changed after laser annealing, where the resistivity of the SiGe fiber with boron coating dropped by over two orders of magnitude. | https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/2624525 |
The place of art in the world has always been debated. Whether it’s a luxury of the elites or a universal language that can push humanity forward, its mainstream importance is often obscured by an air of exclusivity. But the subtle ways in which art can impact people and shape events occasionally come into sharp focus. In his latest novel, Benchere in Wonderland, Steven Gillis finds the intersections between art and society on both the micro and macro levels.
Michael Benchere, a famous artist, is deep in grief following the death of his wife and partner, Marti, when he decides to go to the Kalahari. There, he plans to complete the final project he and Marti undertook, a combination of his artistic eye and her engineering skill. The sculpture he plans to build, rising a few hundred feet over the desert, draws international attention. Soon after he arrives, Benchere is surrounded by a growing community of fans and supporters, all hoping to be close to the act of creation.
But as unrest sweeps Africa, Benchere is absorbed by protesters and becomes a symbol of resistance. Meanwhile, the over two hundred people who have joined him in the Kalahari become restless, and Benchere is confronted by a couple who hope to capitalize on what they see as an incredible economic opportunity. Struggling to maintain his vision and independence, Benchere’s hand is increasingly forced by strife both international and closer to camp, until his espoused neutrality threatens his legacy and he must decide where he and his art stand.
Gillis is able to fold a great deal into the story, jumping between continents and characters multiple times within each chapter. The core people who join Benchere in Africa each bring with them a need to get back to themselves, be it his daughter Zooie who is caught up in a dead-end relationship or Linda Darling, a reality tv star at rock bottom. For those characters, art is a means by which they can connect with something more essential in the human experience, clearing their heads and resetting, in a sense. It’s art as therapy, creation as comfort:
Linda Darling proves a pleasant surprise. Resourceful and engaged, she asks for assignments, rolls up her sleeves and dives into each task. Spirited, she jokes with the others, starts a friendship with Harper, follows Benchere’s instructions as she would a director on set. A marked improvement from her first night, when she disappeared after dinner, reemerging stoned. Benchere took her aside then and said, “Listen, Darling, this is not celebrity rehab. You want to get fried, have at it. But if your being here fucks with my work, we’ll put you on the next truck back to Maun.
But soon they are joined by others, including Dancy and Gabriella Mund, entrepreneurs who see the open land surrounding the sculpture as a chance to develop a resort. Benchere’s reputation is far reaching, and his name being attached to a project can bring in a steady flow of capital and tourists. While the Munds hope to capitalize on that potential for themselves, Benchere’s son Kyle is in Providence, Rhode Island, doing much the same but for the public good, working on a set of Benchere-designed rowhouses he hopes to manage as a cooperative to rebuild a largely rundown neighborhood. Littered with mentions of “Benchere originals” and allusions to his wealth, Gillis doesn’t avoid the commercial value of a well-known artist, but he doesn’t paint the impact of such moneyed interest with broad strokes.
One of the larger questions the book asks is what it means for an artist to lose control of his work. Even as the community built up around his latest project begins to segment and collapse, Benchere’s name is co-opted by demonstrators throughout Africa. Protesters build faux-Benchere sculptures from found objects, leaving their work behind when forced to flee by police. Overnight, Benchere goes from a private artist to a seditious political figure, his work representing the hopes of people fighting for their rights. Although he supports the cause personally, Benchere struggles to draw lines between his own work and the works being built in his name, with those around him pulling in either direction:
“These sculptures,” Stern says. “The two you made and then the rest.”
“They muddy the waters.”
“Stir the pot.”
“Give people a reason to rally.”
“ You have to remember,” Rose says, “war is tricky business.”
“For every Mau Mau there’s a dozen disasters.”
“A dozen or more.”
With so many layers of meaning, and Benchere’s memories of Marti a steady thread throughout the story, it’s no small task to keep it all together. But Gillis does just that, and Benchere in Wonderland is a comprehensive view of the role of art in both the personal and political realms. Engaging, moving, and at times funny, it’s a stark reminder of the way art can and does change the world. | http://necessaryfiction.com/reviews/BenchereinWonderlandbyStevenGillis |
At Sonas childcare centre we understand how important it is to establish good healthy eating practices at a very young age. We believe in providing highly nutritious good quality meals and encouraging a positive stimulating environment where children are given the opportunity to interact socially while enjoying their meal.
• We prepare all our home cooked meals on site in our purpose built,
HSE approved kitchen.
• Our menu is varied and we use fresh produce that is sourced locally.
• We have a three week rotating menu in place to ensure that your child receives a variety of nutrients and tastes.
• Our menu follows the food pyramid, which is a definitive guide to ensuring that your child gets all their nutritional needs.
• Our menu is designed to consider all the developmental stages of our child that will carry them from our baby room through to Pre School.
• We fully supervise all snack and mealtimes, to provide a safe and positive but stimulating social environment for your child to enjoy their food.
• We adhere to individual food choices or specific dietary requirement you or your child may have. Any specific Allergy/Cultural dietary requirements can be outlined to our team at registration.
• We keep daily detailed report sheets on what your child has eaten during the day. A copy of this sheet is available on request. | https://mountmellickchildcare.com/hes |
---
abstract: 'Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representation of Ollivier curvature using the graph Laplacian. Using the Laplacian as a generator of a continuous time Markov chain, we connect Ollivier curvature with the heat equation which is strongly related to continuous time random walks. In particular, we prove that a lower bound on the Ollivier curvature is equivalent to a certain Lipschitz decay of solutions to the heat equation. This is a discrete analogue to a celebrated Ricci curvature lower bound characterization by Renesse and Sturm. Our representation of Ollivier curvature via the Laplacian allows us to deduce a Laplacian comparison principle by which we prove non-explosion and improved diameter bounds.'
author:
- 'Florentin Münch, Rados[ł]{}aw K. Wojciechowski'
bibliography:
- 'Bibliography.bib'
title: ' Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds '
---
Introduction
============
Ricci curvature is strongly related to the heat equation. In particular, lower Ricci curvature bounds can be characterized via gradient estimates as in the following theorem by Renesse and Sturm (see [@renesse2005transport Theorem 1.3 and Corollary 1.4]).
\[thm:Sturm\] For any smooth connected Riemannian manifold $M$ and any $K \in {{\mathbb{R}}}$ the following properties are equivalent:
1. $Ric(M) \geq K$.
2. For all $f \in C_c^{\infty}(M)$ and all $t > 0$ $$\|\nabla P_t f \|_\infty \leq e^{-Kt} \|\nabla f \|_\infty.$$
3. For all bounded $f \in C^{Lip}(M)$ and all $t>0$ $$Lip(P_t f) \leq e^{-Kt} Lip(f).$$
4. For all $x,y \in M$ and all $t>0$ $$\begin{aligned}
W(p^x_t,p^y_t) \leq e^{-Kt} d(x,y)
\end{aligned}$$
where $P_t$ denotes the heat semigroup generated by the Laplace-Beltrami operator, $p_t^x$ denotes the heat kernel and $W$ denotes the $L_1$-Wasserstein distance.
We prove that the same holds true on graphs (see Theorem \[thm:gradientGraphs\]). To do so, we employ a new method intertwining the heat semigroup with a cutoff function, which we call the perpetual cutoff method. Our curvature notion will be a generalized Ollivier curvature based on its modification by Lin, Lu and Yau (see [@lin2011ricci]) which we extend to the case of general graph Laplacians. In particular, we now apply this curvature notion to graphs with unbounded vertex degree. For an introduction to Ollivier curvature, see [@ollivier2007ricci; @ollivier2009ricci]. A relation between curvature and the number of triangles in a graph is given in [@jost2014ollivier]. Methods to compute the curvature can be found in [@loisel2014ricci]. Ollivier curvature has been applied to describe the internet topology [@ni2015ricci; @wang2016interference], wireless network theory [@wang2014wireless], economic market analysis [@sandhu2016ricci] and cancer networks [@sandhu2015graph; @tannenbaum2015ricci; @sandhu2015analytical].
The breakthrough paper of Renesse and Sturm mentioned above motivated a generalization of Ollivier curvature to semiproups compatible with Lipschitz functions and Wasserstein metrics. Approaches in this direction have been made in [@bass1981markov; @joulin2007poisson; @joulin2009new; @veysseire2012coarse]). However, the problem suggested by Ollivier (see Problem D in [@ollivier2010survey]), namely, if a lower curvature bound implies non-explosion (also known as stochastic completeness), is still open. Non-explosion in this context means that the process remains in the state space for all time. We prove non-explosion for all locally finite graph Laplacians with Ollivier curvature decaying not faster than $-\log R$ (see Theorem \[thm:StochComplete\]). Therefore, this theorem can be seen as an initial step to solve Ollivier’s problem in a general setting. Moreover, we prove that the curvature decay rate $-\log R$ is optimal.
One of the main observations of our paper is that, on graphs, the limit expression for Lin, Lu and Yau’s Ollivier curvature simplifies to the limit-free expression $$\begin{aligned}
\kappa(x,y) = \inf_{\substack {f\in Lip(1) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f\end{aligned}$$ where $\nabla_{xy} f = \frac{f(x)-f(y)}{d(x,y)}$, $d$ is the usual combinatorial graph distance, $Lip(1)$ denotes the functions with Lipschitz constant 1 with respect to this metric and $\Delta$ is the graph Laplacian (see Theorem \[thm:nablaDelta\]). Furthermore, the curvature admits another limit-free expression in terms of transport costs (see Proposition \[pro:CharTransport\]) which simplifies to give an explicit formula in the case of combinatorial graph (see Theorem \[thm:TransprtCombinatorial\]). These simplifications give the starting point for proving the semigroup characterization of a lower Ricci curvature bound in analogy to the work of Renesse and Sturm. Our gradient estimates for the continuous time heat equation seem to be the first result of this kind for Ollivier curvature. Indeed, these gradient estimates have been previously used as a globally defined Wasserstein curvature bound (see [@joulin2007poisson Definition 2.1]).
In contrast to Ollivier curvature, there are various gradient estimates on graphs under Bakry-Emery curvature bounds [@horn2014volume; @gong2015properties; @munch2014li; @lin2015equivalent; @bauer2015li] and under entropic Ricci curvature [@erbar2016poincar Theorem 3.1]. Using a modification of Ollivier curvature, gradient estimates have been established for continuous time Markov processes in [@veysseire2012coarse]. In Section \[sec:MarkovProcesses\], we prove that this modification coincides with our curvature notion on locally finite graphs with a lower curvature bound (see Corollary \[cor:MarkovChains\]).
The control of the Lipschitz constant of the semigroup yields stochastic completeness for all graphs with a constant lower Ollivier curvature bound (see Lemma \[lem:StochComplete\]). However, as already mentioned above, one can get even better results by employing different techniques. To do so, we first establish a Laplacian comparison principle which seems to be the first of its kind for any discrete Ricci curvature notion. The simplest version (Theorem \[thm:LaplaceCompare\]) states that under the assumption of a lower curvature bound $K \in {{\mathbb{R}}}$, we have $$\begin{aligned}
\Delta d(x_0,\cdot) \leq {\operatorname{Deg}}(x_0) -Kd(x_0,\cdot)\end{aligned}$$ where ${\operatorname{Deg}}(x_0)$ is the weighted vertex degree of a fixed vertex $x_0$. This Laplacian comparison can be extended to the case of decaying curvature (see Theorem \[thm:LaplaceComparisonNonConst\]). Via the Laplacian comparison, we compare the curvature of a given graph to the curvature of the birth-death chain associated to the graph (see Corollary \[cor:CurvCompare\]). Birth-death chains are Markov processes on weighted path graphs. For our purposes, we identify the birth-death chain with its associated weighted path graph, see Section \[sec:OllivierBirthDeath\]. The comparison to birth-death chains allows us to reduce many problems to the case of weighted path graphs where the curvature can be easily calculated (see Theorem \[thm:line\]).
Using the Laplacian comparison principle and Khas’minskii’s criterion (see [@huang2011stochastic]), we will prove stochastic completeness under the assumption that the Ollivier curvature does not decay to $-\infty$ faster than $-\log R$ (see Theorem \[thm:StochComplete\]). This result is remarkable when comparing to known stochastic completeness results for graphs which use the Bakry-Emery curvature instead and require a constant lower bound as well as additional assumptions, such as a non-local completeness condition and a lower bounded vertex measure (see [@hua2017stochastic Theorem 1.2]). As such, this article may also give inspiration to transfer the new methods presented here to discrete Bakry-Emery theory. As another application of the Laplacian comparison principle, we can prove finiteness and improved diameter bounds. For diameter bounds under uniformly positive Ollivier curvature, see [@lin2011ricci; @ollivier2009ricci]. Diameter bounds under a positive average Ollivier curvature can be found in [@paeng2012volume]. Diameter bounds under uniformly positive Bakry-Emery curvature are proven in [@liu2016bakry; @fathi2015curvature; @horn2014volume]. In this article we show that if the vertex degree is bounded and the curvature decays not faster than $1/R$, then the graph is finite (see Theorem \[thm:ImprovedDiamBound\] and Corollary \[cor:finite\]). Surprisingly, uniformly positive Ricci curvature alone does not imply finiteness (see Example \[Ex:positiveCurvInfiniteDiam\]). However, if we additionally assume a lower bound on the vertex measure, then uniformly positive Ricci curvature indeed implies finiteness (see Corollary \[cor:FiniteDiamBoundedMeasure\]).
Discussion and comparison to manifolds
--------------------------------------
The reader familiar with the manifold case might be surprised at the optimal curvature decay rates: $- \log R$ for stochastic completeness and $1/R$ for compactness in the case of graphs.
In [@grigor1999analytic Theorem 15.4], it is shown that, for manifolds, the optimal curvature decay rate for stochastic completeness is $-R^2$ which was proven in [@varopoulos1983potential; @hsu1989heat]. One tempting explanation for the discrepancy of the decay rate between manifolds and graphs is the choice of the metric. Frequently, intrinsic metrics introduced in [@frank2014intrinsic] are used to describe the geometry of graphs with unbounded vertex degrees and to give analogues to results on manifolds (see, for example, [@bauer2012cheeger; @huang2013note; @folz2015volume; @keller2015intrinsic]). However, we give an example of a stochastically incomplete graph with curvature decaying like $-(\log \sigma)^{1+{\varepsilon}}$ with respect to an intrinsic metric $\sigma$, even if the curvature is defined by incorporating the intrinsic metric, see Example \[ex:incompleteIntrinsic\].
The optimal decay rate on manifolds to guarantee compactness is $C/R^2$ for some constant $C$. Interestingly, for $C>\frac{n-1}4$, compactness holds, but for $C \leq \frac{n-1}4$, non-compact manifolds are known with the corresponding Ricci curvature decay (see [@cheeger1982finite; @holcman2005boundary]). The discrepancy of the decay rate between manifolds and graphs here also cannot be explained via intrinsic metrics since we assume a bounded vertex degree for our result and, therefore, the combinatorial distance is intrinsic up to a factor. Hence, it might be interesting to ask for the deeper reasons for these two discrepancies.
Before introducing the setup and notations, we give a brief summary of the subsequent sections. In Section \[sec:Ollivier\], we prove the limit-free simplifications of the curvature formula and compute the curvature of combinatorial graphs and birth-death chains. In Section \[sec:GradEstimates\], we introduce the perpetual cutoff method and non-linear cutoff semigroups which turn out to perfectly intermesh with Ollivier curvature, yielding the desired gradient estimate for the semigroup. In Section \[sec:LaplaceCompare\], we present the Laplacian comparison theorem and, as applications, we prove a birth-death chain reduction, stochastic completeness and improved diameter bounds which lead to our finiteness results. Finally in Section \[sec:MarkovProcesses\], we prove that on graphs with a lower curvature bound, our curvature notion coincides with the curvature introduced in [@veysseire2012coarse].
Setup and Notation
------------------
A triple $G=(V,w,m)$ is called a *graph* if $V$ is a countable set, $w:V^2 \to [0,\infty)$ is symmetric and zero on the diagonal and $m:V \to (0,\infty)$. In the following, we only consider *locally finite* graphs, i.e., for every $x \in V$ there are only finitely many $y \in V$ with $w(x,y) >0$. We call $V$ the *vertex set* with elements of $V$ called *vertices*, $w$ the *edge weight* and $m$ the *vertex measure*. We will write $x \sim y$ if $w(x,y)>0$ and say that $(x,y)$ is an *edge* in the graph. We say that $G$ is a *combinatorial graph* if $w(x,y) \in \{0,1\}$ for all $x,y \in V$ and if $m \equiv 1$.
We define the *graph Laplacian* $\Delta: {{\mathbb{R}}}^V \to {{\mathbb{R}}}^V$ via $$\Delta f(x) := \frac 1 {m(x)} \sum_{y\in V} w(x,y)(f(y) - f(x)).$$ We define the function spaces $$\begin{aligned}
C(V)&:=\{f:V \to {{\mathbb{R}}}\}= {{\mathbb{R}}}^V, \\
\ell_\infty(V)&:=\{f \in C(V): f \mbox{ is bounded}\}, \\
C_c(V) &:= \{f \in C(V): f \mbox{ is finitely supported}\},\end{aligned}$$ all endowed with the supremum norm $\|\cdot \|_\infty$. We let $\ell^p(V,m)$ denote the $\ell^p$ spaces with respect to $m$, that is, $\ell^p(V,m) = \{ f \in C(V) : \sum_{x \in V} |f(x)|^p m(x) <\infty\}.$ We let $${\operatorname{Deg}}(x) := \frac1 {m(x)} \sum_{y \in V} {w(x,y)}$$ denote the *vertex degree* and let ${\operatorname{Deg}}_{\max} := \sup_x {\operatorname{Deg}}(x) \in (0,\infty]$. We remark that the Laplace operator is bounded on $\ell_\infty(V)$ and $\ell^p(V,m)$ for any $p\geq 1$ if and only if ${\operatorname{Deg}}_{\max} < \infty$ (see [@keller2010unbounded Theorem 11], [@haeseler2011laplacians Theorem 9.3]). In this case, we will say that $G$ has *bounded vertex degree*. For a non-negative $f \in \ell_\infty(V)$, we denote by $P_t f$ the smallest non-negative bounded continuous solution $u(x,t)$ to the heat equation $$\begin{aligned}
\begin{cases}
\begin{tabular}{rll}
$\Delta u(x,t)$ & $=\partial_t u(x,t)$ & $x \in V$, $t\geq0$ \\
$u(x,0)$ & $= f(x)$ &$x \in V$.
\end{tabular}
\end{cases}\end{aligned}$$ A proof of the existence and uniqueness of $P_t f$ and further details can be found in [@wojciechowski2008heat; @keller2012dirichlet]. Note, in particular, that $P_{s+t} f = P_s P_t f$ which is referred to as the *semigroup property* and that $P_t$ is *positivity preserving*, i.e., $P_t f\geq0$ for $f\geq0$. A graph is called *stochastically complete* or *non-explosive* if $P_t \mathbf 1 = \mathbf 1$ for all $t>0$ where $\mathbf 1$ is the constant function which is 1 on $V$.
We define the *combinatorial graph distance* $d$ on $V\times V$ via $d(x,y):= \inf\{n:x=x_0\sim \ldots \sim x_n=y\}$. A graph is said to be *connected* if $d(x,y) < \infty$ for all $x,y$ in $V$. We will always assume that graphs are connected. We write $B_r(x) = \{ y \in V \ | \ d(x,y) \leq r \}$ and $S_r(x) = \{ y \in V \ | \ d(x,y)=r \}$. We note that $G$ is connected if and only if $P_t$ is a positivity improving semigroup, that is, $P_t f >0$ if $f \geq 0$ for $t>0$, see [@keller2012dirichlet].
We write $f \in Lip(1)$ if $|f(x)-f(y)| \leq d(x,y)$ for all $x,y \in V$. The *Wasserstein distance* $W(\mu,\nu)$ for probability measures $\mu$ and $\nu$ on $V$ is given by $$\begin{aligned}
W(\mu,\nu) &:= \sup_{f \in Lip(1) \cap \ell_\infty(V)} \int f d\mu - \int f d\nu \\
&= \sup_{f \in Lip(1) \cap \ell_\infty(V)} \sum_{x \in V} f(x) (\mu(x) - \nu(x)).\end{aligned}$$ We note that the supremum is well defined due to the boundedness of the functions and that it suffices to take the supremum over functions in $Lip(1)$ when the measures are finitely supported. Equivalently (see e.g. [@villani2003topics Theorem 1.14]), the Wasserstein metric can be defined as $$W(\mu,\nu) := \inf_{\rho} \sum_{x,y \in V} \rho(x,y) d(x,y)$$ where the infimum is taken over all $\rho: V^2 \to [0,1]$ which satisfy $\sum_{y \in V} \rho(x,y) = \mu(x)$ and $\sum_{x \in V} \rho(x,y) = \nu(y)$ for all $x,y \in V$. We call such a $\rho$ a *coupling* between $\mu$ and $\nu$.
Ollivier curvature and graph Laplacians {#sec:Ollivier}
=======================================
Ollivier curvature is a powerful and easy to calculate tool used to study analytic and geometric properties of a space. Until now, Ollivier curvature for graphs has only been used in the case of bounded Laplacians. In this section, we extend the definition of Ollivier curvature to the case of unbounded graph Laplacians. We then present a strikingly simple expression for calculating the curvature using only the Laplacian as well as a formula involving transport costs. Along the way, we illustrate how to calculate the curvature in a variety of situations including graphs without cycles, combinatorial graphs and birth-death chains.
For ${\varepsilon}>0$, we let $$\begin{aligned}
m_x^{\varepsilon}(y) := 1_y(x) + {\varepsilon}\Delta 1_y (x)\end{aligned}$$ which is a finitely supported probability measure and, in particular, non-negative if ${\varepsilon}$ is sufficiently small. This can be seen as $$\begin{aligned}
m_x^{\varepsilon}(y) = \begin{cases}
1 - {\varepsilon}{\operatorname{Deg}}(x) &: y=x \\
{\varepsilon}w(x,y)/m(x) &: \mbox{otherwise}.
\end{cases}\end{aligned}$$ In particular, $$\begin{aligned}
\int f dm_x^{\varepsilon}= \sum_{y \in V} f(y) m_x^{\varepsilon}(y) = (f + {\varepsilon}\Delta f) (x).\end{aligned}$$ We remark that $m_x^{\varepsilon}$ can be seen as a first order approximation to the heat kernel $P_{\varepsilon}1_x$. This connection will be further explored in Section \[sec:MarkovProcesses\].
In the case of the normalized Laplacian, that is, when $w: V^2 \to \{ 0,1\}$ and $m(x) = d_x := \# \{y \sim x \}$, for $\alpha := 1-{\varepsilon}$ one has $$\begin{aligned}
m_x^{\varepsilon}(y) = \begin{cases}
\alpha &: y=x \\
(1-\alpha)/d_x&: y \sim x \\
0 &: \mbox{otherwise}
\end{cases}\end{aligned}$$ which corresponds to the definition of Lin, Lu and Yau (see [@bauer2011ollivier; @lin2011ricci]). Note that, in this case, ${\operatorname{Deg}}=1$ so that the normalized Laplacian is always a bounded operator.
Following the standard definition, we let, for $x \not = y$ $$\begin{aligned}
\kappa_{\varepsilon}(x,y) := 1 - \frac{W(m_x^{\varepsilon},m_y^{\varepsilon})}{d(x,y)}\end{aligned}$$ where $W$ denotes the Wasserstein distance. In [@bourne2017ollivier] it is shown that for the normalized Laplacian and $x \sim y$, the function $\kappa_{\varepsilon}(x,y)$ is concave and piecewise linear in ${\varepsilon}\in [0,1]$ with at most three linear parts. Analogous to Lin, Lu and Yau, one can prove the existence of $$\begin{aligned}
\kappa(x,y) := \lim_{{\varepsilon}\to 0^+} \frac 1 {\varepsilon}\kappa_{\varepsilon}(x,y)\end{aligned}$$ by which we extend Lin, Lu and Yau’s curvature definition to arbitrary graph Laplacians.
Using standard theory, the curvature $\kappa(x,y)$ is uniquely determined by the induced subgraph $B_1(x) \cup B_1(y)$ for $x \sim y$. We write $Ric(G) \geq K$ if $\kappa(x,y) \geq K$ for all $x,y$. We remark that to show $Ric(G) \geq K$ it suffices to show that $\kappa(x,y) \geq K$ for adjacent vertices $x \sim y$ as in [@lin2011ricci].
As a first example, we mention that it is well-known that, in the case of the normalized Laplacian, Abelian Cayley graphs have non-negative Ollivier curvature (see e.g. [@lin2014ricci Theorem 2]). We will give further examples later in this section.
Bypassing the limit
-------------------
Our first aim is to express the curvature without the limit which turns out to be surprisingly simple. To do so, we introduce the notation of the gradient $$\nabla_{xy} f :=
\frac{f(x) - f(y)}{d(x,y)}$$
for $x\neq y \in V$ and $f \in C(V)$ and the associated Lipschitz constant $$\|\nabla f \|_\infty := \sup_{x \neq y} |\nabla_{xy} f| = \sup_{x \sim y} |\nabla_{xy} f| \in [0,\infty].$$ For $K \geq 0$, we let $Lip(K) = \{f \in C(V): \|\nabla f \|_\infty \leq K \},$ that is, the set of functions with Lipschitz constant $K$ or $K$-Lipschitz functions. We are now prepared to present our limit-free curvature formula.
\[thm:nablaDelta\] Let $G=(V,w,m)$ be a graph and let $x \neq y$ be vertices. Then, $$\begin{aligned}
\kappa(x,y) = \inf_{\substack {f\in Lip(1) \cap C_c(V) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f.\end{aligned}$$
By definition, one has
$$\begin{aligned}
W(m_x^{\varepsilon},m_y^{\varepsilon}) &= \sup_{f \in Lip(1)} \sum_z f(z)(m_y^{\varepsilon}(z) - m_x^{\varepsilon}(z))\\
&= \sup_{f \in Lip(1)} [(f(y) + {\varepsilon}\Delta f(y)) - (f(x) + {\varepsilon}\Delta f(x))]\\
&= d(x,y)\sup_{f \in Lip(1)} \nabla_{yx}(f + {\varepsilon}\Delta f).\end{aligned}$$
Hence, $$\begin{aligned}
\frac 1 {\varepsilon}\kappa_{\varepsilon}(x,y)
&= \frac 1 {\varepsilon}\left( 1- \frac{W(m_x^{\varepsilon},m_y^{\varepsilon})}{d(x,y)} \right) \\
&= \frac 1 {\varepsilon}\left( \inf_{f \in Lip(1)} ( 1- \nabla_{yx}(f + {\varepsilon}\Delta f) ) \right)\\
& = \inf_{f \in Lip(1)} \left(\frac 1 {\varepsilon}(1- \nabla_{yx}f) +\nabla_{xy} \Delta f \right)\\
&\leq \inf_{\substack {f\in Lip(1) \cap C_c(V) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f \end{aligned}$$
To prove the other inequality, we first show the existence of a minimizer $f_{\varepsilon}\in Lip(1) \cap C_c(V)$ of the expression $\frac 1 {\varepsilon}(1- \nabla_{yx}f) +\nabla_{xy} \Delta f$ found above for every ${\varepsilon}>0$ satisfying $f_{\varepsilon}(x)=0$. This follows as, for every $f \in Lip(1)$, we construct $\widetilde f \in Lip(1)$ supported on $B_{2r}(x)$ with $r:=d(x,y)+1$ which satisfies $$\begin{aligned}
\frac 1 {\varepsilon}(1- \nabla_{yx}f) +\nabla_{xy} \Delta f = \frac 1 {\varepsilon}(1- \nabla_{yx}\widetilde f) +\nabla_{xy} \Delta \widetilde f. \label{eq:fwidetildefMaxExist}\end{aligned}$$ By adding a constant to $f$, we can assume that $f(x)=0$. This yields that $|f(z)|\leq r$ for all $z \in B_1(x) \cup B_1(y)$ since $f \in Lip(1)$. Let $\phi:V \to {{\mathbb{R}}}$ be given by $$\phi(z) = \left[r \wedge (2r - d(x,z)) \right]_+.$$ Observe that $\phi(z)=r$ for all $z \in B_1(x)\cup B_1(y)$. Therefore, $\widetilde f := -\phi \vee f \wedge \phi$ satisfies (\[eq:fwidetildefMaxExist\]) as it agrees with $f$ on $B_1(x)\cup B_1(y)$. Moreover, $\phi$ and thus $\widetilde f$ are supported on $B_{2r}(x)$. This construction of $\widetilde f$ shows that we can restrict the infimum to functions supported on the compact set $B_{2r}(x)$ which yields the existence of a minimizer $f_{\varepsilon}$ with $f_{\varepsilon}(x)=0$ for all ${\varepsilon}>0$ due to continuity.
Due to the compactness of $B_{2r}(x)$ and since $f_{\varepsilon}(x)=0$ and $f_{\varepsilon}\in Lip(1)$ for all ${\varepsilon}>0$, there exists a sequence ${\varepsilon}_n \to 0$ such that $f_0 := \lim_{n\to \infty} f_{{\varepsilon}_n}$ exists. Since $\frac 1 {\varepsilon}\kappa_{\varepsilon}(x,y) = \frac{1}{{\varepsilon}} (1-\nabla_{yx}f_{\varepsilon})+\nabla_{xy}\Delta f_{\varepsilon}$ and $\lim_{{\varepsilon}\to 0^+} \frac 1 {\varepsilon}\kappa_{\varepsilon}(x,y)$ exists, we get that $\nabla_{yx}f_{\varepsilon}\to 1$ as ${\varepsilon}\to 0^+$. Therefore, $f_0 \in Lip(1) \cap C_c(V)$, $\nabla_{yx}f_0 = 1$ and since $\nabla_{yx} f_{\varepsilon}\leq 1$, we get $$\begin{aligned}
\kappa(x,y) &= \lim_{{\varepsilon}\to 0^+} \frac 1 {\varepsilon}(1- \nabla_{yx}f_{\varepsilon}) +\nabla_{xy} \Delta f_{\varepsilon}\\
&\geq \lim_{n \to \infty} \nabla_{xy} \Delta f_{{\varepsilon}_n} \\
& = \nabla_{xy} \Delta f_0\\
&\geq \inf_{\substack {f\in Lip(1) \cap C_c(V) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f.\end{aligned}$$
Putting together the upper and lower estimates yields $$\begin{aligned}
\kappa(x,y) = \inf_{\substack {f\in Lip(1) \cap C_c(V) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f\end{aligned}$$ as desired.
Following [@bhattacharya2015exact Lemma 2.2], it suffices to optimize over all integer valued Lipschitz functions $f$ which yields the following corollary.
Let $G=(V,w,m)$ be a graph and let $x \neq y$ be vertices. Then, $$\begin{aligned}
\kappa(x,y) = \inf_{\substack {f: B_1(x) \cup B_1(y) \to {{\mathbb{Z}}}\\ f\in Lip(1) \\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f\end{aligned}$$ Moreover, on combinatorial graphs, the curvature $\kappa(x,y)$ is integer valued for all $x\sim y$.
The proof of the first part follows [@bhattacharya2015exact Lemma 2.2]. For the integrality of the curvature in the case of combinatorial graphs, observe that $\nabla_{xy}\Delta f \in {{\mathbb{Z}}}$ whenever $f$ is integer valued, $x\sim y$ and $\Delta$ is the combinatorial graph Laplacian.
We now explicitly calculate the curvature of large girth graphs in our setting by using Theorem \[thm:nablaDelta\].
\[ex:NoCycles\] Let $G=(V,w,m)$ be a graph and let $x\sim y$ be vertices. Suppose that the edge $(x,y)$ is not contained in any 3-,4- or 5-cycles. Then, an optimal 1-Lipschitz function $f$ is given by an extension of $$f(z) = \begin{cases}
0 &: z \sim x \mbox{ and } z \neq y\\
1 &: z=x\\
2 &: z=y\\
3 &: z \sim y \mbox{ and } z \neq x
\end{cases}$$ yielding the curvature $$\kappa(x,y) = 2w(x,y) \left(\frac 1{m(x)} + \frac 1 {m(y)} \right) - {\operatorname{Deg}}(x) - {\operatorname{Deg}}(y).$$
We now give another limit-free expression of our extension of Lin-Lu-Yau’s Ollivier curvature via transport costs.
\[pro:CharTransport\] Let $G=(V,w,m)$ be a graph and let $x_0 \neq y_0$ be vertices. Then, $$\begin{aligned}
\kappa(x_0,y_0) &= \sup_{\rho} \sum_{\substack{x \in B_1(x_0) \\ y \in B_1(y_0)}}\rho(x,y) \left[1 - \frac{d(x,y)}{d(x_0,y_0)}\right] \label{eq:PropTransport} \end{aligned}$$ where the supremum is taken over all $\rho: B_1(x_0) \times B_1(y_0) \to [0,\infty)$ such that $$\begin{aligned}
\sum_{y \in B_1(y_0)} \rho(x,y) &= \frac {w(x_0,x)}{m(x_0)} \qquad \mbox{ for all } x \in S_1(x_0) \mbox{ and} \label{eq:rhoXProp}\\
\sum_{x \in B_1(x_0)} \rho(x,y) &= \frac {w(y_0,y)}{m(y_0)} \qquad \mbox{ for all } y \in S_1(y_0) \label{eq:rhoYProp}.\end{aligned}$$
We remark that $\rho$ is defined on balls, but we only require the coupling property on spheres. We additionally do not assume anything concerning $\sum_{x,y} \rho(x,y)$.
We will write $$F(\rho) = \sum_{x \in B_1(x_0)} \sum_{y \in B_1(y_0)} \rho(x,y) \left[1 - \frac{d(x,y)}{d(x_0,y_0)}\right]$$ for any coupling $\rho$. We wish to show that $\kappa(x_0,y_0) = \sup_\rho F(\rho)$ where the supremum is taken over all couplings $\rho$ satisfying (\[eq:rhoXProp\]) and (\[eq:rhoYProp\]).
Using that $\sum_{x,y}\rho(x,y)=1$ for all couplings considered in the transport definition of $W$, we have $$\begin{aligned}
\kappa_{\varepsilon}(x_0,y_0) = 1 - \frac {W(m_{x_0}^{\varepsilon},m_{y_0}^{\varepsilon})}{d(x_0,y_0)} = 1 - \frac{\inf_\rho \sum_{x,y} \rho(x,y)d(x,y)}{d(x_0,y_0)} = \sup_\rho F(\rho)
\end{aligned}$$ where the supremum is taken over all $\rho: B_1(x_0) \times B_1(y_0) \to [0,\infty)$ such that $$\begin{aligned}
\sum_{y \in B_1(y_0)} \rho(x,y) &= m_{x_0}^{\varepsilon}(x) = 1_x(x_0) + {\varepsilon}\Delta 1_x(x_0) \qquad \mbox{ for all } x \in B_1(x_0) \mbox{ and} \\
\sum_{x \in B_1(x_0)} \rho(x,y) &= m_{y_0}^{\varepsilon}(y) = 1_y(y_0) + {\varepsilon}\Delta 1_y(y_0) \qquad \mbox{ for all } y \in B_1(y_0).
\end{aligned}$$ Dividing $\rho$ by ${\varepsilon}$ yields $$\begin{aligned}
\frac{1}{{\varepsilon}} \kappa_{\varepsilon}(x_0,y_0) = \frac 1 {{\varepsilon}} \left(1 - \frac {W(m_{x_0}^{\varepsilon},m_{y_0}^{\varepsilon})}{d(x_0,y_0)} \right) =
\sup_\rho F(\rho)
\end{aligned}$$ with the supremum taken over all $\rho: B_1(x_0) \times B_1(y_0) \to [0,\infty)$ such that $$\begin{aligned}
\sum_{y \in B_1(y_0)} \rho(x,y) &= \frac 1 {\varepsilon}1_x(x_0) + \Delta 1_x(x_0) \qquad \mbox{ for all } x \in B_1(x_0) \mbox{ and} \label{eq:rhoXProof} \\
\sum_{x \in B_1(x_0)} \rho(x,y) &= \frac 1 {\varepsilon}1_y(y_0) + \Delta 1_y(y_0) \qquad \mbox{ for all } y \in B_1(y_0) \label{eq:rhoYProof}.
\end{aligned}$$
We remark that (\[eq:rhoXProp\]) and (\[eq:rhoYProp\]) hold for all $\rho$ satisfying (\[eq:rhoXProof\]) and (\[eq:rhoYProof\]) as $\Delta 1_x (x_0)= \frac {w(x_0,x)}{m(x_0)}$ for $x \not = x_0$. Therefore, $\frac 1 {\varepsilon}\kappa_{\varepsilon}(x_0,y_0)$ is less than or equal to the right hand side of (\[eq:PropTransport\]).
We now show that if we modify $\rho$ satisfying (\[eq:rhoXProp\]) and (\[eq:rhoYProp\]) appropriately, then the value of $F(\rho)$ in the right hand side of (\[eq:PropTransport\]) does not change and the modified $\rho$ satisfies (\[eq:rhoXProof\]) and (\[eq:rhoYProof\]) which will show that $\frac 1 {\varepsilon}\kappa_{\varepsilon}(x_0,y_0)$ is larger than or equal to the right hand side of (\[eq:PropTransport\]) for small ${\varepsilon}$.
Suppose that $\rho$ satisfies (\[eq:rhoXProp\]) and (\[eq:rhoYProp\]). We define $$\begin{aligned}
\rho_{\varepsilon}(x,y) := \rho(x,y) + 1_{x}(x_0)1_{y}(y_0) \left( \frac 1 {\varepsilon}- \sum_{u,v} \rho(u,v) \right)
\end{aligned}$$ which is non-negative if ${\varepsilon}$ is small. We observe that $$\begin{aligned}
F(\rho)= \sum_{x,y} \rho(x,y) \left[1 - \frac{d(x,y)}{d(x_0,y_0)} \right] = \sum_{x,y} \rho_{\varepsilon}(x,y) \left[1 - \frac{d(x,y)}{d(x_0,y_0)} \right] = F(\rho_{\varepsilon})
\end{aligned}$$ since $\rho_{\varepsilon}(x,y)$ and $\rho(x,y)$ only differ at $(x_0,y_0)$ where the latter factor in the sums vanishes. Moreover, $$\begin{aligned}
\sum_{x \in B_1(x_0)} \sum_{y \in B_1(y_0)} \rho_{\varepsilon}(x,y) = \frac 1 {\varepsilon}.
\end{aligned}$$
We now show that $\rho_{\varepsilon}$ satisfies (\[eq:rhoXProof\]). Since $\frac 1 {\varepsilon}1_x(x_0) = 0$ on $S_1(x_0)$, we see that (\[eq:rhoXProp\]) implies (\[eq:rhoXProof\]) for $x\in S_1(x_0)$. For the remaining case $x=x_0$, equation (\[eq:rhoXProof\]) follows since by (\[eq:rhoXProp\]), $$\begin{aligned}
\sum_{y\in B_1(y_0)} \rho_{\varepsilon}(x_0,y) = \frac 1 {\varepsilon}- \sum_{x \in S_1(x_0)} \sum_{y \in B_1(y_0)} \rho(x,y) = \frac 1 {\varepsilon}- \sum_{x \in S_1(x_0)} \frac{w(x_0,x)}{m(x_0)} = \frac 1 {\varepsilon}+ \Delta 1_{x_0} (x_0).
\end{aligned}$$ Due to an analogous argument, $\rho_{\varepsilon}$ also satisfies (\[eq:rhoYProof\]). Putting everything together proves that $\frac 1 {\varepsilon}\kappa_{\varepsilon}(x_0,y_0)$ equals the right hand side of (\[eq:PropTransport\]) for small ${\varepsilon}$. Taking ${\varepsilon}\to 0^+$ finishes the proof.
Ollivier curvature on combinatorial graphs
------------------------------------------
We now show how the transport cost expression for the curvature simplifies on combinatorial graphs. We first describe the curvature on combinatorial graphs intuitively. We note how this case complements Example \[ex:NoCycles\] which considered the case of graphs with no cycles.
- Given an edge $x \sim y$, we have initial curvature $\kappa(x,y)=2$.
- Every triangle containing $x,y$ increases $\kappa(x,y)$ by one.
- Adding 4-cycles containing $x,y$ does not change $\kappa(x,y)$.
- Adding 5-cycles containing $x,y$ decreases $\kappa(x,y)$ by one.
- Every additional neighbor of both $x$ and $y$ decreases $\kappa(x,y)$ by one.
The following theorem gives a precise expression for the curvature of combinatorial graphs making the above intuition explicit.
\[thm:TransprtCombinatorial\] Let $G=(V,w,m)$ be a combinatorial graph and let $x_0 \sim y_0$ be adjacent vertices. Let $B_{x_0y_0}:= B_1(x_0) \cap B_1(y_0)$, $B_{x_0}^{y_0}:=B_1(x_0) \setminus B_1(y_0)$ and $B_{y_0}^{x_0}:=B_1(y_0) \setminus B_1(x_0).$ Let $\Phi_{x_0y_0} := \{\phi: D(\phi) \subseteq B_{x_0}^{y_0} \to R(\phi) \subseteq B_{y_0}^{x_0} : \phi \mbox{ bijective}\}. $ For $\phi \in \Phi_{x_0y_0}$ write $D(\phi)^c := B_{x_0}^{y_0} \setminus D(\phi)$ and $R(\phi)^c := B_{y_0}^{x_0} \setminus R(\phi).$ Then, $$\begin{aligned}
\kappa(x_0,y_0) = \# B_{x_0y_0} - \inf_{\phi \in \Phi_{x_0y_0}} \left( \#D(\phi)^c + \# R(\phi)^c + \sum_{x\in D(\phi)} [d(x,\phi(x)) - 1] \right).\end{aligned}$$
We remark that $\Phi_{x_0y_0} \neq \emptyset$ since $\Phi_{x_0y_0}$ always contains the empty function.
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To prove the theorem, we first show that the coupling function $\rho$ which gives $\kappa(x_0,y_0)$ via the expression found in Proposition \[pro:CharTransport\] can be assumed to be integer valued for combinatorial graphs. As in the proof of Proposition \[pro:CharTransport\] and since we assume that $x_0 \sim y_0$, we let $$F(\rho)= \sum_{x,y} \rho(x,y)(1 - d(x,y))$$ for $\rho: B_1(x_0) \times B_1(y_0) \to [0,\infty)$. We note that in the case of combinatorial graphs, (\[eq:rhoXProp\]) and (\[eq:rhoYProp\]) become $$\label{eq:TransportCombinatorial}
\sum_{y \in B_1(y_0)} \rho(x,y) = 1 \quad \forall x \sim x_0 \qquad \mbox{ and} \qquad \sum_{x \in B_1(x_0)} \rho(x,y) = 1 \quad \forall y \sim y_0.$$ In particular, as we assume that $x_0 \sim y_0$, we have $\sum_x \rho(x,x_0)= \sum_y \rho(y_0,y) =1$.
\[lem:01lemma\] Let $G=(V,w,m)$ be a combinatorial graph and let $x_0 \sim y_0$. Then, there exists $\rho:B_1(x_0) \times B_1(y_0) \to \{0,1\}$ satisfying such that $\kappa(x_0,y_0) = F(\rho)$. Furthermore, $\rho$ can be chosen to satisfy $\rho(z,z)=1$ for all $z \in B_{x_0y_0}$.
We first show that $\rho$ can be chosen to take values in $\{0,1\}$. Suppose not. Let $\rho$ be a coupling which satisfies such that $\kappa(x_0,y_0) = F(\rho)$ and so that $\rho$ has the minimal number of non-$\{0,1\}$ entries. Denote by $$M=\{(x,y) \in B_1(x_0) \times B_1(y_0):\rho(x,y) \notin \{0,1\}\}.$$ By assumption $M \not = \emptyset$. We first note that $(x_0,y_0) \not \in M$ as, if $(x_0,y_0) \in M$, then we could replace $\rho$ by a coupling whose value at $(x_0,y_0)$ is 0 without changing the value of $F(\rho)$.
By using repeatedly, we can then construct a maximal sequence $S=((x_A,y_A),\ldots,(x_B,y_B))$ in $M$ with $B\geq A \geq 0$ which has the following properties:
1. $x_{2n+1} = x_{2n} \neq x_k$ for all $k \notin \{2n,2n+1\}$.
2. $y_{2n}=y_{2n-1} \neq y_k$ for all $k \notin \{2n,2n-1\}$.
Without loss of generality, we may assume that either $x_B=x_{B-1}$ or $A=B$.
Now, suppose that $y_B \neq y_0$. Then, by there exists $(x_{B+1},y_B) \in M$ with $x_{B+1} \neq x_B$ since $\sum_x \rho(x,y_B) = 1$. Due to the maximality of $S$, we cannot add $(x_{B+1},y_B)$ to $S$ and the only possible reason for this is that there exists $A'<B$ with $x_{A'} = x_{B+1}$ where we choose $A'$ to be maximal. In this case, we replace $S$ by the loop $L=((x_{A'},y_{A'}), \ldots, (x_{B},y_B),(x_{B+1},y_B))$. We proceed analogously if $y_B=y_0$ and $x_A \neq x_0$ and replace $S$ by the loop $L=((x_{A-1},y_A),\ldots,(x_{B'},y_{B'}))$. In case we do not replace $S$ by a loop, the sequence starts with $(x_0,y_A)$ or $(x_A,y_0)$ and ends with $(x_0,y_B)$ or $(x_B,y_0)$.
Given a sequence $S$ or a loop $L$ as constructed above, we can change $\rho$ on $S$ or $L$ while preserving . We do this by letting $\rho_C(x_n,y_n) := \rho(x_n,y_n) + C(-1)^n$ and $\rho_C(x,y) := \rho(x,y)$ otherwise. It is easy to check that also holds for $\rho_C$. The objective function $F$ is linear. Therefore, $F(\rho_C) \geq F(\rho)$ for all negative or all positive $C$. Without loss of generality, we assume that $F(\rho_C) \geq F(\rho)$ for all positive $C$. We choose $C$ maximal such that $\rho_C \geq 0$. Then, there exists $(x,y)$ in the sequence with $\rho_C(x,y) = \rho(x,y)-C = 0$ so that $\rho_C(x,y) \in \{0,1\}$ but $\rho(x,y) \notin \{0,1\}$. This contradicts the minimality of the number of $\{0,1\}$ entries of $\rho$. The contradiction finishes the proof of the first part of the statement.
We now show the furthermore statement, that is, that $\rho$ can additionally be chosen so that $\rho(z,z) =1$ for all $z \in B_{x_0y_0}=B_1(x_0) \cap B_1(y_0)$. Suppose that $\rho(z,z) \neq 1$ for some $z \in B_{x_0y_0}$. Then, $\rho(z,z)=0$.
Case 1. We first assume that $z \in S_1(x_0) \cap S_1(y_0)$. Then, there exists $x_z \in B_1(x_0)$ and $y_z \in B_1(y_0)$ with $\rho(x_z,z)=1=\rho(z,y_z)$ and thus $\rho(x_z,y_z)=0$ by . Define $\widetilde\rho(z,z)=\widetilde\rho (x_z,y_z)=1$ and $\widetilde \rho(x_z,z) = \widetilde \rho (z,y_z)=0$ and $\widetilde\rho(x,y) = \rho(x,y)$ otherwise. Then, $\widetilde \rho$ also satisfies . Moreover, $F(\tilde \rho) = F(\rho) + d(x_z,z) + d(z,y_z) - d(x_z,y_z) \geq F(\rho)$.
Case 2. If $z=x_0$, there exists $x_z\sim x_0$ with $\rho(x_z,x_0)=1$. Now, set $\widetilde \rho(x_0,x_0)=\widetilde \rho(x_z,y_0) = 1$ and $\widetilde \rho(x_z,x_0)=0$ and $\widetilde{\rho}(x,y)=\rho(x,y)$ otherwise. Then, $\widetilde \rho$ also satisfies . Moreover, $F(\tilde \rho) = F(\rho) + 2 - d(x_z,y_0) \geq F(\rho)$. An analogous argument works in the case $z=y_0$.
Therefore, in both cases, $\widetilde \rho$ is also a $\{0,1\}$-valued function satisfying such that $\kappa(x_0,y_0) = F(\widetilde \rho)$ and $\widetilde{\rho}(z,z)=0$. Repeating the argument yields the existence of a $\widetilde \rho$ such that $\widetilde\rho(z,z) = 1$ for all $z \in B_{x_0y_0}$.
One can also prove the integrality of the transport function $\rho$ in Lemma \[lem:01lemma\] by using the theory of linear programming. In particular, the constraint matrix is a submatrix of the constraint matrix of a classical assignment problem and, therefore, totally unimodular. By standard theory and due to the integrality of all parameters, this implies the existence of an integral optimal solution $\rho$.
We are now prepared to prove Theorem \[thm:TransprtCombinatorial\] expressing the curvature for combinatorial graphs via transport costs.
Due to Lemma \[lem:01lemma\], we can assume that the optimizing function $\rho$ satisfying and $\kappa(x_0,y_0)=F(\rho)=\sum_{x,y} \rho(x,y)(1-d(x,y))$ takes values in $\{0,1\}$ and satisfies $\rho(z,z) = 1$ for all $z \in B_{x_0y_0}$. Therefore, $\rho(x,x_0)=0$ for all $x \sim x_0$, $\rho(y_0,y)=0$ for all $y \sim y_0$ and $\rho(z,y)=\rho(x,z)=0$ for all $z \in S_1(x_0) \cap S_1(y_0)$ where $x,y \not = z$. Thus, $$\begin{aligned}
F(\rho) &= \sum_{x,y} \rho(x,y)(1 - d(x,y)) \\
&= \sum_{x\neq x_0, y\neq y_0} \rho(x,y) (1-d(x,y)) + \rho(x_0,x_0) - \sum_{y \in B_{y_0}^{x_0}} \rho(x_0,y) + \rho(y_0,y_0) - \sum_{x \in B_{x_0}^{y_0}} \rho(x,y_0) \\
&=\# B_{x_0y_0} - \sum_{x \in B_{x_0}^{y_0}} \rho(x,y_0) - \sum_{y \in B_{y_0}^{x_0}} \rho(x_0,y)
+ \sum_{x\in B_{x_0}^{y_0}, y \in B_{y_0}^{x_0}} \rho(x,y)(1-d(x,y)). \end{aligned}$$
If $x \in B_{x_0}^{y_0}$, then $x \sim x_0$ so that $\sum_y \rho(x, y)=1$ by . Therefore, as $\rho(x,z)=0$ for all $z \in S_1(x_0) \cap S_1(y_0)$, either $\rho(x,y_0)=1$ or there exists a unique $y \in B_{y_0}^{x_0}$ such that $\rho(x,y)=1$. In the second case, $y \in B_{y_0}^{x_0}$ is unique as $\sum_x \rho(x,y)=1$ by . Hence, $\rho$ can be uniquely associated with a bijection $\phi_\rho \in \Phi_{x_0y_0}$ by letting $$D(\phi_\rho) = \{ x \in B_{x_0}^{y_0} : \mbox{ there exists a unique } y \in B_{y_0}^{x_0} \mbox{ such that } \rho(x,y)=1 \}$$ and $\phi_\rho(x) = y$ for $x \in D(\phi)$.
Note, by the dichotomy above, that $D(\phi_\rho)^c = \{ x \in B_{x_0}^{y_0} : \rho(x,y_0) =1 \}$ and $R(\phi_\rho)^c=\{ y \in B_{y_0}^{x_0} : \rho(x_0,y)=1\}$. Therefore, $$\begin{aligned}
\kappa(x_0,y_0) = F(\rho) &= \# B_{x_0y_0} - \left( \#D(\phi_\rho)^c + \#R(\phi_\rho)^c + \sum_{x \in D(\phi_\rho)} [d(x, \phi_\rho(x))-1] \right) \\
&\leq \# B_{x_0y_0} - \inf_{\phi \in \Phi_{x_0y_0}} \left( \#D(\phi)^c + \# R(\phi)^c + \sum_{x\in D(\phi)} [d(x,\phi(x)) - 1] \right).\end{aligned}$$ On the other hand, if $\phi \in \Phi_{x_0y_0}$, we can reverse the process above to define $\rho_\phi:B_1(x_0) \times B_1(y_0) \to \{0,1\}$ by letting $\rho_\phi(z,z)=1$ for all $z \in B_{x_0y_0}$, $\rho_\phi(x, \phi(x))=1$ for all $x \in D(\phi)$, $\rho_\phi(x,y_0)=1$ for all $x \in D(\phi)^c$, $\rho(x_0,y)=1$ for all $y \in R(\phi)^c$ and $\rho_\phi(x,y)=0$ otherwise. As above, it follows that $\rho_\phi$ satisfies and that $F(\rho_\phi) = \# B_{x_0y_0} - \left( \#D(\phi)^c + \#R(\phi)^c + \sum_{x \in D(\phi)} [d(x, \phi(x))-1] \right).$ Therefore, $$\begin{aligned}
\kappa(x_0,y_0) = \sup_\rho F(\rho) \geq F(\rho_\phi) &= \# B_{x_0y_0} - \left( \#D(\phi)^c + \#R(\phi)^c + \sum_{x \in D(\phi)} [d(x, \phi(x))-1] \right)
\end{aligned}$$ for all $\phi \in \Phi_{x_0y_0}$. Combining the two inequalities completes the proof.
Ollivier curvature on birth-death chains {#sec:OllivierBirthDeath}
----------------------------------------
The curvature of birth-death chains is easy to compute. Moreover, as we will see later, many problems of interest concerning Ollivier curvature can be reduced to the case of birth-death chains.
A graph $G=({{\mathbb{N}}}_0,w,m)$ is called a *birth-death chain* if $$w(m,n) =0 \quad \mbox{ whenever } \quad |m-n|\neq 1.$$
\[thm:line\] Let $G=({{\mathbb{N}}}_0,w,m)$ be a birth-death chain and let $f(r):=d(0,r)=r$. Then for $0\leq r<R$, $$\begin{aligned}
\kappa(r,R) &= \nabla_{rR}\Delta f = \frac{\Delta f(r) - \Delta f(R)}{R-r} \\&= \frac{w(r,r+1) - w(r,r-1)}{(R-r)m(r)} - \frac{w(R,R+1) - w(R,R-1)}{(R-r)m(R)}\end{aligned}$$ where we set $w(r,r-1) :=0$ if $r=0$.
The last equality is a straightforward computation. We now prove the first equality. Due to Theorem \[thm:nablaDelta\], as $f \in Lip(1)$ and $\nabla_{Rr}f=1$, it is clear that $$\begin{aligned}
\kappa(r,R) \leq \nabla_{rR} \Delta f = \frac{\Delta f(r) - \Delta f(R)}{R-r}.
\end{aligned}$$
We will now show the other inequality to complete the proof. Let $g \in Lip(1)$ be such that $\nabla_{Rr}g=1$, i.e., $g(R)-g(r)=R-r$. Therefore, $g(n+1)-g(n)=1$ for all $r \leq n \leq R-1$ so that, in particular, $g(r+1)-g(r) = 1 = g(R)-g(R-1)$. Moreover, $a:=g(r)-g(r-1) \leq 1$ and $b:=g(R+1)-g(R) \leq 1$ since $g \in Lip(1)$. As $$m(r)\Delta g(r) = w(r,r+1) - a w(r,r-1) \geq m(r)\Delta f(r)$$ and $$m(R)\Delta g(R) = bw(R,R+1) - w(R,R-1) \leq m(R)\Delta f(R)$$ it follows that $$\begin{aligned}
\frac{\Delta g(r) - \Delta g(R)}{R-r} \geq \frac{\Delta f(r) - \Delta f(R)}{R-r}.\end{aligned}$$ Therefore, Theorem \[thm:nablaDelta\] yields that $$\kappa(r,R) \geq \frac{\Delta f(r) - \Delta f(R)}{R-r}$$ which implies the claim of the theorem.
\[rem:line\] We note that it is easy to see from the above that $$\kappa(0,r) = \frac{1}{r} \sum_{n=0}^{r-1} \kappa(n,n+1).$$ In particular, $\kappa(r-1,r)=K$ if and only if $\kappa(0,r)=K$ for all $r \geq 1$.
Gradient estimates {#sec:GradEstimates}
==================
Our proof of the gradient estimate of the semigroup under a Ricci curvature bound deeply relies on the maximum principle which requires taking maxima over compact sets. For applying this technique to infinite, and hence, non-compact graphs, we employ a cutoff method. However, standard cutoff techniques like taking Dirichlet boundary conditions on a finite subgraph do not work since the gradient of a function may leave the subgraph. Also cutting off with a finitely supported function after taking the semigroup appears to be not successful since we do not have control over the semigroup before taking the cutoff.
The idea to overcome these difficulties is to deeply intertwine the semigroup with a finitely supported cutoff function. We call this the perpetual cutoff method which will result in a non-linear cutoff semigroup whose general properties we first develop below. For general theory on non-linear semigroups, see e.g. [@barbu1976nonlinear; @kato1967nonlinear; @miyadera1992nonlinear]. We will then apply this general theory to prove our main characterization which connects a lower Ricci curvature bound with a gradient decay of the semigroup.
The perpetual cutoff method
---------------------------
The intuition of the non-linear cutoff semigroup presented below is that it behaves exactly as the heat semigroup whenever the heat does not surpass the cutoff threshold. The name perpetual cutoff method comes from the fact that the cutoff threshold is not only applied once, but perpetually for all times $t>0$.
Let $\phi \in C_c(V)$ be a non-negative function and let $f \in [0,\phi] := \{g \in C_c(V) : 0 \leq g \leq \phi\}$.
For $t\geq0$, we define $$Q_t^\phi f := P_t f \wedge \phi$$ and the *cutoff semigroup* $$P_t^\phi f := \inf_{t_1 + \ldots + t_n = t} Q_{t_1}^\phi \ldots Q_{t_n}^\phi f.$$
We note, by checking cases, that $Q_t^\phi Q_s^\phi \leq Q_{t+s}^\phi$ and, as $P_t$ is positivity preserving, the infimum exists.
Let $W \subset V$ be finite. We will show that $P_t^\phi f$ is a generalization of the semigroup $e^{t \Delta_W}$ with $\Delta_W f := 1_W \Delta (1_W f)$ corresponding to the Dirichlet problem $\partial_t u = \Delta u$ on $W$ and $u=0$ on $V \setminus W$. In particular, $P_t^\phi = e^{t \Delta_W}$ when we take $\phi = 1_W$ as the cutoff function. Furthermore, $P_t^\phi$ solves the heat equation at all vertices $x$ where $P_t^\phi(x) < \phi(x)$.
We collect these and some other useful properties of $P_t^\phi$ in the following theorem. We write $$\overline {\partial_t^\pm} G(t) := \limsup_{h \to 0^\pm} \frac{G(t+h) -G(t)} h$$ and $$\underline {\partial_t^\pm} G(t) := \liminf_{h \to 0^\pm} \frac{G(t+h) -G(t)} h$$ for a function $G$ depending on $t$.
\[thm:Cutoff\] Let $G=(V,w,m)$ be a graph and let $\phi \in C_c(V)$ be non-negative. The family $P_t^\phi : [0,\phi] \to [0,\phi]$ is a nonlinear contraction semigroup with respect to $\|\cdot\|_p$ for all $p \in [1,\infty]$ and $t\geq0$. In particular, for $f,g \in [0,\phi]$ and $s,t \geq0$, we have:
(i) $P_t^\phi P_s^\phi = P_{t+s}^\phi$,
(ii) $\| P_t^\phi f - P_t^\phi g \|_p \leq \|f-g\|_p$,
(iii) $P_0^\phi f = f$,
(iv) $
P_t^\phi f \geq P_t^\psi g \quad \mbox{ whenever } \quad \phi \geq \psi \geq f \geq g,
$
(v) $e^{-t {\operatorname{Deg}}} f \leq P_t^\phi f \leq P_t f$,
(vi) $P_t^\phi f$ is Lipschitz in $t$,
(vii) $
\overline{\partial_t^\pm} P_t^\phi f \leq \Delta P_t^\phi f,
$
(viii) $
\partial_t P_t^\phi f (x) = \Delta P_t^\phi f (x) \quad \mbox{ whenever } \quad P_t^\phi f (x) < \phi(x),
$
(ix) $
P_t^\phi f = e^{t\Delta_W} f \quad \mbox{ whenever } \quad \phi = 1_W \mbox{ for } W \subset V \mbox{ finite}.
$
By definition, $Q_t^\phi$ maps $[0,\phi]$ to $[0,\phi]$, and so does $P_t^\phi$. We prove the semigroup property $(i)$ by observing that $$\begin{aligned}
P_t^\phi P_s^\phi f = \inf_{t_1 + \ldots + t_n = t} Q_{t_1}^\phi \ldots Q_{t_n}^\phi \inf_{s_1 + \ldots + s_m = s} Q_{s_1}^\phi \ldots Q_{s_m}^\phi f &=
\inf_{\substack{t_1 + \ldots + t_n = t \\ s_1 + \ldots + s_m = s}} Q_{t_1}^\phi \ldots Q_{t_n}^\phi Q_{s_1}^\phi \ldots Q_{s_m}^\phi
\\
&= \inf_{t_1 + \ldots + t_{n+m} = t+s} Q_{t_1}^\phi \ldots Q_{t_{n+m}}^\phi f\\
&=P_{t+s}^\phi f.\end{aligned}$$ where the second equality follows from the montone convergence of $Q_t^\phi$ and the third equality follows from $Q_t^\phi Q_s^\phi f \leq Q_{t+s}^\phi f$.
To prove the contraction property $(ii)$, observe that for $p \in [1,\infty]$, $P_t$ is contracting on $\ell^p(V,m)$ so that $$\| Q_t^\phi f - Q_t^\phi g \|_p \leq \| P_t f - P_t g \|_p \leq \|f-g\|_p$$ implying that $$\| P_t^\phi f - P_t^\phi g \|_p \leq \|f-g\|_p.$$ It is clear that $P_0^\phi f = f$ since $f \leq \phi$. This proves $(iii)$.
To prove $(iv)$, observe that $Q_t^\phi f \geq Q_t^\psi g$ whenever $\phi \geq \psi \geq f \geq g$ as $P_t$ is positivity preserving. This property is immediately transmitted to $P_t^\phi$.
To prove $(v)$, i.e., the lower and upper estimate of $P_t^\phi f$, we use $$e^{-t {\operatorname{Deg}}} f \leq Q_t^\phi f \leq P_t f$$ which implies that $$e^{-t {\operatorname{Deg}}} f \leq P_t^\phi f \leq P_t f.$$ as desired.
By using the estimates directly above and applying Taylor’s theorem to $P_t f$ at $t=0$, we can deduce the existence of a constant $C_\phi>0$ such that for all $f \in [0,\phi]$ and $t\geq0$, $$\begin{aligned}
-C_\phi t \leq 1_{{\operatorname{supp}}\phi} (e^{-t{\operatorname{Deg}}} - 1) f \leq P^\phi_{t} f - f \leq 1_{{\operatorname{supp}}\phi} (P_t -1) f \leq C_\phi t\end{aligned}$$ implying that $P_t^\phi f$ is Lipschitz in $t$ by using the semigroup property $(i)$, thus proving $(vi)$.
Furthermore, due to Taylor’s theorem again, there exists a constant $C_\phi'>0$, such that for all $t>0$ and all $f \in [0,\phi]$, $$\frac 1 t \left( P^\phi_{t} f - f \right) \leq 1_{{\operatorname{supp}}\phi} \frac 1 t (P_t f -f) \leq \Delta f + C_\phi't$$ since $1_{{\operatorname{supp}}\phi} \Delta f \leq \Delta f$ for $f \in [0,\phi]$. This directly implies that $$\overline{\partial_t^+} P_t^\phi f = \limsup_{{\varepsilon}\to 0^+} \frac 1 {\varepsilon}\left({P_{{\varepsilon}}^\phi P_t^\phi f - P_t^\phi f} \right) \leq \Delta P_t^\phi f$$ by using the semigroup property and the fact that $P_t^\phi f \in [0,\phi]$. Similarly, $$\overline{\partial_t^-} P_t^\phi f = \limsup_{{\varepsilon}\to 0^-} \frac{1}{-{\varepsilon}} \left( {P_{-{\varepsilon}}^\phi P_{t+{\varepsilon}}^\phi f - P_{t+{\varepsilon}}^\phi f} \right) \leq \limsup_{{\varepsilon}\to0^-} \left(\Delta P_{t+{\varepsilon}}^\phi f - C_\phi' {\varepsilon}\right) = \Delta P_{t}^\phi f.$$ Putting these two inequalities together yields $(vii)$.
In order to do prove $(viii)$, we first define $\Delta^x: C(V)\to C(V)$ via $$\Delta^x f (y) := \begin{cases} \Delta f(x) & \mbox{if } y=x \\
-{\operatorname{Deg}}(y) f(y) & \mbox{otherwise}
\end{cases}$$ and let $P_t^x := e^{t\Delta^x}$. We remark that $\Delta^x$ is an asymmetrization of $\Delta$ and that $P_t^x u$ does *not* give a solution to the Dirichlet problem $\partial_t u(x)=\Delta u(x)$ and $u=0$ on $V\setminus \{x\}$. We also note that $P_t^x$ is positivity preserving.
\[lem:Ptx\] Let $t>0$ and let $f \in [0,\phi]$. If $P_s f(x) \leq \phi(x)$ for $0\leq s \leq t$, then $$P_t^x f \leq P_t^\phi f.$$
Obviously, $P_s^x f \leq P_s f$. Observe that $P_s^x f \leq \phi$ since $P_s^x f(y) = e^{-s{\operatorname{Deg}}(y)}f(y) \leq f(y) \leq \phi(y)$ for $y \not = x$ and since $P_s^x f (x) = P_s f (x) \leq \phi(x)$ by assumption. Hence, $P_t^x f \leq Q_t^\phi f$. Induction over $n$ for $s_1 + \ldots + s_n = t$ yields $$P_t^x f= P_{s_1 +\ldots + s_n}^xf = P_{s_1}^x \ldots P_{s_n}^x f\leq Q_{s_1}^\phi \ldots Q_{s_n}^\phi f \leq Q_t^\phi f.$$ Taking the infimum over all such $s_1, \ldots, s_n$ finishes the proof of the lemma.
We now prove $(viii)$. Since we already proved $(vii)$, it suffices to show that $$\underline{\partial_t^\pm} P_t^\phi f(x) \geq \Delta P_t^\phi f(x).$$ whenever $P_t^\phi f(x) < \phi(x)$.
Due to Taylor’s theorem with Lagrange remainder term, there exists a constant $C_\phi''>0$ such that for all $g \in [0,\phi]$ and all ${\varepsilon}\in(0,t]$, there exists $\delta \in [0,{\varepsilon}]$ such that $$\begin{aligned}
\label{eq:Taylor}
\frac{1}{{\varepsilon}} (P_{\varepsilon}^x g - g)(x) &= \Delta^x g(x) + \frac {\varepsilon}2 \partial_s^2 P_s^x g(x)|_{s=\delta} \nonumber
\\
&\geq \Delta g(x) - C_\phi'' {\varepsilon}. \end{aligned}$$ since $\partial_s^2 P_s^x g(x) = \Delta^x \Delta^x P^x_s g(x)$ is uniformly bounded on $[0,t]\times [0,\phi]$ and $x$ is fixed.
Choose $g=P_t^\phi f$. Since we have assumed that $g(x)=P_t^\phi f(x) < \phi(x)$, by continuity of $P_s$, there exists ${\varepsilon}\in (0,t]$ such that $P_sg(x) \leq \phi(x)$ for all $s \in [0,{\varepsilon}]$. By Lemma \[lem:Ptx\], we then have that $P_{\varepsilon}^x g \leq P_{\varepsilon}^\phi g$ proving that $\underline{\partial_t^+} P_t^\phi f(x) \geq \Delta P_t^\phi f(x)$ by using .
We next prove the same inequality for the left derivative.
We note that for ${\varepsilon}<0$ small enough, we have that $P_{t+{\varepsilon}}^\phi f(x)<\phi(x)$ so by using continuity of $P_s$ as above, we may apply Lemma \[lem:Ptx\] and again to get $$\begin{aligned}
\underline{\partial_t^-} P_t^\phi f(x) &\geq \liminf_{{\varepsilon}\to 0^-} \frac{1}{-{\varepsilon}} \left({P_{-{\varepsilon}}^x P_{t+{\varepsilon}}^\phi f(x) - P_{t+{\varepsilon}}^\phi f(x)}\right) \\
& \geq \liminf_{{\varepsilon}\to 0^-} \left( \Delta P_{t +{\varepsilon}}^\phi f(x) - C_\phi'' {\varepsilon}\right) = \Delta P_{t}^\phi f(x).
\end{aligned}$$
Putting this together with $\underline{\partial_t^+} P_t^\phi f(x) \geq \Delta P_t^\phi f(x)$ and $(vii)$ yields $(viii)$.
We finally prove $(ix)$. Let $\phi=1_W$. First, we suppose that $f\leq \mathbf 1-{\varepsilon}$. Then, $P_t^\phi f$ solves the Dirichlet problem $\partial_t u = \Delta u$ on $W$ and $u=0$ on $V \setminus W $ due to $(viii)$ as $P_sf(x) < 1 = 1_W(x)$ for $x \in W$ and $s \in [0,t]$ implies that $P_s^\phi f(x) < \phi(x)$ for all $x \in W$. This shows that $P_t^\phi f = e^{t\Delta_W} f$ since $e^{t\Delta_W} f$ is the unique solution to the Dirichlet problem.
For a general function $f \in [0,\phi]$, the claim follows by approximation since both $P_t^\phi$ and $e^{t\Delta_W}$ are contraction semigroups with respect to $\|\cdot\|_\infty$. This proves $(ix)$ and finishes the proof of the theorem.
Cutoff semigroups and Ricci curvature
-------------------------------------
Using the above observations, we can deduce a Lipschitz decay of the cutoff semigroup under lower curvature bounds. We observe that the cutoff semigroup $P_t^\phi$ defined on $[0, \phi]=\{g \in C_c(V) : 0 \leq g \leq \phi \}$ canonically extends to functions $f: V \to [0,\infty)$ via $P_t^\phi f := P_t^\phi (f \wedge \phi)$. In particular, $P_0^\phi f = f \wedge \phi$ whenever we do not assume that $f \leq \phi$.
\[lem:CutoffRicci\] Let $G=(V,w,m)$ be a graph with $Ric(G) \geq K$. Let $f:V \to [0,1]$ be non-constant, $T>0$ and $\phi:V \to [0,1]$ be compactly supported such that $\|\nabla \phi\|_\infty < \|\nabla f\|_\infty (1 \wedge e^{-KT})$. Then, for $t \in [0,T]$, $$\|\nabla P_t^\phi f\|_\infty \leq e^{-Kt} \|\nabla f\|_\infty.$$
Without loss of generality, we can assume that $\kappa(x,y) > K$ for all $x,y \in V$ instead of $\kappa(x,y) \geq K$. Furthermore, as we assume that $f$ is non-constant, it follows that $\|\nabla f\|_\infty>0$.
For $t \in [0,T], x,y \in V$ with $x \sim y$ we define $$\begin{aligned}
F(t,x,y) := e^{Kt} \nabla_{yx} P_t^\phi f.\end{aligned}$$ We aim to show that $F \leq \|\nabla f\|_\infty$. Suppose not.
Since the support of $P_t^\phi f$ is contained in the finite support of $\phi$, the continuous function $F$ attains its maximum $F_{\max}$ at some $(t_0,x_0,y_0)$ where $y_0$ is in the support of $\phi$. Therefore, $F(t_0,x_0,y_0) = F_{\max} > \| \nabla f \|_\infty$.
Since $$F(0,x_0,y_0) = \nabla_{y_0x_0}(f \wedge \phi) \leq \|\nabla f\|_\infty \vee \|\nabla \phi\|_\infty = \|\nabla f\|_\infty < F(t_0,x_0,y_0),$$ we obtain that $t_0>0$. Furthermore, observe that $$P_{t_0}^\phi f(x_0) < \phi(x_0)$$ since otherwise $$\nabla_{y_0x_0} P_{t_0}^\phi f \leq \nabla_{y_0x_0} \phi < \|\nabla f\|_\infty (1 \wedge e^{-KT}) \leq \|\nabla f\|_\infty e^{-Kt_0}$$ which would imply that $F(t_0,x_0,y_0) < \|\nabla f\|_\infty$.
This yields $\partial_t P_{t}^\phi f(x_0)|_{t=t_0} = \Delta P_{t_0}^\phi f(x_0)$ due to Theorem \[thm:Cutoff\] $(viii)$. Moreover at $y_0$, Theorem \[thm:Cutoff\] $(vii)$ gives that $\overline{\partial_t^-} P_{t}^\phi f(y_0)|_{t=t_0} \leq \Delta P_{t_0}^\phi f(y_0)$. Subtracting yields $$\begin{aligned}
\underline{\partial_t^-} \nabla_{x_0y_0} P_t^\phi f |_{t=t_0} \geq \nabla_{x_0y_0} \Delta P_{t_0}^\phi f.\end{aligned}$$
Observe that $\|\nabla P_{t_0}^\phi f\|_\infty \leq F_{\max}e^{-Kt_0}$ and $\nabla_{y_0x_0}P_{t_0}^\phi f = F_{\max}e^{-Kt_0}$ due to maximality. Hence, due to Theorem \[thm:nablaDelta\], we get that $F_{\max}e^{-Kt_0} \cdot \kappa(x_0,y_0) \leq \nabla_{x_0y_0} P_{t_0}^\phi f.$ Therefore, by our curvature assumption, $$\begin{aligned}
\underline{\partial_t^-} \nabla_{x_0y_0} P_t^\phi f |_{t=t_0} \geq \nabla_{x_0y_0}\Delta P_{t_0}^\phi f \geq F_{\max}e^{-Kt_0} \cdot \kappa(x_0,y_0) > F_{\max}e^{-Kt_0} K.\end{aligned}$$ Thus, $$\begin{aligned}
\overline{\partial_t^-} F(t_0,x_0,y_0) &= \overline{\partial_t^-} \left(e^{Kt}\nabla_{y_0x_0} P_{t}^\phi f\right)|_{t=t_0} \\
&= KF_{\max} - e^{Kt_0} \underline{\partial_t^-} \nabla_{x_0y_0} P_t^\phi f |_{t=t_0} \\
&< KF_{\max} - KF_{\max} =0.\end{aligned}$$
Due to maximality in time of $F$ at $(t_0,x_0,y_0)$, since $t_0>0$, we have $\overline{\partial_t^-} F(t_0,x_0,y_0) \geq 0$ which contradicts the above inequality. Hence, $F\leq \|\nabla f\|_\infty$ which finishes the proof.
\[lem:RicImpliesGradient\] Let $G=(V,w,m)$ be a graph with $Ric(G) \geq K$. Let $f : V \to [0,1]$ be non-constant. Then, for all $t>0$, $$\|\nabla P_t f\|_\infty \leq e^{-Kt} \|\nabla f\|_\infty.$$
Let $T>0$. We prove the statement for all $t \in [0,T]$ which will prove the lemma. Let $W_1 \subset W_2 \subset \ldots$ be finite subsets of $V$ such that $\bigcup W_n = V$. Let $\phi_n :V \to [0,1]$ be functions such that $\phi_n = 1$ on $W_n$ and such that $\|\nabla \phi_n\|_\infty < \|\nabla f\|_\infty(1 \wedge e^{-KT})$. Let $x\neq y \in V$ and $t \in [0,T]$. For all $n \in {{\mathbb{N}}}$, Lemma \[lem:CutoffRicci\] yields $$\nabla_{xy} P_t^{\phi_n} f \leq e^{-Kt} \|\nabla f\|_\infty.$$ Due to Theorem \[thm:Cutoff\] $(ix)$, we have $e^{t\Delta_{W_n}}= P_t^{1_{W_n}}$ on $[0,1_{W_n}]$, and since $1_{W_n} \leq \phi_n$, Theorem \[thm:Cutoff\] $(iv)$ yields $$e^{t\Delta_{W_n}}f = P_t^{1_{W_n}} f\leq P_t^{\phi_n} f \leq P_t f.$$ Since $e^{t\Delta_{W_n}}f$ converges to $P_t f$ pointwise as $n \to \infty$, we infer that $$\nabla_{xy} P_t f = \lim_{n \to \infty} \nabla_{xy} P_t^{\phi_n} f \leq e^{-Kt} \|\nabla f\|_\infty.$$ Now the claim follows immediately since $x,y$ and $t$ are arbitrary.
Using semigroup methods, we can now show that a lower curvature bound implies stochastic completeness. We want to point out that we will later independently prove stochastic completeness under even weaker assumptions using the Laplacian comparison principle (see Theorem \[thm:StochComplete\]).
\[lem:StochComplete\] If $G=(V,w,m)$ is a graph with $Ric(G) \geq K$, then $G$ is stochastically complete.
We note that the proof closely follows the proof of stochastic completeness under a Bakry-Emery curvature bound in ([@hua2017stochastic Theorem 1.2]).
Let $\eta_i:V \to [0,1]$ be non-constant such that $\eta_i \to 1$ pointwise and $\|\nabla \eta_i\|_\infty \to 0$ as $i \to \infty$. Then for all $x\neq y$ and $t>0$, Lemma \[lem:RicImpliesGradient\] implies that $$\begin{aligned}
\nabla_{xy} P_t \mathbf{1} = \lim_{i \to \infty} \nabla_{xy} P_t \eta_i \leq \lim_{i \to \infty} e^{-Kt}\|\nabla \eta_i\|_\infty =0.\end{aligned}$$ Hence, $\|\nabla P_t \mathbf{1}\|_\infty = 0$ which implies stochastic completeness as $P_0 \mathbf{1}=\mathbf{1}$.
Semigroup characterization
--------------------------
Using Theorem \[thm:nablaDelta\] and Lemma \[lem:RicImpliesGradient\], we now give a heat semigroup characterization of lower curvature bounds.
\[thm:gradientGraphs\] Let $G=(V,w,m)$ be a graph and let $K\in{{\mathbb{R}}}$. The following statements are equivalent:
(1) $Ric(G) \geq K$.
(2) For all $f \in C_c(V)$ and all $t>0$ $$\|\nabla P_t f\|_\infty \leq e^{-Kt}\|\nabla f\|_\infty.$$
(3) For all $f \in \ell_\infty(V)$ and all $t>0$ $$\| \nabla P_t f \|_\infty \leq e^{-Kt} \| \nabla f \|_\infty.$$
(4) $G$ is stochastically complete and for all $x,y \in V$ and all $t>0$ $$\begin{aligned}
W(p^x_t,p^y_t) \leq e^{-Kt} d(x,y)
\end{aligned}$$ where $p_t^x := \frac m {m(x)} P_t 1_x$ denotes the heat kernel.
We note that stochastic completeness is needed to state (4) since the Wasserstein distance $W$ is only defined on probability measures and $p_t^x$ is a probability measure only in the case of stochastic completeness.
We first prove $(3) \Leftrightarrow (4)$.
For all bounded $1-$Lipschitz functions $f$, we have $$\begin{aligned}
\int f dp^x_t - \int f dp^y_t = \sum_{z \in V}f(z)\left(p_t^x(z)-p_t^y(z)\right) = P_t f(x) - P_t f(y) \label{eq:Wassertstein-Lipschitz}.\end{aligned}$$ By definition, assertion $(4)$ is equivalent to $$\begin{aligned}
\int f dp^x_t - \int f dp^y_t \leq e^{-Kt} d(x,y)\end{aligned}$$ for all bounded Lipschitz functions which is equivalent to assertion $(3)$ due to (\[eq:Wassertstein-Lipschitz\]). It is also clear that $(3)$ implies stochastic completeness by noting that $(3)$ implies that $\| \nabla P_t \mathbf{1} \|_\infty =0$.
The implication $(1)\Rightarrow (3)$ follows from Lemma \[lem:RicImpliesGradient\] if $f$ is non-constant and Lemma \[lem:StochComplete\] if $f$ is constant.
The implication $(3)\Rightarrow (2)$ is trivial.
We finally prove $(2) \Rightarrow (1)$. Fix $x\sim y \in V$. By Theorem \[thm:nablaDelta\], it suffices to show that $$\inf_{\substack {f\in Lip(1) \cap C_c(V)\\ \nabla_{yx}f=1}} \nabla_{xy} \Delta f \geq K.$$ Let $f \in Lip(1)\cap C_c(V)$ be such that $\nabla_{yx}f =1$. By assertion $(2)$, we have $$\nabla_{yx} P_t f \leq e^{-Kt}.$$
Hence, by taking the time derivative at $t=0$, $$\begin{aligned}
\nabla_{xy} \Delta f = - \partial^+_t \nabla_{yx} P_t f |_{t=0}
= \lim_{t\to 0^+} \frac 1 t \left( \nabla_{yx} f - \nabla_{yx} P_t f \right)
\geq \limsup_{t\to 0^+} \frac 1 t \left(1 - e^{-Kt} \right)
= K\end{aligned}$$ which proves assertion $(1)$ of the theorem since $f$ is arbitrary.
Laplacian comparison principle {#sec:LaplaceCompare}
==============================
The classical Laplacian comparison theorem on manifolds compares the Laplacian of the distance function on the manifold to that of a model space with constant curvature. This means, for a given Riemannian manifold $M$ with Ricci curvature bounded from below by $K$ and for the model space $H$ with constant Ricci curvature $K$, one has $$\Delta^M d(x_0^M,\cdot) \leq \Delta^H d(x_0^H,\cdot).$$ For a survey of comparison geometry of Ricci curvature on manifolds see [@zhu1997comparison].
We give a discrete analogue of the above theorem in the sense that we upper bound the Laplacian of the distance function. As a replacement of a model space, we will associate a birth-death chain to a given graph having the same sphere measure (see Section \[sec:LaplaceCompareAndLineGrpahs\]). We will also introduce a new quantity called the sphere curvature which depends only on the distance to a fixed vertex instead of considering all curvatures between neighbors.
We first give a discrete Laplacian comparison principle without a model space by explicitly estimating $\Delta d(x_0,\cdot)$. Even though the proof is a one-liner in light of Theorem \[thm:nablaDelta\], the following discrete Laplacian comparison theorem, and its extension to the case of decaying curvature, turns out to be a foundation of a variety of applications, such as results concerning stochastic completeness and improved diameter bounds.
\[thm:LaplaceCompare\] Let $G=(V,w,m)$ be a graph. Let $x_0 \in V$ and suppose that $\kappa(x_0,\cdot) \geq K$ for some $K\in {{\mathbb{R}}}$. Then, $$\begin{aligned}
\Delta d(x_0,\cdot) \leq {\operatorname{Deg}}(x_0) - Kd(x_0,\cdot).
\end{aligned}$$
Let $y \in V$, $y \not=x_0$, and set $f:=d(x_0,\cdot)$. Note that $f \in Lip(1)$ and $\nabla_{yx_0}f=1$ so that due to Theorem \[thm:nablaDelta\], we have $$\begin{aligned}
K \leq \kappa(x_0,y) \leq \nabla_{x_0y} \Delta f = \frac{\Delta f(x_0)- \Delta f(y)}{d(x_0,y)} = \frac{{\operatorname{Deg}}(x_0) - \Delta f(y)}{d(x_0,y)}.
\end{aligned}$$ Rearranging yields the claim.
We next give a Laplacian comparison principle for decaying curvature. To do so, we need to measure the minimal curvature in terms of the distance to some fixed vertex $x_0$.
\[def:sphereCurvature\] Let $x_0 \in V$ be a fixed vertex. By abuse of notation, we denote $S_{r} := S_{r}(x_0)$ and $B_r := B_r(x_0)$. For $r\geq 1$, we let the *sphere curvatures* be given by $$\begin{aligned}
\kappa(r) := \min_{y\in S_{r}} \max_{\substack{x \in S_{r-1} \\ x\sim y}} \kappa(x,y).\end{aligned}$$
We remark that $$\kappa(r) \geq \min_{x,y \in B_r} \kappa(x,y)$$ which describes the the curvature decay in a simpler way. However, for all of our results it will suffice to have a lower bound on $\kappa(r)$.
\[thm:LaplaceComparisonNonConst\] Let $G=(V,w,m)$ be a graph, $x_0 \in V$ and $f:= d(x_0,\cdot)$. Then, $$\begin{aligned}
\Delta f \leq \Phi(f)
\end{aligned}$$ with $$\begin{aligned}
\Phi(R) := {\operatorname{Deg}}(x_0) - \sum_{r=1}^{R} \kappa(r)
\end{aligned}$$ for $R\geq1$ and $\Phi(0) = {\operatorname{Deg}}(x_0)$. The inequality is sharp for birth-death chains where we take $x_0=0$ so that $f(r)=d(0,r)=r$.
Note, in particular, that if $G$ is a graph with $Ric(G)\geq K$ and $H$ is a birth-death chain with $Ric(H)=K$ satisfying ${\operatorname{Deg}}_G(x_0)={\operatorname{Deg}}_H(0)$, then $$\Delta^G d(x_0, x) \leq \Delta^H d(0,R)$$ for all $x \in S_R$. This makes the analogy to the statement concerning manifolds mentioned above precise.
We prove the result via induction over the radius $R$. The claim is clear for $R=0$ since $\Delta f(x_0) = {\operatorname{Deg}}(x_0)$. Let $R>0$ and let $y \in S_R$. Let $x \in S_{R-1}$ with $x\sim y$ be such that $\kappa(x,y)$ is maximal on $\{ (z,y) \ | \ z \in S_{R-1}, z \sim y \}$. Due to the definition of $\kappa(R)$ and Theorem \[thm:nablaDelta\], we have $$\begin{aligned}
\kappa(R) \leq \kappa(x,y) \leq \nabla_{xy}\Delta f= \Delta f(x) - \Delta f(y).
\end{aligned}$$ By the induction assumption, we have $$\begin{aligned}
\Delta f(x) \leq {\operatorname{Deg}}(x_0) - \sum_{r=1}^{R-1} \kappa(r).
\end{aligned}$$ Rearranging and combining these yields $$\begin{aligned}
\Delta f(y) \leq \Delta f(x) -\kappa(R) \leq {\operatorname{Deg}}(x_0) - \sum_{r=1}^{R} \kappa(r)
\end{aligned}$$ which proves the first statement.
For birth-death chains, due to Theorem \[thm:line\], we have for $r\geq 1$, $$\begin{aligned}
\kappa(r) = \kappa(r-1,r) = \Delta f(r-1) - \Delta f(r).
\end{aligned}$$ Summing this up yields $$\begin{aligned}
\Delta f(R) ={\operatorname{Deg}}(0) - \sum_{r=1}^R \Delta f(r) = \Phi(R) = \Phi(f(R)).
\end{aligned}$$ for all $R\geq 1$ which finishes the proof.
Curvature comparison and associated birth-death chains {#sec:LaplaceCompareAndLineGrpahs}
------------------------------------------------------
We now prove that the Laplacian comparison principle is compatible with the transition to birth-death chains.
\[def:AssociatedlineGraph\] Let $G=(V,w,m)$ be a graph with $x_0 \in V$ called the root vertex and let $S_r:=S_r(x_0)$. We define the *associated birth-death chain* $ \widetilde G = ({{\mathbb{N}}}_0,\widetilde w, \widetilde m)$ via $$\begin{aligned}
\widetilde m(r) &:= m(S_r) \qquad \mbox{ and }\\
\widetilde w(r,r+1)&:= w(S_r,S_{r+1}) := \sum_{\substack{x \in S_r \\y \in S_{r+1}}} w(x,y).
\end{aligned}$$
\[thm:AssociatedLaplaceComparisonLine\] Let $G=(V,w,m)$ be a graph, $x_0 \in V$ and $f:=d(x_0,\cdot)$. Let $\widetilde G$ be the associated birth-death chain with Laplacian $\widetilde \Delta$ and $\widetilde f:=d(0,\cdot)$. Let $\Phi :{{\mathbb{N}}}_0 \to {{\mathbb{R}}}$ be a function. Then, $$\begin{aligned}
\Delta f \leq \Phi(f) \qquad \mbox{ implies } \qquad \widetilde \Delta \widetilde f \leq \Phi(\widetilde f).
\end{aligned}$$
We first note that $\Delta f(x_0) = {\operatorname{Deg}}(x_0)=\widetilde{{\operatorname{Deg}}}(0) = \widetilde{\Delta}\widetilde{f}(0)$.
Next, we let $r \in {{\mathbb{N}}}$ and integrate $\Delta f \leq \Phi(f)$ over the sphere $S_r:=S_r(x_0)$. For $x \in S_r$, we note that $m(x) \Delta f(x)= \sum_{y \in S_{r+1}} w(x,y) - \sum_{y \in S_{r-1}} w(x,y) \leq m(x) \Phi(r)$ so that $$\begin{aligned}
\Phi(r)\widetilde m(r) = \Phi(r) m(S_r) &= \sum_{x \in S_r} \Phi(r) m(x) \\&\geq \sum_{x \in S_r} \left( \sum_{y \in S_{r+1}} w(x,y) - \sum_{y \in S_{r-1}} w(x,y) \right) \\
&=\widetilde{w}(r,r+1) - \widetilde w(r,r-1).
\end{aligned}$$ Hence, $$\begin{aligned}
\widetilde \Delta \widetilde f(r) = \frac{\widetilde{w}(r,r+1) - \widetilde w(r,r-1)}{\widetilde{m}(r)} \leq \Phi(r) = \Phi(\widetilde f(r))
\end{aligned}$$ which finishes the proof.
Combining this with the sharp Laplacian comparison for birth-death chains allows us to compare the curvature between a graph and its associated birth-death chain.
\[cor:CurvCompare\] Let $G=(V,w,m)$ be a graph, $x_0 \in V$ be a root vertex and $\kappa(r)$ be the sphere curvatures with respect to $x_0$. Let $\widetilde G= ({{\mathbb{N}}}_0, \widetilde w, \widetilde m)$ be the associated birth-death chain with root vertex $\widetilde x_0 = 0$ and sphere curvatures $\widetilde \kappa(r) = \widetilde \kappa(r,r-1)$. Then, $$\begin{aligned}
\sum_{r=1}^R \widetilde \kappa(r) \geq \sum_{r=1}^R \kappa(r).
\end{aligned}$$
Let $f:=d(x_0,\cdot)$ on $G$ and $\widetilde f := d(0,\cdot)$ on $\widetilde G$. Let $$\begin{aligned}
\Phi(R) := {\operatorname{Deg}}(x_0) - \sum_{r=1}^{R} \kappa(r) \qquad \mbox{ and} \qquad
\widetilde\Phi(R) := \widetilde {\operatorname{Deg}}(0) - \sum_{r=1}^{R} \widetilde \kappa(r) .
\end{aligned}$$ Due to Theorem \[thm:LaplaceComparisonNonConst\], we have $$\begin{aligned}
\Delta f \leq \Phi(f) \qquad \mbox{ and } \qquad \widetilde \Delta \widetilde f = \widetilde \Phi(\widetilde f)
\end{aligned}$$ Now, Theorems \[thm:AssociatedLaplaceComparisonLine\] yields $$\begin{aligned}
\widetilde \Phi(\widetilde f) = \widetilde \Delta \widetilde f \leq \Phi(\widetilde f)
\end{aligned}$$ so that $\widetilde \Phi(R) \leq \Phi(R)$. The fact that $\widetilde {\operatorname{Deg}}(0) = {\operatorname{Deg}}(x_0)$ completes the proof.
One might be tempted to think that the sphere curvatures can also be compared without summation, i.e., $\widetilde \kappa(r) \geq \kappa(r)$ for all $r$. But this turns out to be wrong as demonstrated by the following example.
Let $G=({{\mathbb{Z}}},w,m)$ with root $x_0 = 0$ be given by $$\begin{aligned}
w(z,z+1) := m(z) := 2^z\end{aligned}$$ and $w(m,n)=0$ if $|m-n| \neq 1$. It is easy to see using the same techniques as in the proof of Theorem \[thm:line\] that $G$ has curvature $\kappa(r)= \kappa(r-1,r) = \Delta f(r-1) - \Delta f(r) = 0$ everywhere.
The associated birth-death chain $\widetilde{G} =({{\mathbb{N}}}_0, \widetilde w, \widetilde m)$ is then given by $$\begin{aligned}
\widetilde w(n,n+1) &= 2^n + 2^{-n-1} \mbox{ for } n\geq 0 \\
\widetilde m(n)&=2^n + 2^{-n} \quad \mbox{ for } n\geq 1 \qquad \mbox{ and } \qquad \widetilde m(0)=1.\end{aligned}$$ Let $\widetilde f := d(0,\cdot)$ on $\widetilde G$. Thus, for $n\geq 1$, $$\widetilde \Delta \widetilde f(n) = \frac{\widetilde{w}(n,n+1) - \widetilde{w}(n,n-1)}{\widetilde{m}(n)} = \frac{2^{n-1} - 2^{-n-1}}{2^n + 2^{-n}}$$ which is strictly increasing in $n$. Hence for $r\geq 2$, $$\widetilde \kappa (r) = \widetilde \Delta \widetilde f (r-1) - \widetilde \Delta \widetilde f (r) < 0.$$
Stochastic completeness
-----------------------
To prove stochastic completeness, we will use the Khas’minskii criterion on graphs established by Huang in [@huang2011stochastic Theorem 3.3] which we restate now using our notation.
\[thm:Huang3.3\] Let $G=(V,w,m)$ be a graph. If there exists a non-negative function $f \in C(V)$ with $$\begin{aligned}
f(x) \to \infty \mbox{ as } {\operatorname{Deg}}(x) \to \infty\end{aligned}$$ satisfying $$\begin{aligned}
\Delta f \leq \Psi(f)\end{aligned}$$ outside of a set of bounded vertex degree for some positive, increasing function $\Psi \in C^1([0,\infty))$ with $$\int_0^\infty \frac{dr}{\Psi(r)} = \infty,$$ then $G$ is stochastically complete.
Combining the Laplacian comparison with the Khas’minskii’s criterion using $f = d(x_0, \cdot)$ yields an optimal stochastic completeness result.
\[thm:StochComplete\] \
(i) If $G=(V,w,m)$ is a graph with $$\kappa(r) \geq -C \log r$$ for some constant $C>0$ and large $r$, then $G$ is stochastically complete.
(ii) For ${\varepsilon}>0$, let $G_{\varepsilon}=({{\mathbb{N}}}_0, w,m)$ be a birth-death chain with $m\equiv1$ and $$w(R,R+1) = 1 + \sum_{r=1}^R\sum_{k=1}^r \left(\log k\right)^{1+{\varepsilon}}.$$ Then $G_{\varepsilon}$ is stochastically incomplete and satisfies $$\kappa(r) \geq - (\log r)^{1+{\varepsilon}}$$ for all $r\geq 2$.
We note that the second statement shows that the first statement is optimal in the sense that the decay rate $-\log r$ cannot be replaced by the faster decay rate $-( \log r )^{1+{\varepsilon}}$.
For the proof of $(i)$, let $f := d(x_0,\cdot)$. Using the Laplacian comparison, Theorem \[thm:LaplaceComparisonNonConst\], we have $$\Delta f \leq \Phi(f)$$ with $$\Phi(R) = {\operatorname{Deg}}(x_0) - \sum_{r=1}^{R} \kappa(r) \leq \Psi(R)\in O(R \log (R))$$ since $-\kappa(r) \in O(\log(R))$, where $\Psi \in C^1([0,\infty))$ is some positive increasing function to which we can apply the Khas’minskii’s criterion. In particular, $$\int_0^\infty \frac {dr}{\Psi(r)} = \infty,$$ so that Theorem \[thm:Huang3.3\] yields stochastic completeness as desired.
To prove $(ii)$, we let $f :=d(0,\cdot)$. We first observe that for $R\geq1$, $$\begin{aligned}
\Delta f(R) = w(R,R+1) - w(R,R-1) = \sum_{k=1}^{R} (\log k)^{1+{\varepsilon}}.\end{aligned}$$ Since $G_{\varepsilon}$ is a birth-death chain, Theorem \[thm:line\] yields $$\kappa(R) = \kappa(R-1,R) = \Delta f(R-1) - \Delta f(R) = -(\log R)^{1+{\varepsilon}}$$ for $R \geq 2$ as desired.
Since $\iint (\log x)^{1+{\varepsilon}} \in \Theta(x^2 (\log x)^{1+{\varepsilon}})$, by definition of $w$, we have $$\begin{aligned}
w(R,R+1) \in \Theta(R^2 (\log R)^{1+{\varepsilon}}). \label{eq:wThetaR}\end{aligned}$$
Observe that as $G$ is a birth-death chain, it is weakly spherically symmetric with respect to $x_0 = 0 \in V= {{\mathbb{N}}}_0$ in the sense of [@keller2013volume Definition 2.3]. Hence, due to [@keller2013volume Theorem 5], we know that $G$ is stochastically complete if and only if $$\sum_r \frac{r+1}{w(r,r+1)} = \infty.$$ Due to (\[eq:wThetaR\]), we have $$\frac {r+1}{w(r,r+1)} \in \Theta\left( \frac 1 {r (\log r)^{1+{\varepsilon}}}\right)$$ and since $$\sum_r \frac 1 {r (\log r)^{1+{\varepsilon}}} < \infty$$ we have $$\sum_r \frac{r+1}{w(r,r+1)} < \infty$$ which implies stochastic incompleteness.
As mentioned in the introduction, the optimal curvature decay rate on Riemannian manifolds is of the order $-r^2$. As the use of intrinsic metrics has resolved various discrepancies between the manifold and graph settings in the past, one might think that using an intrinsic metric $\sigma$ instead of the combinatorial graph metric might give stochastic completeness when assuming $\kappa(r) \geq -C \sigma(0,r)^2$ in line with the manifolds case. This turns out to be wrong as we give an example of a stochastically incomplete graph with $\kappa(r) \sim -(\log \sigma(0,r))^{1+{\varepsilon}} $ for an intrinsic metric $\sigma$ where $f(n) \sim g(n)$ means $cf(n)<g(n)<Cf(n)$ for all $n \in {{\mathbb{N}}}$ and some $C>c>0$.
We recall that a metric $\sigma$ on $V$ is called *intrinsic* if $$\Delta \sigma(x,\cdot)^2(x)=\frac 1 {m(x)}\sum_{y \in V} w(x,y)\sigma(x,y)^2 \leq 2$$ for all $x \in V$. For various uses the intrinsic metrics in the graph setting, see [@keller2015intrinsic].
\[ex:incompleteIntrinsic\] Let $G=({{\mathbb{N}}}_0,w,m)$ be a birth-death chain with $m(r)=2^r$ and $w(r-1,r)=(\log r)^{1+{\varepsilon}} \cdot r \cdot 2^r$ for ${\varepsilon}>0$. By Theorem \[thm:line\], we obtain that $\kappa(r) \sim -(\log r)^{1+{\varepsilon}}$. Moreover, one can check that $$\sigma(r,R) := \sum_{k=r}^{R-1} {\operatorname{Deg}}_+(k)^{-1/2}$$ gives an intrinsic metric where ${\operatorname{Deg}}_+(r) := w(r,r+1)/m(r) \sim r(\log r)^{1+{\varepsilon}}$. In particular, $\sigma(0,r) \sim \sqrt{r/(\log r)^{1+{\varepsilon}}}$ and, thus, $\kappa(r) \sim -(\log \sigma(0,r))^{1+{\varepsilon}}$.
An objection to the example above is that the definition of the spherical curvature $\kappa$ depends on the combinatorial graph distance function $d$. However, in analogy to Theorem \[thm:nablaDelta\], we can also define a curvature $\kappa_\sigma$ with respect to the intrinsic metric $\sigma$ via $$\kappa_\sigma(x,y)= \inf\left\{\nabla_{xy}^\sigma \Delta f : \nabla_{yx}^\sigma f=1, \; \|\nabla^\sigma f\|_\infty = 1 \right\}$$ where $\nabla_{xy}^\sigma f := \frac{f(x)-f(y)}{\sigma(x,y)}$. On birth-death chains and intrinsic path metrics $\sigma$, for $x<y$ this simplifies to $$\kappa(x,y)= \nabla_{xy}^\sigma\Delta \sigma(0,\cdot).$$ In our example, we have $$\Delta \sigma(0,\cdot)(r) \sim \sqrt{{\operatorname{Deg}}_+(r)} \sim \sqrt{r \cdot (\log r)^{1+{\varepsilon}}}$$ and, by using the mean value theorem to estimate the difference, $$\begin{aligned}
\kappa_\sigma(r,r+1) &=\sqrt{{\operatorname{Deg}}_+(r)} \cdot \left(\Delta \sigma(0,\cdot)(r)- \Delta \sigma(0,\cdot)(r+1) \right) \\
&\sim - \sqrt{{\operatorname{Deg}}_+(r)} \cdot \sqrt{\frac {(\log r)^{1+{\varepsilon}}}r} \\
&\sim - (\log r)^{1+{\varepsilon}}.\end{aligned}$$ In particular, we also have $\kappa_\sigma(r) \sim -(\log \sigma(0,r))^{1+{\varepsilon}}$.
We are left to show stochastic incompleteness. Due to [@keller2013volume Theorem 5], $G$ is stochastically complete if and only if $$\sum_r \frac{m(\{1,...,r\})}{w(r,r+1)} = \infty.$$ However, $$\frac{m(\{1,...,r\})}{w(r,r+1)} \sim \frac 1{r (\log r)^{1+{\varepsilon}}}$$ which is summable. Therefore, $G$ is stochastically incomplete.
Improved diameter bounds
------------------------
We prove that a graph with bounded degree and sphere curvatures decaying not faster than $1/R$ must be finite (Corollary \[cor:finite\]). We also show that this decay rate is optimal (Theorem \[thm:FiniteOptimal\]). For various diameter bounds on finite graphs see [@paeng2012volume].
On the other hand, we show that in the case of unbounded degree, even a uniform positive lower curvature bound does not imply finiteness (see Example \[Ex:positiveCurvInfiniteDiam\]). In contrast, if we assume that the measure is bounded from below, then a uniform positive lower curvature bound implies finiteness even in the case of unbounded degree (see Corollary \[cor:FiniteDiamBoundedMeasure\]).
As a warm-up, we start with the following diameter bound from [@lin2011ricci Theorem 4.1] transferred to our setting.
\[prop:FiniteDiamBoundedDegree\] Let $G=(V,w,m)$ be a graph and let $x, y \in V$ with $x \not = y$. If $\kappa(x,y)>0$, then $$d(x,y) \leq \frac{{\operatorname{Deg}}(x) + {\operatorname{Deg}}(y)} {\kappa(x,y)}.$$
It is easy to see that $W(1_x,1_y) = d(x,y)$. Furthermore, observe that for sufficiently small ${\varepsilon}$, $$W(1_x,m^{\varepsilon}_x) = {\varepsilon}{\operatorname{Deg}}(x).$$ This follows as $W(1_x,m_x^{\varepsilon}) = \sup_{f \in Lip(1)} - {\varepsilon}\Delta f(x) \leq {\varepsilon}{\operatorname{Deg}}(x)$ with equality for $f=1_x$. Hence, by the triangle inequality, $$\begin{aligned}
W(m_x^{\varepsilon}, m_y^{\varepsilon}) &\geq W(1_x,1_y) - W(1_x,m^{\varepsilon}_x) -W(1_y,m^{\varepsilon}_y) \\
&= d(x,y) - {\varepsilon}({\operatorname{Deg}}(x) + {\operatorname{Deg}}(y)).\end{aligned}$$
Thus, $$\begin{aligned}
\kappa_{\varepsilon}(x,y)
&= 1 - \frac{W(m_x^{\varepsilon}, m_y^{\varepsilon})}{d(x,y)} \\
&\leq {\varepsilon}\cdot \frac{ {\operatorname{Deg}}(x) + {\operatorname{Deg}}(y)} {d(x,y)}.\end{aligned}$$ This yields the claim since $\kappa(x,y) = \lim_{{\varepsilon}\to 0^+} \frac{1} {{\varepsilon}}\kappa_{\varepsilon}(x,y)$.
In particular, if the degree is bounded and the curvature is uniformly positive, then the graph is finite. More specifically, if we let ${\operatorname{diam}}(G)= \sup_{x,y \in V} d(x,y)$ denote the diameter of $G$, then if ${\operatorname{Deg}}(x)\leq M$ and $Ric(G)\geq K>0$, then $${\operatorname{diam}}(G) \leq \frac{2M}{K}.$$ We now improve this result in the sense that we only lower bound the sphere curvatures, which allows for some negative curvature, and consider part of the vertex degrees. For a fixed vertex $x_0 \in V$, we let for $x \in S_r:= S_r(x_0)$, $${\operatorname{Deg}}_{\pm}(x) = \frac{1}{m(x)} \sum_{y \in S_{r\pm1}} w(x,y)$$ denote the *outer* and *inner* degree of $x$. Using the Laplacian comparison principle for non-constant curvature, we immediately obtain the following improved diameter bound.
\[thm:ImprovedDiamBound\] Let $G=(V,w,m)$ be a graph with $x_0 \in V$. If $S_R \not = \emptyset$ for $R>0$, then $$\begin{aligned}
\sum^R_{r=1} \kappa(r) \leq {\operatorname{Deg}}(x_0) + \min_{x \in S_R} \left({\operatorname{Deg}}_-(x) - {\operatorname{Deg}}_+(x) \right).
\end{aligned}$$ In particular, if $\min_{x \in S_r} \left({\operatorname{Deg}}_-(x) - {\operatorname{Deg}}_+(x) \right) \leq M$ and $\kappa(r) \geq K >0$ for all $r\geq1$, then $${\operatorname{diam}}(G) \leq \frac{2({\operatorname{Deg}}(x_0) +M)}{K}.$$
We recall that the Laplacian comparison, Theorem \[thm:LaplaceComparisonNonConst\], gives that $$\Delta f(x) \leq {\operatorname{Deg}}(x_0) - \sum_{r=1}^R \kappa(r)$$ for $x \in S_R$ where $f(x) = d(x,x_0)$. Now, the first statement follows as $\Delta f(x) = {\operatorname{Deg}}_+(x) - {\operatorname{Deg}}_-(x)$ by an easy calculation. The second statement is an immediate consequence of the first statement and the triangle inequality.
The theorem immediately gives us the following corollary.
\[cor:finite\] If $G=(V,w,m)$ is a graph with bounded degree, then $$\limsup_{R \to \infty} \sum_r^R \kappa(r) < \infty.$$ Consequently, there is no infinite graph with bounded vertex degree satisfying $$\limsup_{R \to \infty} \sum_r^R \kappa(r)=\infty.$$
We show that the results above are optimal in the sense that whenever we have a given summable positive sequence $k_r$, we can find an infinite graph with bounded degree and summable sphere curvatures $\kappa(r)$ larger than $k_r$.
\[thm:FiniteOptimal\] For every positive sequence $(k_r)_{r \in {{\mathbb{N}}}}$ such that $ \sum_r k_r < \infty$ there exists an infinite graph $G=(V,w,m)$ with bounded degree such that $$\kappa(r) \geq k_r \qquad \mbox{ and } \qquad \sum_r \kappa(r) <\infty.$$
We define a birth-death chain $G=({{\mathbb{N}}}_0,w,m)$ inductively with $w$ symmetric and $m$ satisfying $m(0)=1$, $w(0,1) = 2 \sum_{i>0} k_i$ and for $r\geq 1$, $$\begin{aligned}
m(r) = \frac{w(r,r-1)}{k_{r+1}}\qquad \mbox{ and} \qquad w(r,r+1) = 2m(r) \sum_{i>r} k_i.
\end{aligned}$$ Note, in particular, that $\frac{w(r,r-1)}{m(r)} = k_{r+1}$ while $\frac{w(r-1, r)}{m(r-1)}= 2 \sum_{i >r-1}k_i$.
Due to Theorem \[thm:line\], for $r>1$, $$\begin{aligned}
\kappa(r) &= \kappa(r-1,r) \\&= \frac{w(r-1,r)-w(r-1,r-2)}{m(r-1)}- \frac{w(r,r+1)-w(r,r-1)}{m(r)} \\
&= 2 \sum_{i>r-1} k_i - k_r - 2 \sum_{i>r} k_i + k_{r+1} \\
&=k_r + k_{r+1} \geq k_r
\end{aligned}$$ which also shows that $\sum_r \kappa(r)<\infty$. Similarly, $\kappa(1)= 2k_1 + k_2 \geq k_1$.
It is left to show that the graph has bounded degree. We have $$\begin{aligned}
{\operatorname{Deg}}(r) = \frac{w(r,r-1)}{m(r)} + \frac{w(r,r+1)}{m(r)} = k_{r+1} + 2 \sum_{i>r}k_i \leq 3 C
\end{aligned}$$ with $C:= \sum_r k_r < \infty$ by assumption. This finishes the proof.
\[Ex:positiveCurvInfiniteDiam\] In contrast to Theorem \[thm:ImprovedDiamBound\], we now show that there exist graphs with uniformly positive curvature which are infinite. We note that all such graphs must have unbounded vertex degree.
We construct an infinite birth-death chain $({{\mathbb{N}}}_0,w,m)$ such that $\kappa(x,y) = K>0.$ We first let $w(r,r+1)$ be strictly positive and decreasing in $r \in {{\mathbb{N}}}_0$. By Theorem \[thm:line\] and Remark \[rem:line\], it suffices to find a choice of measure $m$ such that $\kappa(0,r)=K$, that is, for $f=d(0,\cdot)$ $$\Delta f(r) = \Delta f(0) - Kr = {\operatorname{Deg}}(0)-Kr.$$
Choose $m(0)$ such that ${\operatorname{Deg}}(0) < Kr$ for all $r\geq 1$. For this it suffices that $m(0) > \frac{w(0,1)}{K}$. Then, for $r\geq 1$, choose $$m(r) := \frac{w(r,r-1) - w(r,r+1)}{Kr - {\operatorname{Deg}}(0)}$$ guaranteeing $$\Delta f(r) = \frac 1 {m(r)} (w(r,r+1) - w(r,r-1)) = {\operatorname{Deg}}(0) - Kr.$$ We remark that $m(r) >0$ since $w(r,r+1)$ is strictly decreasing.
Finiteness of the measure
-------------------------
In this section, we show that a suited positive lower bound on the curvature implies finite measure, that is, $m(V):= \sum_{x\in V} m(x) < \infty$.
Let $G=(V,w,m)$ be a graph. If $$\liminf_{R\to \infty}\sum_{r=1}^R \kappa(r)>{\operatorname{Deg}}(x_0),$$ then $m(V) < \infty$.
We first show that it suffices to prove the theorem for birth-death chains. Let $\widetilde G = ({{\mathbb{N}}}_0,\widetilde w, \widetilde m)$ be the birth-death chain associated to $G$. Due to Corollary \[cor:CurvCompare\], we also have $$\liminf_{R\to \infty}\sum_{r=1}^R \widetilde\kappa(r)>{\operatorname{Deg}}(x_0)$$ where $\widetilde \kappa(r)$ are the sphere curvatures of $\widetilde G$. Assuming that the theorem is proven for birth-death chains, we obtain that $m(V)=\widetilde m({{\mathbb{N}}}_0) < \infty$ which would finish the proof.
Now we prove the theorem for birth-death chains. Let $f=d(0,\cdot)$. Due to Theorem \[thm:LaplaceComparisonNonConst\] and since $\liminf_{R\to \infty} \sum_r^R \kappa(r)> {\operatorname{Deg}}(0)$, we get $$\limsup_{R \to \infty} \Delta f(R) = \limsup_{R \to \infty} \left({\operatorname{Deg}}(0) - \sum_{r=1}^R \kappa(r) \right) = {\operatorname{Deg}}(0) - \liminf_{R \to \infty} \sum_{r=1}^R \kappa(r) <0$$ so that there exists ${\varepsilon}>0$ and $R>0$ such that $\Delta f(r)\leq -{\varepsilon}$ for all $r\geq R$. This implies that $${\varepsilon}m(r) \leq w(r,r-1) - w(r,r+1)$$ for $r\geq R$. Summing up, we obtain $${\varepsilon}\sum_{r=R}^\infty m(r) \leq w(R,R-1)$$ which yields the finiteness of the measure of the birth-death chain. This finishes the proof.
The theorem immediately gives the following corollary.
\[cor:FiniteDiamBoundedMeasure\] Let $G=(V,w,m)$ be a graph. If $\liminf_{R\to \infty}\sum_{r=1}^R \kappa(r) = \infty$, then $m(V)$ is finite. If, additionally, $\inf_{x \in V} m(x) >0$, then $G$ is finite.
Combining this with Corollary \[cor:finite\] we get the following dichotomy.
Let $G=(V,w,m)$ be a graph and suppose that $$\kappa(r)\geq \frac{C}{r}$$ for some $C>0$ and all large $r$. Then either $G$ is finite or $G$ is infinite with unbounded vertex degree and finite measure.
Ricci curvature for continuous-time Markov processes {#sec:MarkovProcesses}
====================================================
In this section, we compare our curvature notion to the curvature defined in [@veysseire2012coarse] for continuous time Markov processes, which generalize both locally finite graphs and Riemannian manifolds. To make the comparison clear, we recall our curvature definition $$\kappa(x,y) = \lim_{t \to 0^+} \frac 1 t \left(1 - \frac{W(m_x^t, m_y^t)}{d(x,y)} \right)$$ where the discrete time Markov kernel $m_x^t$ with laziness parameter $t \in(0,\infty)$ is given by $$\int{ f dm_x^t} = (f + t \Delta f)(x).$$ Note that $m^t$ is only non-negative if the vertex degree is bounded and if $t$ is sufficiently small. By abuse of notation, we call $m^t$ a Markov kernel in any case.
The idea to define curvature in [@veysseire2012coarse] is to replace the measure $m_x^t$ by the continuous time heat kernel $p_x^t$ which has already appeared in Theorem \[thm:gradientGraphs\] and is given by $$\int{ f dp_x^t} := P_t f (x).$$ We note that this is equivalent to $$p_x^t (y)= P_t 1_y(x) = \frac{m(y)}{m(x)}P_t 1_x (y).$$
Due to Taylor’s theorem, it is reasonable to hope that $m_x^t$ is a good approximation for $p_x^t$ as $t \to 0^+$. Criteria for this approximation will be investigated in the next subsection.
Corresponding to [@veysseire2012coarse Definition 6], the coarse Ricci curvature on stochastically complete, continuous time Markov processes is defined by $$\overline \kappa(x,y) := \limsup_{t \to 0^+} \frac 1 t \left( 1 - \frac{W(p_x^t, p_y^t)}{d(x,y)} \right)$$ and $$\underline \kappa(x,y) := \liminf_{t \to 0^+} \frac 1 t \left( 1 - \frac{W(p_x^t, p_y^t)}{d(x,y)} \right).$$ We recall that $\overline \kappa$ and $\underline \kappa$ do not coincide in general (see e.g. [@veysseire2012coarse Example 8]). Furthermore, the above definition only makes sense in the stochastically complete case since, otherwise, $p_x^t$ is not a probability measure and, therefore, the Wasserstein distance is not well defined.
The main result of [@veysseire2012coarse] is the equivalence of the lower curvature bound $\overline \kappa(x,y) \geq K$ and the Wasserstein contraction property $$W(p_x^t,p_y^t) \leq d(x,y)e^{-Kt}.$$ We note that the same statement for a lower bound on $\kappa$ was shown in Theorem \[thm:gradientGraphs\]. We will show in Corollary \[cor:MarkovChains\] that $\kappa=\underline\kappa=\overline \kappa$ when assuming that $\kappa$ is uniformly bounded from below. Therefore, the result in [@veysseire2012coarse] combined with this equality gives an alternative proof of Theorem \[thm:gradientGraphs\].
Discrete and continuous time Markov kernels
-------------------------------------------
We will next give conditions guaranteeing that discrete and continuous time Markov kernels approximate each other.
As a convenient notation, we extend the definition of the semigroup $P_t$ to possibly unbounded non-negative functions.
For $f\geq 0$, we define $$P_t f := \sup_{\substack{0\leq g\leq f \\ g \in \ell_\infty(V)}} P_t g.$$
The aim of this subsection is to prove that $W(p_x^t,m_x^t)=O(t^2)$ if and only if $P_t d(x,\cdot)<\infty$ for small $t>0$. As a first step, we show a uniform boundedness property of the semigroup when applied to unbounded functions.
\[lem:uniformFinitePt\] Let $x \in V$ and let $f \geq 0$. If $P_T f(x) < \infty$ for some $T>0$, then
(i) $
\sup_{t \in [0,T]} P_t f(x) < \infty.
$
(ii) $P_t f < \infty$ for all $t < T$.
We first prove $(i)$. Let $t \in [0,T]$ and let $g \in \ell_\infty(V)$ be such that $0\leq g \leq f$. Then, $$P_t g(x) \leq e^{(T-t){\operatorname{Deg}}(x)}P_T g(x) \leq e^{T{\operatorname{Deg}}(x)} P_T f(x) < \infty$$ independently of $t$ and $g$. Taking the supremum over $t \in [0,T]$ and $g$ yields $(i)$.
We now prove $(ii)$. Let $t<T$ and $y \in V$. Let $g \in \ell_\infty(V)$ be such that $0\leq g \leq f$. Observe that $P_T g(x) \geq P_{T-t}1_y(x) \cdot P_t g(y)$ where $P_{T-t}1_y(x)>0$ due to connectedness. Hence, taking the supremum over $g$ yields $$P_t f(y) \leq \frac{P_T f(x)}{P_{T-t}1_y(x)} < \infty$$ due to assumption. This proves $(ii)$ and finishes the proof of the lemma.
We now characterize when the ball measure $m_x^t$ approximates the heat kernel measure $p_x^t$ as $t \to 0^+$.
\[pro:PtdANDWpm\] Suppose that $G=(V,w,m)$ is stochastically complete and let $x \in V$. The following statements are equivalent:
(1) $P_t d(x,\cdot) < \infty$ for some $t>0$.
(2) $P_t f < \infty$ for all $f\geq 0$ with $\|\nabla f \|_\infty < \infty$ and some $t>0$.
(3) $W(p_x^t,m_x^t) = O(t^2)$ as $t \to 0^+$.
(4) $W(p_x^t,m_x^t) < \infty$ for some $t>0$.
We remark that the above properties also play an important role as a standing assumption in [@joulin2007poisson] denoted by $P_t (x,\cdot) \in \mathscr{P}_1(E)$.
The implication (1) $\Rightarrow$ (2) follows since $f \leq d(x,\cdot) + f(x)$ implies that $P_t f \leq f(x) + P_td(x,\cdot) < \infty$.
The implication (2) $\Rightarrow$ (1) is obvious.
We now show that (1) $\Rightarrow$ (3). Due to Kantorovich duality, we have $$\begin{aligned}
W(p_x^t,m_x^t) &= \sup_{f \in Lip(1) \cap \ell_\infty(V)} \sum_{y \in V} f(y) \left( p_x^t(y) - m_x^t(y) \right) \\
&= \sup_{f \in Lip(1) \cap \ell_\infty(V)} \sum_{y\in V} f(y) \left( P_t 1_y(x) - 1_y(x) - t \Delta 1_y(x) \right)\\
&= \sup_{f \in Lip(1) \cap \ell_\infty(V)} \left( P_t f - f - t \Delta f \right)(x).\end{aligned}$$ When optimizing, we can assume that $f(x)=1$ without loss of generality. Since $f \in Lip(1)$, replacing $f$ by its positive part does not change the values of $f$ on $B_1(x)$ and does not decrease the values on $V\setminus B_1(x)$. Since $p_x^t - m_x^t$ is non-negative on $V \setminus B_1(x)$, the objective function $\sum_y f(y) \left( p_x^t(y) - m_x^t(y) \right)$ is not decreased when replacing $f$ by its positive part. Therefore, we can assume that $f(x)=1$ and $f\geq 0$ when optimizing. This gives $$\begin{aligned}
\label{eq:WPepsmEps}
W(p_x^t,m_x^t) & = \sup_{\substack{f \in Lip(1) \cap \ell_\infty(V) \\ f\geq 0, f(x)=1}} (P_t f - f - t \Delta f)(x). \end{aligned}$$
Therefore, let $f \in Lip(1) \cap \ell_\infty(V)$ with $f\geq0$ and $f(x)=1$. Then, $0 \leq f \leq 1 + d(x,\cdot).$ Due to Lemma \[lem:uniformFinitePt\] $(i)$ and by assumption, there exists $C>0$ such that $P_t(1 + d(x,\cdot)) \leq C$ on $B_2(x)$ for all $t \in [0,T]$. Thus, we also have $0 \leq P_t f \leq C$ on $B_2(x)$ for all $t \in [0,T]$. This yields the existence of $C'$ independent of $f$ and $t \in [0,T]$ such that $$|\Delta \Delta P_t f| \leq C'.$$
Due to Taylor’s theorem, there exists $\delta \in [0,t]$ such that $$\begin{aligned}
(P_t f - f - t \Delta f)(x) = \frac {t^2} 2 \Delta \Delta P_\delta f(x) \leq \frac {t^2} 2 C' = O(t^2).\end{aligned}$$ Putting this together with (\[eq:WPepsmEps\]) proves that (1) $\Rightarrow$ (3).
The implication (3) $\Rightarrow$ (4) is obvious.
We now show that (4) $\Rightarrow$ (1). Let $f = d(x,\cdot)+1$ and $f_n := f \wedge n \in \ell_\infty(V)$. Due to (\[eq:WPepsmEps\]), we have $$\infty > W(p_x^t,m_x^t) \geq (P_t f_n - f_n -t \Delta f_n)(x)$$ yielding $$P_t f(x) = \sup_n P_t f_n(x) \leq W(p_x^t,m_x^t) + f(x) + t \Delta f(x) <\infty.$$ Thus by Lemma \[lem:uniformFinitePt\] $(ii)$, $P_s f < \infty$ for all $s < t$ as desired. This finishes the proof.
Another Ricci curvature characterization
----------------------------------------
We now prove that on locally finite graphs with Ricci curvature bounded from below, our definition of $\kappa$ coincides with $\overline{\kappa}$ and $\underline{\kappa}$ as defined in [@veysseire2012coarse Definition 6]. This will yield another characterization of lower Ricci curvature bounds by combining with [@veysseire2012coarse Theorem 9].
As a preparation, we show the subexponential behavior of non-negative $\lambda$-subharmonic functions under the heat equation.
\[lem:subharmonicPt\] Let $f \geq 0$ satisfy $\Delta f \leq \lambda f$ for some $\lambda >0$. Then, $P_t f \leq e^{\lambda t} f$.
Let $W \subset V$ be finite. Let $f_W:= f 1_W$ and let $e^{t\Delta_W}$ be the semigroup corresponding to $\Delta_W$ with $\Delta_W g := 1_W \Delta (g 1_W)$ representing Dirichlet boundary conditions. First, we observe that $\Delta_W f_W \leq \lambda f_W$ since $\Delta f \leq \lambda f$ and $f \geq 0$.
Let $\phi := e^{t\left(\Delta_W-\lambda \right)} f_W$. Then, $$\partial_t \phi = e^{-\lambda t} \left( \Delta_W -\lambda \right) e^{t\Delta_W} f_W = e^{-\lambda t} e^{t\Delta_W} \left( \Delta_W -\lambda \right) f_W \leq 0$$ showing that $\phi(t)=e^{-\lambda t}e^{t \Delta_W} f_W \leq f_W = \phi(0)$. Since $e^{t_W} f_W \to P_t f$ pointwise as $W\to V$, we obtain the desired claim that $P_t f \leq e^{\lambda t} f$.
We remark that the step in the proof above where we take Dirichlet boundary conditions is necessary to ensure that $\Delta P_t f = P_t \Delta f$ which generally only holds true on the domain $D(\Delta) \subseteq \ell^2(V)$ on which $\Delta$ is self-adjoint.
We next prove that a lower Ricci curvature bound implies that $P_t d(x,\cdot)<\infty$.
\[lem:Ptd\] Let $x \in V$ and $f := d(x,\cdot)$. If $Ric(G) \geq -K$ for some $K >0$, then $$P_t f \leq e^{Kt} (f + {\operatorname{Deg}}(x)/K)<\infty.$$
Due to the Laplacian comparison principle, Theorem \[thm:LaplaceCompare\], we have that $$\Delta(f + {\operatorname{Deg}}(x)/K) = \Delta f \leq K(f + {\operatorname{Deg}}(x)/K).$$ Thus, Lemma \[lem:subharmonicPt\] yields $$P_t f \leq P_t (f + {\operatorname{Deg}}(x)/K) \leq e^{Kt} (f + {\operatorname{Deg}}(x)/K)$$ as desired.
We now present the main theorem of this section.
\[thm:MarkovChains\] Let $G=(V,w,m)$ be a stochastically complete graph. Suppose that $P_t d(x_0,\cdot) < \infty$ for some $x_0 \in V$ and some $t>0$. Then, for all $x \neq y$, $$\kappa(x,y) = \lim_{t \to 0^+} \frac 1 t \left(1 - \frac{W(p_x^t,p_y^t)}{d(x,y)} \right) = \overline\kappa(x,y) = \underline \kappa(x,y).$$
Let $x \neq y \in V$. Due to the triangle inequality and Proposition \[pro:PtdANDWpm\], we have $$W(m_x^t,m_y^t) = W(p_x^t,p_y^t) + O(t^2)$$ as $t\to 0^+$. By definition, $$\begin{aligned}
\kappa(x,y) = \lim_{t\to 0^+} \frac 1 t \left(1 - \frac{W(m_x^t,m_y^t)}{d(x,y)} \right) &=\lim_{t\to 0^+} \frac 1 t \left(1 - \frac{W(p_x^t,p_y^t) + O(t^2)}{d(x,y)} \right)\\
&=\lim_{t \to 0^+} \frac 1 t \left(1 - \frac{W(p_x^t,p_y^t)}{d(x,y)} \right).\end{aligned}$$ This finishes the proof.
Since both stochastic completeness and $P_t d(x_0,\cdot)<\infty$ are implied by a lower Ricci curvature bound (see Theorem \[thm:StochComplete\] and Lemma \[lem:Ptd\]), we immediately obtain the following corollary.
\[cor:MarkovChains\] Let $G=(V,w,m)$ be a graph with $Ric(G) \geq K$ for some $K \in {{\mathbb{R}}}$. Then, for all $x \neq y$, $$\kappa(x,y) = \lim_{t \to 0^+} \frac 1 t \left(1 - \frac{W(p_x^t,p_y^t)}{d(x,y)} \right).$$
Combining with [@veysseire2012coarse Theorem 9], we immediately obtain that $W(p_t^x,p_t^y) \leq e^{-Kt} d(x,y)$ whenever $Ric(G)\geq K$. We remark that this gives an alternative method for proving Theorem \[thm:gradientGraphs\].
Acknowledgments {#acknowledgments .unnumbered}
---------------
F.M. wants to thank the German National Merit Foundation for financial support. R.K.W. gratefully acknowledges financial support from PSC-CUNY Awards, jointly funded by the Professional Staff Congress and the City University of New York, and the Collaboration Grant for Mathematicians, funded by the Simons Foundation. R.K.W. would also like to thank Fudan and Hokkaido Universities for their generous hospitality while parts of this work were completed. Furthermore, both authors want to thank the Harvard University Center of Mathematical Sciences and Applications for their hospitality.
Florentin Münch,\
Department of Mathematics, University of Potsdam, Potsdam, Germany\
Currently: Center of Mathematical Sciences and Applications, Harvard University, Cambridge MA, USA\
`[email protected]`\
\
Rados[ł]{}aw K. Wojciechowski,\
York College and the Graduate Center of the City University of New York, New York, USA\
`[email protected]`
| |
The pandemic has forced all types of therapy online, including treatment for children with ASD. Although the transition was driven by necessity during the pandemic, it has raised crucial questions about the future. Can technological platforms meet the needs of children with ASD? How can they be adapted? What advantages and disadvantages do they present? Professor Ofer Golan sat down with psychologist Dr. Gili Segall to hear her perspective after a year of working remotely.
Gili Segall is a New York- and Israeli-licensed clinical psychologist, and a clinical assistant professor in the department of child and adolescent psychiatry at the NYU Grossman School of Medicine – NYU Langone Health, where she works with families as a clinical psychologist. She received her doctorate in Clinical Psychology from Bar-Ilan University in Israel, where her Ph.D. thesis analyzed how parents talk to their children and the relationship between certain aspects of the parent-child dialogue and childrens’ emerging beliefs and perceptions.
Ofer Golan is a clinical psychologist, associate professor, and the head of the Autism Research Lab at Bar-Ilan University, Israel. His research focuses on socio-emotional functioning in autism, including social communication, emotion recognition, expression and regulation, and ways to develop them through evidence-based interventions. He is the founder and the clinical advisor of two Israeli clinical centers which provide evidence-based diagnosis and intervention services for children, adolescents, and adults with autism and their families, in addition to training clinicians and disseminating evidence-based interventions nationwide.
Segall moved most of her therapy sessions online at the onset of the pandemic, when the New York area was sheltering in place. The transition wasn’t easy and led to significant challenges. “There are children that I evaluated initially through telehealth, but when I went the extra mile and invited them to the office, I was able to see aspects of social interactions that I was not aware of online. Technology can mask some of the issues that we encounter when we meet face to face,” she explains.
However, despite the challenges the online work has posed for some of her clients, for others, Segal was surprised to discover, it has actually been an advantage. “It’s more of like a focused stimulus, not necessarily having to absorb certain other sensory stimulations like sounds or smells or a more complex experience of sitting with someone in a room…when you’re more comfortable in a setting then you are often better able to get something out of the interaction in that setting.”
Another key advantage is accessibility. Since the majority of children with ASD do not live in close proximity to a treatment center, before the pandemic, they could not take advantage of the expertise that a clinician like Segall offers. “If you had told me a year and a half ago that I would be working completely online, I would have probably said ‘no way. It’s something that I didn’t feel very comfortable with, and wasn’t familiar with,” says Segall. “But I can actually see that technology can be very effective and very helpful and make our interventions much more accessible. My office is in the center of Manhattan, but now I can work with people from all over New York, New Jersey, and Connecticut, and even Florida, which is amazing. These are individuals and families that may not have treatment options near them, but they can access the support through online platforms.”
One of the concerns professionals have about incorporating technology into treatment is that it may discourage, rather than encourage human interaction. Segal sees it differently—since parents are not trained in therapy techniques, supporting them through technology can actually be a way to increase their involvement in the therapeutic processes. “I think that the work with parents is a very crucial aspect of working with children because you can definitely help the parent do this process, become more aware of their children’s experience, become a better social coach their children in their daily routines. There’s a very strong connection between how parents support and help their children and the children’s outcomes.”
Not only is parental involvement beneficial for the children, but the parents benefit as well. “I was involved in a program called Peers for Peace that helped kids increase their ability to actually engage in play dates. We found that helping the parents become social coaches for their children decreased their parental stress, which, in turn, had a positive impact on the children’s mental states. It helped parents feel that they might understand their children and their emotional experience a bit better”.
EmotiPlay is a research-based tool that helps therapists, teachers, and parents teach children with autism to understand emotion, a life skill that impacts almost every element of their lives. | https://www.emotiplay.com/blog/telehealth-for-kids-with-asd-advantage-or-disadvantage/ |
Real quilts are designed with 1/4 inch seam allowances. What do Queen of Stitching blocks use?
Although many of the techniques we use have been modernized to deliver what we call “Next Generation” piecing or quilting - Queen of Stitching designers adhere to “Traditional Quilt” values.
As a rule - the distances you will find between our fabric placement guides and our stitching lines are 1/4”. Occasionally, depending on the angle of the placement or the position of the placement, our instructions may call for using a fabric strip that when sewn into position, extends more than 1/4” beyond the adjoining stitching line. The decision to use a bit of extra fabric is intentional as it makes correctly positioning the fabric strip easier. The extra fabric will make no difference in your final result.
If you find the idea of occasionally exceeding the 1/4” directive to be troubling - by all means trim the excess fabric back to 1/4’ after you have stitched the adjoining placement guide. | https://queenofstitching.com/faqs/36-do-queen-of-stitching-blocks-use-14inch-seam-allowances |
It was a great decade for music and culture.
In the decade after the Great Depression, America was a thriving, thriving nation with the greatest musical tradition of all time.
It was the golden age of jazz, gospel, blues, country, and more, and America was enjoying its most prosperous decade in modern history.
But in the 1940s, a different story unfolded.
The war in Europe brought an end to the postwar world order that had governed the United States for more than three decades.
The Cold War ended, and the Cold War began.
America was torn apart and the United Kingdom was partitioned and America declared itself a republic.
The great American musical traditions were swept away.
America was still on its feet.
The Beatles, the Rolling Stones, the Bee Gees, the Doors, the Beach Boys, and many others were still around, performing and performing well, producing some of the greatest albums of all times.
But the music of the time had changed, and it had been replaced by more commercialized, commercial music that had a commercial appeal.
American music had become, in the words of a popular 1960s song, the new American pop.
The 1950s, the decade that was defined by the rise of the Beatles and the Beatles concert, saw the birth of the first American pop star, Elvis Presley.
Elvis was the most popular rock star of the 1950s and had a big fan base that had never before seen anything like it.
Elvis’s concerts, his popularity, and his celebrity made him an international superstar.
Elvis and his rock and roll inspired millions of people around the world to come see him.
The American public, however, was still divided on the issue of whether to support the military-industrial complex, which had been waging a war on drugs and poverty, and whether to continue supporting the war on terror, which was being waged against the Soviet Union.
It is this split that has been reflected in the popularity of Elvis Pressey and the rise and popularity of his rock star image in America.
The country was divided.
There was a sense that America had gone back to the old days, that America was still the same country that we were in 1945.
The only difference was that the world was now divided into countries that controlled drugs and guns and the world that controlled America.
In the 1950 to 1960s, American pop music was a big part of the American economy.
It produced a big number of albums, which, for a while, seemed to be a big draw in the marketplace.
American pop artists were successful in the United Nations and on the radio, which provided the only source of exposure for artists.
The public was becoming increasingly educated and had grown up in a post-war world, where the Cold war had ended and the new Cold War was underway.
Elvis Presay was one of the major pop stars of the era.
He had a lot of hits and his popularity was so great that many people, including President Kennedy, who had been a big Elvis fan, became fans of Elvis.
The song, “All I Want for Christmas,” was a song that everyone listened to.
The album, Elvis’s Rock and Roll, which came out in 1957, was a huge success.
The album was recorded in many different cities around the country and, in fact, many of the songs were recorded on tape in the studio.
Elvis would sometimes sing the songs to himself.
The songs had such a tremendous appeal that the public began to buy the albums, believing that they were the real deal.
The popularity of the albums led to a great deal of popularity for Elvis and the pop culture of the 1960s and early 1970s.
The pop stars and the artists, including Elvis, were celebrities.
The media was very popular and it was also very important to the pop stars that they had the public in their pocket.
The country was in the midst of a great period of economic growth.
The government was in full swing.
There were a number of new television shows and movies.
The economy was growing and many people were earning more money than they had ever done in their lives.
But for all of the great music, the music industry had been losing money for a number, and some of it had even gone bust.
Elvis, who made a lot more money as a performer than as a singer, had been doing so well that he was able to pay the bills and even pay his own mortgage.
He was not happy with this situation, and he went to the media and said that he had a plan to change the way he was getting paid.
The plan was to take away all the money from the artists and replace it with a small sum of money.
This was a major change in the way the industry worked.
Elvis said that there was going to be no more money going to the artists. | https://inipainting.com/archives/219 |
Thursday, July 5, 2012
Time for UK to default?
Devaluing the Pound Isn’t a Solution, It’s Default
Official net U.K. debt excluding the effect of financial interventions such as bank bailouts is about 1 trillion pounds ($1.57 trillion), or 36,000 pounds per household. A recent analysis of U.K. pension accounts by Ros Altmann of Saga Group Ltd., an enterprise focusing on financial services for those aged over 50, estimated that the 5 trillion-pound to 7 trillion- pound cost of U.K. unfunded state pensions amounts to at least an additional 180,000 pounds per household. The U.K. government must either default or modify unfunded promises if it is to resolve those debts. Devaluing the pound would be one way to achieve that.
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Question #32146
Man17089984
Recent Question/Assignment
Deakin College
SIT105 – Critical Thinking and Problem Solving
Assignment 1 Trimester 2 2017
This assignment is to be completed individually. It is worth 15% of overall marks. It is due on or before 11.30pm Friday August 11.
Objectives:
There are two main goals for the assignment. The first is to demonstrate your understanding of some of the terms used in “Critical Thinking”. The second goal is to undertake research with correct referencing.
Tasks:
• Carefully read all the questions
• Provide answers to all four questions in a word processed document. Use the marking criteria for each question as a guide.
• Submit your assignment in the assignment submission tool available in Moodle (Week 6). Late submissions will be penalised (see SIT105 Unit Outline for further details re late submission).
(Note: The assignment is marked out of 30 marks but is worth 15 marks i.e. 15%)
Question 1 [10 marks]
Choose any two of the following terms. For each of your chosen terms find three sources that provide information about the term and create ONE (single) definition from those sources. Also provide an example, preferably from IT systems development. Each definition with example should be no more than 200 words. (Note: Do not use Wikipedia, blogs or your prescribed text as your source).
• Divide and Conquer
• Hypothesis Testing
• Brainstorming
• Reduction
• Research
• Trial and Error
The Harvard style of referencing is to be used. For more information refer to Deakin guide to referencing at http://www.deakin.edu.au/students/study-support/referencing
Marking Guide for Question 1
• Appropriate sources selected
• Sources correctly referenced and in Harvard style
• Paraphrasing done well
• Examples provide appropriate information for the chosen terms
Question 2 [10 Marks]
Read the adapted and heavily edited excerpt from the article below and identify all the statements, claims and arguments presented and provide an issue based on each identified claim.
You should answer this question using a table like the one shown below. Column 1 indicates whether you have identified a statement, claim, argument or issue. [Note: Only identify a statement as a statement if it is not a claim or argument].
Column 2 contains the text of that statement, claim or argument, and issue. Use the colour-coding as shown in the table and keep the order of the text in Column 2 the same as the actual text of the article.
Type Text
Claim She swiftly glued a canary yellow certificate on my windscreen
Issue Did she quickly glue the canary yellow certificate on my windscreen?
Statement How you going, mate?
Argument
Premise/Conclusion My car was on high-beam because there was something wrong with the dip-switch
Reliability
by Margaret Rouse September 2005
Reliability is an attribute of any computer-related device that consistently performs according to its specifications. It has long been considered one of three related attributes that must be considered when making, buying, or using a computer product or component. Reliability, availability, and serviceability - RAS, for short - are considered to be important aspects to design into any system. In theory, a reliable product is totally free of technical errors; in practice, however, vendors frequently express a product's reliability quotient as a percentage. Evolutionary products are usually considered to become increasingly reliable, since it is assumed that bugs’ have been eliminated in earlier releases. For example, IBM's z/OS (an operating system for their S/390 server series), has a reputation for reliability because it evolved from a long line of earlier MVS and OS/390 operating system versions.
Marking Guide for Question 2
• All statements correctly identified
• All claims correctly identified
• All arguments correctly identified
• Premise and conclusion correctly identified for each argument
• All issues listed
Question 3 (6 Marks)
Draw a full truth table to determine the validity of the following argument and provide reasons why the argument is valid or invalid:
(R v S)? ~P
P ? ~R
S ? (~S ?P)
----------------
R
Marking Guide for Question 3
• All variables identified correctly and the number of rows determined correctly
• All columns identified and included in the table
• All premises listed correctly
• T/F values listed correctly for preliminary columns
• T/F values computed correctly applying the right rule to each cell
• Correct conclusion
Question 4 (4 Marks)
Draw a short truth table to determine the validity of the following argument and clearly explain the steps and provide reasons why the argument is valid or invalid:
(L ? ~S) v ~W
P v ~L
~S
-----------------------
S ? (W ? P)
Marking Guide for Question 4
• Variables identified correctly
• Premises and conclusion listed correctly
• T/F values computed correctly applying the right rule to each cell
• Correct conclusion
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Mixed martial arts fighter Ronda Rousey is the most dominant female athlete alive, right?
Maybe not. A few other women have a legitimate claim to that title. And if Serena Williams can smash her way through the US Open to wrap up a Grand Slam – an actual, honest-to-goodness, calendar-year Grand Slam, the first since Steffi Graf did it in 1988 – then let’s change that answer to “no”.
Williams already holds all four major titles and is gunning for her fourth straight US Open.
A title at Flushing Meadows would be her fifth straight major (she already holds all four titles) and her fourth straight US Open. She will have won nine of the last 14 majors. The last 15 times she has been in a tournament final, she has walked away with the trophy. Her 2015 record: 48-2.
Rousey’s claim to the Most Dominant Female Athlete title is certainly strong – by one reckoning, she’s the most dominant athlete, period, male or female. She has a habit of quickly dispatching her opponents. Eight of her 12 professional fights have lasted less than a minute. Fellow Olympic medallist (for wrestling) Sara McMann made it to the 66-second mark and was the first fighter to lose by some manner other than armbar.
The quick fights put Rousey in a Catch-22 situation. We don’t know how she handles adversity because she simply hasn’t faced any since her judo career, in which she was a legit Olympic medallist but not an undefeated machine. In mixed martial arts, Rousey is so efficient that she’s rarely tested.
Williams is in less of a hurry. She sometimes seems to be using the first set as a warm-up. Then she turns it up to a level no one can touch.
And Williams, unlike Rousey, loses every once in a while. That’ll happen in a career of more than 850 matches. In the weekly grind of the World Tennis Association (WTA) tour, Williams sometimes isn’t at her best.
When she’s healthy and dialled in, though, she’s as unbeatable as Rousey. No one can handle her serve, which has topped 206km/h this year. No one can outslug her from the baseline.
Rousey has beaten seven of the top 10 in the current Ultimate Fighting Championship rankings. Williams has beaten the rest of the WTA top 10 many times over. Several players have beaten her once in several tries. Last week in Cincinnati, Williams improved her career record against Halep to 6-1 and against No 7 Ana Ivanovic to 9-1. She’s 10-1 against Caroline Wozniacki, 5-1 against Petra Kvitova. She has never lost to Lucie Safarova or Karolina Pliskova.
The only player in the top 10 with two wins against the world’s No 1 is Maria Sharapova. As of 2004, the Russian star had a 2-1 record against Williams. Since then? Williams has won 17 straight.
That dominance shows in the WTA rankings. Williams has 12?721 points, more than double those of No 2 Simona Halep (6?130). Her Romanian rival moved up to second by reaching the final in Cincinnati last weekend but fell to Williams’s onslaught of 15 aces and 83 winners.
Little wonder that ESPNW gave Williams the No1 1 seed in its online bracket for the Best Female Athlete Ever. Rousey is in the competition but isn’t seeded.
So if you’re judging Williams vs Rousey, the question is which you prefer – someone who competes a few times a year and ploughs through anyone placed in front of her, or someone who has been winning consistently since her teens and is having the best year of her career at age 33? | https://mg.co.za/article/2015-09-03-serena-williams-wins-by-a-knockout |
Fried egg rolls stuffed with minced pork, crab, onion, egg and lots of other goodies. Served with our special sauce.
Crispy deep fried chicken and vegetable dumpling served with our special sauce.
Fried rice with delicious house sauce.
Special Thai thin noodles pan-fried with egg, peanuts and garnished with fresh bean sprouts.
Choice of protein in coconut milk with Thai eggplant, bamboo shoots, mixed vegetables, bell pepper, green bean and sweet basil leaves, flavored with chili and red curry paste.
There’s still more delicious dishes to discover.
Whether you're craving a quick bite or looking to treat yourself to a delicious meal, you can now order your Simply Thai Bistro favorites for takeout or delivery. | http://www.simplythaibistrowheaton.net/ |
Midnight Jasmine is part of the Scents of the Silk Road series, a range of candles and diffusers inspired by the scents, stories and designs of the ancient Silk Road.
This jar candle is made from 100% beeswax and finished with four cotton wicks. It is presented in a no-plastic gift box.
The artwork for Midnight Jasmine was created by Hong Kong artist Charlie Wong for Carroll&Chan.
Burns for approximately 55 hours.
Made by hand in Hong Kong. | https://carrollandchan.com/shop/beeswax-candles/large-jar-candles/midnight-jasmine-4-wick-large-beeswax-jar-candle/ |
In 1902, Albert Einstein gifted a book, Karl Pearson’s The Grammar of Science, to his colleagues to start a conversation about the universe. Expanding on that conversation, we invite a variety of experts to share the stories behind landmark advancements and discoveries in the fields of science and technology. Recorded in front of a live audience at the 1888 Center, this educational program is designed as a series of brief explorations into our natural world and the human ability to manipulate it. In partnership with Ingram Micro and Chapman University.
The 1888 Podcast Network is a curated collection of educational and entertaining podcasts. Each program is designed to provide a unique platform for industry innovators to share stories about art, literature, music, history, science, or technology.
Subscribe to our monthly newsletter to receive information on upcoming books, podcasts, programs, and other news related to 1888. You may also connect with us via Facebook, Instagram, and Twitter. For partnership requests, please submit your inquiry using our contact form. | http://1888.center/the-grammar-of-science-and-technology/ |
Art is an interdisciplinary subject and various kinds of subjects are included in arts. Few of the subjects are as follows:
- Philosophy-It is the study of general and fundamental problems concerning matters such knowledge, existence, values, Reason, mind and language. This subject answers different questions regarding nature and human being. Different sub-divisions of Philosophy are as follows-
- Epistemology- Studying knowledge
- Metaphysics- Study of basic questions regarding human life
- Ethics- Studying the perfect way to live
- Aesthetics- Study of nature and its beauty
- Reasoning and logic
Various new subjects are now studied which are considered to be part of arts group. Such as film studies, Queer studies, women’s studies and cultural studies.
- History: Thestudy of past as it is described in written documents. Events occurring before written record are known as pre-history
- Political Science: It is a social science also known as government, which deals with the system of governance, and the analysis of political activities, political thoughts and political behaviour.
- Economics: Social science concerned chiefly with description and analysis of the production, distribution and consumption of goods and services. Economics focuses on the behaviour and interactions of the economic agents.
- Sociology and Anthropology- Sociology is the scientific study of the society, including patters of relationships, social interaction and culture. Anthropology is the study of various aspects of humans within past and present societies.
- Languages- A student of arts need to learn a few languages of the country and all other language for academic and further life
The requirement for Arts Assignment Assistance
As arts are an interdisciplinary subject, hence students need to study a combination of various subjects. Hence, they need to gather complete knowledge of each of the subjects concerned. It becomes difficult for them while writing assignments in each of the subjects. Hence, they need the assistance from Arts Assignment Help service that provides support with various kinds of subject of arts
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Assistance from Bestassignmentexperts.com
The usefulness and benefits of our service is as follows:
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- Best quality and plagiarism free assignments. | https://www.bestassignmentexperts.com/arts-assignment-help-services-online |
In 1863 Charles Darwin published an article enthusing about a new concept in natural history: protective mimicry. The term had been coined by Henry Walter Bates, an entomologist who had observed numerous uncanny resemblances between different species of insect in the Amazon. Bates argued that such mimicry was a survival strategy; insects vulnerable to predators evolved to resemble other species that predators knew to be distasteful. With each generation, the individuals that more closely resembled the model species were more likely to be left alone, while those who deviated from this model were more likely to be eaten, rendering the mimicry ever-more perfect. Bates claimed that other species' uncanny resemblances to vegetation or stones had developed in the same way, as random variations helped certain individuals to survive while their more conspicuous brethren died out (see Bates 1862). In the following years, naturalists would argue for the existence of many other forms of biological mimicry, from camouflaged predators to plants that tricked insects into spreading their pollen by mimicking the appearance of nectar.1 Biological mimicry confounded mechanistic models of animal life. Earlier naturalists had noticed uncanny resemblances between different species, and between organisms and other natural objects, but often explained them away as either unimportant coincidences or proofs of the creator's love of symmetry and patterns (Komarek 1998, 24–28). Such explanations framed the animal world as senseless, acquiring meaning only through the intelligent perceptions of human observers. Conversely, biological mimicry suggested [End Page 63] that nonhuman perceptions and interpretations were fundamental to nature's processes. Darwin queried rhetorically, "Why to the perplexity of naturalists has Nature condescended to the tricks of the stage?" (1863, 220–21). Part of the reason for naturalists' "perplexity" had been that they approached animals as rigid machines rather than sentient beings that inhabited not only physical environments but also fields of mutual perception and semiosis. | https://muse.jhu.edu/article/655413 |
The concept of the actor as a scenographic instrument is often associated with the new perceptions and expressionism that stems out of the modern and contemporary theatre, probably because it represents such an antithetical approach to the naturalistic perspective of 19th century theatre.
In this context, perhaps the individual who has helped best in defining and proposing this new concept was Robert Wilson. His complex combination of language, movement or lighting into a unitary support framework for the artist was a cornerstone of what the concept stands for. This paper aims both to examine the concept itself and present Robert Wilson's vision on it.
The first notable element about the new concept pertaining to the role and expression of the artist is that it comes to oppose the "naturalistic concept of the actor as imitator of human behavior"
. The emphasis previously placed on physical characteristics and elements, deriving from the need not only to relate best to the character being played and the only tool used to denote expression on stage, is now complemented successfully by other elements.
The interest is no longer solely towards reproducing the character played, but moves more towards placing the actor is a more collective reality and, thus, amplifying his capacity of expression through additional means. The belief of the concept of the actor as a scenographic instrument is that placing him in this larger framework will maximize his representation on stage.
Following this introduction to the concept, one needs to point out that this new perspective on the artist goes hand in hand with the idea of symbolism. In its simplest form and explanation, the artist no longer bases his performance solely on his own person, but uses auxiliary elements to move the representation to a different integrative level. In order to do that, he uses symbols and allusions, innuendos and objects on the stage, all in a symbolist framework. This is why the new concept for the artist transforms his representation from a linear one to a complex, more detailed performance.
Ultimately, the difference between a naturalistic artistic performance and the concept of the actor as a scenographic instrument is also the difference between two different perceptions of reality. One is the realist, naturalist perception, which is meant to capture in detail the elements of reality, emphasizing the way these are reflected in physical or psychological reflections. The other is a symbolical interpretation, according to which there does not need to be a naturalistic connection, as long as there is another element that can offer that connection on a symbolist level, based on individual perception.
Because of the need for a higher integration of different elements (in 1915, Craig mentioned no less than eight such compositional elements, including movement, light, painted faces and facial expressions
) into a unitary stage representation, the role of the director increases as compared to the naturalistic approach, as he takes over the task of putting together all the elements, place them alongside the actor's performance and ensure that the synergetic effect of this action is a positive one. If one complements the role of the actor with props, music, stage effects or dance, one needs to ensure that all these come together not necessarily in a logical way, but a way in which the audience receive the appropriate message that the artist and the director wish to pass along.
On these general considerations comes the work of director and playwright Robert Wilson, considered by many one of the leading theater artists of the 20th century. The subsequent part of this paper will refer to how some of the elements that have been discussed previously are used by Wilson to create the concept of the artist as a scenographic instrument.
The best place to start is with the artist's movement or, as Wilson occasionally puts it, with the lack of artist's speech or language. The issue at hand here is that, contrary to the naturalist concept for artistic expression, as previously shown, other elements play a similarly important, if not more important, expressionistic role. In this case, it is the movement of the artist on stage that can express feelings, denote sensations or states of mind or play. As Holmberg points out referring to Wilson's concept, "they don't understand the weight of a gesture in space. A good actor can command an audience by moving one finger"
Movement can occasionally be associated with language in Wilson's interpretation or, as it often happens, with the lack of language. The message that Wilson passes on is that language by itself is useless in providing a full expressionist value to the work of theatre. Words need to be placed alongside other props that the actor can provide, not in the least his movement on stage. Movement and language will eventually come together as a unitary expression of the artistic credo.
It is important to note that reference to movement and language includes reference to the lack of movement or language, to which Wilson adheres repeatedly in his work. There is no dichotomy here between the concepts. The explanation is that Wilson perceives language and movement in their entirety, with different particular values, such as the existence of such actions or, if the case requires it, the absence. In both cases, he is basing the artistic act on the capacity of expression that these additional elements bring to the stage.
The importance of the combination between language and movement in Wilson's conception of the actor as a scenographic instrument is shown in the way he perceives Marlene Dietrich's comments to someone who has mentioned her cold way of acting. Her reply shows that "you didn't listen to my voice" and "And that was so true. The voice could be very hot and erotic, while her movements could be icy-cold. She turned to me and said, "The difficulty is to place the voice with the face"
Wilson underlies here the importance he gives not to the language and movement as separate elements of his concept of the actor, but rather as their role together, the synergy the two can create. An actress like Marlene Dietrich understands that the two elements can work very well together even if they seem to go into different directions at some point of the actor's play.
Wilson's belief that movement is the central element in his conception of the artist as a scenographic instrument can be seen from the way he perceives movement in his interpretation of time and space. According to Wilson, "time is a line that goes to the centre of the earth and goes to the heavens. Time and space make the basic architecture of everything"
. There are several important things that can be understood from this statement.
First of all, in Wilson's artistic conception, the actor is only part of an environment entirely founded on temporal and spatial coordinates. His own interpretation of the role evolves in this environment and this is not something from which he can derive. Second, because of the importance of time and space, the way the actor moves on the two coordinates is essential to his success in the role. This is way, as previously pointed out, movement does not necessarily always means a dynamic action, but, occasionally, simply standing (temporal coordinate) in a certain part of the stage (spatial coordinate). By standing still, the actor still acts in the given framework as part of the scenography conceived by the director.
Another important element in Wilson's work is the lighting in stage. At first glance, the lighting is able to bring forth a certain element on stage and, thus, emphasizing a particular part of the artistic act, depending on what the director wants. Quite often in Wilson's work, lighting takes on a different dimension as well, other than that of an element in the overall concept of the actor as a scenographic instrument. The lighting underlines and emphasizes the work of the actor, but, at the same time, it has its own scenario to play and, occasionally, it has a life of its own rather than an on-off effect on the stage. The key characteristics for lighting in Wilson's work is that it is continuous and, from that perspective, it has a continuous role in his artistic creation.
Finally, there are the props. As the other elements mentioned, the props complete the evolution of the actor on stage to transform it completely from a naturalistic approach to one that is fully integrated and entirely expressionistic. The relationship between all these elements is obvious in what Wilson says on his choice between wooden and aluminum chairs: "I want wood chairs. If we make them out of aluminum, they won't sound right when they fall over and hit the floor. They'll sound like metal, not wood. It will sound false. Just make sure you get strong wood. And no knots"
Here one better understands how… [END OF PREVIEW]
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ABNL Limited was incorporated in 1992, as an Oil and Gas servicing company, with a vision to provide professional engineering and technical services to the Oil and Gas exploration and production companies in Nigeria and the West African sub-region. Since its incorporation the company has grown in leaps and bound, expanding its business horizon but still within the Oil and Gas sector.
ABNL is a 100% owned indigenous firm, with a shareholding in excess of 100 million Naira. Its’ directors are individuals of proven integrity, who over the years have acquired the requisite experience in the Oil and Gas business, as well as in the banking and finance sector. The integrity of the directors and the staff of the company have been a priceless contributive factor in the growth and wellbeing of the company.
We are recruiting to fill the position below:
Job Title: Contracts Lead II
Location: Lagos
Employment Type: Contract
Main Functions
- Manages portfolio of agreements and new requests, enabling efficient and effective purchase of goods and services (from purchase to pay) with a focus on business value, on-time delivery, and cycle time
- Works with moderate work direction and is skilled and knowledgeable to the position.
- This position could be described as Senior Procurement Associate / Specialist / Contracts Manager who, on top of level 1 duties, might be managing contracts with high complexity or deep business knowledge.
Tasks and Responsibilities
- Negotiates contract pricing and terms & conditions directly with suppliers while working with internal stakeholders, seeking lowest total system costs and appropriate mitigation of supply and legal risk.
- Executes PtP (Procure-to-Pay) processes in full compliance with Sourcing & Acquisition Handbook and CIMS (Controls Integrity Management System) catalogs.
- Fully leverages Procurement processes and tools to ensure most effective procurement method is utilized.
- Maintains agreement portfolio: scope updates, pricing, Exhibits, amendments.
- Ensures contract compliance and utilization – monitors supplier performance, troubleshoots issues, etc.
- Ensures transactional efficiency of agreements by leveraging systems.
- Identifies business value and other opportunities within the portfolio.
- Develops and maintains internal and external relationships to meet business line expectations.
- Provides fit-for-risk process improvements.
- Influences development and implements Category strategic and commercial guidance; shares portfolio specific market intelligence to Category Networks.
Skills and Qualifications
- Bachelor’s / Master’s Degree with a minimum of 8 years experience.
- Behavioral Skills: Analytical, applies learning, communicates effectively, creates business value & competes to win in marketplace, makes sound decisions.
- Functional Skills: PtP knowledge, system utilization & efficiency, total system cost, apply controls mindset, application of business & procurement technical knowledge, service excellence, contract development, contract management, negotiating and influencing, relationship management.
Application Closing Date
3rd September, 2022.
Method of Application
Interested and qualified candidates should send their CV / Resume to: [email protected] using the Job Title as the subject of the email. | https://vacancies.website/contracts-lead-ii-at-abnl-limited/ |
Please refer to the specific study period for contact information.
Overview
|Year of offer||2019|
|Subject level||Undergraduate Level 1|
|Subject code||GEND10001|
|Campus|
Parkville
|Availability|
Semester 2
|Fees||Subject EFTSL, Level, Discipline & Census Date|
The world is gendered - but what is gender? We know gender is fundamental to the way in which we see ourselves and others, and how our communities and institutions are organised, but why? Why do gender norms and stereotypes emerge? What effects do they have on our lives? Drawing on feminist and queer theory, transgender studies, masculinity studies, and a range of disciplines across humanities and social sciences, this subject introduces students to the major concepts in gender studies, including: biological determinism, cultural essentialism, social constructionism, power and inequalities, sexuality, and queering categories of difference. Using a variety of case studies from social media, politics, sport, fashion, film, and music, the course will analyse how sex, gender, age, ethnicity, race, class, politics and social movements intersect to influence our understanding of sex, gender, and culture.
Intended learning outcomes
On completion of the subject students should:
- Demonstrate an introductory knowledge and understanding of contemporary gender theories;
- Have developed a foundational appreciation of the significance of gender in contemporary culture;
- Have laid the foundations of relevant research skills including use of the library, e-research skills, and appropriate referencing and presentation of written work;
- Appreciate national and international debates on specific contemporary issues and complex problems connected with sex and gender in contemporary societies;
- Grasp the importance of an independent approach to knowledge that uses rigorous methods of inquiry and appropriate theories and methodologies that are applied with intellectual honesty and a respect for ethical values;
- Have developed a foundation of relevant knowledge and methodologies, both critical and theoretical, on which to base further studies in Gender Studies.
Generic skills
On completion of this subjects students will :
• be skilled in critical thinking and analysis;
• cultivate written communication skills;
• develop an understanding of cultural and social contexts;
• be skilled at managing time and resources effectively.
Eligibility and requirements
Prerequisites
None
Corequisites
None
Non-allowed subjects
|Code||Name||Teaching period||Credit Points|
|GEND10001||Sex, Gender and Culture: An Introduction||
|
Semester 2
|12.5|
Core participation requirements
The University of Melbourne is committed to providing students with reasonable adjustments to assessment and participation under the Disability Standards for Education (2005), and the Assessment and Results Policy (MPF1326). Students are expected to meet the core participation requirements for their course. These can be viewed under Entry and Participation Requirements for the course outlines in the Handbook.
Further details on how to seek academic adjustments can be found on the Student Equity and Disability Support website: http://services.unimelb.edu.au/student-equity/home
Assessment
Description
- A reflective Journal, equivalent to 1500, due throughout the semester (40%)
- A 1500 word research essay due in week 6 (35%)
- A 1000 word take home exam due in the examination period (25%)
Hurdle requirement: Class attendance is required for this subject; if you do not attend a minimum of 80% of classes without an approved exemption you will not be eligible for a pass in this subject.
Note: Assessment submitted late without an approved extension will be penalised at 10% per day. In-class tasks missed without approval will not be marked. All pieces of written work must be submitted to pass this subject.
Dates & times
- Semester 2
Principal coordinator Hannah Mccann Mode of delivery On Campus — Parkville Contact hours A total of 36 hours: A 90 minute lecture and a 90 minute tutorial a per week. Total time commitment 170 hours Teaching period 29 July 2019 to 27 October 2019 Last self-enrol date 9 August 2019 Census date 31 August 2019 Last date to withdraw without fail 27 September 2019 Assessment period ends 22 November 2019
Time commitment details
Time commitment totals 170 hours.
Further information
- Texts
Prescribed texts
A Subject Reader will be available.
- Subject notes
This subject is compulsory for students undertaking the major or minor in Gender Studies.
- Related Handbook entries
This subject contributes to the following:
Type Name Major Gender Studies
- Breadth options
This subject is available as breadth in the following courses:
- Available through the Community Access Program
About the Community Access Program (CAP)
This subject is available through the Community Access Program (also called Single Subject Studies) which allows you to enrol in single subjects offered by the University of Melbourne, without the commitment required to complete a whole degree.
Entry requirements including prerequisites may apply. Please refer to the CAP applications page for further information.
- Available to Study Abroad and/or Study Exchange Students
This subject is available to students studying at the University from eligible overseas institutions on exchange and study abroad. Students are required to satisfy any listed requirements, such as pre- and co-requisites, for enrolment in the subject. | https://handbook.unimelb.edu.au/2019/subjects/gend10001/print |
TECHNICAL FIELD OF THE INVENTION
BACKGROUND OF THE INVENTION
DETAILED DESCRIPTION
The present invention relates generally to microelectromechanical systems (MEMS) devices. More specifically, the present invention relates to a MEMS device with an impacting mass structure for enhanced resistance to stiction and damage from mechanical shock.
Microelectromechanical Systems (MEMS) devices are widely used in applications such as automotive, inertial guidance systems, household appliances, protection systems for a variety of devices, and many other industrial, scientific, and engineering systems. Such MEMS devices are used to sense a physical condition such as acceleration, pressure, or temperature, and to provide an electrical signal representative of the sensed physical condition.
Suspended movable microstructures such as plates and beams are commonly used in the manufacturing of various microelectromechanical systems (MEMS) MEMS devices. These suspended movable microstructures can be adversely affected during normal use by excessive external forces, such as mechanical shock. A mechanical or physical shock is a sudden acceleration or deceleration caused, for example, by impact, drop, kick, and so forth. This mechanical shock can cause severe reliability problems in the structure of a MEMS device.
The suspended microstructures of MEMS devices typically have relatively large surface areas with high stiffness. However, the suspension springs for such suspended microstructures may have relatively low stiffness depending upon the application. For example, some accelerometers are designed to include highly compliant (i.e., low stiffness) suspension springs so that the suspended microstructures will move a detectable amount under conditions of 1 g magnitude or less. In addition, the microstructures are fabricated a few microns off their supporting substrate. The combination of these characteristics makes MEMS devices susceptible to surface forces which can deflect the suspended movable microstructures vertically toward vertical motion stops and/or the supporting substrate. Additionally or alternatively, the suspended movable microstructures can deflect laterally toward surrounding structures or lateral motion stops. If the deflection force is sufficiently strong, the movable member can come into contact with and temporarily or permanently adhere to the underlying substrate or the lateral structures causing false output signals and/or device failure. This unintentional adhesion of a movable structure is referred to as stiction. Stiction can occur both during MEMS device fabrication and during normal use.
Embodiments disclosed herein entail microelectromechanical (MEMS) devices with enhanced resistance to stiction and damage when subjected to mechanical shock. In particular, embodiments entail a secondary structure extending from a suspended movable element. The secondary structure includes a spring element adapted for movement so that a secondary mass of the secondary structure will impact the movable element when the MEMS device is subjected to mechanical shock. The term “secondary structure” used herein refers to a projecting member coupled to the movable element. In addition, the term “impact” used herein refers to movement of the secondary structure relative to the suspended movable element such that the secondary structure forcefully strikes the movable element.
In general, when a large enough force (e.g., mechanical shock) is applied to the movable element, the movable element will move until it comes into contact with appropriately placed motion stops thus halting movement of the movable element. In such an event, a stiction event is possible in which the movable element adheres to the motion stops.
In accordance with embodiments described herein, the additional force beyond what is needed to cause a stiction event is used to push the secondary structure into a state ready for recoil and impact with the movable element in a direction that is likely to dislodge a potentially stuck movable element when the movable element is struck by the secondary. Furthermore, some of the energy from the mechanical shock may be absorbed by the spring element in order to limit or prevent breakage to internal structures of the MEMS device. Thus, a MEMS device that includes the secondary structure may be less likely to fail when subjected to mechanical shock, thereby enhancing long term device reliability.
FIGS. 1 and 2
FIG. 1
FIG. 2
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Referring to , shows a top view of a MEMS device in accordance with an embodiment, and shows a side view of MEMS device . In this example, MEMS device is a two layer capacitive transducer having a “teeter-totter” or “see saw” configuration. MEMS device includes a movable element or plate, referred to as a proof mass , suspended above a substrate . In an embodiment, proof mass may be flexibly suspended above substrate by one or more spring members, for example, rotational flexures situated at elevated attachment points via an anchor coupled to the underlying substrate .
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FIG. 2
Rotational flexures enable rotation of proof mass about a rotational axis under z-axis acceleration, represented by an arrow , relative to substrate . The accelerometer structure of MEMS device can measure two distinct capacitances between proof mass and two sense plates and that are symmetrically located relative to rotational axis in order to determine differential or relative capacitance. The side view of MEMS device in represents a condition in which proof mass is in a neutral position, i.e., an initial position or a position that proof mass returns to when it is not being subjected to a measurable z-axis acceleration .
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Although only a single anchor and a pair of rotational flexures is shown, those skilled in the art will recognize that proof mass may be flexibly suspended above substrate by a different anchor and spring configuration than that which is shown. Additionally, the anchor and spring configuration need not be physically located on rotational axis . Instead, an alternative anchor and spring configuration may include multiple anchors and springs that are appropriately spaced to form a virtual rotational axis between pairs of spring members. Furthermore, although the embodiments discussed herein pertain to accelerometer structures, it should be understood that the following discussion applies equivalently to other MEMS devices having movable parts that could be damaged when subjected to a mechanical shock.
FIGS. 1 and 2
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Certain features within are illustrated using various shading and/or hatching to distinguish different elements of MEMS device . For example, substrate is illustrated using light shading so that it is distinguishable through apertures or openings extending through proof mass . The different elements within MEMS device may be produced utilizing current and upcoming surface micromachining techniques of deposition, patterning, etching, and so forth. Accordingly, although different shading and/or hatching may be utilized in the illustrations, the different elements within MEMS device may be formed out of the same material, such as polysilicon, single crystal silicon, and the like.
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The elements of MEMS device (discussed below) may be described variously as being “attached to,” “attached with,” “coupled to,” “fixed to,” or “interconnected with,” other elements of MEMS device . However, it should be understood that the terms refer to the direct or indirect physical connections of particular elements of MEMS device that occur during their formation through patterning and etching processes of MEMS fabrication.
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Proof mass includes a first section between rotational axis and a first end of proof mass , and a second section between rotational axis and a second end of proof mass . First section exhibits a first length between rotational axis and first end . Likewise, second section exhibits a second length between rotational axis and second end . In an embodiment, second length is greater than first length . Thus, rotation can occur about rotational axis in response to z-axis acceleration because the weight of second section is greater than the weight of first section . Thus, the second section resembles the “heavy end” of a “teeter-totter” or “see saw” proof mass configuration.
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The terms “first” and “second” used herein do not refer to an ordering or prioritization of elements within a countable series of elements. Rather, the terms “first,” “second,” and so forth are used herein to distinguish similar or related elements, such as the sections and of proof mass , the ends and of proof mass , the lengths and of proof mass , and so forth for clarity of discussion.
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FIG. 2
MEMS device may include a number of vertical motion stops . In general, each vertical stop includes a post unit coupled to substrate and passing through an opening (best seen in ) extending through in proof mass . In the illustrated embodiment, MEMS device includes one of vertical stops having post unit directed through first section of proof mass and one of vertical stops having post unit directed through second section of proof mass . Alternative embodiments may include more or less than the two vertical stops shown. Each vertical stop includes a cap having a greater diameter than the diameter of opening .
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FIGS. 1 and 2
FIG. 2
FIGS. 3-6
A secondary structure is coupled to proof mass and is suspended above substrate . In the illustrated embodiment, proof mass includes an aperture extending through a thickness of proof mass and defined by inner side walls . Secondary structure resides in aperture and is spaced apart from inner side walls of proof mass by a gap . Although secondary structure is visible in both of , secondary structure is expressly represented in (as well as in ) by rightwardly and downwardly directed wide hatching.
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FIG. 2
Secondary structure includes a spring element and a secondary mass . Spring element is positioned in aperture , and a first end of spring element is coupled to, i.e. formed to extend from, one of inner side walls of proof mass . Secondary mass includes a stem (visible in ) and a cap . Stem is coupled, i.e. formed to extend from, to a second end of spring element and resides in aperture . Stem is oriented approximately perpendicular to a surface of proof mass . Cap is coupled to stem and resides above surface of proof mass such that a portion of cap overlies surface to yield a gap between surface of proof mass and cap .
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As will be discussed in greater detail below, the spring members, i.e., rotational flexures , that suspend proof mass above substrate , are configured to undergo a first deflection amount in response to a unit of force, e.g., z-axis acceleration , so that proof mass rotates about axis of rotation in response to z-axis acceleration. Additionally, spring is configured to undergo a second deflection amount in response to the same unit of force, e.g., z-axis acceleration . In an embodiment, the second deflection amount of spring is less than the first deflection amount of rotational flexures .
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In an embodiment, the magnitude/amount of deflection in response to a unit of force that each of spring and rotational flexures are capable of can be at least partially established by the spring stiffness of each of rotational flexures and spring . For example, spring may be configured to have a spring stiffness that is greater than a collective spring stiffness of the spring members, i.e., rotational flexures , that suspend proof mass above substrate . In an exemplary embodiment, the spring stiffness of spring may be at least one order of magnitude (i.e., about ten times) greater than a collective spring stiffness of rotational flexures . The stiffness of a spring is generally a measure of its resistance to deformation. Thus, a stiffer spring requires greater force to deform it than a spring that is more compliant, i.e., less stiff.
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It should be understood that the spring stiffness needed for spring in an embodiment is additionally related to the mass (i.e., weight) of stem and cap of secondary mass relative to the mass (i.e., weight of proof mass ). In other words, the spring stiffness needed for spring is related to how much mass (i.e. weight) spring is supporting. It is the combination of the spring stiffness and the weight of a particular mass that rotational flexures or spring is supporting that determines how much the spring will deflect, i.e., a deflection amount, in response to a unit of force.
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In an embodiment, the collective spring stiffness of rotational flexures in concert with the weight of proof mass , enables flexures to twist in response to z-axis acceleration within a particular sensing range, for example, between 0 and 8 g's, so that proof mass rotates about rotational axis at accelerations within the sensing range.
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FIGS. 3-6
However, due to its greater stiffness in concert with the weight of secondary mass , spring is prevented from deflecting appreciably, i.e., twisting or bending, within the particular sensing range of MEMS device . Instead, spring must be subjected to a significantly greater force, e.g., mechanical shock, at levels of hundreds or thousands of g's before it will deflect appreciably. The operation of MEMS device , and particularly with respect to secondary structure , will be described in connection with the ensuing discussion of .
FIG. 3
FIGS. 3-6
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shows a side view of the MEMS device subjected to a force, i.e., z-axis acceleration , greater than a predetermined sensing range for MEMS device . When MEMS device is subjected to z-axis acceleration , proof mass will rotate about rotational axis . In the example presented in , proof mass rotates such that second section of proof mass moves upwardly, while first section of proof mass moves downwardly. It should be readily apparent however, that z-axis acceleration may be directed opposite to that which is shown. Accordingly, proof mass would rotate in the opposite direction.
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FIG. 2
The predetermined sensing range for a MEMS device, such as MEMS device , is typically much less than the motion range for proof mass . Thus, under conditions of z-axis acceleration that are greater than the predetermined sensing range, surface of proof mass may momentarily contact a bottom surface of one or more caps of vertical motion stops as proof mass rotates about rotational axis . Proof mass will subsequently return to its neutral position, shown in , following application of z-axis acceleration . Vertical stops can limit movement of proof mass under greater than normal z-axis acceleration so that proof mass is less likely to become damaged. However, spring of secondary structure is prevented from deflecting appreciably, i.e., twisting or bending, when z-axis acceleration is within or near the predetermined sensing range for MEMS device . Accordingly, secondary structure that includes secondary mass having cap and spring does not change or affect the normal operation of MEMS device under relatively low z-axis acceleration .
FIG. 4
FIG. 2
FIG. 4
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shows a side view of MEMS device subjected to a mechanical shock , represented by a heavy arrow. Mechanical shock is a z-axis acceleration () that is considerably greater than the sensing range of MEMS device . For example, mechanical shock may be orders of magnitude greater than the sensing range of MEMS device . When subjected to mechanical shock , proof mass will again rotate about rotational axis until surface of proof mass forcefully collides with bottom surface of one or more caps of vertical motion stops . In the example presented in , the collision of proof mass against bottom surface of one of caps is represented by a star element . That is, star element is not a physical component of MEMS device , but rather star element is used to represent a forceful contact or collision of proof mass with motion stops that could potentially cause physical damage to MEMS device .
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FIG. 4
Mechanical shock applied to proof mass is strong enough to use up the range of motion of proof mass such that proof mass stops against bottom surface of one or more caps . When a conventional MEMS device that does not include secondary structure is subjected to, for example, mechanical shock , any additional force from mechanical shock placed against caps from proof mass can result in a stiction event in which proof mass is temporarily or permanently adhered to caps , or alternatively, caps may break or shear off. With the inclusion of secondary structure in MEMS device , the additional force from mechanical shock beyond what is needed to push proof mass against caps of vertical motion stops is instead used to push secondary mass into a state ready for recoil. That is, as shown in , spring flexes, twists, or bends, as represented by a clockwise curved arrow such that cap springs upwardly away from surface of proof mass .
FIG. 5
FIG. 4
FIG. 5
FIG. 4
FIG. 4
FIG. 5
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shows a side view of MEMS device in which secondary structure rebounds in response to mechanical shock () and the resulting collision of proof mass with cap of vertical motion stop . The condition illustrated in approximately instantaneously follows the condition illustrated in and is a direct response to the condition illustrated in . As represented in , spring has used up its range of motion and springs back, i.e., rebounds, in the opposite direction relative to its initial movement as represented by a counterclockwise curved arrow .
FIG. 6
FIG. 6
FIG. 5
FIG. 2
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shows a side view of MEMS device in which cap of secondary structure impacts the movable element, i.e., proof mass , of MEMS device . The condition illustrated in approximately instantaneously follows the condition illustrated in . The rebound effect of spring results in cap impacting surface of proof mass with a force, represented by an arrow , in a direction that is likely to dislodge or move the potentially stuck proof mass when it is struck by cap of secondary structure . This direction may be opposite to the direction of mechanical shock . Thus, immediately following the impact of cap against surface , proof mass can return to its neutral position illustrated in .
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FIG. 4
Accordingly, secondary mass of secondary structure is adapted to impact proof mass in response to the motion of proof mass when proof mass is subjected to a force, e.g., mechanical shock (), that is considerably greater than the sensing range of MEMS device . Should MEMS device sustain high z-axis acceleration, e.g., mechanical shock , the impact of secondary mass against proof mass will make sustained stiction less likely.
FIG. 7
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shows a top view of a MEMS device in accordance with another embodiment. Previous discussion was directed to teeter-totter style sensors for detection of a physical condition perpendicular to a plane of the sensor. However, principles of the present invention need not be limited to z-axis sensors. Instead, the present invention can be readily adapted for sensors that detect a physical condition in a direction parallel to a plane of the sensor.
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MEMS device includes a movable element, in the form of a proof mass , suspended above a substrate by one or more spring members . In an embodiment, spring members are interconnected between proof mass and substrate , and enable substantially linear motion of proof mass relative to substrate . In this example, proof mass with spring members can move substantially parallel to substrate in response to an x-axis acceleration, as represented by an arrow .
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Proof mass includes movable fingers extending from a body of proof mass . Each movable finger is located between a pair of fixed fingers that are fixed, or stationary, relative to substrate . The accelerometer structure of MEMS device can measure distinct capacitances between each movable finger and its corresponding pair of fixed fingers . These capacitances are a measure of an external physical condition such as x-axis acceleration . Although only two movable fingers and two pairs of fixed fingers are shown, those skilled in the art will readily recognize that MEMS device can include any number of movable fingers and fixed fingers in accordance with particular design criteria.
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FIG. 1
MEMS device may include a motion stop structure embodied as a fixed frame coupled to and extending above substrate that at least partially encircles proof mass . In addition, or alternatively, MEMS device may include lateral motion stops (not shown) that extend through apertures in proof mass . The lateral motion stops may be similar in design to vertical motion stops (), discussed above. Fixed frame and/or lateral motion stops function to limit excessive movement of proof mass in response to x-axis acceleration that is greater than the sensing range for MEMS device .
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A secondary structure extends from proof mass and is suspended above substrate . Secondary structure includes a secondary mass and a spring element interconnected between proof mass and secondary mass . Thus, secondary structure is a projecting member that is supported only at one end, i.e., the interconnection of spring element with proof mass . In an embodiment, spring element may be a compression spring that exhibits a stiffness that is greater than a combined stiffness of spring members . In alternative embodiments, secondary structure may include more than one spring element interconnected between proof mass and secondary mass that collectively exhibit a stiffness that is greater than a combined stiffness of spring members .
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FIGS. 8 and 9
In an embodiment, the collective spring stiffness of spring members , in concert with the weight of proof mass , enables spring members to deflect in response to x-axis acceleration within a particular sensing range, for example, between 0 and 8 g's, so that proof mass moves laterally, i.e. parallel to the surface of substrate , and movable fingers change position with respect to fixed fingers . However, due to its greater stiffness in concert with the weight of secondary mass , compression spring is prevented from deflecting appreciably, i.e., compressing, within the particular sensing range of MEMS device . Accordingly, secondary structure that includes secondary mass and compression spring does not change or affect the normal operation of MEMS device under relatively low x-axis acceleration . Instead, compression spring must be subjected to a significantly greater force, e.g., a mechanical shock, at levels of hundreds or thousands of g's before it will deflect appreciably. The operation of MEMS device , and particularly with respect to secondary structure , will be described in connection with the ensuing discussion of .
FIG. 8
FIG. 7
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shows a partial top view of MEMS device subjected to a mechanical shock , represented by a heavy arrow. Mechanical shock is an x-axis acceleration () that is considerably greater than the sensing range of MEMS device . For example, mechanical shock may be orders of magnitude greater than the sensing range of
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MEMS device . When subjected to mechanical shock , proof mass will move substantially parallel to the surface of substrate until proof mass forcefully collides with a portion of fixed frame . Mechanical shock applied to proof mass is strong enough to use up the range of motion of proof mass such that proof mass stops against fixed frame . In the example presented in , the collision of proof mass against fixed frame is represented by star elements . That is, star elements are not physical components of MEMS device , but rather star elements are used to represent a forceful contact or collision of proof mass with fixed frame that could potentially cause physical damage to MEMS device .
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FIG. 8
When a conventional MEMS device that does not include secondary structure is subjected to, for example, mechanical shock , any additional force from mechanical shock placed against fixed frame from proof mass can result in a stiction event in which proof mass is temporarily or permanently adhered to fixed frame , or alternatively, fixed frame and/or proof mass may break or be otherwise damaged. With the inclusion of secondary structure in MEMS device , the additional force from mechanical shock beyond what is needed to push proof mass against fixed frame is instead used to push secondary mass into a state ready for recoil. That is, as shown in , compression spring stretches, as represented by a rightward pointing arrow , such that secondary mass springs laterally outwardly away from proof mass .
FIG. 9
FIG. 9
FIG. 8
FIG. 9
FIG. 7
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shows a partial top view of MEMS device in which secondary mass of secondary structure impacts proof mass . The condition illustrated in approximately instantaneously follows the condition illustrated in . As represented in , compression spring has used up its range of motion and springs back, i.e., rebounds, in the opposite direction relative to its initial movement. The rebound effect of compression spring results in secondary mass impacting an outer side edge of proof mass with a force, represented by a leftward pointing arrow , in a direction that is likely to dislodge or move the potentially stuck proof mass when it is struck by secondary mass . This direction may be opposite to the direction of mechanical shock . Thus, immediately following the impact of secondary mass against proof mass , proof mass can return to its neutral position illustrated in .
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Accordingly, secondary mass of secondary structure is adapted to impact proof mass in response to the motion of proof mass when proof mass is subjected to a force, e.g., mechanical shock (), that is considerably greater than the sensing range of MEMS device . Should MEMS device sustain high x-axis acceleration, e.g., mechanical shock , the impact of secondary mass against proof mass will make sustained stiction less likely.
FIG. 10
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shows a top view of a MEMS device in accordance with yet another embodiment. Like MEMS device (), MEMS device can detect a physical condition in a direction parallel to the plane of MEMS device . MEMS device includes a movable element, in the form of a proof mass , suspended above a substrate by one or more spring members . In an embodiment, spring members are interconnected between proof mass and substrate , and enable substantially linear motion of proof mass relative to substrate . In this example, proof mass with spring members can move substantially parallel to substrate in response to x-axis acceleration .
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Proof mass includes movable fingers extending from a body of proof mass . Each movable finger is located between a pair of fixed fingers that are fixed, or stationary, relative to substrate . The accelerometer structure of MEMS device can measure distinct capacitances between each movable finger and its corresponding pair of fixed fingers . These capacitances are a measure of an external physical condition such as x-axis acceleration . Although only two movable fingers and two pairs of fixed fingers are shown, those skilled in the art will readily recognize that MEMS device can include any number of movable fingers and fixed fingers in accordance with particular design criteria.
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MEMS device may include a motion stop structure embodied as a fixed frame coupled to and extending above substrate that at least partially encircles proof mass . In addition, or alternatively, MEMS device may include lateral motion stops (not shown) that extend through apertures in proof mass . Fixed frame and/or the lateral motion stops function to limit excessive movement of proof mass in response to x-axis acceleration that is considerably greater than the sensing range for MEMS device .
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In the embodiment of , a secondary structure is coupled to proof mass and is suspended above substrate . More particularly, proof mass includes an aperture extending through the thickness of proof mass and defined by inner side walls . Secondary structure resides in aperture and is spaced apart from inner side walls of proof mass by a gap . Placement of secondary structure within aperture may achieve savings in terms of the overall size of MEMS device relative to MEMS device .
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Secondary structure includes a spring element and a secondary mass . Spring element is positioned in aperture , and a first end of spring element is coupled to, i.e. formed to extend from, one of inner side walls of proof mass . Thus, secondary structure is a projecting member that is supported only at one end, i.e., the interconnection of spring element with proof mass . In an embodiment, spring element may be a compression spring that exhibits a stiffness that is greater than a combined stiffness of spring members .
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FIGS. 1) and 96
FIG. 7
FIGS. 11 and 12
Like MEMS devices ( (), the collective spring stiffness of spring members in concert with the weight of proof mass enables spring members to deflect in response to acceleration, e.g., x-axis acceleration , within a particular sensing range, for example, between 0 and 8 g's, so that proof mass moves laterally, i.e. parallel to the surface of substrate , and movable fingers change position with respect to fixed fingers . However, due to its spring stiffness in concert with the weight of secondary mass , compression spring is prevented from deflecting appreciably, i.e., compressing, within the particular sensing range of MEMS device . Accordingly, secondary structure that includes secondary mass and compression spring does not change or affect the normal operation of MEMS device under relatively low x-axis acceleration . Instead, compression spring must be subjected to a significantly greater force, e.g., a mechanical shock, at levels of hundreds or thousands of g's before it will deflect appreciably. The operation of MEMS device , and particularly with respect to secondary structure , will be described in connection with the ensuing discussion of .
FIG. 11
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shows a partial top view of MEMS device subjected to mechanical shock . Again, mechanical shock is an x-axis acceleration () that is considerably greater than the sensing range of MEMS device . For example, mechanical shock may be orders of magnitude greater than the sensing range of MEMS device . When subjected to mechanical shock , proof mass will move substantially parallel to the surface of substrate until proof mass forcefully collides with a portion of fixed frame . Mechanical shock applied to proof mass is strong enough to use up the range of motion of proof mass such that proof mass stops against fixed frame . In the example presented in , the collision of proof mass against fixed frame is also represented by star elements . That is, star elements are not a physical components of MEMS device , but rather star elements are used to represent a forceful contact or collision of proof mass with fixed frame that could potentially cause physical damage to MEMS device .
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With the inclusion of secondary structure in MEMS device , the additional force from mechanical shock beyond what is needed to push proof mass against fixed frame is instead used to push secondary mass residing in aperture into a state ready for recoil. That is, as shown in , compression spring stretches, as represented by a rightward pointing arrow , such that secondary mass springs laterally outwardly away from the attachment point of compression spring to one of inner side walls .
FIG. 12
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shows a partial top view of MEMS device in which secondary mass of secondary structure impacts proof mass . The condition illustrated in approximately instantaneously follows the condition illustrated in . As represented in , compression spring residing in aperture has used up its range of motion and springs back, i.e., rebounds, in the opposite direction relative to its initial movement. Aperture is shaped such that one of inner side walls forms an abutment edge . The rebound effect of compression spring results in secondary mass impacting abutment edge of proof mass with a force, as represented by a leftward pointing arrow , in a direction that is likely to dislodge or move the potentially stuck proof mass when it is struck by secondary mass . Thus, immediately following the impact of secondary mass against abutment edge , proof mass can return to its neutral position illustrated in .
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FIG. 8
Accordingly, secondary mass of secondary structure is adapted to impact proof mass in response to the motion of proof mass when proof mass is subjected to a force, e.g., mechanical shock (), that is greater than the sensing range of MEMS device . Should MEMS device sustain high x-axis acceleration, e.g., mechanical shock , the impact of secondary mass against proof mass will make sustained stiction less likely.
In summary embodiments of the invention entail microelectromechanical (MEMS) devices with enhanced resistance to damage from mechanical shock. In particular, embodiments entail a secondary structure extending from a suspended movable element, i.e., proof mass. The secondary structure includes a spring element adapted for movement in response to motion of the proof mass when the MEMS device is subjected to mechanical shock. This movement causes the secondary structure to impact the proof mass in a direction that would be likely to dislodge a potentially stuck proof mass. Furthermore, some of the energy from the mechanical shock may be absorbed by the spring element in order to limit or prevent breakage to internal structures of the MEMS device. Thus, a MEMS device that includes the secondary structure may be less likely to fail when subjected to mechanical shock, thereby enhancing long term device reliability.
Although the preferred embodiments of the invention have been illustrated and described in detail, it will be readily apparent to those skilled in the art that various modifications may be made therein without departing from the spirit of the invention or from the scope of the appended claims. For example, embodiments of the invention may be adapted to provide failure protection from mechanical shock to MEMS devices having two or more sense directions, such as a MEMS device capable of both a z-axis sense direction perpendicular to a plane of the sensor, and an x-axis and/or y-axis sense direction parallel to the plane of the sensor. Additionally, the types of spring element and the particular configuration of the mass structure can be varied from that which is shown herein. And in still other embodiments, a MEMS device may have more than one secondary structure in order to achieve an appropriately sufficient rebound effect.
BRIEF DESCRIPTION OF THE DRAWINGS
A more complete understanding of the present invention may be derived by referring to the detailed description and claims when considered in connection with the Figures, wherein like reference numbers refer to similar items throughout the Figures, and:
FIG. 1
shows a top view of a MEMS device in accordance with an embodiment;
FIG. 2
FIG. 1
shows a side view of the MEMS device of ;
FIG. 3
FIG. 1
shows a side view of the MEMS device of subjected to a force greater than a predetermined sensing range for the MEMS device;
FIG. 4
shows a side view of the MEMS device subjected to mechanical shock;
FIG. 5
shows a side view of the MEMS device in which a secondary structure rebounds in response to the mechanical shock;
FIG. 6
shows a side view of the MEMS device in which the secondary structure impacts a movable element of the MEMS device;
FIG. 7
shows a top view of a MEMS device in accordance with another embodiment;
FIG. 8
FIG. 7
shows a partial top view of the MEMS device of subjected to mechanical shock;
FIG. 9
FIG. 7
shows a partial top view of the MEMS device of in which a secondary structure impacts a movable element of the MEMS device;
FIG. 10
shows a top view of a MEMS device in accordance with yet another embodiment;
FIG. 11
FIG. 11
shows a partial top view of the MEMS device of subjected to mechanical shock; and
FIG. 12
FIG. 11
shows a partial top view of the MEMS device of in which a secondary structure impacts a movable element of the MEMS device. | |
Expressions Dance's Tumbling/Acrobatic Arts program is a professional curriculum designed to promote excellence in flexibility, strength, balance, limbering, tumbling, and partner/group stunting. This 12-level program will help dancers learn the necessary exercises and skills to technically master tumbling skills that are commonly used in dance, such as: aerials, back handsprings, aerial walkovers and layouts. Dancers will simultaneously gain the grace, flexibility, balance, and strength to seamlessly, beautifully, and safely perform all skills on a stage or hard floor—away from Acro/Tumbling mats.
Attire: Tank top or leotard, black dance shorts. Hair must be pulled into a ponytail or bun. | https://www.dancetoexpress.com/hip-hop-1/ |
It was Christmas Eve 1922. The place was Grabtown, just outside the town of Smithfield. The event was the birth of a little girl to a farming family in Johnston County.
While still a little girl, the father lost the family farm. There were some family problems and poverty was setting in. The little girl's mother moved to Smithfield and opened a small boarding home for school teachers.
Growing up was not easy for this little girl. As a teenager she wanted to drop out of school because she had to wear the same sweater to class every day. She did, to her credit, finish high school and enrolled in Atlantic Christian College (now Barton College) in Wilson.
While attending college she made several trips to visit her sister in New York City. On one such visit her brother-in-law, a photographer, took a picture of her and posted it in the window of his studio on 5th Avenue. From that photo her life took a turn to stardom. It would bring her fame and fortune. It would also see her marry three of America's heart-throbs.
By now you may be asking "who is this girl from Grabtown?" This plain Jane from Johnston County? Well, the marquee on Broadway said it all: "The snows of Kilimanjaro" starring the beautiful Ava Gardner. Although she had no acting ability, she had the looks and that beauty captured Hollywood and the world. It didn't take her long to add talent to her beauty and she soon became a leading lady all over the world.
Ava Gardner married Mickey Rooney in 1942. After a short marriage she divorced Rooney and married the great band leader, Artie Shaw. That union only lasted a short time. In 1951 Ava married Frank Sinatra. That marriage also lasted only a few years.
Ava Gardner, the little girl from Grabtown, died January 25, 1990 and was buried in Smithfield - the place she always called home. | https://www.ncspin.com/north-carolina-minute-the-little-girl-from-grabtown |
We spent a memorable afternoon with Maria Azevedo Coutinho Vasconcelos e Souza, an aristocratic octogenarian who was a close friend of the great poet Sophia de Mello Breyner Andersen.
Sophia liked to eat well and was a great cook. One day, she gave Maria a handwritten cookbook with her favorite recipes. This little book shows Sophia’s attention to detail and joie de vivre.
In the first page, Sophia lays out some general advice:
1 – smell everything before cooking;
2 – use small amounts of salt and pepper; they mask the natural taste of the ingredients;
3 – salt the fish just before cooking;
4 – be faithful to the nature and the truth of every flavor. | https://saltofportugal.com/2015/11/09/sophias-cookbook/ |
Self awareness is critical to personal and professional development. The Myers-Briggs Type Indicator ( MBTI ) is a time-tested technique that leverages Jungian psychological types to help individuals understand their strengths and weaknesses and open up the opportunities that lie before them.
The theory behind MBTI is this: individuals have preferences for different ways of thinking and acting, just as individuals have a preference for right-handedness or left-handedness. No preference is of itself better or worse than another, but understanding your preferences gives you valuable insight into the way that you engage (or avoid) the tasks that face you in your personal and professional lives.
While the theory may be simple, the administration of MBTI requires interpersonal skills and life experience that can't be learned from a book. Thorp & Associates are experienced MBTI administrators. They have used MBTI to drive change in numerous organizations, and they can do the same for yours. Contact Thorp & Associates to find out how we can use the Myers-Briggs Type Indicator to help your organization. | http://thorpandassociates.com/myers-briggs.php |
As the use of oxaliplatin in the treatment of gastrointestinal, gynecologic, and other cancers continues to grow, so too does the incidence of hypersensitivity reactions (HSRs) connected to the drug. Because the development of HSRs may require patients to discontinue oxaliplatin even if it’s effective against their cancer, oncology nurses need to be able to prevent or minimize reactions whenever possible.
In their article in the February 2019 issue of the Clinical Journal of Oncology Nursing, Rogers et al. gave an overview of how to prevent, recognize, and manage HSRs from oxaliplatin so that patients have the best chance of continuing successful treatment.
Oxaliplatin HSRs
Oxaliplatin is associated with several toxicities, including peripheral sensory neuropathy, cytopenias, gastrointestinal toxicity, fatigue, and HSRs. Incidence rates for HSRs have been reported to vary from 2%–25%, and some study reports indicated HSR-related treatment discontinuation rates of 21%. The risk of grade 3–4 reactions is about 1.6% and life-threatening reactions is about 1%.
Rogers et al. explained that HSRs typically result from an immunoglobulin E (IgE)/mast cell-mediated action and that symptoms can involve cutaneous, respiratory, cardiovascular, and other systems. (See sidebar for a list of specific symptoms.) Cutaneous symptoms occur in about 45% of HSRs, respiratory in about 42%, and cardiovascular/anaphylaxis rarely.
Because they are IgE mediated, oxaliplatin HSRs occur only after repeated exposure to the drug. Rogers et al. reported that the reactions usually manifest during the seventh to ninth infusion, but some studies cite two peaks in the third and sixth cycle. Additionally, HSR risk increases if patients experience an oxaliplatin-free interval and then are re-exposed to the drug.
Risk factors include female gender, pre-existing allergies to other drugs, lactate dehydrogenase level, and higher neutrophil count but lower monocyte count (for grade 3–4 reactions).
Strategies to Manage HSRs
Prevention is the preferred strategy, Rogers et al. said. Premedications should include antiemetics (e.g., 5-HT3 antagonist with dexamethasone), histamine blockers (i.e., H1 and H2), and corticosteroids. Some studies have shown that intense premedication regimens with high-dose dexamethasone are associated with a lower incidence of HSRs and that they increase the median number of oxaliplatin cycles from 9 to 12.
Treatment choices will vary depending on an HSR’s severity but could include stopping oxaliplatin infusion and administering corticosteroids, histamine blockers, IV normal saline, and oxygen. For severe anaphylaxis, quickly administering epinephrine is critical. Infusion nurses should be prepared to administer those medications, assemble emergency equipment if needed, and assess vital signs every two to five minutes until patients are stabilized.
Resuming Oxaliplatin After HSRs
Although some study reports indicate that oxaliplatin can be resumed the same day if HSRs are mild, most practices will not rechallenge patients on the same day, Rogers et al. said. If HSRs are significant, prescribing physicians should determine whether patients can be rechallenged. However, studies show that oxaliplatin rechallenges are less successful than with other drugs: 50% of patients receiving platinum compounds will experience another HSR despite premedication.
In the case of grade 1–2 HSRs, a rapid drug desensitization (RDD) procedure may enable patients to temporarily tolerate oxaliplatin. Using a predetermined protocol, nurses administer sequentially increasing doses of the drug until the therapeutic amount is reached. Typically, the protocol uses dexamethasone premedication followed by high-dose dexamethasone plus histamine (H1 and H2) blockade. See sidebar for a standardized RDD for oxaliplatin. Because the desensitization is transient, RDD must be followed for every subsequent oxaliplatin administration.
One tool that may help providers decide which patients are most appropriate for a rechallenge is skin testing (e.g., patch, prick, intradermal) to determine mast cell reactivity. Negative skin test results indicate that HSRs are non-IgE mediated and RDDs will be less effective. Researchers reported sensitivity rates ranging from 26%–100%, with patch tests being least specific but prick and intradermal tests producing more reliable results.
Oncology Nurses’ Role in Managing HSRs
Nurses must be aware of HSR symptoms and their expected timing so that they can promptly intervene if HSRs occur. Standard nursing care plans should include stopping oxaliplatin and starting normal saline, administering medications (e.g., dexamethasone, diphenhydramine, famotidine, epinephrine for anaphylaxis), taking vital signs, obtaining a 12-lead echocardiogram for chest pain, and notifying the primary oncology provider. Nurses also need to educate patients about the reaction and help manage any anxiety.
For more information about oxaliplatin HSRs and the opportunity to earn 0.5 CNE contact hours and ILNA points (free for ONS members), refer to the full article by Rogers et al.
Questions regarding the information presented in this article should be directed to the Clinical Journal of Oncology Nursing editor at [email protected]. Photocopying of this article for educational purposes and group discussion is permitted. | http://voice.ons.org/news-and-views/for-oxaliplatin-hypersensitivity-reactions-prevention-is-the-best-strategy-but-heres |
Following the Charlie Hebdo shooting in Paris on January 8th 2015, Prime Minister David Cameron of the United Kingdom asked American president Barack Obama to increase pressure on American Internet companies to work more closely with British intelligence agencies, in order to deny potential terrorists a “safe space” to communicate. Further, Prime Minister Cameron pushed for co-operation to implement tighter surveillance controls. Under new proposals, messaging apps will have to either add a backdoor to their programs, or risk a potential ban within the UK. “Are we going to allow a means of communications which it simply isn’t possible to read?”Cameron asked in early 2015 while campaigning, in reference to apps such as WhatsApp, Snapchat, and other encrypted services. “My answer to that question is: ‘No, we must not'” he continued.
Prime Minister Cameron argued that the Paris attacks, including the one on satirical newspaper Charlie Hebdo, underscored the need for greater access. “The attacks in Paris demonstrated the scale of the threat that we face and the need to have robust powers through our intelligence and security agencies in order to keep our people safe,” Cameron said. These comments came following the threat and attack on Sony Pictures, which came after the intended release of The Interview, a political satire comedy film poking fun at the leader of North Korea Kim Jong-Un. Taking up the themes of The Interview,Obama recently accused North Korea of orchestrating the cyber attack on the film studio Sony Pictures due to the film’s subject matter pertaining to North Korea.
Some have interpreted Mr. Cameron’s words as meaning that the government is seeking to ban an important technology that underpins the global economy. SSL (Secure Sockets Layer) and TLS (Transport Layer Security) encryption protects financial details when people shop or bank online while so-called end-to-end encryption such as PGP (Pretty Good Privacy) help keep personal messages private.”Encryption is mathematics, not technology. It can’t be suppressed by law,” Matthew Bloch, managing director of internet hosting firm Bytemark, told the BBC.
It’s simply not possible to ban strong encryption within a country and software that uses strong encryption from crossing its borders. Further, preventing people from installing the software they want on the computing devices they own is an impossible feat. Countries like Iran, Syria, Pakistan, Russia, Kazakhstan, and Belarus have tried it and failed. China has tried before and is trying again. I wonder if Cameron is aware of the kind of company he is associating himself with, saidBruce Schneier, a security technologist and CTO of Resilient Systems, Inc. However, as it stands now, the government has “no intention” of introducing legislation to weaken encryption, minister for internet safety and security Baroness Shields told the House of Lords in the wake of the TalkTalk cyber attack debacle.
We once lived in a time where all we ever did was share, communicate, search, view, and enjoy with limited care for safety and privacy. With the world becoming more connected and the need for privacy ever more important, this has become a battle of privacy rights. Plots of terrorism and extremists should be subject to a due and just trial; however, all other individuals who want to protect their affairs, events, PINs, pictures, videos, searching habits, and the conversations they have should be allowed to use encrypted sources like BBM, WhatsApp and the updated iOS. A warrant for access to a back door is still access, and asking for access breaches any potential thought of whatever attempt at encryption was and is if organizations and institutions are subject to handing over mass communication of encrypted data to be a data mine for the government. Enough excuses and piggybacking on potential threats. Be in the know. Stand up for your right to online security and ask your local government representative for their stance on the topic.
About Jitesh Chauhan
A student of life with a passion for people, communication, and privacy. | https://www.privacyshell.com/government-cryptobackdoors |
On 21 June 2022, Centexbel and Sioen presented the RESERVIST project at the TechTextil Forum Conference in Frankfurt (Germany).
The presentation focused on the production of sustainable textile protective equipment in case of spiking demand times.
The COVID-19 crisis has shown that several types of protective and medical equipment, products and services are crucial, including the ability to deploy them rapidly. Because of cost factors, these products, or at least critical components for them, are often produced outside the EU. Once things normalize, there will be a large pressure to return to that situation.
The RESERVIST project aims to set up alternative lines called ‘reservist cells’, that can be activated within 48 hours, in order to switch easily from a day-to-day production line to one that focuses on products and services needed in time of crisis.
If you are interested in the presentation, the PowerPoint is available here. | https://cov-reservist.eu/reservist-at-the-techtextil-forum-conference/ |
Though it shines in moments, like a ring discovered in a dark cave, Synchronicity Theatre’s The Hobbit is a quirky, uneven piece that doesn’t succeed at telling its story clearly. It’s onstage at the Midtown theater through February 23.
Sure, aside from the younger children at which it’s primarily aimed, most theatergoers will already know the story of Bilbo Baggins, either because of the J.R.R. Tolkien novel or the six Peter Jackson-directed movies set in the fantasy realm of Middle-earth. But, with this telling and its cast of five playing nearly every role in the story, it’s sometimes very confusing to follow the unexpected journey as it goes there and back again.
Bilbo Baggins is a hobbit who lives in a neighborhood called The Shire, where he says he’s perfectly content with a full belly and a quiet life. Yet a wizard named Gandalf, sensing that Bilbo might be fit for adventure, lands at the hobbit’s doorstep with a proposition. The wizard recruits him to join an army of dwarves as they attempt to reclaim their homeland in the Lonely Mountain from a dragon named Smaug, who hoards the treasure he stole from them. Along the way, Bilbo discovers how to be savvy, confident and resourceful, and even finds a magic ring that helps him disappear and become a better thief.
This Hobbit’s biggest strengths are its dedicated actors, the acrobatics and the fight choreography. Director Jake Guinn and the Havoc Movement Company deserve credit for creating a show full of impressive practical stunts. When characters flip their way onstage or enter wearing stilts, it enlivens the narrative. As the dwarves tumble their way into Bilbo’s peaceful existence, the actors dangle from rafters and flip their way across the stage.
The climactic Battle of the Five Armies is electric and thrilling, full of swords and combat. In those moments, you may forget you’re watching only five people on a smallish stage –it’s the best scene in the show.
Clearly, the performers are lively and fully committed. Brooke Owens makes a charming Bilbo, particularly when she’s fussy and contrary. She’s charged with engaging the kids in the audience and guiding them — in narration — through this complicated world-building. It’s not an easy job, and Owens quickly manages to get her young audience’s trust and loyalty. She has a tougher job as her Bilbo gains the ring, becoming more conflicted and less trustworthy. Act 2 asks her to carry the focus and narration behind much of the story, and confusion follows. Blame the script though, not the performer.
Kids in the opening-night audience cheered loudest for Bilbo, and the casting of Owens allowed girls to see themselves as the hero of a favorite story.
Ash Anderson plays multiple roles and steals every scene, both as Gandalf the Wizard and the vicious Gollum — the hissing, weird creature that Bilbo encounters in a cave. The Gollum scene is an Act 1 highlight, a two-performer battle of wits. Anderson crouches and slithers, eyes costumed in goggles, low stage lights casting the character in eerie shadows. In every telling of The Hobbit, Gollum always goes away too quickly.
Actors Ryan Vo and Tennison Barry bring goofy energy to the other characters. They play two of the named dwarves and, in a running gag, point out to the audience when counting any of the others. Benedetto Robinson, primarily playing Thorin the Dwarf King, is arrogant and prideful. His character’s motivations and moods change frequently in Act 2, however, and it’s not always easy to figure out why.
The Greg Banks script is the main problem. Too much of it relies on the audience having prior knowledge of the material. The world-building it provides is clunky. There has to be a more engaging way to tell this story with these characters. Instead, many important points — like The Arkenstone — feel name-dropped without a sense of what they are or why they matter.
There’s music here, too, and it’s clumsy. Syncopated narration arrives randomly alongside the regular narration. It doesn’t delve into the characters’ inner desires or add any layers to the story.
And instead of a moment where we fully see Smaug in all his glory, we get a claw, an eye and a megaphone and are asked to imagine the rest. The climax of the entire Smaug adventure happens offstage while the characters watch and describe what they see.
Middle-earth would have been a tremendous sight. The Hobbit, unfortunately, tells much more than it shows. | https://www.artsatl.org/review-synchronicitys-inventive-hobbit-comes-up-short-in-its-storytelling-and-score/ |
by Forum House Publishing Company in Toronto .
Written in English
Edition Notes
|Series||Notes on general anthropology, 545|
|Classifications|
|LC Classifications||GN739 .B3|
|The Physical Object|
|Pagination||150 p.|
|Number of Pages||150|
|ID Numbers|
|Open Library||OL5010495M|
|LC Control Number||76541085|
A Culture of Growth documents the cultural shifts that permitted the interrogation of nature that then flowered into scientific advances. This book offers us an optimistic vision: a great expansion of communication preceded our modern prosperity, and we can expect this to happen again."—Angus Deaton, Nobel Laureate in Economics. In this groundbreaking book, one of the world's most celebrated economic historians argues that it was a culture of growth, specific to early modern Europe and the European Enlightenment, that laid the foundations for the scientific advances and pioneering inventions that would bring about explosive technological and economic development. "An extremely important work that explains what is meant by 'the new cultural history.' It successfully explores the central ideas of this line of research, and it shows how this growing new field relates to developments in such other disciplines as anthropology. The book is uncommonly readable."—Elvin Hatch, University of California, Santa. A Culture of Growth book. Read 19 reviews from the world's largest community for readers. During the late eighteenth century, innovations in Europe trigg /5(19).
Joel Mokyr, A Culture of Growth: The Origins of the Modern Economy. Princeton, NJ: Princeton University Press, xiv + pp. $35 (cloth), ISBN: Reviewed for by Claude Diebolt, Department of Economics, University of Strasbourg. I enjoyed this new book by Joel Mokyr, which is praiseworthy for its elegance and. As far as I’m aware, this is the most comprehensive book on third culture kids: their common traits, experiences, and thoughts. Written by David C. Pollock and Ruth E. Van Reken, Third Culture Kids: Growing Up Among Worlds provides great insight into the . G. Ricuperati, in International Encyclopedia of the Social & Behavioral Sciences, 5 Circulation and Practices: the Growth in Literacy. If one examines this process from the point of view of cultural history and of general transformation of intellectual models, European and now world-wide circulation of ideas can be measured through parameters which are apparently external to intellectual. Reading Level: Ages Audience: 1st Grade Delivery: Choice read for SSR/Group read Diversity Focus: Contemporary multicultural representation To Teach: Children's books with actual photographs are excellent to weave into the can present an accurate, realistic, and contemporary portrayal of a culture and it's people. This book in particular is excellent because it represents.
Paleolithic technology, culture, and art. Organizing paleolithic societies. Practice: Paleolithic life. Practice: The origin of humans and early human societies. Next lesson. The Neolithic Revolution and the birth of agriculture. Sort by: Top Voted. History and prehistory. Knowing prehistory. A Culture of Growth: My argument is that in Europe between the educated elite developed a culture and a set of institutions that was more suitable for intellectual innovation and the accumulation of useful knowledge than before. They came. The history section of IAS Prelims Exam GS Paper I Syllabus comprises questions from Indian art, culture, Ancient, Medieval and Modern Indian History. In this page, we give you the entire history syllabus for IAS prelims exam including ancient, medieval and modern history syllabus for UPSC exam. Human development is the process of a person's growth and maturation throughout their lifespan, concerned with the creation of an environment where people are able to develop their full potential, while leading productive and creative lives in accordance with their interests and needs. gender equality and cultural liberty, are also central. | https://jozuceqysosyx.chevreschevalaosta.com/pre-history-and-cultural-growth-book-18743ls.php |
The goal of this study was to examine the effects of a coaching intervention on teachers’ ability to implement academically responsive instruction through flexible instructional arrangements in self-contained classrooms for students who are Deaf and hard of hearing (DHH). A secondary goal of the study was to determine the impact of the implementation of flexible instructional arrangements on students’ academic engagement within instructional arrangements. Three teachers at a center school for the Deaf received differentiated coaching to learn how to implement the indicators of flexible instructional arrangements. Teachers were coached on 12 operationalized indicators using individual approaches that met the needs, learning styles, and preferences of each teacher. A changing criterion design replicated across teachers was used to examine the impact of the coaching intervention on teachers’ implementation of the indicators, as well as the impact of flexible instructional arrangements on students’ active engagement. Results show that coaching had an impact on all three teachers’ implementation of flexible instructional arrangements. As teachers mastered the indicators of flexible instructional arrangements requiring coaching, change occurred in their implementation of instructional arrangements. Students’ active engagement increased and passive engagement decreased when they participated in less whole class instruction and spent more time in small group and child-managed arrangements. After no longer receiving coaching, teachers maintained the implementation of flexible instructional arrangements and students continued to demonstrate higher levels of active engagement as compared to baseline. Limitations and implications for future practice and research are discussed. | https://repository.arizona.edu/handle/10150/627746 |
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