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This is definitely an experimental recipe… one that ended up tasting great, but, trust me, looked a bit funky.
I got introduced to Ube in the form of ice cream from Mitchell’s Ice Cream here in San Francisco. I was ready to finally try an Ube cupcake recipe, so I went to Pacific Super, an awesome Asian grocery on Alemany Street not too far from where I live, and bought nearly everything Ube.
I reconstituted Ube powder per the instructions on the packet, but didn’t like the taste very much at all. I boiled and mashed some fresh purple yams and they tasted much better. It was an easy decision to skip the powdered and go with the fresh.
One thing I didn’t do, that I noticed the few recipes on the net did, is add food coloring. After my red velvet experience, I have been anti-food-coloring-in-batter. In this case, I think I would have been better off to relax my newfound prejudice…
The yams were a deep purple blue, which was great, but when mixed with the yellow egg yolks and other ingredients, the batter ended up a ghastly green. Maybe a little red food coloring would have balanced it out. On the other hand, I can say that the ghastly green is natural! Maybe a half cup of smashed beets might do the trick, if anyone cares to experiment even further.
One definite keeper for me is the topping of peanuts crushed with a good amount of salt. This is something I stole from a caterer friend. The salty-sweet taste is very satisfying.
Ube Cupcakes
24 regular cupcakes / 350 degree oven
3-4 small to medium purple yams (ube)
4 large eggs, room temperature
2 cups sugar
1 cup oil, grapeseed or vegetable
1/2 cup macapuno, chopped
1 teaspoon vanilla
3 cups all-purpose flour
1 teaspoon baking powder
1 teaspoon baking soda
1/2 teaspoon salt
1 cup buttermilk milk
1. Peel and slice the yams into inch thick slices. Transfer into a medium pan, rinse, then cover with cool water. Bring to boil and simmer until tender. Mash with a fork
2. Crack eggs into a large bowl. Beat with a whisk until yellows and whites are combined.
3. Gradually add sugar and whisk until combined.
4. Add oil and whisk until combined. Add chopped macapuno and vanilla and stir with a wooden spoon to combine.
5. Sift together flour, baking powder, baking soda, and salt in another bowl.
6. Add about a third of the flour mixture to the wet mixture and mix to combine.
7. Add about one half of the buttermilk and mix to combined.
8. Repeat above, alternating flour and buttermilk and ending with the flour mixture.
9. Scoop into cupcake papers about half to two-thirds full (depending on whether you want flat or domed cupcakes. Note that these cupcakes will don’t shrink. Two-thirds full will result in domed cupcakes.)
10. Bake for 22-25 minutes until a cake tester comes out clean.
Bubble Buttercream
3 cups water
1/4 cup tapioca balls
1 cup (2 sticks) butter, room temperature
5-6 cups powdered sugar, sifted
1 teaspoon vanilla
1. Boil the water in a medium saucepan (I used the water from the yams).
2. Add the balls to the boiling water and boil for 30 minutes, stirring occasionally.
3. Turn off heat and let the balls steep in the water for another 30 minutes.
4. Transfer balls to a sieve and rinse with cool water to remove starch. Transfer to a bowl and cover with cold water until ready to use.
5. Beat butter in an electric mixer until soft.
6. Add 3 cups of the sifted sugar and beat until combined.
7. Drain tapioca balls and measure out 1/2 cup. Fold into the frosting.
8. Add remaining sugar, 1 cup at a time, stirring until combined.
Note: This frosting was very soft and almost looked a little curdled. It definitely has a not so pretty texture, but it tasted fine. The tapioca balls are a novelty and perhaps not necessary. There is the added interest of the chewy texture, but I am not sure it’s worth the sacrifice in the visual appeal. If you are looking for something picture perfect, definitely skip this recipe. It needs work.
Assemble
1. Pipe frosting onto cooled cupcakes.
2. Crush about a 1/3 cup of unsalted peanuts with about a teaspoon of salt. Sprinkle over cupcakes. | http://cupcakeblog.com/?p=51 |
Those new students, many of whom are deemed at-risk or have special education needs, will require the hiring of four special education teachers and five other staff members.
But Superintendent Russell Mayo said even with those new hires, the district will save nearly a half-million dollars compared to what it previously paid the charter school.
In addition to four special education teachers, the district is seeking to hire four para-professionals and one behavior specialist.
These nine permanent hires will cost the district an estimated $493,032, according to Deborah Hartman, the district's director of special education.
But those costs will be offset by the savings from the $951,752 the district had previously paid Vitalistic for those services, she said.
"These positions were not accounted for in the budget, but you can see how it's compensated for because of the additional money is coming back," Mayo said. "So it's a net gain."
Hartman said the district may see even further cost savings because they will no longer need to provide special transportation for some of the former Vitalistic students.
The Allentown School Board's education committee unanimously voted to recommend the new hires. The full board will vote on the measure Dec. 20.
Vitalistic will permanently close its doors on Jan. 25. The 111 Dewberry Ave. school had been the target of inquiries by the Allentown and Bethlehem Area school districts, which had threatened to revoke its charter.
Vitalistic officials have long denied the allegations against them and originally planned to fight any charter revocation, but announced in October they would close due to a lack of finances.
Eighty-eight of Vitalistic's 93 students were from the Allentown School District, all of whom are from kindergarten to grade three, Hartman said.
Hartman said 20 of the students have special education needs, 16 need evaluation and 12 are considered at-risk.
Not all 88 students will necessarily return. Hartman said it will depend on how many of the parents enroll them in the district, a process that has already begun for many of the households.
"There are still some parents who are unaware that Vitalistic is closing," Hartman said. "We will be reaching out to those parents and giving them information."
Mayo said if less students come back to the district than are anticipated, they may hire less than the four teachers and five staffers being requested.
The new students will be largely scattered throughout the district's 15 elementary schools, although the Mosser Elementary School will get the largest impact at 21 new kids, Hartman said.
There was little discussion among the school board committee about Vitalistic or the new hires, but board President Robert Smith Jr. said he was pleasantly surprised the district would saves money through the changes.
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Also asked, Can an innocent person fail a polygraph test?
The first reason is that an innocent person can fail a polygraph test. ... A second reason why you shouldn't take a polygraph test unless your lawyer advises doing so, is that polygraph results are generally inadmissible in court.
In this manner, How accurate is a polygraph test 2020?. Despite claims of 90% validity by polygraph advocates, the National Research Council has found no evidence of effectiveness. ... The American Psychological Association states "Most psychologists agree that there is little evidence that polygraph tests can accurately detect lies."
Herein, Are polygraphs 100 percent?
Consequently, most courts do not admit polygraph evidence. ... The American Polygraph Association, which sets standards for testing, says that polygraphs are "highly accurate," citing an accuracy rate above 90 percent when done properly. Critics, however, say the tests are correct only 70 percent of the time.
How accurate are polygraphs Really?
There have been several reviews of polygraph accuracy. They suggest that polygraphs are accurate between 80% and 90% of the time. This means polygraphs are far from foolproof, but better than the average person's ability to spot lies, which research suggests they can do around 55% of the time.
How often do polygraphs give false positives?
correct innocent detections ranged from 12.5 to 94.1 percent and averaged 76 percent; false positive rate (innocent persons found deceptive) ranged from O to 75 percent and averaged 19.1 percent; and. false negative rate (guilty persons found nondeceptive) ranged from O to 29.4 percent and averaged 10.2 percent.
Can you trick a lie detector test?
A simple way to cheat the polygraph is to deliberately distort your physiological readings when telling the truth, such as by biting your tongue, or imagining an embarrassing incident in the past.
Can anxiety disorder cause false positives on a polygraph?
The answer: sort of. Dr. Saxe explains: “The fundamental problem is that there is no unique physiological response to lying. So, yes, anxiety plays a role, as do medications that affect heart rate and blood pressure.”
How can I pass a polygraph?
So here's how you beat the test: Change your heart rate , respiratory rate, blood pressure and sweat level while answering control questions. Send your control lies off the charts. By comparison, your answers to the relevant questions (whether they are truths or falsehoods) will seem true.
What is the most accurate lie detector test?
Best Lie Detector: EyeDetect | 97-99% Accuracy with Polygraph. Combine EyeDetect with Polygraph to get 97-99% outcome confidence. The two combined are the best lie detector test: highest accuracy.
Will a polygraph examiner tell you if you failed?
Most courts might not allow the results of a polygraph examination. ... Basically, many polygraph examiners will claim the subject “failed the polygraph” and push to have the subject change the story. They will claim your brain is suppressing the truth to protect the subject from feeling ashamed or guilty.
Can I take a lie detector test to prove my innocence?
If criminal investigators ask you to take a polygraph test, it's safe to assume they are trying to gather evidence, usually against you. Occasionally, a suspect will ask to take a test in order to establish his innocence. You are never under any legal obligation to take a lie detector test in a criminal investigation.
Can a narcissist pass a polygraph test?
But narcissists and sociopaths have been known to pass polygraph tests with ease. That's because they believe that what they are doing is right for them, regardless of cultural or social norms or harm done. They don't see themselves as wrongdoers.
What disqualifies you on a polygraph?
You will be asked about the following topics during a typical police polygraph or CVSA: Shoplifting or theft of money or merchandise from employer. Illegal drug trafficking or dealing. Illegal drug or medication use, including steroids.
How do you stay calm during a polygraph?
Tice says it's also easy to beat a polygraph while telling a real lie by daydreaming to calm the nerves. "Think of a warm summer night... or drinking a beer, whatever calms you. You're throwing them off," he says.
What medications affect a lie detector test?
When considering the effect of drugs on the polygraph, the Federation of American Scientists reported that “the tranquilizer, meprobamate (“Miltown”), permits subjects who are being deceptive to increase their ability to avoid detection in a polygraph examination.” This drug and other anti-anxiety medications or ...
Can you lie and pass a polygraph?
A 2004 report on the validity of polygraphs by the British Psychological Society found that the tests are likely to produce more false positives than false negatives. That means that more innocent people will wrongly fail the test than guilty people will wrongly pass.
Is it hard to pass a polygraph test?
A polygraph test or lie detector test is designed to analyze physiological reactions to questions to determine whether or not a subject is being truthful. ... Fortunately for them, it's not that hard to beat a lie detector test. The first step to passing the test is understanding how it works.
What questions can you ask in a lie detector test?
...
Ten Commonly Asked Questions
- Is your name Sandy Hill? ( ...
- Are you 43 years old?
- Do you suspect anyone of selling drugs? ( ...
- Is your cat's name Josie?
- Were you born in 1956?
Can a polygraph give false positives?
Polygraph tests aren't psychic tools that can tell beyond any doubt that someone is lying. They sometimes produce false results. A false positive can occur when someone telling the truth triggers the device, which then may indicate he's lying even if he isn't.
Can anxiety affect a lie detector test?
A polygraph test, in essence, measures one thing: anxiety. "All these physiological measures are simply associated with fear and anxiety," Saxe says. "And people are anxious sometimes when they're telling the truth, and they can be not anxious sometimes when they're lying.
Does caffeine affect a polygraph?
Many applicants worry that something like caffeine will hinder their performance on the polygraph. But it's more likely to hurt you if you drink a cup of coffee every morning, and then skip it the morning of the polygraph. The same goes with prescription medications.
How do you lie and not get caught?
- DO: Maintain your baseline. Stay calm. ...
- DON'T: Swallow hard. Swallowing hard is a giveaway. ...
- DO: Breathe normally. Inhale, exhale. ...
- DON'T: Touch your skin. ...
- DO: Lean in. ...
- DON'T: Shorten the syntax of words. ...
- DO: Try not to sweat. ...
- DON'T: Say "I don't lie"
Can you fail a lie detector test and still be telling the truth?
You can fail the test simply because you don't quite understand the question, or over-analyze the question each time, even if the examiner gave you clarification multiple times. ... You tell the examiner, and they just say it's not something to worry about, that the question does not refer to them.
How much does a lie detector test cost?
How much does a private polygraph test cost? Trained polygraph examiners administer lie detector tests for a fee. The typical cost is between $200 and $2,000. The specific cost usually increases with the length of the test. | https://moviecultists.com/are-polygraphs-100-accurate |
Interfacial reaction in the synergistic extraction rate of Ni(II) with dithizone and 1,10-phenanthroline.
The kinetic synergistic effect of 1,10-phenanthroline (phen) on the extraction rate of Ni(II) with dithizone (HDz) into chloroform was studied by means of a high-speed stirring method combined with photodiode-array spectrophotometry. The initial extraction rate of the adduct complex NiDz(2)phen depended upon the concentrations of both HDz and phen, suggesting the formation of NiDzphen(+) as the rate-controlling step. When [HDz] < [phen], the initial extraction of NiDz(2)phen competed with the formation of an intermediate complex, which was adsorbed at the interface and assigned most probably to NiDzphen(+)(2). The intermediate complex was gradually converted to NiDz(2)phen at a later stage of the extraction. The rate constants for the formation and consumption of the intermediate were determined, and the kinetic mechanism in the synergistic extraction was discussed.
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CROSS-REFERENCE TO RELATED APPLICATIONS
FIELD OF THE INVENTION
BACKGROUND
SUMMARY
BRIEF DESCRIPTION OF THE DRAWINGS
DETAILED DESCRIPTION
This application is a continuation that claims priority under 35 U.S.C. §120 of Patent Cooperation Treaty Application Serial No. PCT/KR2003/001589 filed on Aug. 7, 2003, entitled “ROLL-UP ELECTRONIC PIANO” which claims priority to Korean Patent Application Serial No. 20-2002-0023572 filed on Aug. 7, 2002, entitled “ROLL-UP ELECTRONIC PIANO.”
The present invention relates to an electronic piano including a roll-up keyboard and, more particularly, to a roll-up electronic piano whose tone is similar to that of a real piano and is much smaller and lighter than existing electronic pianos. The electronic piano according to the present invention may comprise a border member that includes a piezoelectric material capable of performing on/off functions, a keyboard whose cover is made of a silicon material and is capable of being folded/unfolded, and a control part that controls and amplifies piezoelectric signals, which can be separated from the keyboard, and can perform wire/wireless send/receive functions.
Children may individually have practical training using keyboard instruments similar to a piano in a music class. A typical personal keyboard instrument for use by children includes a hard keyboard with about two octaves that is played through insufflating air by mouth. As an example, Korean utility model patent publication No. 88-13187 discloses an electronic piano using an electronic circuit design with an IC (integrated circuit). However, it is impossible for this electronic piano to control the volume of sound according to the strength of force pressing on the keyboard. Moreover, the electronic piano has only on/off functions. More advanced keyboard instruments may be connected to a computer, but these instruments also include simple on/off functions.
In addition, the above-mentioned keyboard instruments are cumbersome to carry because of the large size thereof and provide little help to further a musical education due to poor effects of usage and function. In other words, it is difficult for the existing keyboard instruments to sound similar in tone to that of a real piano and to represent the strength of the sound.
U.S. Pat. No. 6,259,006 discloses an electronic piano providing some means capable of coping with the above-identified problems. The electronic piano is portable and foldable, has a small size and is relatively lightweight, and can be connected to a computer. Moreover, the tone of the electronic piano is similar to that of a real piano. However, the electronic piano has a problem that when the keyboard is folded with the power on, the keys pressed may generate a noise. Moreover, if the user does not turn off the power and rolls up the keyboard carelessly, the keyboard may continue to generate the noise for a long time without being perceived by the user because the electronic piano has headphones instead of loudspeakers. Thus, this may have an electrically damaging effect on the inside electric devices of the electronic piano.
FIG. 1
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The above-mentioned U.S. Patent has another problem in its method of generating sounds. A real piano does not have a vibrato effect, or a tremulous or pulsating effect produced by minute and rapid variations in pitch. However, in an electronic piano without a time period for a piezoelectric material to perceive input, if a user first presses a key of the keyboard and, then, changes the pressure of his/her finger, the vibrato effect may occur. In addition, if the user presses slowly and deeply on the keyboard, the pressure on the piezoelectric material gradually increases and, therefore, the volume of sound also gradually increases because an electric current increases gradually. Conversely, if the user slowly relaxes the pressure of his/her finger, the pressure on the piezoelectric material gradually decreases and, therefore, the volume of sound also gradually decreases because an electric current gradually decreases. This may become a considerable problem in reproducing a tone of a real piano. As shown in , a real piano generates sound by a hammer () hitting a string () when a user presses a key (). Thus, when the user presses a key once, the hammer hits the string once and the vibrato effect cannot occur. The vibrato effect is different from tremolo as a rapid repetition of a note. For example, the vibrato of string instruments occurs by means of fast movement of a finger on a string and the vibrato of wind instruments, by means of controlled breathing.
The electronic piano according to the above-mentioned U.S. Patent, which has an imperfect operating construction of piezoelectric material, can embody a similar tone to that of a real piano to some degree, but cannot embody a sound effect that is produced by a hammer hitting a string. Moreover, although the above-mentioned U.S. Patent can embody a tone of a real piano through using a simple contact method instead of using a piezoelectric material, the simple contact method cannot control the strength of the tone. In other words, it cannot embody sound effects according to the strength of pressing a key. The U.S. Patent can damp or sustain a sound by using pedals, but cannot embody an effect strengthening the sound.
Another disadvantage of the conventional foldable electronic pianos is that a user may not play the electric piano in a location having a narrower width than the length of the keyboard. Generally, a desk in a classroom has a length of about 60 cm and, therefore, the keyboard with more than four-octaves whose keys have the same size as those of a real piano endows some trouble in rolling it out on the desk. In other words, some section of the keyboard drops downward from the top of the desk, and, therefore, it is difficult to utilize all the keys of the electric piano.
Accordingly, the present invention is directed to a new roll-up electronic piano that substantially obviates one or more limitations and disadvantages of the related art.
It is an object of the present invention is to provide a roll-up electronic piano having a small size attributable, at least in part, to a foldable keyboard. The electronic piano can produce sounds according to the strength and the duration of pressure in pressing a key and can embody a similar tone to that of a real piano by preventing the vibrato effect. Still further, the electronic piano can be controlled through wired or wireless communications and be played even in a narrow space because it includes a removable structure capable of separating the keyboard from the control part, and can be used conveniently in a music class for children and students due to its small size and lightweight.
To achieve the objects and other advantages of the present invention, as embodied and broadly described herein, the invention provides a roll-up electronic piano comprising a keyboard made of piezoelectric material and a control part which can control and amplify electrical piezoelectric signals.
The keyboard of the electronic piano according to the present invention produces a signal by electromotive force through a piezoelectric polymer film such as poly vinylidene fluoride (hereinafter referred to as “PVDF”) or a piezoelectric fiber when a user presses a key of the keyboard. The piezoelectric polymer film generates a voltage according to the strength and the duration of impact. The generated signals are controlled and amplified in the control part with a sound chip and thereafter directed to a speaker.
The keyboard of the present invention has 4-8 octaves, and can be rolled up to form a cylinder. The keyboard comprises a coupling member for connection with the control part. The coupling member is made of flexible material that can mechanically withstand strains produced by folding and unfolding operations. In addition, white keys and black keys of the keyboard are designed in accordance with the standard configuration of a real piano. Inside the keyboard is a piezoelectric polymer film such as PVDF or a piezoelectric fiber that generates a piezoelectric electromotive force corresponding to each key of the keyboard. The covering of the keyboard, except the coupling member, is made of flexible rubber. Therefore, the electronic piano of the present invention can be easily rolled up and is portable. The control part comprises an electronic circuit design including a microprocessor and a sound chip.
In the electronic piano of the present invention, a noise is not generated although the keyboard is rolled up, the power on, and the keys are depressed because the border member of the keyboard part comprises a second piezoelectric material other than the piezoelectric material contained inside the keyboard. The second piezoelectric material generates a voltage when it is bent. If the piezoelectric material is bent excessively and the resulting voltage is higher than an adequate voltage, the control part produces a signal based on the voltage which controls the electrical power. Therefore, the roll-up electronic piano can obviate noise produced from depression of the keys of the keyboard when folded without requiring the power to be turned off.
The roll-up electronic piano can produce a similar tone to that of a real piano because the present invention can eliminate the vibrato effect. One of the general characteristics of the piezoelectric material is a change in voltage due to a change of resistance by pressure. Based upon the change in voltage, the volume of sound generated from a sound chip changes. An electronic piano according to a prior art may sound like a real piano tone indicative of a hammer hitting a string, but only when a key is pressed using a uniform force. Although a user may not become aware of the difference while playing simple musical notes, he/she may notice the difference when playing complex musical notes. For example, if a user's first finger presses a first key while a second finger continuously presses a second key, it is difficult for the first finger to continuously press the first key with uniform force. To obviate this problem, the keyboard of the electric piano according to the present invention has a function to detect the start of applying pressure and a function to detect a change in voltage in the piezoelectric material for a fixed time, thereby preventing the vibrato effect. That is, the start of transformation of the piezoelectric material at more than a fixed pressure is perceived as the contact of a finger on a key, and a voltage immediately after the key is pressed at a desired force is obtained while a change in voltage generated thereafter is not obtained. Then, if the pressure on the key is diminished to less than a fixed pressure and the piezoelectric material is restored to the original state, the keyboard finishes generating a sound. Therefore, in playing the electronic piano, an unnecessary vibrato effect can be prevented.
The roll-up electronic piano according to the present invention can be played effectively in a narrow space because it includes a function controlling the number of octaves of the keyboard thereby eliminating the inconvenience of detaching the keyboard.
In addition, the roll-up electronic piano is divided into the control part and the keyboard and can be controlled by a wire or a wireless method. When the electronic piano is divided into the control part and the keyboard, signals according to a change in voltage generated from the keyboard is transformed to a digital signal and the digital signal along with a unique ID value (a unique number of an apparatus) are transmitted to the control part. The control part identifies the ID value received and generates sounds according to the digital signals received.
The covering of the keyboard is made of a flexible material such as rubber or, more preferably, silicon rubber. The silicon rubber has good thermal endurance, cold resistance, and moisture resistance.
If a (printed circuit board) PCB is used instead of the piezoelectric material and the wireless function is omitted, the roll-up electric piano with sufficient functions can be produced at a low cost.
The roll-up electronic piano according to the present invention, may be connected to peripheral devices such as a personal computer that can store and reproduce a record of playing as a music file by transforming analog signals into digital signals.
FIG. 1
is a diagram illustrating a principle of sound generation in a real piano;
FIG. 2
is a block diagram describing an exemplary electronic piano according to the present invention;
FIG. 3
a
is a cross-sectional view of a keyboard according to an exemplary embodiment of the present invention;
FIG. 3
b
is a cross-sectional view of another exemplary keyboard in accordance with the present invention;
FIG. 3
c
is a cross-sectional view of a still further exemplary keyboard in accordance with the present invention;
FIG. 4
illustrates an exemplary removable structure of a roll-up electronic piano in accordance with the present invention;
FIG. 5
illustrates an exemplary roll-up electronic piano with pedals in accordance with the present invention;
FIG. 6
illustrates an exemplary exterior of a roll-up electronic piano in accordance with the present invention;
FIG. 7
illustrates, in a cross-sectional view, an exemplary folded state of an exemplary roll-up electronic piano in accordance with the present invention;
FIG. 8
illustrates, in a perspective view, another exemplary folded state of an exemplary roll-up electronic piano in accordance with the present invention; and
FIG. 9
is a perspective view of an even further exemplary roll-up electronic piano in accordance with the present invention.
Reference will now be made to the exemplary embodiments of the present invention that are illustrated in the accompanying drawings.
FIG. 2
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Referring to , a keyboard () outputs a signal attributable to an electromotive force using poly vinylidene fluoride (PVDF) or a piezoelectric fiber with piezoelectric characteristics based on the strength and duration of pressure exerted by a user. That is, while an electric current is applied into the PVDF, the PVDF generates a voltage according to the strength and the duration of the pressure. The outputted signal is amplified in the control part () having a sound chip () and then, sound comes out through a speaker (). The sound chip may provide hundreds of tones.
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In addition, the sound chip () according to the present invention can provide the same sound effect as a real piano. The various magnitudes of sound generated when a key of a real piano is pressed at various pressures are logged and the logged data is compared to a voltage from the piezoelectric material (), which is generated when a key of keyboard () of the electronic piano is pressed. Then, the logged data corresponding to the voltage is stored in the sound chip () to generate the same magnitude of sound with that of the real piano.
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FIG. 3
a
In detail, inside the keyboard () is the piezoelectric material perceiving pressure, which is enclosed by a conducting pattern film, and an insulator is formed to enclose the conducting pattern film. Referring to , upper and bottom conducting pattern films () are positioned on and underneath the piezoelectric material (), and upper and lower insulators () are positioned to enclose the conducting pattern films (). Then, a shock-absorbing member () is positioned on top of the upper insulator (). The covering of the keyboard () is made of silicon rubber (). The insulators () suppress static electricity that may be generated on the surface of the keyboard ().
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In the above-mentioned keyboard structure, the voltage generated when a key of the keyboard () is pressed is controlled by a microprocessor () of the control part (). The signal controlled by the microprocessor () is converted into sound through the sound chip (), an amplifier, and the speaker. When a user presses a key of the keyboard (), pressure is applied to the silicon rubber () and the pressure is transmitted to the shock-absorbing member (). The shock-absorbing member () does not transmit the pressure to the insulator () if the pressure is less than a predetermined value. That is, in case of merely placing a finger on the keyboard the shock-absorbing member (), made of a soft material, absorbs the pressure to suppress the generation of a sound. In addition, the shock-absorbing member () may be formed in various densities and thicknesses to simulate the touch of a real piano. If the shock-absorbing member () is excessively thin or soft, the keyboard may reduce the simulated feeling of pressing a key of a real piano. If the shock-absorbing member () is excessively thick, the keyboard may not be easily rolled up. If the shock-absorbing member () is excessively hard, more pressure may be required to press the key than for a real piano. Thus, the shock-absorbing member () should be made of a material with an adequate density and thickness.
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If adequate pressure is applied on the shock-absorbing member (), the insulator () disposed underneath the shock-absorbing member () is pressed and then, the conducting pattern film () underneath the insulator () is pressed to transmit the pressure to the piezoelectric material (). The conducting pattern film () is a device operative to transmit the change in voltage of the piezoelectric material () to the control part. The pressure transmitted transforms the piezoelectric material (), thereby changing the resistance of the piezoelectric material and also changing the voltage.
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Then, the microprocessor () in the control part generates a first signal. Even after the microprocessor generates the first signal, the pressure continues to increase until it reaches a desired strength. If the desired strength is transmitted completely, the transformation rate of the piezoelectric material () is diminished and the microprocessor perceives a decrease in the change in voltage and resistance. A second signal is generated after detecting the desired strength.
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The time interval between the first and second signals corresponds to a period starting when a user first presses a key of a real piano to when he/she finishes to press the key with the desired strength. That is, the time interval corresponds to a period starting when a user presses a key of a real piano to when a hammer in the real piano hits a string. The microprocessor () controls the sound chip so as not to generate sound for the time interval between the first and the second signal. The microprocessor () generates sound after the second signal is created. In addition, to prevent the vibrato effect brought about by a minute change in pressure after completely pressing to the desired strength, the magnitude of the change in voltage corresponding to the second signal is compared to the data stored in the sound chip () beforehand to generate only a single adequate sound. Therefore, a single sound is generated for each press of a key. The sound generated by the second signal is set to decrease according to the strength of the pressure exerted by a user, just like a real piano whose sound slowly decreases while the user continues to press the key. In addition, the sound is set up to stop sounding immediately after the user removes his finger from the key, just the same as a real piano.
Accordingly, the present invention can prevent variable electronic sound that may be generated from existing electronic keyboards, and generate the clear sounds of a real piano. Examples of the variable electronic sounds include a crescendo effect where the volume of sound gradually increases during a time interval between the first and second signal, a vibrato effect that may be generated because of a user's fingers pressing a first key while continuously pressing another key, and a decrescendo effect where the volume of sound gradually decreases during an instant time interval in which the user's finger is removed from the key.
The covering of the keyboard is made of a flexible rubber, preferably, silicon rubber so that the keyboard can be rolled up. Silicon rubber has good heat resistance, cold resistance, and moisture resistance. In addition, a pattern imprinted on the surface of the silicon rubber stands well. The present invention includes reforming the surface of the silicon rubber to facilitate mounting portions of the silicon rubber to one another.
A method of reforming the silicon rubber comprises the steps of injecting an inert gas into a vacuum chamber, converting the inert gas into a plasma state by applying plasma potential so that the inert gas can move very rapidly, and forming fine grooves on the surface of the silicon rubber using the inert gas in its plasma state. An adhesive infiltrates into the fine grooves so that the upper plate of the keyboard covering made of silicon rubber can adhere to the lower plate of the keyboard covering. By reforming of the surface of the silicon rubber, an inexpensive general adhesive can be used to connect the upper and lower plates of the keyboard covering instead of an expensive adhesive adapted for bonding to silicon rubber. The reformed silicon rubber can maintain adhesive force with repeated folding and unfolding operations.
The shape of a keyboard is printed on the upper plate of the keyboard covering. Here, another advantage of the reformed silicon rubber with fine grooves is that when white keys and black keys are printed on the upper plate of the keyboard covering, the number of printing sequences can be markedly reduced because ink can easily infiltrate into the silicon rubber through the fine grooves. For example, in using a typical silicon rubber, white ink is applied three times and black ink is applied twice during printing. On the other hand, in using the reformed silicon rubber, white ink is applied twice and black ink is applied once during printing. In addition, a good printing quality can be achieved even with general inks instead of expensive inks.
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Inside the border member of the keyboard () is a second piezoelectric material. The second piezoelectric material is operative to obviate the keyboard from generating a loud noise when it is rolled up with the power on. If the second piezoelectric material is bent excessively to more than a particular angle due to a folded keyboard or a careless usage, it is transformed to change its resistance, thereby changing the voltage and, ultimately, the power is turned off. In another exemplary embodiment, the voltage generated from the piezoelectric material may turn off the power or the power may be turned on when the piezoelectric material is unfolded to less than a particular angle. Such a power on/off method includes on/off only by a power switch; on/off by a power switch and off by the second piezoelectric material in the border member; on/off by a power switch and on/off by the second piezoelectric material in the border member; and, on/off only by the second piezoelectric material in the border member.
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The roll-up electronic piano according the present invention can be played in a narrow space because the control part () has a button that can control an active range of the keyboard (). The keyboard () may have an arbitrary octave range, preferably 3-8 octaves. The control part () can choose the active range of the keyboard () according to the availability of space. For example, the active range of the keyboard may be selected from 1-8 octaves, based on an octave unit or a key unit, starting from the control part (). The active range of the keyboard may also be arbitrarily selected regardless of the vicinity of the control part based on the octave unit or the key unit. Therefore, the roll-up electronic piano can be played conveniently in a narrow space with a simple operation of a button compared to the inconvenience of a conventional electronic piano whose keyboard part may be detached or attached on occasion.
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Referring to , the roll-up electronic piano of the present invention has a removable structure where the control part and the keyboard are separated and can be controlled through a wired or wireless method. In controlling the electronic piano through a wireless method, there are several additional elements that may be included such as a separate power supply unit for the keyboard, A/D converter which converts changes in voltage generated from the keyboard () into a digital signal, a signal processing part that adds a unique ID value (i.e., a unique number of an apparatus) to the digital signal, and a transmitting part that transmits the signal with the unique ID value. The power supply unit, A/D converter, signal processing part, and transmitting part may be installed in a coupling member (). The control part () also has a receiver that receives the signal from the keyboard (). The signal received in the control part is perceived after the microprocessor () identifies the unique ID that is included in the signal to prevent errors that may occur when a plurality of people play electronic pianos simultaneously.
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The roll-up electronic piano of the present invention may comprise different components according to the wired or wired/wireless simultaneous method. Using the wired method in which the keyboard () is coupled to the control part (), the sound is generated by an electric analog signal. Using the wired/wireless simultaneous method, the sound is generated by a process of converting the analog signal into the digital signal. Therefore, the necessary components are adequately selected according to which signal is selected from the digital and analog signals.
By using the digital signal, the electronic piano can be connected to peripheral devices such as a personal computer via a USB connection in order to record the playing as a music file. In addition, using the wireless method that produces a digital signal, software may be used as the control part in addition to or instead of hardware. For example, a program used as the control part can be downloaded from an Internet website. In this method, a receiver, which receives signals from the keyboard, should be included. In circumstances where a personal computer is used in place of the control part, the program used as the control part may be downloaded from an Internet website and, then, a receiver coupled to a USB receiving board associated with the computer receives the signals transmitted from the keyboard to generate sound.
When personal telecommunications devices such as a cellular phone and a personal digital assistance (PDA) are used as the control part, a program used as the control part may be downloaded from an Internet website and, then, a receiver coupled to an interface in the personal telecommunications device receives the signals transmitted from the keyboard to-generate sound. In addition, by installing the program used as the control part in external telecommunications equipment, the roll-up electronic piano may be utilized more effectively.
The roll-up electronic piano may have a simpler structure using a PCB (printed circuit board). The PCB is relatively inexpensive and can provide a simple constitution although it may not embody the strength of sound compared to the method using piezoelectric material and a microprocessor. For example, the roll-up electronic piano comprises a keyboard including two plates of PCB film and silicon rubber and a control part that includes a simple sound chip without a microprocessor.
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Referring to , a plurality of protrusions () made of insulating material are formed at a lower surface of an upper PCB () to maintain a distance between the upper () and a lower () PCB when a key of the keyboard is depressed. When a key is depressed, pressure is first applied to the keyboard covering made of silicon rubber () and transferred to a shock-absorbing member (). If the pressure is less than an adequate value, the shock-absorbing member may not transfer the pressure received to the insulator (). That is, if a user lightly touches the keyboard with his/her finger, the shock-absorbing member made of a soft material absorbs the pressure to suppress the generation of sound. If the pressure transmitted to the shock-absorbing member is more than an adequate value, the pressure is transferred to the insulator () and the PCB in sequence. The upper and lower PCBs are connected to the sound chip through a metal interconnection for each key. The PCBs ( and ) can maintain flexibility and elasticity during repeated folding and unfolding operations and generate sound by an electric current that flows when the upper PCB comes in contact with the lower PCB.
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In another exemplary embodiment, the electronic piano may comprise the piezoelectric material and the PCB together. Referring to , a keyboard () comprises a shock-absorbing member (), insulator (), piezoelectric material (), an upper PCB (), and a lower PCB (). Such a keyboard structure may simplify functions of the microprocessor () and the sound chip () compared to the method using only electrical signals and detecting the strength of pressure exerted upon a key. With respect to a correlation between the piezoelectric material () and the microprocessor (), if only the piezoelectric material () is used, generation of sound is controlled by pulse signals according to time using a first and second signals. However, if the piezoelectric material and the PCB are used together, the first signal is not required. That is, only one signal is generated because the voltage generated from the piezoelectric material () is reduced after an electric current flows between the upper and the lower PCBs that replaces both first and second signals. As soon as the upper PCB becomes in contact with the lower PCB, the magnitude of the voltage from the piezoelectric material is perceived and a signal to determine the volume of sound is generated.
A real piano may include two or three pedals. The pedals control the strength and the length of sound being related to movement of the user's fingers. There are several types of pedals such as a damper pedal, a soft pedal, a sostenuto pedal, and a muffler pedal. The damper pedal raises all the dampers (the felt pads which rest on the strings to stop the sound) and lets all the strings vibrate without having to hold the keys down. If a user presses down on the damper pedal, the dampers are removed from the strings and sustain the vibration of the strings long after the user removes his/her finger from a key, thereby providing a larger volume of sound, an abundant timbre, and a rich tone. If the user presses down on the soft pedal, hammers move toward the strings to shorten the distance that the hammers move to hit the strings, thereby producing soft sounds. The sostenuto pedal is a type of selective sustain pedal found in acoustic grand pianos. It sustains only the sounds of keys that were pressed at the time the pedal was engaged-and all other notes -remain unchanged. If the user presses down on the muffler pedal, the felt pads are positioned between the hammers and the strings so that the hammers hit the strings behind the felt pads thereby substantially reducing the volume of sound.
FIG. 5
Referring to , the various functions of pedals in a real piano can be embodied in the roll-up electronic piano of the present invention by connecting the pedals to the sound chip. The pedals can be selectively set up in accordance with types of pianos, for example, grand pianos or upright pianos and be connected to the control part.
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illustrates, in a plane view (), a left side view (), and a front view () of the exterior of the roll-up electronic piano according to the present invention. In the plane view (), the keyboard () comprises four octaves with a length of 690.5 mm and a width of 170.0 mm and, therefore, it has an adequate size for use in a music class. In the front view (), the keyboard () comprises white keys () with a height of 0.5 mm, black keys () with a height of 0.5 mm from the white keys, and a keyboard pad () with a height of 2.5 mm. Therefore, the keyboard has thickness of 3.5 mm so as to be easily foldable and portable. The height and length of other parts, as well as the height of black keys (), can be arbitrarily adjusted. The coupling member () that connects the keyboard () with the control part () can firmly maintain the coupling with repeated folding and unfolding operations. The control part () has an exemplary length of 120.0 mm, a width of 170.0 mm, and a height of 38.0 mm. The white keys and black keys are designed in accordance with the standard configuration of a real piano. Inside the keyboard () is a piezoelectric polymer film to create a piezoelectric electromotive force corresponding to each key of keyboard. The control part () comprises a circuit including an electronic circuit design and a button controlling the volume of sound.
Particularly, a standard volume corresponding to the volume of a real piano is marked around the volume button, thereby maximizing the effect of musical education. In addition, a music stand may be additionally positioned on the border member of the keyboard at a location where the music stand does not obstruct the function of the second piezoelectric material.
The sound chip of the roll-up electronic piano can produce hundreds of tones. The sound chip has a database for tones and sound effects of hundreds of instruments such as pianos, guitars, flutes, saxophones, violins, mandolins, harps, and so on as well as data related to tones and the strength of sound according to the pressure exerted while playing a real piano. Particularly, it is important to set up additional functions according to tones selected. For example, when selecting a violin tone, an additional function is set up in the microprocessor to not generate a signal due to the change of voltage in the piezoelectric material as in the electronic piano because a real violin generates sound not by hitting strings with hammers but by rubbing strings with a fiddle bow.
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illustrates, in a cross-sectional view, a folded state of the roll-up electronic piano according to the present invention. Inside the keyboard () part, except the coupling member (), is a soft piezoelectric polymer film or a piezoelectric fiber and the keyboard pad () is made of silicon rubber so that the keyboard can be easily rolled up.
FIG. 8
FIG. 9
illustrates, in a perspective view, a folded state of the roll-up electronic piano according to the present invention. is a perspective view of the roll-up electronic piano according to the present invention.
The roll-up electronic piano according to the present invention can simultaneously generate six notes or more and can be arbitrarily designed to utilize 3-4 octaves in a music class and 8 octaves for general use. | |
Contract letter are legal document designed for depicting the appeal for an agreement. This kind of document has legal clauses and needs to be reviewed before forwarding to the opposite party. Contract letter can be written contractor for home improvement or to the organization for supplying raw material. Contract letter needs to cover all the points before availing or giving any service.
The date in the letter is very important for any contract. The purpose of the contract needs to be included while drafting the letter.
All the legal clauses and related terms and sections should be specified clearly with consent of a legal advisor. This is because; any contrariety may lead to legal implications or termination of the contract.
Include all the required points in the letter like number hours a worker will work, benefits to the worker or salary to worker and agreement duration.
Contract letters can become well accepted if it could be presented with a tone of gesture. | https://www.letters-home.com/contract-letter/contract-letters/ |
Table of Contents
What Could Turtles Eat? Animal-based food sources for turtles can include processed pet foods like drained sardines, turtle pellets, and trout chow.
You can also feed them cooked chicken, beef, and turkey.
Live prey can include moths, crickets, shrimp, krill, feeder fish, and worms.
What do you feed a pet turtle? Fresh Foods to Feed Your Pet Turtle
Source Protein: Boiled eggs, mealworms, snails, crickets, earthworms.
Vegetables: Corn, beans, beets, carrots, peas, squash, yams.
Greens: Carrot tops, lettuce, collard greens, kale, spinach.
Fruits: Apples, grapes, strawberries, cantaloupe, banana, kiwi, mango, tomato.
What snacks can turtles eat? Cut up pieces of apple, lettuce, blueberries, corn or any number of other produce can make great treats for your turtle. Make sure the pieces are small enough to be bite sized, and remove any and all seeds. Little bits of meat. Your turtle can eat little bits of cooked ground beef, chicken, or pork from the table.
What food kills turtles? Foods to Never Feed Your Box Turtle
The leaves of rhubarb, potato and tobacco plants.
Avocado peel, seeds and leaves.
Tomato leaves and vines.
Poison ivy.
What Could Turtles Eat – Related Questions
What can you feed turtles in a pond?
What to Feed Turtles in a Pond
Can turtles eat bread?
No. Turtles are omnivores (although some are strictly carnivores/vegetarians) and eat a variety of fish, insects, worms, vegetables and plants, frogs, etc. Bread should NOT be part of their diet as their stomachs cannot digest bread and some of the other foods we eat, such as dairy products.
Can turtles eat bananas?
Yes. Turtles can eat bananas and in fact, most owners found them to be quite a favorite. While this is fantastic when you think about saving on costs of fruits, it is important to consider what the nutritious benefits of the banana would be and any effect these may have on your pet’s behavior or health.
Can turtles eat popcorn?
Feeding and Nutrition :: Popcorn
Can turtles eat apples?
Yes! Turtles can eat apples; however, it needs to be a rare treat. Apples are high in sugar and acid content, and if one does decide to feed their pet turtle apples, there are many things to consider beforehand.
Can turtles eat pasta?
Pasta. Just like with the bakery products, they can eat them but no kind of pasta is good for the health of the turtle. So don’t feed your turtle pastas.
Do turtles like dirty water?
Keeping Your Turtle’s Water Clean. One of the most important things you have to do to keep your turtle healthy and happy is keep the water in its tank clean and fresh. If we don’t keep up with it, your turtle’s water will very quickly get dirty and smelly, and your turtles will become ill.
Can turtles eat rice?
As much as possible, offer your turtle high quality, meat-based dry foods rather than those made mostly of wheat, corn or rice.
You may have to break large kibble into smaller pieces – try to feed your turtle pieces that are smaller than the distance between his eyes.
What is the lifespan of turtles?
Even so, if an individual survives to adulthood, it will likely have a life span of two to three decades. In the wild, American box turtles (Terrapene carolina) regularly live more than 30 years. Obviously, sea turtles requiring 40 to 50 years to mature will have life spans reaching at least 60 to 70 years.
What should you not feed turtles?
If it’s created for humans to eat, avoid giving it to your turtle at all times. Avoid feeding your turtle raw meats, feeder fish and fruits, except in limited cases. Kale, romaine lettuce and anacharis or waterweed are the best vegetable and plant staple foods. Use turtle pellets as your protein staple.
Can turtles eat tuna?
As pets, they need to be fed animal proteins and commercial turtle foods. They accept cooked chicken, tuna, mollusks, snails, mudpuppies, shrimp, krill, crayfish, crickets, mealworms, superworms, and small fish.
Do wild turtles eat carrots?
Yes! Carrots are an excellent vegetable for turtles! If you are feeding a wild turtle or a pet turtle, you should grate the carrots into bite-size pieces so they can eat them.
Should I feed my turtle everyday?
They need vitamin and calcium supplements about three times a week and should be fed every day. They’re typically the most active during mornings and afternoons, so those are good times to feed your turtle.
Do turtles eat cucumbers?
It has minimal nutrition, but yes, you can occasionally give a slice of cucumber. None of my turtles has ever wanted to eat it when I’ve offered it, though.
Can turtles eat peanut butter?
In the wild, they like to hunt and bring home live worms, snails, roly poly bugs and creepy crawlies and all taste delicious to a turtle. If you are not sure, keep the snails in a bucket with a mesh lid and feed them peanut butter for a few days. If they are not dead they are ok to feed to your turtles.
Can turtles eat grapes?
Yes, turtles can eat grapes. However, it’s not the healthiest food for your turtle. Grapes contain a large amount of sugar and are poor in minerals. Feed your turtle with grapes only on rare occasions.
Can turtles eat lettuce?
Turtle can eat lettuce, and they may even prefer lettuce to other leafy greens. However, when giving lettuce to your turtle, avoid the iceberg lettuce and also aim at providing your turtle with the most nutritious types of foods. Lettuce should just come once in a while. | https://neeness.com/what-could-turtles-eat/ |
Chock-a-block toy box birthday cake
Serves:
Ingredients
- 2 x 340g packets golden buttercake mix + ingredients to bake the cake
- 1 1/2 quantities Buttercream icing
- 2 x 170g blocks Kit Kat Classic milk chocolate break
- 1 red jube
- 1 cup assorted lollies (jelly snakes, jubes, Jaffas, freckles, Mentos, jelly dinosaurs, jelly babies, licorice allsorts)
- You will also need :1 x 22cm-square cardboard box lid
- 1 x 28cm-square cake board
- small childrens party toys (such as yo-yos, spiky balls, dinosaurs, Duplo blocks)
Method
Preheat oven to 180C (160C fan-forced). Grease a 7cm-deep, 22cm-square cake pan. Line base and sides with baking paper.
Prepare cakes, following packet directions. Pour batter into prepared pan. Smooth top with a spatula. Bake for 1 hour or until a skewer inserted into the centre of cake comes out clean. Stand in pan for 5 minutes before turning onto a wire rack to cool.
Meanwhile, make buttercream icing. Using food colouring, tint chocolate brown. Using sharp scissors, cut cardboard lid in half.
Place cake on a flat surface. Level top of cake, if necessary. Using a serrated knife, cut cake in half crossways. Place one cake half on cake board. Spread top with 1/2 cup icing. Sandwich with cake top. Thinly spread top and sides of cake with 1 cup icing. Spread top and sides of cardboard lid with 1/2 cup icing.
Place on a baking tray lined with baking paper. Refrigerate cake and lid for 30 minutes or until firm.
Meanwhile, break Kit Kat into pieces. Using a pair of small sharp scissors, trim red jube into a heart shape.
Reserve 1/3 cup icing. Spread remaining icing over top and side of cake. Position heart-shaped jube on the top of the front panel of toy box.
Line edges of the toy box with Kit Kat pieces, trimming to fit, if necessary. Place two Kit Kat pieces vertically on each back corner of toy box (see pic). Starting from the bottom, arrange Kit Kat pieces in 3 rows on back of toy box (this is to help support the lid).
Arrange 3/4 of the lollies and toys on top of toy box. Position lid over lollies. Refrigerate for 30 minutes to allow the lid to set in place.
Arrange remaining lollies and toys in and around toy box.
Find more Birthday Cake recipes:
- Balloon birthday cake
- Birthday cake recipes
- Bouncing balls birthday cake
- Chocolate owl birthday cake
- Chocolate-dipped strawberry flower birthday cake
- Cupcake tower birthday cake
- Dino birthday cake
- Easy lolly birthday cake recipe
- Flower pot birthday cake
- Handbag birthday cake
- Hollywood starlet birthday cake
- Ice cream cone birthday cake
- Island birthday cake
- Make-up compact birthday cake
- Marshmallow flower birthday cake
- Mermaid treasure birthday cake
- My Little Pony birthday cake
- Pirate ship birthday cake
- Princess birthday cake
- Simple marshmallow birthday cake recipe
- Soccer field birthday cake
- Spaceship birthday cake
Serving Suggestions
Note
- To make this into a pirate or treasure chest cake you can replace the tiny toys with gold coins and and gold coloured beads.
- You can also make it into a girls jewellery box and fill it with costume jewellery.
- You could bake this cake and freeze the pieces so they are easy to ice.
- Recipes by Kathy Knudsen & Nadia French.Photography by Andrew Young & Sam McAdam-Cooper Styling & food preparation Nadia French. | https://kidspot.co.nz/recipe/chock-a-block-toy-box-birthday-cake/ |
They say that an apple a day keeps the doctor away for us humans—but can dogs eat apples? Here’s what pet parents need to know.
Filter by Pet Type:
Bananas are a delicious and healthy treat for humans, but can dogs eat bananas? Are bananas good for dogs? Here’s what you need to know.
Can dogs eat blueberries? Blueberries are a delicious snack for humans, but are they safe for dogs to eat too? We've got the answer.
Watermelon is a summer favorite for humans, but can dogs enjoy it too? Find out everything about dogs and watermelon.
We’ve got some tips and advice from our experts on when to clean your dog's ears, what to use, and how to your pup to sit still while you do it.
Puppy vomit is a serious matter. Find out more about why you need to be worried when your puppy throws up, and how to make him or her feel better.
If your senior dog has diarrhea, don’t panic: Fortunately, there are several ways to address the uncomfortable problem of old dog diarrhea.
There are a number of reasons that could be behind an old dog throwing up, from eating too many treats to kidney disease.
Learn about the fruits that are safe for dogs, as well as how to prep them.
BeChewy gives you the details on vegetables for dogs: which ones are healthy for canines, plus the best ways to prepare them.
When do puppies lose their teeth, and how can pet parents help their dog through the process? Read our guide to puppy teething symptoms and solutions.
Learn more about what you should expect from a young dog in heat. | https://be.chewy.com/bewell/health-nutrition/page/3/ |
CROSS-REFERENCE TO RELATED APPLICATION
BACKGROUND OF THE INVENTION
SUMMARY OF THE INVENTION
DETAILED DESCRIPTION OF THE INVENTION
This application claims the benefit of priority of U.S. provisional application No. 61/968,194, filed Mar. 20, 2014, the contents of which are herein incorporated by reference.
The present invention relates to beehives and, more particularly, to a beehive insulating cover.
A beehive is an enclosed structure in which some honey bee species live and raise their young. Natural beehives are naturally occurring structures occupied by honeybee colonies, such as hollowed-out trees, while domesticated honeybees live in man-made beehives, often in an apiary. These man-made structures are typically referred to as “beehives.”
Modern honey beekeeping utilizes beehives that do not provide the insulating value of beehives found in the wild. Current hive covers only use passive heat. If the temperature drops drastically or if there is not enough radiant sunlight the hive could die by freezing. Honey bees must keep the innermost part of the hive at least 80 degrees. Covers may not be enough to keep these hives protected.
As can be seen, there is a need for an improved bee hive cover.
In one aspect of the present invention, a method of warming a man made beehive comprises: providing a heating pad comprising: an outer insulating layer; an inner insulating layer; at least one heating cable disposed in between the outer insulating sheet and the inner insulating sheet; electric wiring connecting the at least one heating cable to a power source; wrapping the heating pad around an outer surface of the man made beehive so that the inner insulating layer is adjacent the outer surface; and powering the at least one heating cable via the power source.
In another aspect of the present invention, a beehive cover comprises: at least one heating pad comprising: an outer insulating layer; an inner insulating layer; at least one heating cable disposed in between the outer insulating sheet and the inner insulating sheet; electric wiring connecting the at least one heating cable to a power source; and a waterproof cover covering the at least one heating pad.
These and other features, aspects and advantages of the present invention will become better understood with reference to the following drawings, description and claims.
The following detailed description is of the best currently contemplated modes of carrying out exemplary embodiments of the invention. The description is not to be taken in a limiting sense, but is made merely for the purpose of illustrating the general principles of the invention, since the scope of the invention is best defined by the appended claims.
The present invention may include a DC powered heat tape system for heating Langstrom style beehives in cold weather. The DC powered exterior heat tape pads cover the outside surfaces of the beehive reducing wind-chill and providing minimal heating of the hive. Providing a heat source on the outside of the hive does not interfere with bee behavior during winter months and can help the bees maintain needed temperatures.
FIGS. 1 through 8
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Referring to , the present invention includes a beehive cover. The beehive cover includes at least one heating pad . The heating pad includes an outer insulating layer , and an inner insulating layer , . At least one heating cable is disposed in between the outer insulating layer , and an inner insulating layer , , and the layers , are sealed together. The layers , may be sealed together by tape , an adhesive, or any mechanical fastener. Electrical wiring connects the heating cables to a power source . A waterproof cover covers the heat pad . The beehive cover is then secured to the outer sides of a beehive in order to keep the beehive warm.
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In certain embodiments, the present invention includes a series of pads , . As illustrated in , the present invention includes a pair of heating pads and a pair of insulating pads . The insulating pads may include the same layers , of the heat pad without the heated cables . A first insulating pad covers a first side of the beehive , a first heating pad covers a second side of the beehive , a second insulating pad covers a third side of the beehive , and a second heating pad covers a fourth side of the beehive . The first insulating pad may be connected to the first heating pad , the first heating pad may be connected to the second insulating pad , the second insulating pad may be connected to the second heating pad , and the second heating pad may be connected to the first heating pad . The insulating pads and the heating pads may be secured within the waterproof covers . The waterproof covers may be secured to one another by hook and loop fasteners .
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As mentioned above, each of the heating pads includes an outer insulating layer , and an inner insulating layer , . Each of the outer insulating layer , and the inner insulating layer , may include a polyester batting sheet and a flexible foil sheet . The polyester batting sheets may surround the heating cables and the flexible foil sheets may surround the polyester batting sheets . The waterproof cover covers the entire heating pad and may prevent the heating cables and insulating layers , from becoming wet. The waterproof cover may be a canvas material or other waterproof material.
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As illustrated in , the present invention may include two heating cables suspended in each heating pad . The two heating cables are connected to wiring . The wiring may include low voltage heating cable termination kits that are connected to an auto/trailer connection . The auto/trailer connection may be connected to a power source via a 10-Male/Female connector . In certain embodiments, the present invention may utilize a fuse box to prevent fires. A bimetal thermal switch may be used to turn the heating cables on and off when a certain temperature threshold is detected. A light may be electrically connected to the power source . The light is turned on when power is supplied to the at least one heating cable .
In an embodiment of the present invention, heat cables may terminate on one end with cable termination kits. The heat cable wire leads may be connected to male connectors resulting in two six inch heat cables. One side of a 12V hour wire auto/trailer connector may be connected to female connectors and then connected to male connectors that are attached to the heat cables. This electrical apparatus may be placed between two polyester battings which is then placed between two insulated panels using electrical tape to secure in place. Duct tape may be used to seal the panel. This process produces two 17.5 inch electrical panel inserts which are placed into a two canvas slip covers. Two polyester battings are placed between two insulated panels. Duct tape is used to seal the panel. This process produces two 14.5 inch non-electric insulating pads which are placed into two canvas slip covers. The alternating side of the 12V auto/trailer connector is fitted with a 12V bimetal thermal switch and a 10 amp fuse kit. The remaining wire leads from the 12V auto/trailer connector are fitted with connectors.
The pads fit the sides of a deep Langstrom hive type box providing passive thermal insulation against cold weather. They connect using Velcro® connections. When the electrical leads are connected to a 12V source the pads with internal heat tapes provide heating once the thermal switch is triggered. The switch is triggered when the outside temperature drops to about 40 degrees or lower. The thermal switch remains on until the outside temperature rises to about 50 degrees.
Weak beehives often succumb to weather when the bees are unable to stop the hive heating process long enough to eat. By providing a minimal heat source for honey beehives during cold weather, enough heat can be maintained to assist the bees with survival but not too much heat which can change bee behavior. The pads also provide a wind break and more insulation to the hive which helps in moderate cold periods.
It should be understood, of course, that the foregoing relates to exemplary embodiments of the invention and that modifications may be made without departing from the spirit and scope of the invention as set forth in the following claims.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
is a perspective view of the present invention shown in use demonstrating overall component configuration and external electrical configuration;
FIG. 2
is a top view of an embodiment of the present invention demonstrating the component configuration around a beehive;
FIG. 3
is a detail view of the wiring and heat cables of an embodiment of the present invention;
FIG. 4
FIG. 3
is an exploded view of the electrical heating pad incorporating the electrical component from ;
FIG. 5
is a perspective view of part of the electrical heating panel with a cover omitted;
FIG. 6
is a perspective view of the electrical heating panel and the cover;
FIG. 7
is a rear view of the electrical heating panel with the cover showing rear hook and loop fasteners; and
FIG. 8
is an exploded view demonstrating operational components of the insulating pad. | |
When our memories are erased and we are in amnesia, we have no context for the Electromagnetic Signals that we are being sent during the Ascension Cycle, because we cannot translate them.
So how does our body translate those energy signals of consciousness memory coming from other Timelines?
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The human nervous system and our Lightbody processes the consciousness memories from other timelines in the form of receiving energetic signals first.
If a person is disconnected from their heart and totally enmeshed in the 3D reality, they will not sense or interpret the signal.
If a person is connected and awakening to higher sensory perception, they will sense and try to interpret the signal, and this may come in as a flood of memories, data streams or emotional sensations.
People commonly refer to these as downloads.
Human Nervous System And Lightbody
Translation of Consciousness Memories
When we try to interpret the signal that we have received, we are using the level of consciousness we have access to, to decode it and translate it into some kind of experience.
The event will be translated into something that only has meaning to us personally, something which was recorded in some kind of pre-conscious processing.
This means within some unconscious level or conscious mind level, something was recorded in the storage area of our memories.
The association given to the signal will be related to the memories that are held in our body, the Mental Map that had been created thus far during life experiences, and the things that are very unique to each person.
So usually the signal will be interpreted by comparing the meaning that we have associated within our own memories, to the memories that we have been given access to.
Bring to mind an energetic waveform coming in with information from other timelines or higher dimensions.
As the signals are coming in, the body goes what’s this?
As the signal is coming in, the signal will be interpreted through the memories that are present in the aura of your lightbody and the associations that have been formed from those accumulated memories.
For many people on the earth, the signal is ignored or not sensed at all, because it cannot be interpreted through the Negative Ego and the 3D belief systems that have been conditioned in a materialistic society.
Lightbody
Translation of Consciousness Memories
Developing Links of Association to Memories
However, if we are open and resonant to receive these energy signals, the memory may now translate into some mental images that we can perceive.
When we are communicating with Consciousness or energy, much of the time the translation will come in through images and feelings.
There may be images and colors, sometimes its sensations and feelings, maybe a word or phrases will come through.
Now when a person creates those memory meanings, it means there is an association that has been made in the brain, there is a neural link.
That Neural Link increases more links that associate meanings, which activate more of the neural Synapses in our brain to receive matched signals.
This creates links of association in the neural synapses of the brain, which in turn increases higher sensory perception by linking new sensations of feelings with the brain’s thought processing.
Link
Translation of Consciousness Memories
As we develop more and more of our higher sensory perception, it develops links of association to access more consciousness memories, and it gets easier to actually interpret the meaning we have made through our feeling sensations.
The more we pay attention to our intuitive feelings and the more we ask for truth and for God’s help in developing our discernment, the easier this process becomes.
As we regain more of our memories we regain our cellular knowing, which can perceive and translate these energy signals as communications that exist all around us.
Consciousness Memory
Translation of Consciousness Memories
However, we must remain aware that these signals are intentionally distorted and interfered with by the NAA so that we do not receive them correctly, so that we cannot develop strong Discernment with the information that we do receive.
Each person will interpret signals and consciousness memories that they receive into some context or association that have specific meaning to them.
When we create those meaning links with memory, it means that person has developed associations to these memories that now have some context for that person.
What that signifies is that person has created a new association, there’s a link to process information that was not there previously.
That link to associate memories increases links to activate in our brain.
When this happens, we are creating links of association in the neural synapses of our brain to recognize information or language that was not previously available. We are moving beyond the mind slides and breaking through the memory wipes.
This in turn increases the higher sensory perception energy receivers to connect our emotional experiences with the mental body.
This sensory-feeling and mental-thought form merge improves the way the brain processes complex sensory experiences, so the person can better synthesize feelings in order to more easily describe them and understand what it is that they are actually experiencing.
Consciousness Memory – Toward Ascension
Translation of Consciousness Memories
As an example, a person may have gone through their life not having any interest to explore UFO’s or the existence of extraterrestrials impacting the Earth, because they had no direct memory meaning assigned to them even existing.
Maybe they were led to believe the whole topic was absurd, filled with crazy people.
Then they have an experience, such as a direct encounter with nonhuman entities, and now all of the sudden they have a deep interest in this topic.
They want to understand what has happened to them, and now they comes across extraterrestrial phenomena everywhere in their life, when they had never even noticed it before.
They have built the links of association that give meaning to this context, when it was not there before.
When a person has no comparable memory because they’ve been memory wiped, and there is no consciousness memory meaning available in their mental map, because unapproved topics and knowledge is withheld from them, they may not awaken and remember the truth.
Maybe their body is trying to interpret the signal from consciousness memories in the Galactic History, that translates into some kind of life experience and it is being interpreted incorrectly.
Lightbody Activation
Translation of Consciousness Memories
Humanity perceives things incorrectly because we do not have a memory association or the correct context, because we have been given a false history and are being deceived.
And this is what the negative alien game is to control the timelines.
By controlling humanity into believing in a false reality that is based upon a false history that was written for the 3D control matrix, they keep humanity feeding into their AI program.
More by Lisa Renee
Indoctrination and Capture of the Educational System – Lisa Renee
It may already be obvious that the educational and academic systems are captured and have been designed to target children and young minds with psychological manipulation tactics and the indoctrination of the controller narratives. Additionally, the educational system is based upon the text books provided for and approved by the elite that reinforce the falsified histories that dumb students down and mold them to grow into adulthood as obedient and predictable employees or worker slaves.
Atlantian Conspiracy – Border Guards of Falsified History – Lisa Renee
Apparently, humanity has undergone several more cycles of destruction of our historical records subsequent to the Atlantian Flood, which is the pact of the Luciferian Covenant. After the Christos Mission project connected with Yeshua returning the true Essene records back to humanity, the NAA introduced pernicious religion in the counterfeit name of Christ.
The Science of Cymatics – Lost Knowledge Part 5 – Lisa Renee
To rediscover the profound nature of sound technologies that were used prior to the fall and lost knowledge of our advanced human civilization, we will be led to the importance of Cymatics. Cymatics is the study of the relationship between geometric patterns which emit frequencies that are formed by sound vibrations and how these sound waves can directly influence and create forms in physical matter.
Rediscovering Conscious Sound – Lost Knowledge Part 4 -Lisa Renee
Within the energy matrix of all creation, God source first expresses itself as an eternal field of sound current. Holy Mother and Holy Father, as conscious sound and conscious light, expanding and contracting the sound fields to become light fields and cycling back the light fields to become the sound fields continuously.
Moon As A Weapon – Lost Knowledge Part 3 – Lisa Renee
The lunar matrix of false magnetism transmitted from the moon is designed to emit artificial waves in the geometric shape of eight lines of reflective symmetry in octagonal cymatic shapes, and the internal angles of any octagon is 1080 degrees. The octagon shape has a 1080 Hz frequency that can be doubled harmonically to form into and transmit Black Cube geometries with a 2160 Hz frequency.
War Over Free Energy – Lost Knowledge Part 2 – Lisa Renee
Many times, it has been discussed that the angelic humans on the planet Earth are in the midst of a war over energy which is a war over consciousness, as well as a war over timelines, while most of the human population are asleep to this fact. Through many wars and genocidal campaigns that have taken place over many lifetimes, including alien hybridization breeding programs to infect our original blood and DNA, the intruding races gradually took away more and more of our natural energy resources and blank slated our history during the end of major astrological cycles.
The Lost Knowledge Of Human Civilization – Lisa Renee
As the ascension and disclosure experiences are becoming more amplified, we are slowly starting to recover The Lost Knowledge of Human Civilization, which reveals highly refined periods of living in harmony with natural laws, to promote beautiful and healing environments supportive of consciousness expansion and unity intelligence. This is a momentous time for human liberation. Yet, it is also a time of awakening into greater realizations of betrayal that may be prevalent in many situations as they are being revealed to us.
Ascension and Trinity Gate Load Out – Lisa Renee
Ascension is the divine birthright of all living souls on this planet and as promised by the Christos Founders of the Emerald Covenant serving the Law of One, it is open and available to everyone irrespective of any belief system or religious affiliation.
Trinity Gates And Transit Stations – Lisa Renee
Recently, it has been made clear to me that there has been a Guardian and group mission to help restore access to the Trinity Gates, because these particular portals are the most easily accessible for the majority of the human race that have not activated their higher DNA or expanded their consciousness bodies during the Ascension Cycle.
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After battling them in my yard for a year now, I’ve decided that gophers are evil. They sneak up underground and consume the tender roots of my flowers and trees. They have so many tunnels under my sandy soil that I’m afraid one of these days my riding lawnmower will fall into a sinkhole. I can only imagine that my lawnmower’s front tires will be in a trench, the back tires up in the air, and I will be sprawled on the ground as if being thrown from a horse! I guess I can be somewhat thankful that dirt shows up in mounds on top of the ground so at least I know gophers are close by. It’s hard to do battle with an enemy that you can’t see.
The Bible tells me my greatest enemy, Satan, is a “thief that comes to steal, kill, and destroy.” (John 10:10) Like the gophers in my yard, he likes to dig tunnels and eat at my roots. He can’t have victory over my eternity, so he goes after my joy, my wife, my family, my relationships, my health, my job, etc. He’s trying to get me to take my eyes off of Jesus by keeping me occupied chasing tunnels and leveling mounds of dirt. Make no mistake; he is a real enemy with real power. However, the Holy Spirit lives in me and His power is superior. Jesus is my Lord and I acknowledge Him as the Son of God and my personal Savior. 1 John 4 tells me that God’s children have overcome the world and verse 4 tells me, “He who is in you is greater than he who is in the world.” In addition, in the second part of John 10:10 Jesus says, “I have come that they may have life, and that they may have it more abundantly. Therefore, I’m standing of the promises of God! | http://www.fbcmwc.org/satan-the-spiritual-gopher-reeces-pieces-10-4-2018/ |
The Karma data integration tool allows users to semantically model tabular data (i.e. spreadsheets) in a visual environment. This makes it easy to understand the best way to map data to ontologies and provides a visual comparison to the VIVO-ISF Ontology Diagrams This approach is especially useful for new VIVO adopters and those who prefer not to create and use their own scripts.
It's probably most common for data to be provided in spreadsheet format, which can be very simple to transform into RDF if each column of every row refers to attributes of the same entity, usually identified by a record identifier. The process becomes more complicated if different cells in the same row of the spreadsheet refer to different entities. This page includes example tabular datasets and screenshots of models created using Karma.
Modeling organizations, people, and positions
The following spreadsheet of organizations (one organization per row) is very easy to load into a VIVO:
You can readily imagine representing the information about each organization – id, name, contact information, and web site address – in additional columns. The Unique Resource Identifier (URI) used by VIVO to identify each organizational unit can be generated by using the org_ID and the institutional VIVO namespace. This is the starting point of creating the basic structure of your VIVO data.
The next step is to store the information about people affiliated with those organizational units, and finally, their positions within those units.
A spreadsheet of people data typically looks like this:
In this spreadsheet the person identifier is called UID (your institution will have a different name for this identifier) and is the unique identifier for a person at your institution, by which that person is uniquely identified in databases at the your institution. NETID (your institution will have a different name for this identifier) is another identifier for a person, often used as a username for logging into university systems. The other columns are self explanatory.
In Karma, the model of this data set containing people information is shown in the image below:
A spreadsheet of people’s positions in the organization will look like this:
As one can notice in this spreadsheet we use the UID of the person and the org_ID of the organizational unit to connect the person with that unit and create the RDF statement containing information about that person’s position.
The model of this data set containing information about people's positions within an institution is shown in the image below:
Modeling Academic Articles
A spreadsheet of academic articles is more complicated:
In this spreadsheet we have two important identifiers to connect the person with the article he/she authored: the articleID and the UID. Article ID can be any unique identifier assigned to the article. A model of this data set containing information about academic articles is shown in the image below:
Modeling Grants
Grants also involve making a number of connections, plus adding two classes that are likely not in your spreadsheet, vivo:AdministratorRole and vivo:PrincipalInvestigatorRole (and maybe vivo:CoPrincipalInvestigatorRole).
You must also assign a unique URI to these extra classes, shown below as URIs with the '_role' suffix in the gold columns. Below, the National Science Foundation has been modeled as vivo:GovernmentAgency. If you have a wide variety of funding organization types on a single spreadsheet source, you may want to create a separate spreadsheet and model for your funding organizations, or generalize the type to vivo:FundingOrganization. Likewise with the people modeled as vivo:FacultyMember below, you may generalize to foaf:Person.
Using PyTransform to create URI
Karma's PyTransform capability allows you to transform your source data using Python. A common use of PyTransform is to create additional unique URIs based off an existing one. The URIs with the '_role' suffix above were created using this Python code:
return getValue("AdminDepartmentID")+'_role'
More information on PyTransform is available here within Karma's documentation.
One example when you will need to use the PyTransform option is to create the position URI in order to create the needed triples for representing each person position within his/her institution. As you can see above in the people's position data example there are few columns that you need the values from to create the correct triples for the position. In the drop down menu found on each column you select the PyTransform option and in the window you type:
return "http://vivo.northwestern/position/n"+getValue("UID")+"_"+getValue("org_ID")+"_"+getValue("position_type").replace(" ","_")
In this case the first part of the URI is the Northwestern University namespace which you need to change to your own namespace, such as http://vivo.school.edu/individual/n and then select values from three columns as shown above. Selecting values from three columns is necessary to identify positions uniquely, since most likely you have people at your institution that have multiple appointments and this allows you to create separate triples for each of their multiple positions within your institution.
Another example when you will need to use the PyTransform option is when you create the authorship URI for modeling the publications data as shown above. To do that you would want to select the PyTransform window found on the drop down menu on each column. Once you open the PyTransform window you type in the following:
return "http://vivo.northwestern.edu/authorship/n"+getValue("ID")+getValue("uuid").replace(" ","_")
The first part is your namespace and you would want to change that with your own namespace. The "ID" and the "uuid" are the names of the columns from which we have decided to create the authorship URI and they represent the article unique ID and the person unique ID respectively. | https://wiki.duraspace.org/display/VIVO/Using+the+Karma+data+integration+tool |
First and foremost, this is not going to be similar to other RPGs on this site. There are set actions you can perform each turn, and locations to visit each turn. There also is, as of right now, no player limit. If it gets to be overwhelming overtime, I will place a cap, but until then, registration will always be open. Now, with these few claims and disclaims out of the way, let's introduce the game.
The Story
Everybody was going about their usual lives. What usual means to you I'll leave up to the imagination, as everyone has different backstories, but essentially, nothing out of the ordinary in their everyday life. As they return home for a well deserved rest, a mysterious robed man appears in their place. As you reach out to him, he quickly moves away, almost as if hovering away...as he hovers away, he chants a incantation and before you are aware of what's happening, you pass out.
The next time you awake, you have awoken in a field with an unspecified amount of people. The field appears to have multiple junctions branching out away from it, but staring you right in the face is a giant tower. The tower is stretching into the clouds above, seemingly neverending. You somehow grasp the feeling that there is nothing natural about this world, and that the unexpected...should be expected.
How to Register
To register, you only need to state your name that you want in the game (if no name is given, username will be used instead), and your class. At the beginning of the game, you're limited to what classes you can choose. But as you progress, more and more classes will open up to you. Here are your base stats, without any class:
Name: FlamingRok (Lv. 1)
EXP: 0/100
Class: Villager
HP: 10/10
SP: 5/5
ATK: 1
DEF: 0
MAG: 1
SPD: 1
Coins: 0
Warrior: +1ATK, -1SPE
A warrior can dish out physical pain to the enemies, but will be
vulnerable to magic.
Healer Mage: +5MP, -1ATK
Meant as more of a support, this mage is sufficient at healing, but
can't dish out too much pain.
Blood Mage: +1SPE, -5HP
The definition of a glass cannon. Will deal great damage, but can't take too many hits.
Thief: +1SPD, -1SPE
Quick on their feet, the thief will excel at moving quickly around the map. Magic however, will tear them apart.
Villager: N/A
The default character. Completely average, nothing remarkable or necessarily weak about this guy.
Q: You're stealing sprites! Thief!
A: This is a forum game. I'm not making any profit off of this, and neither are you. Don't ask this actually. Seriously. I'll kick whoever legitimately asks this question.
How to Play
Knowing how to sign up is great and all, but how do you play the game? It's simple, but don't think this game'll be easy later on! The game will be played on a grid-like map, similar to many RPG video games, like Pokemon, Fire Emblem, Dragon Warrior, etc.
Each turn, you have three actions (you start off with three that is). When you start off, you will have three basic actions that you may perform. They follow as:
Move: Moves one space adjacent to your position.
Examine: Examines the space you are standing on. This can lead to items, enemies, or absolutely nothing. Examine also means talk, as in talking to people, though obviously you'll be talking instead of examining people. That'd be weird.
Attack: Performs a damaging action. To start off, you will only be able to perform a basic attack that deals 1ATK damage to an enemy adjacent from you.
Now let's talk stats. The stats correspond to the following:
Lv: Short for level. Every new level you reach, you put a level into one of your stats.
EXP: Short for experience points. When you reach 100, you gain a new level. The higher your level, the slower the EXP gain will be.
HP: Stands for hit points. Get to zero, you die, and respawn at your nearby town. Die without reaching a town and you'll respawn at the entrance to the tower.
SP: Stands for skill points. These will let you perform more advanced attacks later on. When this is at zero, you'll be stuck with your basic attack.
ATK: Short for attack, this is your physical attack stat. Your basic attack uses this, and the higher your ATK is, naturally the more damage you do.
DEF: Short for defense, this stat will reduce the damage of physical attacks. The higher your DEF is, the less damage you'll receive from physical attacks.
MAG: Short for magic, this is both your magic attack AND defense. That means the higher this stat is, not only will you deal more magic damage, but you'll also take less magic damage. Think the special stat in Pokemon Gen 1.
SPD: Short for speed, this determines the movement order when everyone moves. The person with the highest speed moves first, and if two users are tied for speed, the post order will determine the movement order. Keep this in mind while co-operating. Also, every 5 points in SPD gains an extra action for your turn.
Coins: Currency. Allows you to buy fancy things inside shops.
Well, that should about wrap things up. If you have any questions, please post about them in the thread. I'll begin the game when...4 players have signed up. Remember: there is no limit here as of right now.
I will join.
Name: Pixel
Class: Warrior
HP: 10/10
SP: 4/4
ATK: 2
DEF: 0
MAG:1
SPD: 1
Coins: 0
Do i fit the qualifacations?
hello?
earth to FlamingRok?
do you copy?
If I understand this correctly, this character should work:
Name: Yellowcat
Level: 1
EXP: 0/100
Class: Healer Mage
HP: 10/10
SP: 5/5
ATK: 0
DEF: 0
MAG: 6
SPD: 1
Coins: 0
You must be logged in to post a reply!
analytics and serving ads. | https://armorgames.com/community/thread/12594926/the-nonsensical-rpg |
A Milk Paint finish is unique and can range from a solid color to a stain, it all depends on how much water you add! It can be used on any porous surface without a primer. It will never chip or peel on porous surfaces as it soaks in and binds with the substrate.
Mix 1 part water to 1 part milk paint powder. Mix for 1-3 minutes until water is absorbed into the powder and the mixture is relatively smooth. Water to powder ratio may be adjusted to achieve desired consistency. Test your consistency on a spare piece of wood. Your mixture should provide good one coat coverage. If your mixture is too thin it will be more transparent and stain like. Add more powder to thicken and stir again. Excess can be stored in an airtight container in the refrigerator for 5-7 days. Stir frequently while using, as the pigments will settle to the bottom of the container. Typically 1-2 coats is required. Allow 30 minutes drying time between coats. Finish with Miss Mustard Seed’s furniture wax, hemp oil or our water based poly tough coat. | https://gardenhousestudio.com/product/grain-sack-miss-mustard-seed-milk-paint/ |
Established in 1970, the LINDA FARROW brand of luxury eyewear rose quickly to acclaim. Originally a fashion designer herself, Linda was one of the first to treat sunglasses as fashion, her finger always on the pulse of the times. A tireless experimenter, Farrow pioneered many of the shapes and styles that remain au courant today. Her pioneering use of shape helped move eyewear into high-fashion territory, with avant-garde styles favored by stars like Yoko Ono.
After a twenty year hiatus, the brand was revived in 2003 by Farrow’s son Simon following the discovery of a vast archive of vintage sunglasses in the family’s London warehouse. Today, LINDA FARROW is renowned for collaborating with some of the world’s most acclaimed designers – Area, Y/Project, The Attico, Dries van Noten amongst many others.
PR SAMPLES COMING SOON
"Making sure you have a strong culture, where people are aligned and similar in their viewpoints and their working mentality and methodology, I think is massively important." | https://doors.nyc/collections/linda-farrow |
You’re probably interested to know how much it’s going to cost to realize your custom-software dream. The short answer is, we have no idea … yet.
Design/Build
We tend to work on large, complex, long-term projects. We’ve taken a page from the construction industry: separating the design from the build.
How this works in practice is we’ll come up with a budget for a Design phase, which will include interviews with your user base, your stakeholders, your IT contributors, and anyone else with skin in this game. Then we’ll produce a requirements document and a proposal for what we think it will take to build what you’re asking for.
Why We Hate Estimating
As much as we’d all like to believe that we can accurately predict the future, estimating is in reality an inexact science. (Perhaps we should go as far as to say it’s more art than science.) Since our company is staffed by perfectionists, this is our least favorite part of the job because we are never right. It hurts to admit it, but it’s true!
There’s a reason nobody has come up with a computer program that produces 100%-accurate project estimates: people are involved … and people are unpredictable. So here’s what we do to come up with your initial estimate: we consider our experience with previous projects of similar scope and attempt to capture as many known factors as possible. We then consider the possible risks associated with the project, and estimate how those risks could affect the timeline, scope, and budget.
We get better and better at estimating the longer we do it, but we can’t predict the external forces that will change the feature set, budget and/or timeline of your project. Here are a few we’ve encountered:
- Scope creep. As you begin to visualize the possibilities of the new product, you may decide to include more features or make changes to the original design.
- Aha moments. As more information about your processes is revealed, we may need to revise things that we have already worked on to refine them to your new perspective.
- Client delays. Stuff like sickness, vacation, personnel turnover, shifting workload priorities, the guy who never returns our phone calls.
- Our delays. We get sick (and so do our kids), another client has an emergency that requires our attention, some aspect of your project takes longer to produce than we anticipated.
- IT issues. The internet backbone goes out the day before your project is scheduled to roll out, a server hard drive fails, Microsoft issues a security update that breaks everything, the SSL cert didn’t get ordered.
Once these factors are considered, shuffled, rearranged and thrown in the air like tea leaves, we’ll give you a ballpark estimate for budgeting and scheduling purposes. This way, we can get started on the work necessary to refine the estimate. As we get further into the process, we will continue to talk to you about how things are progressing to decide if we need to make adjustments to the original timeline, scope, or budget.
Estimating is the first phase of your project where Think 360™ comes to bear. First, we make sure we’re covering your stated needs [Balance]. Then we plan for the inevitable shifting landscape of features and timeline as the project progresses [Evolution]. And we encourage you to budget for service and maintenance in the years after your new system goes live [Sustainability]. | https://www.streamline-studio.com/cost/ |
Because Pierce County remains at very high risk for transmitting the Delta variant (see COVID Act Now data here), and since the sanctuary is the only space that we can currently make compliant with UUA, CDC, State, and County guidelines, TUUC’s indoor gathering guidelines allow for small group gatherings of up to 8 participants to take place in our sanctuary only. Restrooms will be accessible so long as the guidelines posted in the restrooms are followed. The rest of the building’s offices, rooms, and basement level remain closed at this time.
Small Group Indoor Gathering Guidelines (Approved September 16, 2021)
Please follow these guidelines when gathering together indoors:
- At this time, the only location we can authorize for any church group gathering is the sanctuary of our building.
- Take care of yourself and others by staying home if you are not feeling well, or if you may have been exposed to COVID in the past 14 days.
- Participants age 3 and older must wear masks at all times inside the building.
- All gatherings must take place in the sanctuary only, with HVAC system turned on, all windows open, fans in front of windows blowing fresh air, and HEPA air purifiers within the group’s circle of chairs turned on.
- Please limit the size of your group to 8 participants (including facilitators) and arrange chairs 6 feet apart. Maintain 6’ social distancing throughout the gathering.
- Groups wishing to have more than 8 people will need to obtain permission from staff.
- Please limit the gathering to 2 hours or less.
- Please refrain from eating or drinking inside the building.
- Please follow posted washroom use guidelines.
- Singing even while masked is not considered safe at this time. No singing inside the church, but do consider hand clapping, movement, rhythm instruments, etc.
- Participants should add their name and contact information to the group leader’s sign-in sheet so they can be contacted in case of exposure.
- Wash hands frequently; use hand sanitizer; use sanitizing wipes on shared objects and surfaces.
For all TUUC gatherings, we ask that a leader be identified who will take responsibility to:
- Ensure all participants have access to hand sanitizer.
- Keep a list of participants’ contact information:
- Ask participants to contact the leader if they test positive for COVID-19 within 14 days after the event.
- Contact participants if someone who attended tests positive during that 14-day period.
- Write a brief recap of the event for the ENews – when appropriate – to inform and invite people to participate in future gatherings. Send to Church Administrator Libby Ball by the Tuesday after the event. Photos are welcome! Any participants pictured and easily identifiable should authorize use of their image in church publications including online and on social media.
- Share any insights from the gathering with the Coming Together Team ([email protected]).
We ask that all members and friends of TUUC bear in mind that these guidelines:
- Are a living document – as COVID situations change in our county, the guidelines will change.
- Take into account the current recommendations from the UUA, the CDC, and the Tacoma/Pierce County Health Department.
- Are aimed at minimizing the risk of COVID transmission to the most vulnerable among us.
- Honor that our congregants have varying levels of comfort when participating in face-to-face gatherings.
- Support what we value most about coming together in person — one another’s well-being, safety, and company.
- Encourage all members of our community to get vaccinated against COVID-19 to serve the greater good of public health and safety. And, we respect each person’s/family’s liberty to determine their personal vaccination preferences.
We are a beloved community of soulful people covering a spectrum of ages, genders, sexual preferences, beliefs, desires, talents, and more. We are a beloved Unitarian Universalist faith community guided by seven core principles and congregational covenants that challenge us to interact respectfully with care towards each other and the world. And we are a beloved community that desires to come together in-person after a long interruption, and to do so as safely as possible during this pandemic as we practice loving kindness, patience, humility, and grace.
During this season of the Delta variant, our highest priority is to create an environment that feels as safe as possible for as many people in our congregation as possible. We understand, and encourage the congregation to understand, that given the limited size of our facility and air systems our program offerings will continue to be adjusted or limited in scope.
The Coming Together Team offers these guidelines for how we gather, which we hope will help align the congregation’s expectations and practices for gathering together with our Unitarian Universalist principles, our congregational covenants, and the current limitations of our church facilities and grounds. | https://www.tahomauu.com/indoor-gathering-guidelines-for-small-groups-approved-september-16-2021/ |
More than 150 Hikers Rescued from Oregon Wildfire
They were wearing flip-flops and bathing suits after a short 1.9 mile hike in and a Saturday spent cooling off in the Columbia River Gorge’s Punchbowl Falls. But the bang of a firecracker in the lush but dry forest turned Labor Day weekend into a hellish survival story for 150 hikers.
The Eagle Creek Trail is one of the Gorge’s most popular hikes, located on the Oregon side of the Columbia River near the town of Cascade Locks. Beyond Punchbowl Falls, the trail is a well-worn backpacking route to Tunnel Falls and eventually Wahntum Lake, deep in Mount Hood National Forest. It wasn’t unusual to have so many visitors at Punchbowl Falls last Saturday around 4:30 p.m when one hiker watched a group of teenagers allegedly tossing fireworks over a nearby cliff. Soon after, the forest began to burn. (Oregon State Police said that the 15-year-old suspect had been found and was cooperating with the investigation.)
By the time the dry mosses and pines had fully erupted, 150 hikers were still at Punchbowl, now cut off from the trailhead by a steadily growing blaze. One hiker found a cell phone signal, and rescue helicopter was able to drop a crude note tied to a rock: “Stay put. We see you. Danger.” Eventually another note gave the group directions to head in the other direction toward Wahntum Lake, some 14 miles away.
By 10 p.m., they were met by a Forest Service ranger who hiked in from the other side. As ash and embers flew overhead, igniting secondary blazes around them, they hiked through the night to just past Tunnel Falls, where they bedded down until morning.
Most of the hikers, who had planned on a short day trip, had no extra food or snacks, let alone overnight supplies, until roughly 3 a.m. when a group of firefighters arrived on scene. By first light on Sunday, they were moving again, heading the 11 miles to awaiting busses. The first groups began shuffling out of the woods around 10 a.m., with the last loading onto busses after 1 p.m.
According to the Hood River County Sheriff's Office, all hikers were accounted for and only one was taken to the hospital, for non-critical exhaustion and dehydration. The remainder were reunited with their families after the drive back to their original trailhead.
By Wednesday morning, the Eagle Creek fire had merged with the nearby Indian Creek fire and grown to over 30,000 acres along the Columbia River, from near Crown Point to Cascade Locks. Embers carried by the strong gorge winds also crossed the river, igniting smaller blazes on the Washington side. Officially, it is zero percent contained.
Popular Multnomah Falls is at risk and roughly 45 trails and 9 campgrounds in the Columbia River Gorge National Scenic Area and Mount Hood National Forest have been closed due to the fire. Meanwhile, residents in Portland—roughly 40 miles from Cascade Locks—have reported ash and debris falling continuously since Monday. | https://www.backpacker.com/news-and-events/hikers-rescued-from-oregon-wildfire |
Planning Your Home Kitchen Design
Before you begin planning your home kitchen design, you should first consider the number of users in your household. If you live in a multi-generational household, you should plan for several different types of seating at the kitchen island. In addition to counter seating, you should consider dimmer switches to control the intensity of the light. Lastly, you should consider task lighting in the kitchen ceiling. You can even use a combination of task and general lighting to create an atmosphere that is both comfortable and efficient.
When you are designing your kitchen layout, it is important to remember that the main goal is not to make the space look old or outdated. Each element should be durable and aesthetically pleasing, so you can splurge on the elements you want without spending more than you have to. For example, an island may be more functional than a traditional countertop space, but it will increase the space in the room while maintaining the same level of functionality.
The kitchen of Chef Mike Friedman, owner of the Baltimore restaurant The Red Hen, blends classic and contemporary design. He has used marble counters and a farmhouse sink, but his home kitchen is bright and airy. To navigate, he uses an arrow key to navigate. Adding a wood-burning fireplace adds a cozy feel to the kitchen. It is not uncommon to find a wood-fired oven in the kitchen of a celebrity chef, but this type of stove is not for everyone.
If you have a small kitchen, you can install a walk-in pantry cupboard. Not only does this save space in your kitchen, but it also has the benefit of giving you a spacious pantry. A small breakfast nook in a L-shaped kitchen might even be possible, depending on the size of the area. You can also consider placing a butler’s pantry in the corner of your kitchen. In this way, your kitchen can accommodate two sinks.
The layout of your kitchen is a crucial element of designing a functional and beautiful space. You must consider the flow of food between kitchen appliances and sinks, as well as the location of potential gathering spots. If you’re building a new house, your home kitchen design should take this into account, as well as the social setting. A home kitchen design can be either limited or expansive depending on how you position your appliances and cabinets. The layout of your kitchen can make or break a renovation.
When planning a kitchen layout, you should consider the work triangle. The work triangle is a concept developed in the 1920s as a standard for residential kitchen design. It involves establishing a clear path between the food preparation area and the cleaning area. Each leg of the triangle should be at least 610mm wide, and each kitchen counter or bar counter should have leg clearances of 380mm or less. Having a work triangle in your kitchen design is an excellent way to avoid many common kitchen layout blunders and errors. | https://homegardenview.com/planning-your-home-kitchen-design/ |
We recently took Ian on his first-ever outing to the park. Okay, this isn’t strictly true. He’s been to the park with us before, but he was so young he pretty much lay there like a slug. This was his first time to go when he could enjoy it at all.
Hannah and I walked for a bit when we first got there, and when Hannah’s brothers got there, I started playing frisbee with them, while she kept walking with him. It was also the first time he’s been to the park where he’s been big enough to sit upright in his stroller, and doesn’t have to be strapped into his car seat. That’s a big plus for him… he used to have only two choices of what to look at: Hannah (or me), or the sky.
We figured we’d try the playground, and were pleasantly surprised that, even at eight months, he enjoys at least some of what it has to offer. He took to the swings especially.
We tried going down the slide with him, but we’re both too heavy for gravity to overcome that pesky old friction, as demonstrated by the following diagram:
So that may have to wait a little while so he can appreciate it for himself. All the same, good times. I can’t wait until he starts running around, and doing things like monkey bars, slides, etc. In the meantime, I’m content with him liking the swings. | http://fleastack.com/convincing_john/?p=69 |
maximize your opportunity for a successful rescue? Have you considered the challenges of unique locations and different environments? Could you activate a detailed response plan within minutes that would minimize wasted time and keep potential rescue within the “Golden Hour”? First Aid only supports life, rescue planning SAVES lives.Farley Kautz is the current director and owner of NwBestCPR, a First Aid and CPR training company based in the greater Seattle area. He previously owned and operated Adventure Pursuits, Inc., an outdoor adventure guiding company conducting three to four-week backcountry adventures in Colorado, Utah, Washington, and Alaska. Farley has more than thirty years of experience creating emergency awareness plans for adventure travel and wilderness adventure trips.
Recent Webinars
GIS-based Vulnerability Assessment of Upland Forests in the Cedar River Watershed
Wed, Nov 28, 2018 9:00 AM – 10:00 AM PST
Register Here: https://register.gotowebinar.com/register/9056766733127931395
Presented by Rolf Gersonde. Climate change presents new challenges for ecological restoration. The recovery of ecological functions, either through reducing disturbance or by actively promoting ecosystem development is put into question as climate change is likely to alter ecosystem development and composition with uncertain outcome for ecological functions. In the diverse landscape of the Cascade Range, climate impacts are going to vary depending on topography and ecosystem composition. While exposed sites are likely to experience stronger climate impacts and have greater uncertainty regarding ecosystem recovery, other sites (climate refugia) are likely to be less impacted or will be altered more slowly. To aid forest and aquatic restoration at the landscape scale in the Cedar River Municipal Watershed, we conducted a vulnerability analysis of ecosystems to guide ecological restoration efforts at the landscape scale and adapt to projected climate change. We identified elements of climate exposure and ecosystem sensitivity that could be spatially represented and scaled. The elements were combined in an additive model to result in a landscape representation of climate vulnerability. Adding a spatial filter of areas where climate impacts would have greater effect on management goals and adding operational constraints enabled us to identify priority areas for conservation measures to restore late-successional forest habitat and ecosystem resilience. This approach could be adapted to other landscapes and management goals and offers managers a tool to prioritize restoration efforts in an uncertain future.
Soil Bioengineering for the Restoration of Steep and Unstable Slopes and Riparian Areas
Tue, Dec 11, 2018 9:00 AM – 10:00 AM PST
Register Here: https://register.gotowebinar.com/register/7869941787395129347
Presented by Dave Polster and SER Northwest. Soil bioengineering is the use of living plant materials to perform some engineering function. In some cases, other materials are included. Soil bioengineering systems can be used to treat steep slopes and to provide stability to unstable sites. Soil bioengineering treatments use pioneering species that initiate the natural successional processes associated with the region in which they are applied. This means that in the long run, soil bioengineering systems promote the successional movement of the ecosystem towards later successional stages. Soil bioengineering systems can be used to stabilize sites that conventional systems would cost millions of dollars to stabilize. In addition, since the soil bioengineering systems promote the natural successional development of the site, there is a long term recovery of the site that does not occur with traditional treatments. In addition unlike traditional treatments, soil bioengineering systems promote the sequestration of Carbon thus help with the current climate crisis.
RECENT WEBINARS
The SER-NW 2019 Graduate Student Colloquia: Day 1
Mon, Feb 25, 2019 11:00 AM – 12:00 PM PST
Register Here: https://register.gotowebinar.com/register/2715752335870807297
The SER-NW 2019 Graduate Student Colloquia: Day 2
Emma MacDonald – Community Science for the 21st century, a tool of Environmental Justice Emma MacDonald is currently a graduate student within IslandWood and Antioch University’s Urban Environmental Education program. This program is a novel approach to traditional Environmental Education pedagogy, emphasizing environmental leadership, social justice, and expanding place-based experiential learning to include the built environments of our cities. Emma has a background in conservation research and ecological restoration through several positions across Oregon, Washington, and Hawaii. Emma’s webinar presentation will focus on utilizing community science (formerly known as citizen science) as a tool for the environmental justice movement; mobilizing communities to become involved with all aspects of planning, research, and implementation of results to effect positive and sustainable change. Scott Davis – Mapping Urban Ecosystems: an Asset Management Approach to Environmental Stewardship Urban natural spaces go beyond just parks, such as storm water detention ponds, urban creek systems, boat slips, and vegetated reservoirs. The positive function and value provided by urban ecosystems is often overlooked or minimized as a result of existing degradation and disconnection. This project developed an effective process for mapping and evaluating ecosystem assets on public property in an urban environment. The project focuses primarily on the evaluation and digital mapping of a) vegetation , b) tree canopy, c) habitat functions, and d) management needs. This pilot project is being conducted on property owned by Seattle Public Utilities (SPU), a public utility operated by the City of Seattle that provides fresh water delivery, solid waste management, as well as drainage and waste water management. The project includes field data collection, map and inventory creation, field data analysis, and recommendations for the study sites, including an in-depth recommendation for one highlighted site.
The SER-NW 2019 Graduate Student Colloquia: Day 3
Wednesday, Feb 27, 2019 9:00am-10:30am PST
Register Here: https://register.gotowebinar.com/register/4965692875394182145
Wetland Re-vegetation: Tools, Techniques and Best Practices
October 24, 2018, 9:00-10:00am PT
Register Here: https://attendee.gotowebinar.com/register/3688481473265972739
Speakers:
Sage-grouse Habitat Conservation Through Prisons
September 28, 2018, 9:00-10:00am PT
Register Here: https://register.gotowebinar.com/register/8906729573830608387
Post-fire Restoration in the Great Basin: Challenges, Opportunities, and a Call to Make Adaptive Management Real
Monday, August 20, 2018
9:00AM Pacific time
The vast sea of sagebrush-steppe rangelands that supported iconic wildlife and many ecosystem services has been heavily impacted by exotic plant invasions and altered wildfire, motivating one of the largest restoration and rehabilitation efforts globally. Members of the Great Basin Chapter of SER will describe the efforts, past and future, from scientific and management perspectives, and address the needs and prospects for an adaptive management approach.
Matt Germino is a research ecologist with the US Geological Survey, Boise ID, whose focuses on basic and applied aspects of plant-soil interactions in restoration and has been conducting research in the adaptive management framework
Cindy Fritz is an Emergency Stabilization and Burned Area Rehabilitation specialist with the Bureau of Land Management, Boise ID, and has led the fire rehabilitation efforts on thousands of acres for over 20 years.
Jeanne Chambers is a research ecologist with Rocky Mountain Research Station, USDA Forest Service, Reno NV who has led the development of science and application of resistance and resilience concepts in sagebrush ecosystems and effective research-management partnerships.
Dave Pyke is a research ecologist with the US Geological Survey, Boise ID who has led the science of assessing post-fire restoration treatment effectiveness along with rangeland monitoring approaches.
Register here. | https://chapter.ser.org/northwest/events/webinars/ |
An Age-Old Question
The aging population; what it means on a local level.
Andy Rooney once summarized the dilemma of aging by quipping that “the idea of living a long life appeals to everyone, but the idea of getting old doesn’t appeal to anyone.”
Appealing or not, our demographic patterns reveal that more people are living longer than ever — with potentially profound implications for our country and our region. We live in an unequivocally aging population, defined by demographers as one in which older residents constitute an increasing proportion of the total populace.
The average life expectancy in the United States, according to the Centers for Disease Control, increased from 69.7 years to 78.7 years between 1960 and 2010. In 2010, roughly 40.2 million Americans—13 percent of the country’s population—were 65 or older, compared to 16.6 million (9 percent of the overall population) in 1960. The median age of the country’s population rose from 29.5 in 1960 to 37.2 in 2010.
In the period between 2000 and 2010 alone, the number of Americans 65 and over increased by 15 percent, or more than 5 million people; the number of residents ages 85 and older grew by 1.2 million, or 30 percent. By comparison, the overall population of the country grew by 10 percent, the number of residents under the age of 20 grew by 3 percent, between 20 and 35 by 6 percent, and between 35 and 55 by 4 percent.
The general reasons for these trends are well understood. For one, the generation approaching and surpassing the 65-year threshold is historically large, products of the post-World War II surge in births. Secondly, advances in medicine and long-term care have extended life expectancy and lowered mortality rates. And finally, birth rates have, on the whole, declined consistently over the past 30 years, from just under 17 births per 1,000 residents in the early 1990s to 12.5 in 2014.
Statewide and locally, our recent population shifts have been complicated by Hurricane Katrina, but we still mirror the national landscape. Between 2000 and 2010, Louisiana’s population increased by 1 percent, whereas the population ages 65 and over grew by 8 percent and the population 85 and up rose by 12 percent. Conversely, the number of residents between the ages of 35 and 55 declined by 5 percent, and the under 20 population declined by 8 percent.
In the New Orleans metro area, we have nearly 15,000 more residents ages 65 and older than prior to Katrina, an increase of 10 percent. Meanwhile, the number of residents in nearly every age grouping under 55 has decreased significantly, the exception being the fact that we have 16,000 (10 percent) more residents between the ages of 25 and 34 than before the storm.
What do these trends mean for our country and our region? First, it is important to realize that nearly every country in the developed world has similar demographic trajectories, and that the United States actually fares better than many nations. The reason is that despite the fact that our birth rates are not robust enough to avoid overall population decline, our immigration rates provide valuable and meaningful replenishment of our working-age and younger populations. It is imperative that we maintain policies and employment opportunities, both nationally and regionally, which accommodate constructive numbers of new Americans.
Secondly, political and business leaders must engage with one another to foster economic and workforce policies that account for these undeniable demographic circumstances. Many economists and think tanks have proposed thoughtful and diverse solutions—from phased retirement programs to trade policy adjustments to ensuring the viability of entitlement programs for the elderly—that merit consideration and implementation. Even as we address the economic and political problems that are often more immediate and more conspicuous, this longer-term and less obvious demographic reality requires exigent attention.
Robert Edgecombe is an urban planner and consultant at GCR Inc. He advises a wide range of clients on market conditions, recovery strategies, and demographic and economic trends. | https://www.bizneworleans.com/an-age-old-question/ |
Accounting – an explanation of the profit and loss account
From the secular point of view, what is the income statement? We examine the various elements of the profit and loss account: revenue, costs of goods sold, expenses and net income. Revenue reporting is useful because it is the story of business activity in order to have the budget for future operations and the risk of future cash flows being assessed. The profit and loss account is also recognized as a profit and loss account.
Due to the nature of the profit and loss account, the operations reflect the "30 June 2006" or "the year ending 31 December 2006". This is different from the balance sheet reflecting a certain date. Revenue statements include so-called "temporary" accounts and the balance includes "permanent" accounts. Temporary reports, such as sales revenues and expenses, are "closed", the net income / loss is determined and this net amount is made in the equity of an owner. The accounts will end at the end of a period, be re-opened and reused for the next period.
Revenue Statement Revenue Reduced by Selling Goods, Less Expense, Is Equal to Net Income or Loss. Revenues are sales of merchandising products; what are you selling? Sell goods? Are you selling services? This sales price is the number of items sold. Sales generally appear as net sales, and sales-related adjustments include sales allowances, revenue and allowances.
If the business sells the goods, the next part of the profit and loss statement corresponds to the price of the goods sold. If the store sells the services, then this part will not be. Since this is a large part of the expenditure for the retail facility, while it is a cost, it will be separate from other spending. The business needs to know how much inventory started and how much inventory was at the end of the period. You must also know how much inventory you have purchased over the period. There are many ways of inventory valuation, such as Fifo (first, first out), Lifo (last, first out), average cost, specific identification, etc. With respect. Given the high level of look at income statement, it is important to note now that the subjectivity of inventory methods makes it more art than science. The start of the inventory and the goods purchased are the same as those available for sale; the goods intended for sale minus the finishing inventory of the goods sold.
Expenditures are cash outflows for business. Some expenditures can easily be identified such as rental or mortgage loans, utility services, office payments, stocks, etc. These are called selling and administrative costs. Selling costs include costs related to the sale of goods, such as salespersons, shipping, freight, advertising, etc. Research and development costs are also valid. If you own the building, the vehicle or the equipment you have an amortization charge. This only means that if you have a device that lasts for a few years, you can write some of the cost of that asset for years as depreciation. Similarly to inventory costs, there are many ways of subjective determination of depreciation, such as straight lines, accelerated depreciation methods, etc. Thus, not only one possible answer can be given to determine depreciation costs.
To determine net income or loss, revenue should be deducted from the cost of the goods sold, less the costs. If this number is positive then it is a net income. If this number is negative then it is a net loss. This amount is excluded for capital access, for example, to the capital account of an owner for the sole proprietor or shareholder's assets.
Expenditures and / or revenue outside of normal business activity should be included at a separate stage. For example, a shop is a shoe store and sells one of their buildings or part of a vacant item that creates a cash inflow. This is not what you expect from a shoe store. In order for the profit and loss account to be comparable annually, this special income should be shown in a separate chapter above the net income.
So we reviewed the income statement at a high level, determined revenue components, the cost of sales, expenses, and net income. We have mentioned areas such as inventory valuation and depreciation, where different methods can be used to determine the various financial amounts. Businesses need to carefully select their methods and stick to their consistency. It is not completely impossible to change these valuation methods, but special disclosure would be required. Once we understand the basics of the profit and loss account, this will help us understand income statements from different companies, regardless of their nature. business. | http://portnokomis.com/accounting-an-explanation-of-the-profit-and-loss-account/ |
Powering a light bulb
The following question was asked recently by a concerned citizen.
How many litres of oil would be needed to run a 100-watt electric bulb consistently for one year? Similarly, how many kg of coal to accomplish the same thing?
On the surface of it, the answer could be calculated by any reasonably smart high-school physics student. We suspect however, given the source of the question, what was being sought was a deeper, more fundamental answer that goes to the core of the energy crisis that Jamaica now faces.
So, first, here's the easy part. A barrel of oil, often referred to as barrel of oil equivalent (BOE) contains approximately 1.7 MWh of energy. Generation plants in the current JPS system extract approximately 35 per cent of the energy content of a barrel of oil (measured by the average heat rate) and convert it into electricity.
Transmission and distribution losses take away another 23 per cent of this energy before it gets to the customer's premises. The light bulb ends up receiving 27 per cent (0.46 MWh) of the energy from the barrel of oil. Burning a 100W incandescent bulb for 24 hours a day and 365 days a year (not advisable) requires 876 KWh of energy (roughly equivalent to half-barrel of oil).
Very inefficient
Incidentally, the typical 100W incandescent bulb is very inefficient, converting less than 20 per cent of the energy consumed into visible light, the rest being dissipated as heat so that the amount of useful energy consumed from the barrel of oil in this scenario is really only five per cent.
Nevertheless, the initial answer to our question is that given the current inefficiencies in the electricity production and distribution system, it requires just about two barrels of oil to keep the light bulb burning continuously for a year.
This calculation holds whether the fuel source is oil, gas or coal. It will take 396 kg of coal and 333 litres of LNG to keep the light bulb burning for one year.
Using nominal trading prices for each fuel type indicates the relative costs. So, hypothetically speaking, all other things being equal (of course they're not but the simplification suits the exercise), using LNG instead of oil would cost roughly half the amount to burn the light bulb, and using coal would be about 1/7th the cost.
The earlier analysis indicates that regardless of fuel, the Jamaican electricity production and distribution system only delivers approximately 27 per cent of the fuel purchased to the end consumer.
Consumers with poor energy conservation practices such as continuously burning incandescent bulbs, or inefficient building air-conditioning systems, contribute further to this energy waste. Jamaica's oil bill in 2010 was 122 per cent of all export earnings.
A significant percentage of this oil bill is used to produce electricity and, on average, we waste 73 per cent of this costly commodity in delivering electricity to consumers.
What is even more perverse about this situation is the well-known fuel pass-through clause in the electricity tariff structure that requires the consumer to pay for the cost of fuel, regardless of how inefficient the procurement or conversion processes become. Is there a better way? Is there a model of electricity production and delivery that can begin to seriously impact on this seemingly perpetual, but clearly unsustainable situation.
Fundamental conversation needed
While the predominant local energy debates about LNG versus coal, and fuel diversity, are important issues; getting Jamaican industry to a competitive 10-15 US cents/KWh will require a more fundamental conversation about the structure of the industry and the way that usable energy is extracted from a barrel of oil or a tonne of coal.
We believe that there is a different industry model that could potentially suit small island states like Jamaica.
A model that takes advantage of new- generation technologies and more efficient industrial structures capable of delivering electricity at lower cost. A model that can extract more usable energy from the barrel of oil, the litre of LNG or the kg of coal to burn the light bulb while running the air conditioning and the water heater.
We believe such a model of electricity sector reform could potentially impact many of the prevailing issues and concerns currently being contemplated, such as:
- providing more competitive industrial and commercial rates in the medium - long term
- increasing the opportunities for domestic private-sector investment/participation in the electricity sector
- providing increased fuel diversity as a natural consequence of investor-determined choices and risks
- considerably reducing transmission/ distribution losses as a result of a more balanced distribution of supply and demand
- creating practical opportunities for increasing the share of renewables in the supply of electricity.
Space doesn't permit a more detailed examination of this conceptual model in this article. Suffice to say that it will require a deep commitment to the national interest and the willing participation of all stakeholders in the electricity sector, including the JPS, the OUR and GOJ, the local private sector, academia and consumers.
All have a role to play in helping to return sanity to a very troubled sector. We will complete the presentation and examination of this conceptual model in subsequent papers. | https://jamaica-gleaner.com/gleaner/20120423/news/news2.html |
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You may find that you sometimes lose patience with yourself. You want to think, feel, or act differently than you do; and so your inclination is to tell yourself to just be different in those ways. When this doesn’t happen, you become frustrated and try harder. Rather than making progress, you just end up being harsher with yourself. Despite your intentions, this approach won’t help.
What you are failing to take into account is the part of you that’s not ready to change. Whatever its reason is, it will probably just feel intimidated by your self-bullying. So, you need to approach it gently.
To clarify, consider the following scenario: You come across an abandoned child (or dog) in an alley. He cowers fearfully in a corner. You want to help him, so you approach him slowly and with a quiet, reassuring voice. With time and patience, you can probably win his trust and guide him to help.
This is the same approach that you need to take with yourself. So, using this analogy, do the following:
Identify a self-criticism: Think about a trait or situation that prompts you to be self-critical.
Imagine the victim in you: See the part of yourself receiving this criticism as a hurt or scared child (perhaps you at a younger age). Try to really connect with what that part of you is feeling.
Practice self-compassion: Choose to be gentle and reassuring with him. You might find it comforting to imagine hugging that part of you, or just placing your hand on his shoulder.
Take time to practice this exercise. Repeat it. And, just as you can calm, reassure and embolden a frightened child or stray dog with kindness and patience, you can be the same loving force in your own life. And with this force, you will find that you feel good about yourself, are happier in your life, and have the resilience to persist in your goals.
Dr. Leslie Becker-Phelps is a clinical psychologist in private practice and is on the medical staff at Somerset Medical Center in Somerville, NJ. She also writes a blog for WebMD (The Art of Relationships) and is the relationship expert on WebMD’s Relationships and Coping Community.
If you would like email notification of new blog postings by Dr. Becker-Phelps, click here.
Making Change blog posts are for general educational purposes only. They may or may not be relevant for your particular situation; and they should not be relied upon as a substitute for professional assistance. | https://www.psychologytoday.com/blog/making-change/201307/nurture-personal-growth-self-compassion |
Filling:
- 1 cup creamy cashew butter, natural no oil added
- 3 tbsp coconut flour
- 1 ½ tbsp coconut sugar
- ½ tsp vanilla extract
- Preferred jam/jelly
Coating:
- 3 (3oz each) vegan chocolate bars, chopped
- 3 tbsp coconut oil
Instructions
- Line a mini muffin tray with liners or coat with coconut oil. In a medium size mixing bowl, stir together cashew butter, coconut flour, coconut sugar and vanilla extract and set aside.
- In a microwave-safe bowl, add chopped chocolate and coconut oil. Microwave for 30 seconds, then remove and stir. Repeat process until chocolate has melted, about 90 seconds total. Finish by separating melted chocolate into two bowls.
- Spread the first bowl of chocolate evenly between your 24 mini cups. Once added, shake the pan to ensure the bottom is evenly coated. Move to the freezer for chocolate to harden, about 10 minutes.
- Remove chocolate cups from freezer and add 1.5 – 2 teaspoons of cashew filling to each mini cup. Top with a small spoonful of preferred jam/jelly.
- Lastly, repeat the process with the second bowl of chocolate, spreading evenly between the 24 mini cups to ensure the filling is fully covered. Again, shake the pan to move around the chocolate if necessary.
- Move cups to the freezer to set, another 10-15 minutes. Cups can be stored in an airtight container in the fridge for ~1-2 weeks or in the freezer for long term storage. | https://bestofvegan.com/cashew-butter-jelly-cups/ |
What Makes a Good Lesson Plan?
Sep 30, 2014 Lesson Planning 5389 Views
DO ALL TEACHERS HAVE LESSON PLANS?
Amongst teachers, the question of whether or not one should always have a detailed lesson plan is up for debate. Some strongly believe that the possession of a detailed written plan hinders the ability of a teacher to be flexible and really respond to their students’ needs as they arise. They say that lesson plans can result in mechanistic and predictable lessons. Moreover, the more experience one gets in teaching, the less need there is for an actual written document. That is not to say that a teacher does not plan what they are going to teach, but that they feel less need to formally write it down. Experienced or not, however, a teacher should always plan what they are going to teach. There are few who would be confident enough to begin a class without the foggiest idea of what they are actually going to present for the next hour. However, amongst new TEFL teachers, you will find that you do benefit from, and want to have, a formally written lesson plan to guide you through your paces and keep the lesson organised in your head. Indeed, your colleagues and your students will expect such professionalism and preparation from you, and if you are ever monitored by an observer they will ask for a lesson plan for reference. How you use a lesson plan, once it is written, is up to you. There are some teachers who, once in the classroom, pay less attention to their plan as they respond to ad hoc situations and student questions in a lesson. Others will follow their plans quite closely, although always allowing room for flexibility. Whatever you decide to do, there are real benefits from writing a lesson plan for the trainee teacher.
WHY SHOULD A TRAINEE TEFL TEACHER HAVE A LESSON PLAN?
When you begin a career as a teacher, the whole experience from undertaking a course to eventually arriving at teaching your first class can be a little daunting. There is a lot of information to absorb, plenty of theory and pragmatic advice to receive, as well as a bank of language to teach. Most trainee teachers have never taught in front of a class before. Nerves can run high, and a teacher can fret in such a situation. The best way to allay nerves and fears is to be prepared. The more you think about a lesson before you give it, the better you will be in its delivery. This is because you will have allowed yourself the opportunity to decide on the aim of the lesson and how to achieve it: i.e. what your students will be learning, why they should be learning it and how they are going to do so. You can then anticipate any problems which may arise and think of potential solutions. In short, a lesson plan allows you to set things straight in your head – to structure a lesson logically and clearly. The result of this process will be making you a more confident and relaxed teacher who can deliver a lesson with a clear direction and point. Students will only be thankful for this – none want a chaotic and messy language class. Moreover, a lesson plan allows you to review aspects of the lesson before it has been given, which you can then adjust if necessary. For instance, if from your lesson plan, it looks as though you have not given enough time to free practice, then you can scale back other activities as necessary to boost the former’s proportion. In a similar way, with a lesson plan you can judge estimated timings of the various stages and activities so that you ensure you do not run out of activities before the lesson is finished, or indeed run out of time before your last stage is over. From this, you should always have extra activities up your sleeve, or be ready to adapt your activities to save time.Overall, you can assess if the lesson seems student-centred enough. The key to TEFL is giving your students enough time to practice using English, as this will probably be the only time during the week they can practise the language. A lesson plan should reflect this and allow for STT (Student Talking Time) as much as possible. That STT should be balanced between pair work, group work and whole-class work, which again can be reflected in the lesson plan. Therefore, a lesson plan is a highly useful document which allows you to plan ahead a more effective lesson.
WHAT MAKES A GOOD LESSON PLAN?
In order to write a good lesson plan, the first thing a teacher needs to think about is flexibility. There must be a careful balance to ensure the plan has an organised structure yet one which can be adapted as and when needed. For example, this could mean extending the presentation stage of a lesson if it becomes clear the students’ have not grasped the form or meaning of a particular grammar structure, or shortening controlled activities that seem to be boring students in favour of a livelier activity in the free production stage. Timings are very important to teachers, but they need to be flexible. If something is taking longer than anticipated, a teacher has to adjust their plan to focus on delivering the main aim of the lesson, whilst dropping activities that may have supported only secondary, subsidiary aims. Quite simply it is of little use having a strict and entrenched plan in an environment that is heavily influenced by the behaviour, needs and desires of students. Without question students need to be guided, and benefit from a clear, well-thought out lesson – the result of a lesson plan – but that lesson cannot be so inflexible that it cannot be changed when required. A lesson plan is meant to be a helpful guide; a reference for the teacher to use during class – but it is not meant to be an absolute blue-print to follow without fail. There will be many on-the-spot decisions a teacher will have to make in order to respond to students’ needs yet achieve the main aim of the lesson. These will alter the plan. Plans are guides, not blue prints. In-built flexibility is paramount. However, as plans are guides, it is very important that they should be laid out in a clear and organised manner so that a teacher can refer to it with ease in a lesson. A teacher should be able to quickly scan the document, understand where they are and what they need to cover next. Similarly, a plan should be detailed enough that if another teacher were to cover your lesson for whatever reason, they would be able to on the basis of your plan. As a result, a plan should be structured, informative and neat. The most effective lessons are organised around appropriate, attainable and well-defined aims which fit into the wider schedule of work based around the course syllabus. These aims would have considered the language level of the class, the students’ interests and learner motivations. Once the aim has been decided, the teacher must research what target language they will present based on what they assume the students’ already know and what they should know by the end of the lesson. At this point, it is critical a teacher anticipates the problems which may arise in the classroom when presenting and practicing the target language and come up with potential solutions. This will prevent any unwelcome surprises and allow the teacher to react most effectively in the classroom. The presentation stage and activity stages should then be arranged according to these aims and their estimated timings and mode of interaction written down. Stages should flow in a logical sequence from one to the other, and students should understand what they are doing and why they are doing it. In other words the students should be able to appreciate the overall progression and point of a lesson. Activities planned should be varied, fun and stimulating with changes of pace and different types of interaction accounted for. Timings should be estimated in order for a teacher to judge whether she is likely to finish on time or not, or if activities should be added or left off. Finally the teacher will review what language materials and visual aids they might need to bring to class to support their teaching. If all this is considered in the writing of a lesson plan, then the teacher has most likely prepared what will be a wellthought out lesson. A flexible plan which is clear, informative and logical in terms of its pre-plan and procedural stages is the best foundation for an effective lesson. It really is one of the most important tools with which a teacher can arm herself with to combat the potential for confusion in TEFL. At this point, it would be useful to assess in more depth each separate component of a typical lesson plan to give you the confidence to start thinking about how you might want to write plans yourself. Remember, there is no right or wrong way to write a plan – really it is up to the individual style and preference of the teacher. However what follows below is a layout with proven efficacy. To read more about lesson plans, why dont you try out our course at www.teachteflfirst.com. | http://eslarticle.com/pub/lesson-planning/108283-What-Makes-a-Good-Lesson-Plan.html |
Astronomers have used NASA’s Chandra x-ray Observatory to discover a powerful jet from a very distant supermassive black hole that is being illuminated by the oldest light in the universe.
This discovery shows that black holes with powerful jets may be more common than previously thought in the first few billion years after the Big Bang. The light detected from this jet was emitted when the universe was only 2.7 billion years old, a fifth of its present age.
In this period of the universe, the intensity of the cosmic microwave background radiation or CMB left over from the Big Bang was much greater than it is today.
Because we’re seeing this jet when the Universe was less than three billion years old, the jet is about 150 times brighter in X-rays than it would be in the nearby Universe,
said Aurora Simionescu ~ lead author of the study published in The Astrophysical Journal Letters.
This jet was found in the system known as be B3 0727+409, is at least 300,000 like years long. Many long Jets emitted by supermassive black holes have been detected in the nearby universe, but no one is exactly sure how these jets give off X-rays. In B3 0727+409, it appears that the cosmic microwave background is somehow being boosted x-ray wavelengths.
Astronomer say that as the electrons in the jet fly from the black hole at close to the speed of light they moved through the ‘sea’ of CMB radiation and collide with microwave photons boosting the energy of the photons up to the x-ray band allowing it to be detected by Chandra.
We essentially stumbled onto this remarkable jet because it happened to be in Chandra’s field of view while we were observing something else,
said co-author Lukasz Stawarz.
This implies that the electrons in the B3 0727+409′ jet must keep moving at nearly the speed of light for hundreds of thousands of light-years. Electrons in black hole’ jets are usually quite brighter at radio wavelengths so typically these things are found using radio observations.
The discovery of the jet here is special though, because so far almost no radio signal has been detected from this object while it’s easily see in the x-ray image.
So far, astronomers have identified very few Jets distant enough that their x-ray brightness is amplified by the CMB as clearly as in this system:
If bright X-ray jets can exist with very faint or undetected radio counterparts, it means that there could be many more of them out there because we haven’t been systematically looking for them.
The phenomenon also might explain more about conditions in the early Universe.
Supermassive black hole activity, including the launching of jets, may be different in the early Universe than what we see later on,
according to co-author Teddy Cheung.
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Q:
How to show that a rule of inference is sound or complete?
I've been reading patrick suppes’ an introduction to logic, and the rules of inference mentioned in it, like universal specification, conditional proof, etc. haven't been proven to be sound or complete in that book. So how are you supposed to prove these rules?
A:
First of all, you won't show that a single rule of inference is sound and complete, but rather that the system of rules as a whole is.
Proving soundness
Γ ⊢ A ⇒ Γ ⊨ A
-- which essentially states that your system doesn't produce nonsense --
is relatively easy: One proves the soundness of each axiom and rule by demonstrating that the axiom is a tautology or, respectively, that if all premises of the rule are true, then the conclusion is true as well; then uses the induction principle to say that if each of the individual rules are fine, the soundness will be preserved when plugging them together to a larger derivation.
Proving completeness
Γ ⊨ A ⇒ Γ ⊢ A
-- which states that every logical inference will be captured by the system --
is trickier. The usual proof proceeds as follows:
Proving by contradiction:
Assume Γ ⊨ A, but not Γ ⊢ A.
Then Γ ∪ {¬A} is not satisfiable (= has no model), but consistent (= no contradiction can be derived from it).
However, it can be shown that any consistent set of formulas is also satisfiable. (*)
Contradiction, so Γ ⊨ A ⇒ Γ ⊢ A.
The proof of (*) is the complicated part.
For propositional logic, one goes by extending a consistent set of formulas to a maximally consistent one, i.e. one where no more formulas can be added without making the set inconsistent; the proof that this consistent extension is possible makes use of the fact that the set of formulas of propositional logic is enumerable. Maximally consistent sets have certain properties, eventually making it possible to show that consistent sets of formulas are satisfiable, i.e., (*).
For predicate logic, where quantifiers need to be taken care of, one starts by constructing from the assumed-to-be-consistent set of premises its so-called theory, which is a set of formulas that is closed under deduction, i.e. everything that can be derived from the set and everything that can be derived from those derived things and so on, is included in the theory.
Theories are then transformed into so-called Henkin theories, in which, essentially, for every existential claim ∃xA(x) there ought to be a name c referencing a concrete object (= a witness) that satisfies that property, A(c).
Lindenbaum's lemma then states that consistent theories can be turned into complete consistent theories, which are theories that decide every formula, i.e. for every formula of the language, a complete theory can either prove the formula or its negation. (Note that this is parallel to the notion of maximal consistency from above.)
For these complete consistent Henkin theories the model existence lemma holds, which states that a consistent set of formulas has a model.
After limiting the language back to the one before henkinization, (*) follows as a corollary.
A detailled proof of soundness and completeness can be found:
- for sequent calculus: e.g. in H.-D. Ebbinghaus/J. Flum/W. Thomas, Mathematical Logic
- for natural deduction: e.g. in D. van Dalen, Logic and structure
- for Hilbert-style calculus: e.g. in H. Enderton, A mathematical introduction to logic
The proofs of soundness for the indvidual rules of course will look different for different rule systems, but the proof idea as a whole is the same across all these presentations.
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Of sustainable cities
IBN-e-Khaldun declared towns as places that bulwark their residents from all pernicious things and are pure of all kinds of adulterants.
Present-day megalopolises, on the contrary, offer a different story. Today, cities look like asphalt jungles. With the supersising of the populace, small towns are turning into mega burgs. 1n 1950, only eighty-three cities with a population of over a million existed.
By 2025, the number is likely to be over thirty mega-cities. Most of these places are fast losing their sustainability which in terms of climate, transport, livelihood and food is an alien concept in many developing societies.
Sustainability is the puissance to draw a stasis between consumption and availability of resources, meeting the current needs without exhausting the ecosystem. Modern metropolises are home to environmental degradation, widespread inequality, urban flooding, food insecurity and slum areas, making cities devoid of the features of sustainability.
Various factors have led us to this situation. First, present-day cities clamor for electricity produced through fossil fuels. Transport facilities also beckon greenhouse gas emissions. This sector is liable for twenty-three percent of C02 emissions.
Mega structures in cities produce almost 39% of carbon dioxide emissions, according to the US Green Building Council.
Hence, the use of fossils has dispossessed cities of their environmental sustainability. Second, modern metropolises rely on food imports from the adjoining areas and are not sustainable enough to feed the ballooning population.
To fulfil the requirements for food, livestock farming has become necessary for feeding the population and is a driver of deforestation. Third, cities produce an elephantine amount of waste, and waste collection mechanisms in cities also add up to greenhouse gas emissions.
The disposal of waste in the surroundings of cities also invites the anathema of air and water pollution to mega towns. Fourth, the imminence of urban flooding has become a reality for modern cities.
Feckless sewage systems and unplanned cities are a cause of urban flooding. Fifth, slums around cities are a breeding ground for poor human development, depriving people of economic sustainability.
According to the UN, between 2014 and 2018, the proportion of the urban population living in slums worldwide increased from 23 percent to 24 percent, translating to over one billion slum dwellers.
Indeed, sustainable cities ensure parity and equality in opportunities for all people. But developing countries do not believe in devolution of power to the grass-root level, which jacks up inequality among the social strata, making cities unsustainable. In fact, modern cities have turned into wicked places owing to the prevalence of these factors. Therefore, resilient and self-sustaining cities are the need of the hour.
The idea of sustainable cities demands serious engrossment, and only this concept can cope with environmental, social, and economic challenges.First, we need a sustainable bond with the environment. For this, the concept of biophilic cities needs to be adopted. Professor Tim Beatley at the University of Virginia shared this idea.
The concept advocates close contact between citizens with nature and bio-diversity. Modern cities can produce fruits and vegetables for local consumption by using available space in the streets, parks, and roof-tops.
Singapore has incorporated this model, conceiving concept of urban farming. Preferring the use of clean and green energy, such as solar, tidal, wind, and thermal energy can diminish the baleful effects of carbon emissions.
Governments must prioritize energy efficiency to minimize energy consumption to make cities sustainable.
For example, Sweden has a per capita gross domestic product almost equivalent to the USA but uses 40 percent less energy per capita owing to its ability to adopt more energy-efficient policies.
Shifting buildings to clean and green energy by installing solar panels can achieve this goal. The concept of sponge cities can help address the menace of urban flooding, and improving the sewage system of cities can also counter the threat.
Rainwater preservation by households can restrict urban floods and minimize the over-consumption of the water table.For good measure, an innovative shift in the transport model can transform our cities’ environment.
In this context, green public transport can help to reduce the levels of CO2 emissions. Local transport in cities can also slough off the waste. It will reduce carbon emissions and increase the efficiency of the local government in cleaning the city.
De-carbonization of energy is the need of the hour, but it demands agreement from all the global stakeholders to accomplish the milestone. Affordable housing schemes can solve the problem of slums.
The devolution of power to the grass-root level can make cities sustainable. The formation of local bodies can wipe out inequalities in the communities by providing equal opportunities for all citizens to play a role in the community’s advancement.
Last, population control is a prerequisite for reducing resource burden and ensuring sustainability. Sustainable cities thrive on the concept of sustaining population, economy, and ecosystem. However, our cities are developing unsustainable characteristics and it is the need of the hour to address the causes that are responsible for the degradation of our cities.
Indeed, destiny favours those nations that take steps to safeguard their habitat because this is the only way forward to secure a jewel-like future for our generations.
—The writer is CSS Officer, based in Sargodha. | https://pakobserver.net/of-sustainable-cities-by-waqar-hassan/ |
The last dialog box will have a field indicating the data should be imported to the import Orders table. A new dialog will ask if you want to save the import steps.We don’t want to, so leave it unchecked and then click Close.The most basic statement you can write with SQL is called a SELECT statement.In the above SQL statement we are retrieving the fields ID, Invoice No, Date and Organisation ID from an Invoices table.Now only blank values in that field will be updated. For each remaining sheet in the workbook, repeat the import process and update the order date.
We can use SQL to sum, count or average values stored in the database.Description: An error will occur when attempting to update a Microsoft Access file that has been loaded into Visual LANSA as an 'OTHER' file and contains more than 127 fields. | https://www.stk-ural.ru/updating-a-query-in-ms-access-302.html |
You will be working on urgently needed projects on the Emmanuel College campus. Arrival is requested to be between 3:00p.m. and 7:00p.m. oon Sunday the 9th of June. You will be assigned lodging so yooou can move in for the week.
Orientation for all participants will be at 8:00 PM on Sunday for approximately one hour (location to be determined).
EC MISSION 2019 begins Sunday afternoon, June 9 to Saturday, June 15.
The EC MISSION director (Barry Wood) will confer with Emmanuel personnel to determine the exact projects we will undertake in July. Of course, the number of people who sign up to attend EC MISSION for the summer of 2019 will determine what we will be able to accomplish. There will be openings for the first 25 EC MISSION volunteers.
TYPICAL DAY - We will work and eat together during the day (minimum 7 hours each day). In the evenings, we will fellowship, take advantage of Emmanuel’s athletic facility (swimming, fitness center, gym, bowling), and/or relax. There may be several times we will gather to encourage and debrief one another as the week progresses.
LODGING - Lodging (Look at Roberson Hall portion of video) will be provided on campus in one of Emmanuel’s newer dormitory suites with private rooms and baths. There is a sitting room and kitchenette in each 4 bedroom suite. Meals will be provided either in the school cafeteria or another designated area.
COSTS - A cost of $150 per person or $250 for married couples will be required.These fees will cover your meals, lodging, and Athletic Center activities. In addition, a Construction Materials Donation is requested, no minimum or maximum amount. You will be contacted on how to pay the fees. NOTE: If you are from the Franklin County area and will not need to be housed in the EC dorm, the fees are as follows: $75 per person per week or $125 per married couple.
INSURANCE - EC MISSION will provide for you (at no additional cost), a supplemental Accident Insurance Policy. It is important that you realize this is not a major medical policy. All participants should have their own primary health insurance carrier. We can provide a copy of the coverage of this accident policy at your request.
See STEP 4 for how to get involved! | http://www.ec.edu/ecmission-step3 |
Child Labour and The Industrial Revolution Essay Samples and Topic Ideas
Revolution, global average surface temperatures have not increased because all of the emitted CO2 has been absorbed by the oceans” is...
- Words: 4400
- Pages: 16
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revolution era. Wealthy Romans would visit famous hot springs and coastal resorts. A custom of grand tour existed from around 1660. The Europeans, mainly from the upper classes would travel around Europe to places like Geneva, Paris, Rome, and Berlin. Leisure activities, as well as leisure time, became popular among the middle class consisting of industrial revolution high classes (Butler, 2014). These were factory owners, traders, wealthy farmers and machine owners, just to mention a few. It was during this time (1758) when Cox & Kings, the first travel company was formed, and it was mainly for factory owners (Getz, 2008). Travelling for pleasure, steadily evolved till it became a trend to...
- Words: 550
- Pages: 2
revolution as one’s ability to do work without enduring fatigue. However, due to changes and evolution in one’s physical fitness, the term is defined as the state of having a good health due practical exercises and proper nutrition (Maffetone, 8). It measures the well-being and the capacity to perform occupations and daily activities. According to Maffetone, fitness gives a standard of the body’s ability to function effectively and efficiently either in work or leisure activities (7). The physical fitness also measures the ability of the body to resist diseases and the capacity to deal with emergency situations efficiently. Physical fitness enhances the interrelation of human fitness and the...
- Words: 825
- Pages: 3
Revolution. The report, put together by the Children's Employment Commission, contains extensive information regarding the ages of the children employed and the number of hours they worked. Conclusion: It is my opinion that this source can be relied upon to evaluate information about the role of children during the Industrial Revolution, and in particular regard to their work in the coal mines. It is a useful source, which when utilized with other sources, can provide sufficient information in detail. Works Cited Children’s Employment Commission. Mines Report....
- Words: 275
- Pages: 1
revolution. With more energy radiating down on the earth compared to back up into space, the earth has continued to heat up. As the atmospheric heat continues to move up, it has been able to hold more water vapor, and hence strengthening the earth’s hydrological cycle (Tang and Taikan 47). With the extra energy, more water is moved from the subtropical regions and pushed to wetter regions in the sub - polar, and which has resulted in strong droughts and the same strong storms. This may seem like an oversimplification of the climate change, which is even not supported by observed data. The reverse may occur, the dry may grow wetter and the wetter regions might become even extra dry. The analysis of...
- Words: 275
- Pages: 1
revolution dating 8000 years ago (Sanderson 1991). The population increased as there was the surplus production of crops such as wheat, barley, rice and corn alongside other animal products. Individuals opted to settle with the intention of developing large scale farming. Cities emerged as the population increased thus necessitating the need to have a place where a multiethnic society would meet for trading. The development of cities brought educators, artisans, religious leaders and merchants together with each taking a given role depending on their specialization. Conflict arose when people settled as neighboring comminutes sought to enlarge their territories. Security measures were beefed up to...
- Words: 1100
- Pages: 4
revolution gave an opportunity for women in society to choose to work e away from their homes. Question 4 Various changes characterized the slavery system and plantation of the crops in 1800 and 1860. The industrial revolution is viewed as one of the major cause of these changes that had taken place during this period. The introduction of the machines to the tasks that were handmade was one of the major characteristics of the industrial revolution. The textile industry became advanced, and the introduction of various power tools increased the level of production in the industry. This act of introducing machines in the industry was one of the major changes that had taken place during this period.... | https://topicsessay.com/essay-samples/child-labour-and-the-industrial-revolution/ |
Hey guys,
I’m currently trying to learn the Gentry Stein trick drift, but can’t land the chopstick. I’m making sure that the sting is on the tip of my index finger and thumb of my non-throw hand, but every time I attempt to land it, the string slides up towards my palm. Any advise?
Hey guys,
Have you learned the thumb mount because it helped me a lot in the trick?
Try and find Paul Dang’s advice on chopsticks.
Basically, open your hand like you are holding a couple boxes of cereal to give it a sort of “square” shape. This helps keep the strings from sliding.
Also what can help is having the strings go over all the tips of your chopsticks hand (not just over the index finger, as in the standard chopsticks mount) …I think Gentry says this in the tutorial.
Lastly, it can help to also use your fingers to pinch the string. For example, for your chopsticks hand have the string not go completely over all your fingers. Instead, have the bottom string go between your ring finger and pinky finger (for example) and pinch the string a bit between those fingers. This will help really stabilize the strings on your chopsticks hand. | https://forums.yoyoexpert.com/t/chopsticks-help/80983 |
Large, single, bright pink flowers with a red centre and attractive green foliage with a tidy habit.
Highlights:
Jack Clark hybrid. Evergreen. Half-hardy. Attracts bees and butterflies.
Application:
Great for containers, gardens and borders.
Situation:
Plant in well drained soil in a sunny position.
Care & Height:
Height to 1.5m and width to 1m. Cut back in late spring to encourage bushiness. Water deeply once a week during hotter months. Frost tender.
This plant appears in the following categories;
Containers, Coastal, Flowers, Foliage Feature, | http://www.colorworxnursery.co.nz/online/plant_information.csn?productid=2086 |
June 12, 2008 19:52 ET
VANCOUVER, BRITISH COLUMBIA--(Marketwire - June 12, 2008) - Yangtze Telecom Corp. (TSX VENTURE:SMS) today reported its financial results for the first quarter ended March 31, 2008. During the quarter, the Company operated through a subsidiary in China - 95%-owned Hunchun VAS Technology Company Ltd. (HCVAS) and a subsidiary in Mongolia - NBC Co. Ltd. (NBC). All dollar amounts are in Canadian dollars unless otherwise noted.
Revenue for the quarter ended March 31, 2008 was $9,996,000, consisting of $9,981,000 of sales from HCVAS and $15,000 from NBC. There was no comparative sales in 1Q 2007 for both companies, since HCVAS was acquired in December 2007 and NBC was still in the startup phase in early 2007. The revenue of HCVAS was all derived from software licenses for the biometric fingerprint car security system.
The cost of sales for HCVAS was about 84% of gross revenue which included 82.5% of revenue sharing with the software developer and software amortization of $103,000. The selling, general and administration expenses for the three months period ended March 31, 2008 amounted to $972,000, of which $734,000 represented accrued warranty expenses and distribution channel development costs. HCVAS is located in the Northeast region of China, which attracts private enterprises through certain government incentives including rebates of value added taxes and profit tax exemptions. Including a tax rebate of $1,512,000, HCVAS recorded a profit for the quarter of $2,168,000, before minority interest, and $2,059,000, net of minority interest.
The cost of sales of NBC was $101,000, consisting of $19,000 in amortization of intangible assets - licenses, $12,000 in cost of set-top boxes, $5,000 in operating expenses and an amortization expense of $65,000 for TV broadcast equipment. The G & A expenses for the quarter were $58,000, compared with the quarterly average for 2007 of $66,000. The Mongolian wireless digital TV operations incurred a loss of $144,000 for 2008.
The G & A expenses for the Yangtze Vancouver head office were $190,000 for the quarter, compared with a quarterly average of $181,000 for 2007. The slight increase was largely accounted for by higher filing fees arising from the initial share issuance with respect to the HCVAS acquisition.
The net profit for the Company in the first quarter of 2008 was $1,726,000, net of minority interest, a significant improvement from the net loss of $1,683,000 in 2007. The balance in the accumulated "other comprehensive gain/loss", a component of shareholders' equity, changed from a loss of $465,000 at year end of 2007 to a gain of $198,000 as at March 31, 2008, representing an unrealized gain of $663,000 from the impact during the quarter of the depreciation in Canadian dollars from 7.391 to 6.8213 against the Company's RMB denominated fixed assets and equity. The earnings per share for the 1st quarter in 2008 were 3.8 cents, compared with a loss of 0.5 cent for the same period in 2007.
"We are very pleased that the Company reported a significant turnaround in profitability in the first full quarter since the acquisition of HCVAS," said Kevin He, CEO of Yangtze Telecom. "We will continue to review and improve our existing businesses while remaining vigilant about acquisition opportunities of leading businesses that offer unique growth potential in China."
Yangtze Telecom's 2008 first quarter financial statements and Management's Discussion and Analysis are accessible on the Company's web site (www.yangtzetelecom.com) and on SEDAR (www.sedar.com).
About Yangtze Telecom Corp.
Yangtze Telecom Corp. conducts its business via two operating subsidiaries - NBC Co. Ltd. based in Ulaanbaatar, Mongolia and Hunchun VAS Technology Company Ltd. based in Jilin, China. NBC operates a wireless digital TV network and offers TV subscription service in Ulaanbaatar. HCVAS licenses technologies in biometric fingerprint car security system and Push-to-talk (PTT) handsets.
For further information on Yangtze Telecom Corp., please visit the Company's web site at www.yangtzetelecom.com. | http://www.marketwired.com/press-release/yangtze-telecom-announces-first-quarter-2008-financial-results-tsx-venture-sms-868196.htm |
The expected hope that this year will be different from the previous years, in terms of flight disruptions during Christmas season due to the harmattan haze has been dimmed, as THISDAY learnt that there could be flight cancellations this season, due to bad weather, despite the improvement on aeronautical facilities.
In the previous years, daylight flights were cancelled due to low visibility occasioned by the harmattan, but the Nigerian Airspace Management Agency (NAMA) said that due to the installation of facilities, the upgrade of equipment, there would be improvement this year, as less number of flights would be cancelled.
Former Managing Director of NAMA, Captain Fola Akinkuotu told THISDAY recently that modern aeronautical equipment, including Category 3 Instrument Landing System (ILS) could significantly improve visibility that aircraft can land in thick haze.
“I believe we have ILS in virtually all our airports and definitely in all the major airports. ILS has categories, Category 3, of course, is the best but they are expensive. And when you do a cost benefit analysis, even globally, they don’t have category 3 in every airport. But the major airports, it will be beneficial to have ILS category 3 in all the major airports, which is what we are trying to do,” he said.
However, industry expert and the Secretary General of Aviation Round Table (ART), Group Captain John Ojikutu is not so optimistic. He told THISDAY that harmattan would have impact on flights landings day and night this year if there are no compliance to the periodic maintenance on the critical safety landing aids such as the ILS that require periodic maintenance of their calibrations every six months and the VOR (Voice Ominidirectional Radio Range) for 12 months; saying that the Nigerian Civil Aviation Authority (NCAA) is expected to verify these alongside NAMA.
“I just hope we won’t have the experience of last year this year,” he said.
Ojikutu also noted that shortage of air traffic controllers (ATC) would definitely cause stress on the few that are available to do the work.
But top official of NAMA told THISDAY that this year there would be less flight disruptions due to weather because the agency has deployed landing aids that would guide flights landing and takeoff, even in low visibility.
“Right now we don’t have weather issues. If there is flight delays or cancelation the airlines will have to explain because weather is relatively okay now. This year should be a better year because most of the airports in Nigeria today have Instrument Landing System (ILS), which is about navigation,” the official said.
According to him, NAMA deployed R-NAV equipment, which is a method of instrument flight rules navigation that allows an aircraft to choose any course within a network of navigation beacons, rather than navigate directly to and from the beacons. This can conserve flight distance, reduce congestion, and allow flights into airports without beacons.
The agency has also deployed Performance Based Navigation (PBN), which is the International Civil Aviation Organisation (ICAO) equipment that specifies that aircraft required navigation performance and area navigation systems defined in terms of accuracy, integrity and availability. So under RNAV, the equipment used to achieve the navigation accuracy is specified. Under PBN, the Required Navigation Performance is specified and it is up to the operator to achieve that performance.
But the modern equipment, THISDAY learnt, must have corresponding equipment in the aircraft for optimum performance and some of the aircraft operating on domestic routes do not have such on-board equipment. Also, the Nigerian Civil Aviation Authority (NCAA) gives approval for the weather minima of every airport. So, while some aircraft can land at low visibility at the airports, they must follow the weather minima specified and approved by NCAA. Few years ago, NCAA approved 300 meters Runway Visual Range (RVR) for 14 airports from 800 meters RVR.
“We have done what we call R-NAV procedures in all the airports in Nigeria, including the smallest airport, Osubi. This is to enable airlines use our navigational facilities. We have done Performance Based Navigation and some airlines that have modern aircraft are benefitting from it. The landing aids at the airports are doing well; so if there is flight cancellation, don’t attribute it to NAMA,” the official said.
THISDAY also learnt that there are other factors that hamper flight operations beyond weather during the Christmas season and this include daylight airports that use Visual Flight Rule (VFR) and are shut down by 6:00 pm because there is no airfield lighting and many airports are under that status.
“Some airlines schedule flights to these airports late and may not have time to officially request for extension of time so they can land and when they do this, ATC will not give them start up to leave the airport of departures o they would be forced to cancel the flight. Besides, when there is no airfield lighting the ILS usage may not be maximized. For example, for you to fully utilise Category 3 ILs, you must have Category 3 airfield lighting. The Minister of Aviation directed that the Lagos and Abuja airports should be installed with Category 3 airfield lighting and they are fully in operation,” the official further said.
THISDAY gathered that in 2018, flight operations did not take place on December 21, 22 and 23, as airlines had to reschedule their flights at huge cost and enormous discomfort to passengers. Similarly, in 2019, there were disruptions but not as severe as the previous year and in 2020 there were delays. Harmattan haze did not disrupt flight like in the past years, but due to weather changes, it is unpredictable if this year could get worse and that is why industry observers said that efficient landing aids are the answer to the intemperate weather changes. | https://www.thisdaylive.com/index.php/2022/12/02/despite-improvement-on-aeronautical-facilities-harmattan-may-hamper-flight-operations-at-christmas-says-expert/ |
Guangzhou Jame Printing Co., Ltd, a Grade-A comprehensive printing enterprise with its own factory occupies an area of 13,800 square meters, was established in January, 1997. We specialize in printing and have over 15 years experience in this area. At present, its products include book, brochure, booklet, catalogue, calendar, envelope, flyer, folder, label & sticker, letterhead, magazine, notepad, paper bag, packing box, poster and so on. | http://www.cnpackingmachines.com/suppliers/jame/ |
Now 1st April has passed it is worth recording that on that day the Imperial Poona Yacht Club held what was possibly the first virtual annual dinner of any yacht club. While many less imperious sailing and yacht clubs have been forced to adopt virtual committee meetings in these troubled times, Poona went one step further to hold its All Fools' Tiffin with the assistance of Zoom.
Twenty-two sahibs from six countries and four continents attended, serving their own curries and beverages while regaling each other of tales on and off the water. During the party a cannon was fired a number of times on the shores of Oyster Bay, New York, while DJ Mark Covell provided the music from Hampshire. The earliest riser was Paddy Oliver from the club's Southern Hemisphere Imperial Thinkers who joined for an early curry breakfast in Melbourne and the last to bed was solo circumnavigator Dilip Donde who joined the dinner at 12 midnight Goan time and left at 2am.
As veteran yachting journalist John Chamier wrote in The Field in 1979: "The yachting world is divided into three factions: those who believe the Imperial Poona Yacht Club exists, those who may be termed the 'come-on-pull-the-other one' lot, and my wife".
It is therefore a great joy to be able to announce that on 1st April 2020 the Club's 75th anniversary history, 75 Years of Balls (ISBN 9780950917955), was published in its second edition and is available worldwide in both print and eBook form from all major book sellers. | https://www.sail-world.com/news/227955/Imperial-Poona-Yacht-Club-virtual-Tiffin |
Q. 284.0( 5 Votes )
Explain how, the total energy a swinging pendulum at any instant of time remains conserved. Illustrate your answer with the help of a labeled diagram.
Answer :
A swinging pendulum is a perfect example to show the conservation of energy. It shows the transformation of potential energy into kinetic energy and kinetic energy back into potential energy without any energy loss. In a pendulum, the law establishes that, when the ball is at its highest point, all the energy is potential energy and there is zero kinetic energy. At the ball's lowest point, all the energy in the ball is kinetic and there is zero potential energy. The total energy of the ball is the sum of the potential energy and kinetic energy.
Initially, the bob of the pendulum is at the mean position (B). When we draw the pendulum bob to one side (Extreme position A), we raise the bob to a little height end give it potential energy. This is the energy transferred by work done by hand. As at the extreme position, the bob has only PE, it tends to move down. The P.E decreases and K.E increases. At the lowest (mean) position, the bob has got K.E. Due to this it moves to the other side. Now, its K.E decreases and P.E increases. At the extreme positions A and C, all energy is in the form of potential energy and therefore it tends to move down. Thus the bob oscillates. At all other intermediate positions, energy of the pendulum is partly potential and partly kinetic. But, the total energy of the pendulum remains conserved.
Rate this question :
A microphone converts:
A. Electrical energy into sound energy in ordinary telephone
B. Microwave energy into sound energy in a mobile phone
C. Sound energy into mechanical energy in a stereo system
D. Sound energy into electrical energy in public address systemLakhmir Singh & Manjit Kaur - Physics
A ball falls to the ground as show2n below:
A potential energy = 80 J
Kinetic energy = 0
B Kinetic energy = 48 J
C = potential energy = 0
(a) What is the kinetic energy of ball when it hits the ground?
(b) What is the potential energy of ball at B?
(c) Which law you have made use of in answering this question?Lakhmir Singh & Manjit Kaur - Physics
The hanging bob of a simple pendulum is displaced to one extreme position B and then released. It swings towards centre position A and then to the other position C. In which position does the bob have: (i) Maximum potential energy?
(ii) Maximum kinetic energy?
Give reasons for your answer.Lakhmir Singh & Manjit Kaur - Physics
Name five appliances or machines which use an electric motor.Lakhmir Singh & Manjit Kaur - Physics
A bulb lights up when connected to a battery. State the energy change which takes place:
(i) In the battery.
(ii) In the bulb.Lakhmir Singh & Manjit Kaur - Physics
An object is falling freely from a height x. After it has fallen a height, it will possess :
A. Only potential energy
B. Only kinetic energy
C. Half potential and half kinetic energy
D. Less potential and more kinetic energyLakhmir Singh & Manjit Kaur - Physics
Which one of the following statements about power stations is not true? | https://goprep.co/explain-how-the-total-energy-a-swinging-pendulum-at-any-i-1nle6o |
Windows do not count as broken space unless the window is a ceiling to floor unit. When you are determining where the outlets must be according to code, you don’t consider a window panel a break when placing outlets. Therefore, outlets can be installed below a window.
How low can you install an outlet?
Electrical receptacle outlets on branch circuits of 30 amperes or less and communication system receptacles shall be located no more than 48 inches (1219 mm) measured from the top of the receptacle outlet box nor less than 15 inches (381 mm) measured from the bottom of the receptacle outlet box to the level of the
Where should electrical outlets be placed?
Ensure that there are electrical outlets in the following areas to ensure that your house is well-wired for convenience.
- Your Kitchen. The kitchen holds many small and large electric appliances. …
- The Family Room or Bar. …
- Your Living Room. …
- Bathrooms. …
- Install Some Outdoor Outlets. …
- Your Home Office. …
- All Bedrooms. …
- Garage.
How do you add an outlet where there is none?
Quote from the video:
Quote from Youtube video: And take a look at where that outlet is and see where we can tap off of that outlet to add an outlet in here the main thing when adding an outlet.
Where are show window receptacles required?
A.
The National Electric Code Article 210.62 requires at least one receptacle outlet shall be installed directly above a show window for each 3.7 linear m (12 linear ft) or major fraction thereof measured horizontally at its maximum width.
Why do electricians install outlets upside down?
Electricians may position the outlet in an upside-down position so that you can quickly identify the switch-controlled receptacle. Since it stands out visually to most people right away – it provides convenience to the occupants to easily remember which outlet is switch controlled.
Jan 5, 2020
What is code for installing electrical outlets?
The US National Electrical Code, Section 210.52, states that there should be an electrical outlet in every kitchen, bedroom, living room, family room, and any other room that has dedicated living space. They must be positioned at least every twelve feet measured along the floor line.
Is it code to install outlets upside down?
The National Electrical Code (NEC) doesn’t require a certain direction. The NEC allows outlets to be installed with the ground plug hole facing up, down or sideways. It’s up to you, there is no standard electric outlet orientation. So that means there really is no such thing as upside down outlets.
May 23, 2022
Should outlets be installed ground up?
The outlet should be oriented with the ground pin up because this pin is longer and the plastic around the plug is meatier, so it will help to keep the plug inserted in the outlet.
Apr 3, 2009
Why are hospital outlets red?
The red outlets (sometimes referred to as sockets) in hospitals and medical facilities indicate that they are on emergency backup power. The bright red color helps nurses, doctors, and hospital staff quickly and clearly identify where to plug in critical equipment during an emergency situation.
How many outlets should be in a room?
According to US regulations, no single point measured along the floor line of the room walls should be more than 6 ft away from an electrical outlet. This means that a standard, 12 x 14 ft room needs at least 4 to 6 electrical outlets, depending on the wall space.
Oct 7, 2021
What is the height for electrical outlets?
Standard Height for Outlet Boxes
The standard height for wall outlet boxes is about 12 inches from the top of the floor covering to the bottom of the receptacle box (or 16 inches to the top of the box).
Jun 22, 2022
How far does an outlet have to be from a corner?
The National Electrical Code requires that you have an electrical outlet within 6 feet of the corner of the wall and at least 12 feet from the same wall. That’s the minimum requirement for living space.
Which most accurately describes the layout of residential electrical receptacles?
Which most accurately describes the layout of residential electrical receptacles? Lay out convenience receptacles so that no point on a wall is more than 12 feet from an outlet.
Is it legal to have an outlet in a closet?
The 2008 National Electric Code forbids open or partially enclosed lamps and pendant lights in closets, per section 410.16(B). It doesn’t have any restriction on outlets in closets.
What is the difference between an outlet and a receptacle?
An outlet is defined as “A point on the wiring system at which current is taken to supply utilization equipment”. A receptacle however has always been something that an attachment plug is connected to. With the introduction of new products comes the need to revise certain code definitions.
What is the maximum distance between receptacles on the same wall?
The maximum spacing between receptacles, according to the National Electric Code (NEC), has been set at 12-feet since 1956–with no point along a wall being more than 6-feet from a receptacle. The logic behind that number is that an appliance with a standard length cord could then be plugged-in anywhere along the wall.
Oct 9, 2018
Is a receptacle required in a bedroom on a 3 ft wall space behind the door?
So a three foot wall behind a door swing requires a receptacle but, in most cases, you just have to count the length of wall behnd the door swing towards the spacing requirement. As long as it is not more than six feet from a receptacle to the edge of the door jamb, it’s acceptable.
Apr 9, 2020
What is the electrical 6 12 rule?
The primary rule is what’s known as the 6’/12′ rule. NEC 210-52 states the following (abbreviated for easier digestion): Receptacles are needed in every room of a home such that no point on a wall is over 6′ from an outlet. This means that you need an outlet within 6′ of a doorway or fireplace. | http://paspolini.studio/en/can-i-install-a-receptacle-outlet-below-a-window/ |
This year is all about healing the heart, Scorpio. It’s time to leave negative attitudes and stoic facades at the door and let others see the real, more vulnerable you.
Percy Freedman is not grieving. Absolutely not, take that back at once. No, he’s entirely sure that selling his dead aunt’s home and leaving the neighbors he’s known for years is the sane thing to do. Who in their right mind would keep the house that smells like all the hugs he’ll never have again?
Nobody, that’s who.
Well, except his cul-de-sac neighbors. They all seem to think some paint and new furniture will clean the emotional slate. They all want him to stay.
Even his nemesis, Callaghan Glover.
Especially his nemesis, Callaghan Glover.
Lured into a game of Sherlock Gnomes, Percy finds himself hanging out with his neighbors more than might be considered healthy. Along with juggling new and surprising verbal grenades from Cal, and his burgeoning friendship with Gnomber9, Percy is starting to wonder if selling might have been the grief talking after all . . .
That’s right, Scorpio. With a little patience, heartbreak might be a thing of the past . . .
Todd's rating:
This second book in Anyta's 'Signs of Love' series is frenemies-to-lovers at its finest.
Percy is thoroughly convinced that everyone will leave him. Always. So not only is he afraid of his attraction to Callaghan because Cal is straight, but also because he fears putting himself out there (again), only to be rejected in the end.
Callaghan isn't afraid of his burgeoning attraction to his long-time nemesis, per se, he just needs some time to wrap his head around the attraction and see if it's something he truly wants to pursue.
When Percy's beloved aunt dies, he returns to sell her house and begin processing his grief. What he doesn't account for, though, is that his neighbors are conspiring against his plans for a hasty departure. Well, one neighbor in particular, mostly. His nemesis, Cal.
The snark and banter in this second book felt much more subtle than it did in Theo and Jamie's story. Also, this story is *much* more of a super-slow burn in the romance and steamy times department. I mean, the first kiss didn't even happen until nearly 70% in, so yeah. Slooooooooow burn.
So while I truly love both Percy and Cal, this more subdued romance left me missing a bit of the quick one-two verbal jabs that were so prevalent in the interactions that we saw between Theo and Jamie as their relationship developed.
One of my favorite aspects of this book is when it finally clicks with Percy that you can move on and recover from loss without necessarily leaving your old life behind entirely, as he'd initially thought he'd need to do. That sometimes the best way to get past your grief is to embrace those around you that you love, and love you in return -- if you'll only open up your heart and let them.
I'd rate this one at a solid 4.25 stars and highly recommend it to any fans of the first book, "Leo Loves Aries."
My ARC copy of the book was provided by the author in exchange for a fair, unbiased review.
Get the book:
Thanks for visiting our blog and I hope you enjoyed my review!
Buylinks are provided as a courtesy and do not constitute an endorsement of or affiliation with this book, author or bookseller listed. | https://www.myfictionnook.com/2017/08/arc-review-scorpio-hates-virgo-signs-of.html |
Teapot Hill is a short 5.5km (out and back) hike in Cultus Lake Provincial Park, in Chilliwack, British Columbia. As you follow the trail to the top, you can find dozens and dozens (we counted over 100 during our visit) of teapots and teacups hidden throughout the hike. Teapot Hill is rated easy and has a net elevation gain of around 250m. The trail takes around two hours but could take longer if you really enjoy searching for teapots.
When you reach the top, you will find views of Cultus Lake. The views are mostly obstructed by trees, but you will still find a pretty enough view of the lake before you reach the absolute top of the trail. However, the teapots and teacups are the real gems of this trail.
Teapot Hill is a wonderful hike for people of all ages and abilities. Keep in mind that it is called Teapot Hill for a reason. Some people find the uphill a little tiring. Take your time and enjoy all the fun teapots and teacups around you, and you should complete it just fine.
This hike is also a wonderful activity option if you’re staying at a Cultus Lake Campground and want something unique to break up your time.
Table Of Contents
Why The Name “Teapot Hill”?
In the 1940s, the area was called Teapot Hill by a logger that found a teapot on the hill. In recent years, someone began leaving teapots on the trail for others to find and the popularity of the hiking trail skyrocketed.
Take a look at some of the teapots and teacups that we found:
Searching For Teapots
We all had a great time searching for teapots and teacups. Some are very obvious, and some you need to be looking up to find. It is a great way to get kids interested in hiking and to inspire people to get outdoors and enjoy nature. There was so much laughter and racing along the trail to see who could find more teapots. It especially gave the kids a reason to keep moving and turned into a fantastic active day in the fresh air.
Stay On The Trail
From time to time the teapots are cleaned up by city workers, so there is a chance that you may hit the trail before the teapots get re-populated again. They are removed because of breakage, creating a trashy mess, and making a safety issue to those using the trail. Be respectful and do not touch any of the teapots and teacups. Just admire them. We lucked out with how many we found!
Also, stay on the trail so you do not damage any sensitive plants that need to not be stomped all over so that they can thrive. The park is host to a rare species of orchid and wandering off the trail could cause you to unknowingly trample them in your efforts to hide or find teapots.
Related Story – The Slesse Memorial Trail: Hiking To One Of Canada’s Worst Aviation Disasters
The Teapot Hill Trail
The trail starts at the parking lot for the Teapot Hill trail. You will see a well-worn trail start climbing a hill. Follow the trail and it soon turns into a large gravel road. You will first pass a small B.C. Hydro shed on your right, and then an outhouse and map of the area on your left.
Continue to climb the trail, surrounded by gorgeous moss-covered trees. The next notable trail view is the pretty creek you will pass on your right. Take a minute to stop and enjoy the sound of the rushing water. After an additional 500m, the trail will begin to narrow.
After roughly 15 minutes you will reach a trail on the left called “Horse Trail”. Do not stop or turn off there. Continue down the same road you have been wandering down and you will eventually see the sign for Teapot Hill, that points for you to go to the right. The trail is clearly marked. The path will again narrow, but it is well-worn and easy to follow.
After turning right at the Teapot Hill sign, this is where you will start to find the majority of the teapots. Five minutes in, the path will narrow even more for a bit of a climb, before leveling out. You will want to especially watch your kids as this narrow section has a drop-off along the left side.
The trail then widens again as you approach the two different viewpoints of Cultus Lake. You have reached the end of the trail when you reach the second viewpoint, which is fenced.
When you’re done enjoying the teapots and views, retrace your steps back to the parking lot.
Related Story – Flood Falls Near Hope Is An Easy Hike For An Impressive Waterfall
When To Go
The trail is accessible year-round but would be difficult to find the teapots and teacups if under snow. The best time to use the trail is May to October.
Bring Your Dog
Dogs are allowed on the Teapot Hill trail. Dogs must be kept on a leash and please properly clean up after them.
Directions To Teapot Hill
Google Map Directions from Vancouver, BC to Teapot Hill.
On the Trans-Canada Highway/ BC-1, head East towards Hope. Take Exit 104 and follow the signs for Cultus Lake; there is adequate signage. You will come to a T in the road. Turn right onto Tolmie Road, and then left onto No. 3 Road.
When you reach the next T-intersection, turn left onto Yarrow Central Rd. Continue onto Vedder Road until you reach the roundabout. Exit right onto Columbia Valley Hwy. Continue down the road along the campsites and public beaches. On your left, just past the Clear Creek Campground, you will see the sign and parking lot for Teapot Hill.
Parking At Teapot Hill
The trail begins at a small parking lot that can only fit about ten cars. If the lot is full, you will have to park along the road. It is a remarkably busy trail, so you will want to arrive early.
Pin It For Later! | https://worldadventurists.com/teapot-hill-hike-in-chilliwack/ |
Local View: Collaboration is needed to protect bats and have responsible forest management
I’ve always admired forest bats, their voracious appetite for insects, and the echolocation and aerial acrobatics they use to catch those insects. But they also creep me out a bit on the rare occasion when they enter my home; there is the threat of rabies.
But recently I discovered that a study by the University of Calgary found that far less than 1 percent of forest bats carry rabies, and if you act responsibly, the threat of contracting rabies from bats is extremely small.
“Bats are rarely aggressive, even if they’re being chased, but they may bite in self-defense if handled,” Bat Conservation International reported. “Only a small percentage of bats (about one-half of 1 percent overall) have rabies, but anyone bitten by a bat should immediately seek medical consultation.”
So, 99.5 percent of bats don’t carry rabies. It’s good to know we have little to fear from forest bats.
However, there is growing anxiety regarding a particular forest-dwelling bat. The northern long-eared bat is being proposed as federally endangered by the U.S. Fish and Wildlife Service per the Endangered Species Act. The primary reason for the potential listing is white nose syndrome, a fungal disease decimating long-eared bat populations in eastern states.
Based on interim documents and the experience in other states with the endangered Indiana bat, significant restrictions on forest management could be imposed here, especially in midsummer when forest bats are roosting in trees. These restrictions would have negative
effects on our northern forest-dependent communities, as the forest industry can’t operate just half of the year. Adding confusion is that there is no indication responsible forest management has caused negative effects to forest bats.
The primary areas of concern related to the impacts on forest management appear to be direct bat mortality from timber-harvest activity in summer and the loss of habitat due to clear cutting.
Minnesota has more than 17 million acres of forestland, and if timber-harvest rates were at the maximum sustainable rate (about 5.5 million cords per year), about
1.5 percent of the forest would be affected annually. Due to our vast wetlands, less than half of that harvest would occur in summer. But the reality is that in the past five years the harvested amount has been about one-half of that volume each year, meaning that more than 99.5 percent of the forest is not affected each summer. The odds of bat mortality from timber harvesting are very small. But that small fraction amounts to nearly 1 million cords and many millions of dollars in economic impact each year.
As our forests age, there are more acres becoming suitable as bat habitat than being lost to intensive harvesting or clearing. Minnesota’s forest-management guidelines recommend retaining habitat features such as dead standing “snag” trees, clumps of live trees and cavity trees, as they are important for a number of wildlife species, including the northern long-eared bat. Forestry audits indicate a very high level of compliance relative to wildlife tree retention. So even after harvest there are features that make most clear-cut sites suitable for roosting bats.
Just as we have little to fear from forest bats, they have little to fear from forest management; in fact, the diverse forests created by responsible forestry practices have
resulted in many millions of acres of bat habitat in our northern forests. Typically, habitat management is the approach used to maintain healthy and viable wildlife populations; but it doesn’t seem to be a consideration regarding this proposed Endangered Species Act listing. Land managers and loggers are unfairly portrayed as being part of the problem and not part of a solution.
I’d suggest that the Fish and Wildlife Service take a collaborative approach in our habitat-rich region by empowering land managers to continue maintaining forest bat habitat via responsible forest-
management practices. Loggers would have the duty of completing these plans on the ground. I believe land managers, loggers and conservationists could rally around the charge of keeping our local bat populations thriving through forest-habitat management. This would allow the limited resources of the Fish and Wildlife Service to focus on stopping the real threat to the northern long-eared bat: white nose syndrome.
If successful, this approach would be a win-win for northern long-eared bats and for our local economy; and perhaps our local healthy bat population eventually could repopulate the rest of the nation.
I realize this thought is a bit naïve. The likely scenario is that limitations to timber harvesting will cause economic hardships; the northern long-eared bat will be demonized as the cause of the problem; and loggers, biologists and conservationists will be at odds: conflict instead of collaboration. It would not be the bat’s fault or the fault of Fish and Wildlife biologists. Blame would belong to the one-size-fits-all inflexibility of the Endangered Species Act.
The sad reality is restrictions to forest management designed to protect individual bats would cause more economic harm to our local forest-dependent communities than the protection it would provide to northern long-eared bat populations.
It seems like we should have learned lessons from the ill-fated conflicts regarding the spotted owl in the West. Will our experience be “same story/different place” or can we join together to protect the northern long-eared bat and our forest-based economy?
Mark Jacobs of McGregor is the Aitkin County land commissioner responsible for managing 223,000 acres of public lands certified as “well-managed” since 1997 by the U.S. Forest Stewardship Council. | http://www.duluthnewstribune.com/opinion/local-view/3291341-local-view-collaboration-needed-protect-bats-and-have-responsible-forest |
# Azad Kooh
Azad Kooh (Persian: آزادکوه) is one of the highest peaks in the central Alburz Range in Mazandaran province north of Iran. In Persian Kooh means mountain and Azad means free, so Azad Kooh can be translated to The Free Mountain. This name is probably chosen by local people because of Azad Kooh's cone like shape and the fact that it is not connected to its surrounding peaks.
## Climbing Azad Kooh
Azad kooh is not a technical climb and you won't need any technical gear to climb the mountain in summer. There are two routes for climbing this peak, the easier and shorter one is from Kalaak-e-Baalaa village and the longer one is from Vaarange Rud village. The former takes 1–2 days and the latter 2–4 days. There is no shelter on either route and you need to carry your own tent if you are planning to camp. Azad Kooh is not such an easy climb in the winter, there will be a considerable amount of snow and the potential danger of avalanche in either route. | https://en.wikipedia.org/wiki/Azad_Kooh |
Content of the material
- How much can you earn as a freelance writer?
- Practice patience
- Video
- Breaking Into Magazine Writing
- The state of freelance writing in 2022
- Dealing with revisions and scope creep
- Should you become a freelance writer?
- Realistic Expectations for a Freelance Writers Salary
- 4. Freelance Medical Writer Salary
- Types of Freelance Writing Work
- How Much Should Freelance Writers Charge Per Article?
- What can influence freelancers’ pay rates?
- Traditional education may not matter much.
- Full-time freelancers make more than part-time freelancers.
- A freelancer’s age affects their earnings.
- Is freelancing worth it?
How much can you earn as a freelance writer?
As you likely suspect, this really depends. I’ve heard of some freelancers that earn upwards of $250,000 each year. And, of course, there are others who treat freelance writing as a side gig and as a way to earn some extra cash—so, they make anywhere from $10,000 and up each year.
I’m a big believer in income transparency for freelancers, which is why I break down my freelance numbers in each of my year-end recaps. I share:
- My gross income
- My total expenses
- How many total clients I worked with
- How many clients were new versus existing
If you’re interested, here’s my 2019 recap and here’s my 2018 recap.
But, how much can you expect to earn as a freelance writer? Let’s dig into some details.
Practice patience
If there’s one thing I can tell you for certain, it’s this: Anybody who promises that they’re going to teach you how to earn thousands of dollars freelancing in a matter of weeks is probably lying to you.
Like building any business, earning a solid living as a freelance writer is going to involve an investment in time and a real commitment to patience.
Need proof? Let’s do this thing:
- 2014: This was my first year in business as a freelance writer (granted, I started in July—halfway through the year). I earned roughly $5, 300 before taxes.
- 2015: This was technically my second year, freelancing. But, it was my first full year. I earned roughly $32,000 before taxes.
- 2016: I earned just under $80,000—not including those big checks I wrote to the government.
- 2017: I earned right around $102,000.
- 2018: My gross income came in around $98,000. That’s right—it was less than the prior year, and I still survived.
- 2019: I earned about $118,000. It’s proof that freelancing has highs and lows.
See? Freelancing is something that builds on itself—it’s absolutely not going to happen overnight. I’ve invested years in making a living that I’m proud of.
I know, getting started is rough and oftentimes disheartening. But, if you keep trucking along, the snowball really starts rolling. Once you have a solid portfolio under your belt, you can chase down bigger clients.
Once you land a few gigs with those bigger clients, your name really gets out there—meaning even bigger, higher-paying clients are more likely to approach you.
So, basically, the lesson is this: Be prepared to put in the work and invest the time. There’s absolutely no way around that.
Video
Breaking Into Magazine Writing
One high-paying writing niche that’s a little tougher to break into is magazine writing.
Writing for a magazine can net you anywhere for $0.10-$2 per word. That can add up to a lot of money. If you’re successful in this niche you can make a lot more versus freelance blogging. The only downside is that your work may be a little less consistent.
If you want to break into magazine writing one of the best places to begin your search is the Writer’s Market. This book contains hundreds of different publications that will pay you for your words. Not only that, but the Writer’s Market has the name of editors to pitch as well submission guidelines and pay.
This is one book I’ve had on my shelf for a couple of years now and is a wealth of information.
The state of freelance writing in 2022
Freelance writers have been important business assets for years. When COVID-19 emerged and businesses and consumers moved online, freelance writers became even more in demand.
To understand how many freelancers arose through a pandemic, 16% of freelancers started less than a year ago as of November 2021. Some 19% reported freelancing for between one and two years. Though the majority of freelancers (65%) have been writing professionally for between two and 10 years.
When we dove into freelance writing niches, one industry was more popular than others.
Some 27% reported working in software. Another 14% wrote for agencies, and 12% wrote for ecommerce.
Understanding what niches writers work in leads us to the next section. We wanted to know how much people were changing. So, we asked freelance writers to disclose their yearly earnings.
More than half of freelance writers earn less than $30K per year 💔
The next income level was $31K and $50K, with 18% of freelance writers claiming that was their annual freelance income for 2020. When you combine the percentages of these two groups, almost three quarters of freelance writers make less than $50K per year.
That suggests there is room for business improvement in a majority of freelance writer’s businesses. Think about your own operations. Where can you provide more value for clients to charge more?
Of the remaining writers in the survey, 9% said they made six figures. Some 5% reported making between $100K and $125K, and 4% earned over $125K in 2020.
When it comes to time spent versus income as a freelance writer, the longer you’ve been around, the more you earn.
It’s probably no surprise that people bringing in larger incomes have been in business for longer. If we look at those who’ve been freelance writing for less than a year, the vast majority (91%) earn less than $30K.
To reach the six-figure salaries, it seems you need to have been working on your business and growing it for at least a couple of years. Of those earning over $100K, all have been freelance writing for at least two years, with 65% writing for more than six years.
💰 Read about how Peak Freelance’s very own Elise Dopson grew her freelance writing business to six figures over three years.
Dealing with revisions and scope creep
Clients asking for revisions is a normal part of writing. It’s highly unlikely that you’ll nail things the first time around. Different clients have different expectations.
We asked our writers how many rounds of revisions they would typically include within their regular rate before charging extra.
Almost all writers (98%) include at least one round of revisions within their standard rate. It’s a fairly even split between those including one round (46%) and two rounds (44%). Some 8% will include three or more rounds within their standard fee.
Revisions may be routine, but what do you do if you start work on a project and realize it is far more extensive than expected?
Responses from our writers varied widely on the topic of scope creep. When the situation happens, most writers (44%) would ask the client for an increase in the budget as soon as it becomes clear the work is more considerable. Some 8% would wait until they submitted a draft to ask for an increase.
Worryingly, almost a third would not say anything at all, and absorb the cost of scope creep—likely to avoid the awkward conversation with the client.
There were a lot of comments on this question. Most people tend to make decisions based on the specific circumstances.
Many felt that making an error with quoting would be their fault, so they should be the ones to absorb the costs. But if the project scope had changed, they would review the fee:
“If I’ve misjudged the scope of the project, that’s my fault and I work to the agreed rate. If, however, the client moves the goalposts or adds to the scope after we’ve agreed the fee, I tell them straight away and revise the fee.”
For a lot of freelancers, approaching the difficult conversation of scope creep depends on the relationship they have with the client:
“It all depends on the client. I request a change for long-term clients. For clients I work with sporadically, I tend not to ask for anything additional.”
While many wouldn’t ask for an increase on a current project, they would mention the issue so that clients were aware future projects would cost more.
“If it’s a one off project I’d absorb it at my loss. If there’s potential for an ongoing relationship I’d be honest and say I underquoted on this one, so they’re aware future jobs may cost more.”
Should you become a freelance writer?
First, you need to find your reason for becoming a freelance writer. Without a solid reason to pull you through the long hours, it can be difficult to make your dream a reality. The benefits of becoming a freelance writer can be absolutely amazing.
You have the opportunity to build your freelancing business from the ground up. You have the option to choose how much you are willing to work, when you are able to work, where you will work and how much you will charge for your services. However, you need to determine why these benefits matter to you.
Would you use the newfound flexibility to spend more time with your kids? To pursue a new passion in your free time? Or to make your own schedule while you travel the world? You’ll need to understand how these benefits will play out in your life.
Find something to hold on to. As you start to build your business, you’ll likely need to put in long hours on top of your regular job. As you get closer to the tipping point of being able to afford to quit your regular job, the longer those hours might get to you. You’ll need to draw on the reason why you are doing this to find the self-discipline to carry on.
Realistic Expectations for a Freelance Writers Salary
Now that we’ve thrown some figures your way let’s get down to brass tacks.
While this probably isn’t what you want to hear, the official answer to “How much can I make as a freelancer?” is, “It depends.”
Means, medians, and averages don’t mean a darn thing if you’re not putting in the work in earnest. Freelancing is one of those careers where “You get what you put in” is entirely and unequivocally accurate.
It’s also prudent to note that especially if you’re just starting out, you probably won’t be pulling in mega-high rates. Even if only for a short time, newbie freelancers will have to “pay their due” while they build both their reputation and portfolio.
The goal should be for growth over time and year after year.
Thankfully, Contena can help you find higher paying clients with ease. Padding your portfolio with profitable clients can help boost your yearly salary by leaps and bounds.
The more you put into your business, the more you’ll get out of it. Plain and simple. Having these figures should be a point of reference, rather than a benchmark set in stone.
4. Freelance Medical Writer Salary
If you want to land medical writing jobs, having a background like nursing or being a paramedic, or having a job in healthcare will make the transition to freelancing easy.
This extra experience and skill level will help you make more money than an average health writer.
According to Payscale, the average medical writer salary is around $75,384/year.
This is the type of writing job where you can leverage your existing network of doctors and other nurses to help you land that first profitable job.
Types of Freelance Writing Work
Next, you'll need to understand that different types of writing have vastly different earning potentials.
Online writing, such as web content, blogging, SEO writing, and content mill work, generally pays the least. Feature writing online can pay more than general online writing as listed above, but usually still less than print.
In the middle of the payment spectrum is newspaper writing and associated functions (copyediting, etc.) and some social media work (posting, campaigns, planning, and production). Note that newspapers are struggling these days, which is partly why they hire freelance writers.
Marketing-related writing pays higher than general content creation. This includes copywriting, sales pages, press releases, ads (pay-per-click ads or advertorials), and email content. Other business writing, such as white papers, brochures, and position statements, generally pays better than articles and other online writing. Working on pieces that are typical for non-profits may also fall in this better-than-average area.
Ghostwriting books can pay very well, although there are some ghostwriting jobs that pay as low as $15 per 1,000 words, which isn't a lot.
Writing that requires you to have specific knowledge or a skill set pays better than general writing. This includes technical writing, medical and scientific writing, health and wellness writing, and financial writing.
How Much Should Freelance Writers Charge Per Article?
So, the nitty-gritty: how much should you earn for your work? First of all, don’t compare freelance income to an employed writer’s salary. Freelancers aren’t just writing, we’re running a freelance writing business, so it makes sense to value that added responsibility.
Remember that freelance writers also have more expenses, including paying about twice as much in taxes. All this means a freelance writer’s income should be significantly higher than a writer with traditional employment, although it can take time to build up to it.
The fact is that there’s a massive range of prices freelance writers charge. You’ll regularly see freelance writing rates ranging anywhere from $.05 to $1 per word or more. A good rule of thumb for a new writer is to start around that lower end, perhaps $.06-$.08 per word, and gradually move up from there as you improve your freelance writing skills.
Keep in mind that some writers translate their per-word rate into a per-article rate. For instance, you might want to make $.10 per word, so you price a 1000-word article at $100, even if the exact word count ends up anywhere between 950 to 1050 words. (Always try to stay within the agreed word count for each article. Some clients may not be willing to pay extra if you write more words) Regardless of how you price projects, always be clear about the per-project rate and payment terms from day one. Life is easier for freelancers and clients alike when everyone’s on the same page rather than waiting and bickering about prices when the invoice arrives.
What can influence freelancers’ pay rates?
Many factors impact how much freelancers earn. According to Payoneer, aside from obvious aspects like location, industry and years of experience, the freelancer’s age, gender, education level and availability can impact pay rates.
Here’s a deeper look at factors that influence freelancers’ earning potential.
Traditional education may not matter much.
The 2022 Payoneer study revealed that freelancers with a college degree earned, on average, $24 per hour, while high school graduates earned $22 per hour. These numbers differed from the 2020 survey, where freelancers with a college degree earned $19 compared to $22 for freelancers with a high school education.
Still, the fact that there’s not a wide gap in pay between college graduates and high school graduates indicates that companies that hire freelancers favor experience and client reviews over formal education, according to Jonny Steel, vice president of marketing at Payoneer.
“Not everyone has the same access to traditional higher education, but the beauty of the internet is that as long as you’re connected, you have access to an almost infinite amount of knowledge,” Steel said. “People can learn skills from watching online tutorials, reading e-books and following the latest trends on top-notch blogs. So when it comes to making an impact on the freelance marketplaces, it matters a lot less where you acquired your skills from, as long as you can do the job.”
Full-time freelancers make more than part-time freelancers
According to MBO Partners, the number of U.S. full-time freelancers grew 25% in 2021 to 3.4 million, and it is likely to increase. As Payoneer suggests, on average, full-time freelancers make $3 more per hour than those who engage in freelance work on the side. Additionally, workers who choose to freelance full time report a significantly higher work-life satisfaction rate.
The decision of whether to commit full time to the freelancing lifestyle depends on many factors. For example, workers at the beginning or end of their career are more likely to freelance exclusively, and those in the industries like finance and QA are much more likely to use freelancing as a side gig.
A freelancer’s age affects their earnings
Considering that technology advancement and shifting work paradigms contribute to the freelancing surge, it’s not surprising that younger workers are more likely to embrace those trends and spearhead the freelancer movement. Millennials and Gen Z represent a vast majority of the freelance workforce worldwide.
However, when it comes to compensation, experience is still rewarded with higher earnings. Freelancers over the age of 55 earn more than twice as much as their 18-to-24-year-old counterparts.
Did you know?: Despite increased representation of women among the freelance workforce in recent years, there is a persistent gender pay gap, with women making an average of $23 per hour versus $28 per hour for men.
Is freelancing worth it?
The short answer is that it depends. While freelancing offers benefits like flexibility and the ability to work remotely, it comes with financial risks and the need for an initial investment to establish yourself in the market. Studies indicate that many freelancers earn more than their employed peers, but the rate varies greatly by industry, specialization, region, experience level and other factors.
There are many opportunities for writers to make a living freelancing. The challenges are determining your niche, embracing your passions and applying your skills. If you’re thinking of becoming a freelancer, research your field’s opportunities and challenges, invest in upskilling, and have a backup plan in case of unexpected financial emergencies.
Nicole Fallon contributed to the writing and reporting in this article. | https://koronavirusa.site/finance/how-much-do-freelance-writers-make-per-article/ |
Wood cuts and where they come from
The term “quarter-sawn oak” refers to oak lumber that has been “quartercut”. In other words, quarter-sawn oak is not a separate species, but merely oak that has been cut from the log in a particular way. The actual species of oak, such as red oak, white oak, overcup oak, etc., is a completely separate issue.
In the illustration below (left side) you’ll see where each of the three basic cuts of lumber comes from in relation to the log. Quite simply, flat-sawn lumber comes from anywhere in the log where the growth rings on the ends of each board are roughly parallel to the wide surface of the board. We are used to seeing the “U”-shaped growth ring patterns on the ends of boards and this, essentially, represents flat-sawn lumber. Similarly, rift-sawn lumber comes from an area of the log yielding growth rings roughly 45 degrees to the large surface of the board. And quarter-sawn lumber has vertical growth rings, or rings that are perpendicular to the large surface. There is nothing magical about these terms.
There are many ways to saw up a log, but the growth ring orientation on the ends of the boards determines whether the lumber is flat-sawn, rift-sawn, or quarter-sawn. One would assume then, that the only way to know what cut of lumber you have in front of you is to look at the end grain. Although this is the easiest way, especially with rough-sawn lumber (which is hard to get a clear look at), you can also tell a great deal from the grain patterns on the surface of a board. This is set out in the illustration below.
Fig. 3 shows that, in addition to growth rings, which look like concentric circles, there are also other anatomical features in a tree that radiate out from the centre like spokes on a wheel. They are called medullary rays, or just “rays”, for short.
They conduct sap horizontally in the tree and some can store carbohydrates until other growing cells need them. The tricky thing about rays is that, although all trees have them, they are not always visible to the naked eye. If you’d like to see them, one of the best and most common examples is oak. Take a look at the ends of an oak board or an oak log and you will see the rays. But look at the ends of a pine board or cherry board and you won’t see them. Rays are still present in pine and cherry, but they are so small as to be visible only under magnification. In fact, rays are so thin in all species of softwoods that they are only visible under a microscope. But in hardwoods, some rays are highly visible and some are invisible to the naked eye. Fig. 1 shows you how to identify the 3 basic cuts of lumber based solely on surface grain pattern, assuming that you don’t have access to the ends of the board in question. The first example on the left is clearly flat-sawn, based on the end grain. On the surface, though, notice the beautiful grain pattern known as “cathedrals”. The angle at which the flat growth rings intersect the surface produces a series of arches. This is one of the features for which flat-sawn lumber is desired, although it can be too “busy” looking, depending on the piece being built and your goals. It is a bold look that ought to be used carefully in already bold species like oak. The second example in fig. 1 shows you what rift-sawn lumber looks like on the surface. There are no cathedrals, but only straight lines. As each growth ring intersects the surface, it produces just a straight line along the length of the board. This can also be desirable for a more subtle, or “quiet” look, in a finished piece of furniture. For example, if you have gorgeous cathedrals on the panel of a frame-and-panel door, you might want the frame to have more subtle grain patterns so as not to distract from the panel itself. The third example in the diagram shows quarter-sawn lumber. The growth rings still produce just straight lines on the surface. However, notice the other markings on the surface. These are caused by cross-sections of the medullary rays (now parallel to the wide surface of the board) intersecting the surface at irregular intervals. It can be absolutely stunning and is known as “ray fleck figure”, or simply “ray fleck”. Remember, though, that rays are not visible in all species and, even if they are, they can be very small. In red and white oak, they are huge and add tremendous interest to the piece. In cherry and maple, they are small and add just a hint of interest. In pine, you won’t see any ray fleck at all since the rays are invisible in softwoods.
This can create a little confusion. If I give you a rift-sawn board and a quarter sawn board in a species where the rays are invisible, it will be almost impossible to know which cut you have in front of you without access to the ends. In both situations, you will see just straight lines on the surfaces. The lines will be closer together in quarter-sawn lumber than in rift-sawn lumber of the same species, but line spacing can also vary by subspecies, growing conditions, age of the tree, etc. So this is a tough thing to identify based on grain pattern alone. However, remember as well that a flatsawn board can often have a quarter-sawn surface on its edges and a quarter-sawn board will have a flat-sawn surface on its edges. This means that a flat-sawn board will often have ray fleck figure on its edges, while the edges of quarter-sawn boards will usually display cathedrals.
Remember that flat-sawn and quarter-sawn cuts are exact opposites of each other. Each is the other cut just turned 90 degrees. So if you see just straight lines on the surfaces but cathedrals on the edges, then you know the board is quarter-sawn. If you see just straight lines on all surfaces and edges, then you know it’s rift-sawn. I hope this has clarified what all of these terms mean. Realize, too, that a given board, depending on width, will often have two cuts of lumber within it. If you look at a flat-sawn board that has distinctly “U”-shaped growth rings on the end grain, it technically is rift-sawn at the outer areas and is only truly flat-sawn at its centre. Combinations of the various cuts are often involved in a single board as well.
In our next issue, Hendrik will discuss the relationship between moisture content and relative humidity and show you how each of the three basic cuts of wood react to changes in moisture content. | https://staging.canadianwoodworking.com/techniques_and_tips/wood-cuts-and-where-they-come-from/ |
---
abstract: 'In this work, we investigate how and to which extent a quantum system can be driven along a prescribed path in Hilbert space by a suitably shaped laser pulse. To calculate the optimal, i.e., the variationally best pulse, a properly defined functional is maximized. This leads to a monotonically convergent algorithm which is computationally not more expensive than the standard optimal-control techniques to push a system, without specifying the path, from a given initial to a given final state. The method is successfully applied to drive the time-dependent density along a given trajectory in real space and to control the time-dependent occupation numbers of a two-level system and of a one-dimensional model for the hydrogen atom.'
author:
- 'I. Serban, J. Werschnik and E.K.U. Gross'
bibliography:
- 'paper.bib'
title: 'Optimal control of time-dependent targets'
---
Introduction
============
Given a quantum-mechanical system, which laser pulse is able to drive the system from state A to state B in a finite time-interval? Which laser pulse maximizes the density in a certain given region in space by the end of the pulse? Questions of this kind are addressed by optimal control theory (OCT) in the context of nonrelativistic quantum mechanics. OCT as a field of mathematics dates back to the late 1950s and is widely applied in engineering. One of the most famous examples in engineering is the reentry problem of a space vehicle into the earth’s atmosphere (see, e.g., [@SB2]). The application of OCT to quantum mechanics started in the 1980s [@BS90book; @HTC83; @PDR88; @K89]. Due to the enormous progress in the shaping of laser fields [@WLPW92], the control of chemical reactions became within reach. Experiments using closed loop learning (CLL) [@JR92] delivered highly convincing results [@B97; @A98; @LMR2001; @D2003].
Calculated pulse shapes may be employed directly in the experimental setup, e.g., as an initial guess for CLL genetic algorithms. Perhaps more important, the theory can be used to decipher the control mechanism embedded in the experimental pulse shapes [@B2004].
The optimal control schemes [@TKO92; @ZBR98; @MT2003] employed so far in theoretical simulations and the experimental applications have been designed to reach a predefined target at the end of a finite time-interval. Little is known about controlling the path the quantum system takes to the desired target, i.e., controlling the trajectory in real space or in quantum number space. To our knowledge, three methods have been proposed so far: A fourth-order Euler-Lagrange equation to determine the envelope of the control field has been derived in Ref. [@GGB2002]. This work, however, is restricted to very simple quantum systems.
Another very elegant method, known as tracking, has been proposed by the authors of Ref. [@ZR2003] and Ref. [@S2003]. Despite its tremendous success, this method bears an intrinsic difficulty: One has to prescribe a path that is controllable, otherwise singularities in the field appear, because of the one-to-one correspondence between the control field and the given trajectory. In practice, this may require a lot of intuition.
The third method is an optimal control scheme for time-dependent targets [@OTR2004]. Basically, it combines optimal control schemes for time-independent targets [@ZBR98; @ZR98] with an extension to Liouville-space [@O2001]. The schemes are generalized by introducing two new parameters like in Ref. [@MT2003] and then extended to include time-dependent targets. The new method is monotonically convergent and, in contrast to tracking, does not require a large amount of intuition, i.e., choosing controllable objectives. Furthermore, the method is not restricted to two-level systems. While Ref. [@OTR2004] presents the monotonically convergent algorithm, the full power of the method has not been exploited as yet. The challenge is the control of a truly time-dependent target represented by a positive-semidefinite, explicitly time-dependent operator. In Sec. II, we describe the general theory along with some examples of such operators. In particular, we discuss the control of occupation numbers in time and the indirect optimization of the dipole operator. The iterative procedure and some numerical details are explained in Sec. III. The results are presented in Sec. IV.
Theory
======
We consider an electron in an external potential $V({\bf r})$ under the influence of a laser field. Given an initial state $\Psi({ \bf r },0)=\phi({ \bf r })$, the time evolution of the electron is described by the time-dependent Schrödinger equation with the laser field modeled in the dipole approximation (length gauge), $$\begin{aligned}
i\frac{\partial}{\partial t}\Psi({ \bf r },t)&=&\widehat{H}\Psi({ \bf r },t),\label{1SE}\\
\widehat{H}&=&\widehat{H}_0-\hat{{\boldsymbol \mu}}{\boldsymbol \epsilon}(t),\\
\widehat{H}_0&=&\widehat{T}+\widehat{V}
$$ (atomic units are used throughout: $\hbar = m =e =1 $). Here, $\hat{{\boldsymbol \mu}}=(\hat{\mu}_x,\hat{\mu}_y,\hat{\mu}_z) $ is the dipole operator and ${\boldsymbol \epsilon}(t)=(\epsilon_x(t),\epsilon_y(t), \epsilon_z(t))$ is the time-dependent electric field. The kinetic energy operator is $\widehat{T}=-\nabla^2/2$.
Our goal is to control the time evolution of the electron by the external field in a way that the time-averaged expectation value of the target operator $\widehat{O}(t)$ is maximized. Mathematically, this goal corresponds to maximizing the functional $$\begin{aligned}
\label{eq:J1}
J_1[\Psi]&=&\frac{1}{T} \int_0^T \!\! dt \,\, \langle \Psi(t)|\widehat{O}(t)|\Psi(t)\rangle,
$$ where $\widehat{O}(t)$ is assumed to be positive-semidefinite.
We want to keep the meaning of the operator $\widehat{O}(t)$ as general as possible at this point. A few examples will be discussed at the end of this section.
Let us define $$\begin{aligned}
\widehat{O}(t)&=&\widehat{O}_{1}(t)+ 2 T \delta(t-T) \, \widehat{O}_{2},
$$ so we can also include targets in our formulation that only depend on the final time $T$ [@ZBR98; @ZR98; @K89]. The functional $J_1[\Psi]$ will be maximized subject to a number of physical constraints. The idea is to cast also these constraints in a suitable functional form and then calculate the total variation. Subsequently, we set the total variation to zero and find a set of coupled partial differential equations [@K89; @PDR88]. The solution of these equations will yield the desired laser field ${ \boldsymbol \epsilon }(t)$.
In more detail, optimizing $J_1$ may possibly lead to fields with very high, or even infinite, total intensity. In order to avoid these strong fields, we include an additional term in the functional which penalizes the total energy of the field, $$\begin{aligned}
J_2[{ \boldsymbol \epsilon }] &=& - \alpha\int_0^T \!\! dt \,\, { \boldsymbol \epsilon }^2(t).
$$ The penalty factor $\alpha$ is a positive parameter used to weight this part of the functional against the other parts.
The constraint that the electron’s wave-function has to fulfill the time-dependent Schrödinger equation is expressed by $$\begin{aligned}
J_3[{ \boldsymbol \epsilon },\Psi,\chi]&=&
- 2 \Im \int_{0}^{T}\!\! dt \,\, \left\langle\chi(t) \left| \left(i\partial_t
-\widehat{H}\right) \right| \Psi(t)\right\rangle
$$ with a Lagrange multiplier $\chi({ \bf r },t)$. $\Psi({ \bf r },t)$ is the wave function driven by the laser field ${ \boldsymbol \epsilon }(t)$.\
The Lagrange functional has the form $$\begin{aligned}
J[\chi,\Psi,{ \boldsymbol \epsilon }] = J_1[\Psi] + J_2[{ \boldsymbol \epsilon }] + J_3[\chi,\Psi,{ \boldsymbol \epsilon }].
$$ Setting the variations of the functional with respect to $\chi$, $\Psi$, and ${ \boldsymbol \epsilon }$ independently to zero yields $$\begin{aligned}
\alpha \epsilon_j(t) &=& -\Im\langle\chi(t)|\hat{\mu}_j|\Psi(t)\rangle, \label{eq:field} \qquad j=x,y,z\\
0 &=& \left( i \partial_t - \widehat{H} \right) \Psi({ \bf r },t), \label{eq:SE}\\
\Psi({ \bf r },0) &=& \phi({ \bf r }),\label{eq:SE_init}\\
&&\left(i\partial_t - \widehat{H}\right)\chi({ \bf r },t) + \frac{i}{T}\widehat{O}_{1}(t) \Psi({ \bf r },t)=\nonumber\\
\label{eq:INHSE}
&&i\left(\chi({ \bf r },t)-\widehat{O}_{2}(t) \Psi({ \bf r },t)\right)\delta(t-T).
$$ Equation (\[eq:field\]) determines the field from the wave function $\Psi({ \bf r },t)$ and the Lagrange multiplier $\chi({ \bf r },t)$.
Equation (\[eq:SE\]) is a time-dependent Schrödinger equation for $\Psi({ \bf r },t)$ starting from a given initial state $\phi({ \bf r })$ and driven by the field ${ \boldsymbol \epsilon }(t)$. If we require the Lagrange multiplier $\chi({ \bf r },t)$ to be continuous, we can solve the following two equations instead of Eq. (\[eq:INHSE\]): $$\begin{aligned}
\label{eq:INHSE2}
\left(i\partial_t - \widehat{H}\right)\chi({ \bf r },t)&=&-\frac{i}{T}\widehat{O}_{1}(t)\Psi({ \bf r },t),\\
\label{eq:INHSE3}
\chi({ \bf r },T) &=&\widehat{O}_{2} \Psi({ \bf r },T),
$$ To show this we integrate over Eq. (\[eq:INHSE\]), $$\begin{aligned}
\nonumber
&& \lim_{\kappa\to 0}\int_{T-\kappa}^{T+\kappa}\!\!\!\!\!\!dt\left[\left(i\partial_t - \widehat{H}\right)\chi({ \bf r },t) + \frac{i}{T}\widehat{O}_{1}(t) \Psi({ \bf r },t)\right]\\
\label{eq:proof1}
&=&\lim_{\kappa\to 0}\int_{T-\kappa}^{T+\kappa}\!\!\!\!\!\!dt\; i\left(\chi({ \bf r },t)-\widehat{O}_{2}(t) \Psi({ \bf r },t)\right)\delta(t-T).
$$ The left-hand side of Eq. (\[eq:proof1\]) is 0 because the integrand is a continuous function. It follows that also the right side must be 0, which implies Eq. (\[eq:INHSE3\]). From Eqs. (\[eq:INHSE3\]) and (\[eq:INHSE\]) then follows Eq. (\[eq:INHSE2\]).
Hence, the Lagrange multiplier satisfies an inhomogeneous Schrödinger equation with an initial condition at $t=T$. Its solution can be formally written as $$\begin{aligned}
\label{eq:SOL_INHSE}
\chi({ \bf r },t)&=&\widehat{U}_{t_0}^{t}\chi({ \bf r },t_0)-
\frac{1}{T}\int_{t_0}^{t} \!\! d\tau \,\, \widehat{U}_{\tau}^{t}\left(\widehat{O}_{1}(\tau) \, \Psi({ \bf r },\tau)\right),
$$ where $U_{t_0}^{t}$ is the time-evolution operator defined as $U_{t_0}^{t}=\mathcal{T}\exp\left(-i \int_{t_0}^{t} \!\! dt' \,\, \widehat{H}(t')\right)$.
The set of equations that we need to solve is now complete: Eqs. (\[eq:field\]), (\[eq:SE\]), (\[eq:SE\_init\]), (\[eq:INHSE2\]), and (\[eq:INHSE3\]). To find an optimal field ${ \boldsymbol \epsilon }(t)$ from these equations we use an iterative algorithm which is discussed in the next section.
In principle, we are not restricted to a single particle. The derivation and the algorithm can be generalized to many-particle systems, but except for a few model systems the numerical solution of the many-particle time-dependent Schrödinger equation is not feasible.
We conclude this section with a few examples for the target operator $\widehat{O}(t)$.
#### Final-time control.
Since our approach is a generalization of the traditional optimal control formulation given in [@K89; @ZBR98; @ZR98], we first observe that the latter is trivially recovered as a limiting case by setting $$\begin{aligned}
\widehat{O}_1(t) = 0, \qquad \widehat{O}_2 = \widehat{P} = | \Phi_f \rangle \langle \Phi_f|.
$$ Here $\Phi_f$ represents the target final state which the propagated wave function $\Psi({ \bf r },t)$ is supposed to reach at time $T$. In this case, the target functional reduces to [@K89; @ZBR98] $$\begin{aligned}
J_1 = \langle \Psi(T) | \widehat{P} | \Psi(T) \rangle = | \langle \Psi(T) | \Phi_f \rangle| ^2 .
$$ The target operator may also be local, as pointed out in Ref. [@ZR98]. If we choose $\widehat{O}_1(t) = 0$ and $\widehat{O}_2 = \delta({ \bf r }-{ \bf r }_0)$ (the density operator), we can maximize the probability density in ${ \bf r }_0$ at $t=T$, $$\begin{aligned}
\label{eq:loc_op}
J_1 = \int \!\! d{ \bf r } \,\, \langle \Psi(T)| \widehat{O}_2 |\Psi(T) \rangle = n({ \bf r }_0,T).
$$ Numerically, the $\delta$ function can be approximated by a sharp Gaussian function.
#### Wave-function-follower:
The most ambitious goal is to find the pulse that forces the system to follow a predefined wave function $\Phi({ \bf r },t)$. If we choose $$\begin{aligned}
\widehat{O}_{1}(t) &=& |\Phi(t)\rangle\langle\Phi(t)|, \\
\widehat{O}_{2}&=&0,
$$ the maximization of the time-averaged expectation value of $\widehat{O}_t^{(1)}$ with respect to the field ${ \boldsymbol \epsilon }(t)$ becomes almost equivalent to the inversion of the Schrödinger equation, i.e., for a given function $\Phi({ \bf r },t)$ we find the field $ { \boldsymbol \epsilon }(t)$ so that the propagated wave function $\Psi({ \bf r },t)$ comes as close as possible to the target $\Phi({ \bf r },t)$ in the space of admissible control fields. We can apply this method to the control of time-dependent occupation numbers, if we choose the time-dependent target to be $$\begin{aligned}
| \Phi(t) \rangle &=&a_0(t)e^{-i\mathcal{E}_0t}|0\rangle +a_1(t)e^{-i\mathcal{E}_1t}|1\rangle +a_2(t)e^{-i\mathcal{E}_2t}|2\rangle +\ldots\:\:,\\
\hat{H}_0|n\rangle&=&\mathcal{E}_n|n\rangle,\\
\label{eq:td_op}
\widehat{O}_{1}(t)&=& | \Phi(t) \rangle \langle \Phi(t) |.
$$ The functions $|a_0(t)|^2, |a_1(t)|^2, |a_2(t)|^2, \ldots$ are the predefined time-dependent level occupations which the optimal laser pulse will try to achieve. In general, the functions $a_0(t),a_1(t),a_2(t), \ldots$ can be complex, but as demonstrated in Sec. IV, real functions are sufficient in this case to control the occupations in time. For example, if in a two-level-system the occupation is supposed to oscillate with frequency $\Omega$ we could choose $a_0(t) = \cos(\Omega t)$ and, by normalization, $a_1(t)=\sin(\Omega t)$. This defines the time-dependent target operator (\[eq:td\_op\]).\
#### Moving density.
The operator used in Eq. (\[eq:loc\_op\]) can be generalized to $$\begin{aligned}
\widehat{O}_{1}(t)&=&\delta({ \bf r }-{\bf r }_0(t)),\\
J_1 &=& \frac{1}{T} \int_0^T \!\! dt \,\, \langle \Psi(t) | \delta({ \bf r }-{ \bf r}_0(t)) | \Psi(t) \rangle \nonumber\\
&=& \frac{1}{T} \int_0^T \!\! dt \,\, n({\bf r}_0(t),t).
$$ $J_1$ is maximal if the field is able to maximize the density along the trajectory ${\bf r }_0(t)$.
Algorithm and numerical details
===============================
Equipped with the control equations (\[eq:field\]), (\[eq:SE\]), and (\[eq:INHSE2\]) we have to a find an algorithm to solve these equations for ${\boldsymbol \epsilon}(t)$. In the following, we describe such a scheme which is similar to Ref. [@OTR2004]:
[l c c l c c c c ]{} & \^[(1)]{}(0) & & \^[(1)]{}(T) & & & &\
& & & & &\
& & & \^[(k)]{}(T) & & \^[(k)]{}(0) & &\
& & & & &\
& & & & &\
& & & & & \^[(k+1)]{}(0) & & \^[(k+1)]{}(T) .\
The laser fields used for the propagation are given by $$\begin{aligned}
\label{feld1}
\widetilde{\epsilon}_j^{(k)}(t) &=& (1-\eta)\epsilon_j^{(k)}(t)
- \frac{\eta}{\alpha}\Im\langle\chi^{(k)}(t)|\hat{\mu}_j|\Psi^{(k)}(t)\rangle,\\
\label{feld2}
\epsilon_j^{(k+1)}(t) &=& (1-\gamma)\widetilde{\epsilon}_j^{(k)}(t)
- \frac{\gamma}{\alpha}\Im\langle\chi^{(k)}(t)|\hat{\mu}_j|\Psi^{(k+1)}(t)\rangle \qquad j=x,y,z.
$$ The initial conditions in every iteration step are $$\begin{aligned}
\Psi({\bf r},0)&=& \phi({\bf r}),\\
\chi({\bf r},T) &=& \widehat{O}_2 \Psi({\bf r},T).
$$ The propagations in brackets are necessary only if one wants to avoid the storage of the time-dependent wave function and Lagrange multiplier. Note that the main difference between this iteration and the schemes used in [@MT2003] is that one needs to know the time-dependent wave function $\Psi({\bf r},t)$ to solve the inhomogeneous equation (\[eq:SOL\_INHSE\]) for the Lagrange multiplier $\chi({\bf r},t)$. Depending on the operator $\hat{O}_1(t)$, if the inhomogeneity is space- and time-dependent, this may require an additional time propagation.
The choice of $\eta$ and $\gamma$ completes the algorithm. $\gamma = 1$ and $\eta = 1$ correspond to the algorithm suggested in [@ZR98], while the choice $\gamma = 1$ and $\eta = 0$ is analogous to the method used in [@K89] with a direct feedback of $\Psi^{(k)}({\bf r},t)$. Further choices are discussed in Refs. [@MT2003; @OTR2004].\
In the following, we demonstrate the application of our algorithm to two different kinds of time-dependent targets, namely the control of occupation numbers and the control of a local operator. The first example is a two-level system consisting of states $|0 \rangle$, $|1 \rangle$ with a resonance frequency of $\omega_{01} = \omega_0 - \omega_1 = 0.395$ and the dipole matrix element $P_{01} = \langle 1 | \hat{\mu} |0 \rangle = 1.05$. The second system is a 1D model for hydrogen [@SE91], that has a “soft” Coulomb potential, $$V(x) = - \frac{1}{\sqrt{x^2 + 1}}.
$$ This type of potential has been used extensively to gain qualitative insights in the behavior of atoms in strong laser pulses [@LGE2000; @KLEG2001].
The parameters $\omega_{01}$ and $P_{01}$ of the two-level system are chosen to be identical with the lowest excitation energy and the corresponding dipole matrix element of 1D hydrogen.
The solution of the optimal control Eqs. (\[eq:field\]), (\[eq:SE\]), (\[eq:INHSE2\]), and (\[eq:INHSE3\]) requires the integration of the time-dependent Schrödinger equation with and without inhomogeneity.\
In the case of the two-level system, one may diagonalize the Hamilton operator analytically and therefore calculate the infinitesimal time-evolution operator directly.
The time-dependent Schrödinger equation for the 1D hydrogen model is solved on a grid, where the infinitesimal time-evolution operator is approximated by the second-order split-operator technique (SPO) [@FMF76], $$\begin{aligned}
\nonumber
\widehat{U}_{t}^{t+\Delta t}&=&\mathcal{T}\exp\left(-i \int_{t}^{t+\Delta t} \!\! dt' \,\, \widehat{H}(t')\right)\\
\nonumber
\label{eq:spo2nd}
& \approx & \exp(-\frac{i}{2}\, \hat {T}\,\Delta t) \exp(-i\, \hat {V}(t)\,\Delta t) \nonumber \\
& &\exp(-\frac{i}{2}\, \hat {T}\,\Delta t) + O(\Delta t^3).
$$ For the inhomogeneous Schrödinger equation (\[eq:INHSE3\]), the infinitesimal time evolution of $\chi(x,t)$ is given by, $$\begin{aligned}
&&\chi(x,t+\Delta t) \nonumber\\
&=&\widehat{U}_{t}^{t+\Delta t}\left(\chi(x,t)-
\frac{1}{T}\int_{t}^{t+\Delta t}\!\!\!\!\!\!\!\!d\tau\,\,
\widehat{U}_{\tau}^{t}\left(\widehat{O}_1(\tau)\Psi(x,\tau)\right)\right)\\
&\simeq& \widehat{U}_{t}^{t+\Delta t}\left(\chi(x,t)-\Delta t\frac{1}{T} \widehat{O}_1(t)\Psi(x,t)\right),
\end{aligned}$$ where we found the above, lowest-order approximation of the integral to be sufficient.\
Following the scheme described above, one needs five propagations per iteration step (if we want to avoid storage of the wave function). Within the 2nd order split-operator scheme each time step requires four fast Fourier transforms (FFT) [@FFTW98], because we have to know the wave function and the Lagrange multiplier in real space at every time step to be able to evaluate the field from Eq. (\[eq:field\]). This sums up to $2*10^6$ FFTs per $10^5$ time steps and iteration. In comparison, optimal control methods for time-independent targets [@MT2003] require four propagations.
Since our hydrogen model can experience ionization, we employ absorbing boundaries to take care of boundary effects (otherwise we will find the outgoing wave incoming from the opposite boundary due to the periodic boundaries introduced by the Fourier transform). The real-space wave function is multiplied with a mask function that falls off like $\cos^{(1/8)}$ at the boundary in every time step.\
Results
=======
### Occupation number control
First, we present the results for the two-level system. The time-dependent target wave function is chosen as $|\Phi(t)\rangle = a_0(t)e^{-i\mathcal{E}_0t}|0\rangle +a_1(t)e^{-i\mathcal{E}_1t}|1\rangle$, where the coefficients $a_0(t)$ and $a_1(t)$ are real and satisfy $a_0^2(t)+a_1^2(t)=1$. $\hat{O}_1=|\Phi(t)\rangle\langle\Phi(t)|$, $\hat{O}_2=0$.
With the parameters $\Delta t = 0.01$, $\alpha$ = 0.05 and the initial guess field $\epsilon_0(t)$ = $10^{-4}$, the algorithm converges to a final value of $J_1$ = 0.9995 with a difference between two consecutive values of the functional $\delta J^{(n,n+1)}\leq 10^{-8}$ . Figure \[3eck\_2lev\] shows the numerical results for the time evolution of the occupation numbers \[Fig. \[3eck\_2lev\_a\]\] and the optimized field \[Fig. \[3eck\_2lev\_b\]\]. Figure \[3eck\_2lev\_c\] illustrates the monotonic convergence of the functional $J_1+J_2$. The agreement between the calculated occupation and the V-shaped target function, as shown in Fig. \[3eck\_2lev\_a\] is quite remarkable. To illustrate the quality of results associated with different values of $J_1$, we have plotted the occupation curves corresponding to $J_1=0.90,0.95,0.99$ in Fig. \[3eck\_2lev\_a\]. Somewhat surprisingly, even if we reach $J_1=0.95$, there is still a sizable difference between the curves. Figure \[3eck\_2lev\_b\] shows the envelopes extracted from the laser fields corresponding to $J_1=0.90,0.95$ as well as the optimal field corresponding to $J_1=0.9995$. The resonance frequency of the system is found within a few steps. Then the algorithm improves the envelope. Figure \[3eck\_2lev\_c\] shows the typical convergence behavior: a rapid improvement of the functional in the first few steps, implying that the difference between two consecutive fields is large (\[eq:field\_diff\]), and a slower convergence for the later steps, meaning that the differences between two steps get smaller \[see Fig. \[3eck\_2lev\_b\]\].
The same problem was solved for the 1D hydrogen model on a grid. We found $J_1 = 0.97$ with $\delta J^{(n,n+1)} \leq 10^{-5} $. The parameters were $\Delta t = 0.005$, $\alpha=1.5$, and the initial choice for the field was again $\epsilon_0(t)$ = $10^{-4}$. From our experience with the two-level system, we expect that the correspondence between the target curve and the optimized curve will not be perfect.
The corresponding numerical results are shown in Fig. \[3eck\_sc\]. Note, that the occupation of higher levels is negligible and that ionization is less than $0.2 \%$. The field is in the weak response regime.\
#### Tracking versus optimal control. {#tracking-versus-optimal-control. .unnumbered}
Zhu and Rabitz showed [@ZR2003] how the exact field necessary to follow a given trajectory can be determined by means of Ehrenfest’s theorem. The exact field, however, may have singularities, i.e., the prescribed trajectory is not controllable with a smooth field. If, like in the next example, the target occupation curves $b_0^2(t)$,$b_1^2(t)$ consist of step functions, the exact field must have $\delta$ peaks and, as a consequence, the tracking method cannot produce any useful results. The optimal control approach followed in this paper finds the best compromise between field energy and overlap with the target, yielding reasonable results such as those shown in Fig. \[step\_2lev\].
At times where $b_0^2(t)$ becomes discontinuous, the field has intense pulses (see Fig.\[step\_2lev\_b\]) consisting of only a few oscillations with the resonance frequency.
The time-dependent occupation numbers in \[Fig. \[step\_2lev\_a\]\] deviate slightly from the target curve. They are “washed out” at the discontinuity points of the target curve. For larger values of the penalty factor we notice that this broadening of the steps is even more pronounced (see Fig.\[steps2\_a\]). In this case, the width of the pulse envelope becomes broader and the maximum field strength lower \[Fig. \[steps2\_b\]\].
### Local operator
A very important quantity to control is the time-dependent dipole moment. This quantity, however, cannot be accessed directly because the dipole operator is not positive semidefinite. As an alternative, we choose the time-dependent density operator, $$\begin{aligned}
\widehat{O}(t)&=& \delta (x-r(t))\label{eq:local_op}\\
\label{eq:delta_approx}
& \approx&\sqrt[4]{\frac{\sigma}{\pi}}e^{-(x-r(t))^2\sigma}.
$$ Intuitively, the dipole moment will roughly follow the curve described by $r(t)$ since $r(t)$ governs the movement of the density.
In the actual computations, we approximate the $\delta$ function by a sharp Gaussian (\[eq:delta\_approx\]).
To test this idea, we first have to choose a reasonable function $r(t)$. For this purpose we solve the time-dependent Schrödinger equation for the 1D hydrogen model with a given laser field $\epsilon(t)$. From the resulting wave function, we calculate the time-dependent expectation value $r(t)= \langle \hat{x} \rangle(t)$. With the function $r(t)$ we then build the target operator (\[eq:delta\_approx\]) and start our optimization with the initial guess $\epsilon_0(t)= 10^{-4}$.\
Figure \[local1\_a\] shows that the expectation value $\langle \hat{x} \rangle_{opt}$ calculated with the optimal field $\epsilon_{opt}(t)$ follows the target $r(t)$ rather closely. We do not obtain a perfect correspondence between $\langle \hat{x} \rangle (t)$ and $r(t)$, but the results clearly demonstrate that the algorithm also works for this type of target and, hence, that the indirect approach to control the dipole moment is appropriate.
As proven in the Appendix, it is also possible to optimize functionals of the type $J_1=\int_0^T dt {\underbrace{\langle\Psi(t)|\widehat{O}(t)|\Psi(t)\rangle}_{I_1}}^n$, $n>1$. Since our integrand $I_1$ is $\leq 1$, the effect will be that $J_1$ carries less weight in the optimization, i.e., the algorithm will try to decrease the field energy. Hence we expect the similarity between the target trajectory $r(t)$ and the calculated expectation value $\langle x\rangle_{opt}$ to be less for increasing $n$. The results for $n=2,3,4$ are shown in Fig. \[hoch\_n\_bild\]. If we build a target functional with the integrand $I_1 \geq 1$ we will find the opposite effect, i.e., $J_1$ will become more important than before. This demonstrates that the parameter $n$ provides a new handle (in addition to the penalty factor $\alpha$) to shift the relative importance of $J_1$ versus $J_2$.
![Comparison for the expectation value $\langle \hat{x}\rangle$ with the target trajectory (squares) for different exponents $n$. []{data-label="hoch_n_bild"}](6.eps){width=".45\textwidth"}
Conclusion
==========
This work deals with the quantum control of time-dependent targets. In Sec. II, we presented explicit examples of positive-semidefinite target operators designed for the control of time-dependent occupations and of the time-dependent dipole moment. We then applied these operators to control the time evolution of a simple two-level system and of a grid model (1D hydrogen). In each case, we find a continuous increase of the value of the functional $J_1 + J_2$. The improvement in the first iteration steps is quite strong, while it takes a large number of iterations to converge the last few percentages. The results also show that a large number of iterations is required to reach perfect agreement with the target trajectories. In the Appendix we prove that by exponentiating the integrand with a positive integer $n>1$, the iteration still converges monotonically. The functional constructed in this way contains two parameters, $\alpha$ and $n$, that allow one to fine-tune the relative importance of $J_1$ and $J_2$. To summarize, we have demonstrated in this work that the quantum control of genuinely time-dependent targets is feasible. In particular, the successful control of the dipole moment in time may open new avenues to optimize high-harmonic generation, which is extremely important for shaping attosecond laser pulses. Work along these lines is in progress.
We would like to thank Stefan Kurth for valuable discussions. This work was supported, in part, by the Deutsche Forschungsgemeinschaft, the EXC!TING Research and Training Network of the European Union and the NANOQUANTA Network of Excellence.
We want to show that the same iteration will converge monotonically also for a functional of the type $$\begin{aligned}
J_1 & = & \int_0^T\!\!\!\!dt\; \langle\Psi(t)|\widehat{O}(t)|\Psi(t)\rangle^{n},\nonumber
$$ where $n>1$, $n\in \mathbbm{N}$. The equation for the Lagrange multiplier then has the form $$\begin{aligned}
(i\partial_t-H)\chi({\bf r},t) & = &
-ni\langle\Psi(t)|\widehat{O}(t)|\Psi(t)\rangle^{n-1}
\widehat{O}(t)\Psi({\bf r},t),\label{hoch_n}\\
\chi({\bf r},T)&=&0.\label{hoch_n_ab}
$$ Now consider $a$ and $b$ as real positive numbers. Then $$\begin{aligned}
a^n-b^n & = & n b^{n-1} (a - b) + \underbrace{a^n + (n - 1) b^n - n b^{n-1} a}_{=A}.
$$ Next we show that $A=a^n+(n-1)b^n-nb^{n-1}a$ is never negative. Defining $a = b + \delta$, $\delta\in[-b, \infty)$, we distinguish between two cases.
Case I: $\delta\in[0, \infty)$, $$\begin{aligned}
&&a^n+(n-1)b^n-nb^{n-1}a\nonumber\\
&& = (b+\delta)^n+(n-1)b^n-nb^{n-1}(b+\delta) \nonumber\\
&& = b^n + n b^{n-1}\delta + \dots + n \delta^{n-1}b \nonumber\\
&& +\delta^n+(n-1)b^n-nb^{n-1}\delta\nonumber\\
&& = \frac{n!}{(n-2)!2!}b^{n-2}\delta^2+ \dots + n \delta^{n-1}b +\delta^n \geq 0.
$$
Case II: $\delta\in[-b, 0)$, $$\begin{aligned}
&&(b + \delta)^n + (n-1)b^n-nb^{n-1}(b+\delta) \nonumber\\
&&=\underbrace{b^n}_{\geq0}\Bigg[\left(1+\frac{\delta}{b}\right)^n\nonumber\\
&&+n-1-n-n\frac{\delta}{b}\Bigg].
$$ To evaluate Case II, we introduce $x= \delta/b$, $x\in[-1,0)$ with, $$\begin{aligned}
f(x)&=&(1+x)^n-nx-1,\nonumber\\
y&:=&x+1,\nonumber\\
y&\in&[0,1),\nonumber\\
f(y)&=&y^n-ny+n-1,\nonumber\\
f(0)&=&n-1,\nonumber\\
f(1)&=&0,\nonumber\\
f'(y)&=&n\underbrace{(y^{n-1}-1)}_{\leq{0}}\nonumber\\
&\Longrightarrow & f(y)\in[0,n-1]\nonumber\\
&\Longrightarrow & f(y)\geq 0.\nonumber
$$ Since $f'(y)<0$, the function $f$ must decrease monotonically from $n-1>0$ to 0, so it cannot become negative. In conclusion, $$\begin{aligned}
a^n - b^n = nb^{n-1}(a-b)+\underbrace{a^n+(n-1)b^n-nb^{n-1}a}_{A > 0}.\label{positiv}
$$ The deviation in $J$ between two consecutive steps is, $$\begin{aligned}
\delta J^{(k+1,k)} & = & J^{(k+1)}-J^{(k)}\nonumber\\
& = &\int_0^T\!\!\!\!dt\;
\Big(\langle\Psi^{(k+1)}(t)|\widehat{O}(t)|\Psi^{(k+1)}(t)\rangle^{n}\nonumber\\
&-&\langle\Psi^{(k)}(t)|\widehat{O}(t)|\Psi^{(k)}(t)\rangle^{n} \nonumber\\
&+& \alpha\left[\epsilon^{(k)}(t)\right]^2-
\alpha\left[\epsilon^{(k+1)}(t)\right]^2\Big).\nonumber\end{aligned}$$ If we identify $a(t) = \langle\Psi^{(k+1)}(t)|\widehat{O}(t)|\Psi^{(k+1)}(t)\rangle$ and $b(t) = \langle\Psi^{(k)}(t)|\widehat{O}(t)|\Psi^{(k)}(t)\rangle$ and use Eq. (\[positiv\]), $$\begin{aligned}
\delta J^{(k+1,k)}& = & \int_0^T\!\!\!\!dt\;\Bigg( A(t) +
\alpha\left[\epsilon^{(k)}(t)\right]^2-
\alpha\left[\epsilon^{(k+1)}(t)\right]^2 \nonumber\\
& + & n \langle\Psi^{(k)}(t)|\widehat{O}(t)|\Psi^{(k)}(t)\rangle^{n-1} \nonumber\\
& & \left(\langle\Psi^{(k+1)}(t)|\widehat{O}(t)|\Psi^{(k+1)}(t)\rangle-
\langle\Psi^{(k)}(t)|\widehat{O}(t)|\Psi^{(k)}(t)\rangle\right)\Bigg),\nonumber\end{aligned}$$ where we have separated the positive term $A(t)=a^n(t)+(n-1)\,b^n(t)-nb^{n-1}(t)\,a(t)$. We define $B(t)=n \langle\Psi^{(k)}(t)|\widehat{O}(t)|\Psi^{(k)}(t)\rangle^{n-1}\langle\delta\Psi^{(k+1,k)}(t)|\widehat{O}(t)|\delta\Psi^{(k+1,k)}(t)\rangle\geq0$ and rewrite $\delta J^{(k+1,k)}$ as $$\begin{aligned}
\delta J^{(k+1,k)}& = &\int_0^T\!\!\!\!dt\; \Big(A(t) + B(t)+
\alpha\left[\epsilon^{(k)}(t)\right]^2-
\alpha\left[\epsilon^{(k+1)}(t)\right]^2\nonumber\\
& + & n \langle\Psi^{(k)}(t)|\widehat{O}(t)|\Psi^{(k)}(t)\rangle^{n-1}
2\Re\langle\Psi^{(k)}(t)|\widehat{O}(t)|\delta\Psi^{(k+1,k)}(t)\rangle\Big),\nonumber\end{aligned}$$ where $\delta\Psi^{(k+1,k)}({\bf r},t)=\Psi^{(k+1)}({\bf r},t) - \Psi^{(k)}({\bf r},t)$. We use the equation for the Lagrange multiplier (\[hoch\_n\]) and obtain $$\begin{aligned}
\delta J^{(k+1,k)}& = & \int_0^T\!\!\!\!dt\; \Big(A(t) + B(t)\nonumber\\
& + & \alpha\left[\epsilon^{(k)}(t)\right]^2-
\alpha\left[\epsilon^{(k+1)}(t)\right]^2 \nonumber\\
& + & 2\Re \left\langle-\left(\partial_t+i\widetilde{H}^{(k)}\right)\chi^{(k)}(t)|\delta\Psi^{(k+1,k)}(t)\right\rangle\Big),
\label{conect2}\end{aligned}$$ where $\widetilde{H}^{(k)}=\widehat{H}_0-\hat{\boldsymbol \mu}\widetilde{\epsilon}^{(k)}$, $$\begin{aligned}
\delta J^{(k+1,k)}& = & \int_0^T\!\!\!\!dt\;\Big(A(t) + B(t) \nonumber\\
& + & \alpha\left[\epsilon^{(k)}(t)\right]^2-
\alpha\left[\epsilon^{(k+1)}(t)\right]^2 \nonumber\\
& + & 2\Im\left\langle\chi^{(k)}(t)|\left(i\partial_t-\widetilde{H}^{(k)}\right)|\delta\Psi^{(k+1,k)}(t)\right\rangle\Big) \nonumber\\
&+& \underbrace{2\Re\langle\chi(t)|\delta\Psi^{(k+1,k)}(t)\rangle|_0^T}_{0}.
\label{hoch_n_conect}
$$ For the last term in Eq. (\[hoch\_n\_conect\]) we used the fact that $\delta\Psi^{(k+1,k)}({\bf r},0)$ = 0 since the initial state for the wave function is fixed and $\chi({\bf r},T)$ = 0 because of Eq. (\[hoch\_n\_ab\]).
We use the time-dependent Schrödinger equation (\[eq:SE\]) for $\Psi^{(k)}$ and $\Psi^{(k+1)}$, where $\widehat{H}^{(k)}=\widehat{H}_0-\hat{\boldsymbol\mu}{\epsilon}^{(k)}$, $$\begin{aligned}
&&\left(i\partial_t-\widetilde{H}^{(k)} \right) \delta \Psi^{(k+1,k)}({\bf r},t) \nonumber\\
&=&\left(i\partial_t-\widetilde{H}^{(k)} \right) \left( \Psi^{(k+1)}({\bf r},t)-\Psi^{(k)}({\bf r},t) \right)\nonumber\\
&=&\left( \widehat{H}^{(k+1)}-\widetilde{H}^{(k)} \right) \Psi^{(k+1)}({\bf r},t) \nonumber\\
\nonumber
&&-\left( \widehat{H}^{(k)} - \widetilde{H}^{(k)} \right) \Psi^{(k)}({\bf r},t) \\
\nonumber
&=&-\left( {\boldsymbol \epsilon}^{(k+1)}(t) - \widetilde{{\boldsymbol \epsilon}}^{(k)}(t) \right) \hat{{\boldsymbol \mu}}({\bf r}) \Psi^{(k+1)}({\bf r},t) \\
&&+\left({\boldsymbol \epsilon}^{(k)}(t)-\widetilde{{\boldsymbol \epsilon}}^{(k)}(t)\right) \hat{{\boldsymbol \mu}}({\bf r})\Psi^{(k)}({\bf r},t).
$$ Consequently, the change in the Lagrange functional becomes $$\begin{aligned}
\nonumber
\delta J^{(k+1,k)} & = & \int_0^T \!\! dt \,\,
\bigg( A(t) + B(t)+ \langle \delta \Psi^{(k+1,k)}(t) | \widehat{O}(t) | \delta \Psi^{(k+1,k)}(t) \rangle \\
\nonumber
&&- \alpha \left( \left[ {\boldsymbol \epsilon}^{(k+1)} \right]^2- \left[ {\boldsymbol \epsilon}^{(k)}(t) \right]^2 \right) \\
\nonumber
&&- 2 \Im \langle \chi^{(k)}(t)| \hat{{\boldsymbol \mu}} |\Psi^{(k+1)}(t) \rangle ( {\boldsymbol \epsilon}^{(k+1)}(t)- \widetilde{{\boldsymbol \epsilon}}^{(k)}(t))\\
&&+ 2 \Im \langle \chi^{(k)}(t)| \hat{{\boldsymbol \mu}} |\Psi^{(k)}(t) \rangle ( {\boldsymbol \epsilon}^{(k)}(t) - \widetilde{{\boldsymbol \epsilon}}^{(k)}(t)) \bigg).
$$ Finally, using Eqs. (\[feld1\]) and (\[feld2\]), we find $$\begin{aligned}
\nonumber
&&\delta J^{(k+1,k)} \nonumber\\
&=& \int_0^T \!\! dt \, \,
\bigg( A(t) + B(t)+ \langle\delta\Psi^{(k+1,k)}(t)|\widehat{O}(t)|\delta\Psi^{(k+1,k)}(t) \rangle \nonumber\\
\nonumber
& - &\alpha \left[ {\boldsymbol \epsilon}^{(k+1)}(t) \right]^2 + \alpha \left[ {\boldsymbol \epsilon}^{(k)} \right]^2 \\
& + & 2\bigg({\boldsymbol \epsilon}^{(k+1)}(t)-(1-\gamma) \widetilde{{\boldsymbol \epsilon}}^{(k)}(t)\bigg) \frac{\alpha}{\gamma}
\nonumber
\bigg({\boldsymbol \epsilon}^{(k+1)}(t)-\widetilde{{\boldsymbol \epsilon}}^{(k)}(t)\bigg)\\
& - & 2\bigg(\widetilde{{\boldsymbol \epsilon}}^{(k)}(t)-(1-\eta){\boldsymbol \epsilon}^{(k)}(t)\bigg) \frac{\alpha}{\eta}
\bigg({\boldsymbol \epsilon}^{(k)}(t)-\widetilde{{\boldsymbol \epsilon}}^{(k)}(t)\bigg)\bigg)\\
\nonumber
& = &\int_0^T\!\! dt \,\, \Bigg( A(t)+B(t) + \langle \delta \Psi^{(k+1,k)}(t)|\widehat{O}(t)|\delta\Psi^{(k+1,k)}(t) \rangle \\
\nonumber
&+& \alpha \left(\frac{2}{\gamma}-1\right) \left( {\boldsymbol \epsilon}^{(k+1)}(t) - \widetilde{{\boldsymbol \epsilon}}^{(k)}(t) \right)^2 \\
\label{eq:field_diff}
&+& \alpha \left(\frac{2}{\eta}-1\right) \left( {\boldsymbol \epsilon}^{(k)}(t) - \widetilde{{\boldsymbol \epsilon}}^{(k)}(t) \right)^2\Bigg).
$$ For $\eta, \gamma \in [0,2]$ (similar to [@MT2003] and [@OTR2004]), the iteration converges monotonically, i.e., $\delta J^{(k+1,k)}\geq 0$. This iteration converges monotonically and quadratically in terms of the field deviations between two iterations.
We emphasize that this proof is true only if the solution of the time-dependent (in)homogeneous Schrödinger equation is exact in each step. Numerical implementations are of course always approximate and, as a consequence, it may happen that the value of the functional $J$ decreases. This, on the other hand provides a test of the accuracy of the propagation method.\
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The Kerala Assembly on Thursday has passed a new law to create farmers welfare board in the state. The board, first of its kind in any Indian state, is aimed at improving the quality of life of the farmers and ensuring better financial stability.
The bill, proposed in the Assembly was referred to the select committee.
Following multiple rounds of discussions and sittings across the state, the bill was submitted and passed in the Assembly with amendments.
FINANCIAL BENEFITS AND PENSION
The bill proposes to set up a farmers welfare board which will ensure attractive financial benefits and monthly pension to the farmers.
Each farmer who posses a maximum of 15 acres of land or who is farming on lease land upto 15 acres is entitled to get the benefits of the welfare board.
Every farmer, above the age of 18 can contribute a minimum of Rs 100 per month to the scheme.
The government will make a contribution of the same amount or a maximum of Rs 250 for each farmer as part of the scheme. If a farmer continuously contributes for a minimum of five years, the farmer will be entitled to a lifetime pension after the age of 60.
The pension amount will be determined based on the contribution and number of years.
A farmer who has contributed to the scheme for 25 years will get a one-time payment based on his contribution to the scheme.
Women farmers or daughters of registered farmers will get financial support for marriage, education and maternity.
'FIRST OF ITS KIND'
State Agriculture Minister VS Sunil Kumar said that this is a first of its kind scheme in the entire country.
"We are trying to give a comprehensive support package to the farmers in the state. Apart from traditional farmers, plantations and farms will also be under this scheme. We have kept an annual income of Rs 5 lakh as the maximum limit to be part of this scheme. Now that the bill has been passed, we hope to setup the board between three to six months," VS Sunil Kumar said. | https://www.indiatoday.in/india/story/kerala-agriculture-farmers-bill-scheme-pension-finance-acres-land-1621302-2019-11-21 |
I first used the C language on a Commodore Amiga back in the early 90s using the DICE compiler. I created a few small programs for myself and friends, mainly mapping software for the play-by-mail games that I used to play at the time (Monster Isalnd, Dungeon Quest, Keys of Medokh etc.), although I did try to create some basic games as well.
Back then I didn't have access to the Internet so nothing ever got published (and probably never actually got to the stage of being publishable anyway). When I bought my first PC I moved onto the Visual Basic / Delphi route of programming and it wasn't until about 2016 that I started thinking about coding in C again.
I opted for C++ as that was integrated into Visual Studio that I had been using for developing in Visual Basic and I thought I'd give a few small problems a go. I created a few programs for solving some Project Euler problems and then came across a course on Udemy that included the basics of C++ and signed up.
My first public release is a very simple word based code-breaking game but hopefully I'll get the chance to create more work soon.
Click on any screenshot to see a larger version.
All images and files on this site are copyright Fractalytic, S.Kennedy.
Please ask if you wish to use anything for your own projects or if you have any questions about anything featured (Contact Me link in menu bar at top of page). Thank you.
Projects
Cows and Bulls v1.00
This is a simple word based code-breaking game created using C++ based on an excellent tutorial in the Unreal Engine Developer course on Udemy by GameDev.tv.
First release: 22/07/18 (v1.00)
Last Updated: no further updates yet
Download size: 20kb
Released July 2018
This is a complete game but currently has only 10 different words of each length to guess so probably won't keep you occupied for too long but I do intend to spend some time improving this and making a more advanced version with better options when I get time.
It works very similar to my Code Breaker game but is a console-based game and so doesn't include any fancy graphics or saving of statistics.
Back to top
Sieve of Eratosthenes v1.00
A simple c++ consolde program that uses the Sieve of Eratosthenes process for creating a list of prime numbers with a user specified maximum limit (up to 150,000). | http://fractalytic.co.uk/cplusplus.htm |
Road sign informs the gradient is 10.9%. Calculate the angle which average decreases.
- Lift
The largest angle at which the lift rises is 16°31'. Give climb angle in permille.
- Climb
For horizontal distance 4.2 km road rise by 6.3 m. Calculate the road pitch in ‰ (permille, parts per thousand).
- Height difference
What height difference overcome if we pass road 1 km long with a pitch21 per mille?
- Cable car
Find the elevation difference of the cable car when it rises by 67 per mille and the rope length is 930 m.
- Cablecar
Funicular on Petrin (Prague) was 408 meters long and overcomes the difference 106 meters in altitude. Calculate the angle of climb.
- Railways
Railways climb 7.4 ‰. Calculate the height difference between two points on the railway distant 3539 meters.
- Road
Between cities A and B is route 13 km long of stúpanie average 7‰. Calculate the height difference of cities A and B.
- Beer permille
In the 5 kg of blood of adult human after three 10° beers consumed shortly after another is 6.6 g of the alcohol. How much is it as per mille?
- Beer
After three 10° beers consumed in a short time, there is 5.6 g of alcohol in 6 kg adult human blood. How much is it per mille?
- Railway
Railway line had on 5.8 km segment climb 9 permille. How many meters track ascent?
- Mountain railway
Height difference between points A, B of railway line is 38.5 meters, their horizontal distance is 3.5 km. Determine average climb in permille up the track.
- Climb in percentage
The height difference between points A and B is 475 m. Calculate the percentage of route climbing if the horizontal distance places A, B is 7.4 km.
- Road - permille
5 km long road begins at an altitude 500 meters above sea level and ends at a altitude 521 ASL. How many permille road rises?
- River
Calculate how many permille river Dunaj average falls, if on section long 957 km flowing water from 1454 m AMSL to 101 m AMSL.
- Slope of track
Calculate the average slope (in permille and even in degrees) of the rail tracks between Prievidza (309 m AMSL) and Nitrianske Pravno (354 m AMSL), if the track is 11 km long.
Do you have an interesting mathematical word problem that you can't solve it? Submit a math problem, and we can try to solve it.
Our permille calculator will help you quickly calculate various typical tasks with permilles. Angle Problems. Per mil Problems. | https://www.hackmath.net/en/word-math-problems/angle?tag_id=16 |
Frequently Asked Questions
Why do moles cause such a problem?
It is believed that a single mole can dig up to 20 meters per day and that their tunnel systems need to be around 150-200 meters long for them to sustain the food intake needed.
Are moles blind?
Although most people assume moles are blind this is not the case. The mole does have poor eye sight but they are able to see although they mainly rely on their sense of smell to navigate and find their food source.
How big are moles?
The average mole weighs between 4-6 ounces with the male mole (boars) are larger than the females (sows). They can grow up to 6 inches in length.
How long do they live?
The average lifespan of a mole is said to be between 3-6 years. They will leave the nesting site from 5 weeks old.
Will they return?
Unfortunately if you have ever had a mole removed from your garden then there is always a possibility that another mole will eventually move and take up residence in the existing tunnels. If the area was suited to one mole then it will be for all.
How often do moles breed?
Moles will have 1 litter a year with the mating season between February and June. The litter can contain up to 7-8 moles but normally about 3-5. Unfortunately it is thought that most moles only have a 50% chance of making adulthood. | http://catchamole.co.uk/faqs/ |
A 200 m long train passes through the 700 m long tunnel so that a time of 1 minute elapses from the entry of the locomotive into the tunnel to the exit of the last wagon from the tunnel. Find the speed of the train.
- The temperature 13
The temperature in Toronto at noon during a winter day measured 4°C. The temperature started dropping 2° every hour. Which inequality can be used to find the number of hours, x, after which the temperature will measure below -3°C?
- Shortcut
The road from the cottage to the shop 6 km away leads either along a straight road that the bicycle can drive at a speed of 18 km/h or by "shortcut". It measures only 3.6 km. But the road from the cottage is all uphill - at a speed of 8 km/h, you can go h
- The car
The car weight 1280 kg, increased its speed from 7.3 m/s to 63 km/h on a track of 37.2 m. What force did the car engine have to exert?
- Škoda cars
At 8:30 am, Škoda 120 started from Prievidza at 60 km/h in the direction of Bratislava. At 9:00 started from Prievidza at the same direction Škoda Rapid at 90 km/h. Where and when will Škoda Rapid catch up Skoda 120?
- Tractor vs car
Behind the tractor, which was traveling at an average speed of 20 km/h, a car started in two hours, which overtook it in 1 hour. At what speed was the car moving?
- The stadion
John can ran around a circular track in 20 seconds and Eddie in 30 seconds. Two seconds after Eddie starts, John starts from the same place in opposite direction. When will they meet?
- Constant Angular Acceleration
The particle began to move from rest along a circle with a constant angular acceleration. After five cycles (n = 5), its angular velocity reached the value ω = 12 rad/s. Calculate the magnitude of the angular acceleration ε of this motion and the time int
- The projectile
The projectile was fired horizontally from a height of h = 25 meters above the ground at a speed of v0 = 250 m/s. Find the range and flight time of the projectile.
- Up and down motion
We throw the body from a height h = 5 m above the Earth vertically upwards v0 = 10 m/s. How long before we have to let the second body fall freely from the same height to hit the Earth at the same time?
- Fighter
A military fighter flies at an altitude of 10 km. The ground position was aimed at an altitude angle of 23° and 12 seconds later at an altitude angle of 27°. Calculate the speed of the fighter in km/h.
- Raindrops
The train runs at a speed of 14 m/s and raindrops draw lines on the windows, which form an angle of 60 degrees with the horizontal. What speed drops fall?
- Skid friction
Find the smallest coefficient of skid friction between the car tires and the road so that the car can drive at a 200 m radius at 108 km/h and does not skid.
- The bridge
A vehicle weighing 5,800 kg passes 41 km/h on an arched bridge with a radius of curvature of 62 m. What force is pushing the car onto the bridge as it passes through the center? What is the maximum speed it can cross over the center of the bridge so that
- Angled cyclist turn
The cyclist passes through a curve with a radius of 20 m at 25 km/h. How much angle does it have to bend from the vertical inward to the turn?
- Rotaty motion
What is the minimum speed and frequency that we need to rotate with water can in a vertical plane along a circle with a radius of 70 cm to prevent water from spilling?
- Motion
The two bodies move in the same direction evenly in a straight line, at speeds of 5 cm/s and 10 cm/s. The movement of the first body started 2 seconds earlier than the movement of the second body, from a point located at a distance of 20 cm from the start
- Two cylinders
Two cylinders are there one with oil and one with an empty oil cylinder has no fixed value assume infinitely. We are pumping out the oil into an empty cylinder having radius =1 cm height=3 cm rate of pumping oil is 9 cubic centimeters per sec and we are p
- Where and when
The truck left Kremnica at 11:00 h at a speed of 60km/h. At 12:30 h, the passenger car started at an average speed of 80km/h. How many kilometers from Kremnica will the passenger car reach the truck, and when?
Do you have an exciting math question or word problem that you can't solve? Ask a question or post a math problem, and we can try to solve it.
Do you want to convert velocity (speed) units? Expression of a variable from the formula - practice problems. Velocity - practice problems. | https://www.hackmath.net/en/word-math-problems/expression-of-a-variable-from-the-formula?tag_id=8 |
We begin by setting up various pieces of background material related to linear algebra. The theory here is of course not explained in the most generality, but simplified and adapted to our discussion. Denote first the set of all real numbers by R. A vector of length n is simply a list of n elements, called vector entries. The notation of a (column) vector v having entries x1, x2, … ,xn is the following:
However, we will only look at vectors whose entries are real numbers (that is x1, x2, … ,xn are all real numbers). As in the case of real numbers, real vectors can be added or multiplied by a real number (called scalar) in a straightforward way:
In a similar way we define the notion of a matrix. An m × n matrix is a rectangular array of entries having hight m and width n. This can also be seen as stacking n vectors of length m one next to the other. Again, take the entries to be real numbers. We denote an m × n matrix A with entries aij by:
This means that on row i and column j we find the real number aij. For example, on the first row, second column lies element a12. Notice that a vector of length m is just a m × 1 matrix and a real number is a 1 × 1 matrix. If m = n then the matrix is called a square matrix, of size n. This is what we will consider from now on. It makes sense now to talk about the diagonal of a matrix, which consists of those elements for which the row number equals the column number ( the elements a11, a22, up to ann).
Notice that the last two matrices are rather "special": they are symmetric. This means that entry on row i and column j is equal to that entry on row j and column i (or otherwise said aij = aji). As an easy exercise, prove that any diagonal matrix (has zeros everywhere, except on the diagonal) is symmetric. The matrix that has ones on the diagonal and zeros everywhere else is called the identity matrix and is denoted by I. Matrix multiplication is done in the following general way:
where the element cij is given by the formula:
Multiplying a matrix with a vector is done as follows:
where every element yi from the resulting vector y is given by the formula
The other two matrix operations, addition and scalar multiplication, are done as in the case of vectors. Adding matrices A and B gives a matrix C which has the entry on row i and column j equal to the sum of the corresponding entries of A and B. Multiplying matrix A with a real number a is the same as multiplying every element of A with a.
One can observe that there is some sort of similarity between matrices A and C in Problem 2. above. The similarity comes from the fact that the entry on row i and column j in matrix A is equal to the entry on row j and column i in matrix C. Of course, the elements on the diagonal are the same. We then say that A is the transpose of C, or equivalently that C is the transpose of A. In general, for a matrix A we denote its transpose by At. More intuitively, given a matrix we find its transpose by interchanging the element at row i, column j with the element at row j, column i. If we do this twice we notice that the transpose of the transpose of a matrix is the matrix itself, or (At)t=A.
We now introduce two important notions in the theory about matrices: eigenvector and eigenvalue. We say that the real number z is an eigenvalue for A if there exists a real vector v of length n such that A·v = z·v. Such a vector v is called an eigenvector corresponding to the eigenvalue z. This is not the most general definition, but it will suffice for our purposes. In general eigenvalues and eigenvectors are complex, and not real. If we assume that A is a (real) symmetric matrix of size n, then we know that it has n real eigenvalues and all eigenvectors are also real. In fact, a matrix of size n can have at most n real eigenvalues.
In order to make these definitions more clear, consider the following explicit example:
Of course, in general a matrix A and its transpose At do not have the same eigenvectors that correspond to the common eigenvalues. For the matrix in the above example, has eigenvalue z = 3 but the corresponding eigenvector is . This follows from the computation below
An important observation is that a matrix A may (in most cases) have more than one eigenvector corresponding to an eigenvalue. These eigenvectors that correspond to the same eigenvalue may have no relation to one another. They can however be related, as for example if one is a scalar multiple of another. More precisely, in the last example, the vector whose entries are 0 and 1 is an eigenvector, but also the vector whose entries are 0 and 2 is an eigenvector. It is a good exercise to check this by direct computation as shown in Example 5.
A matrix is called column-stochastic if all of its entries are greater or equal to zero (nonnegative) and the sum of the entries in each column is equal to 1. If all entries of a matrix are nonnegative, then we say that the matrix itself is nonnegative. Furthermore, a matrix is positive is all its entries are positive (greater than zero) real numbers.
It is easy to see that A is column-stochastic, while At is not. However, the sum of the elements on each row of At is equal to 1. We first show that z = 1 is an eigenvalue for At, with corresponding eigenvector
This is true since At·v = 1·v. Then, from Fact 3, 1 is an eigenvalue for the matrix A as well.
x1, x2, x3, and x4 are all real numbers that we don't yet know. If we multiply term by term we find that
Substituting in the third relation we obtain and so the vector u is of the form
As x1 is just a real number (hence a scalar) we can take x1 = 12 and we have just proved that the vector whose entries are 12, 4, 9, and 6 (from top to bottom) is an eigenvector for A.
In the first part of the previous example we have just shown 1 is an eigenvalue for that particular case. However, this is true for any column-stochastic matrix, as stated below.
Notice also that the eigenvector that we found in the second part of the example above is rather "special" itself. We have chosen x1 = 12 and we obtained an eigenvector with positive entries. If however we choose x1 = -12 then we obtain an eigenvector with negative entries (smaller than 0).
When we are working with positive column-stochastic matrices A it is possible to find an eigenvector v associated to the eigenvalue z = 1 such that all its entries are positive. Hence A·v = v and the entries of v are all positive. | http://pi.math.cornell.edu/~mec/Winter2009/RalucaRemus/Lecture1/lecture1.html |
Author/Opus: This is the 3rd puzzle from guest contributor Craig Kasper, who also contributed these two Sunday Surprises. Craig has recently started a new puzzle blog.
Answer String: Enter the length in cells of each of the shaded segments from left to right for the marked rows, starting at the top. Separate each row’s entry from the next with a comma.
Time Standards (highlight to view): Grandmaster = 1:45, Master = 3:00, Expert = 6:00
Solution: PDF; a solution video is available here.
Note: Follow this link for other classic LITS. If you are new to this puzzle type, here are our easiest LITS to get started on.
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This week’s theme has me unreasonably giddy. Gold Star to Craig and the other authors this week.
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Damn, after entering a false answer I realised I was solving the wrong puzzle or rather solving this puzzle wrong…
If that’s a spoiler please remove this post.
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2 star Star Battle?
I do tend to wonder though, if creating a puzzle that has a unique solution for both sets of rules was intentional or coincidental though. 🙂 (and what are the odds of the latter?)
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Given the rest of the puzzles this week, I would say intentional.
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I saw that Thomas Snyder did a youtube walkthrough of this old puzzle, and wanted to solve it, so made this: | https://www.gmpuzzles.com/blog/2014/05/lits-craig-kasper/ |
Angolan pythons are reddish brown to black with white markings. They have a yellow belly and five heat sensitive pits on each side of the head.
Habitat/Range:
Found in southern Angola and Namibia, this snake prefers rocky outcrops on mountain terrain or brushy plains.
Size:
Length: 6 feet.
Due to civil war in Angola, these snakes are rarely found in captivity.
Least Concern. They are used in the international pet trade.
Conservation Action:
These snakes live in some protected areas.
Burmese Python
Burmese pythons are darkly colored, with reddish to brown and dark cream colored rectangles. They have an arrow shaped marking on the top of their head. Males are usually smaller than females.
Habitat/Range:
These snakes prefer tropic and sub tropic areas of south Asia.
Size:
Length: Burmese Up to 24.5 feet. Weight: Burmese Up to 310lbs.
- Burmese pythons can hold their breath for up to 30 minutes!
- Burmese pythons are an invasive species in Florida and are thriving there, even though they are a vulnerable species in their native habitat.
IUCN lists as vulnerable. The main threat to these snakes is the illegal wildlife trade. Their skins are used for decoration, leather and musical instruments. They are also sold as pets and hunted for meat. They are also affected by habitat degradation such as slashing and burning. In Vietnam it is listed as critically endangered.
Conservation Action:
They are a protected species in Vietnam, China, Thailand, and Indonesia. Listed on CITES Appendix II. Where it is invasive in Florida, measures are being taken to control or eradicate the species.
Common Kingsnake
They are black or dark brown with a white or yellow spot on their scales which lend to their name. They have a yellow belly with black markings.
Habitat/Range:
The speckled kingsnake prefers prairies, brushy areas, forest edges, rocky, wooded hillsides, and the edges of swamps or marshes.
Size:
Length: 3-4 feet.
They are also known as the “salt-and-pepper snake” due to their coloration.
Pet trade, human interference, and habitat loss affect this species. The speckled kingsnake helps us by making sure that other animal’s populations don’t get to be too densely inhabited.
Copperhead
Copperheads have reddish-brown bodies covered in a crossband pattern consisting of tan, copper, and rich brown colors that extend throughout the body. Their heads are triangular and a solid brown color. Each subspecies of copperhead has variations in color and pattern. Males are typically longer than females and juveniles are grayer with a yellow-tipped tail.
Range/Habitat:
The southern copperhead range extends through Massachusetts, westward to Texas and southeastern Nebraska. They can inhabit a variety of habitats including forests, wet woodlands, edges of swamps, stream beds and gulches. They can also be found in and around man-made environments.
Size:
Length: 30-53in.
- Copperheads are a venomous snake, however, their venom is somewhat milder compared to other venomous snakes and rarely results in death in a healthy human.
- The yellow-tipped tail on juveniles is thought to help lure small prey. They will move the tail mimicking caterpillar movements, bringing their prey within striking distance.
- Kingsnakes and opossums are reported to be immune to copperhead venom.
Least Concern. Certain areas of the US are experiencing declines in population and they are considered endangered in Iowa and Massachusetts. Their threats include habitat destruction, invasive plants, insecticide application, and road mortality.
Cottonmouth
This venomous snake enjoys being around water and is named for the white lining of its mouth. They have triangular shaped heads, heat sensing pits which help identify warm blooded animals and is found between the eyes and nostrils. Their eyes have vertically slit pupils. They are rough in texture due to their keeled scales which provide ridges on the scales. They are usually olive-brown with paler bellies.
Habitat/Range:
This snake is found in north central Texas and prefers streams, river floodplains, swamps, wetlands and marshes.
Size:
Length: 2-6 feet.
They vibrate their tails when threatened in order to mimic a rattlesnake.
Least Concern. They are threatened by human activity which includes persecution and habitat loss.
Conservation Action:
In some states, they are protected under native snake laws. They are endangered in Indiana.
Eyelash Palm Pit Viper
Eyelash vipers can be found in a variety of color morphs. They are most commonly olive green but can also be bright yellow, pink, green, silver, dark grey or brown. Faint markings of various colors can be seen on the body. The tips of the tail are yellow or green and their undersides are pale yellow often with darker spots. Habitat directly affects the coloration of these snakes. They are characterized by a prehensile tail used for climbing, a triangular head, and distinctive keeled scales above the eyes that gives them a “browed” appearance. Their scales are ridged to give them a better grip on vegetation. Females tend to be larger than males and juveniles look the same as adults.
Range/Habitat:
Eyelash vipers range from southern Mexico through northwestern Ecuador and western Venezuela. They prefer moist tropical forests but can also be found in woodlands and shrublands both at lower and higher elevations.
Size:
1-2.75 feet.
- Eyelash viper venom is hemotoxic, meaning it destroys red blood cells. The venom also contains procoagulants and haemorrhagins, and affects both the central nervous system and the cardiovascular system.
- Scientists aren’t quite sure what the purpose of the “eyelash” scales are on these snakes. It is thought that they aid in camouflage by breaking up the snakes silhouette and may protect their eyes when moving through thick vegetation.
IUCN has not evaluated this animal and the Convention on International Trade of Endangered Species has removed them from their list. They are most likely threatened by deforestation due to agriculture, urbanization and the timber industry. No fatalities from eyelash vipers has ever been reported. Due to their sometimes yellow coloration, these snakes have accidentally been shipped in banana boxes all over the world.
Garden Tree Boa
They vary in coloration and can be tan to black with yellow to red touches. They have five stripes on their head with yellow, gray, or red eyes and a black tongue.
Habitat/Range:
This species is arboreal and prefers the high humidity of the Amazon rainforest. They may also be found in savannas or dry forests.
Behavior:
These snakes live solitary lives and are capable of being active both during the day and night. They tend to be aggressive and will bite humans, however they are non-venomous.
These snakes have claw-like remnants of vestigial hind limbs.
This species has not been evaluated for the IUCN Red List. They are common in the pet trade, though they can be aggressive. Their populations are not considered threatened.
Green Tree Python
As their name implies, these snakes are a bright green color. On their back, they have a ridge of scales that are white or yellow that form a broken or continuous line down their backs. Their stomachs are generally yellow. These pythons do have heat sensing pits on their upper lips. Juveniles that are yellow in color are found throughout the range and red juvenile morphs are found in parts of Indonesia and New Guinea. The same clutch can have both red and yellow morphs. The juveniles have white blotches edged in black or brown running down their backs. They also have a white streak edged in black that runs from the nostril, through the eye to the back of the head. When they are young, females are may have longer and wider heads than males of similar size. Some adults may never fully change from their juvenile coloring.
Habitat/Range:
Green tree pythons are found in New Guinea, eastern Indonesia, surrounding islands, and the Cape York Peninsula of Australia. They inhabit tropical rainforest. As juveniles they like to hang out in forest edges or near gaps in the canopy, but as adults they are found in closed-gapped canopy.
Size:
Length: 5 feet, Up to 7 feet.
- Green tree pythons have a prehensile tail to aid them in climbing.
- Juvenile snakes change color around 6 months to 1 year in age. This does not have to do with sexual maturity. Instead, it has to do with length. Once the snake reaches a certain length, it can change its feeding habits. As a juvenile, it lived in forest gaps (where lighter coloring would provide better camouflage) where smaller prey lives. As an adult, they inhabit closed-gap canopy (where green provides more camouflage) where larger prey is found.
IUCN has not determined their status as of yet. They are one of the most common pet snakes. Some of these snakes are captive bred, but others are wild caught. Indigenous people in New Guinea hunt this species for food. They are most likely affected by habitat degradation as well.
Conservation Action:
Listed CITES Appendix II. In Australia it is illegal to capture wild green tree pythons or import them from New Guinea.
Macklot's Python
(Savu or Sawu Python)
They have white eyes which are unique. These pythons are dark brown or black in coloration.
Habitat/Range:
These snakes are found only on the island of Sawu in Indonesia.
Size:
Weight: 1-3lbs.
This species has not been evaluated for the IUCN Red List.
Mexican Burrowing Python
Mexican burrowing pythons are dark in color with patches of white scales. However, occasionally almost entirely white snakes can be observed after shedding. They have small eyes, a narrow head, and a shovel shaped snout which is used for burrowing.
Range/Habitat:
The Mexican burrowing python can be found along the Pacific coast of Mexico, down through Guatemala, Honduras, El Salvador, Nicaragua, and Costa Rica. They inhabit a variety of areas including: tropical, moist, and dry forests, mangroves, beaches, as well as dry inland valleys.
Size:
Length: 3-5 feet.
Mexican burrowing pythons are not in the python family. They belong to the Loxocemidae family, which is comprised of only this one species.
IUCN lists as a species of least concern. These snakes are sometimes exported for the pet trade and may be persecuted by humans.
Conservation Action:
They are found in several protected areas throughout their range. Mexican burrowing pythons are protected under Mexican law.
Mexican Moccasin
(Taylor’s Cantil)
Thick bodied with large heads and long, slender tails. The heads have five pale stripes, one vertically on the front of the snout and two laterally on each side of the head. It has black bands that cross the back separated by gray or brown areas that often contain orange. The chin is white or yellow and the stomach is checked with black or gray markings. Juveniles have a yellow, white or pink tail tip.
Range/Habitat:
Taylor’s cantil are native to northeastern Mexico. They inhabit mesquite-grassland, thornforest, and tropical deciduous forests. These snakes are often found near small bodies of water.
Size:
Length: 2-8 feet.
- Taylor’s cantil has only been recognized as a species since 2000.
- Taylor’s cantil is names in honor or American herpetologist Edward Harrison Taylor who focused on Mexican reptiles and amphibians.
IUCN lists as a species of least concern. There is little information on this snake because it is rarely found. Due to its rarity, it is often sought out by collectors. It is also threatened by habitat loss and modification for cattle grazing.
Milksnake
They have orange, black, and white stripes.
Habitat/Range:
Found throughout central East Coast North America, these snakes prefer farmlands, woods, outbuildings, meadows, river bottoms, bogs, rocky hillsides, and rodent runways.
Size:
Length: 2-3 feet.
This species has not been evaluated for the IUCN Red List.
Mole Snake
Brown, gray, or black in color with juveniles having mottled markings which fade with time, these snakes love to burrow and have an aggressive self-defense display. They have a round pupil.
Habitat/Range:
This snake prefers grasslands but is present in nearly all habitats and is found in southern Africa.
Size:
Length: 4.6 feet.
These are one of the few snake species that give birth to live young.
This species has not been evaluated for the IUCN Red List. Though they are said to be aggressive, they supposedly make good pets.
Philippine Pit Viper
The female tend to be larger than the males. They used to be thought to be sexually dichromatic with the females being yellow in color and the males being brown or silver in color, but it is now known that either sex can show a wide variety of colors. Colors can be yellow, brown, silver, orange-yellow, beige, and have spots/bands or no spots/bands.
Habitat/Range:
Found in the Islands the Philippines these snakes are found in a wide variety of habitats as long as they have cover to hide in.
Size:
Length: 3 feet.
- Also known as McGregor’s Pitviper and Batanes Bamboo Pitviper.
- Very little is known about this species.
IUCN lists this as a species of least concern and data deficient. Pressures from the pet trade and habitat loss may be a concern in the future.
Conservation Action:
They occur in some protected areas.
Pine Snake
As their name implies, these snakes are black in color with a small head compared to their bodies. What distinguishes them from black racers is their keeled, or ridged scales. Juveniles are born with blotchy patterns that fade with age.
Range/Habitat:
The black pine snake is from southwestern Alabama, through southern Mississippi, and into southeastern Louisiana. They prefer upland, longleaf pine forests, but may also inhabit pitcher plant bogs and river/stream corridors.
Size:
Length: 4-7.5 feet.
The United States Fish and Wildlife Service lists as a threatened species. The state of Mississippi classifies it as an endangered species. They are threatened by habitat loss due to logging, agriculture, development, and the suppression of natural fires. Their preferred habitat, longleaf pine forests, are one of the most endangered habitats in the nation.
Conservation Action:
The Wildlife Conservancy is attempting to protect and restore longleaf pine forests, carry out controlled burns in the forests, and track and monitor these snakes. The USFWS is looking at protecting areas for this species.
Puff Adder
This snake has a large triangular head with large nostrils that point upwards. They have yellow-brown to light-brown bodies covered in pale-edged, black/brown, v-shaped/u-shaped markings that run the length of the body. Their stomach is white or yellow, sometimes with brown spots.
Habitat/Range:
Puff adders inhabit sub-Saharan Africa and a small part of the Arabian Peninsula. They are found in grasslands and savannahs.
Size:
Length: 3.2-6.2 feet. Weight: 13lbs.
- This snake gets its name from blowing out air when threatened rather than moving away.
- They are one of the fastest snakes in the world if not the fastest. It can strike both forward and to the side in 0.25 seconds!
- A female in a Czech zoo gave birth to 156 young once, the most recorded of any snake species anywhere.
- Although the puff adder isn’t the most venomous species of snake in Africa, it is considered responsible for the majority of snakebite fatalities in Africa. This is most likely due to its wide distribution, its frequent inhabitation of densely populated areas and its aggressive nature.
IUCN has not evaluated this species yet. It is one of the most common and widespread in Africa.
Pygmy Rattlesnake
This snake is named for its small size. It has a small tail with rarely more than a few rattles on it. Coloration varies greatly depending on location. It can have a background color of gray, brown, black, pink or reddish. A dark line runs vertically through the eye and down the face. Dark, circular markings line the back and a thin, reddish-orange stripes runs down the mid-body line. Juveniles resemble adults except for a yellow tipped tail. Facial pits for detecting heat are located on the face.
Habitat/Range:
The pygmy rattlesnake is found throughout the Southeastern United States. They can be found in wet habitats such as floodplains, rice fields, marshes, swamps and forests.
Size:
Length – 1-2 feet.
- There are three different subspecies of the pygmy rattlesnake: Carolina, Dusky and Western.
- While waiting for prey to venture by, they will remain coiled, sometimes for as long as 2-3 weeks!
IUCN lists as a species of least concern. Currently, no major threats are known to exist, but habitat loss may be a threat to some populations.
Conservation Action:
Occur in protected areas. They are protected in North Carolina and Tennessee.
Red Spitting Cobra
As its name implies, this snake is red in color. They have a broad black throat and black tear-drop markings underneath the eyes.
Range/Habitat:
Red spitting cobras inhabit eastern Africa, from Somalia to Tanzania. They are found in savannahs and grasslands.
Size:
Length: 5 feet.
Spitting cobras are able to shoot their venom up to 6 feet! They are also almost always able to hit their target in the eyes, some species being able to hit their target 10 out of 10!
IUCN has not assessed this species as of yet.
Reticulated Python
These snakes are covered in a complex geometric pattern that incorporates a variety of colors. The back is made up of a diamond like pattern, usually surrounded by smaller darker markings with light centers. The head has no markings except a black line running from each eye to the corner of its jaws. Females are often larger than males.
Range/Habitat:
Reticulated pythons can be found throughout Southeast Asia including the Nicobar Islands, Burma through Indochina, and Borneo, Sulawesi, Ceram and Timor in the Malay Archipelago. They inhabit humid tropical rainforests and are usually found near a water source.
Size:
Length: Up to 25+ feet. Weight: Up to 350lbs.
- Females will keep their eggs warm by coiling around them. They will “shiver,” or contract their muscles, to increase the temperature of the eggs. The females will not leave the eggs to eat until they hatch.
- Reticulated pythons are known to eat deer, even with antlers. If the antlers are small enough, they can simply be swallowed and digested. If not, the snake can break the antlers so they lie flat against the body or eat the deer hind quarters first (rare) and partially digest the deer until the antlers fall off before swallowing the head.
- This snake is the longest snake in the world.
IUCN has not evaluated this species. These snakes are killed for their skins, for traditional medicines, and because people fear them. They also are threatened by the Asian tradition of blood drinking and gall bladder removal. Rapid growth rate, and high fertility are the only things helping this snake survive.
Conservation Action:
Listed on CITES Appendix II.
Rock Rattlesnake
As their name implies, they do have a banded pattern. Males usually have a gray background color growing to green around the middle, with black stripes. Females tend to have a uniformly gray background color with black stripes. These snakes typically have 13-21 bands running down the body and 1-6 bands on the tail. Their scales are ridged and they have rattles on their tail like all rattlesnakes. On their face, they have heat sensing pits and vertical pupils. Males are usually larger and have longer tails in proportion to their bodies than females. Newly hatched rattlesnakes have yellow tipped tails, as is common with many snakes.
Range/Habitat:
This snake is found in southeastern Arizona, southwestern New Mexico, the mountains around El Paso Texas, and south to Jalisco Mexico. It is often found in rocky habitats such as slopes, canyons and rock outcrops. It can also be found in semi-desert grasslands and forests.
Size:
Length: 2-2.7 feet.
This snake is one of four subspecies of Rock Rattlesnakes.
IUCN list the rock rattlesnake (Crotalus Lepidus) as a species of least concern. They are affected by poaching, removal and habitat destruction.
Western Rat Snake
As the name implies, this snake’s body is black except for a white chin/throat and white/mottled belly. Occasionally, brown splotches of color can be seen on the body. Juvenile snakes are light gray or tan, with distinct dark brown or black blotches on the back and sides. A black band passes between the eyes and angles down toward the mouth. After a year or two of growth, the color changes to a more uniform black.
Habitat/Range:
Range from New England south through Georgia and west across the northern parts of Alabama, Mississippi, and Louisiana, and north through Oklahoma to southern Wisconsin. Found in a variety of habitats included rocky, wooded hillsides and farmlands.
Size:
Length: 3.5-6 feet.
The black rat snake is just one of several species of rat snake found throughout the United States. Other species include the fox snake, yellow rat snake, corn snake, and gray rat snake.
IUCN lists them as a species of least concern. Their main threats are habitat loss (due to logging and land development) and human persecution.
Conservation Action:
Occur in some protected areas.
White-Lipped Tree Viper
This snake is bright green in color with a lighter, yellowish underside. It is slender with a large triangular head. As its name suggests, it white or yellow colored “lips,” chine and throat. The tail is a contrasting brown color and the eyes are yellow-orange with vertical pupils.
Habitat/Range:
White-lipped pit vipers can be found from Myanmar across southern China, south to Java and Indonesia. This snake can found in a wide variety of habitats such as mountain forests, shrubland, plains, and agricultural areas.
Behavior:
These snakes are often found off the ground in trees or bushes.
This snake is able to be found in abundancy even in habitats that have been greatly altered or degraded.
IUCN lists as a species of least concern. The greatest threat to this snake is persecution by humans. In some areas it is also harvested for food and traditional medicines.
Conservation Action:
Found in some protected areas.
Western Gaboon Viper
Gaboon vipers are very striking snakes. Their body has a base color of brown or grayish-purple. The back is covered in four sided shapes that separated by brown hourglass spaces. The sides of the body have triangular brown or purple markings separated by brown or purple blotches. The underside is light yellow with dark spots. Most of the scales on their body are ridged and keeled. Their head is large and triangular with a dark stripe running down the center and two dark spots above each side of the jaw. Their most distinct characteristic is the horn-like projections on the tip of their nose. Juveniles look the same as the adults.
Range/Habitat:
Gaboon vipers are found throughout sub-Saharan Africa. They inhabit rainforests and other moist, tropical habitats.
Size:
Length: 4-7 feet. Weight: 15.5-22lbs.
- Gaboon vipers are the largest of the vipers.
- Gaboon viper fangs are two inches long (longest fangs of any snake)!
This species has currently not been evaluated by the IUCN. Their population status is unknown but they are not believed to be threatened. They help control rodent populations but can be very harmful to humans.
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---
abstract: |
We study the behaviour of nonnegative solutions to the quasilinear heat equation with a reaction localized in a ball $$u_t=\Delta u^m+a(x)u^p,$$ for $m>0$, $0<p\le\max\{1,m\}$, $a(x)=\mathds{1}_{B_L}(x)$, $0<L<\infty$ and $N\ge2$. We study when solutions, which are global in time, are bounded or unbounded. In particular we show that the precise value of the length $L$ plays a crucial role in the critical case $p=m$ for $N\ge3$. We also obtain the asymptotic behaviour of unbounded solutions and prove that the grow-up rate is different in most of the cases to the one obtained when $L=\infty$.
Keywords: Quasilinear diffusion equations, localized reaction, grow-up.
address:
- 'Raúl Ferreira Departamento de Matemáticas, U. Complutense de Madrid, 28040 Madrid, Spain. e-mail: [[email protected]]{}'
- 'Arturo de Pablo Departamento de Matemáticas, U. Carlos III de Madrid, 28911 Leganés, Spain. e-mail: [[email protected]]{}'
author:
- 'R. Ferreira and A. de Pablo'
title: 'Grow-up for a quasilinear heat equation with a localized reaction in higher dimensions'
---
Introduction {#sect-introduction}
============
We consider non-negative solutions to the following problem $$\label{eq.principal}
\left\{
\begin{array}{ll}
u_t(x,t)=\Delta u^m(x,t)+a(x) u^p(x,t),\qquad & (x,t)\in \mathbb R^N\times\mathbb R^+,\\
u(x,0)=u_0(x),
\end{array}\right.$$ with $m,\,p>0$, $N\ge2$. We refer to [@FerreiradePablo] for the case $N=1$. The initial datum is a continuous, nonnegative and nontrivial function $u_0\in L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$. The reaction coefficient is the characteristic function of the ball of radius $L$, $a(x)=\mathds{1}_{B_L}(x)$.
The existence of a solution to problem , local in time, can be easily achieved. To avoid uniqueness issues when $p<1$ we assume that $u_0$ is strictly positive in $\overline{B_{L}}$. If $T$ is the maximal time of existence of the unique solution then the solution is bounded in $\mathbb{R}^N\times[0,t]$ for every $t<T$.
Problems like are studied mainly when $p>1$ and in the context of blow-up, i.e. when $T$ is finite, and in that case $$\label{bup}
\|u(\cdot,t)\|_\infty\to \infty \quad \mbox{as } t\to T.$$ The case with global reaction, $L=\infty$, has been described by Fujita in the semilinear case $m=1$ in the the seminal work [@Fujita]. It is proved in that paper that there exist two exponents, the *global existence exponent* $p_0=1$ and the so called *Fujita exponent* $p_F=1+2/N$, such that for $0<p< p_0$ all the solutions are globally defined in time, for $p_0<p< p_F$ all the solutions blow up, whereas for $p>p_F$ there exist both, global solutions and blowing-up solutions. The limit cases $p=p_0$ and $p=p_F$ belong, for this problem and respectively, to the global existence range and blow-up range, see also [@Hayakawa]. From this result several extensions have been investigated in the subsequent years, for all values of $m>0$, or with different diffusion operators and reactions; we mention the monographs [@GalaktionovKurdyumovMikhailovSamarski; @QuittnerSouplet] for equation with $L=\infty$.
In the presence of a localized reaction, $L<\infty$, problem has been studied, again in the context of blow-up, in [@BaiZhouZheng; @FerreiradePabloVazquez; @Liang; @Pinsky].
\[teo-exponents\] The global existence exponent and the Fujita exponent for problem are $$\label{exp-p0}
\begin{array}{ll}
p_0=\max\{1,\dfrac{m+1}2\},\quad p_F=m+1,&\quad\text{if } N=1, \\ [3mm]
p_0=p_F=\max\{1,m\},&\quad\text{if } N\ge2.
\end{array}$$
We remark that the fast diffusion case $m<1$ is not covered by those references, but the result is trivial in that range, see Lemma \[lem-exp\].
Our purpose in this work is to study the behaviour of global solutions in the lower interval $p\le p_0$, and to characterize wether they are bounded or not. In the last case we have that holds with $T=\infty$, a phenomenon that is called [*grow-up*]{}. When $L=\infty$ every solution with $p\le p_0=1$ has grow-up, see [@dePabloVazquez]. As Theorems \[teo-global-p0\] and \[teo-GUP\] below show, the situation where $L<\infty$ is much more involved. This is in particular true in the case $p=m$ and $N\ge3$, where there exists a critical length $L^*=L^*(N)$, see , delimiting two very different behaviours.
In the critical exponent case $p=p_0$ the solutions can be global or not, contrary to what happens when $L=\infty$. In fact $p=p_0$ lies in the global existence side if $N=1$, see [@FerreiradePabloVazquez; @BaiZhouZheng] or $N\ge2$ and $m\le1$, see Lemma \[lem-exp\]. We emphasize that the case $p=m>1$, $N\ge3$, is not clear in the bibliography, see [@Liang Theorem 1.1] where the author asserts that the solution blows up for every $L>0$, which is not true when $L$ is small.
\[teo-global-p0\] Let $p=p_0$. The solution to problem is always globally defined in time if $p_0=1$, while if $p_0=m>1$ it is global if and only if $0<L\le L^*$, $N\ge3$.
We now study the global solutions in the range $p\le p_0$. The behaviour depends on the dimension and also on $p$, $m$ and $L$. Comparison with stationary solutions or unbounded explicit solutions will be useful, see Section \[sect-special\]. We include here the case $N=1$ studied in [@FerreiradePablo] for completeness.
\[teo-GUP\] Let $p\le p_0$ and let $u$ be a global solution to problem .
- If $N=1$ then $u$ is unbounded.
- If $N=2$ and the initial value is large then $u$ is unbounded. If in addition $p\le m$ then $u$ is always unbounded.
- Let $N\ge3$.
- If $p< m$ then $u$ is bounded.
- If $p=m$ then $u$ is bounded when $L\le L^*$ (assuming also if $L=L^*$), and $u$ is unbounded when $L>L^*$.
- If $p>m$ then $u$ can be bounded or unbounded depending on the initial value.
![Grow-up regions below $p_0$ in dimensions $N=1$, $N=2$ and $N\ge3$ resp.: $A$, all solutions are unbounded; $B$, there exist unbounded solutions; $C$, there exist bounded and unbounded solutions; $D$, all solutions are bounded.[]{data-label="fig.gup-vs-bdd"}](growup-regions.eps){width="14cm"}
Each border line belongs to the corresponding subset below it, except for the line $p=m\le1$ when $N\ge3$, where the boundedness of the solutions depends on the length $L$. In the critical case $p=m$ and $L=L^*$ when $N\ge3$ we must impose an extra condition on the behaviour of the initial value at infinity, namely $$\label{extrainfinity}
\limsup_{|x|\to\infty}|x|^{\frac{N-2}m}u_0(x)<\infty.$$ Under this condition the solution is always bounded.
We do not know if all the solutions grow up in the parameter range $m<p\le1$ when $N=2$, that is if $B$ is actually $A$ or $C$ in Fig. \[fig.gup-vs-bdd\]. We notice that in the linear reaction case $p=1$ with superfast diffusion $0<m<m_*\equiv \frac{N-2}N$, $N\ge3$, besides unbounded solutions there also exist solutions that vanish identically in finite time, see Remark \[rem-vanish\].
For the unbounded solutions to problem we also characterize the grow-up set. We assume for simplicity that the initial datum is radial. This is not a restriction if $p\le m$ with $N=2$ or if $p=m<1$ with $L>L^*$ when $N\ge3$, since then every solution is unbounded and we may use comparison with a smaller initial datum satisfying those properties. When $m<p\le 1$ we obtain the grow-up set only for radial unbounded solutions with limit infinity at some pint.
\[teo-sets\] Let $u$ be a global unbounded solution to problem . If $m<p\le 1$ assume also that $u$ is radial satisfying $\lim\limits_{t\to\infty}u(x,t)=\infty$ for $|x|=R$ and some $R\ge0$. Then $\lim\limits_{t\to\infty}u(x,t)=\infty$ uniformly in compact sets.
We do not know if in the upper range $m<p\le 1$ there exist unbounded solutions with $\liminf\limits_{t\to\infty}\|u(\cdot,t)\|_\infty<\infty$.
Once the existence of global unbounded solutions is characterized, the main question to deal with is to determine the grow-up rate, that is, the speed at which they go to infinity. An easy upper estimate of the grow-up rate for $0<p\le1$ is given by the solutions of the ODE $U'(t)=U^p(t).$ This gives, $$\label{flat0}
u(x,t)\le\begin{cases} ct^{\frac1{1-p}},& \quad \text{if } p<1, \\
ce^t,& \quad \text{if } p=1.
\end{cases}$$ We call this the *natural rate*.
If $L=\infty$ and $0<p< p_0=1$ the grow-up rate is indeed given by the natural rate, that is, $$u(x,t)\sim t^{\frac1{1-p}},$$ where by the symbol $f\sim g$ we mean $0<c_1\le f/g\le c_2<\infty$. If $p=1$ we can perform a change of variables to eliminate the reaction term as in [@FerreiradePablo], getting in this way an exponential grow-up, $$u(x,t)\sim\begin{cases}
t^{-\frac N2}e^t,& \quad \text{if } m=1, \\
e^{\frac{2t}{N(m-1)+2}},& \quad \text{if } m>1, \\
e^t,& \quad \text{if } m<1.
\end{cases}$$ We write this in weak form as $$\label{log-p=1}
\lim\limits_{t\to\infty}\frac{\log u(x,t)}t=\min\{1,\frac2{N(m-1)+2}\}.$$ We remark that this change is not possible if $L<\infty$.
We prove in this paper that for the case of a localized reaction estimates are not always sharp, that is, the grow-up rate for problem is in most of the cases strictly less than the natural grow-up rate. Let us see this phenomenon heuristically. If we perform the rescaling, for $p\ne 1$, $$\label{rescale}
v(\xi,\tau)=t^{-\alpha} u(x,t),\qquad \xi=xt^{-\beta},\;\tau=\log t,$$ with $\alpha(m-1)-2\beta+1=0$, we have that $v$ is a solution to the equation $$\label{rescaled-eq}
v_\tau=\Delta v^m+\beta\xi\nabla v+b(\xi,\tau)v^p-\alpha v,\qquad$$ where the reaction coefficient becomes $$\label{rescaled-reaction}
b(\xi,\tau)=e^{\gamma\tau}\mathds{1}_{\{|\xi|<Le^{-\beta\tau}\}}(\xi),\qquad \gamma=\alpha(p-1)+1\ge 0.$$ Now choose the natural rate $\alpha=1/(1-p)$. This implies $\beta=(m-p)\alpha/2$ and $\gamma=0$. Therefore, when $p>m$ we have $\beta<0$ and the reaction coefficient tends to 1 in the whole $\mathbb{R}^N$, so the rescaled solution $v$ is supposed to stabilize to the constant $ (1-p)^{\frac1{1-p}}$, at least below the critical Sobolev exponent $p_S=m(N+2)/(N-2)$ . The grow-up rate must then be the natural one. The proof of this fact is our first result. The importance of the critical Sobolev exponent $p_S$ is well known in the characterization of the blow-up rates in superlinear problems, see for instance [@QuittnerSouplet].
\[teo-rates-N>1\] Let $m<p< \min\{p_S,1\}$. If $u$ is a solution to problem with global grow-up then as $t\to\infty$, $$\label{rates11}
u(x,t)\sim t^{\frac1{1-p}}$$ for every $|x|<L$.
If $p=m$, the parameters $\beta$ and $\gamma$ are zero and the rescaled solution $v$ is supposed to stabilize to a (nonconstant) positive stationary profile. This suggest again that the grow-up rate of $u$ is the natural one. But the fact that $x=\xi$ in that case gives that the rate must hold in the whole space.
Let $p=m<1$ and let $u$ be a global unbounded solution of . Then $$u(x,t)\sim t^{\frac1{1-m}}$$ uniformly in compact sets of $\mathbb{R}^N$.
When $p<m$ (which implies $N=2$ in order to have grow-up) the reaction coefficient disappears in the limit and the function $v$ must tend to zero. This gives that the rate should be strictly smaller than the natural one. Following what is done in [@FerreiradePablo] to treat the one dimensional problem, we look at the case when the reaction coefficient tends to a Dirac delta at the origin, which for $N=2$ implies $\gamma=2\beta$. This means $\alpha=0$, which suggests a logarithmic grow-up rate. Thanks to Duhamel’s formula we prove that this is indeed what happens in the case of linear diffusion $m=1$.
\[teo-rates-p<1\] Assume $N=2$, $m=1$, $0<p<1$ and let $u$ be a solution to problem . Then $$\label{rates-p<1}
u(x,t)\sim (\log t)^{\frac1{1-p}}$$ uniformly in compact sets.
The case $p < m\ne 1$ in dimension $N = 2$ will be the subject of a separate work.
Finally when $p=1\ge m$ we cannot perform the previous rescaling and we must try an exponential type change of variables. The argument therefore suggests an exponential grow-up, $\log\|u(\cdot,t)\|_\infty\sim \lambda t$. The main point is that the natural rate $\lambda=1$ is obtained only if $m<1$, whereas when $m=1$ the rate is smaller and it depends on the length $L$.
\[teo-rates-N>1\] Let $p=1$ and $\frac{N-2}{N+2}<m\le1$. If $u$ is a solution to problem with global grow-up then for $t\to\infty$,
1. if $m<1$ $$\label{rates21}
u(x,t)\sim e^t$$ for every $|x|<L$;
2. if $m=1$ and $L>L^*$ there exists a function $\lambda_0=\lambda_0(L)\in(0,1)$ such that $$\label{rates-p=1}
\lim\limits_{t\to\infty}\frac{\log u(\cdot,t)}t=\lambda_0$$ uniformly in compact sets.
The function $\lambda_0(L)$ is increasing in $(L^*,\infty)$ and satisfies $\lim\limits_{L\to L^*}\lambda_0(L)=0$, $\lim\limits_{L\to\infty}\lambda_0(L)=1$, see . Observe also the influence again of the Sobolev exponent $p_S$.
Let us note that for $m<p\le1$ we have obtained the grow-up rate only inside $B_L$. Our last result involves the characterization of the grow-up rate outside $B_L$ and prove that, under certain restrictions, it is different (smaller) from the rate inside the ball. This is particularly outstanding in the case $p=1$, when the solution grows outside like a power, which is much slower than the exponential growth inside. The proof uses comparison with solutions of the pure diffusion equation of a particular self-similar form, see Section \[sect-special\]. The existence of such special solutions will require to consider (not too) fast diffusion, $m^*<m<1$.
\[teo.tasas.fuera\] Let $m<p\le 1$ with $m>m^*$ and $p<p_S$ if $N\ge3$. Assume that there exists $C>0$ such that for $|x|$ large $$u(x,0)
\le C |x|^{\frac{-2}{1-m}}
\qquad \mbox{if } p<1$$ or $$u(x,0)\sim |x|^{\frac{-2}{1-m}}(\log(x)^{\frac1{1-m}} \qquad \mbox{if } p=1.$$ Then, for every $|x|>L$ it holds $$u(x,t)\sim t^{\frac1{1-m}}.$$
The paper is organized as follows: Section \[sect-special\] is devoted to the existence of special solution: stationary solutions, exponential unbounded solutions for the linear equation, and self-similar solutions to the pure fast diffusion equation; Section \[sect-bdd-notbb\] deals with the question of whether the global solutions below the global existence exponent $p_0$ are bounded or not; in Section \[sect-gupset\] we show that the grow-up set is $\mathbb{R}^N$ generically; finally in Section \[sect-guprate\] we study the rate at which the unbounded solutions tend to infinity.
Special solutions {#sect-special}
=================
In this Section we study three families of special solutions, namely stationary solutions, explicit unbounded solutions and self-similar solutions. We first characterize the existence of stationary solutions, both for the Cauchy problem and for the corresponding Dirichlet problem in a ball. We then study the existence of explicit unbounded solutions with exponential growth in the linear equation. We finally construct certain type of self-similar solutions for the pure fast diffusion equation.
Stationary solutions {#sect-stationary}
--------------------
We show here that problem admits stationary solutions for every $N\ge3$ and any $p>0$, independent of the Sobolev exponent $p_S$. We concentrate in radial solutions, $u=u(r)$, $r=|x|$. We also consider later the corresponding Dirichlet problem in a ball. We remark that in this last case the critical Sobolev exponent does play a role.
\[teo-stat\] Problem possesses positive radial stationary solutions only if $N\ge3$. Moreover
1. If $p<m$, for any $k\ge0$ there exist a unique stationary solution such that $\lim\limits_{r\to\infty}u(r)=k$.
2. If $p>m$, there exists a finite value $k^*>0$, such that there exist stationary solutions with $\lim\limits_{r\to\infty}u(r)=k$ if and only if $0\le k\le k^*$ if $p<p_S=\frac{m(N+2)}{N-2}$, or $0< k\le k^*$ if $p\ge p_S$.
3. If $p=m$, there exists a critical length $L^*=L^*(N)$, such that there exist stationary solutions if and only if $L\le L^*$. The solution is characterized by $k=\lim\limits_{r\to\infty}u(r)$, and is unique for any $k>0$ if $L<L^*$, while it is unique up to a multiplicative constant if $L=L^*$, in which case $k=0$.
Putting $w=u^m$ we obtain $w$ matching two functions for $r<L$ and $r>L$, respectively. More precisely, $w$ is given by $$\label{stat-w}
w(r)=
\begin{cases}
Av(A^{\frac{\gamma-1}2}r),\qquad\text{for } 0<r<L, \\
c_1+c_2\phi(r),\qquad\text{for } r\ge L,
\end{cases}$$ where $\gamma=p/m$, $\phi$ is the Green function $$\label{green}
\phi(r)=\begin{cases}
r^{2-N},&\text{ if } N\ne2, \\ \log r,&\text{ if } N=2,
\end{cases}$$ and $v$ satisfies $$\label{stat-rad1}
\begin{cases}
v''+\dfrac{N-1}rv'+v^\gamma=0, \\
v(0)=1,\quad v'(0)=0.
\end{cases}$$
We observe that there exist no nonnegative stationary solution if $N=1$ or $N=2$ since the Green function is unbounded in those dimensions. Let then be $N\ge3$. It is well known that there exists a unique function $v$ solution to defined in a maximal interval $[0,r_0)$, which is positive and decreasing in $0<r<r_0$ and $\lim\limits_{r\to r_0}v(r)=0$, where $r_0<\infty$ if $0<\gamma<\gamma_S=\frac{N+2}{N-2}$, while $r_0=\infty$ if $\gamma\ge \gamma_S$. Moreover $v$ is explicit in the limit case $\gamma=\gamma_S$, namely $v(r)=(1+Br^2)^{\frac{2-N}2}$, $B=\frac1{N(N-2)}$, while $\lim\limits_{r\to\infty}r^{\frac2{\gamma-1}}v(r)=K(\gamma,N)$ if $\gamma>\gamma_S$. See for instance [@GalaktionovKurdyumovMikhailovSamarski Lemma 3.IV.1].
In order to match the two pieces at $r=L$ we have to choose $A>0$ properly. First we must have $A^{\frac{\gamma-1}2}L<r_0$ when $r_0$ is finite. Thus, depending on the sign of the exponent, we see that $v(A^{\frac{\gamma-1}2}L)>0$ for $A$ large if $\gamma<1$, for $A$ small if $1<\gamma<\gamma_S$, and for $L<L_1$ if $\gamma=1$, where $L_1$ is the radius of the ball for which the first eigenvalue of the Laplacian is 1. We then study the matching conditions, $$\label{matching}
\begin{cases}
c_1+c_2L^{2-N}=Av(A^{\frac{\gamma-1}2}L), \\
(N-2)c_2L^{1-N}=-A^{\frac{\gamma+1}2}v'(A^{\frac{\gamma-1}2}L),
\end{cases}$$ and characterize when it is $c_1\ge0$. Observe that $c_2>0$ trivially under the above conditions on $A$ or $L$. We get $$c_1=Av(A^{\frac{\gamma-1}2}L)+\frac L{N-2}A^{\frac{\gamma+1}2}v'(A^{\frac{\gamma-1}2}L)= AF(A^{\frac{\gamma-1}2}L),$$ where $$F(r)=v(r)+\frac1{N-2}rv'(r).$$ We compute $$\begin{array}{l}
F'(r)=\dfrac1{N-2}\big((N-1)v'(r)+rv''(r)\Big)=-\dfrac{rv^\gamma(r)}{N-2}<0 \quad \text{for } 0<r<r_0, \\ [3mm]
F(0)=1>0,\quad \begin{cases}F(r_0)=\dfrac{r_0v'(r_0)}{N-2}<0,&\text{if } \gamma<\gamma_S, \\
\lim\limits_{r\to\infty}F(r)=0,&\text{if } \gamma\ge\gamma_S.
\end{cases}
\end{array}$$ The precise behaviour of $F$ at infinity in this latter cases is $$\left\{\begin{array}{ll}
F(r)=(1+Br^2)^{-\frac N2}, &\mbox{if } \gamma=\gamma_S, \\
F(r)\sim r^{-\frac 2{\gamma-1}}, &\mbox{if } \gamma>\gamma_S.
\end{array}\right.$$ Then, if $\gamma<\gamma_S$ we have that $F$ has a unique root $0<r^*<r_0$, while $F(r)>0$ for every $r>0$ if $\gamma\ge\gamma_S$. Let $A^*=(r^*/L)^{\frac2{\gamma-1}}$ if $\gamma<\gamma_S$, $\gamma\neq1$. We thus have:
- If $\gamma<1$, the function $c_1=c_1(A)$ is positive and increasing in $A\in(A^*,\infty)$, vanishes at $A=A^*$ and satisfies $\lim\limits_{A\to\infty}c_1(A)=\infty$.
- If $1<\gamma<\gamma_S$, then $c_1(A)$ is positive in $A\in(0,A^*)$, and vanishes at $A=0$ and $A=A^*$.
- If $\gamma=\gamma_S$ it is $c_1(A)>0$ for every $A>0$ and $\lim\limits_{A\to\infty}c_1(A)=0$.
- If $\gamma>\gamma_S$ it is again $c_1(A)>0$ for every $A>0$, but $\lim\limits_{A\to\infty}c_1(A)>0$.
- If $\gamma=1$ we have $c_1(A)=AF(L)>0$ for every $A>0$ provided $0<L< L^*\equiv r^*$, $c_1(A)=0$ when $L=L^*$.
We have characterized the existence of the stationary solutions in terms of the value $A=w(0)$, if $\gamma\ne1$, and in terms of the length $L$ when $\gamma=1$. The limit at infinity is $k=\lim\limits_{r \to\infty} w(r)=c_1(A)$. In summary we have obtained that, if $\gamma<1$, for each $0\le k<\infty$ there exists a unique $A=c_1^{-1}(k)\in[A^*,\infty)$, whereas for $\gamma>1$ there exists some $A$ with $c_1(A)=k$ when $0\le k\le k^*$ if $\gamma<\gamma_S$ or when $0< k\le k^*$ if $\gamma\ge\gamma_S$; the maximum value $k^*$ is given by $$k^*=\max\limits_{0<r<A^*}c_1(A),$$ where we put $A^*=\infty$ when $\gamma\ge \gamma_S$.
\[rem-chino\] From this result we immediately deduce that there exist bounded solutions to problem for $N\ge3$ and $p=m$ provided $L$ is small, which contradicts [@Liang].
The stationary solutions are explicit in the case $p=m$ since $v$ can be written in terms of Bessel functions $v(r)=r^{\frac{2-N}2}J_{\frac{N-2}2}(r)$. The matching condition is $$\label{stationary-bessel}
\frac{N-2}2J_{\frac{N-2}2}(L)+L J'_{\frac{N-2}2}(L)=0,$$ and $L^*$ is the first positive root of that equation.
For instance when $N=3$ we have $L^*=\pi/2$, and for $L\le \pi/2$ the solution is any multiple of $$\label{stationary-seno}
w(r)=\left\{\begin{array}{ll}
\dfrac1r\,\sin r,&\text{ for } 0<r<L, \\ [3mm] \dfrac1r(\sin L-L\cos L)+\cos L,&\text{ for } r\ge L.
\end{array}\right.$$
As a byproduct of the above calculations, just looking at negative values of the function $c_1(A)$, we describe the existence of stationary solutions for the Dirichlet problem. That is, we consider, for some $R>L$, the problem $$\label{eq.dirichlet}
\left\{
\begin{array}{ll}
\Delta w+a(x) w^\gamma=0,\qquad & |x|<R,\\
w(x)>0,\qquad & |x|<R,\\
w(x)=0,\qquad & |x|=R.
\end{array}\right.$$ Clearly if $0<R\le L$ and $\gamma<\gamma_S$, $\gamma\neq1$, the solution is given by $w(x)=Av(A^{\frac{\gamma-1}2}|x|)$, $A=\left(\frac{R}{r_0}\right)^{-\frac2{\gamma-1}}$, where $v$ is the solution to , while if $\gamma=1$ the solutions (any multiple of $v$) exist only when $R=L_1$, the length for which the eigenvalue of the Laplacian is 1. We consider here all dimensions $N\ge1$ (we set $L^*(N)=0$ for $N\le2$).
\[estacionarias-Dirichlet\] Problem with $R>L$ possesses bounded positive solutions if $N\le2$ or if $N\ge3$ and $\gamma<\gamma_S=\frac{N+2}{N-2}$. They are radially decreasing. Moreover,
1. If $0<\gamma<1$ they exist for every $R>L$. The value at the origin $w(0)=A=A(R)$ increases with $R$.
2. If $1<\gamma<\gamma_S$ they exist for every $R>L$. The value $A=A(R)$ decreases with $R$.
3. If $\gamma=1$ they exist only if $L^*<L<L_1$ and only for a precise value $R=R(L)>L_1$, which is decreasing in $L$.
In the case $0<\gamma<\gamma_S$, $\gamma\neq1$, it is easy to establish the asymptotics, for $R\to\infty$ $$\label{eq.A-R}
\begin{array}{ll}
A^{1-\gamma}\sim\dfrac R{L},&\text{ if } N=1, \\ [3mm]
A^{1-\gamma}\sim\dfrac1{L^2}\log R,&\text{ if } N=2, \\ [4mm]
A\sim A^*(1+cR^{2-N}),&\text{ if } N\ge3.
\end{array}$$ And in the case $\gamma=1$ we have, as $L\to L^*$, $$\label{eq.R-L}
\begin{array}{ll}
R\sim\dfrac 2{L},&\text{ if } N=1, \\ [3mm]
R\sim Le^{\frac2{L^2}},&\text{ if } N=2, \\ [4mm]
R\sim c(L-L^*)^{-\frac1{N-2}},&\text{ if } N\ge3.
\end{array}$$ See the proof of Theorem \[teo-stat\] for the values of $A^*,\,L^*,\,L_1$ and $r_0$.
\[rem-dirichlet\] We have proved that the Dirichlet problem in a ball corresponding to the equation in has stationary solutions only below the Sobolev exponent $p_S$ if $N\ge3$ or for every $p>0$ if $N\le2$. The solutions to the Dirichlet problem can be used in comparison arguments as subsolutions to the Cauchy problem.
Exponential solutions for the linear equation {#expo-linear}
---------------------------------------------
We look for explicit radial global unbounded solutions in the case $p=m=1$. The length $L$ plays a fundamental role in the existence. We try solutions in the form $$u(x,t)=e^{\lambda t}\varphi_\lambda(|x|)$$ where the profile $\varphi_\lambda$ satisfies two Bessel equations $$\label{two-bessel}
\left\{
\begin{array}{ll}
\varphi''+\frac{N-1}r\varphi'+(1-\lambda)\varphi=0,&\quad\text{if } 0<r<L,\\
\varphi''+\frac{N-1}r\varphi'-\lambda\varphi=0,&\quad\text{if } r>L,\\
\varphi(r)>0,&\quad\text{for } r\ge0,\\
\varphi'(0)=0.
\end{array}
\right.$$
\[prop-lambda0\] Given any $L> L^*(N)$ there exists a unique value $\lambda_0=\lambda_0(L)\in(0,1)$ for which there exists a solution $\varphi_{\lambda_0}\in C^1([0,\infty))$ of . The solution is unique up to multiplicative constants.
The solution of both Bessel equations in give $$\varphi_\lambda(r)=r^{-\nu}\left\{
\begin{array}{ll}
J_{\nu}(\sqrt{1-\lambda}\,r), \qquad & \text{if } 0<r< L,\\
BI_{\nu}(\sqrt{\lambda}\,r)+CK_{\nu}(\sqrt{\lambda}\,r), \qquad & \text{if } r>L.
\end{array}\right.$$ Here $J_\nu$ is the Bessel function of first kind of order $\nu\equiv\frac{N-2}2$, and $I_\nu,\,K_\nu$ are the modified Bessel functions of order $\nu$, respectively of first and second kind. For the case $N=1$ we refer to [@FerreiradePablo]. Denote also by $\eta_{\nu,k}$ the $k$–th root of $J_\nu$. The condition $\varphi_\lambda(r)>0$ implies $L\sqrt{1-\lambda}<\eta_{\nu,1}$ and also that no modified Bessel function of first kind appear, so $B=0$. Recall that $K_\nu>0$, $K_\nu'<0$ and $K_\nu\sim z^{-1/2}e^{-z}$ at infinity. Functions with $B\ne0$ will be useful as subsolutions.
The compatibility conditions at $r=L$ gives the value of $C$ and $\lambda$. First, continuity implies $$C=\dfrac{J_{\nu}(L\sqrt{1-\lambda})}{K_{\nu}(L\sqrt{\lambda})}>0.$$ Now differentiability fixes the value of $\lambda$ in terms of $L$ if there exists a solution to the equation $$\label{lambda-L}
\Phi(\lambda,L)\equiv\sqrt{1-\lambda}\,\dfrac{J_{\nu}'(L\sqrt{1-\lambda})}{J_{\nu}(L\sqrt{1-\lambda})}-
\sqrt{\lambda}\,\dfrac{K_{\nu}'(L\sqrt{\lambda})}{K_{\nu}(L\sqrt{\lambda})}=0.$$ We see next that there always exists a root $\lambda_0=\lambda_0(L)$ if $N=2$, but only for $L$ large if $N>2$.
For $N=2$ we have $$\Phi(0,L)=\frac{J_0'(L)}{J_0(L)}<0,\qquad \Phi(1,L)=-\frac{K_0'(L)}{K_0(L)}>0.$$ There exists a solution for every $L>0$, unique if $L$ is small. As $L$ increases multiple roots appear, due to the zeroes of $J_0(L\sqrt{1-\lambda})$, and we choose the biggest root, $\lambda_0\in(1-\frac{\eta_{0,1}^2}{L^2},1)$, in order to get $\varphi_{\lambda_0}(r)>0$ for every $0\le r\le L$.
When $N>2$ the function $\Phi(\lambda,L)$ satisfies, for $L>0$ small, $$\Phi(\lambda,L)\sim\frac{2\nu}L>0\quad\text{for every } 0<\lambda<1.$$ There exists then no root. On the other hand, $$\Phi(0,L)=\frac{J'_{\nu}(L)}{J_{\nu}(L)}+\frac{\nu}L,\qquad \Phi(1,L)=-\frac{K'_{\nu}(L)}{K_{\nu}(L)}>0.$$ We see that there is a solution $\lambda_0=\lambda_0(L)$ if and only if $L>\overline L^*$, where $\overline L^*\in(0,\eta_{\nu,1})$ is the first root of $\Phi(0,L)$. Observe that this value $\overline L^*$ coincides with the value $L^*$ that appeared in the construction of the stationary solutions, see . Choosing as before the largest root $\lambda_0$ when multiple roots appear, we obtain a function $\lambda_0=\lambda_0(L)$ for $L> L^*$, increasing with $\lim\limits_{L\to (L^*)^+}\lambda_0( L)=0$, $\lim\limits_{L\to\infty}\lambda_0(L)=1$. In fact $\lambda_0\sim1- cL^{-2}$ for $L$ large.
Self-similar solutions of the pure diffusion equation {#subsec-self-similar}
-----------------------------------------------------
We study in this subsection the existence of radial solutions in self-similar form of two special types for the pure diffusion equation $
u_t=\Delta u^m$ for $x\neq0$. We consider fast diffusion $m<1$, but we restrict ourselves to the so called *not too fast* diffusion range, $m>m^*=\frac{(N-2)_+}N$. This solution will be used in comparison arguments in our problem to study the grow-up set for different values of the reaction exponent $p$.
We look for solutions $U=U(r,t)$, $r=|x|$, to the equation $$u_t=\Delta u^m,\qquad x\neq0,$$ of the forms $$\label{eq.self-types}
U(r,t)=t^{\alpha} f(r t^{\beta})\qquad \text{ or }\qquad U(r,t)=e^{\alpha t} f(r e^{\beta t}).$$ We denote those solutions as of types I and II, respectively. In both cases the profile $f$ verifies the equation $$\label{eq.perfil}
(f^m)''+\frac{N-1}\xi (f^m)'=\alpha f+\beta \xi f',\qquad \xi>0,$$ where $f'$ denote $df/d\xi$, and the self-similar exponents satisfy the relation $$\label{eq.delta}
\delta\equiv\alpha (1-m)-2\beta\in \{0,1\}.$$ In fact we have $\delta=1$ for solutions of type I and $\delta=0$ for solutions of type II. As we have said, when using those solutions for comparison we will consider each of those types depending on the value of $p$ in problem . We now obtain solutions of the ODE for all values of $\delta\ge0$. We refer to [@FerreiradePablo] for the case $N=1$.
Let $m_*<m<1$, $\alpha>0$ and $\beta>0$ be three positive parameters such that $\delta=\alpha (1-m)-2\beta\ge 0$. Then, there exists a non-negative decreasing solution $f$ of for $\xi>0$, such that $\alpha f+\beta\xi f'\ge 0$. Moreover, the behaviour of $f$ is given by $$\label{eq.f-0}
f(\xi)\sim
\left\{
\begin{array}{ll}
1,\qquad & \text{if } N\le2,\\
\xi^{-\frac{N-2}m}, & \text{if } N\ge 3,
\end{array}\right.
\qquad \text{as } \xi\sim 0,$$ and $$\label{eq.f-infty}
f(\xi)\sim
\left\{
\begin{array}{ll}
\xi^{\frac{-2}{1-m}},\quad & \mbox{if }\delta>0,\\
\xi^{\frac{-2}{1-m}} (\log(\xi))^{\frac1{1-m}}, & \mbox{if }\delta=0,
\end{array}\right.
\qquad \mbox{as }\xi\to\infty.\qquad$$
We assume $N\ge2$ and introduce the following variables $$X=\frac{\xi f'}{f}, \qquad Y=\frac1{m} \xi^2 f^{1-m}, \qquad \eta=\log\xi.$$ The resulting system is $$\left\{
\begin{array}{l}
\dot{X}=(2-N)X-mX^2+Y(\alpha+\beta X),\\
\dot{Y}=(2+(1-m)X)Y,
\end{array}\right.$$ where $\dot{X}=dX/d\eta$. We look for non-negative decreasing profiles, so we focus on the second quadrant $X<0\,,\, Y>0$.
We first consider $\delta>0$, in which case the critical points are $$A=(0,0), \qquad B=\left(\frac{2-N}{m},0\right), \qquad C=\left(\frac{-2}{1-m},\frac{4-2N(1-m)}{(1-m)\delta}\right).$$ Notice that since $m>m_*$ the critical point $C$ belongs to the second quadrant. Let us define, $$\begin{array}{l}
\displaystyle \Gamma_1=\left\{\frac{-2}{1-m}\le X\le \frac{2-N}{m}\,,\, Y=0\right\},\\[3mm]
\displaystyle \Gamma_2=\left\{X=\frac{-2}{1-m}\,,\, 0\le Y\le \frac{4-2N(1-m)}{(1-m)\delta}\right\},\\[3mm]
\displaystyle \Gamma_3=\left\{\frac{-2}{1-m}\le X\le \frac{2-N}{m}\,,\, Y= \frac{mX^2-(2-N)X}{\alpha+\beta X}\right\}.
\end{array}$$ Note that in $\Gamma_1\cup\Gamma_2$ we have $\dot{X}\le 0$ and $\dot{Y}=0$, while in $\Gamma_3$ we have $\dot{X}= 0$ and $\dot{Y}>0$. Then, if we look at the orbits backward in time, the region $$\Omega=\left\{\frac{-2}{1-m}\le X\le \frac{2-N}{m}\,,\, 0\le Y\le \frac{mX^2-(2-N)X}{\alpha+\beta X}\right\}$$ is invariant. Even more, in this region it holds $dY/d\eta>0$, so if we look for the orbit passing through a point in either $\Gamma_2$ or $\Gamma_3$ the only possibility is that it comes from the point $A$ if $N=2$ or $B$ if $N\ge3$. Therefore, there exists a separatrix orbit connecting the points $A$ and $C$ if $N=2$ and the points $B$ and $C$ if $N\ge 3$, see Fig. \[fig.planofases\].
![The phase plane for $\delta>0$. $N=2$ to the left and $N\ge 3$ to the right.[]{data-label="fig.planofases"}](planofases2d.png "fig:")![The phase plane for $\delta>0$. $N=2$ to the left and $N\ge 3$ to the right.[]{data-label="fig.planofases"}](planofases3d.png "fig:")
This separatrix orbit gives us a decreasing positive trajectory such that $$f_*(\xi) \sim \xi^{-\frac2{1-m}} \quad \mbox{as } \xi\to\infty.$$ For $\xi$ near zero we have $Y\sim e^{2\eta}$ in dimension $N=2$, while for $N\ge 3$ we have $X\sim -(N-2)/m$. Therefore $$f_*(\xi)\sim
\left\{
\begin{array}{ll}
1,\qquad & \text{if } N=2,\\
\xi^{-\frac{N-2}m}, & \text{if } N\ge 3,
\end{array}\right.
\qquad \text{as } \xi\sim 0.$$
Now we consider $\delta=0$. In this case, the critical point $C$ disappears, but we can use the same argument as before, observing that the separatrix orbit connects the point $A$ with the point $(-2/(1-m),\infty)$ for $N=2$ and the point $B$ with the point $(-2/(1-m),\infty)$ if $N\ge 3$. The picture is the analogous to Fig. \[fig.planofases\] with the point $C$ going vertically to infinity. We obtain in this way a decreasing positive solution with the same behaviour as before near the origin, and the behaviour for $\xi$ large $$f_*(\xi)\sim \left(\frac{\log\xi}{\xi^2}\right)^{\frac1{1-m}}.$$
Finally, we observe that in both cases the separatrix orbit lives in $\Omega$ for all the values of the parameter $\xi>0$, which implies $X\ge -2/(1-m)$. Thus, $$\alpha f_*+\beta \xi f_*'= (\alpha+\beta X)f_*\ge \frac\delta{1-m}\,f_*\ge0.$$
As a corollary we obtain the existence of grow-up self-similar solutions to the fast diffusion equation.
\[cor.ss.pme\] For every $m_*<m<1$ there exists two biparametric families of self-similar solutions $\{U_{\alpha,\mu,I},\,\alpha>1/(1-m),\,\mu>0\}$ and $\{U_{\alpha,\mu,II},\,\alpha>0,\,\mu>0\}$, to the equation $\partial_tu=\Delta u^m$ for $x\neq0$, of types I and II respectively. These solutions are radially decreasing in space and increasing in time.
For each $\alpha>1/(1-m)$ in the case of type I, or $\alpha>0$ in the case of type II, we consider the self-similar solution corresponding to the profile $f=f_*$ just constructed with $\beta>0$ satisfying and $$\lim_{\xi\to0^+}\xi^{\frac{(N-2)_+}m}f(\xi)=1.$$ Monotonicity in space follows from the property $f'(\xi)\le0$. As to the monotonicity in time we use the fact that $\alpha f(\xi)+\beta\xi f'(\xi)\ge 0$. Now for each $\mu>0$ we consider the self-similar solution with profile $f_\mu(\xi)=\mu^{\frac2{1-m}}f(\mu\xi)$.
Bounded vs. unbounded solutions {#sect-bdd-notbb}
===============================
By the definition of global existence exponent $p_0$, if $p<p_0$ all the solutions are global, while if $p>p_0$ there exist solutions that blow-up in finite time. The value of $p_0$ is given in . We study in this section two different questions: $(i)$ if the solutions are global or not in the limit case $p=p_0$; and $(ii)$ if the global solutions for $p\le p_0$ are bounded or not.
Before that we observe that the value of $p_0$ in the case $N\ge2$ and $m<1$ is not covered by the literature, though it is easy to see that $p_0=1$ and that it lies in the global solutions side.
\[lem-exp\] Let $m\le1$. Every solution is global if $p\le1$, while there exist blow-up solutions when $p>1$. Thus $p_0=1$.
The case $p\le1$ follows by comparison with the supersolution $$\overline u(t)=Me^t,\qquad M=\|u_0\|_\infty.$$
On the other hand, for $p>1$ we can apply Kaplan’s method to obtain blow-up solutions if the initial value is large. The method works precisely because $m\le1<p$. To that purpose let $(\lambda_1,\varphi_1)$ be the first eigenvalue and eigenfunction of the Laplacian in the ball $B_L$, normalized such that $\int_{B_L}\varphi=1$. Let $J(t)=\int_{B_L}u\varphi$. We have $$J'(t)\ge-\lambda_1\int_{B_L}u^m\varphi_1+\int_{B_L}u^p\varphi_1\ge-\lambda_1(\int_{B_L}u\varphi_1)^m+(\int_{B_L}u\varphi_1)^p.$$ If the initial value is large so as to satisfy $\int_{B_L}u_0\varphi>\lambda_1^{\frac1{p-m}}$, then $J'(t)\ge CJ^p(t)$, which means that $J$ (and thus $u$) blows up in finite time.
Let us now concentrate in the limit case $p=p_0$, and study if the solutions are global or not. By the above lemma we only have to consider the case $m>1$. The unidimensional case is solved in [@FerreiradePabloVazquez], and the solutions are global. The case $N\ge2$ is considered in [@Liang], but the proof of blow-up presented in that paper fails when $N\ge3$ and $L$ is small, precisely by the existence of stationary solutions, see Theorem \[rem-chino\].
\[teo-bup-p=m\] Assume $p=m$ and let $u$ be the solution to problem .
1. If $N=2$ then $u$ blows up in a finite time if $m>1$ and it is global unbounded if $m\le 1$.
2. For $N\ge 3$ the behaviour of $u$ depends on $L$:
1. $u$ is global and bounded if $L\le L^*$, assuming also the behaviour when $L=L^*$;
2. For $L>L^*$ the function $u$ blows up in a finite time if $m>1$ and it is global unbounded if $m\le 1$.
Moreover, the global unbounded solutions grow up in some ball of positive radius.
Let $N\ge 3$ (the case $N=2$ is similar to the case $N\ge 3$ and $L>L^*$) and let us consider the function $g_A=w^{1/m}$ where $w$ es defined in . By Theorem \[teo-stat\] we have that for $L<L^*$ taking $A$ large $g_A$ is a supersolution, bigger than $u_0$ at $t=0$ and thus bigger than $u(\cdot, t)$ at any time. This is clear when $L<L^*$, since the stationary solution is strictly positive. If $L=L^*$ we use the behaviour .
On the contrary, when $L>L^*$ the function $g_A$ vanishes at some point $R>0$ independent of $A$. Then, taking $A$ small enough $u$ is a supersolution of the problem $$\label{eq.dirichlet3}
\left\{
\begin{array}{ll}
w_t=\Delta w^m+a(x) w^m,\qquad & |x|<2R, t>0,\\
w(x,t)=0,\qquad & |x|=2R, t>0,\\
w(x,0)=w_0(x),
\end{array}\right.$$ where $w_0(x)=g_A(x)$ for $|x|\le R$ and $w_0(x)=0$ in $R\le |x|\le 2R$. Notice that if the initial datum $u_0$ were not positive, by the penetration property of the solutions to the pure diffusion equation without reaction, there exists a time $t_0$ such that the support of $u(\cdot,t_0)$ contains the ball $B_R$ and then taking $A$ small enough $u(x,t_0)\ge w_0(x)$. Thus, again by comparison $u(x,t+t_0)\ge w(x,t)$ for $t\ge 0$.
We claim that $w$ is unbounded in $B_R(0)$, and moreover it blows up in a finite time if $m>1$. This then gives that $u$ is global unbounded when $m\le1$ (at least in $B_R(0)$), and blows up if $m>1$.
In order to prove the claim we note that problem has no stationary solution, see Theorem \[estacionarias-Dirichlet\]. Moreover, since $w_0$ is a radial decreasing function which satisfies $\Delta w_0^m+a(x)w_0^m\ge 0$, we get a radial decreasing solution which is increasing in time. This implies that the solution can not go to zero and then it must be unbounded. Indeed, let us consider the Lyapunov functional $$\label{eq.lyapunov}
E_w(t)=\frac12 \int_{|x|<2R} |\nabla w^m|^2\,dx-\frac12 \int_{|x|<2R} a(x) w^{2m}\,dx.$$ It is nonincreasing, $$\label{eq.lyapunov-decrece}
E_w'(t)=-\frac{4m}{(m+1)^2} \int_{|x|<2R} \left((w^{\frac{m+1}2})_t\right)^2\,dx\le0,$$ and also $E_w$ is bounded from below provided $w$ is bounded. Therefore, by standard arguments $w$ converges (up to a subsequence of times) to an stationary solution. Ruled out the possibility to go to the only stationary solution, the trivial one, this implies that $w$ is unbounded, that is, $$\limsup_{t\to T} \|w(\cdot,t)\|_\infty =\infty\quad \mbox{for some } T\le\infty$$ Even more, $w$ is unbounded in $B_R(0)$, since if we suppose that $w(x,t)$ is bounded for $|x|=R_1<R$, then for $M$ large enough $w$ is a subsolution to $$\left\{
\begin{array}{ll}
z_t=\Delta z^m+z^m, \quad & |x|<R_1,\, 0<t<T,\\
z(x,t)=M, & |x|=R_1,\, 0<t<T,\\
z(x,0)=M, & |x|<R_1.
\end{array}\right.$$ On the other hand, $g_A$ is a stationary supersolution for $A$ large. Then, by comparison $w$ is bounded. This contradiction implies that $w$ is unbounded in $B_R(0)$.
On the other hand, using the concavity argument of [@LS] we obtain that the function $$J(t)=\frac{1}{m+1}\int_{|x|<2R} w^{m+1}(x,t)\,dx$$ satisfies $$\begin{array}{rl}
\displaystyle J'(t)&\displaystyle=\int w^m w_t=\int w^m\Delta w^m+\int a(x)w^{2m}=-2E_w(t), \\
\displaystyle(J'(t))^2&\displaystyle=\left(\int w^{\frac{m+1}2}w^{\frac{m-1}2}w_t\right)^2\le\frac4{(m+1)^2}\int w^{m+1}\int \left((w^{\frac{m+1}2})_t\right)^2, \\
\displaystyle J''(t)&\displaystyle=-2E'_w(t)=\frac{8m}{(m+1)^2} \int \left((w^{\frac{m+1}2})_t\right)^2,
\end{array}$$ and finally $$(J'(t))^2\le \frac{m+1}{2m}J(t)J''(t).$$ Since $J(t)$ is unbounded $J'(t_0)>0$ at some time $t_0$. Moreover $E_w$ is decreasing, thus $J'(t)>0$ for every $t>t_0$. Therefore we can integrate the above inequality to get $$\label{eq.J'}
J'(t)\ge C J^{\frac{2m}{m+1}}(t).$$ Let us observe that if $m>1$ the exponent $\frac{2m}{m+1}>1$, then $J$ (and therefore $w$) blows up in a finite time $T<\infty$.
This completes the proof of Theorem \[teo-global-p0\].
\[rem.tasaBR\] Notice that, since $w$ is radially nonincreasing, inequality for $m<1$ gives the lower estimate $u(0,t)\ge ct^{\frac1{1-m}}$. As we will see this estimate can be extended to the whole $\mathbb{R}^N$, see Lemma \[lem-tasa-p=m\].
We now consider exponents $p\le p_0$ and prove Theorem \[teo-GUP\]. We start with an easy result.
\[teo.m-menor-1\] Let $u$ be a global solution with $p< m$. Then $u$ is bounded if and only if $N\ge3$.
If $N\ge3$ we use the fact that there exist large stationary solutions. For $N=2$ we argue by contradiction, assuming $u(x,t)\le K$ for every $x\in\mathbb{R}^2$ and $t>0$. We consider then the Dirichlet problem $$\label{eq.dirichlet2}
\left\{
\begin{array}{ll}
w_t=\Delta w^m+a(x) w^p,\qquad & |x|<R,\;t>0,\\
w(x,t)=0,\qquad & |x|=R, \;t>0,\\
w(x,0)=w_0(x),& |x|<R,
\end{array}\right.$$ for some $R>0$. By we can take $R=R_1$ large such that the corresponding stationary solution $W_{R_1}$ satisfies $W_{R_1}(0)>K$. On the other hand, if we take $R=R_0<R_1$ and $w_0(x)=W_{R_0}(x)$ for $0\le |x|\le R_0$, $w_0(x)=0$ for $R_0\le |x|\le R_1$, we obtain a bounded increasing in time solution $w$ to problem . By standard arguments $\lim_{t\to\infty}w(x,t)=W_{R_1}(x)$.
On the other hand if $R_0$ is so small in order to have $w_0(x)\le u_0(x)$ for $|x|<R_1$, comparison implies $w(x,t)\le u(x,t)$ for $|x|<R_1$ and every $t>0$. This is a contradiction.
\[rem-pmenorm\] Notice that for $N=2$ and $p< m$ the solution $u$ must be unbounded in any ball. Indeed, by comparison we can assume that $u$ is radial. Arguing as in the case $p=m$, if $u$ is bounded on $|x|=R_1$ for some $R_1>0$, we can put above $u$ a large stationary solution, so $u$ can not grow-up.
In order to complete the proof of Theorem \[teo-GUP\] it only remains to consider the range $m<p\le1$ and show that there exist unbounded solutions.
\[teo.m-menor-1\] If $m<p\le1$ there exist global unbounded solutions to problem .
Consider the problem, for some $m<q<p_S$ (and $q\le1$), $R>0$, $$\label{eq.CauchyDirichlet}
\left\{
\begin{array}{ll}
z_t=\Delta z^m+a(x) z^q,\qquad & |x|<R,\; t>0,\\
z(x,t)=0,& |x|=R,\; t>0,\\
z(x,0)=z_0(x)& |x|<R.
\end{array}\right.$$ Observe that if $R<L$ the function $\underline z(x,t)=\psi(t)\varphi(x)$ is a subsolution if $\varphi$ is a stationary solution (see Section \[sect-stationary\]) and $\psi$ satisfies $$\psi'=\frac{\psi^q-\psi^m}{\varphi^{1-q}(0)},\quad \psi(0)>1.$$ In fact, since $\psi(t)>1$ for every $t>0$ and $\varphi(x)\le\varphi(0)$ for every $x$, we have $$\begin{array}{rl}
\underline z_t-\Delta \underline z^m-a(x) \underline z^q&=\psi'\varphi-(\psi^m-\psi^q)\varphi^q \\ [3mm] &=-\left(1-\left(\dfrac\varphi{\varphi(0)}\right)^{1-q}\right)(\psi^q-\psi^m)\varphi^q\le0
\end{array}$$ for every $|x|<R$, $t>0$. Since $\psi'\sim\psi^q$, the function $\psi$ tends to infinity and $\underline z$ grows up in $B_R$.
Assume first $m<p<p_S$. Then our solution $u$ is a supersolution to problem with $q=p$. We conclude grow-up for any initial value above $\underline z(x,0)$. If on the contrary $p_S\le p\le1$, we have that our solution is a supersolution to problem for any $q<p_S$ provided that $u(x,t)\ge 1$ in $B_L$. Thus, as before, we have grow-up for large initial data. In order to prove that there exists a solution with $u(x,t)\ge 1$ in $B_L$ we compare with the subsolution $\underline v(x)=\lambda z(x)$, where $z$ is a stationary solution of with some $R>L$ and $\lambda$ is large enough.
\[rem-vanish\] It is well known that if $0<m<\frac{N-2}N$ with $N\ge3$ there exist solutions to the very fast diffusion equation $$v_\tau=\Delta v^m$$ that vanish identically at a finite time $\tau_0$, which depends on the initial value, see for instance [@V2]. Take now $v(\cdot,0)$ be such that $\tau_0<1/(1-m)$. Then $$w(x,t)=e^{t}v(x,\tau),\quad \tau=\frac{1-e^{-(1-m)t}}{1-m},$$ is a supersolution to our problem with $p=1$ and it satisfies $$w(x,t_0)\equiv0 \quad\text{for every } x\in\mathbb{R}^N,$$ where $t_0=\frac1{1-m}\log(1-(1-m)\tau_0)$. Therefore, in the case of linear reaction and superfast diffusion in problem , any initial value $u_0\le v(\cdot,0)$ produces a solution with finite time extinction.
Grow-up set {#sect-gupset}
===========
The main objective of this section is to study if the unbounded global solutions to problem tend to infinity for every point $x\in\mathbb{R}^N$. We first remind that the case $p<m$ (which implies $N=2$) follows directly from Remark \[rem-pmenorm\], since we can put below the solution a subsolution with grow-up set as large as we want. We therefore deal here with the upper range $m\le p\le1$.
If $p>m$ we also assume that the initial value is a radial function and so it is the solution. We denote the solution $u(r,t)$, $r>0$, $t>0$, since no confusion arises. We impose the additional condition in that case $$\label{eq.tasah}
\lim_{t\to\infty} u(R,t)=\infty,\quad \mbox{for some } R\in[0, L].$$ Notice that if $u$ is bounded in $B_L$ (the region where the reaction takes place), then $u$ is bounded.
Next we prove that under the previous hypotheses the grow-up set is the whole $\mathbb{R}^N$, that is, we prove Theorem \[teo-sets\]. We divide the proof into several lemmas.
\[teo-0-eps\] Assume $R>0$ in . Then $\lim\limits_{t\to\infty}u(r,t)=\infty$ for every $r\ge R$.
From , given any $K>0$, there exists $t_K$ a time such that $u(R,t)\ge K$ for all $t\ge t_K$. Now, for any $R_1>R$ we consider the problem $$\left\{ \begin{array}{ll}
w_t=\Delta w^m\equiv r^{1-N} (r^{N-1}(w^m)_r)_r ,\quad& r\in (R,2 R_1), \;t>t_K,\\
w(R,t)=K , & t>t_K,\\
w(2 R_1,t)=0 ,& t>t_K,\\
w(r,t_K)=w_0(r), & r\in(R,2R_1).
\end{array}\right.$$ Taking $w_0(r)\le \min\{u(r,t_K),K\}$ a continuous function which satisfies the boundary condition we get that $u$ is a supersolution, then $u\ge w$ for $t>t_K$. It is easy to see that any solution to this problem converges as $t\to\infty$ to the explicit stationary solution $$h(x)= K \left( \frac{(2R_1)^{N-2}-r^{N-2}}{(2R_1)^{N-2}-R^{N-2}}\right).$$ Thus taking $t$ large enough we can get $u(R_1,t)\ge cK$, which is as large as we please.
\[teo.global.p<1\] Let $p\le 1$ and assume $0<R<L$ in . Then $\lim\limits_{t\to\infty}u(r,t)=\infty$ for every $r\le R$.
As before, for every $K>0$ we consider a time $t_K$ such that $u$ is a supersolution to the problem in the ball $$\left\{ \begin{array}{ll}
w_t=\Delta w^m+ w^p, \quad& 0<r<R,\; t>t_K,\\
w(R,t)=K, & t>t_K,\\
w(r,t_K)=u(r,t_K) ,& 0<r<R.
\end{array}\right.$$ Notice that the flat solution, that is, the solution of $W'=W^p$ with initial datum $W(0)=\inf\limits_{x\in B_R(0)} u(\cdot,t_K)$, is a subsolution for $t\in(T_K,T)$, where $W(T)=K$. Therefore by comparison $u(r,T)\ge K$ for $0\le r\le R$.
We finally consider the case $R=0$ in .
\[teo.globalfast\] Let $0<m<p\le 1$, and assume $\lim\limits_{t\to\infty} u(0,t)=\infty$. Then $\lim\limits_{t\to\infty}u(r,t)=\infty$ for every $r>0$.
We use the Intersection Comparison technique with respect to the family of positive radial stationary solutions $$\left\{\begin{array}{ll}
\Delta w^m+w^p=0, \qquad & 0<r<R.\\
w(R)=0,\;
(w^m)'(0)=0,
\end{array}\right.$$ which are constructed in Subsection \[sect-stationary\]. That is, we study the number of sign changes between $w$ and $u$ in $[0,R]$, $$N(t)=N(w(\cdot,t),u(\cdot,t)).$$ The main result of the Intersection Comparison argument asserts that the number of sign changes between two solutions of a large class of nonlinear parabolic equations, which includes our equation, does not increase in time provided no new intersections appear through the boundary, cf. [@Galaktionov], [@Matano].
First, since $p>m$ there exists $R_0$ small enough such that for $R\le R_0$ the initial intersection number is $N(0)=1$. Also, as $u(R,t)>0$ no new intersection can appear through $r=R$. Now, at $r=0$ we define $t_0$ and $R<R_0$ such that $u(0,t_0)=w(0)$ with $u_t(0,t_0)>0$ (notice that $t_0$ and $R$ exist because $u(0,t)\to \infty$) then $$\begin{array}{rl}
0<u_t(0,t_0)&\displaystyle=\Delta u^m(0,t_0)-\Delta w^m(0,t_0)+u^p(0,t_0)-w^p(0,t_0) \\ [3mm]
&\displaystyle=\Delta( u^m(0,t_0)-\Delta w^m(0,t_0)).
\end{array}$$ This implies that the function $z(r)=u^m(r,t_0)-w^m(r)$ satisfies $z(0)=z'(0)=0$, $z''(0)>0$, so that $z(r)>0$ for $r\in(0,\delta)$ and some $\delta>0$. So not only no new intersections appear through $r=0$ but an intersection is lost at that point at time $t=t_0$, i.e. $N(t_0)=0$. By intersection comparison this implies $N(t)=0$ for $t\ge t_0$.
Therefore $u(r,t_1)> w(r)$ for some $t_1>t_0$. Applying the same argument given in the proof of Theorem \[teo.m-menor-1\] we get that $u$ grows up in $B_R(0)$. Then, by Theorem \[teo-0-eps\] $u$ has global grow-up.
As we have said, if $p<m$ the global grow-up follows from Remark \[rem-pmenorm\]. In the case $p=m$ we first use Theorem \[teo-bup-p=m\] to get that the solutions is unbounded at every point in some ball, and the by Theorem \[teo-0-eps\] the same holds in every ball. Finally if $m<p\le 1$ the result follows by applying Theorems \[teo-0-eps\], \[teo.global.p<1\] and \[teo.globalfast\].
Grow-up rate {#sect-guprate}
============
This section is devoted to study the speed at which the global unbounded solutions to problem tend to infinity.
An easy upper estimate of the grow-up rate for $0<p\le1$ is given by the solutions of the ODE $$U'(t)=U^p(t).$$ This gives, $$\label{flat}
u(x,t)\le\begin{cases} (M^{1-p}+(1-p)t)^{\frac1{1-p}},& \quad \text{if } p<1, \\
Me^t,& \quad \text{if } p=1,
\end{cases}$$ where $M=\|u_0\|_\infty$. We have named this bound the natural rate.
We see next that these estimates are far from being sharp in some cases, and indeed, in the cases when the solution grows up with the natural rate, it does so only in the ball $B_L$, where the reaction applies.
Linear diffusion, sublinear reaction, $m=1$, $p<1$
--------------------------------------------------
We prove here Theorem \[teo-rates-p<1\] by means of Duhamel’s formula, $$\label{duhamel}
u(x,t)=\int_{\mathbb{R}^N}u_0(y)\Gamma(x-y,t)+ \int_0^t\int_{|y|\le L} u^p(y,s)\Gamma(x-y,t-s)\,dy\,ds,$$ where $\Gamma$ is the Gauss kernel, and we consider every dimension $N\ge1$ for completeness. The formal proof is as follows: if $u(x,t)\sim g(t)$ in $B_L$ for $t\ge1$, then $$\begin{array}{rl}
g(t)&\displaystyle\sim\int_1^t\int_{|y|\le L} g^p(s)\Gamma(x-y,t-s)\,dy\,ds \sim\int_1^t g^p(s)
\int_0^{\frac{L^2}{4(t-s)}} r^{\frac{N-2}2}e^{-r}\,dr\,ds\\ [3mm]
&\displaystyle\sim\int_1^t g^p(s)s^{-\frac N2}\,ds.
\end{array}$$ Now, the solution of the resulting differential equation $g'(t)\sim g^p(t)t^{-\frac N2}$ is $$g^{1-p}(t)\sim\begin{cases} 1+t^{\frac{2-N}2},& \text{ if } N\ne2, \\ \log t,& \text{ if } N=2.\end{cases}$$ The case $N=1$ was obtained in [@FerreiradePablo]. In the case $N\ge3$ this also explains why the solution must be bounded.
We then proceed with the detailed proof
Assume $u(x,t)\le g(t)$ for every $|x|\le L$ and some increasing function $g$. Since the first term in is bounded, we have $$\begin{array}{rl}
u(x,t)&\displaystyle\le 2\int_1^t\int_{|x-y|\le 2L} g^p(s)(4\pi(t-s))^{-1}e^{-\frac{|x-y|^2}{4(t-s)}}\,dy\,ds\\ [3mm]
&\displaystyle
\le c\int_1^t g^p(s) \int_0^{\frac{L^2}{t-s}} e^{-r}\,dr\,ds\le c g^p(t)\int_1^t(1-e^{-\frac{L^2}{t-s}})\,ds\\ [4mm]
&\displaystyle\le cL^2 g^p(t)\log t.
\end{array}$$ In the last step we have used L’Hôpital’s rule, $$\lim_{t\to\infty}\frac{\displaystyle\int_1^t (1-e^{-\frac{L^2}{t-s}})\,ds}{\log t}=\lim_{t\to\infty}\frac{\displaystyle\int_{\frac{L^2}{t-1}}^\infty (1-e^{-z})z^{-2}\,dz}{\log t}=L^2.$$ We iterate this estimate starting with $g_1(t)=\nu t^{\frac1{1-p}}$, see . We obtain in this way the sequence $$g_k(t)= c_kt^{\delta_k}(\log t)^{\sigma_k}, \qquad \delta_{k+1}=p\delta_k,\quad \sigma_{k+1}=p\sigma_k+1,\quad c_{k+1}=cL^2c_k^p.$$ We end with the limits, $$\lim_{k\to\infty}\delta_k=0,\quad\lim_{k\to\infty}\sigma_k=\dfrac1{1-p},\quad\lim_{k\to\infty}c_k= (cL^2)^{\frac1{1-p}}.$$
As to the lower estimate, it is clear first that $u(x,t)\ge C_0>0$ in $B_L$ for $t>0$, since we can put below a small stationary subsolution. Assume that we have $u(x,t)\ge g(t)$ for every $|x|<L$, $t>0$. Then the above Duhamel’s formula gives $$u(x,t)
\ge\displaystyle c \int_1^t g^p(s)
\int_0^{\frac{(L-|x|)^2}{4(t-s)}} e^{-r}\,dr\,ds=\int_1^t g(s)(1-e^{-\frac{L^2}{t-s}})\,ds.$$ We now observe that for every $q\ge0$ $$\int_1^t (\log s)^q(1-e^{-\frac{c}{t-s}})\,ds\ge c(\log t)^{q+1},$$ to get, again by iteration, $$u(x,t)\ge c(L-|x|)^{\frac1{1-p}}(\log t)^{\frac1{1-p}}.$$
We have just proved, for instance for $|x|<L/2$, that $u(x,t)\ge h(t)=c(\log t)^{\frac1{1-p}}$. We now extend this estimate to every compact of $\mathbb{R}^2$.
To this purpose we use that $u$ is a supersolution to the problem $$\left\{
\begin{array}{ll}
w_t=\Delta w,\qquad & |x|>L/2,\; t>t_0,\\
w(x,t)=h(t), &|x|=L/2,\; t>t_0,\\
w(x,t_0)=u(x,t_0), & |x|>L/2,
\end{array}\right.$$ and a subsolution can be found explicitely. In fact, for $R>L$ fixed the function $\underline w(x,t)=h(t)\varphi(|x|)^\gamma$, where $\varphi= (1-|x|/R)_+$ do the job provided that both $t_0$ and $\gamma$ are large enough. Indeed, $$\underline w_t-\Delta\underline w=h\varphi^{\gamma-2}\left(\frac{h'}{h}\varphi^2+
\frac{\gamma}{R} \Big(\frac{1}{r}-\frac{\gamma}{R}\Big)\right).$$ Since $h'/h\to 0$ as $t\to\infty$ we can take $t_0$ such that $\varphi^2 h'/h\le 1$ for all $t\ge t_0$. Then for $\gamma$ large enough $\underline w_t-\Delta\underline w\le 0$.
Moreover, $\varphi(r)<(1-L/R)_+<1$, then $\underline w(x,t)<h(t)$ and taking $\gamma$ large enough $\underline w(x,t_0)< u(x,t_0)$.
This means that given any $x_0\in\mathbb{R}^2$ we can define $R=2|x_0|>L$ to have $$u(x_0,t)\ge \underline u(x_0,t)=h(t) 2^{-\gamma}.$$
Linear diffusion, linear reaction, $m=p=1$
------------------------------------------
In this case we also have that the presence of a localized reaction provokes a growth of the solutions that is strictly slower than that of the solutions with global reaction, that is, we prove that the solutions behave for large times like an exponential, but the exponent depends on the length $L$ and is strictly less than 1. We use the explicit radial global unbounded solutions obtained in Subsection \[expo-linear\] in order to establish an estimate of the growth of general solutions.
Let $u$ be a solution to problem where $L>L^*$. By comparison from above and below with the solutions obtained in Theorem \[prop-lambda0\] with different values of $\lambda$, we get that there exists a function $c(s)$, decreasing with $\lim\limits_{s\to0}c_1(s)=0$, such that $$c(\varepsilon)e^{(\lambda_0-\varepsilon)t}\le u(x,t)\le c^{-1}(\varepsilon) e^{(\lambda_0+\varepsilon)t},$$ for every $x\in\mathbb{R}^N$, $t\ge1$, $\varepsilon>0$. We conclude .
The critical line $m=p<1$
-------------------------
In this parameter, we show that the natural grow-up rates are sharp by proving the lower bound.
\[lem-tasa-p=m\] Let $p=m<1$ and let $u$ be a global unbounded solution of . Then $$u(\cdot,t)\ge ct^{\frac1{1-m}}$$ uniformly in compact sets of $\mathbb{R}^N$.
We consider a solution in separated variables, $w(x,t)=\psi(t)\varphi(|x|)$, where $\psi=\psi_\lambda$ satisfies $\psi'=\lambda\psi^m$, and $\varphi=\varphi_\lambda$ is a solution to $$\left\{
\begin{array}{ll}
( \varphi^m)''+\frac{N-1}r(\varphi^m)'+a(r)\varphi^m-\lambda \varphi=0, \quad & r>0,\\
\varphi(0)=1,\ (\varphi^m)'(0)=0.
\end{array}\right.$$ It is easy to check that if $L>L^*$ there exists a limit value $\lambda^*>0$ such that for every $0<\lambda<\lambda^*$ the solution $\varphi$ is positive and decreasing in $[0,R_\lambda)$, with $\varphi(R_\lambda)=0$, and $\lim_{\lambda\to\lambda^*}R_\lambda=\infty$. In fact, by the results in Section \[sect-stationary\] we know that $\varphi$ crosses the axis at some point if $\lambda=0$, so by continuous dependence with respect to $\lambda$ the same holds when $\lambda$ is small. On the other hand, the solution corresponding to $\lambda=1$ satisfies $\varphi(r)=1$ in $0<r<L$ and it increases to infinity for $r>1$. The existence of $\lambda^*$ is now standard.
Let now $x_0\in\mathbb{R}^N$ be any point. We take $\lambda\sim\lambda^*$ so that $R_\lambda>|x_0|$. Comparison in $[0,R_\lambda]$ gives the grow-up rate in the ball $B_{|x_0|}$.
The supercritical case $m< p\le 1$
----------------------------------
As in the previous case we show here that the grow-up rate of our solutions is the natural one, but only inside the ball $B_L$, where the reaction takes place.
\[lem.bola\] Let $m<p\le1$, and also $p<p_S$ if $N\ge3$. Let $u$ be a solution of with global grow-up. Then for every $|x|<L$ it holds $$u(x,t)\ge c \left\{
\begin{array}{ll}
t^{\frac1{1-p}},\quad &\text{if}\quad p<1,\\
e^t, & \text{if}\quad p=1.
\end{array}\right.$$
We compare with the subsolution in separated variables $\underline z$ given in the proof of Theorem \[teo.m-menor-1\] with $R=L$. Let $t_1>0$ be such that $u(x,t_1)\ge\underline z(x,0)$ for $|x|<L$. We obtain $u(x,t)\ge \phi(x)\psi(t-t_1)$, and conclude with the behaviour of $\psi'\sim\psi^p$ as $t\to\infty$.
In the case $p\ge p_S$ we only can compare with a subsolution satisfying $\psi'\sim\psi^q$, $q<p_S$, thus obtaining $t^{\frac1{1-q}}$ as grow-up rate, which is presumed not to be sharp.
The next task is to obtain the grow-up rate outside the ball $B_L$. We show that there the rate is strictly smaller. To that purpose we consider the self-similar solutions constructed in Section \[sect-special\].
Let us consider first the case $p<1$. By Lemma \[lem.bola\] we know that there exits a constant $C_1>0$ and a time $t_0>0$ such that $u(x,t)\ge C_1 t^{\frac1{1-p}}$ for $|x|=L/2$, $t\ge t_0$, so $u$ is a supersolution to the problem $$\left\{
\begin{array}{ll}
w_t=\Delta w^m, \qquad &|x|>L/2,\; t>t_0,\\
w(x,t)=C_1 t^{\frac1{1-p}}, &|x|=L/2,\; t>t_0,\\
w(x,t_0)=u_0(x).
\end{array}\right.$$ Put $\underline w(x,t)=AU(|x|,t-t_0)$, where $U=U_{\frac1{1-p},1,I}$ is the selfsimlar solution of type I given in Theorem \[cor.ss.pme\] with $\mu=1$ and $\alpha=1/(1-p)$, which implies $\beta=\alpha(p-m)/2$. Since $\underline w_t\ge 0$ we get $$\underline w_t-\Delta \underline w^m=(A-A^m)\underline w_t\le 0$$ provided $A<1$. Let us look at the initial time $t=t_0$. If $N=2$ the profile $f$ defining $U$ is bounded, which implies $\underline w(x,t_0)\equiv0$. When $N\ge3$, using the fact that $p< p_S$ implies that $\alpha-\beta(N-2)/m>0$, the behaviour near the origin of $f$, see , gives us $$\underline w(x,t_0)\sim A (t-t_0)^{\alpha-\beta\frac{N-2}m}|x|^{-\frac{N-2}m} \to 0 \quad \text{for } |x|\ge L/2,\;\text{as } t\to t_0.$$ On the other hand, we note that for $|x|=L/2$ it holds $$\lim_{t\to t_0}\frac{(t-t_0)^\alpha f_*((t-t_0)^\beta L/2)}{t^\alpha}=0=
\lim_{t\to \infty}\frac{(t-t_0)^\alpha f_*((t-t_0)^\beta L/2)}{t^\alpha}.$$ Then there exists $A>0$ small such that $$\underline w(x,t)=A (t-t_0)^\alpha f_*(\frac{L}2(t-t_0)^\beta)<C_1 t^{\alpha} \qquad \text{for } |x|=L/2,\; t\ge t_0.$$ Then by comparison $u\ge \underline w$, and thus for $t$ large and $|x|>L$ it holds, using , $$u(x,t)\ge \underline w(x,t)\sim |x|^{\frac{-2}{1-m}} t^{\frac{1}{1-m}}.$$
In order to obtain the upper estimate, we observe that from we have $u(L,t)\le C_2 (t+1)^{\frac1{1-p}}$ for $t\ge 0$. Then, $u$ is a subsolution of $$\left\{
\begin{array}{ll}
w_t=\Delta w^m, \qquad &|x|>L,\; t>0,\\
w(x,t)=C_2(t+1)^{\frac1{1-p}},&|x|=L,\; t>0,\\
w(x,t_0)\ge u(x,t_0).
\end{array}\right.$$ Here we consider the function $\overline w(x,t)=AU(|x|-L,t+1)$, where $U=\widetilde U_{\frac1{1-p},1,I}$ is the one dimensional self-similar solution (that is, $ U_t=( U^m)_{xx}$, $x>0$), with the same exponents as before. First observe that since the profile satisfies $f'\le0$, it is a supersolution to our multidimensional equation. Using now the behaviour at infinity of $u_0$ it is easy to see that for $A$ large enough $\overline w$ is a supersolution to the above problem. Then by comparison we get, for $t$ large and $|x|>L$, $$u(x,t)\le \overline w(x,t)\sim |x|^{\frac{-2}{1-m}} t^{\frac{1}{1-m}}.$$
The case $p=1$ follows in a similar way using this time comparison with a self-similar solution $U=U_{1,1,II}$, that is a solution of type II with $\alpha=1$, $\beta=(1-m)/2$. We only have to take into account that:
i\) By Lemma \[lem.bola\] we have $u\sim e^t$ for $|x|< L$ and $t>t_0$. On the other hand, $u$ is a supersolution of the fast diffusion equation, so $u(x,t)\sim 1 $ for $|x|< L$ and $t\in[0,t_0]$. Therefore, $u\sim e^t$ for $|x|< L$ and $t>0$. Thus, we can repeat the same argument as before with $t_0=0$.
ii\) The comparison at time $t=0$ follows thanks to the behaviour imposed to $u_0$.
Observe finally that for $t\to\infty$ we have, thanks to , $$U(x,t)=e^tf(|x|e^{\frac{m-1}2\,t})\sim |x|^{-\frac2{1-m}}(\log|x|)^{\frac1{1-m}}\,t^{\frac1{1-m}}.$$
Acknowledgments {#acknowledgments .unnumbered}
===============
Work supported by the Spanish project MTM2014-53037-P.
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Effect of Custom-made Probiotic Chocolates on Streptococcus mutans, Plaque pH, Salivary pH and Buffering Capacity in Children - A Randomised Controlled Trial.
To compare the effect of three custom-made probiotic chocolates and conventional chocolates on plaque pH, salivary pH and buffering capacity of saliva in children. The study also evaluated its antimicrobial efficacy against S. mutans. A parallel randomised double-blinded trial was conducted in two phases. For the phase І trial, 90 children were randomly divided into 3 groups: milk (MC), white (WC) and dark chocolate (DC). Salivary pH, plaque pH and buffering capacity were assessed at baseline, 10 min, 30 min and 60 min after consumption of the chocolates. After a washout period of 20 days, the children were assigned to their respective probiotic chocolate groups and the assessments were repeated. In the phase ІІ trial, 60 children were divided into 3 groups (n = 20): probiotic milk (PMC), white (PWC) and dark chocolate (PDC). They were given probiotic chocolates for 5 consecutive days in a week. S. mutans colony count was measured at baseline, post intervention, 15 days and 30 days. All probiotic chocolates were less acidogenic than their counterparts. PWC was found to be the least acido-genic. DC was found to be the least acidogenic among plain chocolates. All probiotic chocolates were effective in reducing the S. mutans colony count. Chocolates can serve as a vehicle for delivering probiotics with the added advantage of making them tooth-friendly.
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Contour integration is closely related to the calculus of residues, a method of complex analysis.
One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods.
Contour integration methods include
- direct integration of a complex-valued function along a curve in the complex plane (a contour)
- application of the Cauchy integral formula
- application of the residue theorem
One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums.
Curves in the complex plane
In complex analysis a contour is a type of curve in the complex plane. In contour integration, contours provide a precise definition of the curves on which an integral may be suitably defined. A curve in the complex plane is defined as a continuous function from a closed interval of the real line to the complex plane: z : [a, b] → C.
This definition of a curve coincides with the intuitive notion of a curve, but includes a parametrization by a continuous function from a closed interval. This more precise definition allows us to consider what properties a curve must have for it to be useful for integration. In the following subsections we narrow down the set of curves that we can integrate to only include ones that can be built up out of a finite number of continuous curves that can be given a direction. Moreover, we will restrict the "pieces" from crossing over themselves, and we require that each piece have a finite (non-vanishing) continuous derivative. These requirements correspond to requiring that we consider only curves that can be traced, such as by a pen, in a sequence of even, steady strokes, which only stop to start a new piece of the curve, all without picking up the pen.
Directed smooth curves
Contours are often defined in terms of directed smooth curves. These provide a precise definition of a "piece" of a smooth curve, of which a contour is made.
A smooth curve is a curve z : [a, b] → C with a non-vanishing, continuous derivative such that each point is traversed only once (z is one-to-one), with the possible exception of a curve such that the endpoints match (z(a) = z(b)). In the case where the endpoints match the curve is called closed, and the function is required to be one-to-one everywhere else and the derivative must be continuous at the identified point (). A smooth curve that is not closed is often referred to as a smooth arc.
The parametrization of a curve provides a natural ordering of points on the curve: z(x) comes before z(y) if x < y. This leads to the notion of a directed smooth curve. It is most useful to consider curves independent of the specific parametrization. This can be done by considering equivalence classes of smooth curves with the same direction. A directed smooth curve can then be defined as an ordered set of points in the complex plane that is the image of some smooth curve in their natural order (according to the parametrization). Note that not all orderings of the points are the natural ordering of a smooth curve. In fact, a given smooth curve has only two such orderings. Also, a single closed curve can have any point as its endpoint, while a smooth arc has only two choices for its endpoints.
Contours
Contours are the class of curves on which we define contour integration. A contour is a directed curve which is made up of a finite sequence of directed smooth curves whose endpoints are matched to give a single direction. This requires that the sequence of curves be such that the terminal point of coincides with the initial point of , . This includes all directed smooth curves. Also, a single point in the complex plane is considered a contour. The symbol + is often used to denote the piecing of curves together to form a new curve. Thus we could write a contour Γ that is made up of n contours as
Contour integrals
The contour integral of a complex function f : C → C is a generalization of the integral for real-valued functions. For continuous functions in the complex plane, the contour integral can be defined in analogy to the line integral by first defining the integral along a directed smooth curve in terms of an integral over a real valued parameter. A more general definition can be given in terms of partitions of the contour in analogy with the partition of an interval and the Riemann integral. In both cases the integral over a contour is defined as the sum of the integrals over the directed smooth curves that make up the contour.
For continuous functions
To define the contour integral in this way one must first consider the integral, over a real variable, of a complex-valued function. Let f : R → C be a complex-valued function of a real variable, t. The real and imaginary parts of f are often denoted as u(t) and v(t), respectively, so that
Then the integral of the complex-valued function f over the interval [a, b] is given by
Let f : C → C be a continuous function on the directed smooth curve γ. Let z : R → C be any parametrization of γ that is consistent with its order (direction). Then the integral along γ is denoted
and is given by
This definition is well defined. That is, the result is independent of the parametrization chosen. In the case where the real integral on the right side does not exist the integral along γ is said not to exist.
As a generalization of the Riemann integral
The generalization of the Riemann integral to functions of a complex variable is done in complete analogy to its definition for functions from the real numbers. The partition of a directed smooth curve γ is defined as a finite, ordered set of points on γ. The integral over the curve is the limit of finite sums of function values, taken at the points on the partition, in the limit that the maximum distance between any two points on the partition (in the two-dimensional complex plane), also known as the mesh, goes to zero.
Direct methods
Direct methods involve the calculation of the integral by means of methods similar to those in calculating line integrals in several-variable calculus. This means that we use the following method:
- parametrizing the contour
- The contour is parametrized by a differentiable complex-valued function of real variables, or the contour is broken up into pieces and parametrized separately
- substitution of the parametrization into the integrand
- Substituting the parametrization into the integrand transforms the integral into an integral of one real variable.
- direct evaluation
- The integral is evaluated in a method akin to a real-variable integral.
Example
A fundamental result in complex analysis is that the contour integral of z−1 is 2πi, where the path of the contour is taken to be the unit circle traversed counterclockwise (or any Jordan curve about 0). In the case of the unit circle there is a direct method to evaluate the integral
In evaluating this integral, use the unit circle |z| = 1 as contour, parametrized by z(t) = eit, with t ∈ [0, 2π], then Template:Nowrap begindz/dt = ieitTemplate:Nowrap end and
which is the value of the integral.
Applications of integral theorems
Applications of integral theorems are also often used to evaluate the contour integral along a contour, which means that the real-valued integral is calculated simultaneously along with calculating the contour integral.
Integral theorems such as the Cauchy integral formula or residue theorem are generally used in the following method:
- a specific contour is chosen:
- The contour is chosen so that the contour follows the part of the complex plane that describes the real-valued integral, and also encloses singularities of the integrand so application of the Cauchy integral formula or residue theorem is possible
- application of the Cauchy–Goursat theorem
- The integral is reduced to only an integration around a small circle about each pole.
- application of the Cauchy integral formula or residue theorem
- Application of these integral formula gives us a value for the integral around the whole of the contour.
- division of the contour into a contour along the real part and imaginary part
- The whole of the contour can be divided into the contour that follows the part of the complex plane that describes the real-valued integral as chosen before (call it R), and the integral that crosses the complex plane (call it I). The integral over the whole of the contour is the sum of the integral over each of these contours.
- demonstration that the integral that crosses the complex plane plays no part in the sum
- If the integral I can be shown to be zero, or if the real-valued integral that is sought is improper, then if we demonstrate that the integral I as described above tends to 0, the integral along R will tend to the integral around the contour R + I.
- conclusion
- If we can show the above step, then we can directly calculate R, the real-valued integral.
Example
Consider the integral
To evaluate this integral, we look at the complex-valued function
which has singularities at i and −i. We choose a contour that will enclose the real-valued integral, here a semicircle with boundary diameter on the real line (going from, say, -a to a) will be convenient. Call this contour C.
There are two ways of proceeding, using the Cauchy integral formula or by the method of residues:
Using the Cauchy integral formula
Note that:
thus
Furthermore observe that
Since the only singularity in the contour is the one at i, then we can write
which puts the function in the form for direct application of the formula. Then, by using Cauchy's integral formula,
We take the first derivative, in the above steps, because the pole is a second-order pole. That is, (z − i ) is taken to the second power, so we employ the first derivative of f(z). If it were (z − i ) taken to the third power, we would use the second derivative and divide by 2!, etc. The case of (z − i ) to the first power corresponds to a zero order derivative—just f(z) itself.
If we call the arc of the semicircle Arc, we need to show that the integral over Arc tends to zero as a → ∞ — using the estimation lemma
where M is an upper bound on |f(z)| along the Arc and L the length of Arc. Now,
So
Using the method of residues
Consider the Laurent series of f(z) about i, the only singularity we need to consider. We then have
(See Sample Laurent Calculation from Laurent series for the derivation of this series.)
It is clear by inspection that the residue is −i/4 (to see this, imagine that the above equation were multiplied by z − i, then both sides integrated via the Cauchy integral formula—only the second term would integrate to a non-zero quantity), so, by the residue theorem, we have
Thus we get the same result as before.
Contour note
As an aside, a question can arise whether we do not take the semicircle to include the other singularity, enclosing −i. To have the integral along the real axis moving in the correct direction, the contour must travel clockwise, i.e., in a negative direction, reversing the sign of the integral overall.
This does not affect the use of the method of residues by series.
Example (II) – Cauchy distribution
The integral
(which arises in probability theory as a scalar multiple of the characteristic function of the Cauchy distribution) resists the techniques of elementary calculus. We will evaluate it by expressing it as a limit of contour integrals along the contour C that goes along the real line from −a to a and then counterclockwise along a semicircle centered at 0 from a to −a. Take a to be greater than 1, so that the imaginary unit i is enclosed within the curve. The contour integral is
Since eitz is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the denominator z2 + 1 is zero. Since z2 + 1 = (z + i)(z − i), that happens only where z = i or z = −i. Only one of those points is in the region bounded by this contour. The residue of f(z) at z = i is
According to the residue theorem, then, we have
The contour C may be split into a "straight" part and a curved arc, so that
and thus
It can be shown that if t > 0 then
Therefore if t > 0 then
A similar argument with an arc that winds around −i rather than i shows that if t < 0 then
and finally we have this:
(If t = 0 then the integral yields immediately to real-valued calculus methods and its value is π.)
Example (III) – trigonometric integrals
Certain substitutions can be made to integrals involving trigonometric functions, so the integral is transformed into a rational function of a complex variable and then the above methods can be used in order to evaluate the integral.
As an example, consider
We seek to make a substitution of z = eit. Now, recall
and
Taking C to be the unit circle, we substitute to get:
The singularities to be considered are at 3−1/2i, −3−1/2i. Let C1 be a small circle about 3−1/2i, and C2 be a small circle about −3−1/2i. Then we arrive at the following:
Example (IIIa) trigonometric integrals, the general procedure
The above method may be applied to all integrals of the type
where P and Q are polynomials, i.e. a rational function in trigonometric terms is being integrated. Note that the bounds of integration may as well be π and -π, as in the previous example, or any other pair of endpoints 2π apart.
The trick is to use the substitution where and hence
This substitution maps the interval [0, 2π] to the unit circle. Furthermore,
and
so that a rational function f(z) in z results from the substitution, and the integral becomes
which is in turn computed by summing the residues of inside the unit circle.
The image at right illustrates this for
which we now compute. The first step is to recognize that
The substitution yields
The poles of this function are at 1 ± √2 and −1 ± √2. Of these, 1 + √2 and −1 −√2 are outside the unit circle (shown in red, not to scale), whereas 1 − √2 and −1 + √2 are inside the unit circle (shown in blue). The corresponding residues are both equal to −i√2/16, so that the value of the integral is
Example (IV) – branch cuts
Consider the real integral
We can begin by formulating the complex integral
We can use the Cauchy integral formula or residue theorem again to obtain the relevant residues. However, the important thing to note is that z1/2 = e1/2·Log(z), so z1/2 has a branch cut. This affects our choice of the contour C. Normally the logarithm branch cut is defined as the negative real axis, however, this makes the calculation of the integral slightly more complicated, so we define it to be the positive real axis.
Then, we use the so-called keyhole contour, which consists of a small circle about the origin of radius ε say, extending to a line segment parallel and close to the positive real axis but not touching it, to an almost full circle, returning to a line segment parallel, close, and below the positive real axis in the negative sense, returning to the small circle in the middle.
Note that z = −2 and z = −4 are inside the big circle. These are the two remaining poles, derivable by factoring the denominator of the integrand. The branch point at z = 0 was avoided by detouring around the origin.
Let γ be the small circle of radius ε, Γ the larger, with radius R, then
It can be shown that the integrals over Γ and γ both tend to zero as ε → 0 and R → ∞, by an estimation argument above, that leaves two terms. Now since z1/2 = e(1/2)Log(z), on the contour outside the branch cut, we have gained 2π in argument along γ (by Euler's Identity, eiπ represents the unit vector, which therefore has π as its log. This π is what is meant by the argument of z. The coefficient of 1/2 forces us to use 2π), so
Therefore:
By using the residue theorem or the Cauchy integral formula (first employing the partial fractions method to derive a sum of two simple contour integrals) one obtains
Example (V) – the square of the logarithm
This section treats a type of integral of which
is an example.
To calculate this integral, one uses the function
and the branch of the logarithm corresponding to .
We will calculate the integral of f(z) along the keyhole contour shown at right. As it turns out this integral is a multiple of the initial integral that we wish to calculate and by the Cauchy residue theorem we have
Let R be the radius of the large circle, and r the radius of the small one. We will denote the upper line by M, and the lower line by N. As before we take the limit when R → ∞ and r → 0. The contributions from the two circles vanish. For example, one has the following upper bound with the ML-lemma:
In order to compute the contributions of M and N we set on M and on N, with 0 < x < ∞:
which gives
Example (VI) – logarithms and the residue at infinity
We seek to evaluate
This requires a close study of
We will construct f(z) so that it has a branch cut on [0, 3], shown in red in the diagram. To do this, we choose two branches of the logarithm, setting
and
The cut of z3/4 is therefore (−∞, 0] and the cut of (3−z)1/4 is (−∞, 3]. It is easy to see that the cut of the product of the two, i.e. f(z), is [0, 3], because f(z) is actually continuous across (−∞, 0). This is because when z = −r < 0 and we approach the cut from above, f(z) has the value
When we approach from below, f(z) has the value
But
so that we have continuity across the cut. This is illustrated in the diagram, where the two black oriented circles are labelled with the corresponding value of the argument of the logarithm used in z3/4 and (3−z)1/4.
We will use the contour shown in green in the diagram. To do this we must compute the value of f(z) along the line segments just above and just below the cut.
Let z = r (in the limit, i.e. as the two green circles shrink to radius zero), where 0 ≤ r ≤ 3. Along the upper segment, we find that f(z) has the value
and along the lower segment,
It follows that the integral of
along the upper segment is −iI in the limit, and along the lower segment, I. | https://en.formulasearchengine.com/wiki/Methods_of_contour_integration |
Cross-Reference to Related Application
TECHNICAL FIELD
BACKGROUND ART
CITATION LIST
Patent Literature
SUMMARY OF INVENTION
Technical Problem
Solution to Problem
Advantageous Effects of Invention
DESCRIPTION OF EMBODIMENTS
This application claims priority to and the benefit of Japanese Patent Application Nos. 2012-098803 and 2012-098804, filed in the Japanese Patent Office on Apr. 24, 2012, the entire contents of which are incorporated herein by reference.
The present invention relates to a musical score playing art of playing music from musical information of an electronically acquired musical score and particularly relates to a musical score playing device and a musical score playing program by which, in a process of determining note values from note shapes written in a musical score, correct note values can be determined even if tuplet symbols are not written.
The present invention also relates to a musical score playing device and a musical score playing program specialized to determine correct note values when tuplet symbols for triplets are not written.
A musical score playing device prepares playing information from musical score information, such as written positions of notes, sound emission starting timings and durations, written positions of bar lines, etc., extracted from a musical score file in PDF or a musical score acquired by a scanner, and plays music automatically in accordance with the playing information. A procedure for extracting musical score information from a musical score and preparing playing information is, for example, described in Patent Document 1.
FIG. 25
A note value (duration of a note) of a triplet, quintuplet, etc., in a musical score differs from the duration indicated by the actually written note. For example, when as indicated in the arrowed portion in the first measure in , a triplet note is denoted by an eighth note, its duration is ⅔ of an eighth note.
Patent Literature 1 Japanese Published Unexamined Patent Application No. H7-129159
FIG. 25
FIG. 25
When, in a case where note values are determined from shapes of written notes to generate playing information from a musical score, there is a tuplet symbol (a numeral indicating a tuplet) that indicates a triplet as in the arrowed portion in the first measure in , the correct note value can be determined as long as the symbol (numeral) can be recognized. However, in many cases, the tuplet symbol is omitted as in the arrowed portion in the third measure in , and in such a case, the note value is determined as a normal eighth note in reading the musical score and preparing the playing information.
Also, even if the tuplet symbols are written, finger numbers, etc., are written in a musical score in many cases, and the finger numbers and the tuplet symbols (numerals indicating the tuplets) must be distinguished in the process of generating the playing information from the musical score and a reading art for this purpose was thus necessary.
The present invention is proposed in view of the above circumstances and an object thereof is to provide a musical score playing device and a musical score playing program by which, even if tuplet symbols are not written in a musical score, playing information having the correct note values can be prepared to enable automatic playing.
1
To achieve the above object, the present invention (claim ) is a musical score playing device comprising:
a prescribed measure time calculating means calculating a prescribed measure time from the meter of a musical composition;
a measure playing time calculating means calculating a measure playing time from sound emission timings and note values of notes and rests within a measure; a comparing means comparing the calculated prescribed measure time and measure playing time; and
a note value correcting means inferring that a tuplet is present within the measure if the prescribed measure time and the measure playing time are not matched and correcting the sound emission timings and note values of the notes and rests; and
wherein the note value correcting means includes
a measure note sequence recording means storing a note sequence within the measure, a grouping means grouping the notes within the measure according to each beat, and a tupleting process means performing a tupleting process of changing the note values of the grouped notes if a playing time of the grouped notes and a reference time of a single beat that is calculated from the note sequence are not matched.
2
1
Claim is the musical score playing device according to Claim , wherein the tupleting process means changes the note values of the grouped notes to perform the tupleting process if the playing time of the grouped notes is longer than the reference time of one beat and there is no note longer than the reference time within the group or if there is a note longer than the reference time among the grouped notes and the playing time exceeds twice the reference time.
3
1
Claim is the musical score playing device according to Claim , wherein the tupleting process means changes the note values of the grouped notes to perform the tupleting process if the playing time of the grouped notes is shorter than the reference time of one beat and the number of sixteenth notes in the group is 3 or the number of thirty-second notes is 5 or 7.
4
2
3
Claim is the musical score playing device according to Claim or , wherein the tupleting process by the tupleting process means changes a total note value for the number of grouped notes to a duration obtained by multiplying a total duration of the note values of the grouped notes by a number set in advance in accordance with the number of notes grouped.
5
4
Claim is the musical score playing device according to Claim , wherein the total note value for the number of notes is changed
to a duration that is 2 times the total duration of the note values of the grouped notes if the number of notes grouped is 3,
to a duration that is 4 times the total duration of the note values of the grouped notes if the number of notes grouped is 5 to 7,
to a duration that is 8 times the total duration of the note values of the grouped notes if the number of notes grouped is 9 to 15, or
to a duration that is 16 times the total duration of the note values of the grouped notes if the number of notes grouped is 17 to 31.
6
Claim is a musical score playing program for making a computer execute
a prescribed measure time calculating step of calculating a prescribed measure time from the meter of a musical composition,
a measure playing time calculating step of calculating a measure playing time from sound emission timings and note values of notes and rests within a measure, a comparing step of comparing the calculated prescribed measure time and measure playing time, and
a note value correction step, in which, if the prescribed measure time and the measure playing time are not matched, it is inferred that a tuplet is present within the measure, a note sequence within the measure is stored, the notes within the measure are grouped according to each beat, and if a playing time of the grouped notes and a reference time of a single beat that is calculated from the note sequence are not matched, a tupleting process of changing the note values of the grouped notes is performed to correct the sound emission timings and note values of the notes and rests.
7
6
Claim is the musical score playing program according to Claim , wherein the tupleting process includes the process of changing the note values of the grouped notes if the playing time of the grouped notes is longer than the reference time of one beat and there is no note longer than the reference time within the group or if there is a note longer than the reference time among the grouped notes and the playing time exceeds twice the reference time.
8
6
3
Claim is the musical score playing program according to Claim , wherein the tupleting process includes the process of changing the note values of the grouped notes if the playing time of the grouped notes is shorter than the reference time of one beat and the number of sixteenth notes in the group is or the number of thirty-second notes is 5 or 7.
9
7
8
Claim is the musical score playing program according to Claim or , wherein the tupleting process changes a total note value for the number of grouped notes to a duration obtained by multiplying a total duration of the note values of the grouped notes by a number set in advance in accordance with the number of notes grouped.
10
9
Claim is the musical score playing program according to Claim , wherein the total note value for the number of notes is changed
to a duration that is 2 times the total duration of the note values of the grouped notes if the number of notes grouped is 3,
to a duration that is 4 times the total duration of the note values of the grouped notes if the number of notes grouped is 5 to 7,
to a duration that is 8 times the total duration of the note values of the grouped notes if the number of notes grouped is 9 to 15, or
to a duration that is 16 times the total duration of the note values of the grouped notes if the number of notes grouped is 17 to 31.
The present invention also relates to a musical score playing device and a musical score playing program specialized to determine correct note values when tuplet symbols for triplets are not written.
11
The present invention (claim ) specialized to determine correct note values when tuplet symbols for triplets are not written is a musical score playing device comprising:
a prescribed measure time calculating means calculating the prescribed measure time from the meter of a musical composition;
a measure playing time calculating means calculating a measure playing time from sound emission timings and note values of notes and rests within a measure;
a first comparing means comparing the calculated prescribed measure time and measure playing time; and
a note value correcting means inferring that a tuplet is present within the measure if the prescribed measure time and the measure playing time are not matched and correcting the sound emission timings and note values of the notes and rests; and
wherein the note value correcting means includes
a target note determining means that successively determines a correction target note,
a tupleting process means performing a tupleting process of changing the note value of each correction target note in the measure to ⅔, and
a second comparing means comparing the prescribed measure time and a corrected measure playing time that is in accordance with the changed note values, and
the tupleting process and the comparison by the second comparing means are repeated for the respective target notes and the tupleting of a triplet is finalized with the changed note values of the respective notes at the point at which the prescribed measure time and the corrected measure playing time are equal.
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11
Claim is the musical score playing device according to Claim , wherein the tupleting process means performs two types of tuplet inference by performing a tupleting process upon changing the note value of a note, having a note value one step greater than the correction target note in the measure, to ⅔ of the note value.
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11
Claim is the musical score playing device according to Claim , wherein the target note determining means performs the tuplet inference by the tupleting process means repeatedly while changing the target note in the order of a sixty-fourth note, thirty-second note, sixteenth note, eighth note, quarter note, and half note.
14
Claim is a musical score playing program comprising:
a prescribed measure time calculating step of calculating the prescribed measure time from the meter of a musical composition;
a measure playing time calculating step of calculating the measure playing time from sound emission timings and note values of notes and rests within a measure;
a tuplet inference step of comparing the calculated prescribed measure time and measure playing time and, if the prescribed measure time and the measure playing time are not matched, inferring that a tuplet is present within the measure;
a target note determining step of determining a correction target note with respect to the notes in the measure; and
a tupleting process step of performing a tupleting process of changing the note value of the correction target note in the measure to ⅔, comparing the prescribed measure time and a corrected measure playing time that is in accordance with the changed note values, and finalizing the tupleting of a triplet with the changed note values of the respective notes at the point at which the prescribed measure time and the corrected measure playing time are equal; and
making a computer perform the tupleting process repeatedly on the respective target notes.
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14
Claim is the musical score playing program according to Claim , wherein, if after the correction target note has been determined in the target note determining step and the note value of the correction target note has been changed in the tupleting process step, the change of the note value has been performed for the first time, the tupleting process is performed upon changing the note value of a note, having a note value that is one step greater than the correction target note in the measure, to ⅔ of the note value.
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14
Claim is the musical score playing program according to Claim , wherein the tupleting process is performed repeatedly while changing the target note in the order of a sixty-fourth note, thirty-second note, sixteenth note, eighth note, quarter note, and half note.
With the musical score playing device and the musical score playing program according to the present invention, even if tuplet symbols are not written in a musical score, if the prescribed measure time and the measure playing time do not match, it is inferred that there is a tuplet portion in the measure and the tupleting process of correcting the sound emission timings and note values of notes and rests is performed to enable playing information having correct note values to be prepared.
If the playing time of grouped notes is longer than the reference time of one beat and there is no note longer than the reference time within the group, or if there is a note longer than the reference time among the grouped notes and the playing time exceeds twice the reference time, or if the playing time of the grouped notes is shorter than the reference time of one beat and the number of notes in the group is 3, the tupleting process is performed by changing the note values of respective notes that have been grouped.
With the tupleting process specialized to triplets, the note values of the correction target notes in a measure are changed to ⅔ and the target notes for which the prescribed measure time becomes equal to the corrected measure playing time are judged to be tuplet notes to be corrected in note value to enable the tupleting process specialized to triplets, which are most frequently used in musical scores, to be performed.
Also, in determining the correction target notes by means of the target note determining means, tuplet inference is performed in the order from a sixty-fourth note to a half note (from small note values) to enable a tupleting process of high efficiency to be performed in musical compositions in which tuplets of notes of small note value tend to be used frequently.
A musical score playing device according to the present invention shall now be described with reference to the drawings.
FIG. 1
is a block diagram of an automatic playing device operating on a computer and incorporating the musical score playing device (musical score playing program) according to the present invention.
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The automatic playing device includes a musical score file storage means storing musical score files and PDF musical score files resulting from scanning of musical scores, a musical score information generating means recognizing a musical score file and generating musical score information, a musical score information storage means storing the generated musical score information, a playing information preparing means preparing playing information from the musical score information, a playing information storage means storing the generated playing information, a musical sound playing means reading the playing information successively and actually playing music, and a musical score display means displaying the musical score files and the musical score information.
4
4
A characteristic arrangement of the present invention is that when the playing information is prepared from the musical score information in the playing information preparing means , inference of tuplets is performed to determine correct note values, and a tuplet inference program that is necessary for this purpose functions as a portion of the playing information preparing means .
FIG. 2
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The hardware of the automatic playing device may be realized by a general purpose information processing device, such as a personal computer, etc. is a block diagram of a hardware arrangement example of the automatic playing device that is constructed on a computer and is arranged by connecting a display , a mouse , a keyboard , a ROM , a RAM , a CPU , an HDD , a disk drive , a MIDI interface , an audio interface , and a network interface to a bus .
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In the HDD of the computer, the musical score playing program, for acquiring a musical score file via the MIDI interface , the audio interface , or the network interface and preparing playing information to perform automatic playing, is installed from a recording medium installed in the disk drive or is downloaded from a predetermined URL via the internet.
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The CPU executes various types of processes (respective steps) in accordance with the predetermined program (musical score playing program) installed or downloaded by the abovementioned procedure and thereby controls the entirety of the musical score playing device. The CPU includes the musical score file storage means , the musical score information generating means , and the musical score information storage means as principal functions to store the acquired musical score information as electronic information and enable display of the stored information on the musical score display means , and includes the playing information preparing means , the playing information storage means , and the musical sound playing means to enable automatic playing of musical sounds in accordance with the playing information generated in accordance with the musical score information.
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The RAM temporarily stores information used in the processes of the CPU .
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The musical score file storage means is arranged from the RAM and the HDD . The musical score file may be acquired from the network interface , etc., as mentioned above or may be acquired by connecting a separate image scanner to the computer.
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The musical score information generating means is arranged from the program stored in the HDD , the CPU that executes the program, the RAM used as a working storage area, etc.
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The musical score information storage means is arranged from the RAM and the HDD .
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The playing information generating means is arranged from the program stored in the HDD , the CPU that executes the program, the RAM used as a working storage area, etc.
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The playing information storage means is arranged from the RAM and the HDD .
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The musical sound playing means includes the musical score playing program stored in the HDD, the CPU that executes the program, the RAM used as a working storage area, a sound source device, the audio interface , etc. The sound source device includes a sound system that includes a D/A converter, an amp, and a speaker.
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The musical score display means is arranged from the program stored in the HDD , the CPU that executes the program, the RAM used as a working storage area, the display , which is a liquid crystal display, etc.
FIG. 3
3
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As shown in , the musical score information storage means includes a page information storage means and a part information storage means . The part information storage means is arranged as a sequence of part information corresponding to the number of parts. Each part information includes tone, reproduction volume, reproduction localization information, etc. In the page information storage means is recorded a sequence of page information corresponding to the number of pages. Each page information includes at least a paragraph information storage means and a staff information storage means .
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a
a,
In the paragraph information storage means is recorded a sequence of paragraph-belonging symbol information that is effective in common for all parts or all staffs belonging to a certain paragraph. For a paragraph-belonging symbol recorded in each paragraph-belonging symbol information a symbol ID unique within the paragraph, symbol category information, symbol type information, parameter sequence in accordance with the symbol category and symbol type, position of symbol within a page, etc., are included. As examples of symbol categories, repeat sign, bar line, etc., can be cited. As examples of symbol types, D.C., D.S., repeat bracket, etc., (category=repeat sign), and single line, double line, begin repeat sign, end repeat sign, double bar line, etc., (category=bar line) can be cited.
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a
a
b,
b
In the staff information storage means is recorded a sequence of staff information corresponding to the number of staffs within a page. Each staff information includes a belonging part ID, belonging paragraph ID, staff-belonging symbol information etc. The staff-belonging symbol information is recorded as a sequence of staff-belonging symbols belonging to the corresponding staff
36
b,
For a staff-belonging symbol recorded in the staff-belonging symbol information a symbol ID unique within the paragraph, symbol category information, symbol type information, OnTime, GateTime, parameter sequence in accordance with the symbol category and symbol type, position of symbol within a page (coordinates having an upper left position of a page as an origin), etc., are included.
As examples of symbol categories, note, rest, time signature, clef, key signature, accidental, etc., can be cited.
As examples of symbol type information, whole note, quarter note, eighth note, sixteenth note, thirty-second note, etc., (category=note), whole note rest, quarter rest, eighth rest, sixteenth rest, thirty-second rest (category=rest), and treble clef, bass clef, etc., (category=clef) can be cited.
As examples of parameters, musical interval (Note No. in MIDI), number of dots, ChordID, TimingNo., beam ID etc., (category=note), number of dots, ChordID, TimingNo., etc., (category ID=rest), ChordID (ID of group of notes sounded at the same timing), TimingNo. (number indicating order of sound emission), GateTime (value indicating duration of a note or rest), OnTime (time from head of measure to start of sound emission), etc., can be cited.
FIG. 4
4
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402
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a
As shown in , the playing information preparing means includes a prescribed measure time calculating means calculating a prescribed measure time from the meter of a musical composition, a measure playing time calculating means calculating a measure playing time from sound emission timings and note values (GateTime) of notes and rests within a measure, a comparing means comparing the calculated prescribed measure time and measure playing time, and a note value correcting means inferring that a tuplet is present within the measure if the prescribed measure time and the measure playing time are not matched and correcting the sound emission timings and note values of the notes and rests.
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a
a
a
The note value correcting means includes a reference time setting means setting a reference time that is the time of a single beat in the musical score, a measure note sequence recording means storing a note sequence within the measure, a grouping means grouping the notes within the measure according to each beat, and a tupleting process means performing a tupleting process of changing the note values of the grouped notes based on a relationship of the playing time of the grouped notes and the reference time of a single beat that is calculated from the note sequence.
FIG. 5
An overall procedure for musical score preparation by the musical score playing device shall now be described with reference to the flowchart of .
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First, the musical score information is generated by the musical score information generating means (step ). The musical score information generating means reads a musical score file from the musical score file storage means and from the writing information contained in the file, the page information, part information, paragraph information, paragraph-belonging symbol information, staff information, and staff-belonging symbol information are generated in accordance with generally-known conventional arts. However, the ChordID, TimingNo., GateTime, and OnTime of the staff-belonging symbol information are provided by the playing information preparing means .
3
The generated musical score information is recorded in the musical score information storage means .
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Thereafter, the playing information is prepared by the procedure of step to step by the playing information generating means .
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FIG. 6
In step , a measure information sequence, such as that shown in , is prepared. Each measure information includes a page number, a paragraph number, the symbol ID of a left bar line, the symbol ID of a right bar line, a prescribed measure time determined from the meter of musical composition, and a measure number corresponding to a measure.
The prescribed measure time is determined by formula (1).
Prescribed measure time=TimeBase×4/Den×Num Formula (1)
480
Here, TimeBase is the number of ticks per quarter note and this shall be in the present embodiment.
401
Den indicates the denominator (length of one beat) of the meter of the musical composition and Num indicates the numerator (number of beats within a measure) of the musical composition meter (prescribed measure time calculating means ).
43
Thereafter, the ChordID, which indicates a group of notes sounded at the same timing, is added (step ). The ChordID is determined by the positions of the notes in the lateral direction and whether or not the notes are in contact with the same stem.
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The TimingNo., which indicates the order of sound emission, is added (step ). The TimingNo. is determined by the ChordID and the position in the lateral direction.
45
The GateTime and OnTime are added (step ). The GateTime is determined by the type of note and number of dots, and the OnTime is determined by the GateTime and TimingNo.
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Step to step are performed in accordance with generally-known conventional arts.
An example of the playing information, ChordID, TimingNo., GateTime, and
FIG. 7
FIG. 8
45
OnTime, obtained for the musical score of by the procedure up to step is shown in . In regard to “Note No.,” numbers are assigned successively from the left to right of an upper staff of the score and then from the left to right of a lower staff. “Category” indicates whether a symbol is a note or a rest. With the present musical score, the symbols are indicated as notes No. 1 to No. 10 from the left side of the upper staff and as notes No. 11 to No. 16 from the left side of the lower staff.
In the present embodiment, time, such as the GateTime, OnTime, etc., is expressed using ticks. A length of a quarter note is defined as 480 ticks.
FIG. 7
FIG. 9
A procedure for calculating the OnTime and the measure playing time shall now be described using the musical score of as an example and with reference to the flowchart of .
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The TimingNo. is initialized to 1 (step ).
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An EndTime sequence is initialized (step ).
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The note No. 1 for which TimingNo.=1 is read (step ).
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The OnTime of the note No. 1 is calculated (step ). OnTime=0 because the TimingNo. is 1.
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The EndTime of the note No. 1 is calculated (step ). The GateTime of the note No. 1 is added to the OnTime of the note No. 1 and thus EndTime=240. This EndTime is added to the EndTime sequence.
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A return to step is performed because a note with the same TimingNo. is present (step ).
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The note No. 11 is read (step ).
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The OnTime of the note No. 11 is calculated (step ). OnTime=0 because the TimingNo. is 1 for this note as well.
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The EndTime of the note No. 11 is calculated (step ). The GateTime of the note No. 11 is added to the OnTime of the note No. 11, and therefore EndTime=480.
The EndTime sequence contains only 240 and does not contain 480, and therefore 480 is also added to the EndTime sequence.
The reading of notes for which TimingNo.=1 is finished, and therefore the TimingNo. is renewed to 2 and the note of the next TimingNo. is read.
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The note No. 2 is read (step ) and the OnTime is calculated (step ). The OnTime of the note No. 2 is the shortest time in the EndTime sequence. In the present case, it is .
95
The EndTime of the note No. 2 is calculated (step ). The GateTime of the note No. 2 is added to the OnTime of the note No. 2, and therefore EndTime=480.
480 is already present in the EndTime sequence, and therefore 480 is not added to the EndTime sequence. Further, there is no other note for which TimingNo.=2, and therefore 240, currently assigned to the OnTime of the note No. 2, is deleted from the EndTime sequence.
The TimingNo. is renewed to 3 and the note of the next TimingNo. is read.
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The note No. 3 is read (step ) and the OnTime is calculated (step ). The OnTime of the note No. 3 is the shortest value in the EndTime sequence. Here, it is 480.
95
The EndTime of the note No. 3 is calculated (step ). The GateTime of the note No. 3 is added to the OnTime of the note No. 3 and the EndTime is thus 720. This value is not present in the EndTime sequence, and therefore it is added to the EndTime sequence. This time, there is still present another note for which the TimingNo. is 3, and therefore the TimingNo. is not renewed and 480 in the EndTime sequence is not deleted from the EndTime sequence.
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The note No. 12 of the same TimingNo. is read in step and the OnTime is calculated (step ). The OnTime of the note No. 12 takes on the value of 480, which is the shortest value in the EndTime sequence.
95
The EndTime of the note No. 12 is calculated (step ). For the note No. 12, OnTime+GateTime=960. This value is not present in the EndTime sequence, and therefore it is added to the EndTime sequence. There is no other note for which TimingNo.=3, and therefore the smallest value in the EndTime sequence is deleted.
By performing the same process, the calculation of the OnTime and renewal of the EndTime sequence are performed for all notes. The largest value remaining in the EndTime sequence when the process is finished for all notes is the measure playing time.
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A characteristic arrangement of the present invention is that tuplet inference in the musical score is performed in step following step to correct the ChordID, TimingNo., GateTime, and OnTime. This portion shall be described in detail later.
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The playing information is prepared from the musical score information for which the tuplet inference was performed in step (step ). The playing information conforms to the MIDI standard format and is arranged from a sequence of the following playing event information.
Note event: Sound emission starting and sound emission stopping events of a note
Control event: Events of setting the volume, localization, etc., in a sound emission channel
Tone event: Event of designating the tone of a sound emission channel
Tempo event: Event of setting a tempo of the musical composition
FIG. 10
A format of the playing event information is shown in .
An event type is a number that identifies an event as a note event, control event, tempo event, etc.
1
Data contains a number that identifies the musical interval in the case of a note event or identifies the volume, localization, etc., in the case of a control event or is a tempo value in the case of a tempo event.
2
Data contains a sound emission strength (with 0 indicating stoppage of sound emission) in the case of a note event and set values of volume, localization, etc., in the case of a control event.
The time information contains the time (ticks) from the start of the musical composition to the generation of the event.
The channel number contains the channel number subject to control of sound emission, volume, etc.
46
FIG. 5
FIG. 11
A general flow of step (tuplet inference) shown in shall now be described with reference to the flowchart of .
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First, the measure information prepared in step is read (step ).
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The note information contained in the measure is read and stored in the note sequence (step ).
The reading concerning notes is performed as follows.
The page information is read based on the page number stored in the measure information.
From the page information, the paragraph information indicated by the paragraph number in the measure information is read.
From the paragraph number, the paragraph-belonging symbol matching the symbol ID of the left bar line in the measure information is read and the lateral direction position of this symbol is set as a measure left end position. Similarly, the paragraph-belonging symbol of the right bar line is read and its position set as a measure right end position.
From the page information, the staff information matching the paragraph number and belonging paragraph ID in the measure information is read. The staff-belonging symbols positioned between the measure left end position and the measure right end position and belonging to the note or rest category are stored in the measure note sequence.
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The measure playing time is calculated from the measure note sequence (step ).
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The measure playing time and the prescribed measure time in the measure information are compared (step ) and if the two are not matched, tuplet inference is performed (step ). If the measure playing time and the prescribed measure time are matched in step , transition to processing of the next measure is performed (step ).
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FIG. 11
FIG. 14
FIG. 13
FIG. 12
A detailed procedure of the tuplet inference in step shown in shall now be described for a case of preparing the playing data shown in from the musical score of with reference to the flowchart of .
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a
FIG. 13
First, the reference time is set by the reference time setting means of the note value correcting means (step ). The reference time is the time of one beat. The musical composition example of is in 4/4 meter, and therefore one beat is a quarter note and is 480 ticks.
In a case of 6/8 meter, etc., the reference time is set to the length of three eighth notes. In this case, the reference time is 720 ticks.
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The TimingNo. sequence is prepared (step ). Here, the numbers 1 to 16, corresponding to the upper staff notes No. 1 to No. 12 and the lower staff notes No. 13 to No. 16, are entered.
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Initialization or renewal of the index of the TimingNo. is performed (step ). At the very beginning, the TimingNo. index is 0.
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The notes of the TimingNo. indicated by the TimingNo. index are read (step ). At this point, the notes of the notes No. 1 and No. 13 are read.
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Of the notes read, the note of smaller note value (GateTime) is selected (step ). Here, the note No. 1 is selected.
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The selected note is added to a beat group (step ).
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Whether or not grouping is completed is judged according to predetermined conditions described later, and if grouping is completed, the next step is entered while if grouping is not completed, a return to step is performed (step ).
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The conditions of completion of grouping in step are as follows.
In the case of a note
There is a beam.
The note is the last note in the measure (A).→Grouping is completed.
Cases besides the above
There is a note of the same beam at a later timing (B).→Continue grouping.
There is no note of the same beam at a later timing (C).→Grouping is completed.
There is no beam (D).→Grouping is completed.
In the case of a rest
The note is the first note of the beat group (E).→Grouping is competed.
Cases besides the above
The note is the last note in the measure (F).→Grouping is completed.
Cases besides the above
A note belonging to the beam and preceding the rest is present in the beat group.
The same beam as that of the note is present after the rest (G).→Continue grouping.
Cases besides the above (H)→Grouping is completed.
Cases besides the above (I)→Continue with grouping.
FIG. 13
In performing grouping on the musical score shown in , first the grouping concerning the note No. 1 is judged. The note No. 1 is a note with a beam and corresponds to a case other than the case of the last note in the measure, a note of the same beam is present at a later timing, and therefore the present case corresponds to the case (B) given above and grouping is continued.
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a
Thereafter, the TimingNo. index is renewed (step ) and the note No. 2 of the TimingNo. 2 is read (step ). There is only one note corresponding to the TimingNo. 2, and therefore the note No. 2 is selected (step ) and added to the beat group (step ). The note No. 2 is a note with a beam and corresponds to a case other than the case of the last note in the measure, there is no note of the same beam at a later timing, and therefore the condition of the case (C) given above applies and grouping is completed in step . The grouping means is arranged from step to step .
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Thereafter, the measure playing time calculating means is used to calculate the playing time of the beat group (step ). In the present case, there are two eighth notes and the playing time is thus 480 ticks.
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The calculated playing time (480 ticks) of the beat group and the reference time (480 ticks, because the present musical score is in 4/4 meter and one beat is a quarter note) are then compared (step ). With the present beat group, the playing time and the reference time are equal and the beat group process is thus completed upon judging that the group is not a tuplet. If the beat group process is completed, the beat group is initialized (step ) and transition to the process of step is performed for grouping of the next beat group.
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Thereafter, the TimingNo. index is renewed (step ) and the notes No. 3 and No. 14 of the TimingNo. 3 are read (step ). The note No. 3, which is smaller in note value, is selected (step ) and added to the beat group (step ). The note No. 3 is a note with a beam and corresponds to a case other than the case of the last note in the measure, there is a note of the same beam at a later timing, and therefore in step , transition to the process of step is performed by the condition of (B).
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The TimingNo. index is renewed (step ) and the note No. 4 of the Timing No. 4 is read (step ) and added to the beat group (step ). The note No. 4 is a note with a beam and corresponds to a case other than the case of the last note in the measure, there is a note of the same beam at a later timing, and therefore in step , transition to the process of step is performed by the condition of (B).
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The TimingNo. index is renewed (step ) and the note No. 5 of the Timing No. 5 is read (step ) and added to the beat group (step ). The note No. 5 is a note with a beam and corresponds to a case other than the case of the last note in the measure, there is no note of the same beam at a later timing, and therefore in step , the condition of (C) applies and the grouping is completed.
68
Thereafter, the playing time of the beat group is calculated (using the measure playing time calculating means) (step ). In the present case, there are three sixteenth notes and the playing time is thus 360 ticks.
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The calculated playing time (360 ticks) of the beat group and the reference time (480 ticks, because the present musical score is in 4/4 meter and one beat is a quarter note) are then compared (step ). With the present beat group, the playing time and the reference time are not equal, and therefore step is entered.
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In step , it is judged whether or not the playing time (360 ticks) of the beat group is shorter than the reference time (480 ticks), and if the playing time is shorter than the reference time, a process for the shorter case is performed (step ).
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240
In step , a process for a case where the playing time of the beat group is an eighth note () is performed.
In the present embodiment, a case of a triplet of sixteenth notes, which is often used in musical compositions, shall be described. Here, the process is performed according to conditions such as the following.
(P) The number of notes in the beat group is 3.
An eighth note processing flag is inverted.
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The tupleting process is performed (step ), the beat group is initialized (step ), and step is entered. The detailed procedure of the tupleting process in step shall be described later.
(Q) The number of notes in the beat group is 2 and the playing time is equal to an eighth note.
The eighth note processing flag is inverted.
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The beat group initializing process is performed (step ) and step is entered.
(R) The number of notes in the beat group is one, the note is an eighth note, and the eighth note processing flag is true.
The eighth note processing flag is set to false.
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The beat group initializing process is performed (step ), and step is entered.
(S) Case not corresponding to any of (P) to (R)
63
Step is entered without performing the beat group initializing process and the next note is added to the beat group.
Although with the present embodiment, the case where there are three sixteenth notes was described, the same process is also performed in a case where there are five or seven thirty-second notes, a case where there are ten sixty-fourth notes, etc.
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FIG. 15
The process of the eighth note processing flag in step is performed to establish a solitary eighth rest after a beam, as in X and Y in , etc., as a beat group.
FIG. 13
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In the present case (note Nos. 3, 4, and 5 in the musical score of ), the condition (number of notes is 3) of (P) above applies, and therefore the eighth note processing flag is inverted (set to false in the present case), the tupleting process is performed (step ), the beat group is initialized (step ), and step is entered.
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404
In step , in which the tupleting process is performed by means of the note value correcting means , the GateTime of the grouped notes are changed (tupleted) by the following procedure.
First, the notes of the same GateTime in the beat group are grouped together as a tuplet group.
The total GateTime value of all notes of the tuplet is set as follows in accordance with the number of notes in the tuplet group.
The number of notes is 3→2 times the GateTime of each note of the tuplet group
The number of notes is 5 to 7→4 times the GateTime of each note of the tuplet group
The number of notes is 9 to 15→8 times the GateTime of each note of the tuplet group
The number of notes is 17 to 31→16 times the GateTime of each note of the tuplet group
Cases where the number of notes is 2, 4, 8, or 16 are exempt from the tupleting process because the note value can be expressed by a normal note (a note that is not a tuplet) in these cases.
In a case of a triple system of a 6/8 meter, etc., the total GateTime value of all notes of the tuplet is set as follows.
The number of notes is 2→1 time the GateTime of each note of the tuplet group
The number of notes is 4 to 5→2 times the GateTime of each note of the tuplet group
The number of notes is 7 to 11→4 times the GateTime of each note of the tuplet group
The number of notes is 12 to 23→8 times the GateTime of each note of the tuplet group
The number of notes is 25 to 47→16 times the GateTime of each note of the tuplet group
In cases of a triple system of a 6/8 meter, etc., cases where the number of notes is 3, 6, 9, or 24 are exempt from the tupleting process because the note value can be expressed by a normal note (a note that is not a tuplet) in these cases.
The GateTime of each note of the tuplet group is calculated by the following formulae.
The GateTime of a note other than the last note of the tuplet group is calculated by formula (2).
GateTime=Total GateTime÷Number of notes in tuplet Formula (2)
The GateTime of the last note of the tuplet group is calculated by formula (3).
GateTime=Total GateTime−(Total GateTime÷Number of notes in tuplet)×(Number of notes in tuplet−1) Formula (3)
The GateTime of just the last note is calculated by formula (3) to accommodate for a case where the total GateTime is not evenly divisible by the number of notes making up the tuplet group.
In the present case, the note No. 3 to note No. 5 are grouped together in a tuplet group and tupleted.
That is, the note No. 3 to note No. 5 are sixteenth notes, and therefore the GateTime (before conversion) of the tuplet group is 120, and the total GateTime, by the calculation method described above, is 240, which is 2 times the GateTime of each note of the tuplet group, because the number of notes is 3. Also, by formula (2) and formula (3), the GateTime of each note after the change is 80.
74
63
After the tupleting process has been performed, the beat group is initialized (step ) and step is entered to perform the process for the next note.
64
66
In the same manner as in the procedure up to now, the note No. 6 and the note No. 7 are read (step ) and registered in the beat group (step ).
67
68
240
After grouping is completed (step ), the playing time of the beat group is calculated (using the measure playing time calculating means) in step . In the present case, there are two sixteenth notes in the beat group and the playing time is thus .
240
480
72
71
The playing time () of the beat group is shorter than the reference time (), and therefore transition to the process of step is performed (step ).
72
75
63
With the process for the shorter case (step ), the condition (the number of notes in the beat group is 2 and the playing time is equal to an eighth note) of (Q) described above applies, and therefore tupleting is not performed, the beat group is initialized (step ), and transition to the process of step is performed.
64
65
66
Thereafter, the note No. 8 and note No. 16 are read (step ). The note No. 8 is selected in step and added to the beat group in step .
67
63
The condition (B) applies in step , and therefore transition to the process of step is performed and the next note is read.
64
66
67
63
The note No. 9 is read (step ) and added to the beat group (step ). The condition (G) applies in step , and therefore transition to the process of step is performed and the next note is read.
64
66
67
The note No. 10 is read (step ) and added to the beat group (step ). The condition (C) applies in step , and therefore the grouping is ended.
68
In step , the playing time of the beat group is calculated. In this case, the playing time is 240×3 and thus 720.
71
69
The playing time and the reference time are compared and transition to step is performed because these are not equal (step ).
71
720
480
76
In step , the playing time () and the reference time () are compared and transition to step is performed because the playing time is longer than the reference time.
76
73
In step , it is checked whether or not the beat group contains a note longer than the reference time (quarter note). There is no such note in the present case, and therefore the tupleting process is performed (step ).
73
480
240
74
63
In the tupleting process (step ), the note No. 8 to note No. 10 are grouped together as a tuplet group. The notes in the tuplet group are eighth notes, the total GateTime is that for the case where the number of notes is 3 and is thus , which is 2 times an eighth note (), and by formula (2) and formula (3), the GateTime of each note in the tuplet group after is 160. When the tupleting process is ended, the beat group is initialized (step ) and a return to step is performed.
64
66
67
68
240
Note No. 11 is read (step ) and added to the beat group (step ). In step , grouping is completed in accordance with the condition (D), and the playing time of the beat group is calculated in step . In the present case, the playing time is .
240
480
69
71
The playing time () and the reference time () are compared in step and step is entered because the two are not equal.
71
240
480
72
In step , the playing time () and the reference time () are compared and step is entered to perform the process for the shorter case because the playing time is shorter than the reference time.
72
63
In step , the condition (none of (P) to (R) applies) of (S) applies, and therefore step is entered.
64
66
67
Note No. 12 is read (step ) and added to the beat group (step ). In step , grouping is completed in accordance with the condition (F).
68
480
The playing time of the beat group is calculated in step . In the present case, the playing time is .
480
480
69
70
63
The playing time () and the reference time () are compared in step and the beat group is initialized (step ) and step is entered because the playing time and the reference time are equal.
63
FIG. 13
FIG. 13
FIG. 16
In step , the process is ended because the process has been completed for all Timing No. (No. 1 to 12) in the musical score of . It can be understood that when tuplet numbers are correctly expressed in the musical score of by performing the tupleting process, the musical score will be as shown in .
81
82
FIG. 12
FIG. 15
Also, step and step in the flowchart of is for accommodating a musical score (dotted note) such as that of Z in .
720
76
71
720
480
76
81
720
480
When a dotted quarter note is read into the beat group, the playing time of the doted quarter note is , which is 1.5 times the playing time of a quarter note. Step is entered from step because the playing time () is longer than the reference time (). From step , step is entered because there is a note () that is longer than the reference time ().
81
720
480
82
82
720
480
63
720
480
In step , the playing time () and the reference time () are compared, and step is entered because the playing time is not twice the reference time. In step , the playing time () and the reference time () are compared, and step is entered to renew the TimingNo. because the playing time () is less than twice the reference time ().
64
67
68
In step , the next eighth note is read and added to the beat group. In step , grouping is completed because the condition (D) applies, and the playing time is calculated in step . In the present case, the playing time is 960 (720+240).
69
81
960
480
81
70
By step to step , the playing time () is twice the reference time (), and therefore from step , step is entered and the process is finally ended without performing tupleting.
With the tupleting process described above, the tupleting process of changing the note values of the grouped notes is performed in the following cases (1) to (3).
76
(1) If the playing time of the grouped notes is longer than the reference time of one beat and there is no note longer than the reference time within the group (in the case of No in step )
72
(2) If the playing time of the grouped notes is shorter than the reference time of one beat and the number of notes in the group is 3 (in the case where the condition (P) is met in step )
82
(3) If there is a note longer than the reference time among the grouped notes and the playing time exceeds twice the reference time (in the case of No in step )
Therefore, with the exception of a case where the playing time of the grouped notes is equal to the reference time of one beat, tuplet inference of a plurality of notes connected by a beam (grouped notes) can be performed to perform processing to correct note values in both the case where the playing time is shorter than the reference time and the case where the playing time is longer than the reference time. In this process, processing can be performed in accordance with any of various tuplets, such as a triplet, quintuplet, septuplet, decuplet, etc.
An embodiment of a musical score playing device that is specialized to triplets in preparing playing information having the correct note values and thereby enables automatic playing shall now be described.
FIG. 1
FIG. 2
FIG. 3
3
As with the musical score playing device described above, the musical score playing device that performs a tupleting process specialized to triplets is arranged from the respective elements of the block diagram of and its hardware arrangement is as shown in . Also, the musical score information storage means is arranged from the respective means of . The functions that the respective arrangements have are the same in content as those of the musical score playing device described above and description thereof shall thus be omitted.
FIG. 17
4
401
402
403
404
b
As shown in , the playing information preparing means of the musical score playing device that performs the tupleting process specialized to triplets includes the prescribed measure time calculating means calculating the prescribed measure time from the meter of a musical composition, the measure playing time calculating means calculating the measure playing time from the sound emission timings and note values (GateTime) of notes and rests within a measure, a first comparing means comparing the calculated prescribed measure time and measure playing time, and the note value correcting means inferring that a tuplet is present within the measure if the prescribed measure time and the measure playing time are not matched and correcting the sound emission timings and note values of the notes and rests.
404
441
442
442
443
b
b
b
b.
The note value correcting means includes a target note determining means that successively determines a correction target note and a tupleting process means changing the note value of each correction target note in the measure to ⅔ and performing a tupleting process. The tupleting process means includes a corrected playing time calculating means calculating the measure playing time from the changed note values, and the prescribed measure time and the corrected measure playing time that is in accordance with the changed note values are compared at a second comparing means
442
443
b
b
Arrangements are made so that the tupleting process by the tupleting process means and the comparison by the second comparing means are repeated for the respective correction target notes and the tupleting of a triplet with the changed note values of the respective notes is finalized when the prescribed measure time and the corrected measure playing time become equal.
4
FIG. 5
FIG. 9
Even in the musical score playing device that performs the tupleting process specialized to triplets, the respective processes of preparation of the playing information by the playing information preparing means () and the calculation of the playing time from the musical score () are performed.
FIG. 11
402
53
In performing the tuplet inference process specialized to triplets (), the measure playing time is calculated from the measure note sequence by the measure playing time calculating means (step ).
401
403
54
55
54
56
b
The measure playing time and the prescribed measure time calculated from the measure information by the prescribed measure time calculating means are compared at the first comparing means (step ), and if the two are not matched, tuplet inference is performed (step ). If the measure playing time and the prescribed measure time are matched in step , transition to processing of the next measure is performed (step ).
55
FIG. 11
FIG. 18
FIG. 20
FIG. 19
With the musical score playing device specialized to triplets, the procedure for tuplet inference differs from that of the musical score playing device described above. The detailed procedure for tuplet inference in step shown in shall now be described with reference to the flowchart of . A case where the playing information shown in is prepared from the musical score of shall be described as an example.
In performing the tupleting process, the tuplet inference is performed repeatedly while changing the target note to be tupleted in the order of a sixty-fourth note, thirty-second note, sixteenth note, eighth note, quarter note, and half note.
The tuplet inference is performed in the order from a sixty-fourth note to a half note because normally in a musical composition, tuplets of notes of small note value tend to be used more frequently than tuplets of notes of large note value.
Also, for a single target note, the tuplet inference is performed twice, that is, once for a case where the note value is changed to that which is one step greater (for example, from a quarter note to a half note) and once for a case where the note value is not changed. By changing the note value to that which is one step greater than that of the target note (for example, from a quarter note to a half note), a triplet arranged from different notes can be judged.
FIG. 19
With the musical score of , there are no applicable notes for cases where the target note is a sixty-fourth note to an eighth note, and therefore the first note that is made a target note is a quarter note.
101
1920
First, the prescribed measure time is calculated (step ). The present musical score is in 4/4 meter and, by formula (1) described above, the prescribed measure time is .
441
102
442
320
103
b
b,
The note that is to be the target note is determined as a quarter note by the target note determining means (step ) and, by the tupleting process means the note value 480 of all quarter notes in the measure is changed to a note value of ⅔, that is, to (step ).
104
105
640
If the change of note values of the target notes is performed for the first time (step ), the note value of a half note, which has a note value that is one step above that of a quarter note, is also changed (step ). The note value of a half note is converted to , which is a note value of ⅔ of the note value 960.
442
106
960
640
1600
b
The measure playing time for the corrected note values is calculated by the corrected playing time calculating means of the tupleting process means (step ). In this case, there are three note values of 320, which makes , and is added thereto so that the measure playing time is .
1600
1920
443
107
108
1600
1920
b
The measure playing time () and the prescribed measure time () are compared by the second comparing means (step ). Step is entered because the measure playing time () does not match the prescribed measure time ().
108
109
103
In step , the change of note value is performed for the first time, and therefore step is entered, the changed note values are returned to the original values and then step is entered.
103
106
105
960
960
1920
1920
107
In step , the note values are changed again from 480 to 320. This is the second time that the note values are changed, and therefore the measure playing time is calculated by the corrected playing time calculating means in step without performing step . In this case, there are three note values of 320, which makes , and (the unchanged note value) is added thereto so that the measure playing time is and the process is ended because this matches the prescribed measure time () (step ).
107
110
108
102
111
In a case of a musical score for which the playing time and the prescribed measure time are not matched in step of the second time, step is entered from step to judge whether or not the process has been completed for all of the types of notes, and if it has been completed, the process is ended. If the process has not been completed for all of the types of notes, the changed note value is returned to the original value and a transition to step is performed (step ) to change the target note and perform the above process again.
FIG. 19
FIG. 20
FIG. 21
With the musical score of , the presence of a triplet is determined by the changing of the note values of the quarter notes, and the playing information of , resulting from tupleting processing by change of the note values of the note No. 1 to note No. 3, is prepared. A musical score, with which the correct tuplet is indicated, is that in which the tuplet symbol “3” is indicated at the triplet of quarter notes as shown in .
FIG. 22
FIG. 23
A case where tupleting is performed on the musical score of to prepare the playing information of shall now be described.
There are no applicable notes for cases where the target note is a sixty-fourth note to an eighth note, and therefore the first note that is made a target note is a quarter note with the present musical score as well.
101
1920
First, the prescribed measure time is calculated (step ). The present musical score is in 4/4 meter and, by formula (1) described above, the prescribed measure time is .
102
480
320
103
The note that is to be the target note is determined as a quarter note (step ) and the note value of all quarter notes in the measure is changed to a note value of ⅔, that is, to (step ).
104
105
640
If the change of note value of the target notes is performed for the first time (step ), the note value of a half note, which has a note value that is one step above that of a quarter note, is also changed (step ). The note value of a half note is converted to , which is a note value of ⅔ of the note value 960.
442
106
b
The measure playing time with the corrected note values is calculated by the corrected playing time calculating means of the tupleting process means (step ).
1280
640
1920
In this case, there are four note values of 320, which makes , and is added thereto so that the measure playing time is .
1920
1920
107
1920
1920
107
The measure playing time () and the prescribed measure time () are compared (step ). The measure playing time () matches the prescribed measure time () (step ), and therefore the process is ended.
FIG. 22
FIG. 23
With the musical score of , the tupleting-processed playing information of is prepared by the change of the note values of the note No. 1 to note No. 5, and therefore the presence of two triplets is determined by the changing of the note values of the four quarter notes and the half note.
FIG. 24
Therefore with a musical score, with which the correct tuplets are indicated, the tuplet symbol “3” is indicated at the triplet of the half note and the quarter note and at the subsequent triplet of quarter notes as shown in .
By the tupleting process described above, the tupleting process specialized to triplets, which are most frequently used in musical scores, can be performed and inference of a triplet of a half note that is not joined by a beam or a triplet of quarter notes, etc., is enabled.
BRIEF DESCRIPTION OF DRAWINGS
FIG. 1
[] is a block diagram of functions of an automatic playing device incorporating a musical score playing device according to an embodiment of the present invention.
FIG. 2
[] is a block diagram of a hardware arrangement example of the automatic playing device.
FIG. 3
[] is a model diagram for describing details of information stored by a musical information storage means.
FIG. 4
[] is a functional block diagram of an arrangement of a playing information preparing means.
FIG. 5
[] is a flowchart of a procedure for preparing playing information by the playing information preparing means.
FIG. 6
[] is a model diagram of an example of a measure information sequence.
FIG. 7
[] is an example of a musical score.
FIG. 8
FIG. 7
[] is a table of playing information obtained from respective notes of the musical score example of .
FIG. 9
[] is a flowchart of a procedure for calculating a measure playing time from a musical score.
FIG. 10
[] is a model diagram of an example of event types.
FIG. 11
[] is a flowchart of an overall flow for performing tuplet inference.
FIG. 12
[] is a flowchart of a detailed procedure for performing tuplet inference.
FIG. 13
[] is an example of a musical score.
FIG. 14
[] is a table of playing information in a case of performing the tuplet inference process on the respective notes of the musical score example.
FIG. 15
[] is an example of a musical score.
FIG. 16
FIG. 13
[] is an example of a musical score in which correct tuplet symbols are indicated for the musical score example of .
FIG. 17
[] is a functional block diagram of an arrangement of a playing information preparing means.
FIG. 18
[] is a flowchart of a detailed procedure for performing tuplet inference.
FIG. 19
[] is an example of a musical score.
FIG. 20
[] is a table of playing information in a case of performing the tuplet inference process on the respective notes of the musical score example.
FIG. 21
FIG. 19
[] is an example of a musical score in which correct tuplet symbols are indicated for the musical score example of .
FIG. 22
[] is an example of a musical score.
FIG. 23
[] is a table of information in a case of performing the tuplet inference process on the respective notes of the musical score example.
FIG. 24
FIG. 22
[] is an example of a musical score in which correct tuplet symbols are indicated for the musical score example of .
FIG. 25
[] is an example of a musical score. | |
Q:
Haskell and Quadratics
I have to write a program to solve quadratics, returning a complex number result.
I've gotten so far, with defining a complex number, declaring it to be part of num, so +,- and * - ing can take place.
I've also defined a data type for a quadratic equation, but im now stuck with the actual solving of the quadratic. My math is quite poor, so any help would be greatly appreciated...
data Complex = C {
re :: Float,
im :: Float
} deriving Eq
-- Display complex numbers in the normal way
instance Show Complex where
show (C r i)
| i == 0 = show r
| r == 0 = show i++"i"
| r < 0 && i < 0 = show r ++ " - "++ show (C 0 (i*(-1)))
| r < 0 && i > 0 = show r ++ " + "++ show (C 0 i)
| r > 0 && i < 0 = show r ++ " - "++ show (C 0 (i*(-1)))
| r > 0 && i > 0 = show r ++ " + "++ show (C 0 i)
-- Define algebraic operations on complex numbers
instance Num Complex where
fromInteger n = C (fromInteger n) 0 -- tech reasons
(C a b) + (C x y) = C (a+x) (b+y)
(C a b) * (C x y) = C (a*x - b*y) (b*x + b*y)
negate (C a b) = C (-a) (-b)
instance Fractional Complex where
fromRational r = C (fromRational r) 0 -- tech reasons
recip (C a b) = C (a/((a^2)+(b^2))) (b/((a^2)+(b^2)))
root :: Complex -> Complex
root (C x y)
| y == 0 && x == 0 = C 0 0
| y == 0 && x > 0 = C (sqrt ( ( x + sqrt ( (x^2) + 0 ) ) / 2 ) ) 0
| otherwise = C (sqrt ( ( x + sqrt ( (x^2) + (y^2) ) ) / 2 ) ) ((y/(2*(sqrt ( ( x + sqrt ( (x^2) + (y^2) ) ) / 2 ) ) ) ) )
-- quadratic polynomial : a.x^2 + b.x + c
data Quad = Q {
aCoeff, bCoeff, cCoeff :: Complex
} deriving Eq
instance Show Quad where
show (Q a b c) = show a ++ "x^2 + " ++ show b ++ "x + " ++ show c
solve :: Quad -> (Complex, Complex)
solve (Q a b c) = STUCK!
EDIT: I seem to have missed out the whole point of using my own complex number datatype is to learn about custom datatypes. I'm well aware that i could use complex.data. Any help that could be given using my solution so far would be greatly appreciated.\
EDIT 2: It seems that my initial question was worded horribly. I'm aware that the quadratic formula will return both (or just the one) root to me. Where I am having trouble is returning these roots as a (complex, complex) tuple with the code above.
I'm well aware that I could use the built in quadratic functions as have been displayed below, but this is not the exercise. The idea behind the exercise, and creating ones own complex number data type, is to learn about custom data types.
A:
Like newacct said, it's just the quadratic equation:
(-b +- sqrt(b^2 - 4ac)) / 2a
module QuadraticSolver where
import Data.Complex
data Quadratic a = Quadratic a a a deriving (Show, Eq)
roots :: (RealFloat a) => Quadratic a -> [ Complex a ]
roots (Quadratic a b c) =
if discriminant == 0
then [ numer / denom ]
else [ (numer + root_discriminant) / denom,
(numer - root_discriminant) / denom ]
where discriminant = (b*b - 4*a*c)
root_discriminant = if (discriminant < 0)
then 0 :+ (sqrt $ -discriminant)
else (sqrt discriminant) :+ 0
denom = 2*a :+ 0
numer = (negate b) :+ 0
in practice:
ghci> :l QuadraticSolver
Ok, modules loaded: QuadraticSolver.
ghci> roots (Quadratic 1 2 1)
[(-1.0) :+ 0.0]
ghci> roots (Quadratic 1 0 1)
[0.0 :+ 1.0,(-0.0) :+ (-1.0)]
And adapting to use your terms:
solve :: Quad -> (Complex, Complex)
solve (Q a b c) = ( sol (+), sol (-) )
where sol op = (op (negate b) $ root $ b*b - 4*a*c) / (2 * a)
Although I haven't tested that code
A:
Since Haskell's sqrt can also handle complex numbers, rampion's solution can even be further simplified:
import Data.Complex
-- roots for quadratic equations with complex coefficients
croots :: (RealFloat a) =>
(Complex a) -> (Complex a) -> (Complex a) -> [Complex a]
croots a b c
| disc == 0 = [solution (+)]
| otherwise = [solution (+), solution (-)]
where disc = b*b - 4*a*c
solution plmi = plmi (-b) (sqrt disc) / (2*a)
-- roots for quadratic equations with real coefficients
roots :: (RealFloat a) => a -> a -> a -> [Complex a]
roots a b c = croots (a :+ 0) (b :+ 0) (c :+ 0)
You can also use this croots function with your own datatype, if you change the types to fit your implementation (and call your root function instead of sqrt).
| |
Additional information:
a. An insurance policy examination showed $1,240 of expired insurance.
b. An inventory count showed $210 of unused shop supplies still available.
c. Depreciation expense on shop equipment, $350.
d. Depreciation expense on the building, $2,220.
e. A beautician is behind on space rental payments and $200 of accrued revenue was unrecorded at the time the trial balance was prepared.
f. $800 of the Unearned Rent account balance was earned by year-end.
g. The one employee, a receptionist, works a five-day workweek at $50 per day. The employee was paid last week but has worked four days this week for which she has not been paid.
h. Three months’ property taxes, totaling $450, have accrued. This additional amount of property taxes expense has not been recorded.
i. One month’s interest on the note payable, $600, has accrued but is unrecorded.
Required:
Based on the additional information, prepare the adjusting journal entries for Bella’s Beauty Salon.
|Rapid Car Services, Inc.
|
Adjusted Trial Balance
For the year ended December 31
|Cash||$33,000|
|Accounts receivable||14,200|
|Office supplies||1,700|
|Vehicles||100,000|
|Accumulated depreciation—Vehicles||45,000|
|Accounts payable||11,500|
|Common stock||1,000|
|Retained earnings||70,900|
|Dividends||40,000|
|Fees earned||155,000|
|Rent expense||13,000|
|Office supplies expense||2,000|
|Utilities expense||2,500|
|Depreciation Expense—Vehicles||15,000|
|Salary expense||50,000|
|Fuel expense||12,000|
|Totals||$283,400||$283,400|
Required:
Prepare the following financial statements for Rapid Car Services, Inc. from the adjusted trial balance. Assume the stockholders did not make any additional investments in the company during the year.
Income Statement
Statement of Retained Earnings
Balance Sheet
|Date||Activities||Units Acquired at Cost||Units Sold at Retail|
|April 1||Beginning Inventory||175 units @ $15.00|
|4||Purchase||150 units @ $16.00|
|7||Sales||160 units @ $30.00|
|10||Purchase||200 units @ $17.00|
|16||Sales||250 units @ $30.00|
|25||Purchase||160 units @ $18.00|
|28||Sales||150 units @ $32.00|
Required:
Determine the cost assigned to ending inventory and cost of goods sold using LIFO with the perpetual inventory system.
From the March 31 bank statement:
NSF: A check from a customer, Cook Co. in payment of their account.
IN: Interest earned on the account.
From the Edwards Company’s accounting records:
Required:
Based on the above information, prepare the2-column bank reconciliation for the Edwards Company for March.
|Administrative salaries expense||$135,000|
|Depreciation expense—Factory equipment||52,400|
|Depreciation expense—Delivery vehicles||36,200|
|Depreciation expense—Office equipment||24,800|
|Advertising expense||22,350|
|Direct labor||268,000|
|Factory supplies used||12,000|
|Income taxes expense||91,500|
|Indirect labor||35,000|
|Indirect material||24,000|
|Factory insurance||15,500|
|Factory utilities||14,000|
|Factory maintenance||7,500|
|Inventories|
|Raw materials inventory, January 1||32,000|
|Raw materials inventory, December 31||28,000|
|Work in Process inventory, January 1||33,780|
|Work in Process inventory, December 31||37,460|
|Finished goods inventory, January 1||56,970|
|Finished goods inventory, December 31||62,000|
|Raw materials purchases||325,000|
|Rent expense—Factory||50,000|
|Rent expense—Office space||24,000|
|Rent expense—Selling Space||24,000|
|Sales salaries expense||97,500|
|Sales||1,452,000|
|Sales discounts||29,000|
Required:
|Alternative 1||Alternative 2|
|Variable costs per unit||$20||?|
|Fixed costs||$200,000||$274,400|
|Selling price per unit||$40||$40|
|Income tax rate||25%||25%|
Required:
(a) Compute the break-even point in units and dollars for both alternatives.
(b) Prepare a forecasted income statement for both alternatives assuming that 30,000 units will be sold. The statements should report sales, total variable costs, contribution margin, fixed costs, income before taxes, income taxes, and net income.
(c) Compute the degree of operating leverage for each alternative. Which alternative would you recommend and why? | https://ecadimi.com/downloads/acc-20364-accounting-business-operations-final-examination/ |
Statement 1: Photons of the same frequency have the same energy and momentum.
Statement 2: When energy is incident on a metal, a part of absorbed energy is used to free the electron from the metal surface and the remaining is converted to thermal energy.
A. Both statements are true.
B. Only statement 2 is true.
C. Only statement 1 is true.
D. Both statements are false.
Asked by kushalv238 | 2nd Aug, 2021, 06:13: PM
Expert Answer:
Answer :- Only Statement-1 is true
Energy of photon = hν ; Momentum of photon p = h/λ = ( h ν ) / c
where h is planck's constant, ν is frequency , λ is wavelength and c is velocity of photon .
Hence photons of same frequency have same energy and momentum . | https://www.topperlearning.com/answer/statement-1-photons-of-the-same-frequency-have-the-same-energy-and-momentum-statement-2-when-energy-is-incident-on-a-metal-a-part-of-absorbed-energy-i%20/xexwquhh |
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).
Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique.
Much of analytic number theory was inspired by the prime number theorem. Let π(x) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The prime number theorem then states that x / ln(x) is a good approximation to π(x), in the sense that the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1:
In two papers from 1848 and 1850, the Russian mathematician Pafnuty L'vovich Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function ζ(s) (for real values of the argument "s", as are works of Leonhard Euler, as early as 1737) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π(x)/(x/ln(x)) as x goes to infinity exists at all, then it is necessarily equal to one. He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near to 1 for all x. Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π(x) were strong enough for him to prove Bertrand's postulate that there exists a prime number between n and 2n for any integer n ≥ 2.
""
"…it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the next objective of my investigation."
Riemann's statement of the Riemann hypothesis, from his 1859 paper. (He was discussing a version of the zeta function, modified so that its roots are real rather than on the critical line.)
Bernhard Riemann made some famous contributions to modern analytic number theory. In a single short paper (the only one he published on the subject of number theory), he investigated the Riemann zeta function and established its importance for understanding the distribution of prime numbers. He made a series of conjectures about properties of the zeta function, one of which is the well-known Riemann hypothesis.
Extending the ideas of Riemann, two proofs of the prime number theorem were obtained independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(s) is non-zero for all complex values of the variable s that have the form s = 1 + it with t > 0.
The biggest technical change after 1950 has been the development of sieve methods, particularly in multiplicative problems. These are combinatorial in nature, and quite varied. The extremal branch of combinatorial theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory, which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.
Developments within analytic number theory are often refinements of earlier techniques, which reduce the error terms and widen their applicability. For example, the circle method of Hardy and Littlewood was conceived as applying to power series near the unit circle in the complex plane; it is now thought of in terms of finite exponential sums (that is, on the unit circle, but with the power series truncated). The needs of diophantine approximation are for auxiliary functions that are not generating functions—their coefficients are constructed by use of a pigeonhole principle—and involve several complex variables. The fields of diophantine approximation and transcendence theory have expanded, to the point that the techniques have been applied to the Mordell conjecture.
Theorems and results within analytic number theory tend not to be exact structural results about the integers, for which algebraic and geometrical tools are more appropriate. Instead, they give approximate bounds and estimates for various number theoretical functions, as the following examples illustrate.
Euclid showed that there are infinitely many prime numbers. An important question is to determine the asymptotic distribution of the prime numbers; that is, a rough description of how many primes are smaller than a given number. Gauss, amongst others, after computing a large list of primes, conjectured that the number of primes less than or equal to a large number N is close to the value of the integral
In 1859 Bernhard Riemann used complex analysis and a special meromorphic function now known as the Riemann zeta function to derive an analytic expression for the number of primes less than or equal to a real number x. Remarkably, the main term in Riemann's formula was exactly the above integral, lending substantial weight to Gauss's conjecture. Riemann found that the error terms in this expression, and hence the manner in which the primes are distributed, are closely related to the complex zeros of the zeta function. Using Riemann's ideas and by getting more information on the zeros of the zeta function, Jacques Hadamard and Charles Jean de la Vallée-Poussin managed to complete the proof of Gauss's conjecture. In particular, they proved that if
This remarkable result is what is now known as the prime number theorem. It is a central result in analytic number theory. Loosely speaking, it states that given a large number N, the number of primes less than or equal to N is about N/log(N).
More generally, the same question can be asked about the number of primes in any arithmetic progression a+nq for any integer n. In one of the first applications of analytic techniques to number theory, Dirichlet proved that any arithmetic progression with a and q coprime contains infinitely many primes. The prime number theorem can be generalised to this problem; letting
There are also many deep and wide-ranging conjectures in number theory whose proofs seem too difficult for current techniques, such as the twin prime conjecture which asks whether there are infinitely many primes p such that p + 2 is prime. On the assumption of the Elliott–Halberstam conjecture it has been proven recently that there are infinitely many primes p such that p + k is prime for some positive even k at most 12. Also, it has been proven unconditionally (i.e. not depending on unproven conjectures) that there are infinitely many primes p such that p + k is prime for some positive even k at most 246.
One of the most important problems in additive number theory is Waring's problem, which asks whether it is possible, for any k ≥ 2, to write any positive integer as the sum of a bounded number of kth powers,
The case for squares, k = 2, was answered by Lagrange in 1770, who proved that every positive integer is the sum of at most four squares. The general case was proved by Hilbert in 1909, using algebraic techniques which gave no explicit bounds. An important breakthrough was the application of analytic tools to the problem by Hardy and Littlewood. These techniques are known as the circle method, and give explicit upper bounds for the function G(k), the smallest number of kth powers needed, such as Vinogradov's bound
Diophantine problems are concerned with integer solutions to polynomial equations: one may study the distribution of solutions, that is, counting solutions according to some measure of "size" or height.
An important example is the Gauss circle problem, which asks for integers points (x y) which satisfy
One of the most useful tools in multiplicative number theory are Dirichlet series, which are functions of a complex variable defined by an infinite series of the form
hence the coefficients of the product of two Dirichlet series are the multiplicative convolutions of the original coefficients. Furthermore, techniques such as partial summation and Tauberian theorems can be used to get information about the coefficients from analytic information about the Dirichlet series. Thus a common method for estimating a multiplicative function is to express it as a Dirichlet series (or a product of simpler Dirichlet series using convolution identities), examine this series as a complex function and then convert this analytic information back into information about the original function.
Euler showed that the fundamental theorem of arithmetic implies (at least formally) the Euler product
Euler's proof of the infinity of prime numbers makes use of the divergence of the term at the left hand side for s = 1 (the so-called harmonic series), a purely analytic result. Euler was also the first to use analytical arguments for the purpose of studying properties of integers, specifically by constructing generating power series. This was the beginning of analytic number theory.
Later, Riemann considered this function for complex values of s and showed that this function can be extended to a meromorphic function on the entire plane with a simple pole at s = 1. This function is now known as the Riemann Zeta function and is denoted by ζ(s). There is a plethora of literature on this function and the function is a special case of the more general Dirichlet L-functions.
In the early 20th century G. H. Hardy and Littlewood proved many results about the zeta function in an attempt to prove the Riemann Hypothesis. In fact, in 1914, Hardy proved that there were infinitely many zeros of the zeta function on the critical line
This led to several theorems describing the density of the zeros on the critical line.
On specialized aspects the following books have become especially well-known:
Certain topics have not yet reached book form in any depth. Some examples are (i) Montgomery's pair correlation conjecture and the work that initiated from it, (ii) the new results of Goldston, Pintz and Yilidrim on small gaps between primes, and (iii) the Green–Tao theorem showing that arbitrarily long arithmetic progressions of primes exist. | https://dir.md/wiki/Analytic_number_theory?host=en.wikipedia.org |
Museo Ferrari (previously known as Galleria Ferrari) is a Ferrari company museum dedicated to the Ferrari sports car marque. The museum is not purely for cars; also on view are trophies, photographs and other historical objects relating to the Italian motor racing industry. The exhibition also introduces technological innovations, some of which had made the transition from racing cars to road cars.
It is located just 300 metres from the Ferrari factory in Ferrari's home town of Maranello, near Modena, Italy.
The museum first opened in February 1990, with a new wing being added in October 2004. Ferrari itself has run the museum since 1995. The total surface area is now 2,500 square metres. The number of annual visitors to the museum is around 180,000.
The exhibits are mostly a combination of Ferrari road and track cars. Many of Ferrari's most iconic cars from throughout its history are present in the museum.(Wikipedia)
|Visitors||46|
|Oldest photo||09/14/2010|
|Newest photo||09/24/2011|
|Alternative titles||
"Ferrari 458 Italia"
|
"Ferrari F40"
"Ferrari 599 GTO"
"Ferrari California"
"Ferrari 458 Spider" | http://www.pointsonamap.com/attraction/box_0_33_11_45/ferrari/2 |
Garden Cottage is in Coldstream, an attractive small town in the Scottish Borders. It is easily accessible from Edinburgh and from the South via the main east coast route, A1.
From the South East: Travel north up the A1 to Berwick-Upon-Tweed. Take the first exit at the Berwick roundabout, signed Coldstream. Follow this road (A697) directly to Coldstream (approx. 15 miles). Drive through the town until you reach a bridge. Take the left turn immediately after the bridge and then the right turn immediately after that. Continue straight down a dirt road and take the right fork. You will see a stable block with a tower over the entrance; Garden Cottage is the only house outside of the stable block.
From the South West: Travel up the M6 to Carlisle, then take the A7 eastwards. At Hawick (pronounced Hoik!) take the A698 to Kelso, follow the bypass, then take a right at the mini-roundabout and follow signs to Coldstream. On the approach to Coldstream you will pass the Health Centre and Hirsel Craft Centre/Golf Course entrances on your left hand side, then you will see a bridge ahead. Turn right immediately before the bridge, and then right again immediately after that. Continue straight down a dirt road and take the right fork. You will see a stable block with a tower over the entrance; Garden Cottage is the only house outside of the stable block.
From Edinburgh and the North: Take the Edinburgh Bypass; at Sheriffhall roundabout take the A720 (new Dalkeith bypass), then take to first sliproad signed to Jedburgh (A68). Follow the A68 southbound to Carfraemill roundabout, there take the first exit onto the A697. Continue through Greenlaw, following signposts to Coldstream. You will pass the Health Centre and Hirsel Craft Centre/Golf Course entrances on your left hand side as you reach the town, then see a bridge ahead. Turn right immediately before the bridge, and then right again immediately after that. Continue straight down a dirt road and take the right fork. You will see a stable block with a tower over the entrance; Garden Cottage is the only house outside of the stable block.
You can also take the A1 south from Edinburgh to Berwick-Upon-Tweed and follow the directions listed ‘from the south’ above.
Google Maps and Google Streetview can also help you find your way!
Take the train to Berwick-Upon-Tweed Station. (This stretch of railway, between Newcastle and Edinburgh, was voted the most scenic in Britain!) Outside of the station there is a bus stop. Take the 67 Bus (which will say Galashiels on the front), to Coldstream. (Approx 45 mins, £2.50 single adult fare). You can check times on www.perrymansbuses.co.uk.
When you reach Coldstream, stay on the bus at the first stop (opposite the Besom Inn). When you have passed the Co-op on your left, press the call button for the driver to stop, he should do so at the stop after the bridge. Take the turning on your side of the road directly after the bridge, then the right turn immediately after that. (Do not go into Lees Mill Drive) Continue to walk straight, down the dirt track alongside the high stone wall, and turn right at the fork. You will see a stable block with a tower over the entrance; Garden Cottage is the only house outside of the stable block, facing the roundabout.
If coming from the West, You can also catch the 67 Bus in Galashiels, St. Boswells and Kelso, to bring you to Coldstream (check times and fares at the Perryman’s site, linked above).
Press the call button as the bus passes the Hirsel Craft Centre entrance (on your left) and the driver should stop before or on the bridge. Cross the road and take the turning directly before the bridge, then the right turn directly after that. Continue to walk down the dirt track and turn right at the fork. You will see a stable block with a tower over the entrance; Garden Cottage is the only house outside of the stable block.
There are bus services from Newcastle to Berwick/Kelso and Edinburgh to Galashiels (where you can then board the 67 to Coldstream) but they are infrequent and take a long time. However, trains we know are expensive, so if the bus is preferable, a well-phrased Google search should bring up times and fares! | http://grahambell.org.uk/getting-here |
Q:
Scala Setting Specific Member Variables to Null
I have a class similar to the following:
class Cat(
val whiskers: Vector[Whiskers],
val tail: hasTail,
val ears: hasEars)
I also have a function which initializes these values based on the contents of a file as so:
val whiskers = initWhiskers()
val cat = new Cat(whiskers = whiskers, tail = initTail(), ears = initEars())
My question is: in some cases, there exist cats with no tails or ears. How do I account for this case by allowing for some of my cat objects to be without ears, and others to be with them?
To clarify:
val cat = new Cat(whiskers = whiskers, tail = null, ears = null)
val cat = new Cat(whiskers = whiskers, tail = initTail(), ears = null)
val cat = new Cat(whiskers = whiskers, tail = initTail(), ears = initEars())
I want all three of the above possibilities to be options. It would all be dependent on whether the file which I'm parsing has ears, or tails, etc.
What I've tried:
Setting values to null (which was terribly unsuccessful).
Making all member variables options (which seemed like a long-winded way of getting null values again).
Any other Options?
Thanks!
A:
In Scala, we use Option monad to wrap things which can be null.
class Cat(
val whiskers: Vector[Whiskers],
val tail: Option[hasTail],
val ears: Option[hasEars]
)
val cat1 = new Cat(whiskers = whiskers, tail = None, ears = None)
val cat2 = new Cat(whiskers = whiskers, tail = Option(initTail()), ears = Option(initEars() )
| |
Q:
How long would it take for an upright rigid body to fall to the ground?
Let's suppose there is a straight rigid bar with height $h$ and center of mass at the middle of height $h/2$. Now if the bar is vertically upright from ground, how long will it take to fall on the ground and what is the equation of motion of the center of mass (Lagrangian)?
A:
To complete the other answers, here's the Lagrangian of the system. Using polar coordinates centred at the base of the rod,
$$\mathcal{L}=T-V$$
$$\mathcal{L}=\frac{1}{2}I\dot{\theta^2}-\frac{1}{2}mgh\cos{\theta}$$
Using the known value for $I$, $\frac{mh^2}{3}$,
$$\mathcal{L}=\frac{1}{6}mh^2\dot{\theta^2}-\frac{1}{2}mgh\cos{\theta}$$
Using Euler-Lagrange, gives us the equation of motion:
$$\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\theta}}=\frac{\partial\mathcal{L}}{\partial\theta}$$
$$\frac{d}{dt}(\frac{1}{3}mh^2\dot{\theta})=\frac{1}{2}mgh\sin{\theta}$$
$$\ddot{\theta}=\frac{3}{2}\frac{g}{h}\sin{\theta}$$
A:
The answer depends on the initial conditions (initial angle, and initial angular velocity). If it starts vertical, then without an initial angular velocity it would take forever to fall.
Also note that as the rotational speed inceases, the bar might lift up from the pivot point due to the centifigual forces. Also once friction is overcome the contact is going to slip. So there are three domains for the solution.
Contact is fixed, forces need to be calculated
Contact is sliding, vertical force needs to be calculated
No longer in contact, no forces applied (other than gravity).
You can try to work out the solutions from the following equations:
Equations Of Motion
$$ F=m\ddot{x} $$
$$ N=m(\ddot{y}+g) $$
$$ \frac{H}{2}\left(F\,\cos\theta+N\,\sin\theta\right)=I_{G}\ddot{\theta} $$
where: $F$ frictional (horiz.) force, $N$ normal contact force, $m$ mass of bar, ($\ddot{x}$, $\ddot{y}$) acceleration of center of gravity, $H$ the total height of the bar, $\theta$ angle of bar from vertical (+=CCW), $I_G$ mass moment of inertia of the bar at the c.g.
Velocity of contact point
$$ vx_{A}=\dot{x}+\frac{H}{2}\left(\dot{\theta}\cos\theta\right) $$
$$ vy_{A}=\dot{y}+\frac{H}{2}\left(\dot{\theta}\sin\theta\right) $$
Acceleration of contact point
$$ ax_{A}=\ddot{x}+\frac{H}{2}\left(\ddot{\theta}\cos\theta-\dot{\theta}^{2}\sin\theta\right) $$
$$ ay_{A}=\ddot{y}+\frac{H}{2}\left(\ddot{\theta}\sin\theta+\dot{\theta}^{2}\cos\theta\right) $$
Friction contdition
$$ F\leq\mu N $$
You will find the equation of motion for case 1 being
$$\ddot{\theta}=\frac{g\sin\theta-\frac{H}{2}\dot{\theta}\cos2\theta}{\frac{I_{G}}{mH/2}+\frac{H}{2}\sin2\theta}$$
Finding the condition where you slip into case 2 and then case 3 involves monitoring the forces $F$ and $N$ and checking when $F>\mu N$ and when $N<0$.
A:
A falling tree is basically an inverted pendulum.
The period of a pendulum of length $h$ for small oscillations is $2\pi \sqrt{h/g}$, with $g$ the acceleration due to gravity, about $10\ m/s$. For an inverted pendulum near the top of its arc, there is no period, but the quantity $\sqrt{h/g}$ does represent a characteristic time scale for this system. The tree will take a few of these characteristic times to fall. $h$ for a tree is an "effective height", and depends on the mass distribution of the tree. If all the mass is at the top, $h$ is the height of the tree. If the tree is uniform, $h$ is $2/3$ the true height.
For a tree with $h = 40\ m$, the characteristic time is $2\ s$. For small angles, the angle the tree makes with the vertical will be multiplied by $e$ in this time. Let's start the tree at $1^{\circ}$ so that it needs to multiply its angle by $90$ to fall. $\ln(90) = 4.5$ so the tree takes about $9\ s$ to fall.
This is mathematically an underestimate because the characteristic time increases slightly as the tree falls, but not too much. Give it a nice round $10\ s$ and you get something that matches the first YouTube video I found.
| |
Q:
Show that $g_n(x)=nf(x+1/n)-nf(x)$ converges uniformly to $f'$
Let $f$ be a real valued, continuous and differentiable function on the real numbers with $f'$ uniformly continuous. Show that $g_n(x)$ with $$g_n(x)=nf(x+1/n)-nf(x)$$ is uniformly convergent to the derivative of $f$.
I cannot figure out how to solve this problem. If anyone could help me for this problem I would be very grateful.
A:
The key is to combine the uniform continuity of $f'$ with the mean-value theorem in order to deduce what is called "uniform differentiability." For any $h > 0$ we can use the MVT to find $0 < k < h$ such that
$$
\frac{f(x+h) - f(x)}{h} - f'(x) = f'(x+k) - f'(x).
$$
For $\epsilon >0$ we pick $\delta >0$ such that $z,y \in \mathbb{R}$ and $|z-y| < \delta$ implies that $|f'(z)-f'(y) | < \epsilon$. Then for $0 < h < \delta$ we use the above to estimate
$$
\left\vert \frac{f(x+h) - f(x)}{h} - f'(x) \right\vert= |f'(x+k) - f'(x)| < \epsilon
$$
since $0 < k < h < \delta$. Since $x$ is arbitrary we find that
$$
0 < h < \delta \Rightarrow \sup_{x } \left\vert \frac{f(x+h) - f(x)}{h} - f'(x) \right\vert < \epsilon.
$$
This is actually a stronger result than what you need. I'll leave it to you to fit the two together.
| |
The Airbus A350 is a long-range, wide-body airliner developed and produced by Airbus. The first A350 design proposed by Airbus in 2004, in response to the Boeing 787 Dreamliner, would have been a development of the A330 with composite wings and new engines. As market support was inadequate, in 2006, Airbus switched to a clean-sheet "XWB" design, powered by Rolls-Royce Trent XWB turbofan engines. The prototype first flew on 14 June 2013 from Toulouse in France. Type certification from the European Aviation Safety Agency (EASA) was obtained in September 2014, followed by certification from the Federal Aviation Administration (FAA) two months later.
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---
abstract: |
We consider a queueing system with $n$ parallel queues operating according to the so-called “supermarket model” in which arriving customers join the shortest of $d$ randomly selected queues. Assuming rate $n\lambda_{n}$ Poisson arrivals and rate $1$ exponentially distributed service times, we consider this model in the heavy traffic regime, described by $\lambda_{n}\uparrow 1$ as $n\to\infty$. We give a simple expectation argument establishing that majority of queues have steady state length at least $\log_d(1-\lambda_{n})^{-1} - O(1)$ with probability approaching one as $n\rightarrow\infty$, implying the same for the steady state delay of a typical customer.
Our main result concerns the detailed behavior of queues with length *smaller* than $\log_d(1-\lambda_{n})^{-1}-O(1)$. Assuming $\lambda_{n}$ converges to $1$ at rate at most $\sqrt{n}$, we show that the dynamics of such queues does not follow a diffusion process, as is typical for queueing systems in heavy traffic, but is described instead by a deterministic infinite system of linear differential equations, after an appropriate rescaling. The unique fixed point solution of this system is shown explicitly to be of the form $\pi_{1}(d^{i}-1)/(d-1), i\ge 1$, which we conjecture describes the steady state behavior of the queue lengths after the same rescaling. Our result is obtained by combination of several technical ideas including establishing the existence and uniqueness of an associated infinite dimensional system of non-linear integral equations and adopting an appropriate stopped process as an intermediate step.
author:
- Patrick Eschenfeldt and David Gamarnik
bibliography:
- 'supermarket.bib'
title: |
Supermarket Queueing System in the Heavy Traffic Regime.\
Short Queue Dynamics
---
Introduction. {#sec:intro}
=============
In this paper we consider the so-called supermarket model in the heavy traffic regime. The supermarket model is a parallel server queueing system consisting of $n$ identical servers which process jobs at rate $1$ Poisson process. The jobs arrive into the system according to a Poisson process with rate $n\lambda_{n}$ where $\lambda_{n}$ is assumed to be strictly smaller than unity for stability. A positive integer parameter $d$ is fixed. Each arriving customer chooses $d$ servers uniformly at random and selects a server with the smallest number of jobs in the corresponding queue, ties broken uniformly at random. The queue within each server is processed according to the First-In-First-Out rule. We denote this system by ${M/M/n\text{-Sup}(d)}$.
The foundational work on this model was done by Dobrushin, Karpelevich and Vvedenskaya [@vvedenskaya] and Mitzenmacher [@mitzenmacher], who independently showed that when $\lambda_{n}=\lambda<1$ is a fixed constant and $d\ge 2$, the steady state probability that the customer encounters a queue with length at least $t$ (and hence experiences the delay at least $t$ in expectation), is of the form $\lambda^{d^{t}}$. Namely it is doubly exponential in $t$. This is in sharp contrast with the case $d=1$, where each server behaves as an $M/M/1$ system with load $\lambda$ and hence the steady state delay has the exponential tail of the form $\lambda^{t}$. This phenomena has its static counterpart in the form of so-called Balls-Into-Bins model. In this model $n$ balls are thrown sequentially into $n$ bins where for each ball $d$ bins are chosen uniformly at random and the bin with the smallest number of balls is chosen. It is well known that for this model the largest bin has $O(\log\log n)$ balls when $d\ge 2$ as opposed to $O(\log n)$ balls when $d=1$. This known as ”Power-of-Two” phenomena.
The development in [@vvedenskaya] and [@mitzenmacher] is based on the fluid limit approximations for the infinite dimensional process, where each coordinate corresponds to the fraction of servers with at least $i$ jobs. By taking $n$ to infinity, it is shown that the limiting system can be described by a deterministic infinite system of differential equations, which have a unique and simple to describe fixed point satisfying doubly exponential decay rate. Some of the subsequent work that has been performed on the supermarket model and its variations can be found in [@mukherjee2015universality; @mukherjee2016efficient; @bramson-2010; @bramson-2012; @bramson-2013; @graham; @luczak-mcdiarmid; @luczak-norris; @Mitzenmacher1999; @vvedenskaya-97; @li; @dieker-suk].
In this paper we consider the supermarket model in the heavy traffic regime described by having the arrival rate parameter $\lambda_{n}\uparrow 1$. The work of Brightwell and Luczak [@luczak] considers the model which is the closest to the one considered in this paper. They also assume that $\lambda_{n}\uparrow 1$, but at the same time they assume that the parameter $d$ diverges to infinity as well. In our setting $d$ remains constant as is the case for the classical supermarket model. More precisely, we assume that as $n$ increases, $d$ is fixed, but $\lambda_{n}=1-\beta/\eta_{n}$ where $\beta>0$ is fixed and $\lim_{n}\eta_{n}=\infty$. Our goal is conducting the performance analysis of the system both at the process level and in steady state. Unfortunately, the fluid limit approach of [@vvedenskaya] and [@mitzenmacher] is rendered useless since in this case the corresponding fluid limit trivializes to a system of differential equations describing the critical system corresponding to $\lambda_{n}=\lambda=1$. At the same time, however, certain educated guesses can be inferred from the case when $\lambda<1$ is constant, namely the classical setting. From the $\lambda^{d^{i}}$ tail behavior which describes the fraction of servers with at least $i$ customers in steady state, it can be inferred that when $\lambda\uparrow 1$, if $i^{*}=o\left(\log_{d}{1\over 1-\lambda}\right)$ then $$\begin{aligned}
\lambda^{d^{i^{*}}}=\lambda^{o\left({1\over 1-\lambda}\right)},\end{aligned}$$ which approaches unity as $\lambda\uparrow 1$. Namely, the fraction of servers with at most $i^{*}$ customers becomes negligible. Of course this does not apply rigorously to our heavy traffic regime as it amounts to first taking the limit in $n$ and only then taking the limit in $\lambda$, whereas in the heavy traffic regime this is done simultaneously. Nevertheless, our main results confirm this behavior both at the process level and in the steady state regime. In terms of our notation for $\lambda_{n}$, we show that when $\omega_{n}$ is an arbitrary sequence diverging to infinity and $$\begin{aligned}
\label{eq:i-n-star}
i^{*}_{n}=\log_{d}{1\over
1-\lambda_{n}}-\omega_{n}=\log_{d}\eta_{n}-\log_{d}\beta-\omega_{n},\end{aligned}$$ (note that the term $\log_{d}\beta$ is subsumed by $\omega_{n}$) the fraction of queues with at most $i^{*}_{n}$ customers is $o(1)$ with probability approaching unity as $n$ increases. The intuitive explanation for this is as above: $$\begin{aligned}
\lambda_{n}^{d^{i^{*}_{n}}}&=\left(1-\beta/\eta_{n}\right)^{d^{-\omega_n}
\over 1-\lambda_{n}} \\
&=\left(1-\beta/\eta_{n}\right)^{\eta_{n}\over\beta d^{\omega_{n}}} \\
&\approx e^{-{1\over d^{\omega_{n}}}} \\
&\rightarrow 1,\end{aligned}$$ as $n\rightarrow\infty$.
We now describe our results and our approach at some level of detail. First we give a very simple expectation based argument showing that in steady state the expected fraction of servers with at least $i$ customers is at least $1 - {\beta \over \eta_n}{d^i -1\over d-1}$. Plugging here the value for $i_{n}^{*}$ given in (\[eq:i-n-star\]) the expression becomes $1-1/d^{\omega_{n}}\rightarrow 1$, as $n\rightarrow\infty$, confirming the claimed behavior in steady state. This immediately implies that the steady state delay experienced by a typical customer is at least $i^{*}_{n}$ with probability approaching $1$ as $n$ diverges. This result is formally stated in Theorem \[thm:steady-state-expectation\].
Our main result concerns the detailed process level behavior of queues, with the eye towards queues of length at most $i_{n}^{*}$. As is customary in the heavy traffic theory, the first step is applying an appropriate rescaling step, and thus for each $i\le i_{n}^{*}$, letting $S^{n}_{i}(t)$ denote the fraction of servers with at most $i$ jobs at time $t$, we introduce a rescaled process $T_{i}^{n}(t)\triangleq
\eta_{n}(1-S^{n}_{i}(t))$. We prove that the sequence $T^{n}=\left(T_{i}^{n}(t), t\ge 0, i\ge 1\right)$ converges weakly to some deterministic limiting process $T=\left(T_{i}(t), t\ge 0,i\ge 1\right)$, provided that the system starts in a state where $T_{i}^{n}(0)$ has a non-trivial limit as $n\rightarrow\infty$, and provided $\eta_{n}$ grows at most order $\sqrt{n}$. We show that the process $T(t)$ is the unique solution to a deterministic infinite system of linear differential equations (given by (\[eq:result-i\]) in the body of the paper).
This is the main technical result of the paper. This result is perhaps somewhat surprising since the processes arising as heavy traffic limits of queueing systems is usually a diffusion, and not a deterministic process as it is in our case. We further show that the unique fixed point of the process $T(t)=(T_{i}(t),
i\ge 0)$ is of the form $\pi_{i}=\pi_{1}(d^{i}-1)/(d-1)$, consistent with our result regarding the lower bound on steady state expectation of $S_{i}^{n}$ discussed above. We also show that this fixed point is an attraction point of the process $T(t)$ and the convergence occurs exponentially fast.
Our main result regarding the weak convergence of the rescaled process $T^{n}$ to $T$ is obtained by employing several technical steps. The first step is writing the term $\left(S_{i-1}^{n}\right)^{d}-\left(S_{i}^{n}\right)^{d}=(1-T_{i-1}^{n}/\eta_{n
})^{d}-(1-T_{i}^{n}/\eta_{n})^{d}$ (which corresponds to the likelihood that the arriving job increases the fraction of servers with at least $i$ jobs), as a sum of a linear function $dT_{i}^{n}/\eta_{n}-dT_{i-1}^{n}/\eta_{n}$ plus the correction term $d g^{\eta_{n}}/\eta_{n}$, where $g^{\eta_{n}}$ is the appropriate correction function, and then showing that this correction has a smaller order of magnitude provided $i\le i_{n}^{*}$. Then, we prove the existence, uniqueness and continuity property of the stochastic integral equation governing the behavior of the rescaled queue length counting process $\left(T_{i}^{n}, t\ge 0, i\ge 1\right)$, up to an appropriately chosen stopping time intended to prevent $T_{i}^{n}$ from “growing too much”. The stopping time utilized in this theorem is similar to the one employed by the authors in a different paper [@eschenfeldt]. Finally, we apply the martingale method by splitting the underlying stochastic processes into one part which is a martingale and the compensating part which has a non-trivial drift. It is then shown that the martingale part is zero in the limit as $n\rightarrow\infty$, thanks to the nature of the underlying rescaling.
The remainder of the paper is laid out as follows: Section \[sec:setup\] defines the model and states our main results. In Section \[sec:steady-state\] we prove our results regarding the steady state regime and prove results regarding the properties of the limiting deterministic process $T(t)$. In Section \[sec:integral\] we will prove Theorem \[thm:const-integral\] regarding the existence, uniqueness and the continuity property of an infinite dimensional stochastic integral equation system governing the behavior of the rescaled process $T^{n}(t)$. In Section \[sec:martingales\] we construct a representation of the system as a combination of martingales and integral terms. Section \[sec:martingale-convergence\] will establish that these martingales converge to zero. This section will also include the conclusion of the proof of Theorem \[thm:const-result\]. Open questions and conjectures are discussed in Section \[sec:Conclusions\].
We close this section with some notational conventions. We use $\Rightarrow$ to denote weak convergence. ${\mathbb{R}}({\mathbb{R}}_{+})$ denotes the set of (non-negative) real values. ${\mathbb{R}}_{\ge 1}$ denotes the set of real values greater than or equal to one. ${{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_+ = {\mathbb{R}}_+ \cup \{\infty\}$ denotes the extended non-negative real line. ${{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1} = {\mathbb{R}}_{\ge 1} \cup \{\infty\}$ denotes the extended non-negative real line excluding reals strictly less than $1$. We equip ${{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1}$ with the order topology, in which neighborhoods of $\infty$ are those sets which contain a subset of the form $\{x > a\}$ for some $a \in {\mathbb{R}}$. Let ${\mathbb{R}}^\infty$ be the space of sequences $x = (x_0,x_1,x_2,\ldots)$. For $x
\in {\mathbb{R}}^\infty$ and $\rho > 1$ we define the norm $${\left|\left|x\right|\right|}_\rho {\,{\buildrel \triangle \over =}\,}\sum_{i \ge 0}\rho^{-i}\left|x_i\right|,$$ where ${\left|\left|x\right|\right|}_\rho = \infty$ is a possibility, and we define the subspace $${\mathbb{R}}^{\infty,\rho} {\,{\buildrel \triangle \over =}\,}\left\{x \in {\mathbb{R}}^\infty {\text{ s.t. }}{\left|\left|x\right|\right|}_{\rho} < \infty\right\}.$$ Note that while ${\left|\left|\cdot\right|\right|}_\rho$ does not induce the standard product topology on ${\mathbb{R}}^\infty$, we will use ${\mathbb{R}}^\infty$ and ${\mathbb{R}}^{\infty,\rho}$ with the product topology unless noted otherwise. Let $D_t = D([0,t],{\mathbb{R}})$ be the space of cadlag functions from $[0,t]$ to ${\mathbb{R}}$. Let $D^\infty_t = D([0,t],{\mathbb{R}}^\infty)$ be the space of cadlag functions from $[0,t]$ to ${\mathbb{R}}^\infty$. For $x \in D_t$ we denote the uniform norm $${\left|\left|x\right|\right|}_t = \sup_{0 \le s \le t} |x(s)|.$$ For $x \in D^\infty_t$ and $\rho > 1$, we define the $\rho$-norm by $${\left|\left|x\right|\right|}_{\rho,t} = \sum_{i \ge 0}\rho^{-i}{\left|\left|x_i\right|\right|}_t,$$ where ${\left|\left|x\right|\right|}_{\rho,t} = \infty$ is a possibility and we define the subspace $$D^{\infty,\rho}_t {\,{\buildrel \triangle \over =}\,}\left\{x \in D^\infty_t {\text{ s.t. }}{\left|\left|x\right|\right|}_{\rho,t} < \infty\right\}.$$ Observe that while ${\left|\left|\cdot\right|\right|}_{\rho,t}$ does not induce the standard product topology on $D^{\infty}_t$, we will use $D^\infty_t$ and $D^{\infty,\rho}_t$ with the product topology unless noted otherwise. For $\eta \in {\mathbb{R}}_+$, let $[-\eta, \eta]^\infty = \{x \in {\mathbb{R}}^\infty {\text{ s.t. }}\forall \text{ } i \ge 0,
|x_i| \le \eta\}.$ Let $D^{\eta}_t = D\left([0,t],[-\eta,\eta]^\infty\right)$ be the space of cadlag functions from $[0,t]$ to $[-\eta,\eta]^\infty$. $D^{\eta}_t$ is also equipped with the standard product topology. For notational convenience, for $\rho > 1$ and $\eta \in {\mathbb{R}}_+$ we define $D^{\eta,\rho}_t {\,{\buildrel \triangle \over =}\,}D^{\eta}_t$.
The model and the main result. {#sec:setup}
==============================
We consider the supermarket model with $n$ exponential rate one servers each with their own queue, and Poisson arrivals with rate $\lambda_n n$ for $\lambda_n < 1$ such that $\lambda_n \uparrow 1$. Specifically, we assume that there exists some sequence $\eta_n$ and constant $\beta > 0$ such that $\eta_n \to \infty$ as $n \to \infty$ and $$\label{eq:lambda-rate}
\lim_{n \to \infty} \eta_n(1-\lambda_n) = \beta.$$ We assume $\eta_n \ge 1$ for all $n \ge 1$. Upon arrival customers select $d \ge 2$ queues uniformly at random with replacement and join the shortest of these queues, with ties broken uniformly at random.
Let $0 \le S_i^n(t) \le 1$ be the fraction of queues with at least $i$ customers (including the customer in service) at time $t$. Then the probability that an arriving customer at time $t$ joins a queue of length exactly $i-1$ is $$(S_{i-1}^n(t))^d - (S_i^n(t))^d.$$ As a result, because the overall arrival rate is $\lambda_n n$, the instantaneous rate of arrivals to queues of length exactly $i-1$ is $$\lambda_n n \left((S_{i-1}^n(t))^d - (S_i^n(t))^d\right).$$ Note that an arrival to a queue of length $i-1$ increases $S_i^n$ by $1/n$, and all other types of arrivals leave $S_i^n$ unchanged. Similarly, a departure from a queue of length $i$ decreases $S_i^n$ by $1/n$ and any other departures leave $S_i^n$ unchanged. The instantaneous rate of departures from queues of length exactly $i$ at time $t$ is $$nS_i^n(t) - nS_{i+1}^n(t).$$ For $i \ge 1$ let $A_i$ and $D_i$ be independent rate-1 Poisson processes. Then we can represent the processes $S_i^n$ via random time changes of these Poisson processes. Specifically, we have $$\begin{aligned}
S_0^n(t) &= 1, \label{eq:s-form-0} \\
S_i^n(t) &= S_i^n(0) + {1 \over n}A_i\left(\lambda_n n
\int_0^t\left(\left(S_{i-1}^n(s)\right)^d-\left(S_i^n(s)\right)^d\right
)ds\right) \notag \\
&\quad\quad - {1 \over n}D_i\left(n \int_0^t\left(S_{i}^n(s) -
S_{i+1}^n(s)\right)ds\right), & i \ge 1. \label{eq:s-form-i}\end{aligned}$$ We let $$T_i^n = \eta_n(1 - S_i^n),$$ where $\eta_n$ is defined by . Observe that $$0 \le T_i^n \le \eta_n.$$ For technical reasons we restrict our choice of $\eta_n$ to those for which there exists constant $Q \ge 0$ such that $$\label{eq:eta-rate}
\eta_n \le Q\sqrt{n},$$ for all $n \ge 1$. That is, we assume $\eta_n = {O\left(\sqrt{n}\right)}$. We further define $\eta_\infty = \infty$. We will prove in Theorem \[thm:const-result\] that under appropriate conditions, $T^n = (T_0^n,T_1^n,\ldots)$ weakly converges to the solution to a certain integral equation, which we first prove in Theorem \[thm:const-integral\] has a unique solution.
For every $\eta \in {\mathbb{R}}_{\ge 1}$ and $x \in {\mathbb{R}}$, we let $$\begin{aligned}
g^\eta(x) &{\,{\buildrel \triangle \over =}\,}{\eta \over d}\left(1-{x\over\eta}\right)^d - {\eta
\over d} + x
\label{eq:g^n}\\
&= {1 \over d} \sum_{l = 2}^d \binom{d}{l}(-1)^l{x^l \over \eta^{l-1}}, \notag\end{aligned}$$ and also let $g^\infty = 0$ for $\eta = \infty$.
Given $b \in {\mathbb{R}}^{\infty,\rho}$, $y \in D^{\infty,\rho}_t$, $\lambda \in {\mathbb{R}}$, and $\eta \in {{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1}$, consider the following system of integral equations: $$\begin{aligned}
T_0(t) &= 0, \label{eq:integral-0}\\
T_i(t) &= b_i + y_i - \lambda d\int_0^t\left(T_i(s) - T_{i-1}(s) -
g^\eta(T_i(s)) +
g^\eta(T_{i-1}(s)) \right)ds \label{eq:integral-i} \\
&\quad\quad + \int_0^t\left(T_{i+1}(s) - T_i(s)\right)ds, \quad\quad i \ge 1.
\notag\end{aligned}$$ An important special case of this system, which will appear as the limiting system in Theorem \[thm:const-result\] below, is when we set $\eta = \infty$ and $\lambda = 1$, and $y = 0$. In this case $b_i = T_i(0)$ for $i \ge 1$. This system is as follows: $$\begin{aligned}
T_0(t) &= 0, \label{eq:result-0}\\
T_i(t) &= T_i(0) - d\int_0^t\left(T_i(s) - T_{i-1}(s)\right)ds \notag\\
&\quad\quad+
\int_0^t\left(T_{i+1}(s) - T_i(s)\right)ds, & i \ge 1 \label{eq:result-i}.\end{aligned}$$ For any $\eta \in {{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1}$, $0 < \alpha < 1/2$, and $\rho > 1$, let $i^* = {\alpha \over 2} \log_\rho \eta$ where $\log_\rho \infty = \infty$. In fact, for our purposes, $\alpha / 2$ can be replaced by any positive number strictly smaller than $\alpha$. Define a stopping time $t^* \in {{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_+$ as follows: $$\begin{aligned}
t^* {\,{\buildrel \triangle \over =}\,}\inf\bigg\{ s \ge 0 :
&\exists\text{ } i {\text{ s.t. }}1 \le i \le i^* \text{ and } |T _i(s)| \ge
\eta^{\alpha} \notag
\\
&\text{ or } \label{eq:stopping-time-full} \\
&\exists\text{ } i {\text{ s.t. }}i > i^* \text{ and } |T_i(s)| \ge \eta + 1
\notag \bigg\}.
\notag\end{aligned}$$ We also define a subset of ${\mathbb{R}}^{\infty,\rho}$ related to this stopping time. Let $$\begin{aligned}
{\mathcal{Z}^{\eta}} {\,{\buildrel \triangle \over =}\,}\bigg\{x \in {\mathbb{R}}^{\infty,\rho} {\text{ s.t. }}&\forall \text{ } 1
\le i \le
i^*, \quad |x_i| \le \eta^{\alpha} \\
&\text{ and } \\
&\forall \text{ } i > i^*, \quad |x_i| \le \eta + 1\bigg\}.\end{aligned}$$ Finally, we define a subset of the product space ${\mathbb{R}}^{\infty,\rho} \times D^{\infty,\rho}_t \times {\mathbb{R}}\times {{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1}$ equipped with the product topology which will allow us to limit our attention to certain parameter values. Specifically, let $$\begin{aligned}
{Z^{\alpha}_{K}} {\,{\buildrel \triangle \over =}\,}\bigg\{(b,y,\lambda,\eta) \in {\mathbb{R}}^{\infty,\rho} \times
D^{\infty,\rho}_t \times {\mathbb{R}}\times {{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1} {\text{ s.t. }}&b + y(0) \in {\mathcal{Z}^{\eta}}, \\
{\left|\left|b\right|\right|}_{\rho} &\le K, {\left|\left|y\right|\right|}_{\rho,t} \le K\bigg\}.\end{aligned}$$ Observe that for $(b,y,\lambda,\eta)$ which are not in ${Z^{\alpha}_{K}}$ for any $K > 0$, we have $t^* = 0$, and that ${Z^{\alpha}_{K}}$ is a closed subset of ${\mathbb{R}}^{\infty,\rho} \times D^{\infty,\rho}_t \times {\mathbb{R}}\times {{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1}$.
Our first result shows that - has a unique solution on the interval $[0,t^*]$ and that it defines a map which satisfies a certain continuity property.
\[thm:const-integral\] For $t \in [0,t^*]$, the system - has a unique solution $T = (T_i, i \ge 0) \in D^{\infty,\rho}_t$. For any $t \ge 0$, defining $$\begin{aligned}
\hat{T}(t) =
\begin{cases}
T(t) & t < t^* \\ T(t^*) & t \ge t^*,
\end{cases} \label{eq:integral-stopped}\end{aligned}$$ we obtain a function $f : {\mathbb{R}}^{\infty,\rho} \times D^{\infty,\rho}_t
\times {\mathbb{R}}\times {{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1} \to D^{\infty,\rho}_t$ mapping $(b,y,\lambda,\eta)$ to $\hat{T} = f(b,y,\lambda,\eta)$. Moreover, when the domain is restricted to ${Z^{\alpha}_{K}}$ for any $K > 0$ equipped with the product topology, $f$ is continuous for every $t \ge 0$.
\[rem:eta=infty\] Note that if $\eta = \infty$ then $t^* = \infty$ and thus $T = \hat{T}$. Further note that for $\eta < \infty$, the definition of $t^*$ implies that in fact either $\hat{T} \in D^{\eta+1,\rho}_t$ or $t^* =
0$. In the latter case $\hat{T} = b + y(0)$ is a constant function.
We prove Theorem \[thm:const-integral\] in Section \[sec:integral\]. We now turn to our main result.
\[thm:const-result\] Suppose $\lambda_n$ satisfies for a sequence $\eta_n$ satisfying for some $Q > 0$. Suppose there exists $\rho > 1$ such that $$\label{eq:const-starting-state}
T^n(0) \Rightarrow T(0) \quad \text{ in } {\mathbb{R}}^{\infty,\rho} \text{ as } n
\to \infty,$$ for some random variable $T(0) \in {\mathbb{R}}^{\infty,\rho}$. Furthermore, suppose $$\label{eq:starting-expectation-bounded}
\limsup_{n \to \infty}{\mathbb{E}}\left[{\left|\left|T^n(0)\right|\right|}_{\rho}\right] < \infty,$$ and there exists $0 < \alpha < 1/2$ such that for all sufficiently large $n$, almost surely $$\label{eq:start-inside}
T_i^n(0) \le \eta_n^{\alpha}, \quad 0 \le i \le i^*.$$ Then for any $t \ge 0$, $$T^n \Rightarrow T \quad \text{ in } D^{\infty,\rho}_t \text{ as } n \to \infty,$$ where $T$ is the unique solution of the system -.
The motivation for the initial condition assumptions and is as follows: as we will see below, we expect that in steady state the limiting system $T$ grows like $T_i = d^i$, so condition can be considered as requiring $T^n(0)$ to be consistent with this behavior, with $\rho > d$. Condition is similar, as $d^{i^*} \approx
\eta_n^{\alpha / 2}$. We prove Theorem \[thm:const-result\] in Section \[sec:martingale-convergence\].
Now we turn to results about the solution $T$ which appears in Theorem \[thm:const-result\].
\[thm:fixed-point\] Consider the system of integral equations given by -. This system has a fixed point $\pi = (\pi_i, i \ge 0)$ given by $$\label{eq:fixed-point}
\pi_i = \pi_1{d^i - 1 \over d -1}.$$ This fixed point is unique up to the constant $\pi_1$.
We also show that this fixed point is attractive:
\[thm:Lyap\] Let $\Phi(t) = \sum_{i \ge 1}d^{-i/2}|T_i(t) - \pi_i|$. If $\Phi(0) < \infty$ then $\Phi$ converges exponentially fast to zero. Specifically, $\Phi(t) \le \Phi(0) e^{-\left(\sqrt{d}-1\right)^2 t}$ for all $t
\ge 0$.
Finally we consider the system in steady state. Via elementary arguments we prove a bound on the expectation of the fraction of short queues in steady state. For $i \ge 0$, let $S_i^n(\infty)$ be the fraction of queues with length at least $i$ in steady state. For the statement below, we assume $\lambda_n$ is given by and $\eta_n \to \infty$ is arbitrary. In particular, the assumption is no longer needed.
\[thm:steady-state-expectation\] For $i \ge 0$ we have $$\label{eq:steady-state-expectation}
{\mathbb{E}}S_i^n(\infty) \ge 1 - \left(1 - \lambda_n\right){d^i -1\over d-1}.$$ As a result, for any sequence $\omega_n$ which diverges to infinity as $n
\to \infty$, the fraction of queues with length at least $\log_d \eta_n -
\omega_n$ approaches unity with probability approaching one as $n \to
\infty$. This further implies that a customer arriving in steady state experiences a delay of at least $\log_d \eta_n - \omega_n$ with probability approaching one as $n \to \infty$.
We prove Theorems \[thm:fixed-point\]-\[thm:steady-state-expectation\] in Section \[sec:steady-state\].
Note that for any sequence $\omega_n$ which diverges to infinity as $n
\to \infty$, the right hand side of diverges to negative infinity for any $i \ge \log_d \eta_n + \omega_n$. Because is only a lower bound this is not useful, but it does suggest that the behavior of $S_i^n$ is best examined for values of $i$ near $\log_d\eta_n$.
The model in steady state. {#sec:steady-state}
==========================
We set the derivative of $T_i$ to zero and and introduce the notation $\pi = (\pi_0,\pi_1,\pi_2,\ldots)$ for the desired fixed point. This leads to the recurrence $$\pi_{i+1} = (d+1)\pi_i - d\pi_{i-1}, \quad\quad i \ge 1,$$ which is solved by $$\pi_i = {1 \over d-1}\left( d \pi_0 - \pi_1 + d^i(\pi_1 -
\pi_0)\right), \quad\quad i \ge 0.$$ By we have $T_0(t) = 0$ for all $t \ge 0$ and thus $\pi_0
= 0$, so this fixed point reduces to $$\pi_i = \pi_1{d^i - 1 \over d - 1}, \quad\quad i \ge 0.$$
Next we prove Theorem \[thm:Lyap\]:
Define $\epsilon_i(t) = T_i(t) - \pi_i$. We have $$\begin{aligned}
{d\epsilon_i \over dt} &= d(\epsilon_{i-1} +\pi_{i-1}) -
(d+1)(\epsilon_i+\pi_i) + (\epsilon_{i+1}+\pi_{i+1}) \\
&= d \epsilon_{i-1} - (d+1)\epsilon_i + \epsilon_{i+1}.\end{aligned}$$
Temporarily assume $\epsilon_i \ne 0$ for all $i$ so the derivative $d\Phi
\over dt$ is well defined. After providing a basic argument under this assumption we will explain how to remove it. Now we have $$\begin{aligned}
{d\Phi \over dt} &= \sum_{i : \epsilon_i > 0}d^{-i/2}\left(d
\epsilon_{i-1} - (d+1)\epsilon_i + \epsilon_{i+1}\right) \\
&\quad\quad - \sum_{i : \epsilon_i < 0}d^{-i/2}\left(d \epsilon_{i-1} -
(d+1)\epsilon_i + \epsilon_{i+1}\right).\end{aligned}$$ Let us now consider the terms involving $\epsilon_i$. We will first consider $i
\ge 2$. There are several cases, depending on the signs of $\epsilon_{i-1},\epsilon_i$, and $\epsilon_{i+1}$. First suppose they are all negative, so the term involving $\epsilon_i$, which we denote $A_i$, is $$\begin{aligned}
A_i &= -d^{-(i-1)/2}\epsilon_i + d^{-i/2}(d+1)\epsilon_i - d^{-(i
+1)/2}d\epsilon_i \\
&= d^{-i/2}\epsilon_i\left(-\sqrt{d} + d + 1 - {d \over \sqrt{d}}\right) \\
&= \left(1 - 2\sqrt{d} + d\right)d^{-i/2}\epsilon_i.\end{aligned}$$ We define $$\delta = 1 - 2\sqrt{d} + d = \left(\sqrt{d} - 1\right)^2$$ and note that the assumption $d \ge 2$ implies $\delta > 0$. Now note that if the sign of $\epsilon_{i-1}$ or $\epsilon_{i+1}$ or both is positive and $\epsilon_i$ remains negative this will simply change the sign of the appropriate coefficient of $\epsilon_i$ from negative to positive, decreasing $A_i$. Thus for all cases with $\epsilon_i$ negative we have $$A_i \le \delta d^{-i/2}\epsilon_i.$$ If all three signs are positive, we have $$\begin{aligned}
A_i &= d^{-(i-1)/2}\epsilon_i - d^{-i/2}(d+1)\epsilon_i + d^{-(i
+1)/2}d\epsilon_i \\
&= d^{-i/2}\epsilon_i\left(\sqrt{d} - d - 1 + {d \over \sqrt{d}}\right) \\
&= -\delta d^{-i/2}\epsilon_i,\end{aligned}$$ and if $\epsilon_{i-1}$ or $\epsilon_{i+1}$ is negative we still have $$A_i \le -\delta d^{-i/2}\epsilon_i.$$ Thus for all $i \ge 2$ we have $$A_i \le -\delta d^{-i/2}|\epsilon_i|.$$ To see that this inequality also holds for $i = 1$ note that for that case we simply omit the first term of $A_i$. Using this result for all $i \ge 1$ we conclude $$\begin{aligned}
{d\Phi \over dt} &= \sum_{i \ge 1} A_i \\
&\le \sum_{i \ge 1}-\delta d^{-i/2}|\epsilon_i| \\
&= -\delta \Phi.\end{aligned}$$ Thus we have $$\Phi(t) \le \Phi(0)e^{-\delta t},$$ and conclude that $\Phi(t)$ converges exponentially.
As in Mitzenmacher [@mitzenmacher], because we are interested in the evolution of the system as time increases, we can account for the $\epsilon_i = 0$ case by considering upper right-hand derivatives of $\epsilon_i$, defining $$\left.{d|\epsilon_i| \over dt}\right\vert_{t = t_0} {\,{\buildrel \triangle \over =}\,}\lim_{t \to
t_0^+}{|\epsilon_i(t)| \over t-t_0},$$ and similarly for ${d\Phi \over dt}$. Now if $\epsilon_i(t_0) = 0$ we have $\left.{d|\epsilon_i| \over dt}\right\vert_{t = t_0} \ge 0$, so we can include the $\epsilon_i = 0$ cases in the above proof with the $\epsilon_i > 0$ case now also including the case with $\epsilon_i = 0$ and ${d\epsilon_i \over dt}
\ge 0$ and similarly for $\epsilon_i < 0$.
From we obtain for $i \ge 1$ $${\mathbb{E}}S_i^n(t) = {\mathbb{E}}S_i^n(0) +
\lambda_n\int_0^t{\mathbb{E}}\left[\left(S_{i-1}^n(s)\right)^d
- \left(S_i^n(s)\right)^d\right]ds
- \int_0^t\left({\mathbb{E}}S_{i}^n(s) - {\mathbb{E}}S_{i+1}^n(s)\right)ds.$$ Assuming $(S_i^n(0), i \ge 0)$ has a steady state distribution, the same applies to $(S_i^n(t), i \ge 0)$, implying ${\mathbb{E}}S_i^n(0) = {\mathbb{E}}S_i^n(t)$. Thus switching to $S_i^n(\infty)$ for steady state version of $S_i^n(t)$, we obtain $$0 = \lambda_n{\mathbb{E}}\left[\left(S_{i-1}^n(\infty)\right)^d
- \left(S_i^n(\infty)\right)^d\right]
- \left({\mathbb{E}}S_{i}^n(\infty) - {\mathbb{E}}S_{i+1}^n(\infty)\right).$$ Because $0 \le S_i^n(t) \le 1$ for all $i \ge 0$ and $t \ge 0$ and $S_{i-1}^n(t) \ge S_i^n$ for all $i \ge 1$ and $t \ge 0$, we have the bound $$\left(S_{i-1}^n(\infty)\right)^d - \left(S_i^n(\infty)\right)^d \le
d\left(S_{i-1}^n(\infty)-S_i^n(\infty)\right),$$ and thus have $$0 \le \lambda_n d\left({\mathbb{E}}S_{i-1}^n(\infty) - {\mathbb{E}}S_i^n(\infty)\right) -
\left({\mathbb{E}}S_{i}^n(\infty) - {\mathbb{E}}S_{i+1}^n(\infty)\right).$$ For $i \ge 0$ define $$\sigma_i {\,{\buildrel \triangle \over =}\,}{\mathbb{E}}S_i^n(\infty) - {\mathbb{E}}S_{i+1}^n(\infty),$$ and observe that $\sigma_i$ is the expected number of queues of length exactly $i$ in steady state. We now obtain the bound $$\sigma_i \le \lambda_n d \sigma_{i-1}, \quad\quad i \ge 1,$$ which implies $$\sigma_i \le \sigma_0d^i\left(\lambda_n\right)^i.$$ We have $S_0^n(t) = 1$ and use Little’s law to observe $S_1^n(\infty) =
\lambda_n$ resulting in $$\sigma_i \le (1 - \lambda_n)d^i\lambda_n^i \le (1 - \lambda_n)d^i, \quad i
\ge 1.$$ Now observe $$\begin{aligned}
{\mathbb{E}}S_i^n(\infty) &= {\mathbb{E}}S_0^n(\infty) - \sum_{j = 0}^{i-1}\left({\mathbb{E}}S_j^n(\infty) - {\mathbb{E}}S_{j+1}^n(\infty)\right) \\
&= 1 - \sum_{j = 0}^{i-1}\sigma_j \\
&\ge 1 - \left(1 -\lambda_n\right)\sum_{j = 0}^{i-1} d^j
= 1 - \left(1-\lambda_n\right) {d^i - 1 \over d - 1}.\end{aligned}$$ This establishes .
Recalling , for $i \le \log_d\eta_n - \omega_n$, we have $${\mathbb{E}}S_i^n(\infty) \ge 1 - \left(1-\lambda_n\right) {\eta_n d^{-\omega_n} - 1
\over d - 1} \to 1,$$ as $n \to \infty$. Since $S_i^n(\infty) \le 1$, this implies that as $n \to
\infty$, $S_i^n(\infty) \to 1$ in probability. Namely, the fraction of queues with length at least $\log_d\eta_n - \omega_n$ approaches one in probability.
Finally observe that the probability of an arriving customer in steady state joining a queue of length at least $i$ is $S_i^n(\infty)^d$. For $i \le
\log_d\eta_n - \omega_n$ because we have $S_i^n(\infty)^d \to 1$ in probability, a customer arriving in steady state experiences a delay of at least $\log_d \eta_n -
\omega_n$ with probability approaching one as $n \to \infty$.
Beyond this elementary bound on the expectation in steady state, Theorem \[thm:const-result\] suggests that $T^n(\infty)$ converges to $\pi$, though formally this is a conjecture because we do not show that the sequence $T^n(\infty)$ is tight. Establishing this interchange of limits would be a potential direction for future work on this system. For the remainder of this section we will suppose the conjecture is true and consider the implications.
First, treating the fixed point as the limit of $T^n(\infty)$, we have $$\pi_1 = \lim_{n \to \infty} \eta_n(1-S_1^n(\infty)).$$ We use Little’s law to replace $S_1^n(\infty)$ by $\lambda_n$ so we have $$\pi_1 = \lim_{n \to \infty} \eta_n(1-\lambda_n) = \beta,$$ and therefore the fixed point becomes $$\pi_i = \beta{d^i - 1 \over d - 1}, \quad\quad i \ge 0.$$
Recall that $T_i(t) = \eta_n\left(1 - S_i(t)\right)$, so this fixed point suggests that in steady state the fraction of servers with at least $i$ jobs can be approximated by $$S_i^n(\infty) \approx 1 - {\beta \over \eta_n}{d^i-1 \over d-1}$$ when $i = \log_d\eta_n - \omega_n$.
![Simulated steady state expectation for [M/M/n(d)]{} with $d = 2$ and $\lambda_n = 1 - \beta n^{-\alpha}$ with $\beta = 2$ and $\alpha = 3/4$. For each line the quantity $\alpha \log_d n$ is rounded to the nearest integer. Each horizontal line indicates $\exp(-\beta d^k / (d-1))$, which is the conjectured limit of ${\mathbb{E}}S^n_{\alpha \log_d n + k}$ as $n$ diverges to infinity.[]{data-label="fig:steady-state"}](graphics/steady-state.eps)
We further conjecture that delay times longer than $\log_d \eta_n + \omega_n$ are unlikely. This is informed by a heuristic analysis of the heavy traffic supermarket model using the fixed $\lambda < 1$ results proved by Mitzenmacher [@mitzenmacher] and Vvedenskaya, et. al. [@vvedenskaya]. For fixed $\lambda < 1$, the system converges to a limiting system which has a unique fixed point at $$\pi_i = \lambda^{d^i-1 \over d-1}.$$ If we simply replace $\lambda$ with $\lambda_n = 1 - {\beta \over \eta_n}$, then we have $$\pi_i = \left(1 -{\beta \over \eta_n}\right)^{d^i-1 \over d-1} \to
\begin{cases} 1 & i \le \log_d\eta_n - \omega_n \\
e^{-{\beta d^k \over d-1}} & i = \log_d \eta_n + k \\
0 & i \ge \log_d \eta_n + \omega_n,
\end{cases}$$ where $\omega_n$ is any sequence diverging to infinity and $k$ is any constant. In particular, this heuristic suggests that in steady state all queues will have length $\log_d \eta_n + {O\left(1\right)}$.
As further evidence for this conjectured behavior, we simulated the system for a variety of values of $n$ to estimate the expectation in steady state. Figure \[fig:steady-state\] shows that the fraction of queues of length at least $i = \log_d \eta_n + k$ remains approximately constant as $n$ increases. Furthermore, these simulated steady state expectations appear to vary around the conjectured limits of ${\mathbb{E}}S_i^n$ for such $i$, with the variation primarily introduced by the necessary rounding of $i$ to an integer value.
Integral representation. {#sec:integral}
========================
We prove Theorem \[thm:const-integral\] in this section. We will make use of a version of Gronwall’s inequality, which we state now as a lemma (see, e.g., pg. 498 of [@ethier-kurtz]).
\[thm:gronwall\] Suppose that $g : [0,\infty) \to [0,\infty)$ is a function such that $$0 \le g(t) \le \epsilon + M\int_0^tg(s)ds, \quad 0 \le t \le T,$$ for some positive finite $\epsilon$ and $M$. Then $$g(t) \le \epsilon e^{Mt}, \quad 0 \le t \le T.$$
We begin by establishing two lemmas related to the function $g^\eta$.
\[thm:g\^m-converge\] Let $0 < \alpha < 1/2$, and $\rho > 1$, and let $\eta^n,\eta \in {{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1}$ be such that $\eta^n \to \eta$, $i^*_n = {\alpha \over 2} \log_\rho \eta^n$, $x_i^n \in D([0,t],[-(\eta^n)^{\alpha}, (\eta^{n})^{\alpha}])$ for $0 \le i \le
i^*_n$ and $x_i^n \in D([0,t],[-\eta^n-1,\eta^{n}+1])$ for $i > i^*_n$. Then for any $i_0 \in {\mathbb{N}}$, $t \ge 0$, $${\left|\left|g^{\eta^n}(x_i^n) - g^{\eta}(x_i^n)\right|\right|}_{t} \to 0, \quad i \le i_0.$$
First suppose $\eta < \infty$. Then for large enough $n$ we have $\eta_n <
\infty$. For $i \ge 0$ we have $$\begin{aligned}
{\left|\left|g^{\eta^n}(x^n_i)-g^{\eta}(x^n_i)\right|\right|}_t &= \sup_{0 \le s \le
t}{1 \over d}\left|\sum_{l = 2}^d\binom{d}{l}(-1)^l x^n_i(s)^l
\left(\left(\eta^{n}\right)^{1-l}-\eta^{1-l}\right)\right| \\
&\le {1 \over d}\sum_{l = 2}^d\binom{d}{l}
\left(\eta^{n}+1\right)^l\left|\left(\eta^{n}\right)^{1-l}-\eta^{1-l}\right|
{\,{\buildrel \triangle \over =}\,}C_n.\end{aligned}$$ Observe that as $\eta^n \to \eta$, $C_n \to 0$, as desired.
Suppose now $\eta = \infty$ and thus $g^\eta = 0$. For $0 \le i \le i^*_n$ we have $$\begin{aligned}
{\left|\left|g^{\eta^n}(x^n_i)-g^{\eta}(x^n_i)\right|\right|}_t
&= {\left|\left|g^{\eta^n}(x^n_i)\right|\right|}_t \\
&\le {1 \over d}\sum_{l =2}^d\binom{d}{l}\left(\eta^{n}\right)^{\alpha l}
\left(\eta^{n}\right)^{1-l} \\
&= {1 \over d}\sum_{l =
2}^d\binom{d}{l}\left(\eta^{n}\right)^{1-l(1-\alpha)}.\end{aligned}$$ As $\eta^n \to \infty$ we have $i^*_n \to \infty$ and thus for large enough $n$, $i_0 < i^*_n$. For $2 \le l \le d$ we have $$\left(\eta^{n}\right)^{1-l(1-\alpha)} \to 0,$$ and thus $${\left|\left|g^{\eta^n}(x^n_i)\right|\right|}_t \to 0.$$
\[thm:g\^m-lipschitz\] For every $\eta \in {\mathbb{R}}_{\ge 1}$, $g^\eta$ is a Lipschitz continuous function with constant $4^d$ on $D^{\eta+2}_t$ equipped with the topology induced by ${\left|\left|\cdot\right|\right|}_{\rho,t}$. That is, for $x^1,x^2 \in D^{\eta+2}_t$, and for $i \ge 0$, $${\left|\left|g^\eta(x^1_i)- g^\eta(x^2_i)\right|\right|}_{t}
\le 4^d {\left|\left|x^1_i-x^2_i\right|\right|}_{t}$$ and $${\left|\left|g^\eta(x^1)-g^\eta(x^2)\right|\right|}_{\rho,t}
\le 4^d{\left|\left|x^1-x^2\right|\right|}_{\rho,t}.$$
Consider the restriction of $g^\eta$ onto $[-\eta-2,\eta+2] \to {\mathbb{R}}$. This function is differentiable and thus is Lipschitz continuous with constant $$\begin{aligned}
\sup_{y \in [-\eta-2,\eta+2]}\left|\dot{g}^\eta(y)\right| &=
\sup_{y \in [-\eta-2,\eta+2]}\left|-\left(1-{y \over
\eta}\right)^{d-1} + 1\right| \le 4^d.\end{aligned}$$ Thus for $y^1,y^2 \in [-\eta-2,\eta+2]$ we have $|g^\eta(y^1)-g^\eta(y^2)| \le 4^d|y^1-y^2|$, which further implies for $x^1,x^2 \in D^{\eta+2}_t$ and for $i \ge 0$ we have $$\begin{aligned}
{\left|\left|g^\eta(x^1_i)- g^\eta(x^2_i)\right|\right|}_t &=
\sup_{0 \le s \le t}|g^\eta(x_i^1(s)) - g^\eta(x_i^2(s))| \\
&\le 4^d\sup_{0 \le s \le t}|x_i^1(s)-x_i^2(s)| \\
&\le 4^d{\left|\left|x_i^1-x_i^2\right|\right|}_t. \end{aligned}$$ This further implies $$\begin{aligned}
{\left|\left|g^\eta(x^1)-g^\eta(x^2)\right|\right|}_{\rho,t} &= \sum_{i \ge
0}\rho^{-i}{\left|\left|g^\eta(x^1_i)- g^\eta(x^2_i)\right|\right|}_t \\
&\le \sum_{i \ge 0}\rho^{-i}4^d{\left|\left|x^1_i-x^2_i\right|\right|}_t\\
&= 4^d{\left|\left|x^1-x^2\right|\right|}_{\rho,t}.\end{aligned}$$
Fix $(b,y,\lambda,\eta) \in {\mathbb{R}}^{\infty,\rho} \times D^{\infty,\rho}_t
\times {\mathbb{R}}\times {{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1}$.
Suppose first $T(0) = b + y(0) \not\in {\mathcal{Z}^{\eta}}$. Then $t^* = 0$ and $\hat{T}(t) = T(0)$ for all $t \ge 0$.
We now suppose $T(0) = b + y(0) \in {\mathcal{Z}^{\eta}}$, and therefore for all $1 \le i \le i^*$ we have $|T_i(0)| \le \eta^{\alpha}$ and for all $i > i^*$ we have $|T_i(0)| \le
\eta + 1$. We will show existence and uniqueness via a contraction mapping argument, showing that the map defined by the right hand side of - is a contraction for small enough $t$. Note that this contraction argument will use the unbounded $\rho$-norm, and uniqueness of the solution with respect to that topology implies uniqueness with respect to the product topology.
We first define the map $\Gamma:D^{\infty,\rho}_t \to D^\infty_t$, where for $x \in D^{\infty,\rho}_t$, $$\begin{aligned}
\Gamma(x)_0(t) &= 0, \\
\Gamma(x)_i(t) &= b_i + y_i(t) - \lambda d\int_0^t\left(x_i(s) - x_{i-1}(s) -
g^\eta(x_i)
+ g^\eta(x_{i-1})\right)ds \\
&\quad\quad + \int_0^t\left(x_{i+1}(s) -
x_i(s)\right)ds, \quad\quad i \ge 1.\end{aligned}$$ Now let $$\begin{aligned}
t^* {\,{\buildrel \triangle \over =}\,}\inf\bigg\{ s \ge 0 :
&\exists\text{ } i {\text{ s.t. }}1 \le i \le i^* \text{ and }
\left|\Gamma(x)_i(s)\right| \ge \eta^{\alpha}
\\
&\text{ or } \\
&\exists\text{ } i {\text{ s.t. }}i > i^* \text{ and } \left|\Gamma(x)_i(s)\right| \ge
\eta + 1 \bigg\}.\end{aligned}$$ and further define $\hat{\Gamma}:D^{\infty,\rho}_t \to D^\infty_t$ by $$\hat{\Gamma}(x)_i(t) = \begin{cases}
\Gamma(x)_i(t) & t < t^* \\ \Gamma(x)_i(t^*) & t
\ge t^*,
\end{cases}$$ for all $i \ge 0$.
By construction, $\hat{\Gamma}(x)_i(t) \in [-\eta-1,\eta+1]$ for all $i \ge 0$, and thus $\hat{\Gamma}: D^{\infty,\rho}_t \to D^{\eta+1}_t$. Further note that if $\eta = \infty$, then $g^\eta = 0$ and $$\begin{aligned}
{\left|\left|\hat{\Gamma}(x)\right|\right|}_{\rho,t} &\le {\left|\left|b\right|\right|}_\rho + {\left|\left|y\right|\right|}_{\rho,t} +
\sum_{i \ge 1}\rho^{-i}\lambda d t\left({\left|\left|x_i\right|\right|}_t + {\left|\left|x_{i-1}\right|\right|}_t\right) \\
&\quad\quad + \sum_{i \ge 1}\rho^{-i} t\left({\left|\left|x_{i+1}\right|\right|}_t + {\left|\left|x_i\right|\right|}_t\right) \\
&\le {\left|\left|b\right|\right|}_\rho + {\left|\left|y\right|\right|}_{\rho,t}
+ \lambda d t\sum_{i \ge 1}\rho^{-i}{\left|\left|x_{i-1}\right|\right|}_t \\
&\quad\quad + t(\lambda d + 1)\sum_{i \ge 1}\rho^{-i} {\left|\left|x_i\right|\right|}_t \\
&\quad\quad + t\sum_{i \ge 1}\rho^{-i}{\left|\left|x_{i+1}\right|\right|}_t.\end{aligned}$$ We bound each of these sums individually. Observe $$\begin{aligned}
\sum_{i \ge 1}\rho^{-i}{\left|\left|x_{i-1}\right|\right|}_t &=
\rho^{-1}\sum_{i \ge 1}\rho^{-(i-1)}{\left|\left|x_{i-1}\right|\right|}_t \\
&= \rho^{-1}\sum_{i \ge 0}\rho^{-i}{\left|\left|x_{i}\right|\right|}_t = \rho^{-1}{\left|\left|x\right|\right|}_{\rho,t}.\end{aligned}$$ Similarly, $$\sum_{i \ge 1}\rho^{-i} {\left|\left|x_i\right|\right|}_t \le \sum_{i \ge 0}\rho^{-i}{\left|\left|x_i\right|\right|}_t = {\left|\left|x\right|\right|}_{\rho,t},$$ and $$\begin{aligned}
\sum_{i \ge 1}\rho^{-i}{\left|\left|x_{i+1}\right|\right|}_t &= \rho \sum_{i \ge 1}\rho^{-(i+1)}{\left|\left|x_{i+1}\right|\right|}_t \\
&= \rho \sum_{i \ge 2}\rho^{-i}{\left|\left|x_i\right|\right|}_t \\
&\le \rho \sum_{i \ge 0}\rho^{-i}{\left|\left|x_i\right|\right|}_t = \rho {\left|\left|x\right|\right|}_t.\end{aligned}$$ Therefore we have $${\left|\left|\hat{\Gamma}(x)\right|\right|}_{\rho,t} \le t\left(\lambda d\rho^{-1} + \lambda d + 1 + \rho\right) {\left|\left|x\right|\right|}_{\rho,t} < \infty,$$ and thus $\hat{\Gamma} : D^{\infty,\rho}_t \to D^{\eta + 1,\rho}_t$.
We now show that for $$t_0 < {1 \over \lambda d(1+\rho^{-1})(1+4^d) + 1 +
\rho},$$ $\hat{\Gamma}$ is a contraction on $D^{\eta+1,\rho}_{t}$ for all $t \le
t_0$. Namely, we claim that there exists $\gamma < 1$ such that for all $t \in [0,t_0]$ and $x^1,x^2 \in D^{\eta+1,\rho}_t$, we have $$\label{eq:contraction-claim}
{\left|\left|\hat{\Gamma}(x^1)-\hat{\Gamma}(x^2)\right|\right|}_{\rho,t} \le
\gamma {\left|\left|x^1-x^2\right|\right|}_{\rho,t}.$$ Let $t \le t_0$ and $x^1,x^2 \in D^{\eta+1,\rho}_t$. We have for $i \ge 1$ $$\begin{aligned}
{\left|\left|\hat{\Gamma}(x^1)_i-\hat{\Gamma}(x^2)_i\right|\right|}_t
&\le \lambda d\int_0^t \bigg(
{\left|\left|x_i^1-x_i^2\right|\right|}_s + {\left|\left|x_{i-1}^1-x_{i-1}^2\right|\right|}_s \\
&\quad\quad\quad + {\left|\left|g^\eta(x^1_i) - g^\eta(x^2_i)\right|\right|}_s
+ {\left|\left|g^\eta(x^1_{i-1}) - g^\eta(x^2_{i-1})\right|\right|}_s\bigg) ds \\
&\quad\quad + \int_0^t\left({\left|\left|x^1_{i+1}-x^2_{i+1}\right|\right|}_s +
{\left|\left|x^1_i-x^2_i\right|\right|}_s\right)ds. \\\end{aligned}$$ By Lemma \[thm:g\^m-lipschitz\], for $\eta < \infty$, $g^\eta$ is Lipschitz when restricted to $D^{\eta+2,\rho}_t$, with constant $4^d$. For $\eta = \infty$, $g^\infty = 0$. Thus we now have $$\begin{aligned}
{\left|\left|\hat{\Gamma}(x^1)_i-\hat{\Gamma}(x^2)_i\right|\right|}_t &\le t \lambda
d(1+4^d) {\left|\left|x^1_{i-1}-x^2_{i-1}\right|\right|}_t \\
&\quad\quad + t\left(\lambda d(1+4^d) + 1\right) {\left|\left|x^1_i-x^2_i\right|\right|}_{\rho,t}
+t {\left|\left|x^1_{i+1}-x^2_{i+1}\right|\right|}_t.\end{aligned}$$ This implies $$\begin{aligned}
{\left|\left|\hat{\Gamma}(x^1)- \hat{\Gamma}(x^2)\right|\right|}_{\rho,t} &\le t\lambda
d(1+4^d) \sum_{i \ge 1}\rho^{-i} {\left|\left|x^1_{i-1}- x^2_{i-1}\right|\right|}_{t} \\
&\quad\quad + t\left(\lambda d(1+4^d) + 1\right) \sum_{i \ge
1}\rho^{-i}{\left|\left|x^1_i-x^2_i\right|\right|}_t \\
&\quad\quad + t\sum_{i \ge 1}\rho^{-i}{\left|\left|x^1_{i+1}-x^2_{i+1}\right|\right|}_t.\end{aligned}$$ Reindexing and bounding these sums individually gives us $$\label{eq:contraction-bound}
{\left|\left|\hat{\Gamma}(x^1)- \hat{\Gamma}(x^2)\right|\right|}_{\rho,t} \le
t\left(\lambda d(1 + \rho^{-1})(1+4^d) + 1 + \rho\right)
{\left|\left|x^1-x^2\right|\right|}_{\rho,t}.$$ Let $$t_0 < {1 \over \lambda d(1 + \rho^{-1})(1+4^d) + 1 + \rho}.$$ Then holds with $$\gamma = t_0 \left(\lambda d(1 + \rho^{-1})(1+4^d) + 1 + \rho\right).$$ By the contraction mapping principle, $\hat{\Gamma}$ has a unique fixed point $\hat{T}$ on $D^{\eta+1,\rho}_{t}$ such that $\hat{\Gamma}(\hat{T}) =
\hat{T}$. This fixed point provides a unique solution $\hat{T}$ to for $t \in [0,t_0]$.
Suppose this fixed solution $\hat{T}$ is such that $t^* < t_0$. Then $\hat{T}$ is uniquely defined for all $t \ge 0$ and the proof is complete. Otherwise, observe for $t \ge 0$ and $i \ge 1$ we have $$\begin{aligned}
T_i(t)
&= T_i(t_0) + y_i(t)-y_i(t_0) \\
&\quad\quad - \lambda d\int_{t_0}^{t}\left(T_i(s) - T_{i-1}(s) -
g^\eta(T_i(s)) + g^\eta(T_{i-1}(s)) \right)ds \\
&\quad\quad + \int_{t_0}^{t}\left(T_{i+1}(s) - T_i(s)\right)ds.\end{aligned}$$ Thus if we define a shifted version of $y$ by $\tilde{y}(u) = y(u + t_0)$, then $T_i(t)$ for $t = u + t_0$ and $u \ge 0$ is the solution to the system $$\begin{aligned}
x_0(u) &= 0 \\
x_i(u) &= T_i(t_0) - y_i(t_0) + \tilde{y}_i(u) \\
&\quad\quad - \lambda d\int_{0}^{u}\left(x_i(s) - x_{i-1}(s) -
g^\eta(x_i(s)) + g^\eta(x_{i-1}(s)) \right)ds \\
&\quad\quad + \int_{0}^{u}\left(x_{i+1}(s) - x_i(s)\right)ds, \quad i \ge 1.\end{aligned}$$ Observe that this is the system - with arguments $$(T(t_0) - y(t_0), \tilde{y}, \lambda, \eta) \in {\mathbb{R}}^{\infty,\rho} \times
D^{\infty,\rho}_t \times {\mathbb{R}}\times {{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1},$$ and furthermore $x(0) = T(t_0) - y(t_0) + \tilde{y}(0) = T(t_0)$ and $T(t_0)
\in {\mathcal{Z}^{\eta}}$ because $t^* \ge t_0$. Thus we can repeat the contraction argument above to find a unique solution $x$ for $u \in [0,t_0]$. This unique $x$ is the unique solution $\hat{T}$ for $t \in [t_0,2t_0]$. If $t^* < 2t_0$, then $\hat{T}$ is uniquely defined for all $t \ge 0$ and the proof is complete. Otherwise, the above extension argument can be repeated to find a unique solution $\hat{T}$ for $[2t_0,3t_0],[3t_0,4t_0], \ldots$. If $t^* < k t_0$ for some $k \ge 3$ then the argument stops there and we conclude $\hat{T}$ is uniquely defined for all $t \ge 0$. Otherwise it may be extended to any $t \ge
0$.
Before proving continuity, we state and prove a lemma bounding the growth of solutions to .
\[thm:uniform-bound\] For any $t \ge 0$, if $x$ is the solution to for arguments $(b,y,\lambda,\eta)$, then $${\left|\left|x\right|\right|}_{\rho,t} \le \left({\left|\left|b\right|\right|}_\rho + {\left|\left|y\right|\right|}_{\rho,t}\right)
e^{\left(\lambda d(1+4^d)\left(1 + \rho^{-1}\right) + 1 + \rho\right)t}$$
We have $$\begin{aligned}
{\left|\left|x_i\right|\right|}_{t} &\le |b_i| + {\left|\left|y_i\right|\right|}_t + \lambda d\int_0^t\left({\left|\left|x_{i-1}\right|\right|}_s + {\left|\left|x_i\right|\right|}_s + {\left|\left|g(x_{i-1})\right|\right|}_s + {\left|\left|g(x_i)\right|\right|}_s\right) ds \\
&\quad\quad + \int_0^t\left({\left|\left|x_i\right|\right|}_s + {\left|\left|x_{i+1}\right|\right|}_s\right)ds, \end{aligned}$$ and thus $$\begin{aligned}
{\left|\left|x\right|\right|}_{\rho,t} &\le {\left|\left|b\right|\right|}_\rho + {\left|\left|y\right|\right|}_{\rho,t}
+ \lambda d(1+4^d)\sum_{i \ge 1}\rho^{-i}\int_0^t\left( {\left|\left|x_{i-1}\right|\right|}_s + {\left|\left|x_i\right|\right|}_s\right)ds \\
&\quad\quad+ \sum_{i \ge 1}\rho^{-i} \int_0^t\left({\left|\left|x_i\right|\right|}_s + {\left|\left|x_{i+1}\right|\right|}_s\right)ds \\
&\le {\left|\left|b\right|\right|}_\rho + {\left|\left|y\right|\right|}_{\rho,t} +
\left(\lambda d(1+4^d)\left(1 + \rho^{-1}\right) + 1 + \rho\right)
\sum_{i \ge 0}\rho^{-i}\int_0^t{\left|\left|x_i\right|\right|}_sds \\
&= {\left|\left|b\right|\right|}_\rho + {\left|\left|y\right|\right|}_{\rho,t} +
\left(\lambda d(1+4^d)\left(1 + \rho^{-1}\right) + 1 + \rho\right)
\int_0^t\sum_{i \ge 0}\rho^{-i}{\left|\left|x_i\right|\right|}_sds \\
&= {\left|\left|b\right|\right|}_\rho + {\left|\left|y\right|\right|}_{\rho,t} +
\left(\lambda d(1+4^d)\left(1 + \rho^{-1}\right) + 1 + \rho\right)
\int_0^t{\left|\left|x\right|\right|}_{\rho,s}ds.\end{aligned}$$ By Gronwall’s inequality (Lemma \[thm:gronwall\]), we have $${\left|\left|x\right|\right|}_{\rho,t} \le \left({\left|\left|b\right|\right|}_\rho + {\left|\left|y\right|\right|}_{\rho,t}\right)
e^{\left(\lambda d(1+4^d)\left(1 + \rho^{-1}\right) + 1 + \rho\right)t}.$$
We now prove continuity for $f$ restricted to the domain ${Z^{\alpha}_{K}}$ for any $K > 0$. Suppose $(b^n,y^n,\lambda^n,\eta^n) \to
(b,y,\lambda,\eta)$ with respect to the product topology, with $(b^n,y^n,\lambda^n,\eta^n) \in {Z^{\alpha}_{K}}$ and since the set ${Z^{\alpha}_{K}}$ is closed, we also have $(b,y,\lambda,\eta) \in {Z^{\alpha}_{K}}$. Suppose $x^n$ is the unique solution to for $(b^n,y^n,\lambda^n,\eta^n)$ and $x$ is the unique solution for $(b,y,\lambda,\eta)$. Let $i^*_n = {\alpha \over 2} \log_\rho \eta_n$ and $i^* = {\alpha \over 2}
\log_\rho \eta$. Let $t^*_n$ and $t^*$ be the stopping times for $x^n$ and $x$, respectively.
Recall that by the definition of ${Z^{\alpha}_{K}}$, we have $x^n(0) = b^n + y^n(0)
\in {\mathcal{Z}^{\eta^n}}$. Note that this, along with the definition of $t^*_n$, implies that $x^n(t) \in {\mathcal{Z}^{\eta^n}}$ for any $t \ge 0$. Similarly, we have $x(t) \in {\mathcal{Z}^{\eta}}$ for any $t \ge 0$. We adopt the simplified notation $g^n {\,{\buildrel \triangle \over =}\,}g^{\eta^n}$ and $g = g^\eta$.
We will show $x^n \to x$ in $D^{\infty,\rho}_t$ equipped with the product topology. Fix $\epsilon > 0$ and $i_0 \in {\mathbb{N}}$.
Recall the map $\hat{\Gamma}$ defined in the proof of existence and uniqueness above. We define $\hat{\Gamma}^n$ and $\hat{\Gamma}$ analogously for $(b^n,y^n,\lambda^n,\eta^n)$ and $(b,y,\lambda,\eta)$, respectively. Define $$t_0 = {1 \over 2\left(1+\lambda\right)d(1+\rho^{-1})(1+4^d) + 1 + \rho},$$ and $$\gamma = t_0 \left(\left(1+\lambda\right)d(1+\rho^{-1})(1+4^d) + 1 + \rho\right) = 1/2.$$ By , for $0 \le t \le t_0$ and $x^1,x^2 \in D^{\infty,\rho}_t$, we have $$\begin{aligned}
{\left|\left|\hat{\Gamma}(x^1)-\hat{\Gamma}(x^2)\right|\right|}_{\rho,t} &\le
t_0\left(\left(1+\lambda\right)d(1+\rho^{-1})(1+4^d) + 1 + \rho\right)
{\left|\left|x^1-x^2\right|\right|}_{\rho,t} \\
&= \gamma {\left|\left|x^1-x^2\right|\right|}_{\rho,t}.\end{aligned}$$ Furthermore, because $\lambda^n \to \lambda$, there exists some $N_\lambda$ such that $\lambda^n < \lambda + 1$ for $n \ge N_\lambda$, and thus for such $n$, $$\begin{aligned}
{\left|\left|\hat{\Gamma}^n(x^1)-\hat{\Gamma}^n(x^2)\right|\right|}_{\rho,t} &\le t_0 \left(\lambda^n d(1+\rho^{-1})(1+4^d) + 1 + \rho\right) {\left|\left|x^1-x^2\right|\right|}_{\rho,t} \\
&\le t_0 \left(\left(1+\lambda\right)d(1+\rho^{-1})(1+4^d) + 1 + \rho\right) {\left|\left|x^1-x^2\right|\right|}_{\rho,t} \\
&= \gamma {\left|\left|x^1-x^2\right|\right|}_{\rho,t}.\end{aligned}$$ Thus for $0 \le t \le t_0$, $\hat{\Gamma}$ and $\hat{\Gamma}^n$ for $n \ge N_\lambda$ are contractions on the space $D^{\infty,\rho}_t$ with coefficient $\gamma = 1/2$. Recall that $x$ and $x^n$ are the fixed points of $\hat{\Gamma}$ and $\hat{\Gamma}^n$, respectively, and that therefore each can be found by repeated iteration of an arbitrary point in $D^{\infty,\rho}_t$. Specifically, we define $$\begin{aligned}
x^0 &= 0 \quad &x^{n,0} &= 0 \\
x^r &= \hat{\Gamma}(x^{r-1}) \quad &x^{n,r} &=
\hat{\Gamma}^n(x^{n,r-1}), \quad r \ge 1.\end{aligned}$$ Then the following inequalities hold: $$\begin{aligned}
{\left|\left|x - x^r\right|\right|}_{\rho,t} &\le {\gamma^n \over 1 - \gamma} {\left|\left|x^1-x^0\right|\right|}_{\rho,t}
= 2^{-r+1}{\left|\left|x^1\right|\right|}_{\rho,t}, \notag\\
{\left|\left|x^n - x^{n,r}\right|\right|}_{\rho,t} &\le {\gamma^n \over 1 - \gamma}
{\left|\left|x^{n,1}-x^{n,0}\right|\right|}_{\rho,t} = 2^{-r+1}{\left|\left|x^{n,1}\right|\right|}_{\rho,t}.
\label{eq:contraction-n}\end{aligned}$$ Observe $$\begin{aligned}
{\left|\left|x^1\right|\right|}_{\rho,t} &= {\left|\left|\hat{\Gamma}(0)\right|\right|}_{\rho,t} =
{\left|\left|b + y\right|\right|}_{\rho,t} \le {\left|\left|b\right|\right|}_{\rho} + {\left|\left|y\right|\right|}_{\rho,t} \le 2K,\end{aligned}$$ and similarly $${\left|\left|x^{n,1}\right|\right|}_{\rho,t} \le {\left|\left|b^n\right|\right|}_{\rho} + {\left|\left|y^n\right|\right|}_{\rho,t} \le 2K.$$
We now argue that there exists $N$ such that for all $n \ge N$ and $i \le i_0$, ${\left|\left|x_i^n - x_i\right|\right|}_t < \epsilon$ for $t \le t_0$. This will establish continuity of $f$ with respect to the product topology for such $t$. Observe $$\label{eq:continuity-breakdown}
{\left|\left|x_i^n - x_i\right|\right|}_{t} \le {\left|\left|x_i^n - x_i^{n,r}\right|\right|}_t + {\left|\left|x_i^{n,r} - x_i^r\right|\right|}_t
+ {\left|\left|x_i^r - x_i\right|\right|}_t.$$ We bound these three terms individually. By , we have $$\begin{aligned}
{\left|\left|x_i^n - x_i^{n,r}\right|\right|}_t &\le \rho^i{\left|\left|x^n-x^{n,r}\right|\right|}_{\rho,t} \\
&\le 2^{-r+1}\rho^i\left({\left|\left|b^n\right|\right|}_\rho + {\left|\left|y^n\right|\right|}_{\rho,t}\right) \\
&\le 2^{-r+2}K\rho^{i_0}.\end{aligned}$$ For $$\label{eq:r-bound}
r > 2 + \log_2{3\rho^{i_0}K \over \epsilon},$$ we have $$\label{eq:bounded-nr}
{\left|\left|x_i^n - x_i^{n,r}\right|\right|}_t < \epsilon / 3 \quad \text{ for all } i \le i_0.$$ Via a similar argument, for $r$ satisfying , we also have $$\label{eq:bounded-limitr}
{\left|\left|x_i - x_i^r\right|\right|}_t < \epsilon / 3 \quad \text{ for all } i \le i_0.$$ Finally, we consider $$\begin{aligned}
{\left|\left|x_i^{n,r+1} - x_i^{r+1}\right|\right|}_t &\le |b_i^n - b_i| + {\left|\left|y_i^n - y_i\right|\right|}_t \\
&\quad\quad + \lambda^n d \int_0^t \big({\left|\left|x^{n,r}_{i-1} - x^r_{i-1}\right|\right|}_s
+ {\left|\left|x^{n,r}_i - x^r_i\right|\right|}_s \\
&\quad\quad\quad + {\left|\left|g^n(x^{n,r}_{i-1}) - g(x^r_{i-1})\right|\right|}_s
+ {\left|\left| g^n(x^{n,r}_i) - g(x^r_i)\right|\right|}_s \big)ds \\
&\quad\quad + |\lambda - \lambda^n|
d\int_0^t\big({\left|\left|x^r_{i-1}\right|\right|}_s + {\left|\left|x^r_i\right|\right|}_s \\
&\quad\quad\quad + {\left|\left|g(x^r_{i-1})\right|\right|}_s + {\left|\left|g(x^r_i)\right|\right|}_s\big)ds \\
&\quad\quad + \int_0^t\left({\left|\left|x^{n,r}_{i} - x^r_i\right|\right|}_s +
{\left|\left|x^{n,r}_{i+1}-x^r_{i+1}\right|\right|}_s\right)ds. \end{aligned}$$ By Lemma \[thm:g\^m-lipschitz\], $g$ is Lipschitz continuous when restricted to $D^{\eta+2,\rho}_t$, so we have $$\begin{aligned}
{\left|\left|g(x^r_{i-1})\right|\right|}_s + {\left|\left|g(x^r_i)\right|\right|}_s &\le 4^d{\left|\left|x^r_{i-1}\right|\right|}_s +
4^d{\left|\left|x^r_i\right|\right|}_s.\end{aligned}$$ Further note the bound $${\left|\left|g^n(x_i^{n,r}) - g(x_i^r)\right|\right|}_s \le {\left|\left|g^n(x_i^{n,r}) - g(x_i^{n,r})\right|\right|}_s
+ {\left|\left|g(x_i^{n,r}) - g(x_i^r)\right|\right|}_s.$$ Using these pieces, crudely bounding integrals for some terms, and rearranging we have $$\begin{aligned}
{\left|\left|x_i^{n,r+1} - x_i^{r+1}\right|\right|}_t &\le |b_i^n - b_i| + {\left|\left|y_i^n - y_i\right|\right|}_t \\
&\quad\quad + t{\left|\left|g^n(x^{n,r}_{i-1}) - g(x^{n,r}_{i-1})\right|\right|}_t
+ t{\left|\left|g^n(x^{n,r}_{i}) - g(x^{n,r}_{i})\right|\right|}_t \\
&\quad\quad + |\lambda - \lambda^n|t(1+4^d)\left({\left|\left|x^r_{i-1}\right|\right|}_t + {\left|\left|x^r_i\right|\right|}_s\right) \\
&\quad\quad + \lambda^n d \int_0^t
\left({\left|\left|x^{n,r}_{i-1} - x^r_{i-1}\right|\right|}_s
+ {\left|\left|x^{n,r}_i - x^r_i\right|\right|}_s \right)ds \\
&\quad\quad + \lambda^n d \int_0^t
\left({\left|\left|g(x^{n,r}_{i-1}) - g(x^r_{i-1})\right|\right|}_s
+ {\left|\left|g(x^{n,r}_i) - g(x^r_i)\right|\right|}_s \right)ds \\
&\quad\quad + \int_0^t\left({\left|\left|x^{n,r}_{i} - x^r_i\right|\right|}_s +
{\left|\left|x^{n,r}_{i+1}-x^r_{i+1}\right|\right|}_s\right)ds. \end{aligned}$$ Recall that for any $t \ge 0$ we have $x^{n,r}(t) \in {\mathcal{Z}^{\eta^n}}$ and thus $\left|x^{n,r}_i(t)\right| \le \left(\eta^n\right)^{\alpha}$ for $i \le i^*_n$, and $\left|x^{n,r}_i(t)\right| \le \eta^n +1$ for $i >
i^*_n$, so the conditions of Lemma \[thm:g\^m-converge\] are satisfied and therefore for $i \le i_0$ we have ${\left|\left|g^n(x_{i}^{n,r}) - g(x_i^{n,r})\right|\right|}_t \to 0$, and similarly ${\left|\left|g^n(x_{i-1}^{n,r}) - g(x_{i-1}^{n,r})\right|\right|}_t \to 0$. This, along with $(b^n,y^n,\lambda^n) \to (b,y,\lambda)$ implies that for any $\delta > 0$ we can choose $N_\delta$ such that for $n \ge N_\delta$ we have $$\lambda_n \le \lambda + 1,$$ $$\eta^n \le \eta + 1,$$ and for all $i \le i_0 + r + 1$, $$\begin{aligned}
&|b_i^n - b_i| + {\left|\left|y_i^n - y_i\right|\right|}_t \\
&\quad + t{\left|\left|g^n(x^{n,r}_{i-1}) - g(x^{n,r}_{i-1})\right|\right|}_t
+ t{\left|\left|g^n(x^{n,r}_{i}) - g(x^{n,r}_{i})\right|\right|}_t \\
&\quad + |\lambda - \lambda^n|t(1+4^d)\left({\left|\left|x^r_{i-1}\right|\right|}_t
+ {\left|\left|x^r_i\right|\right|}_t\right) \quad\quad < \delta.\end{aligned}$$ Observe that $\eta^n \le \eta + 1$ implies $x_i^{n,r} \in D^{\eta + 2,\rho}_t$, so $${\left|\left|g(x_i^{n,r}) - g(x_i^r)\right|\right|}_s \le 4^d{\left|\left|x_i^{n,r} - x_i^r\right|\right|}_s.$$ For $n \ge N_\delta$, we have for all $i \le i_0 + r + 1$ $$\begin{aligned}
{\left|\left|x_i^{n,r+1} - x_i^{r+1}\right|\right|}_t &< \delta +
\int_0^t\left({\left|\left|x^{n,r}_{i} - x^r_i\right|\right|}_s +
{\left|\left|x^{n,r}_{i+1}-x^r_{i+1}\right|\right|}_s\right)ds \\
&\quad\quad + (1+4^d)(1+\lambda) d \int_0^t
\left({\left|\left|x^{n,r}_{i-1} - x^r_{i-1}\right|\right|}_s
+ {\left|\left|x^{n,r}_i - x^r_i\right|\right|}_s \right)ds.\end{aligned}$$ We rewrite this as $$\begin{aligned}
{\left|\left|x_i^{n,r+1} - x_i^{r+1}\right|\right|}_t &< \delta +
C
\int_0^t \max_{i-1 \le j \le i+1}{\left|\left|x^{n,r}_j - x^r_j\right|\right|}_s ds,
$$ where $C = \left(2(1+4^d)(1+\lambda) d + 2\right)$. For $i \le i_0$, this can be expanded as $$\begin{aligned}
{\left|\left|x_i^{n,r+1} - x_i^{r+1}\right|\right|}_t &< \delta + Ct \max_{i-1 \le j \le i+1}{\left|\left|x^{n,r}_j - x^r_j\right|\right|}_t \\
&< \delta + Ct\left(\delta +
Ct\max_{i-2 \le j \le i+2}{\left|\left|x^{n,r-1}_j - x^{r-1}_j\right|\right|}_t\right) \\
&< \delta \sum_{k = 0}^{r}(Ct)^k + (Ct)^{r+1}\max_{(i - r-1) \wedge 0 \le j \le i+r+1}{\left|\left|x_j^{n,0} - x^0_j\right|\right|}_t\end{aligned}$$ Recall that $x^{n,0} = x^0 = 0$, so for all $i \ge 0$ we have ${\left|\left|x_i^{n,0} - x_i^0\right|\right|}_t = 0$, and thus $$\begin{aligned}
{\left|\left|x_i^{n,r+1} - x_i^{r+1}\right|\right|}_t < \delta {(Ct)^{r+1}-1 \over Ct - 1}.\end{aligned}$$ Reindexing gives, for all $i \le i_0$, $${\left|\left|x_i^{n,r} - x_i^{r}\right|\right|}_t < \delta {(Ct)^{r}-1 \over Ct - 1},$$ and thus for $$\delta < {\epsilon \over 3} \cdot {Ct - 1 \over (Ct)^{r}-1 },$$ we have $$\label{eq:bounded-diff}
{\left|\left|x_i^{n,r} - x_i^{r}\right|\right|}_t < \epsilon / 3.$$ Thus by plugging ,, and into , if we choose $r$ and $n$ such that $$r > 2 + \log_2{3\rho^{i_0}K \over \epsilon}, \quad \text{ and } \quad
n \ge \max\left(N_\lambda,N_\delta\right),$$ we have for all $i \le i_0$, $${\left|\left|x_i^n - x_i\right|\right|}_t < \epsilon,$$ establishing the continuity of $f$ for $t \le t_0$.
As in the proof of existence and uniqueness, we define a shifted version of $y$ by $\tilde{y}(u) = y(u+t_0)$ observe that $x(t)$ for $t = u + t_0$ and $u \ge 0$ is the solution $\hat{z}$ to the system $$\begin{aligned}
z_0(u) &= 0 \\
z_i(u) &= x_i(t_0) - y_i(t_0) + \tilde{y}_i(u) \\
&\quad\quad - \lambda d\int_{0}^{u}\left(x_i(s) - x_{i-1}(s) -
g^\eta(x_i(s)) + g^\eta(x_{i-1}(s)) \right)ds \\
&\quad\quad + \int_{0}^{u}\left(x_{i+1}(s) - x_i(s)\right)ds, \quad i \ge 1 \\
\hat{z}(u) &= \begin{cases} z(u) & u < u^* \\ z(u^*) & u \ge u^*, \end{cases} \end{aligned}$$ where $u^*$ is defined analogously to $t^*$ in . Observe that this is the system -, with arguments $$(x(t_0) - y(t_0), \tilde{y}, \lambda, \eta) \in {\mathbb{R}}^{\infty,\rho} \times
D^{\infty,\rho}_t \times {\mathbb{R}}\times {{\mkern 4mu\overline{\mkern-4mu{\mathbb{R}}\mkern-4mu}\mkern 4mu}}_{\ge 1},$$ and furthermore $\hat{z}(0) = x(t_0) - y(t_0) + \tilde{y}(0) = x(t_0)
\in {\mathcal{Z}^{\eta}}$. Also ${\left|\left|\tilde{y}\right|\right|}_{\rho,u} \le {\left|\left|y\right|\right|}_{\rho,u+t_0} < K$ and by Lemma \[thm:uniform-bound\] we have $$\begin{aligned}
{\left|\left|x(t_0)-y(t_0)\right|\right|}_{\rho,t} &\le {\left|\left|y(t_0)\right|\right|}_\rho + {\left|\left|x(t_0)\right|\right|}_\rho \\
&\le K + \left({\left|\left|b\right|\right|}_\rho + {\left|\left|y\right|\right|}_{t_0,\rho}\right)
e^{\left(\lambda d(1+4^d)\left(1 + \rho^{-1}\right) + 1 + \rho\right)t_0} \\
&\le K +
2Ke^{\left((1+\lambda)d(1+4^d)\left(1 + \rho^{-1}\right)
+ 1 + \rho\right)t_0}.\end{aligned}$$ We define $$K_1 {\,{\buildrel \triangle \over =}\,}K +
2Ke^{\left((1+\lambda)d(1+4^d)\left(1 + \rho^{-1}\right)
+ 1 + \rho\right)t_0},$$ so $$(x(t_0) - y(t_0), \tilde{y}, \lambda, \eta) \in {Z^{\alpha}_{K_1}}.$$ A similar construction allows us to define $\tilde{y}^n$ and $\hat{z}^n$, and we have $\tilde{y}^n \to \tilde{y}$ in $D^{\infty,\rho}_u$ for any $u \ge 0$. For sufficiently large $n$ we have $\lambda^n < \lambda + 1$, and thus $$\begin{aligned}
{\left|\left|x^n(t_0)-y^n(t_0)\right|\right|}_{\rho,t} &\le K +
2Ke^{\left(\lambda^nd(1+4^d)\left(1 + \rho^{-1}\right)
+ 1 + \rho\right)t_0} \\
&\le K +
2Ke^{\left((1+\lambda)d(1+4^d)\left(1 + \rho^{-1}\right)
+ 1 + \rho\right)t_0} = K_1,\end{aligned}$$ so $$(x^n(t_0) - y^n(t_0),\tilde{y}^n,\lambda^n,\eta^n) \in {Z^{\alpha}_{K_1}}.$$ Therefore we can repeat the continuity argument above to show $\hat{z}^n \to \hat{z}$ for $u \le t_0$, which implies $x^n \to x$ for $t \in [t_0,2t_0]$. This extension argument can be repeated to prove $x^n \to x$ for $[2t_0,3t_0],[3t_0,4t_0], \ldots$. Thus for any $t \ge 0$ we have $x^n \to x$ in $D^{\infty,\rho}_t$ and thus $f$ is continuous.
Martingale representation. {#sec:martingales}
==========================
We now show that the stochastic process underlying the supermarket system stopped at some appropriate time can be written in a form that exactly matches that of $\hat{T}$ in . This will allow us to use Theorem \[thm:const-integral\] to prove Theorem \[thm:const-result\] in the next section.
Before introducing the stopped variant, we will consider the original supermarket system and show that it can be represented by the equations - for a particular choice of arguments $(b,y,\lambda, \eta)$.
For $i \ge 1$, recall the representation . Given the definition $T_i^n = \eta_n\left(1 - S_i^n\right)$ we can rewrite this as $$\begin{aligned}
T_i^n(t) &= T_i^n(0) - {\eta_n \over n}A_i\left(\lambda_n n
\int_0^t\left(\left(1 - {1 \over \eta_n}T_{i-1}^n(s)\right)^d-\left(1 -
{1 \over \eta_n}T_i^n(s)\right)^d\right)ds\right) \notag\\
&\quad\quad + {\eta_n \over n} D_i\left( n \int_0^t\left(\left(1-{1 \over
\eta_n}T_i^n(s)\right) - \left(1 -
{1\over\eta_n}T_{i+1}^n(s)\right)\right)ds \right) \notag\\
&= T_i^n(0) - {\eta_n \over n}A_i\bigg(\lambda_n n
\int_0^t\bigg(\left(1 - {d \over
\eta_n}T_{i-1}^n(s) + {d \over \eta_n} g^{\eta_n}(T_{i-1}^n(s))\right) \notag\\
&\hspace{0.25\textwidth} -\left(1 - {d
\over \eta_n}T_i^n(s) + {d \over \eta_n}
g^{\eta_n}(T_i^n(s))\right)\bigg)ds\bigg)
\notag\\
&\quad\quad + {\eta_n \over n} D_i\left( {n \over \eta_n}
\int_0^t\left(T_{i+1}^n(s) - T_{i}^n(s)\right)ds \right)
\notag\\
&= T_i^n(0) - {\eta_n \over n}A_i\left(\lambda_n d
{n \over \eta_n}\int_0^t\left(T_{i}^n(s) - T_{i-1}^n(s) - g^{\eta_n}(T_i^n(s))
+ g^{\eta_n}(T_{i-1}^n(s))\right)ds\right) + \notag\\
&\quad\quad + {\eta_n \over n}D_i\left({n \over \eta_n}
\int_0^t\left(T_{i+1}^n(s) - T_i^n(s)\right)ds\right),
\label{eq:T-form-i-no-martingales}\end{aligned}$$ where $g^{\eta_n}$ is defined as in . We now define scaled martingale processes $$\begin{aligned}
M_i^n(t) &= {\eta_n \over n}A_i\left(\lambda_n d
{n \over \eta_n} \int_0^t\left(T_{i}^n(s) - T_{i-1}^n(s) + g^{\eta_n}(T_i^n(s))
- g^{\eta_n}(T_{i-1}^n(s))\right)ds\right) \notag\\
&\quad\quad - \lambda_n d
\int_0^t\left(T_{i}^n(s) - T_{i-1}^n(s) + g^{\eta_n}(T_i^n(s)) -
g^{\eta_n}(T_{i-1}^n(s))\right)ds, \label{eq:martingale-arrival}\\
N_i^n(t) &= {\eta_n \over n}D_i\left({n \over \eta_n}\int_0^t\left(T_{i+1}^n(s)
- T_{i}^n(s)\right)ds\right) - \int_0^t\left(T_{i+1}^n(s) -
T_{i}^n(s)\right)ds. \label{eq:martingale-departure}\end{aligned}$$ Now we can rewrite the system for $i \ge 1$ as $$\begin{aligned}
T_i^n(t) &= T_i^n(0) - M_i^n(t) - \lambda_n d\int_0^t\left(T_i^n(s) -
T_{i-1}^n(s) + g^{\eta_n}(T_i^n(s)) - g^{\eta_n}(T_{i-1}^n(s)\right)ds \notag\\
&\quad\quad + N_i^n(t) + \int_0^t\left(T_{i+1}^n(s) - T_i^n(s)\right)ds.
\label{eq:T-form-i}\end{aligned}$$ This representation matches with $b = T^n(0)$, $y = -M^n
+ N^n$, $\lambda = \lambda_n$ and $\eta = \eta_n$.
Recall that the assumptions of Theorem \[thm:const-result\] include constants $\rho > 1$ and $0 < \alpha < 1/2$. We now define $i^*_n = {\alpha \over 2} \log_\rho \eta_n$ and define a stopping time $$t^*_n = \inf\left\{t \ge 0 : \exists\text{ } i {\text{ s.t. }}1 \le i \le i^*_n
\text{ and } T_i^n(t) \ge \eta_n^{\alpha}\right\}.$$ Note that compared to , this stopping time does not contain terms checking $T_i^n(t) \le -(\eta_n)^{\alpha}$, or $\left|T_i^n(t)\right| \ge \eta_n + 1$. This is because $0 \le T_i^n(t) \le \eta_n$ for all $i \ge 0$ and for all $t \ge 0$, so such conditions are never met. Thus $t^*_n$ is equivalent to the stopping time defined by . We consider the process $\hat{T}^n$ defined by $$\hat{T}_i^n(t) = \begin{cases} T_i^n(t) & t < t^*_n \\ T_i^n(t^*_n) & t
\ge t^*_n.\end{cases}$$
As noted above, the stopped supermarket model $\hat{T}^n$ is the unique solution of the integral equation system described in Theorem \[thm:const-integral\], with arguments $b = T^n(0)$, $y = -M^n + N^n$, $\lambda = \lambda_n$, and $\eta = \eta_n$.
Martingale convergence. {#sec:martingale-convergence}
=======================
Because our sequence of supermarket models indexed by $n$ are all examples of the integral equation system , and Theorem \[thm:const-integral\] shows that this system defines a continuous map from arguments $(b,y,\lambda,\eta) \in {Z^{\alpha}_{K}}$ for some $K > 0$ to the stopped system $f(b,y,\lambda,\eta) = \hat{T}$, we will find the weak limit of the finite system $\hat{T}^n$ by finding the limits of the arguments $(T^n(0), -M^n + N^n, \lambda_n, \eta_n)$. Though we do not use the continuous mapping theorem because we do not have $(T^n(0), -M^n + N^n, \lambda_n, \eta_n) \in {Z^{\alpha}_{K}}$ almost surely for any non-random $K$, the proof will still rely on the continuity of $f$ and the limits of the arguments. Three of these limits are given, as provides $\hat{T}^n(0) \Rightarrow T(0)$, and we have $\lambda_n \to 1$ and $\eta_n \to \infty$. We claim $-M^n + N^n \Rightarrow 0$.
We now prove the following:
\[thm:martingale-limit\] For $M^n$ and $N^n$ as defined in -, if the assumptions of Theorem \[thm:const-result\] hold, then $${\mathbb{E}}{\left|\left|M^n\right|\right|}_{\rho,t}, {\mathbb{E}}{\left|\left|N^n\right|\right|}_{\rho,t} \to 0 \quad \text{ as } n \to \infty.$$ This implies $M^n, N^n \Rightarrow 0$ in $D^{\infty,\rho}_t$ equipped with the product topology
Before proving this proposition, we prove a bound on the unbounded $\rho$-norm of $T^n$ in expectation.
\[thm:const-expectation-finite\] For any $\gamma > 1$ we have $$\label{eq:const-expectation-finite}
{\mathbb{E}}\left[{\left|\left|T^n\right|\right|}_{\gamma,t}\right] \le {\mathbb{E}}\left[{\left|\left|T^n(0)\right|\right|}_\gamma\right]
e^{\gamma t}.$$
By dropping negative terms from we have $$T_i^n(t) \le T_i^n(0) + {\eta_n
\over n}D_i\left({n \over \eta_n}\int_0^tT_{i+1}^n(s)ds\right).$$ Then we have $$\begin{aligned}
{\mathbb{E}}\left[{\left|\left|T_i^n\right|\right|}_t\right] &\le {\mathbb{E}}T_i^n(0) +
{\mathbb{E}}\left[\int_0^tT_{i+1}^n(s)ds\right]
\\
&\le {\mathbb{E}}T_i^n(0) + \int_0^t{\mathbb{E}}\left[{\left|\left|T_{i+1}^n\right|\right|}_s\right]ds.\end{aligned}$$ This implies $$\begin{aligned}
{\mathbb{E}}\left[{\left|\left|T^n\right|\right|}_{\gamma,t}\right] &= \sum_{i \ge
1}\gamma^{-i}{\mathbb{E}}\left[{\left|\left|T_i^n\right|\right|}_t\right] \\
&\le {\mathbb{E}}\left[{\left|\left|T^n(0)\right|\right|}_\gamma\right] + \sum_{i \ge 1}
\gamma^{-i}\int_0^t{\mathbb{E}}\left[{\left|\left|T_{i+1}^n\right|\right|}_s\right]ds \\
&\le {\mathbb{E}}\left[{\left|\left|T^n(0)\right|\right|}_\gamma\right] +
\gamma\int_0^t{\mathbb{E}}\left[{\left|\left|T^n\right|\right|}_{\gamma,s}\right]ds.\end{aligned}$$ We can now apply Lemma \[thm:gronwall\] to conclude $${\mathbb{E}}\left[{\left|\left|T^n\right|\right|}_{\gamma,t}\right] \le {\mathbb{E}}\left[{\left|\left|T^n(0)\right|\right|}_\gamma\right]
e^{\gamma t},$$ as desired.
We first prove the statement for $N^n$. We begin with some observations about $N^n$ and introduce some additional definitions. Let $$\label{eq:tau-N}
\tau^n_i {\,{\buildrel \triangle \over =}\,}\int_0^t\left(T_{i+1}^n(s) -
T_{i}^n(s)\right)ds, \quad i \ge 0.$$ Observe for all $i \ge 0$ $$\label{eq:tau-N-bound}
\tau^n_i \le t{\left|\left|T_{i+1}^n\right|\right|}_t.$$ Now observe $$\begin{aligned}
{\left|\left|N^n_i\right|\right|}_t &= \sup_{0 \le u \le t}\left| {\eta_n \over n}D_i\left({n
\over \eta_n}\int_0^u\left(T_{i+1}^n(s)
- T_{i}^n(s)\right)ds\right) - \int_0^u\left(T_{i+1}^n(s) -
T_{i}^n(s)\right)ds\right| \notag\\
&= \sup_{0 \le s \le \tau^n_i}\left|{\eta_n \over n}D_i\left({n
\over \eta_n}s\right) - s\right| \notag\\
&= \sup_{0 \le u \le {n \tau^n_i \over \eta_n}} {\eta_n \over
n}\left|D_i(u)-u\right|. \label{eq:CLT-setup}\end{aligned}$$
We claim $$\label{eq:kappa-exp-N-short}
\lim_{n \to \infty} \max_{1 \le i \le i^*_n}
{\mathbb{E}}{\left|\left|N_i^n\right|\right|}_{t} = 0.$$ By Lemma \[thm:const-expectation-finite\] with $\gamma = \rho$ we have $${\mathbb{E}}\left[{\left|\left|T^n\right|\right|}_{\rho,t}\right] \le
{\mathbb{E}}\left[{\left|\left|T^n(0)\right|\right|}_{\rho}\right] e^{\rho t}.$$ Recall , fix some $\epsilon > 0$ and let $$C = \left(\limsup_{n \to
\infty}{\mathbb{E}}\left[{\left|\left|T^n(0)\right|\right|}_{\rho}\right]+\epsilon\right) e^{\rho t}.$$ For all sufficiently large $n$ we have $${\mathbb{E}}\left[{\left|\left|T^n\right|\right|}_{\rho,t}\right] \le C.$$ This further implies that for all $1 \le i \le i^*_n+1$ we have $${\mathbb{E}}\left[{\left|\left|T_i^n\right|\right|}_t\right] \le C\rho^{ i} \le C\rho^{i^*_n+1}.$$ Define $$\delta_n = {1 \over \left(\log_\rho \eta_n\right)^2},$$ and define the event $$A = \left\{\exists \text{ } i \le i^*_n+1 {\text{ s.t. }}{\left|\left|T_i^n\right|\right|}_t \ge {1
\over \delta_n}C\rho^{i^*_n+1}\right\}.$$ By Markov’s inequality and the union bound $$\label{eq:T-small-small-i}
{\mathbb{P}}\left(A\right) \le \delta_n (i^*_n+1) = {\alpha \over 2\log_\rho
\eta_n}
+ {1 \over \left(\log_\rho \eta_n\right)^2}.$$ Observe $$\begin{aligned}
{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_{t}\right] &=
{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t
{\mathbbm{1}}\{A\}\right] + {\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t {\mathbbm{1}}\{A^c\}\right].
\label{eq:breakdown}\end{aligned}$$ We bound these two terms separately. We first consider the second term of . Recall and observe $$\label{eq:CLT-setup-A^c}
{\left|\left|N_i^n\right|\right|}_t{\mathbbm{1}}\{A^c\} \le \sup_{0 \le u \le {n \tau^n_i \over
\eta_n}}
{\eta_n \over n}\left|D_i(u)-u\right|.$$ For $0 \le i \le i^*_n$, $A^c$ and imply $$\begin{aligned}
\tau^n_i &\le {t \over \delta_n}C \rho^{i^*_n+1 } \\
&= t\rho C {\eta_n^{\alpha/ 2} \over \delta_n}.\end{aligned}$$ Applying this for we obtain $${\left|\left|N_i^n\right|\right|}{\mathbbm{1}}\{A^c\} \le \sup{\eta_n \over
n}\left|D_i(u)-u\right|,$$ where the supremum is over $$0 \le u \le t\rho C n \delta_n^{-1} \eta_n^{{\alpha \over 2} - 1}.$$ We define $$\begin{aligned}
\nu_n &{\,{\buildrel \triangle \over =}\,}n \delta_n^{-1} \eta_n^{{\alpha \over 2} - 1} \\
&= {n \over \eta_n^{1-\alpha / 2}}\left(\log_\rho \eta_n \right)^2.\end{aligned}$$ Recall that $\eta_n = {O\left(\sqrt{n}\right)}$, so $$\eta_n^{1 - \alpha / 2} = {O\left(n^{1/2-\alpha /4}\right)},$$ so $\nu_n \to \infty$ as $n \to \infty$. Thus we have $$\begin{aligned}
{\left|\left|N_i^n\right|\right|}{\mathbbm{1}}\{A^c\} &\le \sup_{0 \le u \le t\rho C
\nu_n}{\eta_n
\over n}\left|D_i(u) - u\right| \\
&= {\eta_n \sqrt{\nu_n} \over n}\sup_{0 \le u \le t\rho C
\nu_n}{1 \over \sqrt{\nu_n}}\left|D_i(u) - u\right|.\end{aligned}$$ Let $$\gamma_n {\,{\buildrel \triangle \over =}\,}{\eta_n \sqrt{\nu_n} \over n}{\mathbb{E}}\sup_{0 \le u \le
t\rho C \nu_n}{1 \over \sqrt{\nu_n}}\left|D_i(u) - u\right|.$$ Thus ${\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t{\mathbbm{1}}\{A^c\}\right] \le
\gamma_n$ for $1 \le i \le i^*_n$. By the Functional Central Limit Theorem (FCLT), since $\nu_n \to \infty$ we have $$\sup_{0 \le u \le t\rho C
\nu_n}{1 \over \sqrt{\nu_n}}\left|D_i(u) - u\right| \Rightarrow \sup_{0 \le u
\le t\rho C} |B(u)|,$$ where $B$ is a standard Brownian motion. Furthermore, observe that $$\begin{aligned}
{\eta_n \sqrt{\nu_n} \over n} &= n^{-1/2}\eta_n^{1/2+
\alpha /4}\log_\rho\eta_n.\end{aligned}$$ By assumption we have $\eta_n = {O\left(\sqrt{n}\right)}$, so $$\begin{aligned}
\eta_n^{1/2+\alpha/4} &= {O\left(n^{1/4 + \alpha / 8}\right)}. \end{aligned}$$ Recall that $\alpha < 1/2$ so $1/4 + \alpha / 8 < 1/2$ which implies $${\eta_n \sqrt{\nu_n}\over n} \to 0,$$ and thus $\gamma_n \to 0$.
We now consider the first term of . By the Cauchy-Schwarz inequality and , we have $$\begin{aligned}
{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t {\mathbbm{1}}\{A\}\right] &\le
\sqrt{{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t^{2}\right]}\sqrt{{\mathbb{P}}(A)} \end{aligned}$$ By and $T_i(t) \le \eta_n$ for all $i \ge 1$, we have $$\begin{aligned}
{\left|\left|N_i^n\right|\right|}_t &\le \sup_{0 \le u \le nt}
{\eta_n \over n}\left|D_i(u)-u\right| \\
&= {\eta_n \over \sqrt{n}}\sup_{0 \le u \le nt}
{1 \over \sqrt{n}}\left|D_i(u)-u\right|.\end{aligned}$$ Let $$\label{eq:w_n}
w_n {\,{\buildrel \triangle \over =}\,}{\eta_n \over \sqrt{n}}{\mathbb{E}}\left[\sup_{0 \le u \le nt}
{1 \over \sqrt{n}}\left|D_i(u)-u\right|\right],$$ and $$z_n {\,{\buildrel \triangle \over =}\,}{\eta_n^2 \over n}{\mathbb{E}}\left[\left(\sup_{0 \le u \le nt}
{1 \over \sqrt{n}}\left|D_i(u)-u\right|\right)^2\right].$$ Then we have, for all $i \ge 1$ $${\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t\right] \le w_n \quad \text{ and } \quad
{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t^2\right] \le z_n.$$ By the FCLT, we have $$\sup_{0 \le u \le nt}{1 \over \sqrt{n}}\left|D_i(u) - u\right| \Rightarrow \sup_{0 \le u
\le t} |B(u)|,$$ where $B$ is a standard Brownian motion. We now have $$\begin{aligned}
{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t {\mathbbm{1}}\{A\}\right] &\le \sqrt{z_n}
\sqrt{{\alpha \over 2\log_\rho \eta_n}
+ {1 \over \left(\log_\rho \eta_n\right)^2}}.\end{aligned}$$ Because $\eta_n = {O\left(\sqrt{n}\right)}$, as $n \to \infty$ we have $${\eta_n^2 \over n \log_\rho \eta_n} \to 0,$$ and thus $$\begin{aligned}
{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t {\mathbbm{1}}\{A\}\right] &\le
\sqrt{{z_n \alpha \over 2\log_\rho \eta_n}
+ {z_n \over \left(\log_\rho \eta_n\right)^2}} \to 0.\end{aligned}$$
Returning to , we obtain $$\begin{aligned}
{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t\right] &=
{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t{\mathbbm{1}}\{A\}\right] +
{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t {\mathbbm{1}}\{A^c\}\right] \\
&\le \sqrt{{z_n \alpha \over 2\log_\rho \eta_n}
+ {z_n \over \left(\log_\rho \eta_n\right)^2}} + \gamma_n,\end{aligned}$$ and thus $$\lim_{n \to \infty} \max_{1 \le i \le i^*_n}
{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t\right]
\le \lim_{n \to \infty} \sqrt{{z_n \alpha \over 2\log_\rho \eta_n}
+ {z_n \over \left(\log_\rho \eta_n\right)^2}} + \gamma_n = 0,$$ which establishes the claim .
Now consider $$\begin{aligned}
{\mathbb{E}}\left[{\left|\left|N^n\right|\right|}_{\rho,t}\right] &= \sum_{i \ge
1}\rho^{-i}{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t\right] \\
&\le \max_{1 \le i \le
i^*_n}{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t\right]\sum_{1
\le i \le i^*_n}\rho^{-i}
+ \sum_{i > i^*_n}\rho^{-i}{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t\right] \\
&\le {\rho \over \rho - 1}\max_{1 \le i \le
i^*_n}{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t\right] + \sum_{i > i^*_n}\rho^{-i}w_n \\
&= {\rho \over \rho -1}\max_{1 \le i \le
i^*_n}{\mathbb{E}}\left[{\left|\left|N_i^n\right|\right|}_t\right] + {\rho^{-i
^*_n} w_n \over \rho -1}.\end{aligned}$$ Recall the definition of $w_n$ given by and observe that $${\eta_n \over \sqrt{n}}\rho^{-i^*}
= {\eta_n^{1-\alpha/2} \over \sqrt{n}} \to 0,$$ as $n \to \infty$ implies the second term goes to zero. Recalling , we conclude $${\mathbb{E}}\left[{\left|\left|N^n\right|\right|}_{\rho,t}\right] \to 0.$$ This implies the convergence $N^n \to 0$ in $D^{\infty,\rho}_t$ equipped with the product topology.
The argument to show ${\mathbb{E}}\left[{\left|\left|M^n\right|\right|}_{\rho,t}\right] \to 0$ is similar: we redefine $\tau^n_i$ as $$\tau_i^n = \lambda_n d \int_0^t\left(T_{i}^n(s) - T_{i-1}^n(s) +
g^n(T_i^n(s)) - g^n(T_{i-1}^n(s))\right)ds.$$ By Lemma \[thm:g\^m-lipschitz\], $g^{\eta_n}$ restricted to $D^{\eta_n+2}_t$ is Lipschitz continuous with constant $4^d$ and that $\lambda_n \uparrow 1$, so we have the bound $$\begin{aligned}
\tau^i_n &\le td(1+4^d){\left|\left|T_i^n\right|\right|}_t + td4^d{\left|\left|T_{i-1}^n\right|\right|}_t.\end{aligned}$$ This bound replaces and the rest of the argument proceeds essentially identically to the $N^n$ case.
We are now prepared to prove our main result:
We first claim $\hat{T}^n \Rightarrow T$. We established in Section \[sec:martingales\] that $\hat{T}^n = f(T^n(0), -M^n+N^n, \lambda_n, \eta_n)$ where $f$ is the function defined in Theorem \[thm:const-integral\]. In Proposition \[thm:martingale-limit\] we established $-M^n + N^n
\Rightarrow 0$. We prove $f(T^n(0), -M^n+N^n, \lambda_n, \eta_n) \Rightarrow f(T(0),0,1,\infty)$ directly. To do that, we choose a closed set $F \subset D_t^{\infty,\rho}$ and show that $$\begin{aligned}
\label{eq:convergence-claim}
\limsup_n {\mathbb{P}}&\left(f(T^n(0), -M^n+N^n, \lambda_n, \eta_n) \in F\right) \notag \\
&\le {\mathbb{P}}\left(f(T(0),0,1,\infty) \in F\right).\end{aligned}$$ Our approach is to choose some large constant $K > 0$ and consider two cases. Let $A_{K,n}$ be the event $\left\{\max\left({\left|\left|T^n(0)\right|\right|}_{\rho},{\left|\left|-M^n+N^n\right|\right|}_{\rho,t}\right) > K\right\}$ and observe that assumption and Proposition \[thm:martingale-limit\] imply $$\limsup_n{\mathbb{P}}\left(A_{K,n}\right) {\,{\buildrel \triangle \over =}\,}\epsilon_K \to 0 \quad \text{ as } \quad K \to \infty.$$ Let $Y^n = (T^n(0), -M^n+N^n, \lambda_n, \eta_n)$ and $Y = (T(0),0,1,\infty)$. Then $$\begin{aligned}
\limsup_n {\mathbb{P}}(f(Y^n) \in F) &= \limsup_n \bigg({\mathbb{P}}\left(f(Y^n) \in F, A_{K,n}\right) \\
&\quad\quad\quad\quad +
{\mathbb{P}}\left(f(Y^n) \in F, A_{K,n}^c\right)\bigg) \\
&\le \epsilon_K + \limsup_n{\mathbb{P}}\left(f(Y^n) \in F, A_{K,n}^c\right).\end{aligned}$$ Recall - and observe that $M^n(0) = N^n(0) = 0$. Thus $A_{K,n}^c$ and assumption imply $Y^n \in {Z^{\alpha}_{K}}$ for sufficiently large $n$. For such $n$, we have $$\begin{aligned}
{\mathbb{P}}\left(f(Y^n) \in F, A_{K,n}^c\right)
&= {\mathbb{P}}\left(Y^n \in f^{-1}(F), A_{K,n}^c \right) \\
&= {\mathbb{P}}\left(Y^n \in f^{-1}(F) \cap {Z^{\alpha}_{K}}\right).\end{aligned}$$ Because both $F$ and ${Z^{\alpha}_{K}}$ are closed and $f$ is continuous on ${Z^{\alpha}_{K}}$, $ f^{-1}(F) \cap {Z^{\alpha}_{K}}$ is closed. Thus the convergence $Y^n \Rightarrow x$ implies $$\begin{aligned}
\limsup_n{\mathbb{P}}\left(Y^n \in f^{-1}(F) \cap {Z^{\alpha}_{K}} \right)
&\le {\mathbb{P}}\left(Y \in f^{-1}(F) \cap {Z^{\alpha}_{K}} \right) \\
&\le {\mathbb{P}}\left(Y \in f^{-1}(F) \right) \\
&= {\mathbb{P}}\left(f(Y) \in F \right).\end{aligned}$$ Thus $$\limsup_n{\mathbb{P}}\left(f(Y^n) \in F, A_{K,n}^c\right)
\le {\mathbb{P}}\left(f(Y) \in F\right),$$ and $$\limsup_n {\mathbb{P}}\left(f(Y^n) \in F\right) \le \epsilon_K +
{\mathbb{P}}\left(f(Y) \in F\right).$$ Taking the limit $K \to \infty$ on both sides establishes . Thus $$\hat{T}^n \Rightarrow f(T(0),0,1,\infty).$$ By definition, we have $f(T(0),0,1,\infty) = T$. Thus we conclude $$\hat{T}^n \Rightarrow T.$$
By construction, $\hat{T}^n(t)$ and $T^n(t)$ are identical for $t
\in [0,t_n^*]$. Thus it remains to show that for all $t \ge 0$, ${\mathbb{P}}(t_n^*
\le t) \to 0$ as $n \to \infty$. Let $$p_n = {\mathbb{P}}(t_n^* \le t) =
{\mathbb{P}}\left(\exists \text{ } i \le i^*_n {\text{ s.t. }}{\left|\left|T_i^n\right|\right|}_t \ge
\eta_n^{\alpha}\right).$$ We have $$\begin{aligned}
{\mathbb{E}}\left[{\left|\left|T^n\right|\right|}_{\rho,t}\right] &\ge \sum_{1 \le i \le i^*_n} \rho^{-i} {\mathbb{E}}{\left|\left|T_i^n\right|\right|}_t \notag \\
&\ge \rho^{-i^*_n}\sum_{1 \le i \le i^*_n}{\mathbb{E}}{\left|\left|T_i^n\right|\right|}_t \notag \\
&\ge \rho^{-i^*_n}\eta_n^\alpha p_n \notag\\
&= \rho^{-{\alpha \over 2}\log_\rho \eta_n }\eta_n^{\alpha} p_n \notag\\
&= \eta_n^{\alpha / 2} p_n, \label{eq:exp-lower-bound}\end{aligned}$$ We have $\eta_n^{\alpha / 2} \to \infty$ as $n \to \infty$ By Lemma \[thm:const-expectation-finite\] with $\gamma = \rho$ and assumption \[eq:starting-expectation-bounded\], we have $\lim_{n \to \infty}{\mathbb{E}}{\left|\left|T^n\right|\right|}_{\rho,t} < \infty$, so does not diverge to infinity, which implies $p_n \to 0$. This implies $T^n \Rightarrow T$, as desired.
Open questions. {#sec:Conclusions}
===============
In addition to the questions and conjectures already proposed in Section \[sec:steady-state\], another potential future direction would be to characterize the detailed behavior of queues of length $\log_d \eta_n + {O\left(1\right)}$. Not only do we conjecture that in steady state almost all queues are of this type, but we also expect that over finite time the process tracking such queues converges to a diffusion process rather than a deterministic system. The methods used in this paper do not naturally translate to the “intermediate length queue” regime.
| |
In terms of overall gaming performance, the graphical capabilities of the Nvidia GeForce GT 730 v2 are very slightly better than the Intel UHD Graphics 630.
The GT 730 has a 552 MHz higher core clock speed but 7 fewer Texture Mapping Units than the UHD Graphics 630. The lower TMU count doesn't matter, though, as altogether the GT 730 manages to provide 6.3 GTexel/s better texturing performance. This still holds weight but shader performance is generally more relevant, particularly since both of these GPUs support at least DirectX 10.
The GT 730 has a 552 MHz higher core clock speed and 5 more Render Output Units than the UHD Graphics 630. This results in the GT 730 providing 6.1 GPixel/s better pixeling performance. However, both GPUs support DirectX 9 or above, and pixeling performance is only really relevant when comparing older cards.
The UHD Graphics 630 was released over three years more recently than the GT 730, and so the UHD Graphics 630 is likely to have far better driver support, meaning it will be much more optimized and ultimately superior to the GT 730 when running the latest games.
The GT 730 has 1024 MB video memory, but the UHD Graphics 630 does not have an entry, so the two GPUs cannot be reliably compared in this area.
The UHD Graphics 630 has 184 Shader Processing Units and the GeForce GT 730 v2 has 384. However, the actual shader performance of the UHD Graphics 630 is 212 and the actual shader performance of the GT 730 is 346. The GT 730 having 134 better shader performance and an altogether better performance when taking into account other relevant data means that the GT 730 delivers a noticeably smoother and more efficient experience when processing graphical data than the UHD Graphics 630.
The UHD Graphics 630 transistor size technology is 14 nm (nanometers) smaller than the GT 730. This means that the UHD Graphics 630 is expected to run slightly cooler and achieve higher clock frequencies than the GT 730. While they exhibit similar graphical performance, the UHD Graphics 630 should consume less power than the GT 730.
The UHD Graphics 630 requires 15 Watts to run and the GeForce GT 730 v2 requires 25 Watts. We would recommend a PSU with at least 300 Watts for the GT 730, but we do not have a recommended PSU wattage for the UHD Graphics 630. The GT 730 requires 10 Watts more than the UHD Graphics 630 to run. The difference is not significant enough for the GT 730 to have a noticeably larger impact on your yearly electricity bills than the UHD Graphics 630.
|Core Speed||350 MHz||vs||902 MHz|
|Boost Clock||1150 MHz||vs||-|
|Architecture||Generation 9.5||Kepler GK208-301-A1|
|OC Potential||-||vs||Fair|
|Driver Support||-||vs||Good|
|Release Date||01 Sep 2017||vs||18 Jun 2014|
|GPU Link||GD Link||GD Link|
|Approved|
|Comparison|
|1366x768||
|
6.4
|vs||
|
6.4
|1600x900||
|
5
|vs||
|
5.1
|1920x1080||
|
3.6
|vs||
|
3.7
|2560x1440||
|
2.3
|vs||
|
2.5
|3840x2160||
|
1.6
|vs||
|
1.8
|Memory||N/A||vs||1024 MB|
|Memory Speed||-||vs||1250 MHz|
|Memory Bus||-||vs||64 Bit|
|Memory Type||-||vs||GDDR5|
|Memory Bandwidth||-||vs||40GB/sec|
|L2 Cache||0 KB||vs||128 KB|
|Delta Color Compression||no||vs||no|
|Memory Performance||0%||vs||0%|
|Comparison|
|Shader Processing Units||184||vs||384|
|Actual Shader Performance||10%||vs||17%|
|Technology||14nm||vs||28nm|
|Texture Mapping Units||23||vs||16|
|Texture Rate||8.1 GTexel/s||vs||14.4 GTexel/s|
|Render Output Units||3||vs||8|
|Pixel Rate||1.1 GPixel/s||vs||7.2 GPixel/s|
|Comparison|
|Max Digital Resolution (WxH)||-||vs||4096x2160|
|VGA Connections||0||vs||1|
|DVI Connections||0||vs||1|
|HDMI Connections||0||vs||1|
|DisplayPort Connections||0||vs||0|
|Comparison|
|Max Power||15 Watts||vs||25 Watts|
|Recommended PSU||-||300 Watts & 22 Amps|
|DirectX||12||vs||12.0|
|Shader Model||6.4||vs||5.0|
|Open GL||4.6||vs||4.5|
|Open CL||-||vs||-|
|Notebook GPU||no||no|
|SLI/Crossfire||no||vs||no|
|Dedicated||no||vs||yes|
|Comparison|
|Recommended Processor||-||Intel Core i3-4130 3.4GHz|
|Recommended RAM||-||8 GB|
|Maximum Recommended Gaming Resolution||-||1366x768|
|Performance Value|
|Mini Review||Overview Intel UHD Graphics 630 are Integrated Graphics in some of Intel's Coffee Lake Processors. Architecture The Coffee Lake Architecture succeeds the Kaby Lake Architecture and aims for increased CPU and Graphics Performance, within the same Power Requirements. It includes support for DirectX 12.1, OpenCL 2.1 and OpenGL 4.6, as well as DDR4 Memory Support. GPU The central unit initially runs at 300MHz and goes up to 1150MHz in Turbo Mode. The GPU Codenamed GT2 offers 184 Shader Processing Units, 23 TMUs and 3 ROPs Memory Speed The operating memory clock depends on which speed the System's RAM is running and which memory type is installed and is ultimately limited by the highest operating speed that the processor supports. Depending on whether or not the System's RAM is Dual-Channeled, it can access either a 64-bit or 128-bit Memory Interface. Memory Frame Buffer The GPU Shares the System's RAM which varies according to the System. The GPU may reserve up to 1.7GB.||Overview |
GeForce GT 730 v2 is a Middle-Class Graphics Card based on the first revision of the Kepler Architecture.
Architecture
The Kepler Architecture was NVIDIA's big step to power efficiency. Each Stream Multiprocessor (SMX) now hosts 192 Shader Processing Units - against the 48 of older Fermi Architecture, and has been redesigned being now clocked at the same speed of the Central Unit. This means they are more energy efficient and will consequently lead to cooler operating temperatures. However, it also means they are weaker. It can be said that one Fermi SMX is as fast as 2 Kepler SMXs.
Additionally, and not available in all GPUs, Kepler also introduced the Boost Clock Feature. The Boost Clock is an even higher Clock Speed activated when in gaming mode and becomes the effective speed of the GPU.
GPU
It equips a GPU Codenamed Kepler GK208-301-A1 which has 2 Stream Multiprocessor activated and thus offers 384 Shader Processing Units, 32 TMUs and 16 ROPs. The Central Unit is clocked at 902MHz.
Memory
The GPU accesses a 1GB frame buffer of fast GDDR5, through a 64-bit memory interface. The size of the frame buffer is adequate. The Memory Clock Operates at 1250MHz.
Features
DirectX 11.0 Support (11.0 Hardware Default) and support for Optimus, CUDA, OpenCL, DirectCompute, 3D Vision Surround, PhysX, Realtime Raytracing and other technologies
Power Consumption
With a rated board TDP of 25W, it requires at least a 300W PSU and it relies entirely on the PCI Slot for power, meaning no extra connectors are required.
Performance
GeForce GT 730 v2 is a direct re-brand of GeForce GT 640 v2.
Gaming benchmarks put its performance on average with Radeon HD 6670.
System Suggestions
We recommend a Modest Processor (Intel Core i3) and 8GB of RAM for a system with GeForce GT 730 v2. | https://www.game-debate.com/gpu/index.php?gid=4096&gid2=2274&compare=intel-uhd-graphics-630-vs-geforce-gt-730-v2 |
FIELD OF THE INVENTION
The present invention relates generally to systems and processes for improved drag reduction estimation and measurement and more specifically to portable drag reduction analyzers and processes for upscaling drag reduction data for a variety of field applications.
BACKGROUND AND SUMMARY
Drag reducing agents also called, among other names, drag reducers or friction reducers are commonly employed in many applications to, for example, reduce frictional pressure loss during fluid flow in a conduit or pipeline in a variety of applications. A few common applications include, for example, hydraulic fracturing, pipelines, injecting fluid, produced water, water transportation, disposal wells, irrigation or other well systems, and the like. There are a wide variety of drag reducing agents that include, for example, various polymers, fibers, surfactants, and boosters.
Unfortunately, current testing procedures of drag reducing properties of various agents requires large stationary equipment. Moreover, testing results are performed in the laboratory or small-scale flow loop and do not accurately predict drag reduction on a larger scale such as in the field. In particular, typical field conditions are not easily producible in a laboratory or small-scale flow loop. Accordingly, what is needed are systems and methods that more reliably predict or estimate drag reduction on a larger scale in the field. It would further be advantageous if testing systems could be made to be portable for small scale testing in the field. Advantageously, the instant inventions meet those needs and may have additional advantages that can lead to, for example, improved oil and/or gas production, better control of fracturing, and improved well design.
In one embodiment, the invention pertains to a portable apparatus for analyzing drag reduction. The portable apparatus comprises a housing encompassing a pipe with an inlet to receive a fluid; a pump to direct the fluid from the inlet through the pipe to an outlet of the pipe; and one or more instruments to measure one or more fluid properties selected from flow rate, fluid temperature, inlet pressure, outlet pressure, a pressure differential across the pipe, and combinations thereof. The apparatus also comprises a data acquisition system that receives data from the one or more instruments to measure one or more fluid properties wherein the drag reduction of the fluid may be determined from the data. A power supply is coupled to the pump, the one or more instruments, and the data acquisition system.
In another embodiment the invention pertains to a process to measure drag reduction in the field comprising flowing fracturing fluid through the aforementioned portable apparatus for analyzing drag reduction and determining the drag reduction of the fracturing fluid.
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In another embodiment the invention pertains to a process comprising preparing a fluid mixture and pumping the fluid mixture through a wellbore into a formation. The fluid mixture comprises a concentration of a drag reducing agent wherein the drag reducing agent and the concentration of drag reducing agent is selected from agents which have a polymer drag reduction parameter, ΔB, in said formation of from about 40 to about 70 during at least a portion of the process.
In another embodiment the invention pertains to a process of hydraulic fracturing an oil or gas formation comprising obtaining data on one or more potential drag reducing agents by conducting flow loop testing and upscaling obtained flow loop data on the one or more potential drag reducing agents to obtain predicted drag reduction of the one or more potential drag reducing agents in a hydraulic fracturing operation. A drag reducing agent and a concentration of the drag reducing agent is selected based at least in part on the predicted drag reduction. Hydraulically fracturing is conducted by pumping a fluid mixture comprising the selected concentration of the selected drag reducing agent through a wellbore into the formation. The predicted drag reduction is obtained utilizing flow loop data based on one or more parameters selected from the group consisting of polymer relaxation time, shear rate, polymer drag reduction parameter ΔB, pipe diameter, flow rate, pipe roughness, shear stress on pipe, or a combination thereof.
In another embodiment the invention pertains to a process to determine drag resistance comprising obtaining data on viscosity as a function of shear rate, drag reduction percentage as a function of velocity, or both on at least one diameter of conduit on a plurality of drag reducing agents for at least one concentration. Drag reduction of the plurality of drag reducing agents is estimated using polymer drag reduction parameter ΔB, shear rate, and polymer relaxation time. The estimated drag reduction(s) may be stored in an analytical model.
BRIEF DESCRIPTION OF THE DRAWINGS
So that the manner in which the above recited features, advantages, and objects of the present invention are attained and can be understood in detail, a more particular description of the invention, briefly summarized above, may be had by reference to the embodiments thereof which are illustrated in the appended drawings. It is to be noted, however, that the appended drawings only illustrate preferred embodiments of this invention, and are therefore not to be considered limiting of its scope, for the invention may admit to other equally effective embodiments that vary only in detail. In the drawings:
FIGS. 1A-G
illustrate exemplary embodiments of a portable apparatus for analyzing drag reduction.
FIG. 2
is a Moody diagram illustrating example flow-loop testing conditions and field conditions.
FIG. 3
illustrates an exemplary embodiment for upscaling drag reduction data for a variety of field applications.
FIG. 4
is a plot displaying viscosity compared to shear rate for an example drag reducing agent.
FIG. 5
is a plot displaying drag reduction percentages for an example drag reducing agent flowing through a 0.305 inch diameter pipe at a range of flow rates.
FIG. 6
is a plot displaying drag reduction percentages for an example drag reducing agent flowing through a 0.425 inch diameter pipe at a range of flow rates.
FIG. 7
is a plot displaying drag reduction percentages for an example drag reducing agent flowing through a 0.62 inch diameter pipe at a range of flow rates.
FIG. 8
FIGS. 5 through 7
is a plot displaying the drag reduction percentages for the example drag reducing agent shown in compared to shear rate.
FIG. 9
FIGS. 5 through 7
is a plot displaying the drag reduction percentages for the example drag reducing agent shown in compared to a range of Weissenberg numbers.
FIG. 10
illustrates the relationship between drag reduction parameter and relaxation time.
FIG. 11
FIG. 11A
FIG. 11B
illustrates velocity profiles for water () and water with a drag reducing agent ().
FIG. 12
FIGS. 5 through 7
is a Moody diagram illustrating the example drag reducing agent shown in compared to calculated fits using example reduction parameters and relaxation times.
FIG. 13
illustrates estimated pressure drop compared to flow rate for an example drag reducing agent for a field scale hydraulic fracturing operation along with corresponding measured field data.
FIGS. 14 and 15
show plots of field pressure gradient compared to flow rate for polymer relaxation times and ranges of polymer drag reduction parameter ΔB values.
FIGS. 16 and 17
show plots of laboratory pressure gradient compared to flow rate for polymer relaxation times and ranges of polymer drag reduction parameter ΔB values.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
Portable Apparatus for Analyzing Drag Reduction and Associated Processes
Upscaling Processes
Specific Embodiments
Certain terms are used throughout the following description and claims to refer to particular system components. As one skilled in the art will appreciate, different companies may refer to a component by different names. This document does not intend to distinguish between components that differ in name but not function.
In one embodiment the invention pertains to a portable apparatus for analyzing drag reduction. By portable, it is meant that the apparatus may easily be transported from site to site. For example, the portable design of the apparatus can permit the apparatus to be readily transported to often remote locations (e.g., to sites in the field where fluids are being used) and used analyze fluids, particularly those that include one or more drag reducing agents. In addition to improving portability, the relatively small footprint of the apparatus allows for the apparatus to be deployed in a wider variety of locales (e.g., in smaller rooms or facilities that lack the available space for a full-scale analytical laboratory). This allows for once complex analytical operations to be efficiently performed on-site using a single, portable piece of equipment.
In some embodiments, the assembled portable apparatus can have a footprint of 25 square feet or less (e.g., 20 square feet or less, 15 square feet or less, 12.5 square feet or less, 10 square feet or less, 9 square feet or less, 8 square feet or less, 7 square feet or less, 6 square feet or less, 5 square feet or less, or 4 square feet or less). In some embodiments, the assembled portable apparatus can have a footprint of at least 2 square feet (e.g., at least 3 square feet, at least 4 square feet, at least 5 square feet, or at least 6 square feet).
The assembled portable apparatus can have a footprint ranging from any of the minimum values described above to any of the maximum values described above. For example, the assembled portable apparatus can have a footprint of from 2 to 25 square feet (e.g., from 2 to 15 square feet, from 2 to 10 square feet, or from 4 to 8 square feet).
In some embodiments, the assembled portable apparatus can occupy a volume of 35 cubic feet or less (e.g., 30 cubic feet or less, 25 cubic feet or less, 20 cubic feet or less, 15 cubic feet or less, 12.5 cubic feet or less, 10 cubic feet or less, 9 cubic feet or less, 8 cubic feet or less, 7 cubic feet or less, 6 cubic feet or less, 5 cubic feet or less, or 4 cubic feet or less). In some embodiments, the assembled portable apparatus can occupy a volume of at least 2 cubic feet (e.g., at least 3 cubic feet, at least 4 cubic feet, at least 5 cubic feet, at least 6 cubic feet, at least 7 cubic feet, or at least 8 cubic feet).
The assembled portable apparatus can occupy a volume ranging from any of the minimum values described above to any of the maximum values described above. For example, the assembled portable apparatus can occupy a volume of from 2 to 35 cubic feet (e.g., from 2 to 15 cubic feet, from 2 to 10 cubic feet, or from 2 to 8 cubic feet).
The components of the apparatus can be self-contained and assembled within a housing. This allows for the apparatus to be rapidly deployed. For example, setup times can be short enough to allow the portable apparatus to be up and running in a matter of hours. This is significantly shorter than the setup time required for a typical full-scale analytical laboratory.
Typically, the apparatus comprises a housing encompassing a pipe with an inlet to receive a fluid. The pipe may be of any suitable conduit, dimensions, and material so long as it is capable of carrying the fluid from the inlet to an outlet of the pipe. Thus, the specific pipe, dimensions, and material may vary depending upon the desired application, type of fluid, and properties to be analyzed by the apparatus. A suitable range to allow for drag reduction measurements in a portable apparatus for the pipe inner diameter is greater than approximately 0.25 inches and less than approximately 0.75 inches. For example, the portable apparatus can have a single pipe diameter of approximately 0.25 inches, approximately 0.3 inches, approximately 0.4 inches, approximately 0.5 inches, approximately 0.6 inches, or approximately 0.75 inches. In one example, the portable apparatus comprises two pipes of different diameters (e.g., a first pipe having a diameter ranging from approximately 0.25 inches to approximately 0.4 inches such as 0.3 inches and a second pipe having a diameter of approximately 0.5 inches to approximately 0.75 inches such as 0.6 inches). In another example, the portable apparatus three pipes of various diameters (e.g., a first pipe having a diameter ranging from approximately 0.25 inches to approximately 0.35 inches such as 0.3 inches, a second pipe having a diameter of approximately 0.35 inches to approximately 0.55 inches such as 0.4 inches, and a third pipe having a diameter of approximately 0.55 inches to approximately 0.75 inches such as 0.6 inches).
The apparatus typically has a pump to direct the fluid from the inlet through the pipe to an outlet of the pipe. The type of pump is not particularly critical so long as it can appropriately direct the fluid to be tested at a desired flow rate without degrading the drag reducer within the fluid. Examples of pumps that can be utilized include low-shear positive displacement pumps such as progressive cavity pumps commercially available by Moyno and Seepex GmbH. Typically, the pump and/or one or more of the other components of the apparatus is capable of adjusting the flow rate of the fluid. The manner of adjusting the flow rate is not critical so long as it is adjustable over the range of flow rates to be studied or employed. In one embodiment, the pump is able to generate a flow rate of between at least 2 gallons per minute and to less than 30 gallons per minute. If the pump does not directly adjust the flow rate, then the flow rate may be adjusted by, for example, a pressure regulator, a control valve and a vent port, a pressure relief valve, or any combination of these or other devices.
In some embodiments, a pump is not necessary. For example, the portable apparatus can be connected or teed into a main flow line (e.g., injection line in a hydraulic fracturing operation) such that it obtains a slipstream sample of the fluid to be tested. The pressure and flow rate in the main flow line is sufficient to supply the portable apparatus such that fluid properties of the fluid can be measured.
Typically, the apparatus is configured with one or more instruments to measure one or more fluid properties. Such properties may be selected from flow rate, fluid temperature, inlet pressure, outlet pressure, a differential pressure across the pipe, and combinations thereof. Thus, typical instruments may include, for example, a pressure gauge or pressure transducer or pressure indicator transmitter (PIT), thermocouple or temperature probe or temperature indicator transmitter (TIT), a flow meter or flow transducer or flow indicator transmitter (FIT), and/or any combination thereof. In some embodiments, the apparatus may also be configured with additional instruments to measure fluid properties such as a pH probe, a conductivity probe, an Oxidation-Reduction Potential (ORP) probe, and/or any combination thereof.
A data acquisition system is usually provided to receive data from the one or more instruments that measure one or more fluid properties. The data acquisition system is often programmed so that it is capable of determining drag reduction of the fluid from data it receives. The data acquisition system may be any convenient system and may comprise, for example, a processor, a memory, a monitor, and/or a keyboard or other data entry capability. The data acquisition system may be a computer in some embodiments. On the other hand, in some embodiments, the data acquisition system may primarily receive data from the one or more instruments and the data acquisition system may be coupled to a different system, such as a computer, and the computer determines drag reduction of the fluid from the data provided to the computer from the data acquisition system. The computer may comprise a processor, a memory, a monitor, and/or a keyboard or other data entry capability. In these embodiments, the data acquisition system inside the apparatus may be coupled to a computer that is external to the apparatus via a wired (e.g. US cable) or wireless connection, and the computer receives the data from the data acquisition system and the computer determines the drag reduction of the fluid from the data.
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If it is desired to measure more than one fluid simultaneously, then the portable apparatus may comprise a second, a third, or even more additional pipes. The additional pipes can receive fluid from the inlet (e.g., via a tee or multi-way valve) and expel fluid to the outlet (e.g., via a tee or multi-way valve), or alternatively each pipe in the apparatus can be associated with a separate inlet to receive fluid and a separate outlet to expel the fluid from the apparatus. The effluent fluid can be recirculated back to inlet for additional testing, can be reintroduced into the fluid being injected into a subterranean reservoir, can be treated and disposed of as waste, or any combination thereof. The additional pipes, if any, are usually configured with one or more instruments to measure one or more additional pipe fluid properties selected from flow rate, fluid temperature, inlet pressure, outlet pressure, a differential pressure across the second pipe, and combinations thereof. The instruments associated with added pipes may be configured to operably connect to the same or a different data acquisition system. In this manner, data received from the one or more instruments to measure one or more added pipe fluid properties may be used to determine drag reduction from data received.
If desired, the portable apparatus may be operably connected to a hydraulic fracturing system. In this manner, drag reduction of a potential fracturing fluid comprising drag reducing agents (or even lacking such agents) may be determined at any desirable stage in the fracking process. For example, drag reduction of a potential fracturing fluid may be determined prior to pumping the fracturing fluid into a borehole. In this manner one could, for example, employ a portable apparatus downstream, upstream, or both of a blender to determine the effects the blender may have on a potential fracturing fluid.
The processes of using the portable apparatus typically comprise flowing a fluid through the apparatus and then determining the drag reduction. For example, if employed with a hydraulic fracturing system comprising a blender, then one portable apparatus may operably be connected to the hydraulic fracturing system upstream of the blender while a second portable apparatus is operably connected to the hydraulic fracturing system downstream of the blender to determine drag reduction of the fracturing fluid. In some embodiments drag reduction of a first fluid can be compared with drag reduction of a second fluid such that adjustments may be made. For example, one or more hydraulic fracturing parameters could be adjusted based on the comparison. Such parameters include, for example, those selected from drag reducing agent, amount of drag reducing agent, flow rate, proppant, amount of proppant, and combinations thereof. The portable apparatus may also be used in conjunction with the below upscaling embodiments to obtain laboratory data or flow loop data which can then be upscaled.
FIG. 1A
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shows an exemplary portable apparatus for measuring drag reduction. The portable apparatus can comprise a housing encompassing one or more components of the portable apparatus. As shown, pump may direct fluid for which a drag reduction measurement is desired through one or more valves to an inlet of apparatus . The fluid may come from a process line or any other source. For example, the fluid may comprise an injection solution used in a hydraulic fracturing operation. Apparatus may be disposed upstream of a blender (not shown) that mixes the fracturing fluids or downstream of a blender such that fluid exiting apparatus at outlet is disposed just prior to fluid being injected in a wellbore into a formation. Valves can be used to regulate the amount or rate of fluid directed to apparatus .
FIG. 1A
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Various instruments may be utilized to measure one or more fluid properties. The location of each instrument may vary depending upon the desired configuration and testing conditions. depicts temperature indicator transmitter (TIT) to measure a temperature of the fluid, pressure indicator transmitters (PIT) to measure an inlet pressure and/or a pressure differential across pipe , and flow indicator transmitter (FIT) measures a flow rate of the fluid received by apparatus . Typically, temperature indicator transmitter (TIT) , pressure indicator transmitter (PIT) , and flow indicator transmitter (FIT) are disposed proximate to inlet of apparatus such that properties of the fluid entering apparatus can be measured. A pressure indicator transmitter (PIT) is also used to measure a pressure differential across pipe . Drag reduction of the fluid is determined based on the measurements from the instruments.
FIG. 1A
FIG. 1B
FIG. 1B
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While depicts a single pipe to measure fluid properties, a suitable system may have two pipes, or more than two pipes, for analyzing fluid properties. shows another exemplary apparatus for measuring drag reduction having two pipes to measure fluid properties. Here, three-way valves are employed to route the fluid to be tested through various pathways through apparatus from the inlet to the outlet . For example, since apparatus of has two pipes , three-way valves could divert a portion of the fluid entering apparatus to each pipe such that measurements are taken simultaneously in each pipe or three-way valves could divert all the fluid through a single pipe such that only a single measurement is taken (i.e., a pressure differential of fluid flowing through each pipe can be tested simultaneously or separately using pressure indicator transmitters (PIT) and flow indicator transmitters (FIT) ). Note that additional flow indicator transmitters (FIT) are placed along each pipe to allow for simultaneous measurements through each pipe . In some embodiments, the diameters of pipe in the separate lines are the same. In some embodiment, the diameters of pipe in the separate lines are different.
FIG. 1C
FIG. 1B
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shows another exemplary apparatus for measuring drag reduction having three pipes to measure fluid properties. Here, four-way valves are employed to route the fluid to be tested through various pathways through apparatus from the inlet to the outlet . For example, since apparatus of has three pipes , four-way valves can divert a portion of the fluid entering apparatus to each pipe such that measurements are taken simultaneously in each pipe , four-way valves can divert a portion of the fluid through only two of pipes such that two measurements are taken, or four-way valves can divert all the fluid through a single pipe such that only a single measurement is taken (i.e., a pressure differential of fluid flowing through each pipe can be tested simultaneously or separately using pressure indicator transmitters (PIT) and flow indicator transmitters (FIT) ). Note that additional flow indicator transmitters (FIT) are placed along each pipe to allow for simultaneous measurements through each pipe . In some embodiments, the diameters of pipe in the separate lines (e.g., upper, middle, and lower as shown) are the same. In some embodiment, the diameters of pipe in the separate lines (e.g., upper, middle, and lower as shown) are different.
FIG. 1D
FIG. 1D
FIG. 1A
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shows an exemplary apparatus for measuring drag reduction where a pump is not utilized. For example, the portable apparatus can be connected or teed into a main flow line (e.g., injection line in a hydraulic fracturing operation) such that it obtains a slipstream sample of the fluid to be tested. The pressure and flow rate in the main flow line is sufficient to supply the portable apparatus such that fluid properties of the fluid can be measured. All other functionality of apparatus in remains the same as shown in .
FIG. 1E
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shows another exemplary apparatus for measuring drag reduction having two pipes arranged in series to measure fluid properties. Here, measurements can be taken simultaneously in each pipe (i.e., a pressure differential of fluid flowing through each pipe can be tested simultaneously using pressure indicator transmitters (PIT) ). In some embodiments, the diameters of pipe in the separate lines are the same. In some embodiment, the diameters of pipe in the separate lines are different. In additional embodiments, more than two pipes are arranged in series where a pressure differential of fluid flowing through each pipe is tested simultaneously using separate pressure indicator transmitters (PIT) .
FIG. 1F
FIGS. 1A-1C
FIG. 1D
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shows an exemplary apparatus for measuring drag reduction where pump and one or more valves are disposed inside housing of apparatus . Further, while not necessary, an additional pressure indicator transmitter (PIT) is shown prior to outlet to measure outlet pressure. While not depicted in , outlet pressure could additionally be measured by placing a pressure indicator transmitter (PIT) prior to outlet if desired (e.g., to confirm accurate measurements of differential pressure measured across each pipe ). All other functionality of apparatus in remains the same as shown in .
FIG. 1G
FIGS. 1A-1F
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shows an exemplary apparatus for measuring drag reduction where pipe is arranged inside apparatus as a coil. This coil arrangement facilitates a longer pipe length of pipe in which a pressure differential can be measured. While not depicted in , each pipe in these embodiments can also be configured as a coil.
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FIG. 1G
A data acquisition system , which is usually provided to receive data from the one or more instruments (, , ) to measure one or more fluid properties. The data acquisition system may be located within housing of apparatus as shown in , coupled to apparatus , or may be remote if a communication system is provided to send the data from the instruments (, , ) to the data acquisition system . In some embodiments, the data acquisition system can also communicate with pump , valves , or other controllers (not shown) to regulate the amount of fluid, rate of fluid, and/or fluid pathway/route that fluid travels through apparatus between inlet and outlet . A power supply is operably connected to the instruments (, , ), pump , and data acquisition system . The power supply may be a common supply for all the components or separate for some or all of the components. While not depicted in , apparatus can also encompass data acquisition system and power supply within housing similar to .
As previously discussed, testing results performed in the laboratory or flow loop do not accurately predict drag reduction observed in field applications. In general, this is due to the laboratory or flow loop conditions being measured on much smaller equipment than what is used in the field. For example, laboratory or flow loop equipment typically measure drag reduction in pipe sizes less than 1 inch, whereas in field applications fluid is flowing through pipes with a diameter typically around 4 or 5 inches. Flow in laboratory equipment or flow loops is typically around 30 gallons per minute, whereas in field applications it can be 100-150 barrels per minute or higher. Laboratory equipment or flow loops also do not mimic other field conditions as well (e.g., pipe wall roughness). This generally results in a range of Reynolds number produced in the laboratory being much lower (e.g., between 50,000-200,000) compared to those observed in the field (e.g., between 1,500,000-3,000,000).
FIG. 2
is a Moody diagram illustrating example flow-loop testing conditions and field conditions. As shown, the resulting friction factors observed in the laboratory or flow loops are different than those in field applications. Accordingly, current practices of using a Moody diagram to upscale laboratory or flow loop data to predict field performance is not as precise as needed.
FIG. 3
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illustrates process for upscaling drag reduction data for a variety of field applications according to an embodiment of the invention. In step , an analytical model is generated to upscale laboratory data or small-scale flow data from a fluid comprising a drag reducing agent a to predict performance of the fluid under field conditions. The analytical model derives a velocity profile of flow in pipes and determines the fluid velocity (V) versus the pressure gradient (dP/dL). In embodiments, parameters accounted for in the analytical model include pipe diameter, pipe wall roughness, wall shear rate/stress, fluid viscosity, fluid density, fluid velocity, fluid temperature, fluid salinity, a concentration of drag reducing agent within the fluid and associated properties of the drag reducing agent (e.g., relaxation time, molecular weight), or any combination thereof.
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In step , laboratory or flow loop data is measured to populate or calibrate analytical model . In embodiments, this data is measured using portable apparatus previously described. In some embodiments, laboratory or flow loop data comprises yard-scale flow loop data (i.e., measurements using larger sized equipment such as 2-3 inch diameter piping). Fluid can be measured at various concentrations of drag reducing agent. For example, initially a fluid (e.g., water plus other chemicals to be injected in the subterranean reservoir) can be measured at a range of flow rates between 2 and 30 gallons per minute (e.g., in increments of 2 gallons per minute each for 15 to 60 seconds). The drag reducing agent can then be added for a range of concentrations between 0.1 to 5 gallons of drag reducing agent per thousand gallons of fluid (e.g., at concentrations 0.1, 0.25, 0.5, 0.75, 1.0, 1.5, 2.0, 2.5, 3.0 3.5, 4.0, 4.5, and 5 gpt) and for each concentration a pressure drop can be recorded as the flow rate is altered (e.g., increased, decreased, or both). Typically, the pressure drop reaches steady state for a change in flow rate within 60 seconds (e.g., 15-30 seconds). Note that at higher flow rates the flow duration should be closely monitored to ensure degradation of the drag reducing agent is limited while still acquiring steady state pressure drops.
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Once the analytical model is populated or conditioned with the laboratory or flow loop data, it can then be used to upscale the drag reduction percentage and applied to predict, design and/or modify field operations to enhance field performance. Field data can further be used to validate analytical model , as well as, guide testing to obtain additional laboratory or flow loop data . Field data can comprise measurements or estimates of drag reduction based on wellhead pressure, bottom hole pressures, estimated hydrostatic pressure, perforation pressure friction loss, tubing pressure differentials, tubing or pipe lengths, flow rates, or any combination thereof. Analytical model can then be used to predict, design and/or modify field operations to enhance field performance.
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Process can additionally include database to store properties of drag reducing agents (e.g., relaxation time, drag reduction parameters, molecular weight) obtained from laboratory or flow loop data , field data , or a combination thereof. Information contained in database can be used by analytical model to predict, design and/or modify field operations to enhance field performance. In particular, storing properties of drag reducing agents in database can help reduce the need for future testing of fluids comprising a drag reducing agent as this information can be utilized as an alternative to, or in addition to, new laboratory or flow loop data , field data , or a combination thereof.
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In one embodiment, a process comprises first preparing a fluid mixture and pumping the fluid mixture through a wellbore into a formation. The specific composition of the fluid mixture will vary depending upon, for example, the well characteristics and/or desired results. In some embodiments the fluid mixture comprises a concentration of a drag reducing agent. Of course, the concentration and the specific drag reducing agent may vary depending upon the desired results and other parameters. In some embodiments the drag reducing agent is selected from agents which have a ΔBin said formation of from about 40, or from about 45, or from about 50 to about 70, or up to about 65, or up to about 60 during at least a portion of the process. Representative useful processes include, for example, hydraulic fracking, injecting fluid, produced water, transportation of water for reinjection, pipelining fluids in turbulent flow conditions, and/or disposal well processes.
In another embodiment, the present application relates to a process useful in oil and gas industry such as hydraulic fracturing an oil or gas formation. Such processes first typically comprise obtaining data on one or more potential drag reducing agents by conducting flow loop testing. Next, upscaling is conducted using obtained flow loop data on the one or more potential drag reducing agents to obtain predicted drag reduction of the one or more potential drag reducing agents in a process such as hydraulic fracturing. A drag reducing agent and a concentration of the drag reducing agent based at least in part on the predicted drag reduction can be selected, and the selected process (e.g., hydraulically fracturing) may be conducted using the selected concentration of the selected drag reducing agent. That is, in the case of hydraulically fracturing, one may pump a fluid mixture comprising the selected concentration of the selected drag reducing agent through a wellbore into the formation.
Obtaining predicted drag reduction comprises using one or more parameters selected from the group consisting of polymer relaxation time, shear rate, polymer drag reduction parameter ΔB, pipe diameter, flow rate, pipe roughness, shear stress on pipe, or any combination thereof. In some embodiments, one may obtain or measure actual drag reduction during hydraulic fracturing and use it advantageously. For example, predicted drag reduction of the selected drag reducing agent and selected concentration of the drag reducing agent may be compared with actual or measured drag reduction to calibrate the upscaling parameters and models.
In other embodiments, viscosity of the one or more potential drag reducing agents may be measured prior to conducting flow loop testing and then viscosity can be measured again subsequent to conducting flow loop testing. An understanding of the differences may be favorably employed in upscaling.
In another embodiment, the application pertains to a process to determine drag resistance. The process typically comprises obtaining data on viscosity as a function of shear rate, drag reduction percentage as a function of velocity, or both for a plurality of drag reducing agents. Preferably, the data is obtained on at least one diameter or preferably a plurality of diameters of conduit. Such conduit may be any shape but it typically substantially cylindrical such as a tubing or a pipe.
Data may be obtained on at least one concentration, however, a plurality of concentrations may be preferable depending upon the specific drag reducing agent or agents and other parameters employed. In embodiments, data can be obtained for concentrations drag reducing agent of approximately 0.1 gallons per one thousand gallons of water (0.1 gpt), approximately 0.25 gallons per one thousand gallons of water (0.25 gpt), approximately 0.5 gallons per one thousand gallons of water (0.5 gpt), approximately 0.75 gallons per one thousand gallons of water (0.75 gpt), approximately 1 gallons per one thousand gallons of water (1.0 gpt), approximately 1.5 gallons per one thousand gallons of water (1.5 gpt), approximately 2 gallons per one thousand gallons of water (2 gpt), approximately 5 gallons per one thousand gallons of water (5 gpt), or any combination thereof.
Data may be obtained on at least one flow rate, however, a plurality of flow rates may be preferable depending upon the specific drag reducing agent or agents and other parameters employed. In embodiments, data can be obtained for flow rates of approximately 2 gallons per minute, approximately 3 gallons per minute, approximately 4 gallons per minute, approximately 5 gallons per minute, approximately 7 gallons per minute, approximately 9 gallons per minute, approximately 11 gallons per minute, approximately 13 gallons per minute, approximately 15 gallons per minute, approximately 20 gallons per minute, approximately 25 gallons per minute, approximately 30 gallons per minute, or any combination thereof. Generally, flow rates are tested for a time period in order for a pressure drop to reach steady state. Flow duration at higher flow rates should be minimized to limit mechanical polymer degradation while still acquiring steady state pressure drop recordings.
The drag reduction of the plurality of drag reducing agents may then be estimated. Typically, estimating uses parameters such as polymer drag reduction parameter ΔB, shear rate, and polymer relaxation time. If desired, the estimates may be stored in an analytical model which advantageously may be used in the future as estimates and actual field data are accumulated. In this manner the analytical model may even be more accurate in the future. That is, estimated drag reduction of the plurality of drag reducing agents may be compared with actual drag reduction if desired. In this manner the estimated drag reduction(s) in the analytical model may be modified based on the actual drag reductions.
In some embodiments, the process may comprise hydraulically fracturing a well using estimated drag reduction or modified estimated drag reduction to select a drag reducing agent, a concentration of drag reducing agent, or both. For example, the concentration of drag reducing agent may be modified (e.g., increased or decreased) in slickwater pumping stages or linear gel pumping stages to maximize recovery performance. Information may be advantageous for a number of additional reasons including, for example, so that a selection may be to reduce required horsepower or energy during hydraulically fracturing. Alternatively, or additionally the process could be used to design a well or completion (e.g., determine length of lateral, casing size) or enhance operation performance such that if fracking is required, then it will be of less cost, more efficient, more productive, and/or provide other advantages to the designed well. The pumping schedule and design for a hydraulic fracturing operation could also be modified (e.g., pumping rate, perforation design, number of pumps).
The specific drag reducing agent or agents employed are not particularly limited and may include, for example, a polymer, a fiber, a surfactant (e.g., a higher molecular weight surfactant), a booster such as a solvent, and/or any combination thereof.
While the aforementioned upscaling processes for drag reduction agents may be accomplished in a number of ways, the specifics of a representative process are described below.
One may first make a solution with a specific concentration of drag reduction agents. Such a solution may be brine with any salinity and may include other chemicals such as surfactants, scale inhibitors, biocide, corrosion inhibitors. One then measures viscosity versus shear rate on the solution using, for example, a rheometer. This can be plotted as viscosity (e.g., units in centipoise: cP) on the Y-axis versus shear rate (e.g., units in reciprocal seconds: 1/sec) on the X-axis.
FIG. 4
FIG. 4
shows an example of a viscosity versus shear rate plot. In , measurements were taken at room temperature using a double-wall couette style rheometer. The sample contains a concentration of one gallon of drag reducing agent per 1000 gallons of tap water (i.e., 1 gpt). The circles represent the measured points and the line represents a corresponding viscosity model that illustrates an approximate fit of the data. The following equation can be used to model viscosity as function of shear rate:
∞
0
∞
2
(n−1)/2
η=η+(η−η)[1+({dot over (γ)}λ)]
0
∞
where η represents dynamic or shear viscosity such that ηis the zero shear viscosity and ηis the infinite shear viscosity, {dot over (γ)} represents shear rate, λ represents relaxation time, and n is a power law index. One skilled in the art will recognize that the shear rate {dot over (γ)} multiplied by the relaxation time λ is alternatively known as the Weissenberg number (We or Wi), which is a dimensionless number that compares elastic to viscous forces.
The equation to model viscosity as function of shear rate can be transformed to shear stress versus shear rate as follows:
<math overflow="scroll"><mrow><msub><mi>τ</mi><mi>w</mi></msub><mo>=</mo><mrow><msub><mi>η</mi><mi>w</mi></msub><mo></mo><msub><mover><mi>γ</mi><mo>.</mo></mover><mi>w</mi></msub></mrow></mrow></math>
<math overflow="scroll"><mrow><msub><mi>τ</mi><mi>w</mi></msub><mo>=</mo><mrow><mrow><msub><mi>η</mi><mi>∞</mi></msub><mo></mo><msub><mover><mi>γ</mi><mo>.</mo></mover><mi>w</mi></msub></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>-</mo><msub><mi>η</mi><mi>∞</mi></msub></mrow><mo>)</mo></mrow><mo></mo><msup><mrow><msub><mover><mi>γ</mi><mo>.</mo></mover><mi>w</mi></msub><mo></mo><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo>(</mo><mrow><msub><mover><mi>γ</mi><mo>.</mo></mover><mi>w</mi></msub><mo></mo><mi>λ</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow></mrow></math>
w
where τrepresents shear stress at the wall of the pipe.
Pressure drop in a pipe is generally a function of shear stress. Shear rate at the wall can be calculated for any flow rate in a pipe using the above equation with measured viscosity versus shear rate data.
<math overflow="scroll"><mrow><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>p</mi></mrow><mo>=</mo><mfrac><mrow><mn>4</mn><mo>/</mo><msub><mi>τ</mi><mi>w</mi></msub></mrow><mi>D</mi></mfrac></mrow></math>
Using a flow-loop (e.g., 1 or more tube diameters), one can measure flow rate versus pressure drop for a wide range of flow rates. This is generally done quickly by switching to a new flow rate when the pressure stabilizes. Pressure drop can be converted to a drag reduction percentage (DR %) using the equation below.
<math overflow="scroll"><mrow><mrow><mi>DR</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mi>%</mi></mrow><mo>=</mo><mrow><mfrac><mrow><msub><mi>dP</mi><mi>water</mi></msub><mo>-</mo><msub><mi>dP</mi><mi>FR</mi></msub></mrow><msub><mi>dP</mi><mi>water</mi></msub></mfrac><mo>*</mo><mn>100</mn><mo></mo><mi>%</mi></mrow></mrow></math>
Drag reduction percentage (DR %) can then be plotted on the Y-axis with flow rate (e.g., gallons per minute) on X-axis using the measured pressure drop data.
FIGS. 5-7
FIG. 5
FIG. 6
FIG. 7
show plots where a fluid concentration of one gallon of drag reducing agent per one thousand gallons of water (1 gpt) that was run through a flow loop comprising three different piping sizes. In each plot, an initial flow rate of 2 gallons per minute (gpm) was increased to 11 gallons per minute in increments, and then decreased in increments back to 2 gallons per minute. In particular, illustrates results using a 0.305 inch diameter pipe, illustrates results using a 0.425 inch diameter pipe, and illustrates results using a 0.62 inch diameter pipe.
FIG. 8
Drag reduction percentage (DR %) can also be plotted compared to shear rate (1/s) as shown in . Here, the flow rate (Q) is modified to shear rate using the viscosity versus shear rate equation above, i.e.,
<math overflow="scroll"><mrow><msub><mi>τ</mi><mi>w</mi></msub><mo>=</mo><mrow><msub><mi>η</mi><mi>w</mi></msub><mo></mo><msub><mover><mi>γ</mi><mo>.</mo></mover><mi>w</mi></msub></mrow></mrow></math>
<math overflow="scroll"><mrow><msub><mi>τ</mi><mi>w</mi></msub><mo>=</mo><mrow><mrow><msub><mi>η</mi><mi>∞</mi></msub><mo></mo><msub><mover><mi>γ</mi><mo>.</mo></mover><mi>w</mi></msub></mrow><mo>+</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>-</mo><msub><mi>η</mi><mi>∞</mi></msub></mrow><mo>)</mo></mrow><mo></mo><msup><mrow><msub><mover><mi>γ</mi><mo>.</mo></mover><mi>w</mi></msub><mo></mo><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo>(</mo><mrow><msub><mover><mi>γ</mi><mo>.</mo></mover><mi>w</mi></msub><mo></mo><mi>λ</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></msup></mrow></mrow></mrow></math>
FIGS. 5-7
FIG. 8
Accordingly, the results using the flow loop experiment using the various pipe diameters shown in are now plotted in single where the X-axis was modified to shear rate instead of flow rate.
FIG. 9
further illustrates the above results using the Weissenberg number (We or Wi), where the shear rate is multiplied with relaxation time. This transforms the data from the three individual pipe sizes into a dimensionless number, thus collapsing all of the data into a single curve where drag reduction percentage (DR %) is on the Y-axis and Weissenberg number (We or Wi) is on the X-axis. In some embodiments, relaxation time is measured using a rheometer. In another embodiment, the criteria that when We<1, DR %=0 and when We>1, DR % increases until a plateau is reached may be used.
FIG. 9
In order to obtain the plot in an additional parameter referred to herein as drag reduction parameter (ΔB) is utilized. The following equations can be utilized for determining the drag reduction parameter:
<math overflow="scroll"><mrow><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>B</mi></mrow><mo>=</mo><mrow><mi>Δ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><msub><mi>B</mi><mi>max</mi></msub><mo></mo><msub><mi>B</mi><mi>N</mi></msub></mrow></mrow></math>
<math overflow="scroll"><mrow><msub><mi>B</mi><mi>N</mi></msub><mo>=</mo><mrow><mn>2</mn><mo></mo><mrow><mo>[</mo><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><mi>a</mi><mo></mo><mrow><mo>(</mo><mrow><msub><mi>Wi</mi><mi>c</mi></msub><mo>-</mo><mi>Wi</mi></mrow><mo>)</mo></mrow></mrow></msup></mrow></mfrac><mo>-</mo><mrow><msub><mi>Wi</mi><mi>c</mi></msub><mo>/</mo><mn>2</mn></mrow></mrow><mo>]</mo></mrow></mrow></mrow></math>
max
N
w
w
c
c
max
FIG. 10
where ΔBis the maximum drag reduction at Wi>>1, Bis the normalized drag reduction as a function of shear, Wi={dot over (γ)}λ is Weissenberg number, {dot over (γ)}is the wall shear rate, λ is the longest relaxation time, Wiis the coil-stretch transition point (Wi=1), and a is a constant representing a shape of transition. illustrates how ΔB, the relaxation time, and the shape of ΔB can be identified for a fluid having a concentration of 1 gallon of drag reducing agent per one thousand gallons of water.
FIG. 10
Drag Reduction Fundamentals
Once ΔB and relaxation time are known, flow rates for imposed pressure gradients can be estimated for turbulent flow in a pipe by deriving the velocity profile of fluid. In the literature, the velocity profile of turbulent flow in a pipe has been measured and derived for water and water with drag reducing agents. The profile consists of three layers: a viscous sublayer near the wall, a buffer layer, and a log-law layer. It has been demonstrated that drag reducing agents function by increasing the buffer layer thickness, and ΔB from is used to estimate the thickness of the buffer layer for various pipe sizes and flow rates. A detailed background on turbulent flow velocity profiles in pipes can be found in a scientific paper by P. S. Virk. (, AIChE Journal, Volume 21, Issue 4, Pages 625-656, July 1975).
FIGS. 11A and 11B
FIG. 11A
FIG. 11B
illustrate how the buffer layer thickness increases with the addition of a drag reducing agent. In particular, plots the viscous sublayer near the wall and a log-law layer layers for a fluid without a drag reducing agent (ΔB=0) and plots these layers for a fluid with a drag reducing agent (ΔB=30). In both plots, the dashed lines represent a viscous sublayer near the wall, Virk's Asymptote, and a log-law layer. The solid lines represents the updated curve taking this combination into account.
FIG. 12
max
illustrates an example of measured data from three pipe sizes on a Moody plot. The solid lines represent approximate fits to the measured data using a value of one for the maximum drag reduction and relaxation time (i.e., ΔB=1 and λ=1). Here the drag reduction percentage versus flow rate plot measured using a flow-loop was transformed into a non-dimensional Moody plot wherein the Darcy Friction Factor is plotted on the Y-axis and the Reynolds number is plotted on the X-axis.
<math overflow="scroll"><mrow><mrow><mi>Friction</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mi>Factor</mi></mrow><mo>=</mo><mrow><mfrac><mrow><mn>2</mn><mo></mo><mi>D</mi></mrow><mrow><mi>p</mi><mo></mo><msup><mi>V</mi><mn>2</mn></msup></mrow></mfrac><mo></mo><mfrac><mi>dP</mi><mrow><mi>d</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>L</mi></mrow></mfrac></mrow></mrow></math>
<math overflow="scroll"><mrow><mrow><mi>Reynolds</mi><mo></mo><mstyle><mspace width="0.8em" height="0.8ex" /></mstyle><mo></mo><mi>Number</mi></mrow><mo>=</mo><mfrac><mrow><mi>ρ</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>VD</mi></mrow><mi>μ</mi></mfrac></mrow></math>
FIG. 12
max
In particular, utilizes the analytical model and the relaxation time from the preceding step and adjusts the maximum drag reduction until data from the three pipe sizes matches the approximate fit. In this case, a value of one for the maximum drag reduction and relaxation time (i.e., ΔB=1 and λ=1) arrive at approximate fits for the data from each pipe size. Note that the model results in three different curves due to three different diameters of the pipes used in the flow loop.
Once maximum drag reduction and relaxation times are tuned based on the measured flow loop data, the model can be used to upscale to estimate Darcy Friction Factor vs. Reynolds number for any pipe diameter. When a Darcy Friction Factor versus Reynolds number curve is generated, using the equation Friction
<math overflow="scroll"><mrow><mrow><mi>Factor</mi><mo>=</mo><mrow><mfrac><mrow><mn>2</mn><mo></mo><mi>D</mi></mrow><mrow><mi>ρ</mi><mo></mo><msup><mi>V</mi><mn>2</mn></msup></mrow></mfrac><mo></mo><mfrac><mi>dP</mi><mrow><mi>d</mi><mo></mo><mstyle><mspace width="0.3em" height="0.3ex" /></mstyle><mo></mo><mi>L</mi></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></math>
then a pressure drop versus flow rate table can be estimated.
FIG. 13
is a plot of estimated pressure drop versus flow rate for a field scale hydraulic fracturing operation compared with measured field data. In particular, flow-loop data for three fluids having a concentration of drag reducing agent ranging from 0.5-1.5 gpt was used to tune the model for the estimated values. The field casing pressure drop versus flow rate data was estimated by performing a step-down test and monitoring the wellhead pressure. No downhole gauge was available and known methods were utilized to decouple casing pressure drop from perforation pressure drop and near-wellbore pressure drop. The results appear to estimate the field observed pressure drop data well based on measured laboratory flow-loop data.
The preceding steps can be repeated for various concentrations of drag reducing agents. And if one desired to evaluate two or more drag reducing agents, then the preceding steps may be repeated for each drag reducing agent or call of agents being considered.
FIGS. 14 and 15
FIG. 14
FIG. 15
show plots of field pressure gradient compared to flow rate for polymer relaxation times and ranges of polymer drag reduction parameter ΔB values. It can be observed from these diagrams that as the ΔB value increases that after a certain quantity the pressure gradient for a particular pump rate no longer changes. For example, in , for a pipe ID of 6.125 inches and a relaxation time of 0.0005 seconds, the ΔB values after about 60 or 70 appear to be approximately equivalent. Accordingly, adding additional drag reducing agent to the fluid being injected in the field to increase the ΔB value after it reaches a value of about 60 or 70 appears to have no benefit. For comparison, if the relaxation time is increased to 0.02 seconds as shown in for a pipe ID of 6.125 inches, the ΔB values after about 50 or 60 appear to be approximately equivalent (i.e., less drag reducing agent is needed compared to the drag reducing agent with a relaxation time of 0.0005 seconds). Therefore, the amount or type of drag reducing agent can be modified or optimized for field application using ΔB values.
FIGS. 16 and 17
FIGS. 14 and 15
show corresponding plots of laboratory pressure gradient compared to flow rate for polymer relaxation times and ranges of polymer drag reduction parameter ΔB values. Here it is observed that under laboratory or flow loop conditions, ΔB values tend to reach an optimum value that is lower than those observed under field conditions (i.e., for the same pipe diameter and relaxation time the ΔB value needed to achieve maximum drag reduction is lower). Similar to , ΔB values observed in the lab can be plotted to modify or optimize the amount or type of drag reducing agent for a field application. | |
6:00 for a 6:15 start. Meet Our group at the Fort Hill parking lot in East Providence at the beginning of the East Bay Bike Path, for the 2ish mile ride to Burnside.
****Traffic can be crazy on 195 that time of day. Be sure to use you GPS app to keep and eye out for backups. WAZE is a good one to add to you phone.
This is a night ride. Bring your lights and check the weather.
Keep your daughter dry on a bike ride.
OK folks, I’ll be at the Revs game at Gillette Stadium on Sunday. Paul will be leading this ride.
How about if we try our first Spring ride on Sunday 3/24/19 at 10 AM on the EBBP weather permitting???
It’s still early Spring, so let’s be smart about snacks, water and clothing. You know what you need.
Pace: Casual. Let’s face it, It’s early spring. Let’s not overdo it.
The chance of rain, snow squalls and gale force wind gusts, has forced the cancellation of tonight’s ride in Providence. There is a ride next Friday, March 30, in Providence. Details will be posted next week.
45 F and partly sunny for the start of this months full moon ride.
This is a casual ride on the East Bay Bike Path. A good ride for those who are coming out of winter hibernation. The usual average speed is 10-13 MPH.
****If you peter out, it’s an easy return to the start on your own. Just stay on the bike path till you see your car.
Time: Meet at 5:45 for a 6 PM start.
We will ride to India Point Bridge in Providence on the East Bay Bike Path.
The ride, around 22 miles, will finish in the dark. Bring a light.
The weather looks decent for our first full moon ride in a while. This is a casual ride on the East Bay Bike Path. A good ride for those who are coming out of winter hibernation. The usual average speed is 10-13 MPH.
Fort Hill Parking lot Veterans Memorial Parkway across from the golf course.
There are many rides and events in this area for all levels of riders. Usually, the larger the group, the better chance of finding people who ride just like you. Weather it be a casual 10-12 MPH average speed, or the intermediates who average 12-14 MPH on long rides.
None of the rides on the following list are geared to the really fast, head down, race to the finish line bicyclists.
April 14. The Boston Midnight Marathon Bike Ride.
For a modest fee, the group will transport your bike from South Station in Boston.
May 19. The Mattapoisett Land Trust Tour de Creme. Mile options are 5-10-15-25-50. Fee is $1 per mile at registration. This one fill up quickly. Watch our pages for open registration. CLICK HERE for the video.
Ten Speed Spoke bike shop in Newport. Longest Day of the Year Ride. It’s the Thursday closest to the Summer Solstice. Probably on June 20th. CLICK HERE for the video. | https://amidnightrider.com/author/capejohn/page/2/ |
Portfolio Manager Jim Miles talks about H&W’s assessment of value, including:
- How we determine what a quality business is
- The importance of in-house research when evaluating small companies
- Our definition of risk
Portfolio Manager Jim Miles talks about H&W’s assessment of value, including:
________________________________________
All investments contain risk and may lose value. This podcast is for general information only and should not be relied on for investment advice or recommendation of any particular security, strategy, or investment product.
The portfolio manager’s views and opinions expressed in this podcast are as of April 29, 2022. Such views are subject to change and may differ from others in the firm, or the firm as a whole. The portfolio manager’s comments may include estimated and/or forecasted views, which are believed to be based on reasonable assumptions within the bounds of current and historical information. However, there is no guarantee that any estimates, forecasts or views will be realized. The securities discussed herein are not meant to be indicative or reflective of the strategy’s entire portfolio. Rather, such examples are meant to exemplify H&W’s analysis and execution of the strategy. While the securities highlighted were positive contributors to the strategy during the one year ended 3/31/22, not all investments held in the strategy were profitable. In addition, no assumption should be made that the securities highlighted continue to be profitable. Past performance of these securities, or any other investments, is not an indicator of future results. H&W reserves the right to change its investment perspective and outlook and has no obligation to provide revised assessments and opinions.
Investment returns include reinvestment of dividends, interest and capital gains. Valuation is based on trade-date information and stated in U.S. dollars. Net performance results are presented after actual management fees and all trading expenses but before custodial fees. The Small Cap Value strategy’s returns for different time periods and market cycles can result in significantly different performance results. An account’s investment guidelines, timing of transactions, market conditions at the time of investment and other factors may lead to different performance results. The Composite includes all Small Cap Value discretionary accounts. The Small Cap Value strategy seeks capital appreciation primarily through investments in equity securities of small cap companies and may invest in foreign (non-U.S.) securities. Additional performance disclosures are included in the strategy’s GIPS Report.
Holdings and attribution analysis (gross of fees) are based on a representative portfolio of the Small Cap Value (SCV) Strategy. The portfolio is selected based on factors determined by Advisor to be “representative” of the strategy, considering such factors as (but not limited to) investment guidelines/restrictions, time period under Advisor’s discretion, and/or cash flow activities. Each portfolio’s holdings/performance in the strategy vary due to different restrictions, cash flows and other relevant considerations. Contributors to Relative Performance identifies those securities that are largest contributors (or detractors) on a relative basis to the index. Securities’ absolute performance may reflect different results. The Russell 2000 Value Index measures the performance of those Russell 2000 companies with lower price-to-book ratios and lower forecasted growth values.
Securities identified do not represent all securities purchased, sold, or recommended for advisory clients, and may not be indicative of current or future holdings or trading activity. No assurance is made that any securities identified, or all investment decisions by H&W were or will be profitable. Any discussion or view of a particular company is as of the publication date but may be sold and no longer held in the SCV strategy at any time, for any reason, without notice, subsequent to the publication date. Portfolio holdings are subject to change; a complete list of holdings is available upon request at [email protected], subject to the firm’s portfolio holdings disclosure policy.
Investing in smaller, medium-sized and/or newer companies involves greater risks not associated with investing in large company stocks, such as business risk, significant stock price fluctuations and illiquidity. Investing in foreign as well as emerging markets involves additional risk such as greater volatility, political, economic, and currency risks and differences in accounting methods. Investing in equity securities have greater risks and price volatility than U.S. Treasuries and bonds, where the price of these securities may decline due to various company, industry, and market factors.
A value-oriented investment approach involves the risk that value stocks may remain undervalued or may not appreciate in value as anticipated. Value stocks can perform differently from the market as a whole or from other types of stocks and may be out of favor with investors and underperform growth stocks for varying periods of time.
Russell Investment Group is the source and owner of the Russell Index data contained herein (and all trademarks related thereto), which may not be redistributed. The information herein is not approved by Russell. H&W and Russell sectors are based on the Global Industry Classification Standard by MSCI and Standard and Poor’s.
Market Disruption: The global coronavirus pandemic has caused disruption in the global economy and extreme fluctuations in global capital and financial markets. H&W is unable to predict the impact caused by coronavirus pandemic, which has the potential to negatively impact the firm’s investment strategies and investment opportunities.
©2023 Hotchkis & Wiley. All rights reserved. No portion of the podcast may be published, reproduced, transmitted or rebroadcast in any form without the express written permission of H&W.
Past performance is not indicative of future performance. | https://www.hwcm.com/2022/05/03/small-cap-update-with-jim-miles/ |
I’m thinking about organizing a chocolate tasting at work, and hoping other people out there will have done something similar. This is just intended for fun, but I still want to run a proper taste-test protocol, and actually determine the real collective preferences, if there are any. Of course all the raw data would be made public.
A few issues I’m thinking about:
Protocol
The protocol will obviously depend on how many people are interested in participating and on whether I accept participant suggestions (or ask them to bring something — would require some care to make sure people didn’t all bring the same bar of Lindt they got at the mall store).
Obviously, the usual precautions (random order of tasting, identifying marks removed to the extent possible, portions controlled) should be taken. That means that I probably don’t get to include my own preferences, if I’m running the tasting. I’m not sure what the ideal number of samples per round would be, but I think it’s probably no more than six. I’m not sure whether there is an ideal number of tasters per round, or indeed whether the same tasters should do all rounds. Presumably I would need to hold back some of each sample for use in later rounds.
Some sort of palate-cleanser will need to be provided — seltzer is good enough for me, but not for many other people. Something with some fat (milk, maybe) or a non-polar solvent like rum may be good. I might just ask tasters to bring their own.
Evaluation
I’m thinking the tasters should provide both individual evaluations for each chocolate tasted, and an overall ranking of all of the chocolates in each round. The overall ranking in each round can be determined from the individual rankings, with ties broken on the basis of individual scores. How many moved on to the next round would depend on how many samples there were, and the number of rounds. (So with 24 chocolates and three rounds, that might be six quarter-finals of four chocolates each in the first round, with the top two moving on to two semi-finals of six each, and then the top three from each semi moving on to the finals.)
For the individual evaluations, I’m thinking a 1-5 scale (5 being the best) on aroma, texture, sweetness, bitterness, and flavor, plus freeform comments.
Categories
Obviously it is important to compare only, um, comparable chocolates against each other. The ones I’m most interested in doing would be high-end extra-dark eating chocolate, but given the variety of chocolates out there on the market, I can easily see doing any of these categories (but not all at once):
- Unadulterated dark chocolate, at least 60% cacao (optionally, vanilla, lecithin) — might need to split this into 60–70% and over-70% as some people would likely find truly high-test chocolate unpalatable and give it poor ratings
- Semisweet chocolate (anything brown with less than 60% cacao but no dairy ingredients)
- Milk chocolate (for those who insist)
- Chocolate with stuff in it:
- Fruit
- Mint
- Nuts (whole, in pieces, or ground)
- Spices (cassia, ginger, chile — any flavoring other than vanilla)
- Anything else — bacon, caramel, potato chips, wafers, etc.
Some of these categories would be more difficult to organize than others, and obviously anything with animal flesh in it would need to be set aside for the vegetarians. I think I could do all of the “anything else” category from the Vosges catalogue alone, given how many different varieties they do.
If I do this, I’d be happy to include remote submissions of chocolate provided they meet the qualifications for whichever category I end up doing.
Analysis
Obviously I would make all of the data available so other people could analyze it. I’m not a statistician myself and would probably write some simple scripts to help with data reduction, with all of the evaluations stored in a database. (Probably keyed by hand, though, unless someone can point out an existing Free Software application that would allow the users to enter their own evaluations on a mobile device.)
Thoughts?
UPDATE: Moved further discussion to the “Tasters” web in CSAIL FosWiki. | https://blog.bimajority.org/2014/07/10/organizing-a-chocolate-tasting/ |
What Are My Company Record Keeping Requirements?
Operating any commercial enterprise involves keeping books and records. Running a company is no exception. Accurate financial records are a must for all companies. In addition to their commercial utility, companies in Australia are also legally required to maintain written financial records. This article explains your company record keeping requirements and details those books and records you need to keep.
Company Record Keeping Requirements
The Corporations Act 2001 (Cth) (the Act) obliges all companies to maintain written financial records that accurately record and explain its transactions, financial position and performance. These records must permit the preparation and audit of true and fair financial statements.
The Act considers correct record keeping such a fundamental task that it lists it as one of the key responsibilities of directors and company secretaries. The Australian Securities and Investment Commission (ASIC) views breaches of this obligation extremely seriously.
What Do We Mean by Financial Records?
The Act’s definition of financial records includes:
- Documents of prime entry;
- Receipts, invoices, orders for the payment of money, bills of exchange, promissory notes, vouchers and cheques;
- Any working papers or other documents necessary to explain how financial statements were drawn up and any adjustments required in that process.
While electronic records are acceptable, the Act requires that all financial records in electronic format must be convertible into hard copy within a reasonable time. The responsibility for this rests with the individual company even if a third party (such as their accountant) keeps their financial records on their computer system.
The point of recordkeeping from a legal perspective is to ensure that a person can correctly gauge a company’s financial position from the books. This assists transparency and accountability.
In general, the kinds of records and books that a company should keep include:
- Financial statements such as profit and loss accounts, balance sheets, depreciation schedules and taxation returns (for income tax, group tax, fringe benefits tax, business activity statements and all supporting documents);
- General ledger;
- General journal;
- Register of assets;
- Cash records including the cash receipts journal, bank deposit books, cash payments journal, cheque butts, petty cash books;
- Bank account statements, bank reconciliations and bank loan documents;
- Records for sales and debtors including the sales journal, debtors ledger, list of debtors, invoices issued, statements issued, delivery dockets.
- Invoices and statements received and paid;
- Creditors ledger;
- Records for work in progress including job/customer files, stock listings, creditors records;
- Unpaid invoices including correspondence, annual returns and forms for ASIC, records for wages and superannuation;
- Records of all computer back-up discs (back-ups should happen at least once a month);
- Registers (where relevant) for members, options, debenture holders, prescribed interests, charges, unclaimed property;
- Minutes of directors’ meetings;
- Minutes of members’ meetings;
- Deeds for trust, debentures, contracts, agreements (for example, lease agreements); and
- Any inter-company transactions, including guarantees.
The introduction of the Personal Property Securities Register on 30 January 2012 repealed those sections of the Act requiring companies to keep a register of all charges. However, this applies only to charges acquired from that date onwards. Companies must therefore still maintain a record for charges held before that date.
Further, while the Act does not require small proprietary companies to keep financial statements (for example, profit and loss accounts) unless ASIC or shareholders specifically request it, they are nonetheless a valuable commercial tool. For that reason, such companies should seriously consider preparing such statements even though it is not legally necessary to do so.
The Act requires companies to retain all of their financial records for a minimum of seven years.
Of course, this list of records is only a guide. The company record keeping requirements of every company differ because they depend on the company. For example, if your business holds a securities or futures licence you will need to keep a register to that effect.
***
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What Can Directors Learn From Storm Financial Ltd? | https://legalvision.com.au/what-are-my-company-record-keeping-requirements/ |
\
Hrachya B. Nersisyan$^{(1)}$ and Amal K. Das$^{(2)}$\
$^{(1)}$ [*Division of Theoretical Physics, Institute of Radiophysics and Electronics, Alikhanian Brothers St. 2, Ashtarak-2, 378410, Republic of Armenia*]{}[^1]
$^{(2)}$ [*Department of Physics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada*]{}
[**[Abstract]{}**]{}
In this paper we report on our theoretical studies of various aspects of the correlated stopping power of two point-like ions (a dicluster) moving in close but variable vicinity of each other in some metallic target materials the latter being modelled by a degenerate electron gas with appropriate densities. Within the linear response theory we have made a comprehensive investigation of correlated stopping power, vicinage function and related quantities for a diproton cluster in two metallic targets, aluminum and copper, and present detailed and comparative results for three approximations to the electron gas dielectric function, namely the plasmon-pole approximation without and with dispersion as well as with the random phase approximation. The results are also compared, wherever applicable, with those for an individual projectile.
INTRODUCTION
============
The stopping of energetic charged particles in a target material is a problem of long-standing theoretical and experimental interest. Early pioneering theoretical work by Bohr who had a lifelong interest in this problem was followed by Bethe and others. There is by now an extensive literature on this topic. We refer the reader to two recent review articles \[1, 2\]. The problem in its generality is rather complex. However simplified theoretical models have been applied with considerable success in explaining experimental data. Assuming a weak coupling between the energetic particle and a target material, specially a metal which is usually approximated by a degenerate electron gas, a detailed theoretical model has emerged through the works of Bohm, Pines, Lindhard, Ritchie and other authors \[3-5\].
A comprehensive treatment of the quantities related to inelastic particle-solid and particle-plasma interactions, e.g. scattering rates and differential and total mean free paths and energy losses, can be formulated in terms of the dielectric response function obtained from the electron gas model. The results have important applications in astrophysics \[6, 7\] and radiation and solid-state physics \[8-11\], and more recently, in studies of energy deposition by ion beams in plasma fusion targets \[12-16\].
One can think of several situations in which the projectile beam ions may be closely spaced so that their stopping is influenced by their mutual interactions \[14, 15\] and thus differs from the stopping of charged particles whose dynamics is independent of each other. This can happen, for example, in the case of very high density beams, or more realistically, when cluster ions are to be used instead of standard ion beams \[16\].
The stopping of uncorrelated or independent charged particles in a degenerate electron gas (DEG) has been extensively studied in the literature (see, for example, \[3-9, 13\] and other references therein). These studies have been done mostly within the linear response formalism and for the projectile velocity $V$ comparable or greater than $v_F$, the electron Fermi velocity.
The objective of this paper is to make a study of the stopping power (SP) of correlated charged particles in a DEG. The simplest and yet physically relevant case is the SP of an ion pair (a dicluster). We report on a comprehensive investigation, which is mostly numerical, of various aspects of a dicluster stopping in a DEG. Previous theoretical works have considered this problem within the linear response theory \[9-11\] and in a simple plasmon-pole approximation. In our study, again in the linear response formalism, we have used both the plasmon-pole approximation (PPA) as well as the full Lindhard expression for the random phase approximation (RPA) in the DEG dielectric function. As in earlier studies we consider a diproton cluster as a projectile and compare our theoretical results with those of Basbas and Ritchie \[10\] who used PPA and with those obtained in RPA. Whenever applicable, the results are also compared with those for an independent projectile (e.g. those due to Yakolev and Kotel’nikov \[7\]). No RPA results for a diproton cluster SP and related aspects have previously been reported in the literature.
STOPPING POWER
==============
Let us consider an external charge with distribution $\rho _{{\rm ext}}({\bf %
r},t)=Q_{{\rm ext}}({\bf r}-{\bf V}t)$ moving with velocity ${\bf V}$ in a medium characterized by the longitudinal dielectric function $\varepsilon
(k,\omega )$. Within the linear response theory and in the Born approximation the scalar electric potential $\varphi ({\bf r},t)$ due to this external charge screened by the medium is given by \[1\]
$$\varphi ({\bf r},t)=4\pi \int d{\bf k}Q_{{\rm ext}}({\bf k})\frac{\exp
\left[ i{\bf k}({\bf r}-{\bf V}t)\right] }{k^2\varepsilon (k,{\bf kV})}{\bf ,%
}$$
where $Q_{{\rm ext}}({\bf k})$ is the Fourier transform of function $Q_{{\rm %
ext}}({\bf r})$.
The stopping power which is the energy loss of the external charge regarded as a projectile, per unit path length in the medium regarded as a target material, can be calculated from the force acting on the charge. The latter is related to the induced electric field ${\bf E}_{{\rm ind}}$ in the medium. For a three-dimensional medium we have, for the SP,
$$S\equiv -\int d{\bf r}Q_{{\rm ext}}({\bf r}-{\bf V}t)\frac{{\bf V}}V{\bf E}_{%
{\rm ind}}({\bf r},t)=\frac{2(2\pi )^4}V\int d{\bf k}\left| Q_{{\rm ext}}(%
{\bf k})\right| ^2\frac{{\bf kV}}{k^2} {\rm Im} \frac{-1}{\varepsilon (k,%
{\bf kV})}{\bf .}$$
Eq. (2) is applicable to any external charge distribution. We shall apply it to a cluster of two point-like ions having charges $Z_1e$ and $Z_2e$ separated by a variable distance ${\bf R}$. For this dicluster
$$\left| Q_{{\rm ext}}({\bf k})\right| ^2=\frac{e^2}{(2\pi )^6}\left[
Z_1^2+Z_2^2+2Z_1Z_2\cos \left( {\bf kR}\right) \right] .$$
Then the stopping power of a dicluster may be written as
$$S=\left( Z_1^2+Z_2^2\right) S_{{\rm ind}}(V)+2Z_1Z_2S_{{\rm corr}}({\bf R}%
,V),$$
where $S_{{\rm ind}}(V)$ and $S_{{\rm corr}}({\bf R},V)$ stand for individual and correlated SP, respectively. From Eqs. (2) and (3)
$$S_{{\rm ind}}(V)=\frac{2e^2}{\pi V^2}\int_0^\infty \frac{dk}k\int_0^{kV}{\rm %
Im} \frac{-1}{\varepsilon (k,\omega )}\omega d\omega ,$$
$$\begin{aligned}
S_{{\rm corr}}({\bf R},V) &=&\frac{2e^2}{\pi V^2}\int_0^\infty \frac{dk}%
k\int_0^{kV}{\rm Im}\frac{-1}{\varepsilon (k,\omega )}\omega d\omega \\
&&\times \cos \left( \frac \omega VR\cos \vartheta \right) J_0\left( R\sin
\vartheta \sqrt{k^2-\frac{\omega ^2}{V^2}}\right) . \nonumber\end{aligned}$$
$J_0(x)$ is the Bessel function of the first kind and zero order and $%
\vartheta $ is the angle between the interionic separation vector ${\bf R}$ and the velocity vector ${\bf V}$. Eqs. (5) and (6) can also be obtained from the linearized Vlasov-Poisson equations for a two-ion projectile system, as has been done by Avanzo et al. \[14\]. In their study the target material is a dense classical electron gas.
We note that there are two contributions to the SP of a two-ion cluster. The first one, given by the first term in Eq. (4), is the uncorrelated particle contribution and represents the energy loss of the individual projectiles due to their coupling with the target electron gas. The second contribution, the second term in Eq. (4), arises due to a correlated motion of the two ions through a resonant interaction with the excitations of the electron gas. Both terms are responsible for an irreversible energy transfer from the two-ion projectile system to the target electron gas.
In many experimental situations, clusters are formed with random orientations of ${\bf R}$. A correlated stopping power appropriate to this situation may be obtained by carrying out a spherical average over ${\bf R}$ of the $S_{{\rm corr}}({\bf R},V)$ in Eq. (4). We find
$$\overline {S}_{{\rm corr}}(R,V)=\frac{2e^2}{\pi R V^2}\int_0^\infty \frac{dk%
}{k^2} \sin (kR) \int_0^{kV} {\rm Im} \frac{-1}{\varepsilon (k,\omega )}%
\omega d\omega.$$
One may consider an interference or vicinage function, which is a measure of the separation of single-particle contribution from its correlated counter-part, to the stopping power. This function is defined as \[10, 14, 15\]
$$\Gamma ({\bf R},V)=\frac{S_{{\rm corr}}({\bf R},V)}{S_{{\rm ind}}(V)}.$$
Eq. (4) can then be put in the form
$$S=\left( Z_1^2+Z_2^2\right) S_{{\rm ind}}(V)\left[ 1+\frac{2Z_1Z_2}{%
Z_1^2+Z_2^2}\Gamma ({\bf R},V)\right] .$$
$\Gamma ({\bf R},V)$ describes the strength of correlation effects with respect to an uncorrelated situation. The vicinage function becomes equal to unity as $R\rightarrow 0$ when the two ions coalesce into a single entity, and goes to zero as $R\rightarrow \infty $ when the two ions are totally uncorrelated.
THEORETICAL CALCULATIONS OF SP
==============================
The key ingredient in the calculation of stopping power, as outlined in Section II, is the linear response function $\varepsilon (k,\omega )$ of the target material. The latter is modelled in most cases by a dense electron gas neutralized by a positive background - the so-called jellium model. The effect of lattice is not included except perhaps through an effective mass of the electrons. This is a reasonable model for target materials like Cu and Al. $\varepsilon (k,\omega )$ for a dense, and hence a degenerate electron gas, has been calculated in various approximations in the literature. Two of them are: (A) Plasmon-Pole approximation (PPA), and (B) the full random phase approximation (RPA). Actually the plasmon-pole approximation is a simplification of the RPA response function. We shall consider SP in these two approximations.
SP in PPA without plasmon dispersion
------------------------------------
Here we consider the simplest model of the dielectric function of a jellium. In Ref. \[10\] (see also the Ref. \[15\]) a plasmon-pole approximation to $%
\varepsilon (k,\omega )$ for an electron gas was used for calculation of dicluster SP. In order to get easily obtainable analytical results, Basbas and Ritchie \[10\] employ a simplified form which exhibits collective and single-particle effects
$${\rm Im}\frac{-1}{\varepsilon \left( k,\omega \right) }=\frac{\pi \omega _p^2%
}{2\omega }\left[ H\left( k_c-k\right) \delta \left( \omega -\omega
_p\right) +H\left( k-k_c\right) \delta \left( \omega -\frac{\hbar k^2}{2m}%
\right) \right] ,$$
where $H(x)$ is the Heaviside unit-step function, $\hbar \omega _p$ is the plasma energy of the electron gas and the choice $k_c=\left( 2m\omega
_p/\hbar \right) ^{1/2}$ allows the two $\delta $ functions in Eq. (10) to coincide at $k=k_c$ in the $k$-$\omega $ plane.
The first term in Eq. (10) describes the response due to nondispersive plasmon excitation in the region $k<k_c$, while the second term describes free-electron recoil in the range $k>k_c$ (single-particle excitations). This approximate function satisfies the sum rule
$$\int_0^\infty {\rm Im}\frac{-1}{\varepsilon (k,\omega )}\omega d\omega =%
\frac{\pi \omega _p^2}2$$
for all values of $k$.
In this approximation if $V>\left( \hbar \omega _p/2m\right) ^{1/2}\equiv
V_p $
$$S_{{\rm ind}}(\lambda )=\frac{e^2\omega _p^2}{V^2}\ln \left( \frac{2mV^2}{%
\hbar \omega _p}\right) =\frac{\Sigma _0}{\pi \chi ^2\lambda ^2}\ln \left(
\frac{\lambda ^2\sqrt{3}}\chi \right) ,$$
where $\lambda =V/v_F$, $\chi ^2=1/\pi k_Fa_0=\left( 4/9\pi ^4\right)
^{1/3}r_s$; $r_s=\left( 3/4\pi n_0a_0^3\right) ^{1/3}$, $n_0$ is the electron gas density and $a_0=0.53\times 10^{-8}$cm is the Bohr radius. $k_F$ is the Fermi wave number of the target electrons and $\Sigma _0=2.18$ GeV/cm. In our calculations $\chi $ (or $r_s$) serves as a measure of electron density. The result in Eq.(12) agrees exactly with the Bethe SP formula, except that the plasmon energy of the electron gas $\hbar \omega _p$ appears instead of the usual mean atomic excitation energy. Eq. (12) represents the contribution of valence/conduction electrons in a solid to the stopping of an ion.
Using Eq. (10) in Eq. (6), in the high-velocity limit $V>V_p$ (or $\lambda
^2>\chi /\sqrt{3}\equiv \lambda _0^2$) one finds
$$\begin{aligned}
S_{{\rm corr}}(R,\vartheta ,\lambda ) &=&\frac{\Sigma _0}{\pi \chi ^2\lambda
^2}\left\{ \cos \left( \frac{2\chi }{\lambda \sqrt{3}}k_FR\cos \vartheta
\right) \right. \\
&&\times \int_1^{\lambda /\lambda _0}\frac{dx}xJ_0\left( \frac{2\chi }{%
\lambda \sqrt{3}}k_FR\sin \vartheta \sqrt{x^2-1}\right) + \nonumber \\
&&\left. \int_{2\lambda _0}^{2\lambda }\frac{dx}x\cos \left( \frac{x^2}{%
2\lambda }k_FR\cos \vartheta \right) J_0\left( xk_FR\sin \vartheta \sqrt{1-%
\frac{x^2}{4\lambda ^2}}\right) \right\} . \nonumber\end{aligned}$$
If one ion trails directly behind the other ($\vartheta =0$) from Eq. (13) we find
$$\begin{aligned}
S_{{\rm corr}}(R,0,\lambda ) &=&\frac{\Sigma _0}{2\pi \chi ^2\lambda ^2}%
\left\{ \cos \left( \frac{2\chi }{\lambda \sqrt{3}}k_FR\right) \ln \left(
\frac{\lambda ^2\sqrt{3}}\chi \right) +\right. \\
&&\left. {\rm ci}\left( 2\lambda k_FR\right) -{\rm ci}\left( \frac{2\chi }{%
\lambda \sqrt{3}}k_FR\right) \right\} , \nonumber\end{aligned}$$
where ${\rm ci}\left( z\right) $ is the integral cosin function
$${\rm ci}\left(z\right) = -\int_z^{\infty} dx\frac{\cos x}{x}.$$
One sees a characteristic oscillatory behavior for large interionic distance $R$. As discussed in \[16\], fluctuations in the stopping power of a medium for a cluster as separation increases are due to electron density variation in the wake of the leading ion. The wavelength of these fluctuations is $%
\sim 2\pi V/\omega _p$ for high-velocity projectiles.
In the case of randomly oriented clusters from Eq. (7) we find
$$\overline{S}_{{\rm corr}}(R,\lambda )=\frac{\Sigma _0}{\pi \chi ^2\lambda ^2}%
\left[ {\rm si_2}\left( \frac{2\chi }{\lambda \sqrt{3}}\left( k_FR\right)
\right) -{\rm si_2}\left( 2\lambda \left( k_FR\right) \right) \right] ,$$
where
$${\rm si_2}\left(z\right) = \int_z^{\infty} dx\frac{\sin x}{x^2} = \frac{%
\sin(z)}{z} - {\rm ci}(z).$$
SP in PPA with plasmon dispersion
---------------------------------
Plasmons without dispersion are an idealization. In real systems plasmons are expected to undergo a dispersion leading to a $\omega (k)$. The actual dispersion ( in RPA) can be obtained from the linear response function (see Sec. 3.3). Here we shall utilize a dispersion which is valid for small and intermediate values of the wave vector ${\bf k}$. Consequently we write
$${\rm Im}\frac{-1}{\varepsilon \left( k,\omega \right) }=\frac{\pi \omega _p^2%
}{2\omega }\delta \left( \omega -\Omega (k)\right) ,$$
where the dispersion is given by
$$\Omega ^2(k)=\omega _p^2+\frac 35k^2v_F^2+\frac{\hbar ^2k^4}{4m^2}.$$
In this approximation when
$$V>\left( \frac 35v_F^2+\frac{\hbar \omega _p}m\right) ^{1/2}\equiv V_0$$
we have, for ISP and CSP,
$$S_{{\rm ind}}(\lambda )=\frac{\Sigma _0}{\pi \chi ^2\lambda ^2}\ln \frac{%
\lambda ^2-3/5+\sqrt{\left( \lambda ^2-3/5\right) ^2-4\chi ^2/3}}{2\chi /%
\sqrt{3}},$$
$$S_{{\rm corr}}(R,\vartheta ,\lambda )=\frac{\Sigma _0}{\pi \chi ^2\lambda ^2}%
\int_{x_{-}(\lambda )}^{x_{+}(\lambda )}\frac{dx}x\cos \left( \frac{\phi
_1(x)}{2\lambda }k_FR\cos \vartheta \right) J_0\left( \frac{\phi _2(x)}{%
2\lambda }k_FR\sin \vartheta \right) .$$
Here $k_F$ is the Fermi wave number, and
$$\phi _1(x)=\sqrt{x^4+\frac{12}5x^2+16\chi ^2/3},$$
$$\phi _2(x)=\sqrt{4\left( \lambda ^2-\frac 35\right) x^2-\left( x^4+16\chi
^2/3\right) },$$
$$x_{\pm }(\lambda )=\sqrt{2\left[ \lambda ^2-\frac 35\pm \sqrt{\left( \lambda
^2-\frac 35\right) ^2-\frac{4\chi ^2}3}\right] }.$$
In the case of randomly oriented clusters we find
$$\overline{S}_{{\rm corr}}(R,\lambda )=\frac{\Sigma _0}{\pi \chi ^2\lambda ^2}%
\left[ {\rm si_2}\left( k_FRx_{-}(\lambda )\right) -{\rm si_2}\left(
k_FRx_{+}(\lambda )\right) \right] .$$
Stopping power in RPA
---------------------
Now we will derive the analytical expressions for the SP of a dicluster in a fully degenerate ($T=0$) electron gas. For this purpose we use the exact RPA dielectric response function obtained by Lindhard \[4\]
$$\varepsilon (z,u)=1+\frac{\chi ^2}{z^2}\left[ f_1(z,u)+if_2(z,u)\right] ,$$
where
$$f_1(z,u)=\frac 12-\frac 1{8z}\left( U_{+}^2-1\right) \ln \left| \frac{U_{+}+1%
}{U_{+}-1}\right| +\frac 1{8z}\left( U_{-}^2-1\right) \ln \left| \frac{%
U_{-}+1}{U_{-}-1}\right| ,$$
$$f_2(z,u)=\left\{
\begin{array}{l}
\frac \pi {8z}\left( 1-\left( u-z\right) ^2\right) ;\quad |u-1|\leqslant
z\leqslant u+1 \\
0;\quad 0\leqslant z\leqslant u-1, \\
0;\quad z\geqslant u+1, \\
\frac 12\pi u;\quad 0\leqslant z\leqslant 1-u
\end{array}
\right. .$$
Here, as in Refs. \[6, 7, 13, 15\], we have introduced the following notations $z=k/2k_F$, $u=\omega /kv_F$, $U_{\pm }=u\pm z$. With these notations Eqs. (5), (6) and (7) read
$$S_{{\rm ind}}(\lambda )=\frac{6\Sigma _0}{\pi ^2\chi ^2\lambda ^2}%
\int_0^\infty z^3dz\int_0^\lambda \frac{f_2(z,u)udu}{\left[ z^2+\chi
^2f_1(z,u)\right] ^2+\chi ^4f_2^2(z,u)},$$
$$\begin{aligned}
S_{{\rm corr}}(\lambda ,R,\vartheta ) &=&\frac{6\Sigma _0}{\pi ^2\chi
^2\lambda ^2}\int_0^\infty z^3dz\int_0^\lambda \frac{f_2(z,u)udu}{\left[
z^2+\chi ^2f_1(z,u)\right] ^2+\chi ^4f_2^2(z,u)} \nonumber \\
&&\times \cos \left( 2\frac{zu}\lambda k_FR\cos \vartheta \right) J_0\left(
2zk_FR\sin \vartheta \sqrt{1-\frac{u^2}{\lambda ^2}}\right) ,\end{aligned}$$
$$\overline{S}_{{\rm corr}}(R,\lambda )=\frac{3\Sigma _0}{\pi \lambda ^2}\frac{%
a_0}R\int_0^\infty \sin \left( 2k_FRz\right) z^2dz\int_0^\lambda \frac{%
f_2(z,u)udu}{\left[ z^2+\chi ^2f_1(z,u)\right] ^2+\chi ^4f_2^2(z,u)}.$$
In order to evaluate the integrals by $z$ in Eqs. (30)-(32) at $\lambda <1$ (in the low-velocity limit) we split the integration region into two domain: $0\leqslant z\leqslant 1-u$ and $1-u\leqslant z\leqslant 1+u$, where ${\rm {%
Im}}$ $\varepsilon \sim f_2\neq 0$. However at $\lambda >1$ (in high-velocity limit) we need to take into account the region $1\leqslant
u\leqslant \lambda $, $0\leqslant z\leqslant u-1$, where $f_2$ may vanish. The integration in this region includes the excitation of collective plasma modes (plasmons) by fast charged particles. Consequently, although $f_2=0$ the integrals in this region are not equal to zero. A calculation of the collective part of SP is facilitated if we use the following known expression
$$\begin{aligned}
\frac{\chi ^2f_2(z,u)}{\left[ z^2+\chi ^2f_1(z,u)\right] ^2+\chi ^4f_2^2(z,u)%
} &\rightarrow &\pi \delta \left( z^2+\chi ^2f_1(z,u)\right) = \\
&=&\pi \frac{\delta \left( z-z_r(\chi ,u)\right) }{\left| 2z+\chi ^2\frac{%
\partial f_1(z,u)}{\partial z}\right| _{z=z_r(\chi ,u)}}, \nonumber\end{aligned}$$
where $z_r(\chi ,u)$ is the solution of the dispersion equation $\varepsilon
(k,\omega )=0$ in variables $z$ and $u$.
Figure 19 shows the solution $z_r(\chi ,u)$ for various values of $\chi $ (solid line, $\chi =0.5$; dashed line, $\chi =0.15$; dotted line, $\chi
=0.05 $). It may be noted that the integration domain $0\leqslant u\leqslant
\lambda $, $z>u+1$, where $f_2=0$, does not contain the dispersion curve $%
z_r(\chi ,u)$ calculated for metallic densities $\chi \sim 0.5$ ($r_s\sim 2$). Consequently the SP in this region of variables $z$ and $u$ vanishes and there is no plasmon excitation.
Let us consider the low-velocity limit ($V\ll v_F$) of Eqs. (30) and (31). In this limit one can obtain simpler expressions for SP. From Eqs. (30) and (31) we have
$$S_{{\rm ind}}(\lambda )\simeq \frac{\Sigma _0}{\pi \chi ^2}\lambda \int_0^1%
\frac{z^3dz}{\left[ z^2+\chi ^2f(z)\right] ^2},$$
$$S_{{\rm corr}}(\lambda ,R,\vartheta )\simeq \frac{3\Sigma _0}{2\pi \chi ^2}%
\lambda \int_0^1\frac{z^3dz}{\left[ z^2+\chi ^2f(z)\right] ^2}\left[ \Phi
_1(zk_FR)+\Phi _2(zk_FR)\sin ^2\vartheta \right] ,$$
where
$$\Phi _1(\xi )=\frac 1{\xi ^3}\left[ \left( \xi ^2-\frac 12\right) \sin
\left( 2\xi \right) +\xi \cos \left( 2\xi \right) \right] ,$$
$$\Phi _2(\xi )=\frac 1{\xi ^3}\left[ \left( \frac 34-\xi ^2\right) \sin
\left( 2\xi \right) -\frac 32\xi \cos \left( 2\xi \right) \right] ,$$
$$f(z)\equiv f_1(z,0)=\frac 12+\frac{1-z^2}{4z}\ln \frac{1+z}{1-z}.$$
From Eqs. (36) and (37) it follows that $\Phi _1(\xi )\rightarrow 2/3$, $%
\Phi _2(\xi )\rightarrow 0$ at $\xi \rightarrow 0$ and consequently, as expected, $S_{{\rm corr}}(\lambda ,R,\vartheta )\rightarrow S_{{\rm ind}%
}(\lambda )$ when $R\rightarrow 0$. Note that (as is well-known \[4, 13\]) in the low-velocity limit the SP is proportional to the velocity of particle (Eqs. (34) and (35)). Thus the vicinage function $\Gamma (\lambda
,R,\vartheta )$ at $\lambda \ll 1$ depends only on interionic distance $R$ and orientation angle $\vartheta $.
NUMERICAL RESULTS AND DISCUSSION
================================
Using the theoretical results of Secs. 2 and 3, we have made extensive numerical calculations of stopping power (SP) and related quantities. In this section we present detailed numerical results for two target materials, Al and Cu. These two targets have been chosen because of their frequent use in experiments and also because of their different electron densities. In our calculations $\chi $ (or $r_s$) is a measure of electron density.
As a simple but generic example of a projectile, we have considered a diproton cluster for which we present theoretical results for the following quantities of physical interest: stopping power (SP/2), vicinage function (VF), angle-averaged stopping power (ASP/2), angle-averaged vicinage function (AVF) together with the dependence of SP/2 and VF on $R$, the inter-ionic separation distance within the cluster. The reason why SP has been divided by a factor of 2 is that the SP results for a diproton cluster are expected to reduce asymptotically (as $R$ tends to infinity) to those for two uncorrelated protons, the latter being referred to as ISP. ASP has been treated in the same way.
In our calculations of these quantities we have employed the linear response approach which assumes a swift ion-cluster projectile and also that the ion cluster presents a weak perturbation on the target plasma. The validity of the linear response approach to study ion-cluster stopping has been discussed in detail by Zwicknagel and Deutsch \[17\]. We refer the reader to their insightful discussion.
We model the Al and Cu targets by a dense (degenerate) electron gas neutralized by a positive background (the jellium model) with electron densities appropriate for the respective targets. The linear response of the target electron gas, which couples the cluster projectile to the target, is considered at three levels of approximations to the dielectric function $%
\varepsilon (k,\omega )$ as discussed in sections 2 and 3. In the context of stopping power these approximations are subject to the following general remarks: The plasmon-pole approximation (PPA) is valid only in the high velocity regime when the mean velocity $V$ of the cluster is $>v_F$, the Fermi velocity of the target electrons. For $V<v_F$ and for velocities near the threshold of collective mode excitations, this approximation is not adequate. The RPA overcomes this limitation although it cannot account for short-range correlations in the electron gas. Within PPA itself, PPA-1 (without plasmon dispersion) is more limited than PPA-2 (with plasmon dispersion). The figures we present serve as a comparative study of how these levels of approximation affect the various physical quantities related to stopping power.
Figs. 1-4 show cluster stopping power (CSP and its dependence on various quantities of experimental interest). Let us first note that these figures are presented for two specific values, $0$ and $\pi /2$, of the angle $%
\vartheta $. Correlations between the two ions in the dicluster are maximum and minimum, respectively, for these two values of $\vartheta $. The objective is then to see how, for these maximum and minimum configurations, CSP depends on $R$ and $V/v_F$. Fig. 1 shows CSP for Al target with $%
R=10^{-8}{\rm cm}$, as a function of $V/v_F$ for the two above-mentioned values of $\vartheta $, within PPA. The lines without circles correspond to PPA-1 and, with circles, to PPA-2. The angular dependence of CSP is particularly noteworthy. It is seen that in a medium velocity range ($V<2v_F$), CSP has a remarkably higher value for the larger value of $\vartheta $. This is likely due to single-particle excitations in this velocity range. In the higher velocity range, the dicluster wake-field excitations become important and we find that the situation is reverse in the higher velocity range ($V>2v_F$) for which CSP for $\vartheta =0$ is larger than for $%
\vartheta =\pi /2$.
In the low velocity range the difference between PPA-1 and PPA-2 (for both $%
\vartheta =0$ and $\pi /2$) is noticeable while in the high velocity range this difference becomes negligible. This is again due to single-particle excitations in the low velocity range. For comparison, we have also presented the uncorrelated stopping power (ISP).
When we increase the inter-ionic separation distance $R$ from $10^{-8}{\rm cm%
}$ to $5\times 10^{-8}{\rm cm}$, keeping other physical parameters the same, some interesting changes occur, as can be seen from Fig. 2. A noticeable change is that now, for $V<2v_F$, CSP for $\vartheta =0$ is higher than that for $\vartheta =\pi /2$. This sensitivity of CSP to the angle $\vartheta $ as $R$ is varied may be due a combination of factors. The dicluster behaves like a compact project for small $R$, and like an extended projectile for large $R$. This has a bearing on $S_{{\rm corr}}$ given in Eqs. (13) and (22). Correlation effects are expected to be maximum when the two ions are aligned with each other in the direction of propagation of the dicluster projectile motion ($\vartheta =0$) while they decay (at least for $V>v_F$) when $\vartheta $ tends to $\pi /2$, the latter behavior being related to the wake-field due to the leading ion. The oscillation amplitude in $S_{{\rm %
corr}}$ tends to decrease from $\vartheta =0$ to $\vartheta =\pi /2$ (the Čherenkov cone). However when $R$ is small each ion is influenced by the unscreened field of the other ion. For model solid targets, the Čherenkov cone semivertex is $\vartheta _C={\rm arcsin}(\sqrt{0.6}v_F/V)$ \[18\]. $\vartheta _C$ approximately equal to 22.8$^0$, 7.4$^0$, and 0.08$^0$ for $V=2v_F$, $V=6v_F$, and $V=10v_F$, respectively. Consequently in the high velocity range the trailing ion moves inside the Čherenkov cone of the leading ion only for almost aligned diclusters. The behavior of CSP shown in Figs. 1 and 2 reflects these features within the linear response and for PPA. It will be noted that the high values of SP are due to the PPA-1 approximation. PPA-2 decreases these values to a small extent. Later, when we use a more realistic, namely RPA, for the linear response function (Figs. 20 and 21) SP considerably decreases in strength.
Figs. 3 and 4 show SP for Cu, another commonly used metallic target. These figures show patterns similar to those in Figs. 1 and 2, except that CSP and ISP have lower values over the entire range of $V/v_F$. This is because Al has a higher electron density than Cu.
The vicinage function (VF) given by Eq. (8) has been plotted as a function of the beam velocity for Al target in Figs. 5 and 6. This function shows an interplay between $\vartheta $ and $R$ more strikingly than CSP. Figs. 7 and 8 display a similar behavior for Cu target.
As stated in Sec. 2, an average stopping power (ASP) is of experimental interest. Figs. 9 and 10 show ASP for Al and Cu, respectively. ISP is also shown, for comparison. The role of PPA-1 and PPA-2 is now more clearly seen.
In the same spirit we have plotted AVF for Al and Cu in Figs. 11 and 12. The role of $R$ is highlighted in these figures. However it will be noticed that PPA-1 and PPA-2 make practically no distinction for AVF.
We have so far plotted SP or ASP (divided by a factor of 2 in both the cases) vs the beam velocity $V/v_F$, for some values of the separation distance $R$. We now look for some complementary information about SP, and plot SP as a function of $R$ with $V=3v_F$, for Al target. Fig. 13 shows an oscillatory character of SP with respect to $R$. The oscillations are the highest for $\vartheta =0$ and lowest for $\vartheta =\pi /2$. The role of PPA-1 and PPA-2 is clearly seen for $\vartheta =0$. Fig. 14 shows a similar behavior of SP for Cu although the amplitudes are now weaker.
In the same way the vicinage function (VF) is plotted in Figs. 15 and 16, for Al and Cu targets, respectively.
Fig. 17 shows ASP vs $R$ for both Al and Cu targets. The difference between PPA-1 and PPA-2 is negligible and the Cu target has ASP smaller than for Al. Now, there is something interesting about Fig.18 which shows AVF. The difference between PPA-1 and PPA-2 is again negligible. But let us note that data for both Al and Cu lie practically on the same curve! Recalling the definition of VF, Eq. (8) one can see from Eqs. (12), (16), (21) and (26) that AVF has a weak dependence on target density. Also, when $\lambda
=V/v_F>2$, $S_{{\rm ind}}$ does not noticeably depend on PPA-1 and PPA-2. These features combine to lead to the behavior of AVF as seen in Fig. 18.
We have so far presented results for PPA-1 and PPA-2. A more realistic linear response function, namely the exact random phase approximation (RPA) will now be used for the metallic target. The theoretical results for SP etc. have been presented in Sec. 3.3. As part of calculating SP in RPA it is useful to examine the plasmon dispersion obtained through $\varepsilon
(z_r,u)=0$, where $z$ and $u$ have been defined in Sec. 3.3. Fig. 19 displays $z_r(\chi ,u)$ vs $u$ for three electron density parameter values.
Next we present ISP and CSP in RPA, for Al and Cu in Figs. 20 and 21, corresponding to $R=10^{-8}{\rm cm}$ and $R=5\times 10^{-8}{\rm cm}$, respectively. For the sake of a better presentation of the data we have also separately displayed the Al data in Figs. 20a and 21a. The RPA results show that SP and ISP decrease in strength with an improved linear response function. This should be of relevance to experiments.
Next, VF in RPA vs $V/v_F$ is presented in Fig. 22, for Al (lines without circles) and for Cu (lines with circles), corresponding to $R=10^{-8}{\rm cm}
$ and for $\vartheta =0$ and $\pi /2$. This figure may be compared with Figs. 5 and 7. For $V/v_F<2$, the curves for VF in RPA tend toward finite values whereas the VF-curves in PPA do not although the angular trend is similar. A similar contrast may be noted between Figs. 6 and 8, and Fig. 23, corresponding to $R=5\times 10^{-8}{\rm cm}$. Again, these findings are of experimental relevance.
Averaged SP vs. $V/v_F$ in RPA is presented in Fig. 24, for Al (curves without circles) and for Cu (curves with circles) along with ISP, corresponding to $R=10^{-8}{\rm cm}$ and $5\times 10^{-8}{\rm cm}$. This figure may be compared with Figs. 9 and 10. Fig. 24 shows an expected overall decrease in the strength of ASP in RPA.
There are similarities but also some interesting differences if we compare Figs. 11 and 12 with Fig. 25. The latter shows AVF in RPA for Al (curves without circles) and for Cu (curves with circles). The differences are more noteworthy for $R=10^{-8}{\rm cm}$.
SP vs $R$ in RPA is plotted in Fig. 26, corresponding to $V=3v_F$, for Al and Cu in the previously stated scheme. When Fig. 26 is compared with Figs. 13 and 14, the differences between PPA and RPA become particularly striking.
A similar contrast is provided by a comparison of Fig. 27 with Figs. 15 and 16, for VF vs. $R$ in RPA and PPA, corresponding to $V=3v_F$ and for $%
\vartheta =0$ and $\pi /2$. For comparison AVF is also plotted in Fig.27.
This completes our extensive presentation of figures exhibiting various aspects of the stopping power of a diproton cluster in PPA and RPA, for Al and Cu targets.
SUMMARY
=======
In this paper we have presented a comprehensive theoretical study of stopping power (SP) of a dicluster of protons in a metallic target. After a general introduction to SP of a cluster of two point-like ions, in Sec. 2, theoretical calculations of SP based on the linear response theory and using PPA without and with plasmon dispersion and then with RPA are discussed in Sec. 3. The theoretical expressions for a number of physical quantities derived in section lead to a detailed presentation, in Sec. 4, of a large collection of data through figures on correlated stopping power (CSP), vicinage function (VF), average stopping power (ASP) and average vicinage function (AVF) of a diproton cluster projectile for two metallic targets, Al and Cu. Whenever relevant, we have also provided a plot of independent (i.e. single-ion) stopping power (ISP) for comparison.
With the proviso stated in Sec. 4, SP and related quantities have been studied within a linear response formalism; some analytical and all numerical results have been obtained corresponding to three approximations to the dielectric function of the target electron gas-the plasmon-pole approximation (PPA) without dispersion (PPA-1) and with dispersion (PPA-2), and also with the random phase approximation (RPA). To our knowledge this is the most comprehensive calculation of the SP-related physical quantities using all the three dominant approximations to the linear response function. The results we have presented demonstrate that with regard to several physical quantities of primary interest the difference between PPA and RPA is substantial while for others, specially for average quantities, this difference may not be of practical significance.
It will be of interest to go beyond RPA in order to include some short-range correlations in the electron gas and to study how dicluster SP is affected. However calculating the linear response function by including electron energy bands is rather involved and detailed theoretical studies of SP with band structure effects included have not yet been reported in the literature. One can include some aspect of band structure in a rather approximate manner through an effective mass for the electrons.
Another aspect we have not considered in this paper is some effect of disorder in the target medium. In real metals electrons suffer collisions with impurities etc. We intend to address this issue in the context of stopping power in a separate study.
[**ACKNOWLEDGMENT**]{}
It is a pleasure to thank to Dr. G. Zwicknagel for useful discussions. We are grateful to V. Nikoghosyan for technical assistance.
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Fig. 1. SP/2 of a diproton cluster with $R=10^{-8}{\rm cm}$, vs $V/v_F$ for Al target ($r_s=2.07$). $\vartheta =0$ (dotted line), $\vartheta =\pi /2$ (dashed line); ISP (solid line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 2. SP/2 of a diproton cluster with $R=5\times 10^{-8}{\rm cm}$, vs $%
V/v_F$ for Al target. $\vartheta =0$ (dotted line), $\vartheta =\pi /2$ (dashed line); ISP (solid line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 3. SP/2 of a diproton cluster with $R=10^{-8}{\rm cm}$, vs $V/v_F$ for Cu target ($r_s=2.68$). $\vartheta =0$ (dotted line), $\vartheta =\pi /2$ (dashed line); ISP (solid line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 4. SP/2 of a diproton cluster with $R=5\times 10^{-8}{\rm cm}$, vs $%
V/v_F$ for Cu target. $\vartheta =0$ (dotted line), $\vartheta =\pi /2$ (dashed line); ISP (solid line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 5. VF of a diproton cluster with $R=10^{-8}{\rm cm}$, vs $V/v_F$ for Al target. $\vartheta =0$ (solid line), $\vartheta =\pi /2$ (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 6. VF of a diproton cluster with $R=5\times 10^{-8}{\rm cm}$, vs $V/v_F$ for Al target. $\vartheta =0$ (solid line), $\vartheta =\pi /2$ (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 7. VF of a diproton cluster with $R=10^{-8}{\rm cm}$, vs $V/v_F$ for Cu target. $\vartheta =0$ (solid line), $\vartheta =\pi /2$ (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 8. VF of a diproton cluster with $R=5\times 10^{-8}{\rm cm}$, vs $V/v_F$ for Cu target. $\vartheta =0$ (solid line), $\vartheta =\pi /2$ (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 9. ASP/2 of a diproton cluster with $R=10^{-8}{\rm cm}$ (dotted line) and $R=5\times 10^{-8}{\rm cm}$ (dashed line) vs $V/v_F$ for Al target. Solid line, ISP. The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 10. ASP/2 of a diproton cluster with $R=10^{-8}{\rm cm}$ (dotted line) and $R=5\times 10^{-8}{\rm cm}$ (dashed line) vs $V/v_F$ for Cu target. Solid line, ISP. The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 11. AVF of a diproton cluster with $R=10^{-8}{\rm cm}$ (solid line) and $R=5\times 10^{-8}{\rm cm}$ (dotted line) vs $V/v_F$ for Al target. The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 12. AVF of a diproton cluster with $R=10^{-8}{\rm cm}$ (solid line) and $R=5\times 10^{-8}{\rm cm}$ (dotted line) vs $V/v_F$ for Cu target. The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 13. SP/2 of a diproton cluster with $V=3v_F$ vs $R$ for Al target. $%
\vartheta =0$ (solid line), $\vartheta =\pi /2$ (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 14. SP/2 of a diproton cluster with $V=3v_F$ vs $R$ for Cu target. $%
\vartheta =0$ (solid line), $\vartheta =\pi /2$ (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 15. VF of a diproton cluster with $V=3v_F$ vs $R$ for Al target. $%
\vartheta =0$ (solid line), $\vartheta =\pi /2$ (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 16. VF of a diproton cluster with $V=3v_F$ vs $R$ for Cu target. $%
\vartheta =0$ (solid line), $\vartheta =\pi /2$ (dotted line). The lines with and without circles correspond to PPA with and without dispersion, respectively.
Fig. 17. ASP/2 of a diproton cluster with $V=3v_F$ vs $R$ for Al (the lines with square symbols) and Cu (the lines with circles) targets. Dotted and solid lines, PPA with and without dispersion, respectively.
Fig. 18. AVF of a diproton cluster with $V=3v_F$ vs $R$ for Al (the lines with square symbols) and Cu (the lines with circles) targets. Dotted and solid lines, PPA with and without dispersion, respectively.
Fig. 19. Relation between $z_r(\chi ,u)$ and $u$, as obtained from the dispersion equation $\varepsilon \left( z_r,u\right) =0$. Solid line: $\chi
=0.5$, dashed line: $\chi =0.15$, dotted line: $\chi =0.05$.
Fig. 20. SP/2 of a diproton cluster with $R=10^{-8}{\rm cm}$ in a RPA vs $%
V/v_F$ for Al (the lines without symbols) and Cu (the lines with circles) targets. $\vartheta =0$ (dotted line), $\vartheta =\pi /2$ (dashed line); ISP (solid line).
Fig. 20a. SP/2 of a diproton cluster with $R=10^{-8}{\rm cm}$ in a RPA vs $%
V/v_F$ for Al target. $\vartheta =0$ (dotted line), $\vartheta =\pi /2$ (dashed line); ISP (solid line).
Fig. 21. SP/2 of a diproton cluster with $R=5\times 10^{-8}{\rm cm}$ in a RPA vs $V/v_F$ for Al (the lines without symbols) and Cu (the lines with circles) targets. $\vartheta =0$ (dotted line), $\vartheta =\pi /2$ (dashed line); ISP (solid line).
Fig. 21a. SP/2 of a diproton cluster with $R=5\times 10^{-8}{\rm cm}$ in a RPA vs $V/v_F$ for Al target. $\vartheta =0$ (dotted line), $\vartheta =\pi
/2$ (dashed line); ISP (solid line).
Fig. 22. VF of a diproton cluster with $R=10^{-8}{\rm cm}$ in a RPA vs $%
V/v_F $ for Al (the lines without symbols) and Cu (the lines with circles) targets. $\vartheta =0$ (solid line), $\vartheta =\pi /2$ (dotted line).
Fig. 23. VF of a diproton cluster with $R=5\times 10^{-8}{\rm cm}$ in a RPA vs $V/v_F$ for Al (the lines without symbols) and Cu (the lines with circles) targets. $\vartheta =0$ (solid line), $\vartheta =\pi /2$ (dotted line).
Fig. 24. ASP/2 of a diproton cluster with $R=10^{-8}{\rm cm}$ (dotted line) and $R=5\times 10^{-8}{\rm cm}$ (dashed line) in a RPA vs $V/v_F$ for Al (the lines without symbols) and Cu (the lines with circles) targets; ISP (solid line).
Fig. 25. AVF of a diproton cluster with $R=10^{-8}{\rm cm}$ (solid line) and $R=5\times 10^{-8}{\rm cm}$ (dotted line) in a RPA vs $V/v_F$ for Al (the lines without symbols) and Cu (the lines with circles) targets.
Fig. 26. SP/2 of a diproton cluster with $V=3v_F$ in a RPA vs $R$ for Al (the lines without symbols) and Cu (the lines with circles) targets; ASP (solid line), $\vartheta =0$ (dotted line), $\vartheta =\pi /2$ (dashed line).
Fig. 27. VF of a diproton cluster with $V=3v_F$ in a RPA vs $R$ for Al (the lines without symbols) and Cu (the lines with circles) targets; AVF (solid line), $\vartheta =0$ (dotted line), $\vartheta =\pi /2$ (dashed line).
[^1]: E-mail: [email protected]
| |
Chess is not a game of luck or chance. The outcome depends on the choices of the players.
One player has white pieces and another player has black pieces. You can choose who gets which color by flipping a coin.
Order of play
First put all the pieces in their starting locations. Then White makes the first move. White always makes the first move. After White has moved, Black will then make a move. Then White, then Black, then White, then Black, and so on to the end of the game.
General movement rules
- A move is moving a single piece to an empty square on the board.
- If a piece is moved onto a square that has the other player's piece on it that piece is removed. That piece has been captured or taken.
- The pieces can not jump over each other (except for the knight).
- You cannot make a move that puts your king in "check" or passes it though check.
The Pieces
King
The king can move one square at a time in any direction.The king is the most important piece belonging to each player but isn't very powerful.
Check
If a king is about to be captured, that king is in check. If a king is in check then it has to move out of check, move another piece to block the check or capture the piece giving the check. A player cannot move any other piece when they are in check unless it is to eliminate the check.
If the king is placed in check and cannot escape, then it is in checkmate. The first player to checkmate the other player's king wins the game. Each player has one king.
Queen
The queen is the strongest piece, and can move any number of squares in any direction. It can move left and right and up and down and diagonally. Each player has one queen. A queen always starts on its color, next to the king.
Bishop
The bishop can move any number of squares diagonally. Each player has two Bishops, one on white squares and one on black squares. An important thing about the Bishop is that it can only move on the same color squares. Bishops are often good early in the game, but cannot do much late in the game.
Knight
The knight moves in a L-shaped. It moves two squares in one direction up or down or left and right and then one square perpendicular. The knight is the only piece that may jump over or move between other pieces. Each player has two knights.
Rook
The rook can move any number of squares left and right and up and down straight, but not, diagonally. The rook cannot jump over any pieces. If the rook takes the place of another piece, then it has captured it. Rooks start in the corner, and each player has two rooks.
Pawn
Pawns can move one square forward. On a pawn's first move, it can move two squares forward. The pawn can move one square diagonally forward to capture a piece, but it cannot capture pieces by moving straight forward, except with en passant.
Pawn Promotion
If a pawn makes it to the far edge of the board, the pawn is promoted. A promoted pawn can be replaced with any other piece (except king). For example, a player could get any other piece of his/her liking from pawn promotion that has previously been captured.
Special Moves
Castling
Castling is a move involving the king and one of the rooks. It is the only move where more than one piece can be moved in a turn. When castling, the king moves two squares toward the edge of the board, and the rook moves two or three squares to the center. This can only be done if the king and rook have not moved, there are no pieces between them, and the king is not in check. | https://simple.wikibooks.org/wiki/Chess/Content/Playing_The_Game |
Today, I am sharing a bookshelf pull tab tutorial. This is the pull tab I use in my September Bullet Journal Setup 2022. In this post, I am sharing how I set up the bookshelf pull tab. To see how I decorate it and how it comes together, you can check out my September Setup. If you prefer to watch the video, you can check it out here. In the video, I am using a B5 size journal but you can follow along here for an A5 size. (I have bolded A5 to make it easier to catch. Also, I alternated between boxes and blocks, referring to the dot grid squares – they are the same thing.) The pull tab mechanism from this tutorial originally from The Pop Up Channel.
Bookshelf Pull Tab Tutorial Supplies
- Archer and Olive 160 GSM Dot Grid Notebook (I am using a B5 journal but an A5 will work too) (Affiliate link – NML10 for 10% off)
- We ‘R Memory Keepers Art Knife
- Small Cutting board
- Scissors
- Tombow Mono Tape Runner Permanent
- Tombow Mono Tape Runner Removable Adhesive (optional)
- Pencil and ruler
BookShelf Pull Tab Tutorial Steps
1. Drawing the Outline
- Start outlining your bookshelf in pencil – the pull tab will sit along the outer edge of this bookshelf
- Measure out your shelves for your books/ornaments to sit on (these can be spaced evenly or alternating sizes like mine)
- My shelves were 6 boxes high (1.125 inches) alternating with 9 boxes (1.875 inches)
- FOR A5: the shelves measure 5 boxes high (1 inch) alternating with 7 boxes (just over 1.125 inches)
- The outer border of my bookshelf was 2 boxes wide (0.625 inches), and the shelves were 1 box wide (just under 0.25 inches)
- Decide where you want your letters to go – making sure not to draw them too close to the edges of the page
- My letters are roughly 4 blocks tall (just over 0.75 inches) and 2 blocks wide (0.625 inches)
- FOR A5: your letters can be the same size
- Sketch in where your slots will go
2. Creating the Pull Tab
- Grab 2-3 pieces of paper from the back of your notebook (or from the Archer and Olive Dot Grid Notepad if you have one)
- I used 2 pieces but a 3rd piece is helpful in case you make mistakes
- Whether you are in A5 or B5, you will need 5 pull tab arms
Creating the Arms of the Pull Tab
- Grab a piece of paper for measuring your pull tab arms
- Using the page length-wise, you will measure 6 blocks (1.125 inches) for the width
- FOR A5: The width will measure 5 blocks (1 inch)
- Keep the length of the page as we will trim those to size later
- Cut 5 of these size pieces
- Create the flap piece by measuring 8 boxes from the edge of the arm you just cut
- FOR A5: measure 7 boxes (just over 1.125 inches) from this edge
- Draw a line across the width of the arm
- Count 2 more boxes and draw another line. This is what will hold your flap in place
- In between these 2 lines, measure 1.5 boxes (draw a line roughly in the middle of the box) from both edges of the arm, and draw lines marking this point
- Cut along the original lines up until the 1.5 box mark you just made
- Do this on both sides
- Trim a small bit of paper (roughly the width (or half the width) of your scissors) between the flap you just cut and the shorter end of the pull tab (that will be forming our flap)
- This helps to keep the flap from getting stuck
- Take longer portion of the pull tab and trim about 0.5 of a box off both ends of the tab – up until the little arms we created above
- Make a fold between the little flaps and the longer end of the pull tab
- DO NOT fold between the shorter end and the little flaps. It’s won’t work properly if you do
- Repeat steps 5-13 for all the pull tab arms (other than step 5, all other steps are the same for an A5)
Cutting the Slots for the Pull Tab Arms
Now we are going to cut the slots into our page. Please note: a slot is slightly wider than a slit. A slit is one line, so a slot will be a tiny bit wider. For this bookshelf pull tab tutorial, try and make that second line for the slot as close to the first cut as possible.
- The slots in your page need to be the same width as the long end of the pull tab arm
- This ends up being about 4 boxes
- FOR A5: this ends up measuring about 3 boxes wide
- These slots need to sit next to your letters that you sketched out earlier
- I cut my slots in the next block to the left of my letters
- Slip the longer side of the pull tab arm into this slot, unfolding the small arm flaps on the back of the page
- Only the larger flap portion of the pull tab arm should be showing
- Do this for all the shelves on you bookshelf – which should be 5 if you are following this tutorial
Creating the Final Pull Tab Piece
- Cut a slit along the length of your bookshelf – along the edge of your notebook replacing the outside border of your sketched bookshelf
- Take another piece of paper, and fold it in half
- This helps to make the pull tab stronger
- Cut to size to fit the slit you just cut
- Slip one side of the pull tab piece into the slit, roughly 5-6 boxes in
- This is the same for A5
- Make sure that there are an even amount of boxes peeking through the slot to ensure the pull tab piece is even
- NOTE: using removable adhesive, you can glue down the flaps of the pull tab arms to keep them in place while gluing this part
- Start trimming the pull tab arms for gluing onto this pull tab piece
- Trim enough off the ends to sit roughly in the middle of the pull tab piece
- Start gluing these pieces onto the pull tab piece
- Maneuver the second (folded) piece of the pull tab piece into the slot
- Glue this piece down, over top of the pull tab arm pieces
- This prevents the edges of the arms from catching on the slit when you pull it
- Once everything is glued in place, trim the pull tab piece to fit your bookshelf – 2 boxes wide sticking out along the edge
- Only trim this piece when the pull tab is in place – meaning pushed in and the flaps aren’t revealing the letters
- Glue the pull tab pieces together from the edge (if they were trimmed as in the step above)
3. Final Steps for the Bookshelf Pull Tab Tutorial
- If you’ve made it this far, YAY! We are so close to being done.
- Now we are going to glue the bookshelf page down, keeping the pull tabs in place and hiding the mechanisms (the pull tab arms)
- This is possibly the trickiest part because the page can bubble if not glued correctly
- Place glue strips along the top, bottom and outside edge of your page with the mechanisms
- Place glue right up to the slit between the edge of the notebook
- This keeps the pull tab in place
- Making sure your notebook is laying flat, slowly and carefully lift the side with the mechanisms and glue, and shut the whole book, pressing down as you go
Bookshelf Pull Tab Tutorial Final Thoughts
Couple of things to remember, while 160GSM is thicker than the average journal, it still isn’t as ideal as, say, 200GSM card stock, so be gentle when pulling and pushing. If you have any issues with your arms bending funny, just correct the fold with a pen through the slit.
This is a super fun effect that gives a cool wow factor to your cover page. If you have any questions or comments, please do not hesitate to reach out. I know this can be tricky (I practiced it MANY times), so just ask if there are any issues.
I hope you enjoyed today’s tutorial. Again, if you would like to see the finished product, please check out my September bullet journal setup 2022.
Thank you so much for your time today! | https://natashamillerletters.com/bookshelf-pull-tab-tutorial/ |
Torryn Hunter Savage was born May 20, 2020, in Lander to Lela C’Hair and Logan Savage. This #little was 9lbs 10oz and measured 21 inches long.
Congratulations to the family of this new #little!
Submit your own Birth Announcement to County 10 for SageWest Health Care’s series #Littles by using the button below! | https://county10.com/help-us-welcome-this-new-little-torryn-savage/ |
Amateur Transit Light Curves
8206gary Frequent uses of a hair dryer to remove frost from the corrector plate; temp = 28 F, Dew Pt = 22 F (RH = 78%). WWV check of time tags.
8201nave (waiting for permission)
8124srdc The clock was checked & found to be accurate.
7c31gary Air mass curvature is high due to use of a BB-filter and high air mass at the beginning.
Out-of-Transit Light Curves
9222-13-FE2 OOT
The closest transit was at 23.06 UT on 2008.03.24 (i.e., 8.9 hrs before mid-observing session).
These observations were a test of a new optical configuration which accounts for the short duration. Nevertheless, there seems to be mild evidence for 1 mmag variations on an hourly timescale.
Professional Transit Light Curves
Gillon et al (2007) SST observations at 8 micron wavelength, reproduced from Ribas et al (2008). Lowest panel shoes effect of a hypothetical 0.1 degree inclination change that could be produced by perturbations from a 5-Earth mass outer orbit planet in a 2:1 resonant orbit.
V-band, Observatory of Geneva 1.2-meter Euler telescope at La Silla Observatory, Chile (Gillon et al, 2007). Mid-transit at 2007 May 02, 02:41 UT. My measurements of this LC yield depth ~6.5 ± 1.0 mag, length = 0.943 ± 0.064 hr.
Finder Image
Figure F1. Finder image with identifier star numbers (above) and J-K colors (times 100, below) selected stars. GJ 436 has V = 10.68 and Rc = 9.66.
Normally I don't present all-sky photometry results without completing
a second observing session and analysis and verify
compatibility between the two results. In this case
I have no plans for doing this since I doubt that anyone
will use any of these results.
Figure A2 and A3. Color/color scatter diagram showing the location
of GJ 436 (red square) and the 9 nearby stars (gray
squares) in relation to 1259 landolt stars.
If the solution for one filter band had a systematic error it would show
up in these plots as a group offset in the color/color
scatter diagrams involving that filter. For example,
if all B-magnitudes were high by 0.05 magnitude then the
goup of gray squares and the red square would be offset to
the right by 0.05 magnitude. It's possible that such an offset
is present in Fig. A2, but it's clear that greater vaues fo such
an offset are very unlikely. An alternative for explaining the
rightward shift of Fig. A2 gray squares is for there to be instead
a downward shift, or values for V that are to negative by about the
same 0.05 magnitude amount. This is unlikely after inspecting Fig.
A3, where there is no evidence of shifts. Presumably, all 3 bands
(V, R and I) are free of calibration error offsets (unless by some
unlikely circumstance there are offset errors in all 3 bands that excatly
compensate to produce color/color agreement with the Landolot stars).
If it is true that V, R and I are free of calibration offset errors
greater than ~0.03 magnitude, then what should be make of the funny
location for GJ 436 in Fig. A3? I claim that it is inescapable that
GJ 436 has a much greater V-I color than the Landolt stars. Since V
appears to be normal (e.g., Fig. A2), we must conclude that I is anomalous.
In other words, these color/color scatter diagrams show that GJ 436
has an I-magnitude that is brighter than normal by ~0.5 magnitude!
I'll leave it to others to explain how this could be the case.
Note: It won't matter what magnitude you assume for reference stars for
the purpose of obtaining quality light curves. These
estimated values are presented for the purpose of
identifying star colors that "match" GJ 436's color which
can be useful in minimizing extinction related systematic
errors (i.e, LC curvature that's correlated with air mass).
In this table the column for R-band will be the most accurate
since it is based on observations. The other magnitudes for Stars
1 through 6 are based on JK magnitudes. For GJ 436 the B and I magnitudes
are based on color/color correlations for main sequence stars.
All stars in the table are compatible with main sequence color/color
relationships.
References
Gillon et al, 2007, Astron, & Astrophys., "Detection of Transits
of the Nearby Hot Neptune GJ 436" http://babbage.sissa.it/abs/0705.2219
Butler et al, 2004, Astrophys. J. Lett.,"A Neptune-Mass Planet Orbiting the Nearby M Dwarf GJ 436" http://adsabs.harvard.edu/abs/2004ApJ...617..580B
Ribas et al, 2008a, Astrophys. J. Lett., "A ~5_earth Super Earth Orbiting GJ 436?: The Poser of Near-Grazing Transits" http://fr.arxiv.org/abs/0801.3230
Ribas et al, 2008b, IAU 253, Boston, MA, 2008 May 19-23
Alonso et al, 2008, "Limits to the planet candidate GJ 436c" http://arxiv.org/abs/0804.3030
Bean et al, 2008, arXiv:0806.0851v2, http://arxiv.org/abs/0806.0851
Coughlin et al, 2008, preliminary and final (ApJL, pay)
Batygin et al, 2009, "A Quasi-Stationary Solution to Gleise 436b's Eccentricity," preprint: http://arxiv.org/abs/0904.3146
Return to calling web page AXA
WebMaster: Bruce
L. Gary. Nothing on this web page is copyrighted. This site opened: | http://brucegary.net/AXA/GJ436/gj436.htm |
As it states, the length is 4 times the width so we can write this as 4w. Next, we need to know the formula for the area of a rectangle. The formulas is this:
A = l × w
Next, we simply SUBSTITUTE the value for l into the formula to find the area and that will allow us to solve for w. We know that the area is 80cm so we can just write that,
80 = (4w)(w)
Now let’s solve for w.
Step 1: Simplify both sides of the equation.
80=4w^2
Step 2: Subtract 4w^2 from both sides.
80−4w^2=4w^2−4w^2
−4w^2+80=0
Step 3: Subtract 80 from both sides.
−4w^2+80−80=0−80
−4w^2=−80
Step 4ivide both sides by -4.
−4w^2/−4=−80/−4
w^2=20
Step 5: Take square root.
w=√20
w = ~4.5 (rounded to tenth)
now that we know w we can solve for l.
remember. l = 4w
l = 4(4.5)
l = 18
we can check this by calculating the area where l is 18 and w is 4.5. | https://sts-math.com/post_9555.html |
Q:
induction to prove $n^2 - 1$ is divisible by 4 by changing variables
I have to prove $n^2 - 1$ is divisible by $4$, where $n\in\mathbb{O}_{>0}$.
It says, "You cannot prove this by induction on $n$. Rewrite $n^2 - 1$ in terms of a variable on which you can do induction."
Why is it not possible to do this by induction on $n$ and how would I change the variable?
All help is appreciated.
A:
The text you are using takes a pretty narrow view of what it means to prove by induction. The statement is: $1^2-1$ is divisible by $4$, and if $k^2-1$ is divisible by $4$ where $k$ is odd, so is the next odd number ($k+2$). But
\begin{equation*}
(k+2)^2-1 = k^2+4k+4-1 = (k^2-1) + 4(k+1),
\end{equation*}
which is divisible by $4$ since $k^2-1$ is by the inductive hypothesis.
I assume that what the text means is that since we are assuming $n$ is odd, we should instead use the statement $(2k+1)^2-1$ is divisible by $4$ if $k$ is a nonnegative integer. That is proven in much the same way: true for $k=0$, and if true for $k$, then
\begin{equation*}
(2(k+1)+1)^2-1 = 4k^2 + 12k+9-1 = 4k^2+4k+4 + 8k+5 = (2k+1)^2 - 1 + 4(1+2k).
\end{equation*}
| |
The sieve of Eratosthenes is a simple method for finding all prime numbers less or equal to some given upper bound. The algorithm was invented by an ancient Greece mathematician Eratosthenes of Cyrene approximately 200 BC. It is one of the most efficient methods for finding primes which are less then . For higher numbers, more efficient algorithms should be used (Lehmann test, Rabin-Miller test...).
Write down all integers less or equal to the upper bound and higher than 2 (1 is not a prime) and set them as unmarked.
Pick the first unmarked number n in the list and mark it as a prime.
Mark all multiples of n as composite numbers.
If the contains some unmarked number – GOTO: 2.
We can improve the termination condition (step 4) by checking that the first number in the list is higher than the square root of the upper bound – if the number is composite, than it must have at least two divisors and and it must hold, that or . After the termination all unmarked numbers should be marked as primes.
The asymptotic complexity of the sieve of Eratosthenes is , where is the upper bound.
Find all prime numbers less than 20.
Write down all numbers less or equal to 20, starting at 2. | https://programming-algorithms.net/article/41297/Sieve-of-Eratosthenes |
The capacitor shown in the circuit below initially holds a charge q0. The switch is closed at t = 0. Find the charge on the capacitor as a function of time, if R2/4 < L/C. What is the oscillation frequency of the circuit when R --> 0?
We are asked to analyze the transient behavior of an RLC circuit.
Q/C - IR - LdI/dt = 0. I = - dQ/dt. d2Q/dt2 + (R/L)dQ/dt + Q/(LC) = 0.
Let Q(t) = Aexp(-bt). Then b2 - b (R/L) + 1/(LC) = 0.
b = R/(2L) ± (R2/(4L2) - 1/(LC))½.
If R2/4 < L/C, then b = R/(2L) ± i(1/(LC)) - R2/(4L2))½ = α ± iω.
α = R/(2L), ω = (1/(LC)) - R2/(4L2))½. Q(t) = q0exp(-αt)cos(ωt).
When R --> 0 then α --> 0 and ω --> 1/(LC)½.
Consider the circuit below, where C1 is initially charged to 75 V. Assume that C1 = 0.01 F, C2 = 0.003 F, and L = 15 H. Explain how to open and close the switches so as to discharge C1 and charge C2. Starting at t = 0 give explicit times for opening and closing each switch. What is the final voltage across C2?
We are asked to analyze the transient behavior of an RL circuit.
Assume that at t = 0 the switch S1 is closed and the switch S2 stays open.
The equation of the circuit is Q/C1 = -LdI/dt, Ld2Q/dt2 = -Q/C1.
Q = Q1exp(iω1t), where ω1 = 1/(LC1)½.
After time t1 = π/(2ω1) the capacitor C1 will be discharged, all the energy will be stored in the inductor.
Q1 = (0.01*75) = 0.75 C, t1 = 0.6 s, ½Q12/C1 = 28.13 J, Imax = 1.94 A.
At t1 we close S2 and open S1. The equation of the circuit now is Q'/C2 = -LdI/dt, Ld2Q'/dt2 = -Q'/C2.
I' = Imaxexp(iω2(t-t1)), where ω2 = 1/(LC2)½.
At time t2 = π/(2ω2) + t1 the capacitor C2 will hold the maximum charge Q2 and no energy will be stored in the inductor.
At time t2 we open switch S2.
½Q22/C2 = ½LImax2. V2(final) = Q2/C2.
t2 = 0.94 s, Q2 = 0.41 C. V2(final) = 137 V.
A mass m fixed to a spring of spring constant k and immersed in a fluid provides a model for a damped harmonic oscillator. A circuit with inductance L, capacitance C and resistance R provides an electric analog to such an oscillator.
(a) Write out the circuit equation for the LCR circuit and Newton's second law of motion for the damped oscillator. What assumption must be made about the dependence of the damping of the mass on velocity for the two equations to have the same mathematical form?
(b) How are the constants m, k, and b (damping constant) related to the circuit constants L, C and R? To what parameters of the electric circuit are the mechanical quantities x (displacement) and v (velocity) related?
(c) Derive and expression for the displacement and velocity in the limit of weak damping.
(d) What voltages measured in the circuit would give values proportional to the displacement and velocity of the mechanical oscillator?
We are asked to compare the differential equation describing the behavior of a series LRC circuit with the equation of motion for a damped harmonic oscillator.
Damped oscillator: md2x/dt2 = -bdx/dt - kx. We must assume that the magnitude of the damping force is proportional to the speed.
(b) d2Q/dt2 = -(R/L)dQ/dt - Q/(LC), d2x/dt2 = -(b/m)dx/dt - (k/m)x.
R/L --> b/m, 1/(LC) --> k/m.
Possible substitutions: L --> m, R --> b, k --> 1/C.
Q --> x, I = dQ/dt --> v = dx/dt.
(c) Try x = x0exp(at+ d).
a2 + ba/m + k/m = 0. a = -b/2m ± ( b2/4m2 - k/m)½.
x = x0exp(-βt) cos(ωt + d), β = b/2m, ω2 = k/m - β2 = ω02 - β2.
If the damping is weak then x ≈ x0exp(-βt) cos(ω0t + d).
(d) The voltage across the capacitor is proportional to Q --> x, The voltage across the resistor is proportional to I --> v. | http://electron6.phys.utk.edu/PhysicsProblems/E&M/5-AC%20circuits/transients%20RLC.html |
TECHNICAL FIELD
BACKGROUND ART
SUMMARY OF THE INVENTION
DETAILED DESCRIPTION OF EMBODIMENTS
Examples
Preparation
Testing
2) Chippings (CH)>0.25 mm
Micrography
The present disclosure relates to a drill device having a margin with a textured area. The present disclosure also relates to a method for manufacturing a drill device having a margin with a textured area.
In drill devices, such as drills made in one piece, or drill tip inserts, for machining of metallic work pieces, it has been observed that the margin of the drill device is subjected to substantial amount of wear during the drilling operation. To increase the wear resistance, attempts have been made to provide the margin of the drill devices with textured areas. For example, in U.S. Pat. No. 9,144,845 B1 an area of the margin which is distal from the drill tip is provided with a texturing of discrete micro-recesses. However, while the test results of U.S. Pat. No. 9,144,845B1 appear promising there is still a need for improving the wear resistance of drill devices for machining metallic materials.
Thus, it is an object of the present disclosure to provide a drill device which solves or at least mitigate on problem of the prior-art. In particular, it is an object of the present disclosure to provide a wear resistant drill device. Yet a further object of the present disclosure is to provide a drill device having a margin with a textured area having an effective wear resistant pattern. A further object of the present disclosure is to provide a method for manufacturing a wear resistant drill device.
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the drill tip comprises at least a first clearance surface . and;
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the drill body comprises at least a first land . having a margin . and;
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an edge between the first clearance surface . and the margin .; and;
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a textured area comprising a plurality of recesses .,
the textured area extends along at least a portion of the margin ., in direction of the rear portion . of the drill body , from a position of 200 μm from the edge ; or from a position in the least first clearance surface .; or from a position there between,
6
1
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the recesses . extend in a direction Y, the direction Y being 20° from or within 20° from the direction of a longitudinal axis of rotation X of the drill .
According to the present disclosure, at least one of these objects is achieved by a drill device having a drill tip and a drill body having a rear portion ., wherein;
1
Results from practical machining trials with the drill device according to the present disclosure have surprisingly shown that the area of the margin proximate to the tip of the drill is vulnerable to wear in machining operations. It has further surprisingly shown that a textured area extending from the tip of the margin or close thereto in direction of the rear of the drill device has a significant beneficial effect on the wear resistance of the margin and thereby the tool life of the drill device. Additionally, the orientation of the recesses of the textured area has shown to be of importance for the tool life of the drill such that the recesses are preferably to be oriented along or almost along the longitudinal axis of rotation X of the drill .
Further embodiments and alternatives of the drill device according to the present disclosure are disclosed in the appended claims and the detailed description.
The drill device according to the present disclosure will now be described more fully hereinafter. The drill device according to the present disclosure may however be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided by way of example so that this disclosure will be thorough and complete, and will fully convey the scope of the present disclosure to those persons skilled in the art. Same reference numbers refer to same elements throughout the description.
FIG. 1
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shows schematically a drill device according to a first embodiment of the present disclosure. In this case the drill device is a drill which is manufactured in one single piece of cemented carbice. The drill device comprises a drill tip and a drill body , which extends from the drill tip and that has a rear portion .. A drill shank for attaching the drill to a holding device (not shown) extends from the rear portion . of the drill body . The drill body comprises a first land . and a second land . and a first flute . and a second flute . extending between the first and the second land ., .. The first and second lands ., . and the first and the second flutes ., . extend helically along the drill body in direction from the rear portion . towards the drill tip . The first land . has a first margin . and the second land . has a second margin.
FIG. 2
FIG. 1
2
1
2
2
3
2
4
2
1
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4
shows a front view of the drill tip of the drill device shown in . The drill tip has a first cutting lip . and a second cutting lip .. A first clearance surface . extends from the first cutting lip . and a second clearance surface . extends from the second cutting lip ..
FIG. 3
FIG. 1
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is a cross-section of the drill device in showing the first and the second margin ., .. The margins ., . are cylindrical portions of the drill body that extends beyond the lands ., .. The margins ., . support the drill device in the bore hole during drilling.
FIG. 4
FIG. 4
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shows a perspective view of the drill tip . Thus, the margin . has a leading edge . and a trailing edge .. An edge separates the upper end of the margin . from the first clearance surface .. It is appreciated that the second margin . is not visible in but that the second margin . comprises identical leading and trailing edges ., . and an identical edge .
It is further appreciated that the drill device according to the present disclosure may comprise further lands, such as a third and a fourth land. It is further appreciated that each additional land may comprise a margin as described above.
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The drill device described above may be a twisted, or helical, drill provided with chip flutes ., . extending helically around the drill. The drill device may alternatively be provided with straight chip flutes extending in the longitudinal direction X of the drill. The drill device may be provided with a drill tip .
2
3
The drill device may be manufactured from steel or hard metal such as cemented carbide, for example WC/Co. The drill may be manufactured by conventional methods including pressing or extrusion of a mixture of metal, carbide powder and binder followed by sintering. The drill may further comprise a wear resistant coating such as TiN, TiAlN and/or TiAl/TiAlCr applied with PVD, for instance. The coating can be a multi layered coating or a single layer. The coating can alternatively or in addition comprise a CVD coating for example comprising layers of TiN, TiCN, TiAlN and/or AlO. The thickness of said coating is preferably 1-5 μm.
FIG. 5
FIG. 6
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shows a drill device according to a second embodiment of the present disclosure. In this case the drill device is an exchangeable drill tip insert which may be releasable attached to a drill tip holder . The drill device of the second embodiment is shown in detail in and comprises the same features as the drill device of the first embodiment. Thus, a drill tip , a body having a rear portion ., at least a first clearance surface, at least a first land . having at least a first margin . and an edge between the clearance surface . and the margin .. The clearance surface may comprise a corner chamfer. The drill device according the second embodiment further comprises an attachment pin extending from the rear portion . of the body and adapted for attaching the drill device to the drill device holder .
According to the present disclosure, the drill device is provided with a textured area which comprises recesses formed in the surface of the drill device. The textured area thereby extends along the margin, in direction of the rear portion of the drill body from a position of 200 μm from the edge between the margin and the clearance surface or from the clearance surface or from a position between the clearance surface and 200 μm from the edge. The following description applies to both the first and the second embodiments of the drill device according to the present disclosure.
a
FIG. 7
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shows a schematic view of the upper portion of the first margin . of a first alternative of the drill device . Also shown is the edge between the margin . and the clearance surface (not shown). The rotational axis of the drill device is indicated by arrow X. A textured area comprising a plurality of recesses . in the form of elongate, continuous grooves that extends along the margin . from the edge in direction of the rear portion of the drill body (not shown). The direction of extension of the recesses . is indicated by arrow Y. The angle between the direction X and the direction Y is α. The textured area has a predetermined length which is determined parallel with the leading edge . of the margin .. In the described embodiment, the length L of the textured area corresponds to the length of the grooves .. The textured area has further a predetermined width W which is determined parallel to the edge between the margin . and the clearance surface (not shown). In the described embodiment, the grooves . are arranged parallel with the leading edge . of the margin . and parallel with each other. The grooves . are further arranged equidistant from each other and are thus distributed homogenously over the width W of the textured area .
6
1
The predetermined length L and the predetermined width W of textured area may be selected in view the overall design of the drill , the drilling application and the material of the work piece. Preferably, the predetermined length L is selected such that the textured area extends over the portion of the margin that is in contact with the surface of the bore in a drilling operation. Practical trials have shown that good wear resistance is achieved when the predetermined length L is 10-100% or 10-50% or 10-30% or 10-20% of the diameter of the drill device.
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3
9
The predetermined width W of the textured area may be selected such that the textured area is positioned at a distance d from the leading edge . and/or the trailing edge . of the margin. The distance d may be selected in dependency of the dimensions of the drill device and the material of the work piece. For example, the predetermined distance d is 1-25 μm or 1-10 μm or 1-5 μm. Alternatively, the predetermined width W of the textured area may be selected such that the textured area extends from the leading edge . and/or the trailing edge . of the margin.
6
1
1
6
1
1
6
1
1
The orientation of the recesses . may extend in a direction Y, the direction Y being 20° from or within 20° from the direction of a longitudinal axis of rotation X of the drill , i.e. an angle α between X and Y is less than or equal to 20°. Alternativelly the recesses . may extend in a direction Y, such that α is less than or equal to 10° or less than or equal to 5°. The direction Y may be within 10° or within 5° from the direction of a longitudinal axis of rotation X of the drill . The recesses . may extend parallel with a longitudinal axis of rotation X of the drill , i.e. α is zero. The leading edge of a drill may be helically positioned around the outer perifery of the drill at an angle β (the so called helix angle β) relative the longitudinal axis of rotation X of the drill. Recesses aligned with the leading edge is therefore within the scope of the present invention if β is less than or equal to 20°. The technical effect of recesses aligned in accordance with the orientation of the invention is thus that the wear of the margin of the drill is reduced. The wear is caused during sliding between the inner wall of the drilled hole and the surface of the margin during rotation of the drill in the hole and during drilling. The improved tool life can be due to improved lubrication at the sliding interface. An orientation of the recesses in parallel with a longitudinal axis of rotation X of the drill corresponds to the recesses on the margin being oriented perpendicular to the sliding direction of the wall of the drilled hole.
2
1
In one embodiment of the present invention the recesses extend in a direction Y, wherein the angle α, i.e. the angle between Y and the longitudinal axis of rotation X of the drill, is any angle between the direction of the leading edge of the drill, preferably the direction of the leading edge at the drill tip of the drill , and the direction X.
In one embodiment of the present invention wherein the drill is twisted or helical with a helix angle β, the direction Y of extension of the recesses differs from the direction of the leading edge such that the angle α is less than the angle β.
b
FIG. 7
3
3
1
4
3
3
1
6
6
1
3
3
4
6
1
6
3
8
3
3
4
3
3
6
3
8
3
9
3
3
6
3
9
3
8
6
1
1
6
1
4
6
1
3
8
3
9
3
8
3
9
shows a schematic view of the upper portion of the first margin . according to a second alternative of the drill device . Also shown is the edge between the margin . and the clearance surface (not shown). The rotational axis of the drill device is indicated by arrow X. A textured area comprising a plurality of recesses . in the form of elongate, continuous grooves extends along the margin . from the edge in direction of the rear portion of the body (not shown). The direction of extension of the recesses . is indicated by arrow Y. Also in this alternative, the textured area has predetermined length L, determined parallel to the leading edge . of the margin . and a predetermined W, determined parallel to the edge between the margin . and the clearance surface (not shown). Identical with the first alternative, the textured area is arranged at a predetermined distance d (not shown) from the leading edge . and/or the trailing edge . of the margin .. However, the textured area may extend from the trailing edge . to the leading edge . of the margin. The grooves . are in this embodiment oriented parallel with the axis of rotation X of the drill . Thus in the described embodiment, a portion of the grooves . may extend, parallel with axis of rotation X, from the edge and a portion of the grooves . may extend parallel with axis of rotation X from the leading edge . and the trailing edge . of the margin or from a predetermined distance d from the leading edge . and/or the trailing edge .. Practical trials have shown that this orientation of the grooves provides a high increase in wear resistance to the margin.
4
4
4
4
According to an alternative (not shown), the textured area extends along the margin from a position of 200 μm from the edge . That is, the textured area initiates at a position which lies between the edge and the rear portion of the drill device. According to an alternative, the textured area initiates at position between the edge and 200 μm from the edge . For example the textured area initiates at a position of 0-100 μm from the edge and extends along the margin in direction towards the rear portion of the drill device.
FIG. 8
a
FIG. 7
b
FIG. 7
FIG. 8
6
1
6
1
3
3
6
2
6
1
6
1
6
2
6
3
6
1
6
1
6
3
6
1
6
1
6
2
6
3
6
2
6
1
6
3
6
2
6
1
6
shows a schematic cross-sectional view of the grooves . seen perpendicular to the width W of or . Thus, the grooves . formed in the margin . are separated by ridges . of margin material. The grooves . have a depth Z and a top width TW and a bottom width BW. The grooves . are formed in margin such that the top width TW is greater than the bottom width BW. The ridges . have a top surface . which divides two adjacent grooves .. In , the grooves . and the ridges . are depicted as mutually inverted truncated cones. However, in reality the shape of the grooves and the ridges is more complex and involves radiuses. The grooves . are formed in surface of the drill device such that grooves . and ridges . are repeated with a periodicity P. The periodicity P is determined from the end of the top surface . of one ridge . across the adjacent groove . to the corresponding end of the end surface . of the next ridge .. The relationship between top width of the grooves and the periodicity should be TW<P. The periodicity P, the top width TW of the grooves and the depth Z of the grooves . are significant for the wear resistance provided by the textured area . Preferably, the periodicity P is selected in the interval of P=10-300 μm, more preferred P=20-70 μm or 20-30 μm. Preferably, the top width TW is selected in the interval TW=10-300 μm, more preferred TW=20-50 μm. Preferably, the depth Z is selected in the interval Z=1-50 μm, more preferred Z=2-7 μm or 2-5 μm. According to one example P=50 μm, W=30 μm, Z=4 μm. According to one example P=25 μm, TW=25 μm, Z=6 μm. According to one example P=25 μm, TW=25 μm, Z=3 μm.
a
FIG. 9
b
FIG. 9
a
FIG. 7
b
FIG. 9
4
3
3
2
1
1
4
3
3
2
1
6
1
6
6
3
2
1
2
1
4
3
3
6
1
shows a side view of the edge between the margin . and the clearance surface . of the drill device . Seen in a microscopic perspective, the edge have a radius R which joins the margin . and the clearance surface .. shows schematically a front view projection of . The radius R is indicated as the area between the two dashed lines. Thus, the grooves . of the textured area may extend from the boundary between the radius R and the margin . or from the radius R or from the boundary between the radius R and the clearance surface .. Also indicated in is a third alternative of the present disclosure in which the textured area extends from the clearance surface ., over the edge and along the margin . in direction of the rear portion of drill body. This is schematically indicated by the rightmost groove ..
4
Preferably, the textured area extends along the margin to the edge . Practical trials have shown that this configuration provides a high increase in wear resistance to the margin.
4
4
4
The textured area may be applied by a laser beam. Preferably the laser beam is moved along the margin indirection of the edge . The advantage thereof is that when the laser beam reaches the curvature of the edge it will loose focus with the surface of the margin and the forming of grooves will cease. This allows for an easy and effective way of forming textured area of grooves which extend along to the margin to the edge .
Although particular embodiments have been disclosed in detail this has been done for purpose of illustration only, and is not intended to be limiting. In particular it is contemplated that various substitutions, alterations and modifications may be made within the scope of the appended claims.
For example, the textured area has here above been described with reference to two alternatives of elongate grooves. However, it is appreciated that the recesses of the textured area may have different configuration and orientation. For example, the textured area may comprise a plurality of discrete recesses which are distributed with a predetermined periodicity both in direction along the margin and in direction across the margin. It is also possible to foresee other design of continuous grooves. For example grooves that extends in the form of sinus curve.
Moreover, although specific terms may be employed herein, they are used in a generic and descriptive sense only and not for purposes of limitation. Finally, reference signs in the claims are provided merely as a clarifying example and should not be construed as limiting the scope of the claims in any way.
Hereinafter, the drill device according to the present disclosure will be described with reference to examples.
A plurality of 870-1400-14-MM (grade GC2234) exchangeable drill tip inserts with a helix angle of 25° were prepared with textured areas in accordance with the below description.
3
2
Drill tip insert substrates with a composition of 84.74 wt % WC, 1.56 wt % CrC, 0.02 wt % TiC and the rest Co were provided. The drill tip insert substrates were coated with a 2 μm thick TiAlN coating applied with PVD forming drill tip inserts. GC2234 exchangeable drill tip inserts are commercially available from the company Sandvik Coromant AB. The margins of the drill tip inserts were provided with textured areas prior to applying the coating to the drill tip inserts. The drill tip inserts were divided into three groups. Each group was divided into two sets which subsequently were subjected to two separate machining trials (DP1 and DP2). A first comparative group of un-textured drill tip inserts was also prepared. Prior to testing the drill tip inserts were mounted on a drill tip holder made of tool steel.
A first group (15UB08) of drill tip inserts was provided with a textured area comprising elongate grooves extending parallel with the leading edge of the margin.
A second group (15UB06) of drill tip inserts was also provided with a textured area comprising elongate grooves extending parallel with the leading edge of the margin. The grooves of 15UB06 were of the same width depth and positioning as the grooves of 15UB08. However, the distance between the grooves (i.e. the periodicity) was half of 15UB08.
A third group (15UB07) of drill tip inserts was provided with a textured area comprising elongate grooves extending parallel with the axis of rotation of the drill device.
The laser used to produce the texturing on the drill tip inserts was a ps laser system from LMI (Preco) with a Talisker Ultra laser (diod pumped YAG) manufactured by Coherent. The target values for the laser grooves and laser settings for the three groups are presented below.
Target values for the laser grooves are shown in table 1 below.
TABLE 1
Target values for the laser grooves
Parameters
15UB08
15UB06
15UB07
Groove width, TW
25
μm
25
μm
25
μm
Groove depth, Z
4
μm
4
μm
4
μm
Groove
50
μm
25
μm
25
μm
periodicity, P
Distance between
0
μm
0
μm
0
μm
edge and textured
area
Texturing width, W
0.7
mm
0.7
mm
0.7
mm
Texturing length, L
2
mm
2
mm
2
mm
Laser settings for each group are shown in table 2 below.
TABLE 2
Laser settings
Wave length λ
355
nm
Effect, P
2 Watt (50% of max.)
Frequency, f
200
kHz
Velocity, v
400
mm/s
Spot distance, δx = v/f
2
μm
Focal height, Z
−200
μm
No. passings, N
20
The position of the textured area was chosen to cover the area where wear occurs and the grooves were stretched across the edge where the margin surface meets the clearance surface. The intention was to texture the surface as close to the periphery of the edge and towards the rear of the margin. The size of the textured area was determined by the curvature of the drill tip, as the margin has a helix formed surface the focal height will vary, setting a limit for the texturing distance along the margin. The length of the textured area was about 2 mm which is approximately 20% of the total margin length. The width of the textured area was difficult to measure using either LOM or SEM images, due to low magnification (LOM) and due to tilted position giving larger distance (SEM). Total margin width is 0.702 mm and the target of texturing the whole width 0.7 mm was not completely reached.
a
FIG. 10
b
FIG. 10
c
FIG. 10
shows a SEM analyses at large magnification of 15UB06. shows 15UB07 and shows 15UB08.
a
c
FIG. 11-11
shows the profile of the grooves in the textured area of a drill tip of respective sample groups 15UB06, 15UB07 and 15UB08.
The numerical results of the laser trenches and textured area size of the respective sample groups is shown in table 3 below.
TABLE 3
Results of the laser trenches and textured area size
Parameters
15UB06
15UB07
15UB08
Groove width,TW
26.5 ± 1.5
μm
23.0 ± 1.3
μm
25.3 ± 0.4
μm
Groove depth, Z
3.4 ± 0.3
μm
6.2 ± 1
μm
2.8 ± 0.3
μm
Groove periodicity, P
50
μm
25
μm
25
μm
Distance between
0
μm
0
μm
0
μm
edge and textured
area
Texturing width, W
~650
μm
~650
μm
~650
μm
Texturing length, L
~1.8
mm
~1.8
mm
~2.0
mm
The drill tips were tested in unalloyed steel S235JR+N (Livallco Stal AB) in through hole drilling. LOM images were taken and the wear was measured during the test every 50:th-100:th hole. Three different stopping criteria were set:
1) Flank wear (FW)>0.25 mm
3) Margin wear which covers the full upper edge towards the primary clearance
The cutting parameters for testing are shown in table 4.
TABLE 4
Cutting parameters for testing
Machine
Matsuura 2
Drill diameter [mm]
 14
Work piece material
S235JR + N/SS1312/P1.1.Z.HT/CMC01.1
Charge: 13311755D, C-content: 0.16%
Vc [m/min]
120
Fn [mm/rev]
   0.2
ap [mm]
41 (3xD)
Coolant
Internal, 8% emulsion
The two sets of drill tip inserts were subjected to separate machining trials (DP1 and DP2). For comparison, an un-textured drill tip inserts were also included in each machining trial.
The textured variants 15UB07 and 15UB08 were significantly less worn compared to the un-textured reference. 15UB06 had troubles with function, with bad noise and chipping in DP1 and breakage in DP2. Neither did it show the same improvement in wear resistance as 15UB07 and 15UB08. All textured variants show a weakness on the corner and upper edge, close to the flank face, which is worn faster than the rest of the margin. Also, in the area closest to the leading edge of the margin on 15UB07 and 15UB08, the substrate is visible. This is probably due to the fact that the laser tracks did not cover this area.
The tool life was recorded for all tested variants. In Table 5 the tool life of tested variants is presented. The following abbreviations are used in Table 5:
Mw=Margin wear covering the whole upper edge towards flank face;
CH(C)=Chipping on corner;
CH(SE)=Chipping on leading edge of margin;
FW=flank wear.
TABLE 5
Result of performance in tool life test
Test of
tool life
(Nr. of
Reference
Reference
Reference
Invention
holes)
No texture
15UB06
15UB07
15UB08
Test DP1
450 (MW)
400 (CH(SE))
500 (FW, MW)
550 (FW, MW)
Test DP2
350 (MW)
266 (tool
450 (CH(SE))
450 (CH(C))
breakage)
Variant 15UB06 showed the worst performance, with functional problems including bad noise and vibrations, resulting in early chipping in DP1 and tool breakage in DP2. 15UB07 and 15UB08 both show an increase in tool life compared to the un-textured reference (REF LASER). The increase in tool life is not as good as the improvement in margin wear resistance, mainly for two reasons:
1) Flank wear starts to be significant for the tool life at around 550 holes in this test.
2) The corner and the secondary edge still show a fairly high wear rate compared to the rest of the margin.
The drill tips were also examined by Scanning Electron Microscopy SEM.
a
c
FIG. 12-12
a
FIG. 12
b
FIG. 12
c
FIG. 12
a
b
FIGS. 13and 13
a
FIG. 13
b
FIG. 13
b
FIG. 14
a
FIG. 14
c
FIG. 14
shows SEM-pictures of drill tips 15UB06, 15UB07, 15UB08 at the end of the tool life in DP1. In (15UB06) tool life was 400 holes, in (15UB07) tool life was 500 holes and in (15UB07) tool life was 550 holes. shows SEM-pictures of drill tips 15UB07 and 15UB08 at the end of the tool life in DP2. In (15UB07) tool life was 450 holes and in (15UB08) tool life was 450 holes. As comparison, shows SEM-pictures of a non-textured drill-tip (see ) in DP1 having a tool life of 450 holes and shows a non-textured drill tip in DP2 having a tool life of 350 holes.
It is clear from the SEM-pictures that there is a big difference in margin wear between non-textured drill tips and textured drill tips, even though the textured drills have run longer. It is also clear that the orientation of the recesses has a great influence on the tool life. The orientation parallel with the longitudinal axis of rotation (X) of the drill (15UB08) showed the longest tool life.
BRIEF DESCRIPTION OF THE DRAWINGS
FIGS. 1-4
: Schematic drawings of a drill device according to a first embodiment of the present disclosure,
FIG. 5, 6
: Schematic drawings of a drill device according to a second embodiment of the present disclosure.
a
FIG. 7
: A schematic drawing of a textured area according to a first alternative of the present disclosure.
b
FIG. 7
: A schematic drawing of a textured area according to a second alternative of the present disclosure.
FIG. 8
: A schematic drawing of recesses in the margin of a drill device according to the present disclosure.
a
b
FIG. 9, 9
: Schematic drawings of the margin of a drill device according to the present disclosure.
a
c
FIGS. 10-10
: SEM-pictures of drill devices according to the present disclosure.
a
c
FIGS. 11-11
: SEM-pictures of recesses in drill devices according to the present disclosure.
a
c
FIG. 13-13
: SEM-pictures of a first set of drill devices according to the present disclosure after machining tests.
a
b
FIG. 14, 14
: SEM-pictures of a second set of drill devices according to the present disclosure after machining tests.
a
c
FIG. 15-15
: SEM-pictures of a comparative drill device before and after machining tests | |
Technical field of the invention
Background of the invention
Summary of the invention
Brief description of the drawings
Description of illustrative embodiments
Example 1: Comparison between a sequence search as known in the prior art and one in accordance with an embodiment of the present invention
Example 1a: Using a short search string
Example 1b: Using a longer protein as the search string
The present invention relates to the handling of biological sequence information, and more particularly to generating said biological sequence information, e.g. by sequencing and sequence assembly.
Biological sequencing has evolved at a blinding speed in the last decades, enabling along the way the human genome project which achieved a complete sequencing of the human genome already more than 15 years ago. To fuel this evolution, ample technical progress has been required, spanning from advances in sample preparation and sequencing methods to data acquisition, processing and analysis. Concurrently, new scientific fields have spawned and developed, including genomics, proteomics and bioinformatics.
MUIR, Paul, et al. The real cost of sequencing: scaling computation to keep pace with data generation. Genome biology, 2016, 17.1: 53
Fuelled by the postgenomic era's emphasis on data acquisition, this evolution has resulted in the accumulation of enormous amounts of sequence data. However, the ability to organize, analyse and interpret this sequence, to extract therefrom biologically relevant information, has been trailing behind. This problem is further compounded by the magnitude of new sequence information which is still generated on a daily basis. Muir et al. observed that this is sparking a paradigm shift and have commented on the resulting changing cost structure for sequencing and other associated hurdles (.).
Currently, the sequencing methods most often employed are those of the so called 'high-throughput' or 'next generation sequencing' (NGS). In contrast to the first generation of sequencing, NGS is typically characterized by being highly scalable, allowing an entire genome to be sequenced at once. Typically, this is accomplished by fragmenting a larger sequence into smaller fragments, randomly sampling for a fragment, and sequencing it. After sequencing the different fragments, the original sequence can be reconstructed using sequence assembly, in which the sequence fragments are aligned and merged on the basis of their overlapping regions.
Shmilovici et al. (SHMILOVICI, Armin; BEN-GAL, Irad. Using a VOM model for reconstructing potential coding regions in EST sequences. Computational Statistics, 2007, 22.1: 49-69
However, sequencers are not flawless and sequencing errors (such as insertions, substitutions and deletions) can always occur, particularly when a high throughput is sought. If the sequence fragments to be assembled contain errors, this obviously complicates the reconstruction of the original sequence as corresponding areas may no longer overlap. Furthermore, also the errors may propagate into the final sequence, e.g. resulting in mistaken variant calling. Some strategies have been developed for dealing with these sequencing errors, such as disclosed by .). However, no efficient method is currently known to validate directly whether a (fragment) sequence is correct or whether it contains one or more sequence errors.
There is thus still a need in the art for further improvements in sequencing and sequence assembly.
It is an object of the present invention to provide a good way to generate biological sequence information. This objective is accomplished by methods, devices and data structures according to the present invention.
The present invention relates to a method for sequencing a biopolymer or biopolymer fragment, comprising sequencing the biopolymer or biopolymer fragment taking into account information contained in a repository of fingerprint data strings, wherein each fingerprint data string representing a characteristic biological subsequence, the repository comprising at least a first fingerprint data string representing a first characteristic biological sequence of a first length, a second fingerprint data string representing a second characteristic biological subsequence of a second length, and for at least one of the fingerprint data strings, data related to one or more units which can appear directly before or after the characteristic biological subsequence when said characteristic biological subsequence is present in a biological sequence, wherein the first and the second length are equal to 4 or more and wherein the first and the second length differ from one another.
Taking into account information contained in the repository of fingerprint data strings may comprise searching a provisional biological sequence for occurrences of a characteristic biological subsequence represented by a fingerprint data string and subsequently validating or rejecting the provisional biological sequence by, for each occurrence, determining whether or not a monomer appearing directly after the characteristic biological subsequence conforms with data in the repository. Determining whether it conforms with data in the repository may comprise determining whether it conforms with data being data related to a secondary and/or tertiary and/or quaternary structure of the characteristic biological subsequence when said characteristic biological subsequence is present in a biopolymer.
Determining whether it conforms with data in the repository may comprise determining whether it conforms with data being data related to a relationship between the characteristic biological subsequence and one or more further characteristic biological subsequences.
Taking into account information contained in the repository of fingerprint data strings may comprise searching a tail of a partial biological sequence for occurrences of a characteristic biological subsequence represented by a fingerprint data string and subsequently predicting a monomer appearing directly after the characteristic biological subsequence from data in the repository.
Predicting a monomer from data in the repository may comprise predicting a monomer from data related to a secondary and/or tertiary and/or quaternary structure of the characteristic biological subsequence when said characteristic biological subsequence is present in a biopolymer.
Predicting a monomer from data in the repository may comprise predicting a monomer from data related to a relationship between the characteristic biological subsequence and one or more further characteristic biological subsequences.
providing a first biological sequence, the first biological sequence being a biological sequence of a first biopolymer fragment,
providing a second biological sequence, the second biological sequence being either a biological sequence of a second biopolymer fragment or being a reference biological sequence,
aligning the first biological sequence to the second biological sequence by aligning fingerprint markers in the first biological sequence with fingerprint markers in the second biological sequence, the fingerprint markers being associated with the fingerprint data strings representing characteristic biological subsequences, and
merging the first biological sequence with the second biological sequence to obtain an assembled biological sequence.
The present invention also relates to a method for performing a sequence assembly, comprising:
The present invention furthermore comprises a system comprising means for carrying out a method as described above.
The system may be an assembler.
The present invention also relates to a computer program product comprising instructions which, when the program is executed by a computer system, cause the computer system to carry out a method as described above.
The present invention also relates to a computer-readable medium comprising instructions which, when executed by a computer system, cause the computer system to carry out a method as described above.
In one aspect, a repository of fingerprint data strings, each fingerprint data string representing a characteristic biological subsequence, the repository comprising at least: a first fingerprint data string representing a first characteristic biological subsequence of a first length, a second fingerprint data string representing a second characteristic biological subsequence of a second length, and, for at least one of the fingerprint data strings, data related to one or more units which can appear directly before or after the characteristic biological subsequence when said characteristic biological subsequence is present in a biological sequence; wherein the first and the second length are equal to 4 or more and wherein the first and the second length differ from one another.
It is an advantage of embodiments of the present invention that a repository of fingerprint data strings corresponding to characteristic biological subsequences can be provided. It is a further advantage of embodiments that the biological subsequences need not be of a single length, as is the case for e.g. k-mers.
It is an advantage of embodiments of the present invention that further data can be included in the repository, such as data on the unit(s) which may succeed or precede a characteristic biological subsequence, data on the secondary/tertiary/quaternary structure of a characteristic biological subsequence, data on a relationship between fingerprints, etc.
In one aspect, a method for sequencing a biopolymer or biopolymer fragment is described, comprising sequencing the biopolymer or biopolymer fragment taking into account information contained in a repository of fingerprint data strings as defined in any embodiments as described above.
It is an advantage of embodiments of the present invention that the sequencing of biopolymers and biopolymer fragments can be improved (e.g. by decreasing the likelihood of errors or by speeding up the process) by relying on information contained in the repository of fingerprint data strings.
It is an advantage of embodiments of the present invention that a provisionally suggested biological sequence can be validated or rejected. It is an advantage of embodiments of the present invention that errors occurring during sequencing can be reduced.
It is an advantage of embodiments of the present invention that the speed of sequencing can be improved by predicting the next unit in the sequence or by limiting the number of options therefor.
A method for processing a biological sequence is described, the method comprising: (a) retrieving one or more fingerprint data strings from the repository as defined above, (b) searching the biological sequence for occurrences of the characteristic biological subsequences represented by the one or more fingerprint data strings, and (c) constructing a processed biological sequence comprising for each occurrence in step b a fingerprint marker associated with the fingerprint data string which represents the occurring characteristic biological subsequence.
It is an advantage of embodiments that a biological sequence can be relatively easily and efficiently processed. It is a further advantage of embodiments of the present invention that a biological sequence can be analysed in a lexical or even a semantic fashion.
It is an advantage of embodiments that the processed biological sequence can be constructed by replacing therein the identified characteristic biological subsequences by markers associated with the corresponding fingerprint data strings.
It is an advantage of embodiments that the portions of the biological sequence which do not correspond to one of the characteristic biological subsequences can be handled in a variety of ways. It is a further advantage of some embodiments that the biological sequence can be processed in a completely lossless way (i.e. no information is lost by processing). It is a further advantage of alternative embodiments that the biological sequence can be processed in a way that the more important information is distilled in a more condensed format.
It is an advantage that the processed biological sequences may be compressed so that they take up less storage space than their unprocessed counterparts.
It is an advantage of embodiments that that matching portions of the biological sequence to the characteristic biological subsequences is not solely limited to the primary structure, but can also take into account the secondary/tertiary/quaternary structure.
It is an advantage of embodiments that a secondary/tertiary/quaternary structure of a biological subsequence can be at least partially elucidated based on the known secondary/tertiary/quaternary structure of characteristic biological subsequences contained therein. It is a further advantage of embodiments of the present invention that biological sequence design (e.g. protein) design can be assisted or facilitated.
A processed biological sequence is described, obtainable by the method according to any embodiment as described above.
A method for building a repository of processed biological sequences is described, comprising populating said repository with processed biological sequences as defined above.
It is an advantage of embodiments that a repository of processed biological sequence can be constructed and stored.
A repository of processed biological sequences is described, obtainable by the method according to any embodiment as described above.
It is an advantage that the repository of processed biological sequences can be quickly searched and navigated. It is a further advantage that the storage size of the repository may be relatively small, compared to the known databases, by populating it with compressed processed biological sequences.
A method for comparing a first biological sequence to a second biological sequence, comprising: (a) processing the first biological sequence by the method according to any embodiment as described to obtain a first processed biological sequence, or retrieving the first processed biological sequence from a repository as defined in any embodiment as described, (b) processing the second biological sequence by the method as described to obtain a second processed biological sequence, or retrieving the second processed biological sequence from a repository as defined in any embodiment as described, and (c) comparing at least the fingerprint markers in the first processed biological sequence with the fingerprint markers in the second processed biological sequence.
It is an advantage of embodiments that the comparison of biological sequences can be changed from an NP-complete or NP-hard problem to a polynomial-time problem. It is a further advantage of embodiments that comparison can be performed in a greatly reduced time and scales well with increasing complexity (e.g. increasing length of or number of biological sequences). It is yet a further advantage of embodiments that the required computational power and storage space can be reduced.
It is an advantage of embodiments that a degree of similarity can be calculated between biological sequences. It is a further advantage of embodiments that a plurality of biological sequences can be ranked based on their degree of similarity.
It is an advantage of embodiments that a sequence similarity search can be quickly and easily performed (e.g. in polynomial time).
A method for aligning a first biological sequence to a second biological sequence is described, comprising performing the method as described above, wherein step c further comprises aligning the fingerprint markers in the first processed biological sequence with the fingerprint markers in the second processed biological sequence.
It is an advantage of embodiments that compared biological sequences can be easily and quickly aligned (e.g. in polynomial time).
It is an advantage of embodiments that also a plurality of sequences can be easily and quickly compared and aligned. It is a further advantage of embodiments that there is no accumulation of errors during the alignment, as is the case in currently known methods (e.g. based on progressive alignment).
A method for performing a sequence assembly is described, comprising: (a) providing a first biological sequence, the first biological sequence being a biological sequence of a first biopolymer fragment, (b) providing a second biological sequence, the second biological sequence being either a biological sequence of a second biopolymer fragment or being a reference biological sequence, (c) aligning the first biological sequence to the second biological sequence using a method as described above, and (d) merging the first biological sequence with the second biological sequence to obtain an assembled biological sequence.
It is an advantage of embodiments that sequences of biopolymer fragments can be easily and quickly aligned and merged to reconstruct the original biopolymer sequence.
A system is described comprising means for carrying out the method according to any of the embodiments described above.
It is an advantage of embodiments that the methods may be implemented by a variety of systems and devices, such as computer-based systems or a sequencer, depending on the application. It is a further advantage of embodiments of the present invention that the methods can be implemented by a computer-based system, including a cloud-based system.
A computer program product is described comprising instructions which, when the program is executed by a computer system, cause the computer system to carry out the method according to any of the embodiments as described above.
A computer-readable medium is described comprising instructions which, when executed by a computer system, cause the computer system to carry out the method according to any of the embodiments as described above.
Use of a library of biological sequence fingerprints as described above is described, for one or more selected from: processing a biological sequence, building a repository of processed biological sequences, comparing a first biological sequence to a second biological sequence, aligning a first biological sequence to a second biological sequence, performing a multiple sequence alignment, performing a sequence similarity search and performing a variant calling.
Use of a processed biological sequence as defined above is described, or use of a library of processed biological sequences as defined above is described, for one or more selected from: comparing a first biological sequence to a second biological sequence, aligning a first biological sequence to a second biological sequence, performing a multiple sequence alignment, performing a sequence similarity search and performing a variant calling.Particular and preferred aspects of the invention are set out in the accompanying independent and dependent claims. Features from the dependent claims may be combined with features of the independent claims and with features of other dependent claims as appropriate and not merely as explicitly set out in the claims.
Although there has been constant improvement, change and evolution of devices in this field, the present concepts are believed to represent substantial new and novel improvements, including departures from prior practices, resulting in the provision of more efficient, stable and reliable devices of this nature.
The above and other characteristics, features and advantages of the present invention will become apparent from the following detailed description, taken in conjunction with the accompanying drawings, which illustrate, by way of example, the principles of the invention. This description is given for the sake of example only, without limiting the scope of the invention. The reference figures quoted below refer to the attached drawings.
Figs. 1 and 2
are graphs showing expected progress enabled by embodiments of the present invention.
Figs. 3 to 6
are diagrams depicting systems in accordance with embodiments of the present invention.
Figs. 7
10
and are graphs comparing the total length of search results using, on the one hand, a prior art method (dotted line) and, on the other hand, a method in accordance with exemplary embodiments of the present invention (solid line).
Figs. 8
11
and are graphs comparing the Levenshtein distance of search results using, on the one hand, a prior art method (dotted line) and, on the other hand, a method in accordance with exemplary embodiments of the present invention (solid line).
Figs. 9
and 12 are graphs comparing the longest common substring of search results using, on the one hand, a prior art method (dotted line) and, on the other hand, a method in accordance with exemplary embodiments of the present invention (solid line).
In the different figures, the same reference signs refer to the same or analogous elements.
The present invention will be described with respect to particular embodiments and with reference to certain drawings but the invention is not limited thereto but only by the claims. The drawings described are only schematic and are non-limiting. In the drawings, the size of some of the elements may be exaggerated and not drawn on scale for illustrative purposes. The dimensions and the relative dimensions do not correspond to actual reductions to practice of the invention.
Furthermore, the terms first, second, third and the like in the description and in the claims, are used for distinguishing between similar elements and not necessarily for describing a sequence, either temporally, spatially, in ranking or in any other manner. It is to be understood that the terms so used are interchangeable under appropriate circumstances and that the embodiments of the invention described herein are capable of operation in other sequences than described or illustrated herein.
Moreover, the terms before, after, and the like in the description and the claims are used for descriptive purposes and not necessarily for describing relative positions. It is to be understood that the terms so used are interchangeable with their antonyms under appropriate circumstances and that the embodiments of the invention described herein are capable of operation in other orientations than described or illustrated herein.
It is to be noticed that the term "comprising", used in the claims, should not be interpreted as being restricted to the means listed thereafter; it does not exclude other elements or steps. It is thus to be interpreted as specifying the presence of the stated features, integers, steps or components as referred to, but does not preclude the presence or addition of one or more other features, integers, steps or components, or groups thereof. The term "comprising" therefore covers the situation where only the stated features are present and the situation where these features and one or more other features are present. Thus, the scope of the expression "a device comprising means A and B" should not be interpreted as being limited to devices consisting only of components A and B. It means that with respect to the present invention, the only relevant components of the device are A and B.
Reference throughout this specification to "one embodiment" or "an embodiment" means that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention. Thus, appearances of the phrases "in one embodiment" or "in an embodiment" in various places throughout this specification are not necessarily all referring to the same embodiment, but may. Furthermore, the particular features, structures or characteristics may be combined in any suitable manner, as would be apparent to one of ordinary skill in the art from this disclosure, in one or more embodiments.
Similarly, it should be appreciated that in the description of exemplary embodiments of the invention, various features of the invention are sometimes grouped together in a single embodiment, figure, or description thereof for the purpose of streamlining the disclosure and aiding in the understanding of one or more of the various inventive aspects. This method of disclosure, however, is not to be interpreted as reflecting an intention that the claimed invention requires more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive aspects lie in less than all features of a single foregoing disclosed embodiment. Thus, the claims following the detailed description are hereby expressly incorporated into this detailed description, with each claim standing on its own as a separate embodiment of this invention.
Furthermore, while some embodiments described herein include some but not other features included in other embodiments, combinations of features of different embodiments are meant to be within the scope of the invention, and form different embodiments, as would be understood by those in the art. For example, in the following claims, any of the claimed embodiments can be used in any combination.
Furthermore, some of the embodiments are described herein as a method or combination of elements of a method that can be implemented by a processor of a computer system or by other means of carrying out the function. Thus, a processor with the necessary instructions for carrying out such a method or element of a method forms a means for carrying out the method or element of a method. Furthermore, an element described herein of an apparatus embodiment is an example of a means for carrying out the function performed by the element for the purpose of carrying out the invention.
In the description provided herein, numerous specific details are set forth. However, it is understood that embodiments of the invention may be practised without these specific details. In other instances, well-known methods, structures and techniques have not been shown in detail in order not to obscure an understanding of this description.
The following terms are provided solely to aid in the understanding of the invention.
As used herein, a biological sequence is a sequence of a biopolymer defining at least the biopolymer's primary structure. The biopolymer can for example be a deoxyribonucleic acid (DNA), ribonucleic acid (RNA) or a protein. The biopolymer is typically a polymer of biomonomers (e.g. nucleotides or amino acids), but may in some instances further include one or more synthetic monomers.
As used herein, a 'unit' in a biological sequence is an amino acid when the biological sequence relates to a protein and is a codon when the biological sequence relates to DNA or RNA.
As used herein, a biological subsequence is a portion of a biological sequence, smaller than the full biological sequence. The biological subsequence may, for example, have a total length of 100 units or less, preferably 50 or less, yet more preferably 20 or less.
As used herein, some concepts will be illustrated with examples relating to proteins and it will be assumed that the possible monomeric units are the 20 canonical (or 'standard') amino acids. However, it is clear that this is merely to simplify the illustration and that similar embodiments can likewise be formulated with an extended number of amino acids (e.g. adding non-canonical amino acids or even synthetic compounds), or relating to DNA or RNA. In the case of DNA or RNA, a link between the DNA or RNA and proteins can be easily made through the correspondence between codons and amino acids.
As used herein, 'secondary/tertiary/quaternary' refers to 'secondary and/or tertiary and/or quaternary'.
n
n
n
It was surprisingly realized within the present invention that, where it was previously assumed that the primary structure of a biological sequence consists of an essentially independent selection of units, so that there are e.g. m biological sequences of length n based on m possible units (e.g. 20 based on 20 canonical amino acids), this in fact not observed in nature. Indeed, it was discovered that from a certain length onwards, not every theoretical combination is seen. To give but one example: the protein subsequence 'MCMHNQA' is not found in any protein in the public databases. It has been contemplated that this is not a mere hiatus in the databases, but that this absence has a physical and/or chemical origin. Without being bound by theory, to name but one possible effect, the steric hindrance of the neighbouring amino acids (e.g. 'MCMHNQ' in the above example) may prohibit one or more other amino acids (e.g. 'A' in the above example) from binding thereto. As such, once an absent subsequence has been identified, computational studies can be used to validate whether this subsequence could potentially occur or whether its existence is physically impossible (or improbable, e.g. because it's chemically unstable). The 'certain length' referred to above depends on the data set that is being considered, but e.g. corresponds to about 5 or 6 amino acids for the publicly available protein sequence databases (which substantially reflect the total diversity seen in nature). For a more limited set (e.g. a set filtered based on a particular criterion), less than the theoretical maximum of m combinations is already found for a length of about 4 or 5.
Simultaneously, because the subsequence 'MCMHNQA' does not exist, the subsequence 'MCMHNQ' is not merely a random combination of 5 amino acids but gains additional significance; such subsequences will be further referred to as 'characteristic biological subsequences' or 'fingerprints'. Because of the added significance or meaning of these fingerprints, it can be considered that the present invention handles biological sequence information in a more semantic fashion. In general, a characteristic subsequence is characterized by having for the unit directly following (or preceding) it less possible options than the maximum number of units (e.g. less than the 20 canonical amino acids); in other words, at least one of the units cannot follow (or precede) it. However, it is possible to select a stricter definition: e.g. only those subsequences which have 15 units or fewer which can possibly follow it, or 10 or fewer, 5 or fewer, 3, 2 or even 1. Furthermore, it can be chosen to consider each such subsequence as a fingerprint, or to consider only those subsequences as fingerprints which do not already comprise another fingerprint. For example: taking 'MCMHNQ' as a fingerprint, there will be longer subsequences which comprise 'MCMHNQ' and which also have less than the theoretical number of units which can follow (or precede) it; in that case, there is the option to consider both the longer subsequences and 'MCMHNQ' as fingerprints, or to consider only 'MCMHNQ' as a fingerprint.
It was then surprisingly found that a limited set of characteristic biological subsequences can be identified. Furthermore, it was observed that these characteristic biological subsequences strike a balance between, on the one hand, being sufficiently specific so that not every characteristic biological subsequence is found in every biological sequence and, on the other hand, being common enough that the known biological sequences typically comprise at least one of these fingerprints.
Fig. 1
n
n
Based on this discovery, new approaches to handling biological sequence information, in all its different but interrelated stages, can be formulated. These approaches can be considered as being akin to a more lexical analysis of the sequences. The result is schematically depicted in , which shows the complexity scaling of the biological sequence information with an increasing number of units (n). This complexity may be the total number of possible combinations of units, but that in turn also relates to the computational effort (e.g. time and memory) needed for handling it (e.g. for performing a similarity search). The solid curve depicts the number of theoretical combinations assuming all units are selected independently, scaling as m, which also corresponds to the scaling of the currently known algorithms. The dashed curve depicts the number of actual combinations found in nature (as observed within the present invention), where the curve departs from m at around 5 or 6 units and asymptotically flattens off for high n. The dotted line shows the number of sequences which correspond for the first time to a characteristic sequence for which the number of units which can follow it is equal to 1; here 'for the first time' means that longer sequences are never counted if they comprise an already counted fingerprint. Thus, the latter corresponds to the number of fingerprints of length n (as observed within the present invention), when the definition thereof is selected as a subsequence which has only 1 unit which can possibly follow it and which does not already comprise another (shorter) fingerprint (cf. supra).
Fig. 2
depicts the predicted benefits of the present invention in time, where the mark on the bottom axis depicts the present day. Curve 1 shows Moore's law as a reference. Curve 2 shows the total amount of acquired sequencing data. Curve 3 shows the total cost of processing and maintaining said sequencing data. By handling biological sequence information as proposed in the present invention, the total required storage for sequencing data and the total cost of data processing and maintenance are expected to drop as depicted in curves 4 and 5 respectively.
Figs. 3 and 4
In a first aspect, a repository of fingerprint data strings is described, each fingerprint data string representing a characteristic biological subsequence, the repository comprising at least: a first fingerprint data string representing a first characteristic biological subsequence of a first length, a second fingerprint data string representing a second characteristic biological subsequence of a second length, and, for at least one of the fingerprint data strings, data related to one or more units which can appear (e.g. which can realistically appear, such as those combinations which are stable) directly before or after the characteristic biological subsequence when said characteristic biological subsequence is present in a biological sequence; wherein the first and the second length are equal to 4 or more and wherein the first and the second length differ from one another. A repository (e.g. database) of fingerprint data strings 100 is schematically depicted in , which will be discussed in more detail under the second, third and fifth aspect.
In embodiments, the length may correspond to the number of units. In embodiments, the length may be up to 100 or less, preferably 50 or less, yet more preferably 20 or less. In embodiments, the first and the second length may be equal to 5 or more, preferably 6 or more. In embodiments, the first and second characteristic biological subsequences may have a length between 4 and 20, preferably 5 and 15, yet more preferably 6 and 12.
In embodiments, the repository of fingerprint data strings may comprise at least 3 fingerprint data strings which differ in length from one another, preferably at least 4, yet more preferably at least 5, most preferably at least 6. Since the characteristic biological subsequences are not defined by their length, but by the number of possible units which follow (or precede) it, a set of characteristic biological subsequences typically advantageously comprises subsequences of varying lengths. The repository of fingerprint data strings in the present invention differs from e.g. a collection of k-mers (as is known in the art) in that it comprises biological subsequences of varying lengths. Furthermore, a collection of k-mers typically comprises every permutation (i.e. every possible combination of units) of length k; this is not the case for the present repository of fingerprint data strings.
In embodiments, the fingerprint data strings may be protein fingerprint data strings, DNA fingerprint data strings or RNA fingerprint data strings. In embodiments, the characteristic biological subsequence may be a characteristic protein subsequence, a characteristic DNA subsequence or a characteristic RNA subsequence. In embodiments, the repository of fingerprint data strings may comprise (e.g consist of) protein fingerprint data strings, DNA fingerprint data strings, RNA fingerprint data strings or a combination of one or more of these. A characteristic protein subsequence can in embodiments be translated into a characteristic DNA or RNA subsequence, and vice versa. This translation can be based on the well-known DNA and RNA codon tables. Similarly, a protein fingerprint data string can be translated into a DNA or RNA fingerprint data string. In embodiments, a repository of DNA or RNA fingerprint data strings may comprise information on equivalent codons (i.e. codons which code for the same amino acid). This information on equivalent codons can be included in the fingerprint data string as such, or stored separately therefrom in the repository.
In embodiments, the data related to the one or more units which can appear directly before or after the characteristic biological subsequence may comprise the number of possible units, the possible units as such, the likelihood (e.g. probability) for each unit, etc.
In embodiments, the repository of fingerprint data strings mayfurthercomprise additional data for at least one of the fingerprint data strings. In preferred embodiments, said data may be included in the fingerprint data string. In alternative embodiments, said data may be stored separately from the fingerprint data strings.
In embodiments, the additional data may comprise data related to a secondary/tertiary/quaternary structure of the characteristic biological subsequence when said characteristic biological subsequence is present in a biopolymer. In embodiments, the data related to the secondary (ortertiary/quaternary) structure may comprise the number of possible structures, the possible structures as such, the likelihood (e.g. probability) for each structure, etc. In the case of multiple possible secondary/tertiary/quaternary structures for a given characteristic biological subsequence, the repository may in embodiments comprise a separate entry for each combination of the characteristic biological subsequence and an associated secondary/tertiary/tertiary structure. In alternative embodiments, the repository may comprise one entry comprising the characteristic biological subsequence and a plurality of its associated secondary/tertiary/quaternary structures. In embodiments, the secondary/tertiary/quaternary structure may be more relevant for proteins than for DNA and RNA.
In embodiments, the additional data may comprise data related to a relationship between the characteristic biological subsequence and one or more further characteristic biological subsequences. In embodiments, the data related to a relationship between the characteristic biological subsequence may comprise further characteristic biological subsequences which commonly appear in its vicinity, the likelihood for the further characteristic biological subsequence to appear in its vicinity, a particular significance (e.g. a biologically relevant meaning, such as a trait or a secondary/tertiary/quaternary structure) of these characteristic biological subsequences appearing close to one another, etc. In embodiments, the relationship may be expressed in the form of a path between two or more characteristic biological subsequences and may include the order of the characteristic biological subsequences, their inter-distance, etc. In embodiments, the additional data may also comprise metadata useful for building said paths.
Fig. 4
In some embodiments, the additional data may have been retrieved from a known data set; e.g. the secondary/tertiary/quaternary structure of several biological sequences is available in the art. In other embodiments, the additional data may have been may be extracted from a processed biological sequence as defined any embodiment of the fourth aspect or from a repository of processed biological sequences as defined in any embodiment of the sixth aspect. For example, after processing a biological sequence according to any embodiment of the third aspect (or building a repository of processed biological sequences according to any embodiment of the fifth aspect), relationships between the characteristic biological subsequences (e.g. paths) may be extracted and added to a repository of fingerprint data strings of the present aspect; this is schematically depicted in by the dashed arrows pointing from the processed biological sequence 210 and the repository of processed biological sequences 220 to the repository of fingerprint data strings 100.
Fig. 3
In a second aspect, a method for sequencing a biopolymer or biopolymer fragment is described, comprising sequencing the biopolymer or biopolymer fragment taking into account information contained in a repository of fingerprint data strings as defined in any embodiments of the first aspect. schematically shows a sequencer 350 which sequences biopolymer (fragments) 500 using information contained in the repository of fingerprint data strings 100.
In embodiments, the method may comprise sequencing an initial (e.g. provisional or partial) biological sequence.
In embodiments, taking into account information contained in the repository of fingerprint data strings may comprise searching a provisional biological sequence for occurrences of a characteristic biological subsequence represented by one of the fingerprint data strings, and subsequently validating or rejecting the provisional biological sequence by, for each occurrence, determining whether or not a unit appearing directly before or after the characteristic biological subsequence conforms with the data as defined in embodiments of the first aspect. Since the repository can contain data on the units which can appear before or after a fingerprint, this information can be advantageously used to verify whether a provisional biological sequence is agreement therewith. If it is not, the provisional biological sequence can be rejected and redone.
In embodiments, taking into account information contained in the repository of fingerprint data strings may comprise searching a head or a tail of a partial biological sequence for occurrences of a characteristic biological subsequence represented by one of the fingerprint data strings and subsequently predicting a monomer appearing respectively directly before or after of the characteristic biological subsequence from the data as defined in embodiments of the first aspect. Since the repository can contain data on units which are the only possible option for appearing before or after a particular fingerprint, this information can be advantageously used to speed up the sequencing by directly appending that unit to the partial sequence; thereby allowing the actual sequencing to skip past said unit. In embodiments, the repository may contain data on a series of two, three, or more units which together are the only possible option for appearing before or after a particular fingerprint. In this case, the whole series can be advantageously directly appended to the partial sequence; thereby allowing the actual sequencing to skip past these units. Similarly, if the repository indicates that for the observed fingerprint a limited number (but more than 1) of options are possible as further units (e.g. two or three options), this information can still allow the sequencer to more quickly identify the specific unit in the present instance.
In embodiments, the searching may be as described for step b of the third aspect.
Fig. 4
In a third aspect, a method for processing a biological sequence is described, comprising: (a) retrieving one or more fingerprint data strings from the repository as defined in any embodiment of the first aspect, (b) searching the biological sequence for occurrences of the characteristic biological subsequences represented by the one or more fingerprint data strings, and (c) constructing a processed biological sequence comprising for each occurrence in step b a fingerprint marker associated with the fingerprint data string which represents the occurring characteristic biological subsequence. schematically shows a sequence processing unit 310 which processes a biological sequence 200 using a repository of fingerprint data strings 100, thereby obtaining a processed biological sequence 210.
In embodiments, the biological sequence to be processed may be biological sequence of a biopolymer fragment, obtainable by the method for sequencing according to the second aspect.
In some embodiments, the marker may be a reference string. Such a reference string may for example point towards the corresponding fingerprint data string in the repository. In other embodiments, the marker may be the fingerprint data string as such, or a portion thereof.
In embodiments, the biological sequence may comprise: (i) one or more first portions, each first portion corresponding to one of the characteristic biological subsequences represented by the one or more fingerprint data strings, and (ii) one or more second portions, each second portion not corresponding to any of the characteristic biological subsequences represented by the one or more fingerprint data strings. In embodiments, constructing the processed biological sequence in step c may comprise replacing at least one first portion by the corresponding marker. In embodiments, constructing the processed biological sequence in step c may further comprise adding positional information about said first portion to the processed biological sequence (e.g. appended to the marker). In embodiments, constructing the processed biological sequence in step c may comprise leaving at least one second portion unchanged, and/or replacing at least one second portion by an indication of the length of said second portion, and/or entirely removing at least one second portion. When leaving the second portions unchanged, the biological sequence is able to be processed in a completely lossless way.
In embodiments, the processed biological sequence can be formulated in a condensed format. For example, by replacing the characteristic biological subsequences (i.e. first portions) with reference strings and/or by replacing the second portions with either an indication of its length or entirely removing it, a processed biological sequence is obtained which requires less storage space than the original (i.e. unprocessed) biological sequence. Additional data compression can be achieved by making use of paths which can represent multiple fingerprints by their interrelation.
In embodiments, the one or more fingerprint data strings may be in a different biological format than the biological sequences (e.g. protein vs DNA vs RNA sequence information) and step b may further comprise translating or transcribing the characteristic biological subsequences prior to the searching.
In embodiments, the searching in step b may include searching for a partial match or an equivalent match (e.g. an equivalent codon, or a different amino acid resulting in the same secondary/tertiary/quaternary structure). In embodiments, the searching in step b may take into account a secondary/tertiary/quaternary structure of the characteristic biological subsequence. The secondary, tertiary and quaternary are typically more evolutionary conserved and often variation in the primary structure occur which do not change the function of the biopolymer, e.g. because the secondary/tertiary/quaternary structure of its active sites is substantially conserved. The secondary/tertiary/quaternary structure may therefore reveal relevant information about the biopolymer which would be lost when strictly searching for a fully matching primary structure.
In embodiments, the method may comprise a further step d, after step c, of at least partially inferring a secondary/tertiary/quaternary structure of the processed biological subsequence based on the data related to the secondary/tertiary/quaternary structure as defined in embodiments of the first aspect. This at least partial elucidation of the secondary/tertiary/quaternary structure can help to assist and/or facilitate biological sequence design. In embodiments wherein a single primary structure of a characteristic biological subsequence is linked to a plurality of secondary or tertiary or quaternary structures, the secondary/tertiary/quaternary structure may be disambiguated based on the context in which the characteristic biological subsequence is found, such as the characteristic biological subsequences which it is surrounded by. The information needed for such disambiguation may, for example, be found in the repository of fingerprint data strings in the form of data related to a relationship in terms secondary/tertiary/quaternary structure of between the characteristic biological subsequence and one or more further characteristic biological subsequences, as defined in embodiments of the first aspect.
Fig. 4
In a fourth aspect, a processed biological sequence is described, obtainable by the method according to any embodiment of the third aspect. A processed biological sequence 210 is schematically depicted in .
Fig. 4
In a fifth aspect, a method for building a repository of processed biological sequences is described, comprising populating said repository with processed biological sequences as defined any embodiment of the fourth aspect. schematically shows a repository building unit 320 storing a processed biological sequence 210 into a repository of processed biological sequences 220.
Fig. 4
In a sixth aspect, a repository of processed biological sequences is described, obtainable by the method according to any embodiment of the fifth aspect. A repository of 220 is schematically depicted in .
In embodiments, the repository may be a repository of processed biological fragment sequences (i.e. processed biological sequences of biopolymer fragments).
In embodiments, the repository may be a database. In some embodiments, the repository of processed biological sequences may be an indexed repository. The repository may, for example, be indexed based on the fingerprint markers (corresponding to the characteristic biological subsequences) present in each processed biological sequence. In other embodiments, the repository may be a graph repository.
Fig. 5
In a seventh aspect, a method for comparing a first biological sequence to a second biological sequence is described, comprising: (a) processing the first biological sequence by the method according to any embodiment of the third aspect to obtain a first processed biological sequence, or retrieving the first processed biological sequence from a repository as defined in any embodiment of the sixth aspect, (b) processing the second biological sequence by the method according to any embodiment of the third aspect to obtain a second processed biological sequence, or retrieving the second processed biological sequence from a repository as defined in any embodiment of the sixth aspect, and (c) comparing at least the fingerprint markers in the first processed biological sequence with the fingerprint markers in the second processed biological sequence. schematically shows a comparison unit 330 comparing at least a first biological sequence 211 and a second biological sequence 212 to output results 400.
By using characteristic biological subsequences according to embodiments(through the fingerprint markers in the processed biological sequences), the problem of comparing sequences is advantageously reformulated from an NP-complete or NP-hard problem to a polynomial-time problem. Indeed, identifying the fingerprints in a sequence and subsequently comparing sequences based on these fingerprints, which can be considered as a lexical approach, is computationally much simpler than the currently used algorithms (which e.g. compare full sequences based on a sliding windows approach). The comparison can therefore be performed markedly faster and furthermore scales well with increasing complexity (e.g. increasing length of or number of biological sequences), even while requiring less computation power and storage space.
In embodiments, the second biological sequence may be a reference sequence.
In embodiments, step c may comprise identifying whether one or more characteristic biological subsequences (represented by the fingerprint markers) in the first processed biological sequence correspond (e.g. match) with one or more characteristic biological subsequences (represented by the fingerprint markers) in the second processed biological sequence. In embodiments, step c may comprise identifying whether the corresponding characteristic biological subsequences appear in the same order in the first processed biological sequence as in the second processed biological sequence. In embodiments, step c may comprise identifying whether one or more pairs of characteristic biological subsequences in the first processed biological sequence and one or more corresponding pairs of characteristic biological subsequences in the second processed biological sequence have a same or similar (e.g. differing by less than 100 units, preferably less than 50 units, yet more preferably less than 20 units, most preferably less than 10 units) inter-distance.
In embodiments, step c may further comprise comparing one or more second portions of the first processed biological sequence with one or more second portions in the second processed biological sequence. In embodiments, comparing one or more second portions may comprise comparing corresponding second portions (i.e. a second portion appearing in between a neighbouring pair of characteristic biological subsequences in the first processed biological sequence and a second portion appearing in between a corresponding neighbouring pair of characteristic biological subsequences in the first processed biological sequence).
In embodiments, step c may further comprise calculating a measure representing a degree of similarity (e.g. a Levenshtein distance) between the first and the second biological sequence.
In embodiments, the method may be used in a sequence similarity search, by comparing a query sequence with one or more other biological sequences (e.g. corresponding to a sequence database that is to be searched, for example in the form of a repository of processed biological sequences). In embodiments, a degree of similarity may be calculated for each of the other biological sequences. In embodiments, the method may comprise a further step of ranking the biological sequences (e.g. by decreasing degree of similarity). In embodiments, the method may comprise filtering the biological sequences. Filtering may be performed before and/or after step c. For example, filtering may be performed by selecting for comparison only those biological sequences from the database which fit a certain criterion, such as based on the organism or group of organisms which they derive from (e.g. plants, animals, humans, microorganisms, etc.), whether a secondary/tertiary/quaternary structure is known, their length, etc. Alternatively, filtering may be performed after the comparison has been performed, based on the same criteria or based on the calculated degree of similarity (e.g. only those sequences may be selected which surpass a certain threshold of similarity). In contrast to sequence similarity searching in the prior art, where an alignment step is typically required and a measure of similarity is then established therefrom, alignment is not strictly necessary for similarity searching according to embodiments. Indeed, similar sequences can already be found by simply searching for sequences with the same fingerprints (optionally also taking into account their order and their inter-distance), without alignment; this in turn allows to further speed up the search. The above notwithstanding, alignment in accordance with embodiments (cf. the eighth aspect) is also computationally simplified, so that it may be chosen to do an alignment anyway, even if not strictly required.
The method of this aspect thus allows determining (and optionally measuring) the similarity between a first and a second biological sequence. Such a comparison is also a cornerstone in other methods, such as those of the eighth and ninth aspect.
Fig. 5
In an eighth aspect, a method for aligning a first biological sequence to a second biological sequence is described, comprising performing the method according to an embodiment of the seventh aspect, wherein step c further comprises aligning the fingerprint markers in the first processed biological sequence with the fingerprint markers in the second processed biological sequence. schematically shows output results 400 from comparison unit 330 (which is in this case better referred to as 'alignment unit 330') in which biological sequences are aligned by their fingerprint markers.
Alignment in thus also simplified in embodiments, since a good alignment can already be obtained by simply aligning the fingerprints. Once more, this significantly reduces the computational complexity of the problem. Furthermore, in the prior art methods, such as those based on progressive alignment, there is a build-up of alignment errors, as misalignment for one of the earlier sequences typically propagate and cause additional misalignments in the later sequences. Conversely, since it is each time the same discrete set of fingerprint markers which are aligned (or at least attempted to) within one (multiple) alignment, there is no such propagation of errors.
In embodiments, the method may further comprise subsequently aligning corresponding second portions. Aligning the second portions may, for example, be performed using one of the alignment methods known in the prior art. Indeed, since the 'skeleton' of the alignment is already provided by aligning the fingerprint markers, only the alignment in between these markers is left to be fleshed out. Since each of these second portions is typically relatively short compared to the total biological sequence length, the known methods can typically perform such an alignment relatively quickly and efficiently.
Fig. 5
In embodiments, the method may be for performing a multiple sequence alignment (i.e. the method may comprise aligning three or more biological sequences). In embodiments, the method may comprise aligning fingerprint markers in a third (or fourth, etc.) processed biological sequence with fingerprint markers in the first and/or second processed biological sequences. This is schematically depicted in in which alignment unit 330 may also compare and align an arbitrary number of further processed biological sequences 213-216.
In embodiments, the method may be used in variant calling. In the case of sequence alignment between two biological sequences, the variant calling may identify variants (e.g. mutations) between a query sequence and a reference sequence. In the case of a multiple sequence alignment, the variant calling may identify the possible variations (which may include determining their frequency of occurrence) in a set of related sequences; optionally with respect to a reference sequence. Identifying variants may furthermore be performed on the basis of the primary structure, but may also take account of the secondary/tertiary/quaternary structure.
Fig. 6
In a ninth aspect, a method for performing a sequence assembly is described, comprising: (a) providing a first biological sequence, the first biological sequence being a biological sequence of a first biopolymer fragment, (b) providing a second biological sequence, the second biological sequence being either a biological sequence of a second biopolymer fragment or being a reference biological sequence, (c) aligning the first biological sequence to the second biological sequence using the method according any embodiment of the eighth aspect, and (d) merging the first biological sequence with the second biological sequence to obtain an assembled biological sequence. schematically shows a sequence assembling unit 340 outputting assembled biological sequence 510, by first aligning (by their fingerprint markers) and subsequently merging an arbitrary number of biological sequences 500 (comprising of at least a first biological sequence 501 and second biological sequence 502).
In embodiments, the method steps a to d may be repeated so as to align and merge an arbitrary number of biopolymer fragments.
In order to facilitate sequencing, longer biopolymers can be fragmented, since the individual fragments are sequenced faster and more easily (e.g. they can be sequenced in parallel); as is known in the art. Sequence assembly is then typically used to align and merge fragment sequences to reconstruct the original sequence; this may also be referred to as 'read mapping', where 'reads' from a fragment sequence are 'mapped' to a second biopolymer sequence. Depending on the type of sequence assembly that is being performed, e.g. a de-novo assembly vs. a mapping assembly, the second biopolymer sequence may be selected to be a second biopolymer fragment or a reference sequence, as appropriate. Herein, a de-novo assembly is an assembly from scratch, without using a template (e.g. a backbone sequence). Conversely, a mapping assembly is an assembly by mapping one or more biopolymer fragment sequences to an existing backbone sequence (e.g. a reference sequence), which is typically similar (but not necessarily identical) to the to-be-reconstructed sequence. A reference sequence may for example be based on (part of) a complete genome or transcriptome, or may be have been obtained from an earlier de-novo assembly.
In embodiments, the method may comprise a further step (e), after step (d), of aligning the assembled biological sequence to the second biological sequence using the method according any embodiment of the eighth aspect. This additional alignment may be used to perform variant calling of the assembled biological sequence with respect to the second biological sequence (e.g. the reference sequence).
In a tenth aspect, a system comprising means for carrying out the method according to any embodiments of the second, third, fifth, seventh, eighth or ninth aspect is described.
The system may typically take on a different form depending on the method(s) it is meant to carry out. In embodiments, the system may be a sequencer (cf. second aspect), a sequence processing unit (cf. third aspect), a repository building unit (cf. fifth aspect), a comparison unit (cf. seventh aspect), an alignment unit (cf. eighth aspect), a sequence assembling unit (cf. ninth aspect). In embodiments, a generic data processing means (e.g. a personal computer or a smartphone) or a distributed computing environment (e.g. cloud-based system) can be configured to perform one or more of these functions. The distributed computing environment may, for example, comprise a server device and a networked client device. Herein, the server device may perform the bulk of one or more methods, including storing the repository of fingerprint data strings (cf. the first aspect) and the repository of processed biological sequences (cf. the sixth aspect). On the other hand, the networked client device may communicate instruction (e.g. input, such as a query sequence, and settings, such as search preferences) with the server device and may receive the method output. The above notwithstanding, the sequencer is typically a more dedicated device and may typically comprise further technical means for performing a sequencing. However, this does not exclude that the sequencer could be configured to also perform one or more further methods (e.g. a sequence assembly); in which case, the sequencer could for example also be referred to as an assembler. Similarly, the sequencer could be part of a distributed computing environment, where e.g. a client-side sequencing unit performs the physical sequencing and communicates with a cloud-based repository of fingerprint data strings.
In an eleventh aspect, a computer program product is described comprising instructions which, when the program is executed by a computer system, cause the computer system to carry out the method according to any embodiments of the second, third, fifth, seventh, eighth or ninth aspect.
In a twelfth aspect, a computer-readable medium is described comprising instructions which, when executed by a computer system, cause the computer system to carry out the method according to any embodiments of the second, third, fifth, seventh, eighth or ninth aspect.
In a thirteenth aspect, use of a library of biological sequence fingerprints as defined in any embodiment of the first aspect is described, for one or more selected from: sequencing a biopolymer or biopolymer fragment, performing a sequence assembly, processing a biological sequence, building a repository of processed biological sequences, comparing a first biological sequence to a second biological sequence, aligning a first biological sequence to a second biological sequence, performing a multiple sequence alignment, performing a sequence similarity search and performing a variant calling.
In a fourteenth aspect, use of a processed biological sequence as defined any embodiment of the fourth aspect or a library of processed biological sequences as defined in any embodiment of the sixth aspect is described, for one or more selected from: comparing a first biological sequence to a second biological sequence, aligning a first biological sequence to a second biological sequence, performing a multiple sequence alignment, performing a sequence similarity search and performing a variant calling.
In embodiments, any feature of any embodiment of any of the above aspects may independently be as correspondingly described for any embodiment of any of the other aspects.
Aspects of certain embodiments will now be described by a detailed description of several embodiments. It is clear that other embodiments of the invention can be configured according to the knowledge of the person skilled in the art without departing from the true technical teaching of the invention, the invention being limited only by the terms of the appended claims.
Two separate searches were performed based on the search string "AVFPSIVGRPRHQGVMVGMGQKDSY". This corresponds to a relatively short protein sequence of length of 25 units, which could for example be a protein fragment in protein sequencing. Such a search could for example be used after sequencing of the fragment as part of identifying a suitable reference sequencing to use in a sequence assembly with the fragment.
https://blast.ncbi.nlm.nih.gov/Blast.cgi?PROGRAM=blastp&PAGE TYPE=BlastSearch& LINK LOC=blasthome).
The first search was performed using BLAST (Basic Local Alignment Search Tool); more particularly 'Protein BLAST' (available at the url: The following search parameters were used: Database = Protein Data Bank proteins (pdb); Algorithm = blastp (protein-protein BLAST); Max target sequences = 1000; Short queries = Automatically adjust parameters for short input sequences; Expect threshold = 20000; Word size = 2; Matrix = PAM30; Compositional adjustment = No adjustment. BLAST required over 30 seconds for this search, after which 604 search results were returned.
On the other hand, based on the principles of the present invention, it was determined that "IVGRPRHQGVM" is a characteristic biological subsequence (i.e. a 'fingerprint') comprised in the above short protein sequence. As such, the second search was performed in a repository of processed biological sequences based on the search string "IVGRPRHQGVM". This repository was based on the same protein database as used in BLAST (i.e. Protein Data Bank; PDB), which had been previously processed using a repository of fingerprint data strings; i.e. characteristic biological subsequences represented by the fingerprint data strings were identified and marked in a set of publicly available biological sequences. This search returned 661 results. In contrast to BLAST, the time frame needed in this case was only 196 milliseconds. As such, even for such a relatively short sequence, it was observed that the present method was able to reduce the required time by a factor of over 150 compared to the known-art method.
Figs. 7
8
9
Fig. 7
Fig. 8
Fig. 9
We now refer to , and , showing the results of both of these searches (BLAST = dotted line; present method = solid line) in terms of their total length (), their Levenshtein distance () and longest common substring (). For each graph, the search results are shown ordered from low to high with respect to the plotted parameter (i.e. total length, Levenshtein distance or longest common substring). Furthermore, one of the search result, namely the protein sequence 5NW4_V (i.e. the first result listed by BLAST), was selected as a reference with respect to which the Levenshtein distance and the longest common substring were calculated. As can be observed in these figures, the present method yielded, across the full range of search results, a smaller variation in total length (characterized by a relative plateau spanning over a significant portion of the results), a considerably lower Levenshtein distance and a considerably larger longest common substring; compared to the BLAST results. The combination of these suggests that the method of the present invention was able to identify results which are more relevant for the performed search.
The previous example was repeated, but this time a complete protein sequence, 3MN5_A (with a length of 359 units), was searched.
The first search, using BLAST, returned 88 search results.
+4641474444415052415646_1, +495647525052485147564d_1,
+4949544e5744444d454b49_1, +494d464554464e5650414d_1,
+494b454b4c435956414c44_1 and +49474d4553414749484554_1,
On the other hand, based on the principles of the present invention, it was determined that six characteristic biological subsequences (i.e. 'fingerprints') could be found in the sequence 3MN5_A; these were denoted as:
where e.g. '49474d4553414749484554' corresponds to the respective subsequence in hexadecimal format. As such, the second search was performed, in the same repository of processed biological sequences as in the previous example, to find those protein sequences which comprise the same six characteristic biological subsequences in the same order. This search returned 661 results.
Fig. 10
11
Fig. 10
Fig. 11
We now refer to , and 12, showing the results of both of these searches (BLAST = dotted line; present method = solid line) in terms of their total length (), their Levenshtein distance () and longest common substring (Fig. 12). For each graph, the search results are shown ordered from low to high with respect to the plotted parameter (i.e. total length, Levenshtein distance or longest common substring). In this case, the Levenshtein distance and the longest common substring were calculated with respect to the original query sequence 3MN5_A. As can be observed in these figures, the characteristics of the search results for both methods are relatively comparable at the extremes. However, the present method yielded in the intermediate range a plateau of results with little variation in total length, a low Levenshtein distance and a fairly high longest common substring. The combination of these suggests that the method of the present invention was able to identify a larger number of relevant results.
It is to be understood that although preferred embodiments, specific constructions and configurations, as well as materials, have been discussed herein for devices according to the present invention, various changes or modifications in form and detail may be made without departing from the scope and technical teachings of this invention. For example, any formulas given above are merely representative of procedures that may be used. Functionality may be added or deleted from the block diagrams and operations may be interchanged among functional blocks. Steps may be added or deleted to methods described within the scope of the present invention. | |
Can you play guitar chords on the ukulele – and vice versa? The guitar and ukulele are similar instruments in many ways. This often leads people to wonder what their differences are and how transferable are chords and melodies between them.
Although the guitar and ukulele are tuned differently, they are tuned to the same intervals. You can convert guitar chords to ukulele chords with some quick transposing.
To convert guitar chords to ukulele, leave out the bottom two strings and transpose the chord up a fourth. So, a D chord on the guitar becomes a G chord on the ukulele, etc. This is even easier for baritone ukuleles – just play the exact same chord shape.
This post goes over an easy method to translate guitar chords to ukulele, so you can play just about any song you want.
Read More:
- Easy Ukulele Chords for Beginners to Learn First
- 57+ Easy Ukulele Songs for Beginners
- 111+ Easy Acoustic Guitar Songs for Beginners
The Difference Between Guitar and Ukulele Chords
Even though the guitar and ukulele look very similar, they use different tunings. Also, most types of ukuleles have four strings, whereas guitars have six strings. As a result, unless you’re playing, say, a guitalele, chords on the ukulele consist of at most four notes instead of six on the guitar.
The number of strings on the ukulele vs guitar also determines the number and complexity of available chord voicings. There’s also a much wider variety of tunings commonly used on the guitar than on the ukulele.
For example, there’s an enormous range of alternate, ‘open’ tunings common on the guitar, like open D, open G, and DADGAD. Chords played with such tunings are often very different from chords played in standard tuning and are sometimes not even possible to play in standard.
However, most guitar and ukulele chords are based on major or minor triads. So, regardless of which instrument you are playing a given chord on, it will have the same basic foundation.
Guitar Tuning vs. Ukulele Tuning
The fact that guitars and ukuleles have different tunings also impacts how chord shapes are constructed and sound on the two instruments. Standard tuning on the uke is one example of re-entrant tuning, which guitars do not use (in re-entrant tunings, the strings are not tuned in pitch order. On the uke, the high g string is tuned an octave higher than it would be in pitch order).
If we disregard the re-entry, though, ukulele standard tuning uses the same intervals as the thinnest four strings on a guitar. The four thinnest strings on the guitar are D-G-B-E, while the ukulele’s standard string order is G-C-E-A, exactly a fourth up. This makes converting guitar chords to ukulele a straightforward process of transposing them up a fourth.
Baritone Ukulele Tuning
The baritone ukulele’s string order is D-G-B-E, the same notes as the top four strings on a guitar. Baritone ukulele tuning is not a re-entrant tuning, unlike standard ukulele tuning.
You can play guitar chords directly on the baritone uke, apart from the notes on the bottom two strings of the guitar. This similarity makes it simple for guitar players to pick up a baritone uke and jam away right off the bat.
Baritone ukes are also larger than soprano, concert, and tenor ukes, which gives them a different sound.
How to Convert Guitar Chords to Ukulele Chords in Standard Tuning
Using the Same Chord Shapes
It’s possible to use many of the same chord shapes used in guitar music as uke chords. All you need to do is use a guitar chord chart and leave out the bottom two strings of the guitar (the E and A strings, which are the two furthest to the left on a chord chart).
Any chord shape you can play on the remaining four strings can be played on the ukulele. Because standard ukulele tuning features the same notes (with a re-entry) as standard guitar tuning, the chords will be the same on the ukulele but transposed up a fourth.
You can use any chord from the guitar on the ukulele – simply transpose it up a fourth.
Of course, the ukulele only has four strings. Most guitar chord shapes use all six strings, so more complex chord voicings can’t be directly converted to the ukulele.
Some Examples of Converting Guitar to Ukulele Chords
To illustrate how to convert guitar chords to ukulele, let’s go over some example chords. These chords feature the same shape on both guitar and ukulele but are transposed up a fourth when played on the uke.
D Major Guitar Chord = G Major Ukulele Chord
The D chord is one of the most widely used basic chords on the guitar. D is a popular key for songwriters, and the D chord is easy to play on the guitar. It’s also a good choice for illustrating how converting guitar chords to ukulele works since you only play a D chord on the guitar using the top four strings.
But if you take that D major guitar chord shape and play it on the ukulele, you get a G chord instead. G is a fourth higher than D, so transposing a D chord up a fourth gives you a G chord.
E Minor Guitar Chord = A Minor Ukulele Chord
E minor is another widely used chord for the guitar, thanks to how easy it is to play and how resonant it sounds with the guitar’s open low E and high e strings.
Playing that same chord shape on a ukulele gets you an A minor ukulele chord.
Remember, we’re leaving out the low E and A strings. If you only play an E minor chord on the top four strings of a guitar, you’ll only be fretting the D string at the second fret. This translates to fretting the high g string at the second fret on the ukulele, which produces an A minor chord.
G Major Guitar Chord = C Major Ukulele Chord
The G major chord is already very straightforward on the guitar. Take away the bottom two strings, and it becomes even more so since it’s predominantly open strings. The great thing about this voicing is that it’s so easy to learn, which is helpful for beginners looking to expand their repertoire of ukulele chords.
If you take that same G major shape and translate it to the uke, you get a C chord. C is a fourth higher than G, so transposing a G chord up a fourth equals a C chord.
Converting Guitar Chords to the Ukulele in Baritone Tuning
If you’re playing a baritone ukulele, you’ll be glad to know that it uses the same chords as a guitar. The baritone ukulele is tuned to D-G-B-E, the same tuning as the top four strings on a guitar.
You can take any chord shape on the top four guitar strings, play that same shape on the baritone ukulele, and it’s the same chord. For example, to play a G chord, use your ring finger to fret the high e string at the third fret. The G chord is the same shape on guitar, but with the addition of the two lower strings.
If you have both a baritone uke and a ukulele that uses standard tuning, you can choose which instrument you want to use when playing guitar chords. Having both instruments available means you can play songs transposed up a fourth or in their original key just by switching ukes.
How to Convert Ukulele Chords to Guitar (plus an easy trick)
If you’re a ukulele player looking for an easy way to play all the ukulele chords you know on a larger, more resonant instrument, there are ways to jump over to the guitar without learning all new chords.
First off, you can convert ukulele chords to the guitar by transposing down a fourth (the reverse of what we’ve covered above). So, a G becomes a D, and so on.
But there’s an even more straightforward way to do this by using a capo.
Putting a capo on the fifth fret of a guitar transposes the strings up a fourth. With a capo on the fifth fret, the top four strings of your guitar match standard ukulele tuning of G-C-E-A. Leaving out the bottom two guitar strings allows you to play ukulele chords on the guitar.
If you do try this easy trick as a ukulele player, be mindful not to strum the bottom two strings by mistake.
Can You Make Your Guitar Sound Like a Ukulele?
Guitar players often wonder if it’s possible to get a ukulele sound out of your guitar. The guitar has different qualities of timbre and tone than the ukulele. So, while you can play the same notes on the guitar as you can on the ukulele, expect to get quite a different sound from the instrument. | https://acousticbridge.com/guitar-chords-to-ukulele-chords/ |
---
abstract: |
We study the symmetric powers of four algebras: $q$-oscillator algebra, $q$-Weyl algebra, $h$-Weyl algebra and $U({\mathfrak
{sl}}_2)$. We provide explicit formulae as well as combinatorial interpretation for the normal coordinates of products of arbitrary elements in the above algebras.
author:
- 'Rafael Díaz [^1] and Eddy Pariguan [^2]'
bibliography:
- 'sqwa.bib'
title: Symmetric quantum Weyl algebras
---
Introduction
============
This paper takes part in the time-honored tradition of studying an algebra by first choosing a “normal” or “standard” basis for it $\mathbb{B}$, and second, writing down explicit formulae and, if possible, combinatorial interpretation for the representation of the product of a finite number of elements in $\mathbb{B}$, as linear combination of elements in $\mathbb{B}$.
This method has been successfully applied to many algebras, most prominently in the theory of symmetric functions (see [@GR]). We shall deal with algebras given explicitly as the quotient of a free algebra, generated by a set of letters $L$, by a number of relations. We choose normal basis for our algebras by fixing an ordering of the set of the letters $L$, and defining $\mathbb{B}$ to be the set of normally ordered monomials, i.e., monomials in which the letters appearing in it respect the order of $L$.
We consider algebras of the form ${{\rm{Sym}}}^{n}(A)$, i.e, symmetric powers of certain algebras. Let us recall that for each $n\in\mathbb{N}$, there is a functor ${{\rm{Sym}}}^{n}\!:\mathbb{C}\mbox{-}
{{\rm{alg}}}\longrightarrow
\mathbb{C}\mbox{-}{{\rm{alg}}}$ from the category of associative $\mathbb{C}$-algebras into itself defined on objects as follows: if $A$ is a $\mathbb{C}$-algebra, then ${{\rm{Sym}}}^{n}(A)$ denotes the algebra whose underlying vector space is the $n$-th symmetric power of $A$:\
${{\rm{Sym}}}^{n}(A)= (A^{\otimes n})/ \langle a_1\otimes
\dots \otimes a_n -a_{\sigma^{-1} (1)}\otimes \dots \otimes
a_{\sigma^{-1}(n)}: a_i\in A, \sigma\in \mathbb{S}_n\rangle ,$ where $\mathbb{S}_n$ denotes the group of permutations on $n$ letters. The product of $m$ elements in ${{\rm{Sym}}}^{n}(A)$ is given by the rule $$\label{PF}
(n!)^{m-1}\prod_{i=1}^{m}\left(\overline{\bigotimes_{j=1}^{n}
a_{ij}}\right) = \sum_{\sigma \in \{{{\rm{id}}}\} \times
\mathbb{S}_n^{m-1}}\overline{\bigotimes_{j=1}^{n}\left(
\prod_{i=1}^{m} a_{i \sigma^{-1}_i(j)}\right)}$$ for all $(a_{ij}) \in A^{[[1,m]]\times[[1,n]]}$. Notice that if $A$ is an algebra, then $A^{\otimes n}$ is also an algebra. $\mathbb{S}_n$ acts on $A^{\otimes n}$ by algebra automorphisms, and thus we have a well defined invariant subalgebra $(A^{\otimes
n})^{\mathbb{S}_n} \subseteq A^{\otimes n}$. The following result is proven in $\cite{DP}$.
The map $s:{{\rm{Sym}}}^{n}(A) \longrightarrow (A^{\otimes n})^{\mathbb{S}_n}$ given by $$s \left(\overline{\bigotimes_{i=1}^{n} a_i}\right)
=\frac{1}{n!} \sum_{\sigma \in \mathbb{S}_n}
\overline{\bigotimes_{i=1}^{n}a_{\sigma^{-1}(i)}},\ \ \mbox{for
all}\ \ a_i\in A$$ defines an algebra isomorphism.
The main goal of this paper is the study of the symmetric powers of certain algebras that may be regarded as quantum analogues of the Weyl algebra. Let us recall the well-known
The algebra $\mathbb{C} \langle x,y \rangle[h]/ \langle yx-xy-h \rangle$ is called the [*Weyl algebra*]{}.
This algebra admits a natural representation as indicated in the
\[rweyl\] The map $\rho:\mathbb{C} \langle x,y \rangle[h]/ \langle yx-xy-h
\rangle \longrightarrow {{\rm{End}}}(\mathbb{C}[x,h])$ given by $\rho(x)(f)=xf$, $\rho(y)(f)=h\frac{\partial f}{\partial x}$, $\rho(h)(f)=hf$ for any $f\in \mathbb{C}[x,h]$ defines a representation of the Weyl algebra.
The symmetric powers of the Weyl algebra have been studied from several point of view in papers such as [@Al],[@EM],[@Etg2],[@Min2]. Our interest in the subject arose from the construction of non-commutative solitonic states in string theory, based on the combinatorics of the annihilation $\frac{\partial}{\partial x}$ and creation $\hat{x}$ operators given in [@EM]. In [@DP] we gave explicit formula, as well as combinatorial interpretation for the normal coordinates of monomials $\partial^{a_1}x^{b_1}\dots\partial^{a_n}
x^{b_n}$. This formulae allow us to find explicit formulae for the product of a finite number of elements in the symmetric powers of the Weyl algebra. Looking at Proposition \[rweyl\] ones notices that the definition of Weyl algebra relies on the notion of the derivative operator $\frac{\partial}{\partial x}$ from classical infinitesimal calculus. The classical derivative $\frac{\partial}{\partial x}$ admits two well-known discrete deformations, the so called $q$-derivative $\partial_q$ and the $h$-derivative $\partial_h$. The main topic of this paper is to introduce the corresponding $q$ and $h$-analogues for the Weyl algebra, and generalize the results established in [@DP] to these new contexts.
For the $q$-calculus, we will actually introduce two $q$-analogues of the Weyl algebra: the $q$-oscillator algebra (section \[qos\]) and the $q$-Weyl algebra (section \[qweyl\]). Needless to say, the $q$-oscillator algebra also known as the $q$-boson algebra [@Sol], and $q$-Weyl algebra are deeply related. The main difference between them is that while the $q$-oscillator is the algebra generated by $\partial_q$ and $\hat {x}$, a third operator, the $q$-shift $s_q$ is also present in the $q$-Weyl algebra. We believe that $s_q$ is as fundamental as $\partial_q$ and $\hat{x}$. The reason it has passed unnoticed in the classical case is that for $q=1$, $s_q$ is just the identity operator. For both $q$-analogues of the Weyl algebra we are able to find explicit formulae and combinatorial interpretation for the product rule in their symmetric powers algebras.
For the $h$-calculus, also known as the calculus of finite differences, we introduce the $h$-Weyl algebra in section \[hweyl\]. Besides the annihilator $\partial_h$ and the creator $\hat{x}$ operators, also includes an $h$-shift operator $s_h$, which again reduces to the identity for $h=0$. We give explicit formulae and combinatorial interpretation for the product rule in the symmetric powers of the $h$-Weyl algebra.
In section \[uea\] we deal with an algebra of a different sort. Since our method has proven successful for dealing with the Weyl algebra (and it $q$ and $h$-deformations); and it is known that the Weyl algebra is isomorphic to the universal enveloping algebra of the Heisenberg Lie algebra, it is an interest problem to apply our constructions for other Lie algebras. We consider here only the simplest case, that of ${\mathfrak {sl}}_2$. We give explicit formulae for the product rule in the symmetric powers of $U({\mathfrak {sl}_2})$.
Although some of the formulae in this paper are rather cumbersome, all of them are just the algebraic embodiment of fairly elementary combinatorial facts. The combinatorial statements will be further analyzed in $\cite{DP1}$.
Notations and conventions {#notations-and-conventions .unnumbered}
-------------------------
- [$\mathbb{N}$ denotes the set of natural numbers. For $x\in
\mathbb{N}^{n}$ and $i\in\mathbb{N}$, we denote by $x_{<i}$ the vector $(x_1,\dots,x_{i-1})\in \mathbb{N}^{i-1}$, by $x_{\leq i}$ the vector $(x_1,\dots,x_{i})\in \mathbb{N}^{i}$, by $x_{>i}$ the vector $(x_{i+1},\dots,x_{n})\in \mathbb{N}^{n-i}$ and by $x_{\geq
i}$ the vector $(x_i,\dots,x_n)\in \mathbb{N}^{n-i+1}$.]{}
- [Given $(U,<)$ an ordered set and $s\in U$, we set $U_{>
s}:=\{u\in U: u>s\}$.]{}
- [ $|\mbox{
}|:\mathbb{N}^{n}\longrightarrow \mathbb{N}$ denotes the map such that $|x|:=\sum_{i=1}^{n} x_i$, for all $x\in \mathbb{N}^{n}$.]{}
- [For a set $X$, $\sharp(X):=$ cardinality of $X$, and $\mathbb{C}\langle X \rangle\!\!:=$ free associative algebra generated by $X$.]{}
- [Given natural numbers $a_1,\dots,a_n\in
\mathbb{N}$, $\min(a_1,\dots,a_n)$ denotes the smallest number in the set $\{a_1,\dots,a_n\}$.]{}
- [Given $n\in\mathbb{N}$, we set $[[1,n]]=\{1,\dots,n\}$.]{}
- [Let $S$ be a set and $A:[[1,m]]\times [[1,n]]\longrightarrow S$ an $S$-valued matrix. For $\sigma \in (\mathbb{S}_n)^{m}$ and $j\in [[1,n]]$, $A_j^{\sigma}:[[1,m]]\longrightarrow S$ denotes the map such that $A_{j}^{\sigma}(i)=A_{i \sigma^{-1}_i(j)}$, for all $i=1,\dots,m.$]{}
- [The $q$-analogue $n$ integer is ${{\displaystyle}[n]:=\frac{1-q^{n}}{1-q}}$. For $k\in \mathbb{N}$, we will use $[n]_{k}:=[n][n-1]\dots [n-k+1]$.]{}
Symmetric $q$-oscillator algebra {#qos}
================================
In this section we define the $q$-oscillator algebra and study its symmetric powers. Let us introduce several fundamental operators in $q$-calculus (see $\cite{Kac}$ for a nice introduction to $q$-calculus).\
The operators $\partial_q, s_q, \hat{x}, \hat{q}, \hat{h}:
\mathbb{C}[x,q,h] \longrightarrow \mathbb{C}[x,q,h]$ are given as follows $$\begin{array}{lclcc}
\partial_q f(x)=\frac{f(qx)-f(x)}{(q-1)x} &\mbox{} & s_q(f)(x)=f(qx)
& \mbox{ } &\hat{q}(f)=qf\\
\hat{x}(f)=xf &\mbox{ } & \hat{h}(f)=hf &\mbox{ } & \\
\end{array}$$ for all $f\in \mathbb{C}[x,q,h]$. We call $\partial_q$ the $q$-derivative and $s_q$ the $q$-shift.
The algebra $\mathbb{C}\langle x,y\rangle[q,h]/I_{qo}$, where $I_{qo}$ is the ideal generated by the relation $yx=qxy+h$ is called the [*$q$-oscillator algebra*]{}.
We have the following analogue of Proposition $\ref{rweyl}$.
\[rqo\] The map $\rho:\mathbb{C}\langle x,y\rangle[q,h]/I_{qo}
\longrightarrow {{\rm{End}}}(\mathbb{C}[x,q,h])$ given by $\rho(x)=\hat{x}$, $\rho(y)=h\partial_q$, $\rho(q)=\hat{q}$ and $\rho(h)=\hat{h}$ defines a representation of the $q$-oscillator algebra.
Notice that if we let $q\rightarrow 1$, $y$ becomes central and we recover the Weyl algebra. We order the letters of the $q$-oscillator algebra as follows: $q<x<y<h$.\
Assume we are given $A=(A_1,\dots,A_n)\in (\mathbb{N}^{2})^{n}$, and $A_i=(a_i,b_i)\in \mathbb{N}^{2}$, for $i\in [[1,n]]$. Set $X^{A_{i}}=x^{a_i}y^{b_i}$, for $i\in [[1,n]]$. We set $a=(a_1,\dots, a_n)\in \mathbb{N}^{n}$, $b=(b_1,\dots, b_n)\in
\mathbb{N}^{n}$, $c=(c_1,\dots, c_n)\in \mathbb{N}^{n}$ and $|A|=(|a|,|b|)\in \mathbb{N}^{2}$. Using this notation we have\
The [*normal coordinates*]{} $ N_{qo}(A,k)$ of ${{\displaystyle}\prod_{i=1}^{n}X^{A_{i}}}\in \mathbb{C}\langle
x,y\rangle[h,q]/I_{qo}$ are given by the identity $$\label{cqo}
{{\displaystyle}\prod_{i=1}^{n}X^{A_{i}}}=
{{\displaystyle}\sum_{k=0}^{\min} N_{qo}(A,k) X^{|A|-(k,k)} h^{k}}$$ where $\min=\min(|a|,|b|)$. For $k>\min$, we set $N_{qo}(A,k)$ to be equal to $0$.
Let us introduce some notation needed to formulate Theorem $\ref{ncqo}$ below which provides explicit formula for the normal coordinates $N_{qo}(A,k)$ of ${{\displaystyle}\prod_{i=1}^{n}X^{A_{i}}}$. Given $(A_1,\dots,A_n)\in (\mathbb{N}^{2})^{n}$ choose disjoint totally ordered sets $(U_i,<_{i})$, $(V_i,<_{i})$ such that $\sharp(U_i)=a_i$ and $\sharp (V_i)=b_i$, for $i\in [[1,n]]$. Define a total order set $(U\cup V\cup\{ \infty\},<)$, where ${{\displaystyle}U=\cup_{i=1}^{n} U_i}$, $V={{\displaystyle}\cup_{i=1}^{n}V_i}$ and $\infty\not\in U\cup V$, as follows: Given $u,v\in U\cup V\cup\{
\infty\}$ we say that $u\leq v$ if and only if a least one of the following conditions hold $$\begin{aligned}
u\in V_i, v\in V_j\ \mbox{and}\ i\leq j; &\mbox{ }&
u\in V_i, v\in U_j\ \mbox{and}\ i\leq j;\\
u\in U_i, v\in V_j \ \mbox{and}\ i\leq j; &\mbox{}&
u,v\in U_i \ \mbox{and}\ u\leq_i v;\\
u,v\in V_i \ \mbox{and}\ u\leq_i v; \mbox{ } \mbox{ } \mbox{ } \mbox{ }
\mbox{ } &\mbox{ }& v=\infty.\end{aligned}$$ Given $k\in \mathbb{N}$, we let $P_{k}(U,V)$ be the set of all maps $p:V\longrightarrow U\cup\{\infty\}$ such that
- [$p$ restricted to $p^{-1}(U)$ is injective, and $\sharp
(p^{-1}(U))=k$.]{}
- [If $(v,p(v))\in V_i\times U_j$ then $i<j$.]{}
Figure $\ref{fg:crnum}$ shows an example of such a map. We only show the finite part of $p$, all other points in $V$ being mapped to $\infty$.
![Combinatorial interpretation of $N_{qo}$.[]{data-label="fg:crnum"}](qosc.eps){height="1.3cm"}
The value of the [*crossing number map*]{} $c:P_{k}(U,V)\longrightarrow \mathbb{N}$ when evaluated on $p\in
P_k(U,V)$ is given by $$c(p)=\sharp (\{(s,t)\in V\times U| s<t<p(s), t\in p(U_{> s})^{c}\}).$$
\[ncqo\] For any $A=(A_1,\dots,A_n)\in (\mathbb{N}^{2})^{n}$ with $A_i=(a_i,b_i)\in\mathbb{N}^{2}$ for $i\in [[1,n]]$ and any $k\in
\mathbb{N}$, we have that $$N_{qo}(A,k)={{\displaystyle}\sum_{p\in P_k(U,V)} q^{c(p)}}.$$
The proof is by induction. The only non-trivial case is the following $$\begin{aligned}
\label{recur} {{\displaystyle}y \prod_{i=1}^{n}
X^{A_i}}& =& {{\displaystyle}\sum_{k=0}^{\min} q^{|a|-k}
N_{qo}(A,k) X^{|A|-(k,k+1)} h^{k}} \nonumber \\
\mbox{} & \mbox{} & +{{\displaystyle}\sum_{k=0}^{\min} N_{qo}(A,k)
\sum_{i=1}^{a-k} q^{i-1} X^{|A|-(k+1,k)}h^{k}}\end{aligned}$$ where $\min=\min(|a|,|b|)$. Normalizing the left-hand side of $(\ref{recur})$ we get a recursive relation $$N_{qo}(((0,1)A),k)=N_{qo}(A,k)+\sum_{i=1}^{a-k}q^{i-1} N_{qo}(A,k-1).$$ The other recursion needed being $N_{qo}(((1,0)A),k)=N_{qo}(A,k)$. It is straightforward to check that $N_{qo}(A,k)={{\displaystyle}\sum_{p\in
P_k(U,V)} q^{c(p)}}$ satisfies both recursions.
Consider the identity $(\ref{cqo})$ in the representation of the $q$-oscillator algebra defined in Proposition $\ref{rqo}$. Apply both sides of the identity $(\ref{cqo})$ to $x^{t}$ for $t\in \mathbb{N}$, and use Theorem $\ref{ncqo}$ to get the fundamental
Given $(t,a,b)\in \mathbb{N}\!\times\! \mathbb{N}^{n}\!\times\! \mathbb{N}^{n}$ the following identity holds $${{\displaystyle}\prod_{i=1}^{n}[t+|a_{>i}|-|b_{>i}|]_{b_i}=\sum_{p\in P_k(U,V)}
q^{c(p)}[t]_{|b|-k} }.$$
Our next theorem gives a fairly simple formula for the product of $m$ elements in ${{\rm{Sym}}}^{n}(\mathbb{C}\langle
x,y\rangle[q,h]/I_{qo})$. Fix a matrix $A:[[1,m]]\!\times\!
[[1,n]]\longrightarrow \mathbb{N}^{2}$, $(A_{ij})=((a_{ij}),(b_{ij}))$. Recall that given $\sigma \in
(\mathbb{S}_n)^{m}$ and $j\in[[1,n]]$, $A_j^{\sigma}$ denotes the vector $(A_{1
\sigma_{1}^{-1}(j)},\dots,A_{m\sigma_{m}^{-1}(j)})\in
(\mathbb{N}^{2})^{m}$ and $X_j^{A_{ij}}=x_j^{a_{ij}}
y_j^{b_{ij}}$. Set\
$|A_j^{\sigma}|=(|a_j^{\sigma}|,
|b_j^{\sigma}|)$ where ${{\displaystyle}|a_j^{\sigma}|=\sum_{i=1}^{m}
a_{i\sigma^{-1}_i(j)}}$ and ${{\displaystyle}|b_j^{\sigma}|=\sum_{i=1}^{m}
b_{i\sigma^{-1}_i(j)}}$.
For any $A:[[1,m]]\times [[1,n]]\longrightarrow
\mathbb{N}^{2}$, the identity $${{\displaystyle}(n!)^{m-1}\prod_{i=1}^{m} \overline{\prod_{j=1}^{n}X_j^{A_{ij}}}=
\sum_{\sigma,k}\left( \prod_{j=1}^{n} N_{qo}(A_j^{\sigma}
,k_j)\right)\overline{\prod_{j=1}^{n}
X_j^{|A_j^{\sigma}|-(k_j,k_j)}}h^{|k|} }$$ where $\sigma\in \{
{{\rm{id}}}\} \times \mathbb{S}_n^{m-1}$ and $k\in\mathbb{N}^{n}$, holds in ${{\rm{Sym}}}^{n}(\mathbb{C}\langle x,y\rangle[q,h]/I_{qo})$.
Using the product rule given in $(\ref{PF})$, the identity $(\ref{cqo})$ and the distributive property we obtain $$\begin{aligned}
(n!)^{m-1}\prod_{i=1}^{m} \overline{\prod_{j=1}^{n}
X_j^{A_{ij}}}&=&\sum_{\sigma \in {\{{{\rm{id}}}\}
\times \mathbb{S}_n^{m-1}}}\prod_{j=1}^{n} \overline{\prod_{i=1}^{m} X_j^{A_{i \sigma_{i}^{-1}(j)}}}\\
&=&\sum_{\sigma \in {\{{{\rm{id}}}\} \times
\mathbb{S}_n^{m-1}}}\overline{\prod_{j=1}^{n} \left( \sum_{k=0}^{\min_j}
N_{qo}(A_j^{\sigma}
,k) x_j^{|a_j^{\sigma}|-k} y_j^{|b_j^{\sigma}|-k} h^{k}\right)}\\
&=& \sum_{\sigma,k}\left( \prod_{j=1}^{n} N_{qo}(A_j^{\sigma}
,k_j)\right)\overline{\prod_{j=1}^{n}
X_j^{|A_j^{\sigma}|-(k_j,k_j)}}h^{|k|}\end{aligned}$$ where $\min_j=\min(|a_j^{\sigma}|,|b_j^{\sigma}|)$.
Symmetric $q$-Weyl algebra {#qweyl}
==========================
In this section we study the symmetric powers of the $q$-Weyl algebra.\
The $q$-Weyl algebra is given by $\mathbb{C}\langle x,y,z
\rangle[q]/I_q$, where $I_q$ is the ideal generated by the following relations:
$$\begin{array}{ccccc} zx=xz+y &\mbox{ } & yx=qxy & \mbox{} &zy=qyz \\
\end{array}$$
We have the following $q$-analogue of Proposition $\ref{rweyl}$.
\[rpq\] The map $\rho:\mathbb{C}\langle x,y,z \rangle[q]/I_q
\longrightarrow {{\rm{End}}}(\mathbb{C}[x,q])$ given by $\rho(x)=\hat{x}$, $\rho(y)=s_q$, $\rho(z)=\partial_q$ and $\rho(q)=\hat{q}$ defines a representation of the $q$-Weyl algebra.
Notice that if we let $q\rightarrow 1$, we recover the Weyl algebra. We order the letters of the $q$-Weyl algebra as follows: $q<x<y<z$. Given $a\in \mathbb{N}$ and $I\subset [[1,a]]$, we define the [*crossing number of $I$*]{} to be $$\chi(I):=\sharp(\{(i,j): i>j, i\in I, j\in I^{c}\}).$$ For $k\in \mathbb{N}$, we let $\chi_k:\mathbb{N}\longrightarrow \mathbb{N}$ the map given by ${{\displaystyle}\chi_k(a)=\sum_{\begin{array}{c}
{\scriptstyle {\sharp (I)=k}} \\
{\scriptstyle {I\subset [[1,a]]}}\\
\end{array}} q^{\chi(I)}}$, for all $a\in \mathbb{N}$. We have the following
\[qfor\] Given $a,b\in \mathbb{N}$, the following identities hold in $\mathbb{C}\langle x,y,z
\rangle[q]/I_q $
1. [$z^{a}x^{b}={{\displaystyle}\sum_{k=0}^{\min}\chi_k(a)[b]_{k} x^{b-k}
y^{k} z^{a-k} }$, where $\min=\min(a,b)$.]{}
2. [$z^{a}y^{b}=q^{ab}y^{b}z^{a}$.]{}
3. [$y^{a}x^{b}=q^{ab}x^{b}y^{a}$.]{}
$2.$ and $3.$ are obvious, let us to prove $1.$ It should be clear that $${{\displaystyle}{z^{a}x^{b}=\sum_{I\subset
[[1,a]]}[b]_{\sharp(I)}x^{b-\sharp(I)}\prod_{j=1}^{a} f_I(j)}},$$ where $f_I(j)=z$, if $j\notin I$ and $f_I(j)=y$, if $j\in I$. The normal form of ${{\displaystyle}\prod_{j=1}^{a} f_I(j)}$ is $q^{\chi(I)}
y^{\sharp(I)} z^{a-\sharp(I)}$. Thus,\
$$z^{a}x^{b}={{\displaystyle}\sum_{k=0}^{\min}\left( \sum_{I}
q^{\chi(I)}\right) [b]_k x^{b-k}y^{k}z^{a-k}
=\sum_{k=0}^{\min}\chi_k(a)[b]_k x^{b-k}y^{k}z^{a-k}},$$ where $\min=\min(a,b)$, $I\subset [[1,a]]$ and $\sharp (I)=k$.
Assume we are given $A=(A_1,\dots,A_n)\in (\mathbb{N}^{3})^{n}$, where $A_i=(a_i,b_i,c_i)$, for $i\in[[1,n]]$, also $X^{A_{i}}=x^{a_i}y^{b_i}z^{c_i}$ for $i\in[[1,n]]$. We set $a=(a_1,\dots,a_n)\in \mathbb{N}^{n}$, $b=(b_1,\dots,b_n)\in
\mathbb{N}^{n}$, $c=(c_1,\dots,c_n)\in \mathbb{N}^{n}$ and $|A|=(|a|,|b|,|c|)\in \mathbb{N}^{3}$. Using this notation, we have the\
The [*normal coordinates*]{} $ N_{q}(A,k)$ of ${{\displaystyle}\prod_{i=1}^{n}X^{A_i}}\in \mathbb{C}\langle x,y,z
\rangle[q]/I_q$ are given via the identity $$\label{cq}
{{\displaystyle}\prod_{i=1}^{n}X^{A_i}= \sum_{k\in \mathbb{N}^{{n-1}} }
N_{q}(A,k) X^{|A|+(-|k|,|k|,-|k|)}}$$ where $k$ runs over all vectors $k=(k_1,\dots,k_{n-1})\in
\mathbb{N}^{n-1}$ such that $0\leq k_i \leq \min (|c_{\leq i}|-|k_{<i}|,a_{i+1})$. We set $N_{q}(A,k)=0$ for $k\in\mathbb{N}^{n-1}$ not satisfying the previous conditions.
Our next theorem follows from Theorem $\ref{qfor}$ by induction. It gives an explicit formula for the normal coordinates $N_{q}(A,k)$ in the $q$-Weyl algebra of ${{\displaystyle}\prod_{i=1}^{n}X^{A_i}}$.
\[nq\] Let $A,k$ be as in the previous definition, we have $$N_{q}(A,k)={{\displaystyle}q^{\sum_{i=1}^{n-1} \lambda(i)}\prod_{j=1}^{n-1}
\chi_{k_j}(|c_{\leq j}|-|k_{<j}|) [a_{j+1}]_{k_{j}}},$$ where $\lambda(i)=b_{i+1}(|c_{\leq i}|-|k_{\leq
i}|)+(a_{i+1}-k_i)(|b_{\leq i}|+|k_{<i}|)$.
Applying both sides of the identity $(\ref{cq})$ in the representation of the $q$-Weyl algebra given in Proposition $\ref{rpq}$ to $x^{t}$ and using Theorem $\ref{nq}$, we obtain the remarkable
For any given $(t,a,b,c)\in\mathbb{N}\!\times\!\mathbb{N}^{n}\!\times\! \mathbb{N}^{n}\!\times\!\mathbb{N}^{n}$, the following identity holds $${{\displaystyle}\prod_{i=1}^{n}q^{\gamma(i)} [t+|a_{>i}|+|c_{>i}|]_{c_i}}
={{\displaystyle}\sum_{k}q^{\beta(k)}\left(
\prod_{j=1}^{n-1}\chi_{k_j}(|c_{\leq
j}|-|k_{<j}|)[a_{j+1}]_{k_j}\right)[t]_{|c|-|k|}}$$ where $k\in
\mathbb{N}^{n-1}$ such that $0\leq k_i \leq \min (|c_{\leq
i}|-|k_{<i}|,a_{i+1})$,\
$\gamma(i)=b_i(t+|a_{>i}|-|c_{>i-1}|)$, and ${{\displaystyle}\beta(k)=\left(\sum_{i=1}^{n-1}\lambda(i)\right)+(|b|+|k|)(t-|c|+|k|)}$.
Our next theorem provides an explicit formula for the product of $m$ elements in the algebra ${{\rm{Sym}}}^{n}(\mathbb{C}\langle x,y,z
\rangle[q]/I_q)$. Fix $A:[[1,m]]\times
[[1,n]]\longrightarrow \mathbb{N}^{3}$, with $(A_{ij})=((a_{ij}),(b_{ij}),(c_{ij}))$. Recall that given $\sigma\in \mathbb{S}_n^{m}$ and $j\in[[1,n]]$, $A_j^{\sigma}$ denotes the vector $(A_{1\sigma_1^{-1}(j)},\dots,
A_{m\sigma_m^{-1}(j)})\in (\mathbb{N}^{3})^{m}$ and set $X_j^{A_{ij}}=x_j^{a_{ij}} y_j^{b_{ij}} z_j^{c_{ij}}$, for $j\in[[1,n]]$. Set $A_j^{\sigma}=(|a_j^{\sigma}|,
|b_j^{\sigma}|,|c_j^{\sigma}|)$, where ${{\displaystyle}|a_j^{\sigma}|=\sum_{i=1}^{m} a_{i\sigma^{-1}_i(j)}}$ and similarly for $|b_j^{\sigma}|$ and $|c_j^{\sigma}|$. We have the following:
\[jo\] For any $A:[[1,m]]\times [[1,n]]\longrightarrow
\mathbb{N}^{3}$, the identity $${{\displaystyle}(n!)^{m-1}\prod_{i=1}^{m} \overline{ \prod_{j=1}^{n} X_j^{A_{ij}}}=
\sum_{\sigma,k}\left( \prod_{j=1}^{n}
N_{q}(A_j^{\sigma},k^{j})\right)\overline{ \prod_{j=1}^{n}
X_j^{|A_j^{\sigma}|+(-|k^{j}|,|k^{j}|,-|k^{j}|)}}}$$ where $\sigma\in\{{{\rm{id}}}\} \times \mathbb{S}_n^{m-1}$ and $k=(k^{1},\dots,k^{n})\in(\mathbb{N}^{m-1})^{n}$, holds in\
${{\rm{Sym}}}^{n}(\mathbb{C}\langle x,y,z \rangle[q]/I_q)$.
$$\begin{aligned}
(n!)^{m-1}\prod_{i=1}^{m} \overline{ \prod_{j=1}^{n}
X_j^{A_{ij}}}&=&\sum_{\sigma \in {\{{{\rm{id}}}\}
\times \mathbb{S}_n^{m-1}}}\prod_{j=1}^{n}\overline{ \prod_{i=1}^{m} X_j^{A_{i \sigma_{i}^{-1}(j)}}}\\
&=&\sum_{\sigma}\overline{\prod_{j=1}^{n} \left(
\sum_{k=0}^{\min_j}
N_{q}(A_j^{\sigma},k) x_j^{|a_j^{\sigma}|-|k|} y_j^{|b_j^{\sigma}|+|k|} z_j^{|c_j^{\sigma}|-|k|} \right)}\\
&=&\sum_{\sigma,k}\left( \prod_{j=1}^{n}
N_{q}(A_j^{\sigma},k^{j})\right)\overline{ \prod_{j=1}^{n}
X_j^{|A_j^{\sigma}|+(-|k^{j}|,|k^{j}|,-|k^{j}|)}},\end{aligned}$$
where $\min_j=\min(|a_j^{\sigma}|,|b_j^{\sigma}|,|c_j^{\sigma}|)$.
Symmetric $h$-Weyl algebra {#hweyl}
==========================
In this section we introduce the $h$-analogue of the Weyl algebra in the $h$-calculus, and study its symmetric powers. A basic introduction to $h$-calculus may be found in $\cite{Kac}$.\
The operators $\partial_h, s_h, \hat{x}, \hat{h}: \mathbb{C}[x,h]
\longrightarrow \mathbb{C}[x,h]$ are given by $$\begin{array}{ccccccc}
\partial_h
f(x)=\frac{f(x+h)-f(x)}{h} & \mbox{ } & s_h(f)(x)=f(x+h) & \mbox{ }& \hat{x}(f)=xf & \mbox{ } &\hat{h}(f)=hf \\
\end{array}$$for $f\in \mathbb{C}[x,h]$. We call $\partial_h$ the $h$-derivative and $s_h$ the $h$-shift.
The [*$h$-Weyl algebra*]{} is the algebra $\mathbb{C}\langle x,y,z
\rangle[h]/I_h$, where $I_h$ is the ideal generated by the following relations: $$\begin{array}{cccccc}
yx=xy+z & \mbox{ } & zx=xz+zh & \mbox{ } & yz=zy \\
\end{array}$$
The map $\rho:\mathbb{C}\langle x,y,z \rangle[h]/I_h
\longrightarrow {{\rm{End}}}(\mathbb{C}[x,h])$ given by $\rho(x)=\hat{x}$, $\rho(y)=\partial_h$, $\rho(z)=s_h$ and $\rho(h)=\hat{h}$ defines a representation of the $h$-Weyl algebra.
Notice that if we let $h\rightarrow 0$, $z$ becomes a central element and we recover the Weyl algebra. We order the letters on the $h$-Weyl algebra as follows $x<y<z<h$. Also, for $a\in\mathbb{N}$ and $k=(k_1,\dots, k_n)\in \mathbb{N}^{n}$, we set ${{\displaystyle}{a\choose
k}:=\frac{a!}{\prod k_i ! (a-|k|)!}}$. With this notation, we have the
\[hwe\] Given $a,b\in \mathbb{N}$, the following identities holds in $\mathbb{C}\langle x,y,z
\rangle[h]/I_h$
1. [$z^{a}x^{b}={{\displaystyle}\sum_{k=0}^{b} {b \choose k} a^{k} x^{b-k} z^{a} h^{k}}$.]{}
2. [$z^{a}y^{b}=y^{b} z^{a}$.]{}
3. [$y^{a}x^{b}={{\displaystyle}\sum_{\scriptstyle{k\in \mathbb{N}^{a}}}
{b \choose k} x^{b-|k|} y^{a-s(k)}z^{s(k)}h^{|k|-s(k)}}$, where $k\in \mathbb{N}^{a}$ is such that\
$0\leq |k| \leq b$ and $s(k)=\sharp (\{i: k_i\neq 0\})$.]{}
$2.$ is obvious and $3.$ is similar to $1.$ We prove $1.$ by induction. It is easy to check that $z^{a}x=xz^{a}+az^{a}h$. Furthermore, $$\begin{aligned}
z^{a}x^{b+1}&=&{{\displaystyle}\sum_{k=0}^{b} {b \choose k} a^{k} x^{b-k} (z^{a}x) h^{k}}\\
\mbox{}&=&{{\displaystyle}\sum_{k=0}^{b} {b \choose k} a^{k} x^{b+1-k} z^{a} h^{k}+
\sum_{k=1}^{b+1} {b \choose k-1} a^{k} x^{b+1-k} z^{a} h^{k}} \\
\mbox{}&=& {{\displaystyle}\sum_{k=0}^{b+1} {b+1 \choose k} a^{k} x^{b+1-k} z^{a} h^{k}}.\end{aligned}$$
Assume we are given $A=(A_1,\dots,A_n)\in (\mathbb{N}^{3})^{n}$ and $A_i=(a_i,b_i,c_i)$, for $i\in[[1,n]]$. Set $X^{A_{i}}=x^{a_i}y^{b_i}z^{c_i}$ for $i\in[[1,n]]$. We set $a=(a_1,\dots,a_n)\in \mathbb{N}^{n}$, $b=(b_1,\dots,b_n)\in
\mathbb{N}^{n}$, $c=(c_1,\dots,c_n)\in \mathbb{N}^{n}$ and $|A|=(|a|,|b|,|c|)\in \mathbb{N}^{3}$. Using this notation, we have the\
The [*normal coordinates*]{} $ N_{h}(A,p,q)$ of ${{\displaystyle}\prod_{i=1}^{n}X^{A_i}}\in \mathbb{C}\langle x,y,z
\rangle[h]/I_h$ are given via the identity $$\label{cnh}
{{\displaystyle}\prod_{i=1}^{n}X^{A_i}}=
{{\displaystyle}\sum_{p,q} N_{h}(A,p,q) X^{r(A,p,q)}h^{|q|+|p|-s(p)} }$$ where the sum runs over all vectors $q=(q_{1},\dots,q_{{n-1}})\in\mathbb{N}^{n-1}$ such that\
$0\leq
q_{j} \leq a_{j+1}$ and $p=(p_{1},\dots,p_{{n-1}})$ with $p_{j}\in
\mathbb{N}^{|b_{\leq j}|-|s(p_{<j})|}$.\
Also, $r(A,p,q)=(|a|-|p|-|q|,|b|-s(p),|c|+s(p))$, where $s(p)=\sum_{j=1}^{n-1} s(p_{j})$. We set $N_{h}(A,p,q)=0$ for $p,q$ not satisfying the previous conditions.
The condition for $p$ and $q$ in the definition above might seem unmotivated. They appear naturally in the course of the proof of Theorem \[ns\] below, which is proved using induction and Theorem \[hwe\].
\[ns\] Let $A,p$ and $q$ be as in the previous definition, we have $$N_h(A,p,q)={{\displaystyle}\prod_{i=1}^{n-1} {a_{i+1}
\choose q_{i}} {a_{i+1}-q_{i} \choose p_{i}}(|c_{\leq i}|+|s(p_{<i})|)^{q_i} }.$$
Figure $\ref{fg:crossing}$ illustrates the combinatorial interpretation of Theorem $\ref{ns}$. As we try to moves the $z$’s or the $y$’s above the ‘$x$’, a subset of the $x$ may get killed. The $z$’s do not die in this process but the $y$’s do turning themselves into $z$’ s.
Our next result provides an explicit formula for the product of $m$ elements in the algebra ${{\rm{Sym}}}^{n}(\mathbb{C}\langle x,y,z
\rangle[h]/I_h)$. Fix $A:[[1,m]]\times
[[1,n]]\longrightarrow \mathbb{N}^{3}$, with $(A_{ij})=((a_{ij}),(b_{ij}),(c_{ij}))$. Recall that given $\sigma\in \mathbb{S}_n^{m}$ and $j\in[[1,n]]$, $A_j^{\sigma}$ denotes the vector $(A_{1\sigma_1^{-1}(j)},\dots,
A_{m\sigma_m^{-1}(j)})\in (\mathbb{N}^{3})^{m}$ and set $X_j^{A_{ij}}=x_j^{a_{ij}} y_j^{b_{ij}} z_j^{c_{ij}}$, for $j\in[[1,n]]$. Set $A_j^{\sigma}=(|a_j^{\sigma}|,
|b_j^{\sigma}|,|c_j^{\sigma}|)$, where ${{\displaystyle}|a_j^{\sigma}|=\sum_{i=1}^{m} a_{i\sigma^{-1}_i(j)}}$ and similarly for $|b_j^{\sigma}|$ and $|c_j^{\sigma}|$.
\[fph\] For any $A:[[1,m]]\times
[[1,n]]\longrightarrow \mathbb{N}^{3}$, the identity $${{\displaystyle}(n!)^{m-1}\prod_{i=1}^{m} \overline{ \prod_{j=1}^{n}
X_j^{A_{ij}}}= \sum_{\sigma,p,q}\left( \prod_{j=1}^{n}
N_{h}(A_j^{\sigma},p^{j},q^{j})\right) \overline{ \prod_{j=1}^{n}
X_j^{r(A_j^{\sigma},p^{j},q^{j})}} h^{k_j(p,q)}}$$ holds in ${{\rm{Sym}}}^{n}(\mathbb{C}\langle x,y,z \rangle/I_h)$, where $\sigma\in
\{{{\rm{id}}}\}\times \mathbb{S}_n^{m-1}$, $p^{j}$, $q^{j}$ are such that $(A_j^{\sigma},p^{j},q^{j})$ satisfy the condition of the definition above. $r(A_j^{\sigma},p^{j},q^{j})=(|a_j^{\sigma}|-|p^{j}|-|q^{j}|,|b_j^{\sigma}|-s(p^{j}),|c_j^{\sigma}|+s(p^{j}))$ and $k_j(p,q)=|q^{j}|-|p^{j}|- s(p^{j})$.
Theorem \[fph\] is proven similarly to Theorem $\ref{jo}$.
Deformation quantization of $({\mathfrak {sl}}_2^{\ast})^{n}/{\scriptstyle
{\mathbb{S}_n}}$ {#uea}
==========================================================================
We denote by ${\mathfrak {sl}}_2$ the Lie algebra of all $2\times
2$ complex matrices of trace zero. ${\mathfrak {sl}}_2^{\ast}$ is the dual vector space. It carries a natural structure of Poisson manifold. We consider a deformation quantization of the Poisson orbifold $({\mathfrak {sl}}_2^{\ast})^{n}/\mathbb{S}_n$. It is proven in [@Kon] that the quantized algebra of the Poisson manifold ${\mathfrak {sl}}_2^{\ast}$ is isomorphic to $U({\mathfrak
{sl}}_2)$ the universal enveloping algebra of ${\mathfrak
{sl}}_2$, after setting the formal parameter $\hbar$ appearing in [@Kon] to be $1$. Thus we regard $(U({\mathfrak {sl}}_2)^{\otimes
n})^{\mathbb{S}_n}
\cong {{\rm{Sym}}}^{n}( U({\mathfrak {sl}}_2))$ as the quantized algebra associated to the Poisson orbifold $({\mathfrak
{sl}}_2^{\ast})^{n}/\mathbb{S}_n$. It is well-known that $U({\mathfrak {sl}}_2)$ can be identified with the algebra $\mathbb{C}\langle x,y,z \rangle/I_{{\mathfrak {sl}}_2}$ where $I_{{\mathfrak {sl}}_2}$ is the ideal generated by the following relations: $$\begin{array}{ccccc}
zx=xz+y & \mbox{ } & yx=xy-2x & \mbox{ } & zy=yz-2z \\
\end{array}$$ The next result can be found in $\cite{CK}$,$\cite{YS}$.
The map $\rho:\mathbb{C}\langle x,y,z \rangle /I_{{\mathfrak
{sl}}_2} \longrightarrow {{\rm{End}}}(\mathbb{C}[x_1,x_2])$ given by $\rho(x)={\displaystyle x_2 \frac{\partial}{ \partial x_1}}$, $\rho(y)={\displaystyle x_1 \frac{\partial}{ \partial x_1}-x_2
\frac{\partial}{ \partial x_2} } $ and $\rho(z)={\displaystyle x_1 \frac{\partial}{ \partial x_2}}$ defines a representation of the algebra $\mathbb{C}\langle x,y,z
\rangle /I_{{\mathfrak {sl}}_2}$.
Given $s,n\in \mathbb{N}$ with $0\leq s \leq n$, the $s$-th elementary symmetric function ${{\displaystyle}\sum_{1\leq
i_1<\dots<i_s\leq n} x_{i_1}\dots x_{i_s}}$ on variables $x_1,\dots,x_n$ is denoted by $e_{s}^{n}(x_1,\dots,x_n)$. For $b\in \mathbb{N}$, the notation $e_{s}^{n}(b):=e_{s}^{n}(b,b-1,\dots, b-n+1)$ we will used. Given $a,n\in \mathbb{N}$ such that $a\leq n$, we set $(a)_n=a(a-1)\dots
(a-n+1)$.
\[ncsl2\] Given $a,b\in \mathbb{N}$, the following identities hold in $\mathbb{C}\langle x,y,z \rangle /I_{{\mathfrak
{sl}}_2}$
1. [$z^{a}x^{b}={\displaystyle}\sum_{s,k} \frac{(a)_k (b)_k}{k!} e_{k-s}^{k}(-a-b+2k)x^{b-k}
y^{s}z^{a-k}$,\
where the sum runs over all $k,s\in \mathbb{N}$ such that $0\leq s\leq k
\leq
\min(a,b)$.]{}
2. [$z^{a}y^{b}={\displaystyle}{\sum_{k=0}^{b} {b\choose
k}(-2a)^{k}y^{b-k}z^{a} }$.]{}
3. [$y^{a}x^{b}={\displaystyle}{\sum_{k=0}^{a} {a\choose
k}(-2b)^{k}x^{b}y^{a-k}}$.]{}
Formula $1.$ is proved by induction. It is equivalent to another formula for the normalization of $z^{a}x^{b}$ given in [@Kac1]. $2.$ and $3.$ are similar to Theorem \[hwe\], part $1$. Formula $2.$ express the fact that as we try to move the $z$’s above the $y$’s some of the $y$’s may get killed. This argument justify the ${b\choose k}$ factor. The $a^{k}$ factor arises from the fact that each ‘$y$’ may be killed for any of the $z$’s. The $(-2)^{k}$ factor follows from the fact that each killing of a ‘$y$’ is weighted by a $-2$.
Assume we are given $A=(A_1,\dots,A_n)\in (\mathbb{N}^{3})^{n}$ and $A_i=(a_i,b_i,c_i)$, for $i\in[[1,n]]$. Set $X^{A_{i}}=x^{a_i}y^{b_i}z^{c_i}$ for $i\in[[1,n]]$, furthermore set $a=(a_1,\dots,a_n)\in \mathbb{N}^{n}$, $b=(b_1,\dots,b_n)\in
\mathbb{N}^{n}$, $c=(c_1,\dots,c_n)\in \mathbb{N}^{n}$ and $|A|=(|a|,|b|,|c|)\in \mathbb{N}^{3}$. Using this notation, we have the\
The [*normal coordinates*]{} $ N_{\mathfrak{sl}_2}(A,k,s,p,q)$ of ${{\displaystyle}\prod_{i=1}^{n}X^{A_{i}}}$ in the algebra ${{\displaystyle}\mathbb{C}\langle x,y,z
\rangle /I_{{\mathfrak {sl}}_2}}$ are given via the identity $$\label{nc}
{{\displaystyle}\prod_{i=1}^{n}X^{A_{i}}= \sum_{k,s,p,q}
N_{\mathfrak{sl}_2}(A,k,s,p,q) X^{r(A,k,s,p,q)}}$$ where $k,s,p,q\in \mathbb{N}^{n-1}$ are such that $0\leq s_i\leq
k_i\leq \min(|c_{\leq i}|-|k_{<i}|,a_{i+1})$,\
$0\leq p_i \leq
b_{i+1}$, $0\leq q_i\leq |b_{\leq i}|+|s_{<i}|-|p_{<i}|-|q_{<i}|$, for $i\in[[1,n-1]]$. Moreover, $r(A,k,s,p,q)=(|a|-|k|,|b|+|s|-|p|-|q|,|c|-|k| ).$ We set $N_{\mathfrak{sl}_2}(A,k,s,p,q)=0$ for $k,s,p,q$ not satisfying the previous conditions.
Theorem \[cnsl2\] provides an explicit formula for the normal coordinates $N_{\mathfrak{sl}_2}(A,k,s,t)$ of ${{\displaystyle}\prod_{i=1}^{n}X^{A_{i}}}$. Its proof goes by induction using Theorem \[ncsl2\].
\[cnsl2\] With the notation of the definition above, we have $$N_{\mathfrak{sl}_2}(A,k,s,p,q )= {{\displaystyle}(-2)^{|p|+|q|}
\prod_{i=1}^{n-1}\alpha_i \beta_i \gamma_i } {b_{i+1}\choose p_i} (|c_{\leq i}|-|k_{\leq
i}|)^{{p_i}}(a_{i+1}-k_i)^{q_i},$$ where $\alpha_i={{\displaystyle}\frac{(|c_{\leq i}|- |k_{<i}|)_{k_i} (a_{i+1})_{k_i}}{k_i!}}$, ${{\displaystyle}\beta_i= e_{k_i-s_i}^{k_i}(-a_{i+1}- |c_{\leq i}|+|k_{<
i}|+2k_i)}$,\
and ${{\displaystyle}\gamma_i={{|b_{\leq
i}|+|s_{<i}|-|p_{<i}|-|q_{<i}|}\choose q_{i}}}$, for all $i\in[[1,n-1]]$.
Our final result provides an explicit formula for the product of $m$ elements in the algebra ${{\rm{Sym}}}^{n}(\mathbb{C}\langle x,y,z
\rangle /I_{{\mathfrak {sl}}_2})$. Fix $A:[[1,m]]\times
[[1,n]]\longrightarrow \mathbb{N}^{3}$, with $(A_{ij})=((a_{ij}),(b_{ij}),(c_{ij}))$. Recall that given $\sigma\in \mathbb{S}_n^{m}$ and $j\in[[1,n]]$, $A_j^{\sigma}$ denotes the vector $(A_{1\sigma_1^{-1}(j)},\dots,
A_{m\sigma_m^{-1}(j)})\in (\mathbb{N}^{3})^{m}$. Set $X_j^{A_{ij}}=x_j^{a_{ij}} y_j^{b_{ij}} z_j^{c_{ij}}$, for $j\in[[1,n]]$ and $|A_j^{\sigma}|=(|a_j^{\sigma}|,
|b_j^{\sigma}|,|c_j^{\sigma}|)$, where ${{\displaystyle}|a_j^{\sigma}|=\sum_{i=1}^{m} a_{i\sigma^{-1}_i(j)}}$ and similarly for $|b_j^{\sigma}|$, $|c_j^{\sigma}|$, and $k,s,p,q\in
(\mathbb{N}^{m-1})^{n}$.
\[fpsl2\] For any $A:[[1,m]]\times
[[1,n]]\longrightarrow\mathbb{N}^{3}$, the identity $${{\displaystyle}(n!)^{m-1}\prod_{i=1}^{m} \overline{ \prod_{j=1}^{n}
X_j^{A_{ij}}}= \sum_{\sigma,k,s,p,q}\left( \prod_{j=1}^{n}
N_{\mathfrak{sl}_2}(A_j^{\sigma},k^{j},s^{j},p^{j},q^{j})\right)
\overline{
\prod_{j=1}^{n} X_j^{r_j(A_j^{\sigma},k^{j},s^{j},p^{j},q^{j})}} }$$ holds in ${{\rm{Sym}}}^{n}(\mathbb{C}\langle x,y,z \rangle /I_{{\mathfrak
{sl}}_2})$, where $k=(k^{1},\dots,k^{n})\in(\mathbb{N}^{m-1})^{n}$, and similar for $s,p,q$, $r_j(A_j^{\sigma},k^{j},s^{j},p^{j},q^{j})=(|a_j^{\sigma}|-|k^{j}|,
|b_j^{\sigma}|+|s^{j}|-|p^{j}|-|q^{j}|,|c_{j}^{\sigma}|-|k^{j}|$, and $\sigma\in
\{{{\rm{id}}}\}\times \mathbb{S}_n^{m-1}$.
Theorem \[fpsl2\] is proven similarly to Theorem \[jo\].
Acknowledgment {#acknowledgment .unnumbered}
--------------
We thank Nicolas Andruskiewitsch, Sylvie Paycha and Carolina Teruel. We also thank the organizing committee of Geometric and Topological Methods for Quantum Field Theory, Summer School 2003, Villa de Leyva, Colombia.
$$\begin{array}{c}
\mbox{Rafael D\'\i az. Instituto Venezolano de Investigaciones Cient\'\i ficas.} \ \mbox{\texttt{[email protected]}} \\
\!\!\!\!\!\mbox{Eddy Pariguan. Universidad central de Venezuela.} \ \mbox{\texttt{[email protected]}} \\
\end{array}$$
[^1]: Work partially supported by UCV.
[^2]: Work partially supported by FONACIT.
| |
Q:
Solving A Quadratic Equation by Factoring
How to solve the following quadratic equation by factoring?
And what to do when the 2nd degree term has a non-1 coefficient?
$$
3x^2 + 11x - 4 = 0
$$
Essentially, what I'm asking is how can I factor $3x^2 + 11x - 4$ as the product of two linear expressions?
A:
$$3x^2+11x-4=(3x-1)(x+4).$$
Can you agree ?
| |
Predicting and understanding weather has become crucial in a number of industries, including agriculture, autonomous driving, aviation, or the energy sector. For example, weather conditions play a significant role for aviation and logistics companies in planning the fastest and safest route. Similarly, renewable energy companies need to be able to predict the amount of energy they will produce at a given day. As a consequence various weather models have been developed and are being applied all over the world. Unfortunately, these models often require highly specific information about the atmosphere and exact conditions.
For this reason, Meteomatics, a weather API that delivers fast, direct and simple access to an extensive range of global weather, climate projections and environmental data, has reached out to us for help. Their goal: To predict precipitation accurately in regions where data is sparse and they have to rely on satellite imagery. In this blog post, we show how we developed a neural network to predict the amount of rainfall in a given region based on infrared satellite data.
This is part one of a two-part blog:
If you have ever worked with neural networks you know that they can be data hungry. For this reason it’s crucial to set up a data pipeline that allows you to collect, manage, and understand the assembled data. Our collaboration partner, Meteomatics, offers an easy-to-use API which enables us to quickly gather training and ground-truth data. For example, to get an infra-red picture of Europe (coordinates from 65, -15 to 35, 20) on the seventh of July 2021 and at a resolution of 800x600 pixels we can simply make the following query:
We ran a Python script every quarter hour for a few days collecting infra-red images over Europe, North America, and Mexico at different wavelengths. We then locally combined the different images for each timestamp into an RGB image. To make the task easier we masked out stratiform precipitation in a first step. However, as we will see later on, this only has a small effect on the accuracy of the model. We also collected ground-truth data for training and evaluating the accuracy of our model. Note that ground-truth data was only available for Europe and North America. Below you can see a pair of input and ground-truth data over Europe:
Following the notorious “garbage in garbage out” mantra, we wanted to understand and curate the collected data before we trained a machine learning algorithm on it. For this, we used our free-to-use exploration tool Lightly.ai. Lightly enables quick and easy ways to analyze a dataset as well as more in-depth algorithms to pick the most relevant training points. After uploading our dataset to Lightly we immediately noticed a crucial property of the collected data: The images over Europe, North America, and Mexico were visually and semantically separated. This resulted in a simple strategy to test the generalization capacity of the algorithm: If we trained it on the data from Europe and it performed well on unseen data from North America and Mexico, the algorithm would generalize well. Note that if we had picked the training dataset and the test dataset to be very similar, then all we would test is the memory of the neural network.
Another key insight we gained was that there were many small clusters of extremely similar images. This is due to the fact that we collected data over a relatively short period of time. Because of this, there were a lot of similar images in the dataset which made it harder for the model to generalize well. Lightly helped us with removing these redundancies with a method called “coreset sampling” which aims to maximize the diversity of the dataset.
Before curating the dataset with Lightly, we had 1158 images in our training dataset (Europe). After data curation, we are left with 578 images. The validation dataset (North America) consists of 1107 images and the test dataset (Mexico) consists of only 43 images as we began data collection later.
We download the images from Lightly and and now we are ready to do some machine learning. Head to Part 2 to see the results! | https://www.lightly.ai/post/predicting-rain-from-satellite-images-part-1 |
The following are the current most viewed articles on Wikipedia within Wikipedia's 1410s category. Think of it as a What's Hot list for 1410s. More info »
This is a beta release and so the figures may be a day or two out of date. We'd love to get your thoughts.
|Rank||Topic||Wikipedia views|
|1||Ming–Kotte War||34|
|2||1410s||29|
|3||Samogitian uprisings||21|
|4||1410s in art||7|
|5||1410s in poetry||6|
This category has the following 19 subcategories, out of 19 total.
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cont.
A
B
C
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D
E
T
W
The following 4 pages are in this category, out of 4 total. This list may not reflect recent changes (learn more).
|This page uses content from the English language Wikipedia. The original content was at Category:1410s. The list of authors can be seen in the page history. As with this Familypedia wiki, the content of Wikipedia is available under the Creative Commons License.|
This category has the following 8 subcategories, out of 8 total.
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B
C
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C cont.
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C cont.
D
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This category contains only the following page. | http://top-topics.thefullwiki.org/1410s |
Open Area: All of South Dakota except the following:
- Sand Lake National Wildlife Refuge in Brown County, Renziehausen Game Production Area and Game Bird Refuge in Brown and Marshall Counties, Gerken Game Bird Refuge in Faulk County and White Lake Game Bird Refuge in Marshall County are open Dec. 7, 2015-Jan. 3, 2016.
Daily Limit: 3 rooster pheasants
Possession Limit: 15 rooster pheasants, taken according to the daily limit. The limit accrues at the rate of 3 birds a day, and 15 birds may not be possessed until after the fifth day of hunting.
Shooting Hours: 12 Noon, Central Time for the first 7 days of the season; 10 a.m., Central Time, to sunset the rest of the season. NOTE: Central Time is used for opening shooting hours statewide.
Future Opening Dates: The pheasant season traditionally opens on the third Saturday in October. | https://www.pheasanthuntingseasons.com/2016/06/south-dakota-2015-pheasant-hunting.html |
Given:
Oil gravity = 30°API, 0.875 S.G.
Oil flow rate = 5,000 bpd
Inlet oil temperature = 80°F
Water S.G. = 1.04
Inlet BS&W =10%
Outlet BS&W =1%
Solution:
1. Settling Equation. Investigate treating at 80°F, 100°F, 120°F.
2. Retention Time Equation. Plot computations of d and Leff with retention times less than 20 minutes.
The shaded area of Figure 6-13 represents combinations of d and Leff with tr less than 20 minutes.
3. Heat Required
Substituting treating temperature values of 80°F, 100°F, and 120°F and substituting initial oil temperature value of 80°F will yield values of heat required of 0, 0.86, and 1.72 MMBtu/h.
4. Selection. Choose any combination of d and Leff that is not in the shaded area. Read corresponding treating temperature.
Example solutions are:
An economical solution would be a 72-inch-diameter treater with a 20-foot coalescing section and a 0.86-MMBtu/h firetube capacity. Given the nature of empirical design procedures, crude could possibly be treated at 80°F. The additional firetube capacity will allow a temperature of 100°F if required by field conditions. | http://www.oilngasprocess.com/oil-handling-surfacefacilities/crude-oil-treating-system/treater-vessels-sizing/example-sizing-a-horizontal-treater.html |
In my experience, students often hit a roadblock when they see the word asymptote. What is an asymptote anyway? How do you find them? Is this going to be on the test??? (The answer to the last question is yes. Asymptotes definitely show up on the AP Calculus exams).
Of the three varieties of asymptote — horizontal, vertical, and oblique — perhaps the oblique asymptotes are the most mysterious. In this article we define oblique asymptotes and show how to find them.
What is an Oblique Asymptote?
An oblique (or slant) asymptote is a slanted line that the function approaches as x approaches ∞ (infinity) or -∞ (minus infinity). Let’s explore this definition a little more, shall we?
It’s All About the Line
Since all non-vertical lines can be written in the form y = mx + b for some constants m and b, we say that a function f(x) has an oblique asymptote y = mx + b if the values (the y-coordinates) of f(x) get closer and closer to the values of mx + b as you trace the curve to the right (x → ∞) or to the left (x → -∞), in other words, if there is a good approximation,
f(x) ≈ mx + b,
when x gets extremely large in the positive or negative sense.
Still with me? I understand completely if you’re still a little lost, but let’s see if we can clear up some confusion using the graph shown below.
As you can see, the function (shown in blue) seems to get closer to the dashed line. Therefore, the oblique asymptote for this function is y = ½ x – 1.
Finding Oblique Aymptotes
A function can have at most two oblique asymptotes, but only certain kinds of functions are expected to have an oblique asymptote at all. For instance, polynomials of degree 2 or higher do not have asymptotes of any kind. (Remember, the degree of a polynomial is the highest exponent on any term. For example, 10x3 – 3x4 + 3x – 12 has degree 4.)
As a quick application of this rule, you can say for sure without any work that there are no oblique asymptotes for the quadratic function f(x) = x2 + 3x – 10, because it’s a polynomial of degree 2.
On the other hand, some kinds of rational functions do have oblique asymptotes.
Rational Functions
A rational function has the form of a fraction, f(x) = p(x) / q(x), in which both p(x) and q(x) are polynomials. If the degree of the numerator (top) is exactly one greater than the degree of the denominator (bottom), then f(x) will have an oblique asymptote.
So there are no oblique asymptotes for the rational function, .
But a rational function like does have one. Knowing when there is a horizontal asymptote is just half the battle. Now how do we find it? This next step involves polynomial division.
Polynomial Division to Find Oblique Asymptotes
If you’ve made it this far, you probably have seen long division of polynomials, or synthetic division, but if you are rusty on the technique, then check out this video or this article.
The idea is that when you do polynomial division on a rational function that has one higher degree on top than on the bottom, the result always has the form mx + b + remainder term. Then the oblique asymptote is the linear part, y = mx + b. We don’t need to worry about the remainder term at all.
Example Using Polynomial Division
Let’s see how the technique can be used to find the oblique asymptote of .
The long division is shown below.
Because the quotient is 2x + 1, the rational function has an oblique asymptote:
y = 2x + 1.
Hyperbolas
Another place where oblique asymptotes show up is in the graphs of hyperbolas. Remember, in the simplest case, a hyperbola is characterized by the standard equation,
The hyperbola graph corresponding to this equation has exactly two oblique asymptotes,
The two asymptotes cross each other like a big X.
Example Involving a Hyperbola
Let’s find the oblique asymptotes for the hyperbola with equation x2/9 – y2/4 = 1.
In the given equation, we have a2 = 9, so a = 3, and b2 = 4, so b = 2. This means that the two oblique asymptotes must be at y = ±(b/a)x = ±(2/3)x.
More General Hyperbolas
It’s important to realize that hyperbolas come in more than one flavor. If the hyperbola has its terms switched, so that the “y” term is positive and “x” term is negative, then the asymptotes take a slightly different form. Furthermore, if the center of the hyperbola is at a different point than the origin, (h, k), then that affects the asymptotes as well. Below is a summary of the various possibilities.
Final Thoughts
So when you see a question on the AP Calculus AB exam asking about oblique asymptotes, don’t forget:
- If the function is rational, and if the degree on the top is one more than the degree on the bottom: Use polynomial division.
- If the graph is a hyperbola with equation x2/a2 – y2/b2 = 1, then your asymptotes will be y = ±(b/a)x. Other kinds of hyperbolas also have standard formulas defining their asymptotes.
Keeping these techniques in mind, oblique asymptotes will start to seem much less mysterious on the AP exam! | https://magoosh.com/hs/ap/oblique-asymptotes/ |
What is the biggest mountain of the world?
Mount Everest
What is technically the tallest mountain on Earth?
The world’s tallest mountain technically is not Mount Everest. Mount Everest is the tallest mountain above sea level, but if we’re talking sheer height here, base to summit, then the tallest mountain is Mauna Kea on the island of Hawaii. Here’s how it breaks down: Everest stands 29,035 feet above sea level.
What are the 7 highest summits in the world?
This creates several possible versions of the seven summits:
- Everest, Aconcagua, Denali, Kilimanjaro, Vinson, Elbrus, Puncak Jaya (the Messner version)
- Everest, Aconcagua, Denali, Kilimanjaro, Vinson, Elbrus, Kosciuszko (the Bass version)
Is k2 the highest mountain in the world?
K2 (Urdu: کے ٹو, Kai Ṭū), also known as Mount Godwin-Austen or Chhogori (Balti and Urdu: چھوغوری, Chinese: 乔戈里峰), at 8,611 metres (28,251 ft) above sea level, is the second highest mountain in the world, after Mount Everest at 8,848 metres (29,029 ft).
|K2|
|Easiest route||Abruzzi Spur|
17 more rows
What is higher than Mt Everest?
The summit of Mount Everest is higher above sea level than the summit of any other mountain, but Mauna Kea is the tallest when measured from base to summit. When measuring mountain summits by their height above sea level, Mount Everest is the highest mountain in the world.
How many miles is Mount Everest height?
Everest: 142. Height: 29,028 feet, or 5 and a half miles above sea level. This is equivalent to the size of almost 20 Empire State Buildings. Location: part of the Himalaya mountain range; straddles border of Nepal and Tibet.
Has every mountain in the world been climbed?
The mountain most widely claimed to be the highest unclimbed mountain in the world in terms of elevation is Gangkhar Puensum (7,570 m (24,840 ft)). In Bhutan, the climbing of mountains higher than 6,000 m (20,000 ft) has been prohibited since 1994.
Is Mount Everest a peak or mountain?
8,848 m
Is Chimborazo higher than Everest?
The summit of Chimborazo is 20,564 feet above sea level. However, due to the Earth’s bulge, the summit of Chimborazo is over 6,800 feet farther from the center of the Earth than Everest’s peak. That makes Chimborazo the closest point on Earth to the stars.
Which of the 7 summits is easiest?
Seven Summits. The ‘Seven Summits’ are comprised of the highest mountains on each of the seven continents of the Earth; Everest, Aconcagua, Denali, Kilimanjaro, Elbrus, Vinson Massif and Carstensz Pyramid.
How many summits are there on Everest?
The list consists of people who reached the summit of Mount Everest more than once. By 2013, 6,871 summits have been recorded by 4,042 different people. Despite two hard years of disaster (2014 and 2015), by the end of 2016 there were 7,646 summits by 4,469 people.
Is Everest safe to climb?
The first recorded deaths on the mountain were the seven porters who perished in an avalanche in the 1922 British Mount Everest Expedition. While dangerous for the novice climber, the mountain has also claimed the lives of some of the most experienced climbers.
Why is Mount Everest called k2?
K2, Chinese Qogir Feng, also called Mount Godwin Austen, called locally Dapsang or Chogori, the world’s second highest peak (28,251 feet [8,611 metres]), second only to Mount Everest. Montgomerie of the Survey of India, and it was given the symbol K2 because it was the second peak measured in the Karakoram Range.
Why are the Himalayas so tall?
The collision was so violent that India’s plate did not just crumple, it pushed under Asia – raising the land mass high into the sky. Plates collide all around the world, but what happened below Everest is unique. The towering Himalayas are the result. But this is not the only reason why Everest is so tall.
Is k2 more dangerous than Everest?
Although Everest is 237m taller, K2 is widely perceived to be a far harder climb. “It’s a very serious and very dangerous mountain,” adds Sir Chris. “No matter which route you take it’s a technically difficult climb, much harder than Everest.
How can you tell how tall you will be?
Here’s a popular example: Add the mother’s height and the father’s height in either inches or centimeters. Add 5 inches (13 centimeters) for boys or subtract 5 inches (13 centimeters) for girls. Divide by two.
Was Mount Everest always the tallest mountain?
Despite being the highest peak on earth, Everest is NOT the tallest mountain? No – at 8,848 m (29,029 ft), Everest is the highest mountain on Earth – in that it reaches the highest altitude – but the tallest is actually Mauna Kea in Hawaii, USA.
Can Everest be seen from India?
Tiger Hill (2,567 m) is located in Darjeeling, in the Indian State of West Bengal, and is the summit of Ghoom, the highest railway station in the Darjeeling Himalayan Railway – a UNESCO World Heritage Site. It has a panoramic view of Mount Everest and Mount Kangchenjunga together.
Who reached Everest first?
Edmund Hillary
Tenzing Norgay
Who was the youngest person to climb Mount Everest?
Youngest summiters
|Record name||Record||Owner|
|Youngest person to climb Mount Everest||13 years 10 months 10 days old||Jordan Romero|
|Youngest person to climb Mount Everest (2003-2010)||13 years 11 months 15 days old||Malavath Purna (female)|
|Youngest person to climb Mount Everest (2001-2003)||16-year 14 days old||Temba Tsheri|
1 more row
Who was first on Everest?
Edmund Hillary
Tenzing Norgay
Which mountain has killed the most climbers?
Contents
- 1 Mount Everest.
- 2 K2.
- 3 Kangchenjunga.
- 4 Lhotse.
- 5 Makalu.
- 6 Cho Oyu.
- 7 Dhaulagiri I.
- 8 Manaslu.
How many bodies are on Mt Everest?
While these two bodies were removed, scores have not been. More than 200 bodies dot the mountain, according to Smithsonian. Some of them are there per their final wishes. Many climbers wish to remain on the mountain should they perish, much like a captain going down with his ship, BBC reported.
What is the highest city in the world?
Settlements below 4,500 metres (14,800 ft)
|Elevation||Name||Comment|
|3,650 metres (11,980 ft)||La Paz||Population 758,845 (2012)Bolivian seat of government; the highest capital city in the world.|
|3,548 metres (11,640 ft)||Sicuani||Population 42,551 (2007).|
59 more rows
Is Denali taller than Everest?
The highest mountain is determined by measuring a mountain’s highest point above sea level. The tallest mountain is measured from base to summit. Using that measurement, Denali is taller than Mount Everest. Actually, the mountain is growing by about .04 inches (1 millimeter) per year, according to NASA.
How big is the top of Mount Everest?
The 8,848 m (29,029 ft) height given is officially recognised by Nepal and China, although Nepal plans a new survey. In 1856, Andrew Waugh announced Everest (then known as Peak XV) as 8,840 m (29,002 ft) high, after several years of calculations based on observations made by the Great Trigonometric Survey.
Who all climbed Mount Everest?
Edmund Hillary
Tenzing Norgay
Is it possible get taller?
A common height myth is that certain exercises or stretching techniques can make you grow taller. Many people claim that activities like hanging, climbing, using an inversion table and swimming can increase your height. They may lead to small temporary changes in height, but these effects are not lasting.
Is Chimborazo active?
Chimborazo had been thought to be extinct, but new studies have shown that it still is an active volcano. Although there are no historical eruptions, Chimborazo erupted at least 7 times during the past 10,000 years. These eruptions produced pyroclastic surges that reached down to 3800 m elevation.
Is Chimborazo the tallest mountain?
With a peak elevation of 6,263 m (20,548 ft), Chimborazo is the highest mountain in Ecuador. Chimborazo is not the highest mountain by elevation above sea level, but its location along the equatorial bulge makes its summit the farthest point on the Earth’s surface from the Earth’s center. | https://the-biggest.net/buildings/what-is-the-biggest-mountain-in-the-world.html |
Imagine you’re developing a video game where the player has to find magical items, build a weapon, and then attack monsters. This would be relatively straight-forward to create. Just throw some magical items around the map, let the player move around to discover them, and leave it to the player to figure out what combination to put together to build a powerful weapon. Easy.
But, what about the NPC? Once we throw a computer-controlled player into the mix, things get a little complicated. How would you tell the computer which items to collect? Should it just move randomly around, pick up random items that it happens to come across, and build whatever weapon that happens to make? It certainly wouldn’t make the most challenging of NPC characters.
There is a more intelligent approach. Through the use of artificial intelligence planning, you can program the computer to formulate a plan. For “easy” difficulty, the computer could have the goal of building a club. For “hard”, the computer could build a bazooka. But, how do we give the computer this kind of planning intelligence?
In this tutorial, we’ll learn about STRIPS artificial intelligence AI planning. We’ll cover how to create a world domain and various problem sets, to provide the computer with the intelligence it needs to make a plan and execute it, effectively providing a much better gaming experience.
What is STRIPS?
The Standford Research Institute Problem Solver (STRIPS) is an automated planning technique that works by executing a domain and problem to find a goal. With STRIPS, you first describe the world. You do this by providing objects, actions, preconditions, and effects. These are all the types of things you can do in the game world.
Once the world is described, you then provide a problem set. A problem consists of an initial state and a goal condition. STRIPS can then search all possible states, starting from the initial one, executing various actions, until it reaches the goal.
A common language for writing STRIPS domain and problem sets is the Planning Domain Definition Language (PDDL). PDDL lets you write most of the code with English words, so that it can be clearly read and (hopefully) well understood. It’s a relatively easy approach to writing simple AI planning problems.
What can STRIPS do?
A lot of different problems can be solved using STRIPS and PDDL. As long as the world domain and problem can be described with a finite set of actions, preconditions, and effects, you can write a PDDL domain and problem to solve it.
For example, stacking blocks, Rubik’s cube, navigating a robot in Shakey’s World, Starcraft build orders, and a lot more, can be described using STRIPS and PDDL.
Creating a Domain
Let’s start with the example game that we began to describe above. Suppose our world is filled with ogres, trolls, dragons, and magic! Various elements are scattered in caves. Mixing them together creates new magic spells for the player. Let’s see how we can describe this problem in PDDL for an artificial intelligence planning implementation.
We’ll start by defining the domain. All PDDL programs begin with a skeleton layout, as follows:
That’s it. So far, pretty easy. Let’s add a description of the different kinds of “things” in our world.
What we’ve defined in the above PDDL code is five types of things for our domain. We’ll have players and locations, obviously. A player is the user or computer-controlled character. A location is a specific place on the map, such as an area, country, or region. We’ll use this to separate our map into specific regions.
We’ve also defined a “monster” type, which will describe enemies that might be guarding treasure or a region. Also, an “element” type. This will be our ingredient type for making magic spells! Finally, a “chest” type. This is basically a treasure chest that our ingredients will sit inside.
So, the idea is that the player or NPC must visit different areas of the world, find treasure chests, defeat any monsters guarding them, open the treasure chests, collect ingredients, mix them together, build a powerful weapon, then attack the other players. Whew, that sure is a lot! Good thing we have artificial intelligence planning to help us.
Creating Domain Actions
Let’s define a simple action for our domain. We’ll create a “move” action that allows the player to move from one location to another, as long as they border each other.
An action is defined by using the :action command. You then specify any parameters used in the action, which in our case, we’ll need to specify a player and two locations (the current location and the new location). We then specify a precondition. This sets the rules for when this action is valid, given the parameters. For example, we’ll only allow moving to a new location if it borders the player’s current location. Otherwise, it’s too far away. We’ll also only allow moving to a location that is not currently guarded by a monster. If it is, the player will have to attack the monster first.
Preconditions are specified by using simple logical phrases. The phrase “and (at ?p ?l1)” simply means that the player must currently be at location 1. The phrase “border ?l1 ?l2” means that the two locations must border each other. Likewise “not (guarded ?l2)” means that the second location has to be free and clear of monsters.
Let’s Move Around in the World
Now that we have a simple STRIPS PDDL artificial intelligence planning domain, we can test it out with a simple AI planning problem. First, let’s describe our world with a basic problem and ask the AI to figure out the steps to move the player from a starting location to a goal location.
Creating a STRIPS Problem
The above STRIPS AI planning problem, uses the domain that we’ve designed above (containing a “move” action command) to define our world. We’re simply included an NPC and three locations. There is a town where the NPC starts. There is also a field and a castle. The goal, defined by the “:goal” directive, is to have the NPC move to the castle. Seems easy enough.
Solution to the Move-to-Castle Problem
If we run the artificial intelligence AI planning technique STRIPS on the domain and problem above, we get the following solution:
This is exactly correct - and optimal too! It only takes 2 steps to move from the town to the neighboring field, and finally to the neighboring castle.
Let’s take a look at a slightly more tricky example. We’ll throw a monster onto the field, blocking that path. Instead we’ll provide a different way to the castle and see if the AI planning algorithm can figure it out.
Bypassing the Dragon in the Field
Notice that this newly defined STRIPS PDDL problem looks very similar to our first one. The difference here is that we’ve defined a dragon and placed him on the field. Since the field is guarded by a dragon (monster), and our “move” action has a precondition that a location can not be guarded by a monster, we can no longer move directly from the town, to the field, and finally to the castle. Our path to the field is effectively blocked. Especially, since we don’t have an attack action defined! The AI must find another way around.
We’ve actually provided an alternate means to get to the castle, by going through the tunnel in town, across the river, and finally to the castle.
Solution to the Sneak-Past-Dragon-to-Castle Problem
If we run the problem, the artificial intelligence AI planner finds the following solution.
In just 3 steps, the AI can move from the town to the castle, safely bypassing the dragon. What’s even more interesting, is taking a look at the search process that the AI planning algorithm used to find the solution.
As part of the search process, the STRIPS AI planner begins searching at the initial state defined in the problem (player at town). This corresponds to a depth of 0 in the search tree. The AI then proceeds down the child states in the graph of available actions. Somewhere at depth 1, the AI runs into the dragon in the field. At this point, it can not go any further and backtracks to a different branch in the search tree of available actions. It then finds another path at depth 1, using the tunnel instead. Finally, it searches forward at depth 2, finding the river, followed by the castle.
Methods for Searching for Solutions
In the above two examples, we’ve seen how the AI searches through the list of available actions at each state, in order to reach the goal state. But, how exactly does the AI search?
With STRIPS AI planning, a graph can be constructed that contains all available states and the actions that bring you to each state. This is called a planning graph. Here’s an example of what a planning graph might look like (this comes from the birthday dinner domain and problem):
As with most trees and graphs, we can traverse them using a variety of algorithms. For STRIPS artificial intelligence planners, a common method is to use breadth-first-search, depth-first-search, and the most intelligent approach - A* search.
Breadth First Search
Breadth-first-search finds the most optimal solution to a STRIPS problem. It searches from the initial state, and evaluates all child states that are available from valid actions at the initial state. It only evaluates at the current depth, completely checking all states before moving to the next depth level. In this manner, if any of the child states results in the goal state, it can stop searching right there and return the solution. This solution will always be the shortest. However, because breadth-first-search scans every single child state at the current depth level, it could take a long time to search, especially if the tree is very wide (with lots of available actions per state).
Depth First Search
With depth-first-search, the initial state is evaluated and all child states are pushed onto a stack or queue. The AI then moves down the tree to the next child state, then the next child state, all the way down until either a goal is found, or no more child states exist. When it reaches a dead-end, it backtracks until another available child state is found. It repeats this until it gets back to the initial starting state, and then chooses the next child state to head down.
Although depth-first-search might not find the most optimal solution to a STRIPS artificial intelligence planning problem, it can be faster than breadth-first-search in some cases.
A* Search
The most intelligent of the searching techniques for solving a STRIPS PDDL artificial intelligence AI planning problem is to use A search. A is a heuristic search. This means it uses a formula or calculation to determine a cost for a particular state. A state that has a cost of 0 is our goal state. A state that has a very large value for cost would be very far from our goal state.
Similar to breadth-first and depth-first search, A search evaluates all valid actions for a state to determine the available child states. However, here is where it differs. Before selecting the next state, A assigns a cost to each one, based upon certain characteristics of the state. It then chooses the next lowest-cost state to move to, with the idea being that the lowest costing states are the ones most likely to result in the goal.
For example, an easy A* search cost heuristic is to use landmark-based heuristics for searching. In our Magic World example, we want to move from the town to the castle. We know that the character must visit the tunnel and the river in order to reach the castle. Therefore, we could assign a cost of 15 to all states by default. If the AI has visited the tunnel state, we reduce the cost by 5, resulting in a cost of 10 for this state. If the AI has visited the river, we again reduce the cost by 5, resulting in a cost of 5 for this state. Finally, when the AI reaches the castle, we reduce the cost to 0.
The code to implement an A* landmark-based heuristic might look something like this:
A* search usually provides the fastest way for finding a goal state in a STRIPS planning problem.
Getting Crazy with Magic World
Let’s beef-up our magic world domain, by adding a whole bunch of new actions. We’ll give our characters the ability to move, attack, open treasure chests, collect elements, and build a weapon. This will be more interesting! We’ll change our domain to be defined, as follows:
In the above domain, we’ve defined a bunch of valid actions. A user or computer-controlled player can move from one area to the next, as long as that area is not currently guarded by a monster. If it is, the player will need to attack the monster first. Hence, we’ve defined an “attack” action as well.
We’ve also defined an “open” action to allow opening a treasure chest, and two types of treasures. We have an element of type fire and an element of type earth. Both can be collected for building a weapon. Once the player has both elements, he can build the “fireball” spell.
You can see how other types of spells and treasures can be added, by simply defining additional actions. Now, how about the problem?
Building a Fireball Weapon
Now, we’ll define an updated problem of figuring out how to build a fireball weapon in our world. The AI will have to figure out a plan for where to move, who to attack, and what to do, in order to build the weapon. We’ll define our problem, as follows:
In the above PDDL problem, we’ve added a lot to our world. We have a couple of monsters (an ogre and a dragon). Both monsters are guarding locations on the map. We’ve also added two treasure chests, containing magical elements that can be collected. A character will need both elements in order to build a fireball weapon. Let’s see how the AI STRIPS planner solves this.
Solution to the Fireball Weapon Problem
If we run the domain and problem, we get the following optimal solution.
The AI has successfully determined a plan, which involves moving from the town to the field, attacking the ogre at the river, and then moving to the river. It then attacks the dragon in the cave, and then opens the treasure chest in the river (the AI apparently wanted to attack the dragon before opening the treasure chest sitting at its feet - in reality, both actions had an equal depth cost, so the AI simply chose the first one that it found). It then collects the fire element from the treasure chest and moves to the cave. The AI opens the treasure chest in the cave, collects the earth element, and finally builds the fireball weapon.
What would happen if instead of using breadth-first-search, we try running this with depth-first-search? Let’s take a look.
Depth-first search produces a significantly longer set of steps to achieve the goal. It starts off the same as the optimal solution above, but at step 5, instead of opening the treasure in the river and collecting the fire element, the AI instead chooses to move into the cave and open the treasure there first. Interestingly, after opening the treasure in the cave, it then moves back to the river and opens the box there. This is effectively back-tracking. Once again, it repeats its steps of moving back into the cave to collect the element, and back to the river to collect the second element. Laughably, the AI then walks all the way back to town before building the fireball weapon!
This is a good example of the difference between breadth-first and depth-first search with STRIPS AI planning. The AI was simply following straight down a deep path of actions that lead to the goal state. Many other paths likely exist as well, including of course, the optimal path that was found by breadth-first search.
Integrating Automated Planning into Games and Applications
You can see how STRIPS artificial intelligence planning allows the computer to prepare a detailed step plan for achieving a goal. Now, how would you use this with a game?
In the main loop of a game, where the screen is continuously redrawn, there is usually associated logic for moving NPC characters and performing other necessary tasks at each tick. An automated planner can be integrated into this loop to continuously update plans for each NPC character, depending on their goals. Since a player or other NPC character can affect the state of the current world, we would need to update the plan at each tick, so that it reacts to any changes in the current state and updates its plan accordingly.
In this manner, the problem PDDL file could contain a dynamically updated :init section, where the current state of the world is described. The :goal section would remain static, while the :init section changes. At each defined interval, the AI would re-execute the automated planner to produce a new plan, given the state of the world. It would then redirect the NPC character to whichever action is next in the computed plan. The resulting plan may contain less or more action steps to achieve the goal. As the initial state of the problem PDDL changes, so too would the formulated solution plan.
Other Automated Planning Algorithms For Speed
It’s important to maintain search speed as a top priority. Since the automated planner may well be running at a frequently defined time interval, the faster the search can complete, the higher the application response rate. You’ll likely want to use an optimally programmed A* search heuristic.
For faster automated planner searching, you may even want to upgrade to other types of artificial intelligence planners, including GraphPlan and hierarchical task network (HTN) planners.
Trying an Automated Planner Yourself
You can experiment with different STRIPS PDDL domains and problems with the online application Strips-Fiddle. Try any of the example domains or create an account to design your own artificial intelligence planning domains and problems.
For integrating STRIPS-based AI planning into your application or game, you can use the node.js strips library, which supports breadth-first, depth-first, and A* searching. The homepage for the strips library provides some higher-level overview on the library, including an example of the Starcraft domain.
Domains and problems can be loaded from plain text files into the node.js strips library. Running a problem set can be done with the following code:
By default, depth-first-search is used. You can change this to breadth-first-search by adding a boolean parameter to the solve() method, as follows:
You can also specify a cost heuristic to use A* search, as follows:
For more details, see the Starcraft strips example. Give it a try and have fun! | https://www.primaryobjects.com/2015/11/06/artificial-intelligence-planning-with-strips-a-gentle-introduction/ |
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