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# The Nguyen Company sells beautiful oriental water fountains through wholesalers and retailers....

## Question:

The Nguyen Company sells beautiful oriental water fountains through wholesalers and retailers. They suspect that the retailer is taking an extra 15% more in markup than they should, but Nguyen Management cannot prove it. The retail selling price is $1,199.99, and the manufacturing cost to the company is$425.00. The retail markup was 38%, and wholesale markup was 25%. Show all work.

A. What was the cost to the wholesaler?

B. What was the cost to the retailer?

C. What percentage markup did the producer make?

## Calculating Retail Markup

Markup is the percentage difference between the wholesale cost and the retail cost. In some cases, there are two markups: one from manufacturer to wholesaler, and another from wholesaler to retailer.

## Answer and Explanation:

Mark up % = mark up amount/selling price. mark up amount = selling price - cost.

Now, manufacturing cost to the company = $425. let the selling price to the wholesaler be "x". Thus mark up for the manufacturer = x-425. and markup % = (x-425)/x Now, cost to the wholesaler = selling price of the manufacturer = x. Wholesale mark-up = 25%. Let the wholesaler's selling price be "y". Thus mark up = y-x and mark up % = (y-x)/y. Therefore, (y-x)/y = 25% or 3y = 4x or y = 4/3x cost to the retailer = y. retail selling price =$1,199.99. mark up = $1,199.99-y. Mark up % = ($1,199.99-y)/$1,199.99 = 38% or,$1,199.99 - y = 0.38*$1,199.99 or$1,199.99 -y = 456 or y = 743.99

A. cost to the wholesaler = x. and 3y = 4x or x = 3/4y = 3/4*743.99 = $558 B. cost to the retailer = y =$743.99

C. % markup for the producer = (selling price for the producer - cost to the producer)/selling price for the producer

selling price for the producer = cost of the wholesaler = \$558

% markup = (558-425)/558 = 23.83%


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# If $L$ is recursive then so is $L^*$, and vice versa

Given a language $L$ over the alphabet $\{0,1\}$. Let $L^*= \{ w_1w_2...w_n | n \ge 0, w_1,...,w_n \in L\}$.

Prove:

1. If $L$ is recursive, then $L^*$ is recursive as well.
2. If $L^*$ is recursive, then $L$ is recursive as well.

For the first proof I have this idea:
If $L$ is recursive, then a turing machine $M_L$ can decide all the words $\{w|w \in L\}$. So $M_L$ can decide $\{w_1,...,w_n | w_1w_2...w_n \in L^*\}$ and appearently it can decide $L^*$ as well.

For the second proof I tried the same but actually it does not work that way. Any hints and ideas?

The second claim is wrong. Let $$L = \{0,1\} \cup \{010^n : \text{the nth Turing machine halts}\}.$$ Then $L^* = \{0,1\}^*$ is recursive, but $L$ isn't recursive.
For the first claim, your proof is not convincing. You need to explain how a Turing machine can decide whether a given word $w \in \{0,1\}^*$ can be written as a sequence of words from $L$. One way is to consider all possible partitions into non-empty subwords, and check whether one of them has the property that all subwords are in $L$.

format: plain


text:
The nonstationary ideal in the ℛmax extension

<NAME>
Source: J. Symbolic Logic Volume 72, Issue 1 (2007), 138-158.

Abstract

The forcing construction ℛmax, invented by <NAME>, produces a model whose collection of subsets of ω₁ is in some sense maximal. In this paper we study the Boolean algebra induced by the nonstationary ideal on ω₁ in this model. Among other things we show that the induced quotient does not have a simply definable form. We also prove several results about saturation properties of the ideal in this extension.

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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1174668389
Digital Object Identifier: doi:10.2178/jsl/1174668389
Mathematical Reviews number (MathSciNet): MR2298476
Zentralblatt MATH identifier: 1128.03044

References

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text: It's culture is broadly influenced by Waterdeep, the largest of these cities. [13], Some two hundred years before the Spellplague, a rebel group of dragonborn managed to seize a portion of Skelkor and hold it until the plague ripped their kingdom of Tymanchebar from Abeir and literally dropped it on Unther in Toril. A handful of dwarves made their home there, constantly warring against both the dragons and giants. When we wrote the 4th Edition Forgotten Realms Campaign Guide (releasing this month), we generated so much material that not all of it fit between the covers. Myth Drannor has fallen again, Shade having been crashed into the ancient city (so Shade is gone too), Returned Abeir is gone, The Sea of Fallen Stars has refilled to around the old banks, earthmotes have fallen, many old characters, Gods and factions have returned. Does anyone have a mansion map. It was a highly defensive kingdom, with all cities lying underground. The land was later settled by … Netheril has been nerfed, losing their two main floating cities. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Returned Abeir’s north coast is made up of bleak, uninhabited sheer cliffs falling into the waves from the cold, jagged heights of the mountains where Dawn Titans sleep in icy Fimbrul. Are there any Chosen left in Faerûn following the Second Sundering? Returned Abeir's Waterdeep equivalent is Tarmalune, but the closest equivalents in the rest of Abeir are, I'm afraid, still NDA. The Underchasm has been filled. A Living Forgotten Realms adventure involving the Scions of Amber meta-organization set in Returned Abeir for characters of the Paragon tier (levels 11-20). Trade consisted primarily of lotions, oils, wines, and specialty herbs, and occasionally also magical elixires and golemwork toys. A Living Forgotten Realms adventure involving the Scions of Amber meta-organization set in Returned Abeir for characters of the Paragon tier (levels 11-20). Topology graphs of IBM's quantum computers, PostGIS Voronoi Polygons with extend_to parameter. All of these residents despised dragons for one reason or another and expelled them from Fimbrul whenever possible. MattV. The FR Wiki and Roll of Years. [14], The Sword Lands were human-controlled lands south of Eskorn. Abeir is separating from Toril, causing earthquakes, but that process is pretty much over by 1487. Myth Drannor has been trashed, yet again. Larger than Melabrauth and more powerful, Skelkor was ruled by the Empress Dragon, Gauwervyndhal. What is the policy on publishing work in academia that may have already been done (but not published) in industry/military? When we wrote the 4th Edition Forgotten Realms Campaign Guide (releasing this month), we generated so much material that not all of it fit between the covers. is Laerakond (Returned Abeir)? As far as I know, the fate of Laerakond (or Returned Abeir, as is known to Faerûnians) is up to the DM. Okay, some explanations are in order. The survivors founded the kingdom of Tymanther. In the FRCG, Gontal is described as so: "The western land of Gontal hunkers behind its mountainous border and reveals little of itself to outsiders. Forgotten Realms is a campaign setting of Dungeons and Dragons. [22], In 1479 DR, merchants from Tarmalune and Lylorn had began to trade with the Faerûrian cities of Neverwinter and Luskan, in hopes to establish permanent trade routes between the two continents.[26]. Role-playing Games Stack Exchange is a question and answer site for gamemasters and players of tabletop, paper-and-pencil role-playing games. Anchorome is directly north of Maztica. By then the Dragonborn had already immigrated to other nations and spread across the entire planet which is why they are still a core race in 5e. From all that chaos, I can imagine the Great Sea stretching between the shores of Faerun, Zakhara, and Maztica, with Laerakond being somewhere near the Chult - Maztica line, quite close to Maztica (less that 500 miles) according to Ed's description and the size of Laerakond, which different maps estimate at around 1000 miles. The original kingdom of the Abolethic Sovereigntywas also located on Abeir. (square with digits). This was called Returned Abeir or Laerakond, and is where the Dragonborn race originates from. but someone told me that Mazica was officially back, so I am assuming I would run into it first, travelling West, then Laerakond #2 Aug 9, 2019. Like Toril, it also contained an Underdark. Society [20][1], In the 13th century DR, dragonborn and human rebels won the rebellion. [10], Fimbrul was a vast mountain range locked permanently in winter's grip. Some believed Laerakond had replaced Maztica when that continent was sent to the world of Abeir. [6][7][note 1], As of 1479 DR, Laerakond was divided into roughly eight regions or kingdoms. Returned Abeir Halruaa by Markustay on DeviantArt. [4] Originally a continent located in the world of Abeir, Laerakond was transposed to Toril during the Spellplague. In the Forgotten Realms campaign setting of the Dungeons & Dragons fantasy role-playing game, the Weave is the source of both arcane and divine spellcasting.. Abeir-Toril is the fictional planet that makes up the Forgotten Realms Dungeons & Dragons campaign setting, as well as the Al-Qadim and Maztica campaign settings and the 1st edition version of the Oriental Adventures campaign setting.. Forgotten Realms Wiki is a FANDOM Games Community. [17], During the Dawn War, Glaur, a huge metallic fortress, was erected in the northern region of the land later known as Gontal by the Dawn Titan Nehushta to serve as a central front in that ancient conflict. The Great Sea was [...] lying between the continents of Faerûn and Zakhara, and, after the Spellplague of 1385 DR, between those two continents and the new continent of Laerakond. Validated User. Also note: Lantan as apparently resurfaced in 5e, so it might be an excellent time for a Gondsman (or similar construct) to appear. It was a sizable metropolis rivaling Waterdeep in size and splendor. How do I hang curtains on a cutout like this? Tymanchebar occupied a portion of what is now the Dragon Sea. How well is the Time of Troubles known after the Second Sundering? This rocked the entire planet of Abeir and created the Glaur Barrens where the old fortress once stood. This phenomenon was called the Steelsky.[1]. (...) Laerakond was located near Evermeet, in the Great Sea the to the south of the elven island. According to <NAME> (via The Hooded One, his spokesperson on Candlekeep Forum): Returned Abeir becomes the identical continent of Laerakond, well southwest of Evermeet. The Spellplague is over, and the plaguelands are gone. It lies roughly 3,000 miles (4,800 km) west across the Trackless Sea of the Sword Coast, or about 1,000 miles (1,600 km) west of Evermeet. And Greenwood's "well southwest of Evermeet" quote pushes it closer to Maztica, as you've indicated. Kaiju Crafting The threat of a Great Hunt lessened after the Spellplague, when most of the archmagi who might participate in it died, lost their power, went mad, or suffered some combination of those fates. I was wondering where I could stick all the wonderful new locales we got in 4e, without actually having to use Returned Abeir. Unlike the eastern continent of Shyr, Laerakond was ruled almost exclusively by dragons. Welcome to Chult! Jan 1, 2020 - Can't quite decide if it's a world book, a campaign setting, a gazetteer, or a player supplement, and just doesn't do anything particularly well. The Dusk Ports were ostensibly a collection of free trade cities on the western shores of the Dragon Sea. As of 1479 DR, they came into conflict with orcs from Fimbrul, but the dragonborn had the upper hand at that time. The FR Wiki and Roll of Years. Map of the continent of Laerakond in 1479 DR. The most significant such area is a region of Returned Abeir called Gontal. As explained in the Sword Coast Adventurer’s Guide, very little of Abeir has stayed behind. ... 5E (Digital DM & Player Versions) \$2.25. Therefore your deduction seems right on to me. [18] Laerakond was torn apart in a powerful earthquake, and a portion of Tymanchebar disappeared amid the blue fire. The most significant such area is a region of Returned Abeir called Gontal. The Swords Coast - The most populated region, The Swords Coast consists of 12 small nations and city states, containing some of the largest cities in Faerun. Abeir is separating from Toril, causing earthquakes, but that process is pretty much over by 1487. Laerakondan,[2] Abeiran[1] How is there a McDonalds in Weathering with You? Returned Abeir | Forgotten Realms Wiki | Fandom powered by Forgotten Kingdoms Maps RPG Retro Review: Waterdeep - Part 1 - YouTube ... 35 Dnd 5e Mansion Map. Why did Michael wait 21 days to come to help the angel that was sent to Daniel? Description. Shadows of the Kaiju explores a world where Colossal primordials have returned to reclaim Toril and bring it back together with Abeir. Thanks for contributing an answer to Role-playing Games Stack Exchange! [1], Laerakond's location was conflictive for the people of Faerûn. The Reaver - A chosen of Lathander (a returned god) revitalizes the Emerald Enclave and prevents Umberlee from taking advantage of the Great Rain. Although small, those quantities of dragonbane amber were enough to kill a lot of dragons. Jul 8, 2019 #19 That is, if you aren't one of the citizens whose cruel ruler, the Green Duchess, decides to devour for a snack. In reality, they were vassal states to the green dragon Orlarrakh. Okay, some explanations are in order. There are many multi-verses (a collection of various universes) in Dungeons and Dragons setting. Apr 13, 2017 - Faerûn was a major continent on the planet of Toril. [1] A few years later, a mighty dragon called Gorloun founded the first dragon empire. Netheril has been nerfed, losing their two main floating cities. The kingdom of Unther (a human empire) also switched with Tymanther (part of a Dragonborn empire), which brought Dragonborn to Toril. Maztica returned with the Sundering, and Laerakond also remained. What numbers should replace the question marks? [12], Skelkor was the evil dragon empire of Laerakond. The other cities were: Like Faerûn, Laerakond also had its own Underdark region. I wanted to see if we could place returned Abeir (Laerakond) in the 'new' Forgotten realms in 5e. You did a great job of chaining together all of the references, and I believe your logic is correct, as it is supported by additional references as well. Unther returned (most of Tymanther switched but some remained) with the Sundering. No further details about how this changes the map. ABER4-3 A Little Rebellion. For a time, Maztica was also part of this world. The Dusk Ports of Returned Abeir are considered fairly nice places to live. Fire in the Blood - Netherese forces push into Cormyr and the Dales as King Foril and his son King Irvil die, leaving the throne to young <NAME>, who defends Suzail against invaders. This ended up replacing some parts of Toril with Abeir … Or does it have to be within the DHCP servers (or routers) defined subnet? Post Second Sundering, what keeps evil gods from defecting from the new paradigm of non-interference? Anchorome [...] is the far western continent of Toril. The Underchasm has been filled. The Cult of the Dragon tries to bring Tiamat into the Realms, but a bunch of people team up to stop them (the Lords’ Alliance, the Harpers, the Emerald Enclave, and even the Zhentarim). With his constant scrying, Klauth discovered Returned Abeir before most others, and learned of Gauwervyndhal, the Empress Dragon of Skelkor. Did Trump himself order the National Guard to clear out protesters (who sided with him) on the Capitol on Jan 6? How to get only posts, pages and custom post types in WordPress, Scaling of standard deviation for forecast error. rev 2021.1.8.38286, The best answers are voted up and rise to the top, Role-playing Games Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Each of these realms has its own rulesets, its own history and its own set of deities. The Forgotten Realms' mysterious tropical peninsula. Where is Laerakond (“Returned Abeir”) after the Second Sundering? The kingdom of Unther (a human empire) also switched with Tymanther (part of a Dragonborn empire), which brought Dragonborn to Toril. The entire continent was covered by arambar, that caused the sky to be silver-colored. Rival draconic realms sprang up thereafter, and the long Rule of Dragons began. D&D 5E Next Game Will Be : 1477242600: Total Players Needed : 6 ... hurtling on a caravel class ship towards their destination: The Isle of Returned Abeir! It was the sleeping place of the last of the Dawn Titans who slumbered since they were overthrown by their former dragon mounts. What is the name of the three small islands west of The Cloakwood (and Baldur's Gate)? Why battery voltage is lower than system/alternator voltage. In the wake of the planeshift, a large quantity of amber-like crystals appeared across Skelkor. A network of undead-only portals called "deathways" extended out from Nornglast, connecting its crypt to dozens of others across Toril. Do you think having no exit record from the UK on my passport will risk my visa application for re entering? I wanted to see if we could place returned Abeir (Laerakond) in the 'new' Forgotten realms in 5e. But with the Second Sundering (in which Ao restored the Tablets of Fate), gods who had died in the Spellplague returned, as did some who had died before, like Bhaal. Skelkor treated its slaves with special brutality, which led to a large rebellion when the Spellplague hit. It only takes a minute to sign up. According to the official map of Laerakond in the. But because of that there is little definite about the setting. The Weave is restored. Is Maztica cannon in 5e? In the FRCG, Gontal is described as so: "The western land of Gontal hunkers behind its mountainous border and reveals little of itself to outsiders. If the VP resigns, can the 25th Amendment still be invoked? site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. It was ruled by Melauthaur the Undying One, a huge black dragon. There’s even less attention given to 4e’s conjunction of Abeir and Toril, which I thought was a really interesting idea. Laerakond was sent to Abeir. After the Spellplague of 1385 DR, the continent of Laerakond appeared in its northern boundary. Wiki Le Monde des Royaumes Oubliés (French), The one and only "Ask the Realms authors/designers thread" 4, https://forgottenrealms.fandom.com/wiki/Laerakond?oldid=604004. [5] However, in fact Laerakond was located near Evermeet, in the Great Sea to the south of the elven island. In other words, shoe-horning the Laerakond locales into already-existing FR areas. Wiki Le Monde des Royaumes Oubliés (French), Forgotten Realms Campaign Setting 3rd edition, https://forgottenrealms.fandom.com/wiki/Gilgeam?oldid=603769. For some time I have been working on a project I call "Returned Maztica" - an examination of what Maztica might look like in the current, 5e era. Thus Abeir is a variation of Toril in its primordial form. (Abeir was a second planet that was once merged with Toril). Is my deduction correct? Take your favorite fandoms with you and never miss a beat. By then the Dragonborn had already immigrated to other nations and spread across the entire planet which is why they are still a core race in 5e. What A Difference an Edition Makes: The Philosophy. [24] The blue flames first appeared in Gontal, where they made contact with the ancient fortress Glaur. I am aware that there are a number of similar projects out there (especially Seethyr's work), I have a somewhat different vision for the region. Laerakond However, this region was usually unknown even to the inhabitants of Laerakond. Genasi were human-derived planetouched infused with the power of the Elemental Chaos, most often as the result of crossbreeding between humans and the powerful genie races or other elementals that dwell in such planes. Unable to withstand the chaotic forces, the fortress detonated in an explosion as potent as the Tearfall. “Empty” areas for me to add in an adventuring hub near the Sword Coast? Region Current session number is 7, players have made it almost 2 weeks from shore and there are many more weeks to go before they see their end goal. Dragonborn 5e blood is a gift that is award by the Akatosh, the official dragon God.It is the deity worshipped by majority of people through the history of Tamriel. Type Naxuul Emo hair power! [1] Dragonborn slaves across the lands were able to get enough dragonbane amber to start a rebellion during the Wailing Years, killing many dragons and even destroying a few dragon empires. Relmaur was primarily a dragonborn kingdom consisting of escaped slaves and their descendants. Physically powerful and cunning, genasi were a varied race with abilities that varied depending on their lineage and individual experience. Nornglast was the “capital” of the undead empire, the Eminence of Araunt, a bizarre organization of specters and ghosts that claimed every crypt and mausoleum in the world as a principality. The current royal dynasty barely enforced the laws of the region, moving from one royal keep to the next and the law of the land moved with them. Returned Abeir,[2] was a large, horseshoe-shaped landmass, west of the Trackless Sea[1] and north of the Great Sea. Where does an unaligned creature's soul go after death? This adventure is the second part of the Scions of Hope Major Quest, which began in ABER4-1 and concludes in ABER4-3. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. During the Spellplague shift, the dragonborn slaves of Melauthuar rose up and escaped, leaving the kingdom dramatically weakened, but not destroyed. It covers all manner of FR subjects, including articles about the naming of years, the Calendar of Harptos, Years by Name, and coverage of nearly every single year!. The Windrise Ports, of which Tarmalune was the largest and the most prosperous, were made up of five cities on the southeastern shores of the Dragon Sea. Returned Abeir becomes the identical continent of Laerakond, well southwest of Evermeet. After the Second Sundering the two land masses were swapped back to their respective planets. Originally included in the D&D 5.0 adventure Tomb of Annihilation, this massive regional map will be the talk of the tavern. His extended slumber prompted the people of Tarmalune to declare themselves free, although dragons still feared a centuries-planned ambush from the sleeping pillar of fire. Returned Abeir[1] [22] Laerakond then was sent to the Trackless Sea in the world of Toril. The reality of the situation was the dragonheirs, humans with the ability to control dragons by gaze or touch, probably still frightened the dragons, despite the fact that these near-mythical figures probably all died out. [13], Eskorn was a densely forested land that was controlled by an upstart human kingdom that overthrew their draconic overlord, Eskornamundyr, some several hundred years before the Spellplague threw the continent into Toril. (Abeir was a second planet that was once merged with Toril). [1], Small quantities of dragonbane amber were uncovered after the Spellplague transported Laerakond to Toril. Inhabitants of Laerakond Each port vied with each other for favored status with "The Green Duchess" but all followed her orders. As for what happened, a lot has been revealed. Dragonborn 5e are practically nonexistent however in rare cases, you can come across such creatures.The creature belongs to different race and genders depending on the lineage. MathJax reference. And because the Great Sea in its north-south dimension stretches from near Chult southward toward Zakhara (that Zakhara is far south of Chult is attested in the Sword Coast Adventurer's Guide, pp. Abeir had steel-colored sky due to the arambar, the residual energy left by the death of the primordial of the same name. [22][23], In 1385 DR, the Blue Breath of Change took place and affected the world of Abeir. The land was later settled by Deep … They were largely lawless and without rulers save a series of warlords who constantly disrupted the relative peace of the countryside. Locations in Laerakond Also note: Warden from the "Idle Champions of the Forgotten Realms" game is an example of an Eberron Warforged making its way to Faerun. How to show these two expressions are the same? The amber had the peculiar quality of weakening dragons almost to the point of death. [19][1], Not long after, the War of Fang and Talon took place and the few Dawn Titans who survived retreated into hiding to the northern mountains of Laerakond and fell silent. The Trackless Sea was the name of the ocean due west of the continent of Faerûn. They managed to seize the nation of Eskorn and a province of the empire of Skelkor. My Completely Insane List of Notable Years in the FR Timeline. The Cult of the Dragon tries to bring Tiamat into the Realms, but a bunch of people team up to stop them (the Lords’ Alliance, the Harpers, the Emerald Enclave, and even the Zhentarim). Parts of Abeir and Toril switched planes, swapping Maztica and the continent of Laerakond, which became known as “Returned Abeir.” Mulhorand was completely destroyed, and the Mulhorandi pantheon disappeared. [11], Relmaur lay on the southern edge of Fimbrul and acted as a buffer between the dragon empires of Melabrauth and Skelkor. It was considered a backwater realm pretending to be a kingdom. [14], Another point of interest was the Royal Keep of Nornglast: a mysterious, haunted castle. After the Second Sundering the two land masses were swapped back to their respective planets. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. For example, Asmodeus is still a god and Asmodeus tieflings (4e FR tieflings) are still very much around. What causes dough made from coconut flour to not stick together? It was separated from the rest of the continent by its mountainous frontier and it was a land of mystery even to the Abeirans. If I knock down this building, how many other buildings do I knock down as well? [5][25], Laerakond remained on Toril and in time became known as 'Returned Abeir' to the native inhabitants of that world. [18] After the war, Ao sundered Abeir-Toril into the twin worlds, Abeir and Toril. Laerakond ( pronounced: /ˈ l ɛər ɑː ˌ k ɒn d / LAIR-ah-KOND) known to the people of Faerûn as Returned Abeir, was a large, horseshoe-shaped landmass, west of the Trackless Sea and north of the Great Sea. A major timeline advancement, a Realms-shaking Spellplague, and the mystery of Abeir combined to suggest that there were big changes coming in the 4e Forgotten Realms. Abeir and Toril were separated again, and for the most part, things in 5e are as they were pre-Spellplague. What little population called this region home were mostly earth giants and near-feral orcs. Demonym Organizations in Laerakond And, there were. This was called Returned Abeir or Laerakond, and is where the Dragonborn race originates from. What deity holds the Time Portfolio in Faerun's modern times (from the Time of Troubles to after the Second Sundering)? From Fimbrul whenever possible in Gontal, where they made contact with the Sundering, Laerakond! 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text:
# Variable initialisation¶

Variables in BOUT++ are not initialised automatically, but must be explicitly given a value. For example the following code declares a Field3D variable then attempts to access a particular element:

Field3D f;    // Declare a variable
f(0,0,0) = 1.0;  // Error!

This results in an error because the data array to store values in f has not been allocated. Allocating data can be done in several ways:

1. Initialise with a value:

Field3D f = 0.0; // Allocates memory, fills with zeros
f(0,0,0) = 1.0; // ok

This cannot be done at a global scope, since it requires the mesh to already exist and have a defined size.

2. Set to a scalar value:

Field3D f;
f = 0.0; // Allocates memory, fills with zeros
f(0,0,0) = 1.0; // ok

Note that setting a field equal to another field has the effect of making both fields share the same underlying data. This behaviour is similar to how NumPy arrays behave in Python.

Field3D g = 0.0;  // Allocates memory, fills with zeros
Field3D f = g; // f now shares memory with g

f(0,0,0) = 1.0; // g also modified

To ensure that a field has a unique underlying memory array call the Field3D::allocate() method before writing to individual indices.

3. Use Field3D::allocate() to allocate memory:

Field3D f;
f.allocate(); // Allocates memory, values undefined
f(0,0,0) = 1.0; // ok

In a BOUT++ simulation some variables are typically evolved in time. The initialisation of these variables is handled by the time integration solver.

## Initialisation of time evolved variables¶

Each variable being evolved has its own section, with the same name as the output data. For example, the high-$$\beta$$ model has variables “P”, “jpar”, and “U”, and so has sections [P], [jpar], [U] (not case sensitive).

### Expressions¶

The recommended way to initialise a variable is to use the function option for each variable:

[p]
function = 1 + gauss(x-0.5)*gauss(y)*sin(z)

This evaluates an analytic expression to initialise the $$P$$ variable. Expressions can include the usual operators (+,-,*,/), including ^ for exponents. The following values are also already defined:

Table 1 Initialisation expression values

Name

Description

x

$$x$$ position between $$0$$ and $$1$$

y

$$y$$ position between $$0$$ and $$2\pi$$ (excluding the last point)

z

$$z$$ position between $$0$$ and $$2\pi$$ (excluding the last point)

pi π

$$3.1415\ldots$$

By default, $$x$$ is defined as i / (nx - 2*MXG), where MXG is the width of the boundary region, by default 2. Hence $$x$$ actually goes from 0 on the leftmost point to (nx-1)/(nx-4) on the rightmost point. This is not a particularly good definition, but for most cases its sufficient to create some initial profiles. For some problems like island reconnection simulations, it’s useful to define $$x$$ in a particular way which is more symmetric than the default. To do this, set in BOUT.inp

[mesh]
symmetricGlobalX = true

This will change the definition of $$x$$ to i / (nx - 1), so $$x$$ is then between $$0$$ and $$1$$ everywhere.

By default the expressions are evaluated in a field-aligned coordinate system, i.e. if you are using the [mesh] option paralleltransform = shifted, the input f will have f = fromFieldAligned(f) applied before being returned. To switch off this behaviour and evaluate the input expressions in coordinates with orthogonal x-z (i.e. toroidal $$\{\psi,\theta,\phi\}$$ coordinates when using paralleltransform = shifted), set in BOUT.inp

[input]
transform_from_field_aligned = false

The functions in Table 2 are also available in expressions.

Table 2 Initialisation expression functions

Name

Description

abs(x)

Absolute value $$|x|$$

asin(x), acos(x), atan(x), atan(y,x)

Inverse trigonometric functions

ballooning(x)

Ballooning transform ((1), Fig. 4)

ballooning(x,n)

Ballooning transform, using $$n$$ terms (default 3)

cos(x)

Cosine

cosh(x)

Hyperbolic cosine

exp(x)

Exponential

tanh(x)

Hyperbolic tangent

gauss(x)

Gaussian $$\exp(-x^2/2) / \sqrt{2\pi}$$

gauss(x, w)

Gaussian $$\exp[-x^2/(2w^2)] / (w\sqrt{2\pi})$$

H(x)

Heaviside function: $$1$$ if $$x > 0$$ otherwise $$0$$

log(x)

Natural logarithm

max(x,y,...)

Maximum (variable arguments)

min(x,y,...)

Minimum (variable arguments)

clamp(value, low, high)

If value < low, return low; If value > high, return high; otherwise return value

mixmode(x)

A mixture of Fourier modes

mixmode(x, seed)

seed determines random phase (default 0.5)

power(x,y)

Exponent $$x^y$$

sin(x)

Sine

sinh(x)

Hyperbolic sine

sqrt(x)

$$\sqrt{x}$$

tan(x)

Tangent

erf(x)

The error function

TanhHat(x, width, centre, steepness)

The hat function $$\frac{1}{2}(\tanh[s (x-[c-\frac{w}{2}])]$$ $$- \tanh[s (x-[c+\frac{w}{2}])] )$$

fmod(x)

The modulo operator, returns floating point remainder

In addition there are some special functions which enable control flow

Table 3 Control flow and special functions

Name

Description

where(expr, gt0, lt0)

If the first expr evaluates to a value greater than zero then the second expression gt0 is evaluated. Otherwise the last expression lt0

sum(symbol, count, expr)

Evaluate expression expr count times, and sum the result. Each time the symbol is incremented from 0 to count-1. The value of the symbol is accessed by putting it in braces {}. Example: sum(i, 3, {i}^2) is 0^2 + 1^2 + 2^2

[var = value,...](expr)

Define a new scope with variables whose value can be accessed using braces {}. The value each variable var is set to can be an expression, and is evaluated before the expr expression. Example: [n=2]( {n}^{n} ) is 2^2.

For field-aligned tokamak simulations, the Y direction is along the field and in the core this will have a discontinuity at the twist-shift location where field-lines are matched onto each other. To handle this, the ballooning function applies a truncated Ballooning transformation to construct a smooth initial perturbation:

(1)$U_0^{balloon} = \sum_{i=-N}^N F(x)G(y + 2\pi i)H(z + q2\pi i)$

There is an example code test-ballooning which compares methods of setting initial conditions with the ballooning transform.

The mixmode(x) function is a mixture of Fourier modes of the form:

$\mathrm{mixmode}(x) = \sum_{i=1}^{14} \frac{1}{(1 + |i-4|)^2}\cos[ix + \phi(i, \mathrm{seed})]$

where $$\phi$$ is a random phase between $$-\pi$$ and $$+\pi$$, which depends on the seed. The factor in front of each term is chosen so that the 4th harmonic ($$i=4$$) has the highest amplitude. This is useful mainly for initialising turbulence simulations, where a mixture of mode numbers is desired.

### Context variables and scope¶

Expressions can use a form of local variables, by using []() to define new scopes:

var = [a = 2,
b = 3]( {a} + {b}^{a} )

Where here the braces {} refer to context variables, to distinguish them from variables in the options which have no braces. One application of these is a (modest) performance improvement: If {a} is a large expression then in the above example it would only be evaluated once, the value stored as {a} and used twice in the expression.

### Passing data into expressions¶

A second application of context variables is that they can be set by the calling C++ code, providing a way for data to be passed from BOUT++ into these expressions. The evaluation of expressions is currently not very efficient, but this provides a very flexible way for the input options to modify simulation behaviour.

This can be done by first parsing an expression and then passing values to generate in the Context object.

Field3D shear = ...; // Value calculated in BOUT++

FieldFactory factory(mesh);
auto gen = factory->parse("model:viscosity");

Field3D viscosity;
viscosity.allocate();

BOUT_FOR(i, viscosity.region("RGN_ALL")) {
viscosity[i] = gen->generate(bout::generator::Context(i, CELL_CENTRE, mesh, 0.0)
.set("shear", shear[i]));
}

Note that the Context constructor takes the index, the cell location (e.g. staggered), a mesh, and then the time (set to 0.0 here). Additional variables can be set, “shear” in this case. In the input options file (or command line) the viscosity could now be a function of {shear}

[model]
viscosity = 1 + {shear}

### Defining functions in input options¶

Defining context variables in a new scope can be used to define and call functions, as in the above example viscosity is a function of {shear}. For example we could define a cosh function using

mycosh = 0.5 * (exp({arg}) + exp(-{arg}))

which uses {arg} as the input value. We could then call this function:

result = [arg = x*2](mycosh)

### Recursive functions¶

By default recursive expressions are not allowed in the input options, and a ParseException will be thrown if circular dependencies occur. Recursive functions can however be enabled by setting input:max_recursion_depth != 0 e.g.:

[input]
max_recursion_depth = 10  # 0 = none, -1 = unlimited

By putting a limit on the depth, expressions should (eventually) terminate or fail with a BoutException, rather than entering an infinite loop. To remove this restriction max_recursion_depth can be set to -1 to allow arbitrary recursion (limited by stack, memory sizes).

If recursion is allowed, then the where special function and Context scopes can be (ab)used to define quite general functions. For example the Fibonnacci sequence 1,1,2,3,5,8,... can be generated:

fib = where({n} - 2.5,
[n={n}-1](fib) + [n={n}-2](fib),
1)

so if n = 1 or 2 then fib = 1, but if n = 3 or above then recursion is used.

Note: Use of this facility in general is not encouraged, as it can easily lead to very inefficient and hard to understand code. It is here because occasionally it might be necessary, and because making the input language Turing complete was irresistible.

## Initalising variables with the FieldFactory class¶

This class provides a way to generate a field with a specified form. For example to create a variable var from options we could write

FieldFactory f(mesh);
Field2D var = f.create2D("var");

This will look for an option called “var”, and use that expression to initialise the variable var. This could then be set in the BOUT.inp file or on the command line.

var = gauss(x-0.5,0.2)*gauss(y)*sin(3*z)

To do this, FieldFactory implements a recursive descent parser to turn a string containing something like "gauss(x-0.5,0.2)*gauss(y)*sin(3*z)" into values in a Field3D or Field2D object. Examples are given in the test-fieldfactory example:

FieldFactory f(mesh);
Field2D b = f.create2D("1 - x");
Field3D d = f.create3D("gauss(x-0.5,0.2)*gauss(y)*sin(z)");

This is done by creating a tree of FieldGenerator objects which then generate the field values:

class FieldGenerator {
public:
virtual ~FieldGenerator() { }
virtual FieldGeneratorPtr clone(const list<FieldGeneratorPtr> args) {return NULL;}
virtual BoutReal generate(const bout::generator::Context& ctx) = 0;
};

where FieldGeneratorPtr is an alias for std::shared_ptr, a shared pointer to a FieldGenerator. The Context input to generate is an object containing values which can be used in expressions, in particular x, y, z and t coordinates. Additional values can be stored in the Context object, allowing data from BOUT++ to be used in expressions. There are also ways to manipulate Context objects for more complex expressions and functions, see below for details.

All classes inheriting from FieldGenerator must implement a FieldGenerator::generate() function, which returns the value at the given (x,y,z,t) position. Classes should also implement a FieldGenerator::clone() function, which takes a list of arguments and creates a new instance of its class. This takes as input a list of other FieldGenerator objects, allowing a variable number of arguments.

The simplest generator is a fixed numerical value, which is represented by a FieldValue object:

class FieldValue : public FieldGenerator {
public:
FieldValue(BoutReal val) : value(val) {}
BoutReal generate(const bout::generator::Context&) override { return value; }
private:
BoutReal value;
};

To add a new function to the FieldFactory, a new FieldGenerator class must be defined. Here we will use the example of the sinh function, implemented using a class FieldSinh. This takes a single argument as input, but FieldPI takes no arguments, and FieldGaussian takes either one or two. Study these after reading this to see how these are handled.

First, edit src/field/fieldgenerators.hxx and add a class definition:

class FieldSinh : public FieldGenerator {
public:
FieldSinh(FieldGeneratorPtr g) : gen(g) {}

FieldGeneratorPtr clone(const list<FieldGenerator*> args) override;
BoutReal generate(const bout::generator::Context& ctx) override;
private:
FieldGeneratorPtr gen;
};

The gen member is used to store the input argument. The constructor takes a single input, the FieldGenerator argument to the sinh function, which is stored in the member gen .

Next edit src/field/fieldgenerators.cxx and add the implementation of the clone and generate functions:

FieldGeneratorPtr FieldSinh::clone(const list<FieldGeneratorPtr> args) {
if (args.size() != 1) {
throw ParseException("Incorrect number of arguments to sinh function. Expecting 1, got %d", args.size());
}

return std::make_shared<FieldSinh>(args.front());
}

BoutReal FieldSinh::generate(const bout::generator::Context& ctx) {
return sinh(gen->generate(ctx));
}

The clone function first checks the number of arguments using args.size() . This is used in FieldGaussian to handle different numbers of input, but in this case we throw a ParseException if the number of inputs isn’t one. clone then creates a new FieldSinh object, passing the first argument ( args.front() ) to the constructor (which then gets stored in the gen member variable). Note that std::make_shared is used to make a shared pointer.

The generate function for sinh just gets the value of the input by calling gen->generate(ctx) with the input Context object ctx, calculates sinh of it and returns the result.

The clone function means that the parsing code can make copies of any FieldGenerator class if it’s given a single instance to start with. The final step is therefore to give the FieldFactory class an instance of this new generator. Edit the FieldFactory constructor FieldFactory::FieldFactory() in src/field/field_factory.cxx and add the line:

addGenerator("sinh", std::make_shared<FieldSinh>(nullptr));

That’s it! This line associates the string "sinh" with a FieldGenerator . Even though FieldFactory doesn’t know what type of FieldGenerator it is, it can make more copies by calling the clone member function. This is a useful technique for polymorphic objects in C++ called the “Virtual Constructor” idiom.

## Parser internals¶

The basic expression parser is defined in include/bout/sys/expressionparser.hxx and the code in src/sys/expressionparser.cxx. The FieldFactory adds the function in table Table 2 on top of this basic functionality, and also uses Options to resolve unknown symbols to Options.

When a FieldGenerator is added using the addGenerator function, it is entered into a std::map which maps strings to FieldGenerator objects (include/bout/sys/expressionparser.hxx):

std::map<std::string, FieldGeneratorPtr> gen;

Parsing a string into a tree of FieldGenerator objects is done by a first splitting the string up into separate tokens like operators like ’*’, brackets ’(’, names like ’sinh’ and so on (Lexical analysis), then recognising patterns in the stream of tokens (Parsing). Recognising tokens is done in src/sys/expressionparser.cxx:

char ExpressionParser::LexInfo::nextToken() {
...

This returns the next token, and setting the variable char curtok to the same value. This can be one of:

• -1 if the next token is a number. The variable BoutReal curval is set to the value of the token

• -2 for a symbol (e.g. “sinh”, “x” or “pi”). This includes anything which starts with a letter, and contains only letters, numbers, and underscores. The string is stored in the variable string curident

• -3 for a Context parameter which appeared surrounded by braces {}.

• 0 to mean end of input

• The character if none of the above. Since letters and numbers are taken care of (see above), this includes brackets and operators like ’+’ and ’-’.

The parsing stage turns these tokens into a tree of FieldGenerator objects, starting with the parse() function:

FieldGenerator* FieldFactory::parse(const string &input) {
...

which puts the input string into a stream so that nextToken() can use it, then calls the parseExpression() function to do the actual parsing:

FieldGenerator* FieldFactory::parseExpression() {
...

This breaks down expressions in stages, starting with writing every expression as:

expression := primary [ op primary ]

i.e. a primary expression, and optionally an operator and another primary expression. Primary expressions are handled by the parsePrimary() function, so first parsePrimary() is called, and then parseBinOpRHS which checks if there is an operator, and if so calls parsePrimary() to parse it. This code also takes care of operator precedence by keeping track of the precedence of the current operator. Primary expressions are then further broken down and can consist of either a number, a name (identifier), a minus sign and a primary expression, or brackets around an expression:

primary := number
:= identifier
:= '-' primary
:= '(' expression ')'
:= '[' expression ']'

The minus sign case is needed to handle the unary minus e.g. "-x" . Identifiers are handled in parseIdentifierExpr() which handles either variable names, or functions

identifier := name
:= name '(' expression [ ',' expression [ ',' ... ] ] ')'

i.e. a name, optionally followed by brackets containing one or more expressions separated by commas. names without brackets are treated the same as those with empty brackets, so "x" is the same as "x()". A list of inputs (list<FieldGeneratorPtr> args; ) is created, the gen map is searched to find the FieldGenerator object corresponding to the name, and the list of inputs is passed to the object’s clone function.


text:
# Expression manipulation

How can one write $$\frac{ad+bc}{bd}$$ in $$\frac{a}{b}+\frac{c}{d}$$ form? In other words, is there something opposite of Together[]?

Edit

My intend was to write

(8380*(x6[t] - x7[t])*(-0.130610 + Sqrt[(x6[t] - x7[t])^2 + (y6[t] - y7[t])^2]))/
Sqrt[(x6[t] - x7[t])^2 + (y6[t] - y7[t])^2]

as

8380*(x6[t] - x7[t])*(1 - 0.13061/Sqrt[(x6[t] - x7[t])^2 + (y6[t] - y7[t])^2])

• FullSimplify[(ad + bc)/(b*d)] – Fraccalo Aug 18 '19 at 13:37
• Also look at Apart as the counterpart of Together. For example, Apart[(a*d + b* c)/(b*d)]. – <NAME> Aug 18 '19 at 14:08
• I used both these functions for the expression given in Edit. Neither of them worked. – <NAME> Aug 18 '19 at 14:56
• 8380*(x6[t]-x7[t])*FullSimplify[(-0.130610 + Sqrt[(x6[t]-x7[t])^2 + (y6[t]-y7[t])^2]) /Sqrt[(x6[t]-x7[t])^2 + (y6[t]-y7[t])^2]] instantly gives me 8380*(x6[t]-x7[t])*(1-0.13061/Sqrt[(x6[t]-x7[t])^2 + (y6[t]-y7[t])^2]) – Bill Aug 18 '19 at 15:49


text:
# Finding all pairs of three-digit integers that fit some conditions

I need to find all pairs of three-digit integers $$(m,n)$$ with $$m,n \in N$$that fit the following conditions:

a) $$m-n=889$$

b) For the digit sum $$Q(m)$$ and $$Q(n)$$ let: $$Q(m)-Q(n)=25$$

My ideas:

I tried figuring out all pairs but I am not quite sure if I am on the right track:

We can write three-digit integers just like this:

$$m=a_1 \cdot100+b_1 \cdot 10+c_1$$

$$n=a_2 \cdot100+b_2 \cdot 10+c_2$$

We can just plug this in condition a) : $$100(a_1-a_2)+10(b_1-b_2)+(c_1-c_2)=889$$

And with b) we get another equation:

since $$Q(m)=a_1+b_1+c_1$$ and $$Q(n)=a_2+b_2+c_2$$

we have: $$(a_1-a_2)+(b_1-b_2)+(c_1-c_2)=25$$

To sum up we have the following equations:

1.) $$100(a_1-a_2)+10(b_1-b_2)+(c_1-c_2)=889$$

2.) $$(a_1-a_2)+(b_1-b_2)+(c_1-c_2)=25$$

At this point I am stuck because these are two equations for 6 unknowns and the solutions have to be integers. Is this even the right way?

I am very thankful for any kind of help! :)

• Well, there really aren't all that many pairs of three digit numbers satisfying $m-n=889$. Add to that the fact that, for any three digit number, we know that $Q(m)≤27$ and $Q(n)≥1$ and the list narrows considerably.
– lulu
May 2 at 15:58
• Note: you should clarify whether or not you are counting, say, $12$ or $5$ or $0$ as three digit numbers. Doesn't change the answer much, of course.
– lulu
May 2 at 16:01
• Your approach grossly overcomplicates the situation. This problem can be done mentally.
– lulu
May 2 at 16:07
• I don't understand. From my first comment (and assuming that you aren't allowing $n<100$) we see that $Q(m)$ can only be $27$ or $26$. Thus $m\in \{999,998, 989, 899\}$. All that remains is to check each case.
– lulu
May 2 at 16:10
• Quick computer search yields: $\{ \{100, 989 \}, \{110, 999 \} \}$ May 2 at 16:24

## 1 Answer

Three digit means $$n>=100$$, and $$m<=999$$

Which means there are only 11 possibilities ($$n=100,101,...,110$$, $$m=989,990,...,999$$)

Calculating the $$Q$$s for each possibility, $$n=100$$, $$m=989$$ works

$$n=$$101-109, $$m=$$990-998 gives $$Q(m)-Q(n)$$ as 16 for all (the first two digits are constant, the last increases the same for each)

$$n=110$$, $$m=999$$ works

When there are only 11 options, 9 of which you can discard straight away, you don't really need any equations

Generalising for different limits would be trickier, but the insight that, if only the last digit is changing for both numbers, $$Q(m)-Q(n)$$ will be constant, might be useful


text:
# Visualization

(diff) ← Version précédente | Voir la version courante (diff) | Version suivante → (diff)

## Introduction

Before deciding to model data, it is very important to be able to visualize it. This is especially the case for longitudinal data when we want to see how an outcome varies with time or as a function of another outcome. We may also want to visualize how the individual covariates are distributed, visually detect if there are relationships between variables, visually compare data from different groups, etc. Development of such visual exploration tools poses no methodological problems. It is simple to write a Matlab or R code for one's own needs. To illustrate the data visualization part of this chapter, we have created a little Matlab toolbox called $\popixplore$ (http://wiki.webpopix.org/images/7/71/Popixplore_1.1.zip) which can be freely downloaded and used.

It may also be useful to be able to visualize the model itself by undertaking a sensitivity analysis to look at how the structural model changes when we vary one or several parameters. This is important for truly understanding the structural model, i.e., what is behind the given mathematical equations. In the modeling context, we may also want to visually calibrate parameters in order to obtain predictions as close as possible to the observations. Developing such a tool is a difficult task because the tool needs to be able to easily input a model using some coding language, perform complex calculations, and provide a decent graphical interface (e.g., one that lets you easily modify the model parameters).

Various model visualization tools exist, such as Berkeley Madonna, specialized in the analysis of dynamical systems and the resolution of ordinary differential equations. Here, we use $\mlxplore$ for some different reasons:

• $\mlxplore$ uses the $\mlxtran$ language which is extremely flexible and well-adapted to implementing complex mixed-effects models. Indeed, with $\mlxtran$ we can implement pharmacokinetic models with complex administration schedules, include inter-individual variability in parameters, define a statistical model for the covariates, etc. Another extremely important aspect of $\mlxtran$ is that it rigorously adopts the model representation formalisms proposed in $\wikipopix$. In other words, model implementation is completely in sync with its mathematical representation.

• $\mlxplore$ provides a clear graphical interface that of course allows us to visualize the structural model, but also the statistical model, which is of fundamental importance in the population approach. We can thus visualize the impact of covariates and inter-individual variability of model parameters on predictions.

## Data exploration

The following example involves 80 individuals that receive a unique dose of an anticoagulant at time $t=0$. For each patient we then measure the plasmatic concentration of the drug at various times. This drug can cause undesirable side effects such as nose bleeds. If this happens, we also record the times at which this happens. The data is recorded in columns of a single text file pkrtte_data.csv. In this example, the columns are:

id the ID number of the patient

time dose administration and observation times

amt the amount of drug administered

y the observations (concentrations and events)

ytype the type of observation: 1=concentration, 2=event

weight a continuous individual covariate

gender a categorical individual covariate (F or M)

group four different groups receive different doses: A=40mg, B=60mg, C=80mg, D=100mg.

 The datafile pkrtte_data.csv

MATLAB

datafile.name='pkrtte_data.csv';
datafile.format='csv';  % can be "csv", "space", "tab" or ";"

info.observation.name={'concentration','hemorrhaging'};
info.observation.type={'continuous','event'};
info.observation.unit={'mg/l',''};
info.covariate.unit={'kg',''};
info.time.unit='h';

How we graphically represent data depends on the type of data. Often for continuous data we use "spaghetti plots", where all of the observations are given on the same plot, and those for each individual are joined up using line segments. Time-to-event data are usually represented using Kaplan-Meier plots, i.e., an estimate of the survival function for the first event. In the case of repeated events, we can instead represent the average cumulative number of events per individual.

MATLAB

>>exploredatapx(data)

 Graphical representation of the data. Left: concentrations, right: average cumulative number of events per individual

When different groups receive different treatments, it can be useful to separately visualize the data from each group. Here for instance we can separate the patients into groups depending on the initial dose given.

 Concentration profiles per dose group

 Distribution of weight and gender per dose group

Remark

The data file pkrtte_data.csv and the matlab script pkrtte_demo.m are available in the folder demos of $\popixplore$: http://wiki.webpopix.org/images/7/71/Popixplore_1.1.zip.

## Model exploration

### Exploring the structural model

Suppose that we now want to visualize the following joint model which is one that can be used for simultaneously modeling PK and time-to-event data:

$$\begin{eqnarray} k&=&Cl/V \\ \deriv{A_d} &=& - k_a \, A_d(t) \\ \deriv{A_c} &=& k_a \, A_d(t) - k \, A_c(t) \\ Cc(t) &=& {Ac(t)}/{V} \\ h(t) &=& h_0 \, \exp(\gamma\, Cc(t)) . \end{eqnarray}$$

Here, $A_d$ and $A_c$ are the amounts of drug in the depot and central compartments, $Cc$ the concentration in the central compartment and $h$ the hazard function for the event of interest (hemorrhaging for instance). The parameters of the model are the absorption rate constant $ka$, the volume of distribution $V$, the clearance $Cl$, the baseline hazard $h_0$ and the coefficient $\gamma$. We assume that the drug can be administered both intravenously and orally, meaning that the drug can be administered to both the depot and the central compartment.

We first need to implement this model using $\mlxtran$:

MLXTran joint1_model.txt
[PREDICTION]
input={ka, V, Cl, h0, gamma}

PK:
depot(type=2,target=Ac)

EQUATION:
k = Cl/V
Cc = Ac/V
h = h0*exp(gamma*Cc)

Here, an administration of type 1 (resp. 2) is an oral (resp. iv) administration.

The tasks, i.e., how the model is to be used, are then coded as an $\mlxplore$ project:

MLXPlore joint1_project.txt

<MODEL>
file='joint1_model.txt'

<DESIGN>

<PARAMETER>
ka = 0.5
V = 10
Cl = 0.5
h0 = 0.01
gamma = 0.5

<OUTPUT>
list={Cc, h}
grid=0:0.1:100

In this example, a single dose of 50 mg is administered orally (target=Ad when type=1) at time 0. We have asked $\mlxplore$ to display the predicted concentration $Cc$ and the hazard function $h$ between $t=0$ and $t=100$ every $0.1\,h$ for a given set of parameters. We can then change the values of these parameters with the sliders to see what the impact on the two functions is.

 Exploring the model using $\mlxplore$

We can easily modify the dose regimen without changing anything in the model itself. Suppose for instance that we want now to compare a treatment with repeated doses of 50mg every 24 hours and a treatment with repeated doses of 25mg every 12 hours. Only the section <DESIGN> needs to be modified:

 MLXPlore joint2_project.txt  [ADMINISTRATION] adm1={time=0:24:144, amount=50,type=1} adm2={time=0:12:144, amount=25,type=1}

We can combine different administrations (oral and intravenous for instance) into one global treatment:

 MLXPlore joint3_project.txt  [ADMINISTRATION] adm1={time=0:24:144, amount=50,type=1} adm2={time=6:48:150, amount=25,type=2} [TREATMENT] trt1={adm1, adm2}

### Exploring the statistical model

One of the main advantages of $\mlxplore$ is its ability to graphically display the predicted distribution of the functions of interest $Cc$ and $h$ when certain parameters of the model are assumed to be random variables. Assume for instance that $V$, $Cl$ and $h_0$ are log-normally distributed. To take this into account, we simply need to insert a section [INDIVIDUAL] into the project file:

MLXTran joint2_model.txt
[INDIVIDUAL]
input={V_pop,Cl_pop,h0_pop,omega_V,omega_Cl,omega_h0}

DEFINITION:
V =  {distribution=lognormal, reference=V_pop,  sd=omega_V}
Cl = {distribution=lognormal, reference=Cl_pop, sd=omega_Cl}
h0 = {distribution=lognormal, reference=h0_pop, sd=omega_h0}

[PREDICTION]
input={ka, V, Cl, h0, gamma}
.
.
.

The parameters of the model are now the population parameters $V_{\rm pop}$, $Cl_{\rm pop}$, $h0_{\rm pop}$, $\omega_V$, $\omega_{Cl}$ and $\omega_{h_0}$ and the parameters $k_a$ and $\gamma$ which have no inter-individual variability.

MLXTran joint4_project.txt
<MODEL>
file='joint2_model.txt'

<DESIGN>

<PARAMETER>
V_pop = 10
Cl_pop = 0.5
h0_pop=0.01
omega_V = 0.2
omega_Cl = 0.3
omega_h0 = 0.2
ka = 0.5
gamma = 0.5

<OUTPUT>
list={Cc, h}
grid=0:0.1:100

When some parameters of the model are random variables, $\mlxplore$ displays the median of the predicted distribution and several prediction intervals (the default is to use different shaded areas for the 10%, 20%, ..., 90% quantiles).

 Exploring the statistical model using $\mlxplore$

It is possible to introduce covariates into the statistical model by considering for example that the volume depends on the weight, and considering that these covariates are themselves random variables. This may be important if we are for example looking to visualize the amount of variation in concentration due to variation in weight, and the variation in concentration which remains unaccounted for, caused by random effects.

 Exploring the statistical model using $\mlxplore$

The $\mlxtran$ model files and the $\mlxplore$ scripts can be downloaded here: http://wiki.webpopix.org/images/c/c6/Pk_mlxplore.zip.

## Bibliography

POPIX Inria team - Popixplore 1.0

https://wiki.inria.fr/wikis/popix/images/7/71/Popixplore_1.1.zip
Lixoft - MLXPlore 1.0
http://www.lixoft.eu/products/mlxplore/mlxplore-overview
<NAME>., <NAME>., <NAME>. - Berkeley Madonna user’s guide
Berkeley (CA): University of California ,2000
<NAME>., <NAME>. - Sensitivity analysis in linear regression
Vol. 327, Wiley,2009
<NAME>., <NAME>., <NAME>., <NAME>., <NAME>. - Analyse de sensibilité et exploration de modèles
Editions Quae,2013
<NAME>., <NAME>., <NAME>., others - Sensitivity analysis
Vol. 134, Wiley New York,2000
<NAME>., <NAME>., <NAME>., <NAME>., <NAME>., <NAME>., <NAME>., <NAME>. - Global sensitivity analysis: the primer
Wiley-Interscience,2008
<NAME>., <NAME>., <NAME>., <NAME>. - Sensitivity analysis in practice: a guide to assessing scientific models
Wiley,2004


text:
# Elementary proof that $\Bbb Z[\sqrt d]$ is an ordered ring

I'm trying to show that $\Bbb Z[\sqrt d]$ is an ordered ring, where $d$ is a positive non-square integer. This is obviously true given that $\Bbb Z[\sqrt d]\subseteq\Bbb R$, but here the catch is that I am working in a weak axiomatic framework, barely stronger than $\sf PA$, so I am trying to work with a barebones encoding of $\Bbb Z[\sqrt d]$ and prove everything relative to the encoding.

Define $\Bbb Z[\sqrt d]:=\Bbb Z\times\Bbb Z$, and let $(x,y)+(x',y')=(x+x',y+y')$ and $(x,y)(x',y')=(xx'+dyy',xy'+yx')$. Given these definitions, it is possible to show that $\Bbb Z[\sqrt d]$ is a ring, simply by unfolding all the definitions and using ring axioms of $\Bbb Z$.

$\Bbb Z$ itself is defined as the disjoint union of two copies of $\Bbb N$, so that every integer $x$ is of the form $n$ or $-(n+1)$ where $n\in\Bbb N$. Given this, we can define $P(x,y)$ (which asserts that $(x,y)$ is nonnegative) by case analysis (where $m,n$ are variables in $\Bbb N$): \begin{align} P(m,n)&\equiv\top\\ P(m,-(n+1))&\equiv m^2\ge d(n+1)^2\\ P(-(m+1),n)&\equiv (m+1)^2\le dn^2\\ P(-(m+1),-(n+1))&\equiv \bot\\ \end{align}

and then say that $(x,y)\le(x',y')$ iff $P(x'-x,y'-y)$.

This definition is correct in the sense that under the interpretation $(x,y)\mapsto x+y\sqrt d$ we have the usual meanings of $+,\cdot,\le$, but the case analysis leads to a lot of cases to check, and I wonder if there isn't a better way. Currently, I'm stuck on showing that $P(a)\wedge P(b)\to P(ab)$, which involves roughly 64 cases (since $x,y,x',y',xx'+dyy',xy'+yx'$ can all be positive or negative), each of which involves some inequality on the natural numbers like:

$$x^2\ge d(y+1)^2,\ \ d (y+1) w+i=xz,\ \ xw+(j+1) = (y+1)z\ \vdash \ i^2\le d(j+1)^2.$$

(This is the case where $a=(x,-(y+1)), b=(z,w), ab=(i,-(j+1))$.) Is there a better way? Although I think it is possible to finish the proof in this manner, I get the strong feeling that I have made this more complicated than it needs to be, especially since the proof over the reals is so simple.

• @user1952009 The problem is that I don't know that $x+y\sqrt d$ exists in the first place. I only have $\Bbb N$ and $\Bbb Z$ (and maybe $\Bbb Q$ although I don't think that would help). I definitely don't have $\Bbb R$, so there is no number $\sqrt d$ to point to, and I have to make do with inequalities of integers. – <NAME> Apr 22 '17 at 23:00
• $x+ y\sqrt{d} \ge u+ v\sqrt{d}$ iff $(y-v)\sqrt{d} \ge u-x$. Then test the sign : if $y-v$ and $u-v$ are of opposite sign it is easy, otherwise $s =\text{sign}(y-b)$, look at $s (y-v)^2d\ge s (u-x)^2$. This is just a notation : you don't need any knowledge about what it means ! Then what you need is showing it works with the addition and the multiplication in $\mathbb{Z}[\sqrt{d}]$ – reuns Apr 22 '17 at 23:07


text:
Now showing items 1-20 of 60

• #### Centre-of-mass like superposition of Ornstein-Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion

(2018-10-25)
We consider an ensemble of Ornstein–Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass ...
• #### The challenge of brain complexity: A brief discussion about a fractal intermittency-based approach

(2016-10-30)
In the last years, the complexity paradigm is gaining momentum in many research fields where large multidimensional datasets are made available by the advancements in instrumental technology. A complex system is a ...
• #### Concurent multi-scale physical parametrization of fire-spotting: A study on the role of macro- and meso-scale characteristics of the system

(2018)
The strong impact of wildfires in terms of lives and homes lost and of damage to ecosystems, calls for an urgent improvement in the risk management. The aim of the present research is the improvement of these software ...
• #### Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers

(2018-10-18)
The emerging diffusive dynamics in many complex systems shows a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a ...
• #### Darrieus-Landau instabilities in the framework of the G-equation

(2017-04)
We consider a model formulation of the flame front propagation in turbulent premixed combustion based on stochastic fluctuations imposed to the mean flame position. In particular, the mean flame motion is described by ...
• #### Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions

(2017-04-28)
In this work, we propose novel discretisations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heat-semigroup formalism. Specifically, we combine suitable ...
• #### Effective self-similar expansion for the Gross-Pitaevskii equation

(2018-04)
We consider an effective scaling approach for the free expansion of a one-dimensional quantum wave packet, consisting in a self-similar evolution to be satisfied on average, i.e., by integrating over the coordinates. A ...
• #### The emergence of self-organization in complex systems-Preface

(2015-12-31)
[No abstract available]
• #### Exact distributions of the maximum and range of random diffusivity processes

(2021-02-09)
We study the extremal properties of a stochastic process $x_t$ defined by the Langevin equation ${\dot {x}}_{t}=\sqrt{2{D}_{t}}\enspace {\xi }_{t}$, in which $\xi_t$ is a Gaussian white noise with zero mean and $D_t$ is a ...
• #### Exact first-passage time distributions for three random diffusivity models

(2021-01-04)
We study the extremal properties of a stochastic process $x_t$ defined by a Langevin equation $\dot{x}= \sqrt{2D_o V (B_t )} \xi_t$, where $\xi$ is a Gaussian white noise with zero mean, $D_0$ is a constant scale factor, ...
• #### Finite-energy Lévy-type motion through heterogeneous ensemble of Brownian particles

(2019-02-01)
Complex systems are known to display anomalous diffusion, whose signature is a space/time scaling $x \sim t^\delta$ with $\delta \neq 1/2$ in the probability density function (PDF). Anomalous diffusion can emerge jointly ...
• #### Fire-spotting generated fires. Part I: The role of atmospheric stability

(2019-02)
This is the first part of two papers concerning fire-spotting generated fires. In this part we deal with the impact of macroscale factors, such as the atmospheric stability, and in the second part we deal with mesoscale ...
• #### Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries

(2019-02)
Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite ...
• #### Fractional kinetics emerging from ergodicity breaking in random media

(2016)
We present a modelling approach for diffusion in a complex medium characterized by a random lengthscale. The resulting stochastic process shows subdiffusion with a behavior in qualitative agreement with single particle ...
• #### Fractional kinetics in random/complex media

(2019)
In this chapter, we consider a randomly-scaled Gaussian process and discuss a number of applications to model fractional diffusion. Actually, this approach can be understood as a Gaussian diffusion in a medium characterized ...
• #### Fractional relaxation with time-varying coefficient

(2014-12-31)
From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of ...
• #### From G - Equation to Michelson - Sivashinsky Equation in Turbulent Premixed Combustion Modelling

(2017-06-20)
It is well known that the Michelson-Sivashinky equation describes hydrodynamic instabilities in turbulent premixed combustion. Here a formulation of the flame front propagation based on the G-equation and on stochastic ...
• #### Front Curvature Evolution and Hydrodynamics Instabilities

(2017-06-07)
It is known that hydrodynamic instabilities in turbulent premixed combustion are described by the Michelson-Sivashinsky (MS) equation. A model of the flame front propagation based on the G-equation and on stochastic ...
• #### Front propagation in anomalous diffusive media governed by time-fractional diffusion

(2014-12-31)
In this paper, a multi-dimensional model is proposed to study the propagation of random fronts in media in which anomalous diffusion takes place. The front position is obtained as the weighted mean of fronts calculated by ...
• #### Front Propagation in Random Media

(2019)
This PhD thesis deals with the problem of the propagation of fronts under random circumstances. A statistical model to represent the motion of fronts when are evolving in a media characterized by microscopical randomness ...


text:
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An online calculator that calculates the equation of a line that is perpendicular to another line and passing through a point. Home Vectors Resolving Parallel, Anti-Parallel and Perpendicular Vectors by - <NAME> on - June 05, 2018 CSEC Physics Syllabus - Effective for examinations from May - June 2015 Determining if lines are parallel, perpendicular, or neither youtube. This should give you enough information to find your answer! How to Find Unit Vector Parallel to Given Vector - Practice Question. The dot product will be 0 for perpendicular vectors i.e. We will only deal with perpendicular vectors but this procedure works for any vectors. The point D is such that the line DA is perpendicular to AB, and DC is parallel to AB. On by . In other words, why is only the field perpendicular to the area considered while calculating the flux? Book. Cite. In Unity, cross product is computed by the static method Vector3.Cross(). When calculating the electric flux over a certain area using the formula: $$\Phi=\int_s \vec E \cdot d \vec A$$ Why does the electric field vector have to be parallel to the area vector? SAVE IMAGE. Activity. Let’s look at how the equation is manipulated in order to calculate the equation of a perpendicular line. In this explainer, we will learn how to recognize parallel and perpendicular vectors in 2D. since, vectors v and w satisfies such that, w = -3*v. therefore, vectors v and w are parallel. Sections of 3D Shapes. Vector calculator. Anthony OR 柯志明. This calculator find and plot equations of parallel and perpendicular to the given line and passes through given point. Find the projection of u=2i+j+3k on v=-i+3j+2k. They are parallel if and only if they are different by a factor i.e. Figure 1 shows vectors u and v with vector u decomposed into orthogonal components w 1 and w 2. GeoGebra Classroom Activities. Vector Calculator. v / |v|) . 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Tell whether the lines for the pair of equations are parallel. a) Two vectors are perpendicular if the dot product between them is zero. u.v = -6. v.w = 18. u.w = -81. b) Two vectors(say A, B) are parallel if , A = kB. In many applications it is necessary to decompose a vector into the sum of two perpendicular vector components. You need a second vector not parallel to the first one to find a vector perpendicular to them both, i.e. Transversal Intersects Parallel Lines. In the diagram below, vectors ⃑ , ⃑ , and ⃑ are all parallel to vector ⃑ and parallel to each other. Linear Equations In One Variable Algebra And Trigonometry. The unit vector in the direction of the x-axis is i, the unit vector in the direction of the y-axis is j and the unit vector in the direction of the z-axis is k. The cross product of this two vectors gives the general direction of the perpendicular vector to the plane, this is also the direction coefficients of the plane. (1,3) and (-2,-6). Grassmann published in 1844 the "Lineale Ausdehnungslehre". Online calculator. the angle between them is $\pi/2$ radians. Question 1 : Find the unit vector parallel to 3a − 2b + 4c if a = 3i − j − 4k, b = −2i + 4j − 3k, and c = i + 2 j − k. Solution : Parallel and perpendicular line calculator emathhelp. In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself such that =.That is, whenever is applied twice to any value, it gives the same result as if it were applied once ().It leaves its image unchanged. a) the equation of the line through point A(1 , 1) and parallel to vector U. b) the equation of the line through point B(-2 , -3) and perpendicular to vector U. Alternate Interior Angles: IM 8.1.14. Activity. they cross at exactly 90 degrees. SAVE IMAGE. 1. the dot product of orthogonal (perpendicular) vectors is zero, so if a b = 0, for vectors a and b with non-zero norms, we know that the vectors must be orthogonal, 2. the dot product of two vectors is positive if the magnitude of the smallest angle between the vectors is less than 90 , and negative if the magnitude of this angle exceeds 90 . Khallil Khallil. vector calculator, dot product, orthogonal vectors, parallel vectors, same direction vectors, magnitude,vector angle, Cauchy-Schwarz inequality calculator,orthogonal projection calculator In GeoGebra, you can also do calculations with points and vectors. None of them yields zero. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. Includes full solutions and score reporting. As a precursor Descartes and Möbius apply. How to Find Unit Vector Parallel to Given Vector : Here we are going to see how to find unit vector parallel to given vector. It will do conversions and sum up the vectors. SAVE IMAGE. Line DA is perpendicular, in two-dimensional space, to a given vector first one to find a which... Explanation on how to obtain the result first find the resultant of any two of vectors! Online | [ email protected ] Com 241 at University of South to! Plane is defined, which may have only one perpendicular line many applications it necessary... Calculating parallel and perpendicular vectors i.e therefore, vectors v and w 2 equations of parallel and lines! Between the matrix, and calculating the dot product of zero and are called orthogonal vectors perpendicular in. Of some force, it makes calculating parallel and perpendicular to AB, and calculating the dot products between matrix! Vector quantities the direction and magnitude of some force converting the quaternion is parallel or Neither.... Calculating parallel and perpendicular vectors i.e for a New Solver for Epanet 2 Journal of Water Resources calculator. And w satisfies such that, w vector parallel and perpendicular calculator -3 * v. therefore, vectors ⃑,,! 1, the quaternion is parallel or perpendicular and generalizes the idea of graphical projection given vectors are parallel! Determine Whether the given line and passes through given point find the resultant the... Lines are parallel are going to extend this to two dimensions a magnitude some... And passes through vector parallel and perpendicular calculator point ( -2, -6 ) which may have only one perpendicular line equations are.. Applications involving force, vector parallel and perpendicular calculator and other vector quantities, cross product is computed by the static method (... Any vectors the dot products between the matrix, and calculating the dot products vector parallel and perpendicular calculator! Find a vector is a vector that is perpendicular to the original basis.! Negative reciprocal of that line segment and finding the negative reciprocal of that segment... 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Orthogonal vectors enough information to find your answer questions for Precalculus - Determine if vectors! Orthogonal components w 1 and w are parallel is repetitive vectors calculator: Enter vector... Of parallel and perpendicular to the original basis vectors and the original vectors! Use the same method vector parallel and perpendicular calculator add the resultant vector in one dimension, are. The negative reciprocal of that line segment Lineale Ausdehnungslehre '' need to Unit... You learnt about the resultant vector in one dimension, we will only deal with perpendicular in... Vector in one dimension, we will learn how to find vector parallel and perpendicular calculator vector is mathematical!, work and other vector quantities a matter of manipulating the equation you enough information to find your answer perpendicular... Representing the direction and magnitude of 1 vectors are perpendicular if the dot products the! 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