title: Yehudit Arnon

abstract: Yehudit Arnon was an Israeli dancer and choreographer.

text:
# Yehudit Arnon

**Yehudit Arnon** (15 October 1926 – 17 August 2013) was an Israeli dancer and choreographer.

## Biography

Yehudit Arnon was born on 15 October 1926 in Komárno, Czechoslovakia to Ludwig Schischa-Halevy and Elisabeth (Betty) Hoffmann. On 11 June 1944 Arnon and her parents were sent to Auschwitz. Upon arriving, she requested to stay with her sickly mother but instead was sent with the other young people to the Birkenau camp while her mother was sent to the gas chambers. She escaped the concentration camp in May 1945 when the Red Army arrived. After escaping, she reached Budapest and began teaching dance. There, she met Yedidya Ahronfeld, whom she married on 16 June 1946. The couple moved to Palestine in 1948, settling in Kibbutz Ga’aton, Israel where she lived until her death.

Arnon established the Kibbutz Contemporary Dance Company (KCDC) in 1970. She served as the artistic director and CEO until 1996. "In June 1997, Arnon received the Distinguished Artist Award of the International Society for the Performing Arts in recognition of her extraordinary contributions of creative talent and inspiration to the world of dance". Arnon was awarded the Israel Prize in 1998 which is Israel's highest award for her achievements in education and the performing arts during her lifetime.

Yehudit Arnon died on 17 August 2013, at 87 years of age.

## InfoBox

| Yehudit Arnon | |
| --- | --- |
|  | |
| Born | (1926-10-15)15 October 1926
Komárno, Czechoslovakia |
| Died | 17 August 2013(2013-08-17) (aged 86)
Kibbutz Ga'aton, Israel |
| Nationality | Israel |
| Occupation(s) | Dancer, choreographer |
| Known for | Founder of Kibbutz Contemporary Dance Company (KCDC) |
| Awards |  |


title: Palaemonella komaii

abstract: Palaemonella komaii is a species of shrimp in the family Palaemonidae, found in the Indo-Pacific. It was first described by Xinzheng Li and Alexander Bruce in 2006, from specimens collected off Tonga and Fiji. The species epithet honours Tomoyuki Komai.

text:
# Palaemonella komaii

***Palaemonella komaii*** is a species of shrimp in the family Palaemonidae, found in the Indo-Pacific. It was first described by Xinzheng Li and Alexander Bruce in 2006, from specimens collected off Tonga and Fiji. The species epithet honours Tomoyuki Komai.

## InfoBox

| *Palaemonella komaii* | |
| --- | --- |
|  | |
|  | |
|
| Scientific classification | |
| Domain: | Eukaryota |
| Kingdom: | Animalia |
| Phylum: | Arthropoda |
| Class: | Malacostraca |
| Order: | Decapoda |
| Suborder: | Pleocyemata |
| Infraorder: | Caridea |
| Family: | Palaemonidae |
| Genus: | *Palaemonella* |
| Species: | ***P. komaii*** |
| Binomial name | |
| ***Palaemonella komaii***<br>Li & Bruce, 2006 | |
|


title: Ontario Highway 81

abstract: King's Highway 81, also known as Highway 81, was a provincially maintained highway in the Canadian province of Ontario. The winding north–south route connected Highway 2 in Delaware with Highway 21 in Grand Bend, passing through Mount Brydges, Strathroy and Parkhill en route. Highway 81 was first designated in 1936 and extended in 1937, and retained generally the same route throughout its existence until it was transferred to the responsibility of Middlesex County and Huron County in 1997 and 1998. Today the entire route is known as Middlesex County Road 81 and Huron County Road 81.

text:
# Ontario Highway 81

**King's Highway 81**, also known as **Highway 81**, was a provincially maintained highway in the Canadian province of Ontario. The winding north–south route connected Highway 2 in Delaware with Highway 21 in Grand Bend, passing through Mount Brydges, Strathroy and Parkhill en route. Highway 81 was first designated in 1936 and extended in 1937, and retained generally the same route throughout its existence until it was transferred to the responsibility of Middlesex County and Huron County in 1997 and 1998. Today the entire route is known as **Middlesex County Road 81** and **Huron County Road 81**.

## Route description

Highway 81 once served as a connecting route between Highway 2 and Highway 7 before its role was largely supplanted by the completion of Highway 402, which generally parallels the southern half of the route. Beginning at former Highway 2 in Delaware, what is now known as Middlesex County Road 81 travels west through Campbellvale, Mount Brydges and Caradoc, curving slightly to the northwest. The route travels through a large swath of farmland between Caradoc and Strathroy, the latter from which it exits to the north. After crossing Highway 402 at Exit 65, the highway encounters former Highway 22 at Wrightmans Corners.

Continuing north, the highway takes a veering route north and west towards Parkhill, bisecting the communities of Crathie and Bornish. Immediately southeast of Parkhill, Highway 81 and Highway 7 shared a brief concurrency, though neither are provincial highways today.

North of Parkhill, former Highway 81 continues to zig-zag north and west towards Grand Bend on the shores of Lake Huron. A one kilometre (0.6 mi) section of the route between Corbett and Greenway straddles the boundary of Middlesex County and Huron County. Beyond there, the final five kilometres (3.1 mi) lay in the latter. The highway ends approximately one kilometre northwest of Highway 21 at Government Road, a beach access road along the shoreline of Lake Huron.

## History

Highway 81 was first established by the Department of Highways (DHO) in late-1936 to connect Highway 2 at Delaware and Highway 22 at Strathroy. On September 16, 1936, 16.9 kilometres (10.5 mi) of roadway was assumed from Middlesex County by the DHO. The following year, several more roads were assumed on September 1 and numbered as an extension of Highway 81 to Grand Bend, bringing the highway to a length of 70.8 kilometres (44.0 mi).

Originally, the mostly-gravel-surfaced highway passed through the community of Springbank via Springbank Road and Glasgow Street. However, it was relocated to the southwest in 1946 to eliminate several sharp corners along the route. It was already paved between Highway 2 near Delaware and Mount Brydges, as well as between Highway 7 in Parkhill and Moray when it was assumed by the DHO. Paving between Mount Brydges and Strathroy was completed by 1950; the section between Moray and Grand Bend was paved within the next two years. A short section between Strathroy and Highway 22 was paved by 1954. The remainder of the route was paved by 1965, between Highway 22 north of Strathroy and Highway 7 east of Parkhill. In 1969, the Parkhill Dam was constructed, creating the Parkhill Reservoir. As a result of this work, Highway 81 was diverted to the east, removing the slight jog at McGuffin Hills Drive.

As part of a series of budget cuts initiated by premier Mike Harris under his Common Sense Revolution platform in 1995, numerous highways deemed to no longer be of significance to the provincial network were decommissioned and responsibility for the routes transferred to a lower level of government, a process referred to as downloading. Highway 81 was deemed to serve a local function and was transferred to Middlesex and Huron counties in two separate downloads. The section of the route between Delaware and Strathroy was transferred on April 1, 1997. The remainder of the highway, between Strathroy and Grand Bend, was transferred, on January 1, decommissioning the route in the process.

## Major intersections

The following table lists the major junctions along Highway 81, as noted by the Ministry of Transportation of Ontario.

| Division | Location | km | mi | Destinations | Notes |
| --- | --- | --- | --- | --- | --- |
| Middlesex | Delaware | 0.0 | 0.0 | County Road 2 | Formerly Highway 2 |
| | Mount Brydges | 6.1 | 3.8 | County Road 14 (Glendon Drive) |  |
| | Strathroy | 18.2 | 11.3 | Metcalfe Street |  |
| | 19.1 | 11.9 | Victoria Street |  |
| | 22.4 | 13.9 | Highway 402 | Exit 65 |
| | Wrightmans Corners | 23.8 | 14.8 | County Road 22 | Formerly Highway 22 |
| | Crathie | 27.7 | 17.2 | County Road 19 north (Petty Street) – Ailsa Craig |  |
| | North Middlesex | 32.5 | 20.2 | County Road 12 west (Townsend Line) – Arkona |  |
| | 42.0 | 26.1 | County Road 17 (Nairn Road) |  |
| | 44.1 | 27.4 | Highway 7 east (Elginfield Road) |  |
| | Parkhill | 46.2 | 28.7 | Highway 7 west (Elginfield Road) |  |
| | 47.8 | 29.7 | Bethany Street |  |
| | Moray | 52.6 | 32.7 | County Road 24 (McGillivray Drive) |  |
| Middlesex–Huron boundary | Corbett | 57.7 | 35.9 | County Road 5 east (Mount Carmel Drive) |  |
| | Greenway | 59.0 | 36.7 | County Road 5 west (Greenway Drive) |  |
| Huron | South Huron | 63.9 | 39.7 | County Road 10 (Crediton Road) |  |
| | Grand Bend | 68.9 | 42.8 | Highway 21 (Ontario Street) |  |
| | 69.6 | 43.2 | Government Road |  |
| 1.000 mi = 1.609 km; 1.000 km = 0.621 mi
 | | | | | |

## InfoBox

| King's Highway 81 | |
| --- | --- |
| Route information | |
| Maintained by the Ministry of Transportation | |
| Length | 69.6 km (43.2 mi) |
| Existed | September 16, 1936–January 1, 1998 |
| Major junctions | |
| South end | Highway 2 in Delaware |
| Major intersections | Highway 22Sarnia, London
Highway 7 near Parkhill |
| North end | Government Road in Grand Bend |
|  | |
| Location | |
| Country | Canada |
| Province | Ontario |
| Towns | Delaware, Mount Brydges, Strathroy, Parkhill, Grand Bend |
|  | |
| Highway system | |
|  | |
| Highway 80   Highway 82 | |


title: Liberal paradox

abstract:
The liberal paradox, also Sen paradox or Sen's paradox, is a logical paradox proposed by Amartya Sen which shows that no means of aggregating individual preferences into a single, social choice, can simultaneously fulfill the following, seemingly mild conditions:
  - The unrestrictedness condition, or U: every possible ranking of each individual's preferences and all outcomes of every possible voting rule will be considered equally,
  - The Pareto condition, or P: if everybody individually likes some choice better at the same time, the society in its voting rule as a whole likes it better as well, and
  - Liberalism, or L: all individuals in a society must have at least one possibility of choosing differently, so that the social choice under a given voting rule changes as well. That is, as an individual liberal, anyone can exert their freedom of choice at least in some decision with tangible results. Sen's result shows that this is impossible. The three, rather minimalistic, assumptions cannot all hold together. The paradox—more properly called a proof of contradiction, and a paradox only in the sense of informal logic—is contentious because it appears to contradict the classical liberal idea that markets are both Pareto-efficient and respect individual freedoms. Sen's proof, set in the context of social choice theory, is similar in many respects to Arrow's impossibility theorem and the Gibbard–Satterthwaite theorem. As a mathematical construct, it also has much wider applicability: it is essentially about cyclical majorities between partially ordered sets, of which at least three must participate in order to give rise to the phenomenon. Since the idea is about pure mathematics and logic, similar arguments abound much further afield. They, for example, lead to the necessity of the fifth normal form in relational database design. The history of the argument also goes deeper, Condorcet's paradox perhaps being the first example of the finite sort.

text:
# Liberal paradox

The **liberal paradox**, also **Sen paradox** or **Sen's paradox**, is a logical paradox proposed by Amartya Sen which shows that no means of aggregating individual preferences into a single, social choice, can simultaneously fulfill the following, seemingly mild conditions:

1. The *unrestrictedness condition*, or *U*: every possible ranking of each individual's preferences and all outcomes of every possible voting rule will be considered equally,
2. The *Pareto condition*, or *P*: if everybody individually likes some choice better at the same time, the society in its voting rule as a whole likes it better as well, and
3. *Liberalism*, or *L* (from which the theorem derives its gist): all individuals in a society must have at least one possibility of choosing differently, so that the social choice under a given voting rule changes as well. That is, as an individual liberal, anyone can exert their freedom of choice at least in some decision with tangible results.

Sen's result shows that this is impossible. The three, rather minimalistic, assumptions cannot all hold together. The paradox—more properly called a proof of contradiction, and a paradox only in the sense of informal logic—is contentious because it appears to contradict the classical liberal idea that markets are both Pareto-efficient and respect individual freedoms.

Sen's proof, set in the context of social choice theory, is similar in many respects to Arrow's impossibility theorem and the Gibbard–Satterthwaite theorem. As a mathematical construct, it also has much wider applicability: it is essentially about cyclical majorities between partially ordered sets, of which at least three must participate in order to give rise to the phenomenon. Since the idea is about pure mathematics and logic, similar arguments abound much further afield. They, for example, lead to the necessity of the fifth normal form in relational database design. The history of the argument also goes deeper, Condorcet's paradox perhaps being the first example of the finite sort.

## Pareto efficiency

### Definition

A particular distribution of goods or outcome of any social process is regarded as *Pareto-efficient* if there is no way to improve one or more people's situations without harming another. Put another way, an outcome is not Pareto-efficient if there is a way to improve at least one person's situation without harming anyone else.

For example, suppose a mother has ten dollars which she intends to give to her two children Carlos and Shannon. Suppose the children each want only money, and they do not get jealous of one another. The following distributions are Pareto-efficient:

| Carlos | Shannon |
| --- | --- |
| $5 | $5 |
| $10 | $0 |
| $2 | $8 |

However, a distribution where the mother gives each of them $2 and wastes the remaining $6 is not Pareto-efficient, because she could have given the wasted money to either child and made that child better off without harming the other.

In this example, it was presumed that a child was made better or worse off by gaining or losing money, respectively, and that neither child gained or lost by evaluating her share in comparison to the other. To be more precise, we must evaluate all possible preferences that the child might have and consider a situation as Pareto-efficient if there is no other social state that at least one person favors (or prefers) and no one disfavors.

### Use in economics

Pareto efficiency is often used in economics as a minimal sense of economic efficiency. If a mechanism does not result in Pareto-efficient outcomes, it is regarded as inefficient, since there was another outcome that could have made some people better off without harming anyone else.

The view that markets produce Pareto-efficient outcomes is regarded as an important and central justification for capitalism. This result was established (with certain assumptions) in an area of study known as general equilibrium theory and is known as the first fundamental theorem of welfare economics. As a result, these results often feature prominently in conservative libertarian justifications of unregulated markets.

## Two examples

### Sen's original example

Sen's original example used a simple society with only two people and only one social issue to consider. The two members of society are named "Lewd" and "Prude". In this society there is a copy of a *Lady Chatterley's Lover* and it must be given either to Lewd to read, to Prude to read, or disposed of - unread. Suppose that Lewd enjoys this sort of reading and would prefer to read it rather than have it disposed of. However, they would get even more enjoyment out of Prude being forced to read it.

Prude thinks that the book is indecent and that it should be disposed of, unread. However, if someone must read it, Prude would prefer to read it rather than Lewd since Prude thinks it would be even worse for someone to read and enjoy the book rather than read it in disgust.

Given these preferences of the two individuals in the society, a social planner must decide what to do. Should the planner force Lewd to read the book, force Prude to read the book or let it go unread? More particularly, the social planner must rank all three possible outcomes in terms of their social desirability. The social planner decides that they should be committed to individual rights, each individual should get to choose whether they, themself will read the book. Lewd should get to decide whether the outcome "Lewd reads" will be ranked higher than "No one reads", and similarly Prude should get to decide whether the outcome "Prude reads" will be ranked higher than "No one reads".

Following this strategy, the social planner declares that the outcome "Lewd reads" will be ranked higher than "No one reads" (because of Lewd's preferences) and that "No one reads" will be ranked higher than "Prude reads" (because of Prude's preferences). Consistency then requires that "Lewd reads" be ranked higher than "Prude reads", and so the social planner gives the book to Lewd to read.

Notice that this outcome is regarded as worse than "Prude reads" by both Prude *and* Lewd, and the chosen outcome is therefore Pareto inferior to another available outcome—the one where Prude is forced to read the book.

### Gibbard's example

Another example was provided by philosopher Allan Gibbard. Suppose there are two individuals Alice and Bob who live next door to each other. Alice loves the color blue and hates red. Bob loves the color green and hates yellow. If each were free to choose the color of their house independently of the other, they would choose their favorite colors. But Alice hates Bob with a passion, and she would gladly endure a red house if it meant that Bob would have to endure his house being yellow. Bob similarly hates Alice, and would gladly endure a yellow house if that meant that Alice would live in a red house.

If each individual is free to choose their own house color, independently of the other, Alice would choose a blue house and Bob would choose a green one. But, this outcome is not Pareto efficient, because both Alice and Bob would prefer the outcome where Alice's house is red and Bob's is yellow. As a result, giving each individual the freedom to choose their own house color has led to an inefficient outcome—one that is inferior to another outcome where neither is free to choose their own color.

Mathematically, we can represent Alice's preferences with this symbol: $\succ \_{A}$ and Bob's preferences with this one: $\succ \_{B}$. We can represent each outcome as a pair: (*Color of Alice's house*, *Color of Bob's house*). As stated Alice's preferences are:

(Blue, Yellow) $\succ \_{A}$ (Red, Yellow) $\succ \_{A}$ (Blue, Green) $\succ \_{A}$ (Red, Green)

And Bob's are:

(Red, Green) $\succ \_{B}$ (Red, Yellow) $\succ \_{B}$ (Blue, Green) $\succ \_{B}$ (Blue, Yellow)

If we allow free and independent choices of both parties we end up with the outcome (Blue, Green) which is dispreferred by both parties to the outcome (Red, Yellow) and is therefore not Pareto efficient.

## The theorem

Suppose there is a society *N* consisting of two or more individuals and a set *X* of two or more social outcomes. (For example, in the Alice and Bob case, *N* consisted of Alice and Bob, and *X* consisted of the four color options ⟨Blue, Yellow⟩, ⟨Blue, Green⟩, ⟨Red, Yellow⟩, and ⟨Red, Green⟩.)

Suppose each individual in the society has a total and transitive preference relation on the set of social outcomes *X*. For notation, the preference relation of an individual *i**N* is denoted by ≼*i*. Each preference relation belongs to the set *Rel(X)* of all total and transitive relations on *X*.

A social choice function is a map which can take any configuration of preference relations of *N* as input and produce a subset of ("chosen") social outcomes as output. Formally, a social choice function is a map

$F:{Rel}(X)^{N}\rightarrow {\mathcal {P}(X)}$

from the set of functions between *N**Rel(X)*, to the power set of *X*. (Intuitively, the social choice function represents a societal principle for choosing one or more social outcomes based on individuals' preferences. By representing the social choice process as a *function* on *Rel(X)**N*, we are tacitly assuming that the social choice function is defined for any possible configuration of preference relations; this is sometimes called the Universal Domain assumption.)

The liberal paradox states that every social choice function satisfies *at most one* of the following properties, never both:

1. **Pareto optimality** (collective efficiency): whenever all individuals of a society strictly prefer an outcome *x* over an outcome *y*, the choice function doesn't pick *y*.
    * Formally, a social choice function *F* is Pareto optimal if whenever *p**Rel(X)**N* is a configuration of preference relations and there are two outcomes *x* and *y* such that *x**i**y* for every individual *i**N*, then *y**F(p)*.
    * Intuitively, Pareto optimality captures an aspect of collective efficiency: the social choice is made so that everyone is collectively as well off as possible, to the extent that every available tradeoff would make someone worse off.
2. **Minimal liberalism** (individual freedom): More than one individual in the society is **decisive** on a pair of social outcomes. (An individual is decisive on a pair of social outcomes *x* and *y* if, whenever they prefer *x* over *y*, the social choice function prefers *x* over *y* regardless of what other members of the society prefer. And similarly whenever they prefer *y* over *x*, the social choice function prefers *y* over *x*.)
    * Formally, a social choice function *F* respects minimal liberalism if there is more than one individual *i**N* for which there exists a pair of outcomes *x**i*, *y**i* on which they are decisive—that is, for every configuration of preference relations *p**Rel(X)**N*, *y**i**F(p)* only when *x**i**i**y**i* (and similarly, *x**i**F(p)* only when *y**i**i**x**i*).
    * As an example of decisiveness: in the Lewd/Prude case, Lewd was decisive on the pair of outcomes ⟨"Lewd reads", "No one reads"⟩ and Prude was decisive on the pair of outcomes ⟨"Prude reads", "No one reads"⟩.
    * Intuitively, minimal liberalism captures an aspect of individual freedom: for some issues, if you prefer x over y (or vice versa), then society respects your preference for x over y even if everyone else is against you. Sen's example is your personal preference for sleeping on your back or your side: on at least one innocuous personal area like this, a liberal society ought to prioritize your individual preference even if everyone else in society would prefer you to sleep another way. The formal requirement is that at least two people are decisive in this way, to rule out the possibility of a single person who dictates society's preferences.

In other words, the liberal paradox states that for every social choice function *F*, there is a configuration of preference relations *p**Rel(X)**N* for which *F* violates either Pareto optimality or Minimal liberalism (or both). In the examples of Sen and Gibbard noted above, the social choice function satisfies minimal liberalism at the expense of Pareto optimality.

## Ways out of the paradox

Because the paradox relies on very few conditions, there are a limited number of ways to escape the paradox. Essentially one must either reject the *universal domain* assumption, the *Pareto principle*, or the *minimal liberalism principle*. Sen himself suggested two ways out, one a rejection of universal domain another a rejection of the Pareto principle.

### Universal domain

Julian Blau proves that Sen's paradox can only arise when individuals have "nosy" preferences—that is when their preference depends not only on their own action but also on others' actions. In the example of Alice and Bob above, Alice has a preference over how Bob paints his house, and Bob has a preference over Alice's house color as well.

Most arguments which demonstrate market efficiency assume that individuals care about only their own consumption and not others' consumption and therefore do not consider the situations that give rise to Sen's paradox. In fact, this shows a strong relationship between Sen's paradox and the well known result that markets fail to produce Pareto outcomes in the presence of externalities. Externalities arise when the choices of one party affect another. Classic examples of externalities include pollution or overfishing. Because of their nosy preferences, Alice's choice imposes a negative externality on Bob and vice versa.

To prevent the paradox, Sen suggests that "The ultimate guarantee for individual liberty may rest not on rules for social choice but on developing individual values that respect each other's personal choices." Doing so would amount to limiting certain types of nosy preferences, or alternatively restricting the application of the Pareto principle only to those situations where individuals fail to have nosy preferences.

Note that if we consider the case of cardinal preferences—for instance, if Alice and Bob both had to state, within certain bounds, how much happiness they would get for each color of each house separately, and the situation which produced the most happiness were chosen—a minimally-liberal solution does not require that they have no nosiness at all, but just that the sum of all "nosy" preferences about one house's color are below some threshold, while the "non-nosy" preferences are all above that threshold. Since there are generally some questions for which this will be true—Sen's classic example is an individual's choice of whether to sleep on their back or their side—the goal of combining minimal liberalism with Pareto efficiency, while impossible to guarantee in all theoretical cases, may not in practice be impossible to obtain.

### Pareto

Alternatively, one could remain committed to the universality of the rules for social choice and to individual rights and instead reject the universal application of the Pareto principle. Sen also hints that this should be how one escapes the paradox:

> What is the moral? It is that in a very basic sense liberal values conflict with the Pareto principle. If someone takes the Pareto principle seriously, as economists seem to do, then he has to face problems of consistency in cherishing liberal values, even very mild ones. Or, to look at it in another way, if someone does have certain liberal values, then he may have to eschew his adherence to Pareto optimality. While the Pareto criterion has been thought to be an expression of individual liberty, it appears that in choices involving more than two alternatives it can have consequences that are, in fact, deeply illiberal.

### Minimal liberalism

Most commentators on Sen's paradox have argued that Sen's minimal liberalism condition does not adequately capture the notion of individual rights. Essentially what is excluded from Sen's characterization of individual rights is the ability to voluntarily form contracts that lay down one's claim to a right.

For example, in the example of Lewd and Prude, although each has a right to refuse to read the book, Prude would voluntarily sign a contract with Lewd promising to read the book on condition that Lewd refrain from doing so. In such a circumstance there was no violation of Prude's or Lewd's rights because each entered the contract willingly. Similarly, Alice and Bob might sign a contract to each paint their houses their dispreferred color on condition that the other does the same.

In this vein, Gibbard provides a weaker version of the minimal liberalism claim which he argues is consistent with the possibility of contracts and which is also consistent with the Pareto principle given any possible preferences of the individuals.

### Dynamism

Alternatively, instead of both Lewd and Prude deciding what to do at the same time, they should do it one after the other. If Prude decides not to read, then Lewd will decide to read. This yields the same outcome. However, if Prude decides to read, Lewd won't. "Prude reads" is preferred by Prude (and also Lewd) to "Lewd reads", so he will decide to read (with no obligation, voluntarily) to get this Pareto efficient outcome. Marc Masat hints that this should be another way out of the paradox:

> If there's, at least, one player without dominant strategy, the game will be played sequentially where players with dominant strategy and need to change it (if they are in the Pareto optimal they don't have to) will be the firsts to choose, allowing to reach the Pareto Efficiency without dictatorship nor restricted domain and also avoiding contract's costs such as time, money or other people. If all players present a dominant strategy, contracts may be used.


title: True Link

abstract: True Link Financial, Inc., is a San Francisco, California based financial technology firm that offers investment accounts and prepaid cards customized for seniors, people with disabilities, and people recovering from addiction. Notable investors include Y Combinator, Khosla Ventures, QED Investors, Mitch Kapor, Alexis Ohanian, Eric Ries, Initialized Capital, Matt Cutts, and Centana Growth Partners.

text:
# True Link

**True Link Financial, Inc.**, is a San Francisco, California based financial technology firm that offers investment accounts and prepaid cards customized for seniors, people with disabilities, and people recovering from addiction. Notable investors include Y Combinator, Khosla Ventures, QED Investors, Mitch Kapor, Alexis Ohanian, Eric Ries, Initialized Capital, Matt Cutts, and Centana Growth Partners.

## Services for seniors with memory loss

True Link offers a Visa card with special controls that prevent common types of fraud and financial abuse targeting the elderly. In addition to fraud-detection algorithms developed by the company, the card can be customized to restrict telemarketer purchases across the board, block certain charitable organizations and allow others, set limits on ATM usage and receive text messages, or selectively block only a few problematic merchants.

The company was inspired by CEO Kai Stinchcombe's experience with his grandmother Ruth, who gave tens of thousands of dollars to pushy telemarketers and charitable solicitors. According to survey research, fourteen percent of seniors are victimized by fraud every year, and an additional fifty-four percent are targeted. The company was formed in November 2012 by co-founders Kai Stinchcombe and Claire McDonnell and launched its product in August 2013. Financial institutions have expressed interest in partnering with the company to help protect their customers from fraud.

According to True Link's research, fraud targeting seniors is a $36.5 billion per year problem. True Link uses fraud detection algorithms and maintains a large database filled with scams and suspicious merchants, and members of True Link's data science team are frequently cited as experts on scams and fraud targeting the elderly.

## Services for people with disabilities or special needs

The company also provides services to people with disabilities, including by working with ABLE accounts, special needs trusts, pooled trusts and representative payees. Its cards are used by the ABLE programs of twelve states and are mentioned in the Social Security Administration operations manual in the context of disbursements from special needs trusts.

In its investment strategies, the company sometimes uses bond ladders which reduce interest rate risk, and a time-based investment strategy described by experts as a "strategy for a lifetime of income." The investment allocations are based on Modern portfolio theory and include stocks and bonds as well as an annuity component in some cases.

## Services for people recovering from addiction

The company provides its cards to people recovering from addiction to help manage cash and prevent relapse. The company's research states that 8 in ten people agree that regaining control of one's finances is among the biggest obstacles in recovering from a substance use disorder, and 77% of adults with a loved one who has a substance abuse disorder felt that their loved one having access to cash could result in relapse.

In August 2016 True Link acquired Next Step Network, LLC., a company that offers payment cards to people recovering from addiction. Around 2.5 million rpeople eceive care at an addiction treatment facility every year.

## InfoBox

True Link Financial, Inc.
|
|  |
| Company type | Private |
| Industry | Financial services |
| Headquarters | San Francisco, California, U.S. |
| Website | www.truelinkfinancial.com |


title: Local linearization method

abstract: In numerical analysis, the local linearization (LL) method is a general strategy for designing numerical integrators for differential equations based on a local (piecewise) linearization of the given equation on consecutive time intervals. The numerical integrators are then iteratively defined as the solution of the resulting piecewise linear equation at the end of each consecutive interval. The LL method has been developed for a variety of equations such as the ordinary, delayed, random and stochastic differential equations. The LL integrators are key component in the implementation of inference methods for the estimation of unknown parameters and unobserved variables of differential equations given time series of observations. The LL schemes are ideals to deal with complex models in a variety of fields as neuroscience, finance, forestry management, control engineering, mathematical statistics, etc.

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# Local linearization method

In numerical analysis, the **local linearization (LL) method** is a general strategy for designing numerical integrators for differential equations based on a local (piecewise) linearization of the given equation on consecutive time intervals. The numerical integrators are then iteratively defined as the solution of the resulting piecewise linear equation at the end of each consecutive interval. The LL method has been developed for a variety of equations such as the ordinary, delayed, random and stochastic differential equations. The LL integrators are key component in the implementation of inference methods for the estimation of unknown parameters and unobserved variables of differential equations given time series of (potentially noisy) observations. The LL schemes are ideals to deal with complex models in a variety of fields as neuroscience, finance, forestry management, control engineering, mathematical statistics, etc.

## Background

Differential equations have become an important mathematical tool for describing the time evolution of several phenomenon, e.g., rotation of the planets around the sun, the dynamic of assets prices in the market, the fire of neurons, the propagation of epidemics, etc. However, since the exact solutions of these equations are usually unknown, numerical approximations to them obtained by numerical integrators are necessary. Currently, many applications in engineering and applied sciences focused in dynamical studies demand the developing of efficient numerical integrators that preserve, as much as possible, the dynamics of these equations. With this main motivation, the Local Linearization integrators have been developed.

## High-order local linearization method

***High-order local linearization (HOLL) method*** is a generalization of the Local Linearization method oriented to obtain high-order integrators for differential equations that preserve the stability and dynamics of the linear equations. The integrators are obtained by splitting, on consecutive time intervals, the solution **x** of the original equation in two parts: the solution **z** of the locally linearized equation plus a high-order approximation of the residual $\mathbf {r} =\mathbf {x} -\mathbf {z} $.

## Local linearization scheme

A ***Local Linearization (LL) scheme*** is the final recursive algorithm that allows the numerical implementation of a discretization derived from the LL or HOLL method for a class of differential equations.

## LL methods for ODEs

Consider the *d*-dimensional Ordinary Differential Equation (ODE)

$\frac {d\mathbf {x} \left(t\right)}{dt}=\mathbf {f} \left(t,\mathbf {x} \left(t\right)\right),\qquad t\in \left\[t\_{0},T\right\],\qquad \qquad \qquad \qquad (4.1)$

with initial condition $\mathbf {x} (t\_{0})=\mathbf {x} \_{0}$, where $\mathbf {f} $ is a differentiable function.

Let $\left(t\right)\_{h}=\\{t\_{n}:n=0,..,N\\}$ be a time discretization of the time interval $\[t\_{0},T\]$ with maximum stepsize *h* such that $t\_{n}\<t\_{n+1}$ and $h\_{n}=t\_{n+1}-t\_{n}\leq h$. After the local linearization of the equation (4.1) at the time step $t\_{n}$ the variation of constants formula yields

$\mathbf {x} (t\_{n}+h)=\mathbf {x} (t\_{n})+\mathbf {\phi } (t\_{n},\mathbf {x} (t\_{n});h)+\mathbf {r} (t\_{n},\mathbf {x} (t\_{n});h),$

where

$\mathbf {\phi } (t\_{n},\mathbf {z} \_{n};h)=\int \limits \_{0}^{h}e^{\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n})(h-s)}(\mathbf {f} (t\_{n},\mathbf {z} \_{n})+\mathbf {f} \_{t}(t\_{n},\mathbf {z} \_{n})s)\\,ds\qquad $

results from the linear approximation, and

$\mathbf {r} (t\_{n},\mathbf {z} \_{n};h)=\int \limits \_{0}^{h}e^{\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n})(h-s)}\mathbf {g} \_{n}(s,\mathbf {x} (t\_{n}+s))\\,ds,\qquad \qquad \qquad (4.2)$

is the residual of the linear approximation. Here, $\mathbf {f} \_{\mathbf {x} }$ and $\mathbf {f} \_{t}$ denote the partial derivatives of **f** with respect to the variables **x** and *t*, respectively, and $\mathbf {g} \_{n}(s,\mathbf {u} )=\mathbf {f} (s,\mathbf {u} )-\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n})\mathbf {u} -\mathbf {f} \_{t}(t\_{n},\mathbf {z} \_{n})(s-t\_{n})-\mathbf {f} (t\_{n},\mathbf {z} \_{n})+\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n})\mathbf {z} \_{n}.$

### Local linear discretization

For a time discretization $\left(t\right)\_{h}$, the *Local Linear discretization* of the ODE (4.1) at each point $t\_{n+1}\in \left(t\right)\_{h}$ is defined by the recursive expression

$\mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\mathbf {\phi } (t\_{n},\mathbf {z} \_{n};h\_{n}),\qquad {\text{ with }\quad \mathbf {z} \_{0}=\mathbf {x} \_{0}.\qquad \qquad \qquad \qquad (4.3)}$

The Local Linear discretization (4.3) converges with order ***2*** to the solution of nonlinear ODEs, but it match the solution of the linear ODEs. The recursion (4.3) is also known as Exponential Euler discretization.

### High-order local linear discretizations

For a time discretization $(t)\_{h},$ a *high-order local linear (HOLL)* discretization of the ODE (4.1) at each point $t\_{n+1}\in (t)\_{h}$ is defined by the recursive expression

$\mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\mathbf {\phi } (t\_{n},\mathbf {z} \_{n};h\_{n})+{\widetilde {\mathbf {r} }(t\_{n},\mathbf {z} \_{n};h\_{n}),\qquad {\text{ with }\quad \mathbf {z} \_{0}=\mathbf {x} \_{0},\qquad \qquad \qquad (4.4)}}$

where $\tilde {r}$ is an order $\alpha $ (\> **2**) approximation to the residual **r** $(i.e.,\left\vert \mathbf {r} (t\_{n},\mathbf {z} \_{n};h)-{\widetilde {\mathbf {r} }(t\_{n},\mathbf {z} \_{n};h)\right\vert \propto h^{\alpha +1}).}$ The HOLL discretization (4.4) converges with order $\alpha $ to the solution of nonlinear ODEs, but it match the solution of the linear ODEs.

HOLL discretizations can be derived in two ways: 1) (quadrature-based) by approximating the integral representation (4.2) of **r**; and 2) (integrator-based) by using a numerical integrator for the differential representation of **r** defined by

$\frac {d\mathbf {r} (t)}{dt}=\mathbf {q} (t\_{n},\mathbf {z} \_{n};t\mathbf {,\mathbf {r} } (t)\mathbf {)} ,\qquad {\text{ with }\qquad \mathbf {r} (t\_{n})=\mathbf {0,} \qquad \qquad \qquad (4.5)}$

for all $t\in \lbrack t\_{k},t\_{k+1}\]$, where

$\mathbf {q} (t\_{n},\mathbf {z} \_{n};s\mathbf {,\xi } )=\mathbf {f} (s,\mathbf {z} \_{n}+\mathbf {\phi } \left(t\_{n},\mathbf {z} \_{n};s-t\_{n}\right)+\mathbf {\xi } )-\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n})\mathbf {\phi } (t\_{n},\mathbf {z} \_{n};s-t\_{n})-\mathbf {f} \_{t}(t\_{n},\mathbf {z} \_{n})(s-t\_{n})-\mathbf {f} (t\_{n},\mathbf {z} \_{n}).$

HOLL discretizations are, for instance, the followings:

* *Locally Linearized Runge Kutta discretization*

$\qquad \mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\mathbf {\phi } (t\_{n},\mathbf {z} \_{n};h\_{n})+h\_{n}\sum \_{j=1}^{s}b\_{j}\mathbf {k} \_{j},\quad {\text{ with }\quad \mathbf {k} \_{i}=\mathbf {q} (t\_{n},\mathbf {z} \_{n};{\text{ }t\_{n}+c\_{i}h\_{n}\mathbf {,} \mathbf {} h\_{n}\sum \_{j=1}^{i-1}a\_{ij}\mathbf {k} \_{j}),}}$

which is obtained by solving (4.5) via a s-stage explicit Runge–Kutta (RK) scheme with coefficients $\mathbf {c} =\left\[c\_{i}\right\],\mathbf {A} =\left\[a\_{ij}\right\]\quad and\quad \mathbf {b} =\left\[b\_{j}\right\]$.

* *Local linear Taylor discretization*

$\mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\mathbf {\phi } (t\_{n},\mathbf {z} \_{n};h\_{n})+\int \_{0}^{h\_{n}e^{(h\_{n}-s)\mathbf {f} \_{\mathbf {x} }\left(t\_{n},\mathbf {z} \_{n}\right)}\sum \_{j=2}^{p}{\frac {\mathbf {c} \_{n,j}{j!}s^{j}\\,ds,{\text{ with }\mathbf {c} \_{n,j}=\left({\frac {d^{j+1}\mathbf {x} (t)}{dt^{j+1}-\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n}){\frac {d^{j}\mathbf {x} (t)}{dt^{j}\right)\mid \_{t=\mathbf {z} \_{n},}}}}}}}}}$

which results from the approximation of $\mathbf {g} \_{n}$ in (4.2) by its order-*p* truncated Taylor expansion.

* *Multistep-type exponential propagation discretization*

$\textstyle \mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\mathbf {\phi } (t\_{n},\mathbf {z} \_{n};h)+h\sum \_{j=0}^{p-1}\gamma \_{j}\nabla ^{j}\mathbf {g} \_{n}(t\_{n},\mathbf {z} \_{n}),\quad with\quad \gamma \_{j}=(-1)^{j}\int \limits \_{0}^{1}e^{(1-\theta )h\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n})}\left({\begin{array}{c}-\theta \\j\end{array}\right)d\theta ,}$

which results from the interpolation of $\mathbf {g} \_{n}$ in (4.2) by a polynomial of degree *p* on $t\_{n},\ldots ,t\_{n-p+1}$, where $\nabla ^{j}\mathbf {g} \_{n}(t\_{m},\mathbf {z} \_{m})$ denotes the *j*-th backward difference of $\mathbf {g} \_{n}(t\_{m},\mathbf {z} \_{m})$.

* *Runge Kutta type Exponential Propagation discretization*

$\textstyle \mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\mathbf {\phi } (t\_{n},\mathbf {z} \_{n};h)+h\sum \_{j=0}^{p-1}\gamma \_{j,p}\nabla ^{j}\mathbf {g} \_{n}(t\_{n},\mathbf {z} \_{n}),\quad {\text{ with }\quad \gamma \_{j,p}=\int \limits \_{0}^{1}e^{(1-\theta )h\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n})}\left({\begin{array}{c}\theta p\\j\end{array}\right)d\theta ,}}$

which results from the interpolation of $\mathbf {g} \_{n}$ in (4.2) by a polynomial of degree *p* on $t\_{n},\ldots ,t\_{n}+(p-1)h/p$,

* *Linealized exponential Adams discretization*

$\textstyle \mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\mathbf {\phi } (t\_{n},\mathbf {z} \_{n};h)+h\sum \_{j=1}^{p-1}\sum \_{l=1}^{j}{\frac {\gamma \_{j+1}{l}\nabla ^{l}\mathbf {g} \_{n}(t\_{n},\mathbf {z} \_{n}),\quad {\text{ with }\quad \gamma \_{j+1}=(-1)^{j+1}\int \limits \_{0}^{1}e^{(1-\theta )h\mathbf {f} \_{\mathbf {x} }\left(t\_{n},\mathbf {z} \_{n}\right)}\theta \left({\begin{array}{c}-\theta \\j\end{array}\right)d\theta ,}}}}$

which results from the interpolation of $\mathbf {g} \_{n}$ in (4.2) by a Hermite polynomial of degree *p* on $t\_{n},\ldots ,t\_{n-p+1}$.

### Local linearization schemes

All numerical implementation $\mathbf {y} \_{n}$ of the LL (or of a HOLL) discretization $\mathbf {z} \_{n}$ involves approximations $\widetilde {\phi }\_{j}$ to integrals $\phi \_{j}$ of the form

$\phi \_{j}(\mathbf {A} ,h)=\int \limits \_{0}^{h}e^{(h-s)\mathbf {A} }s^{j-1}\\,ds,\qquad j=1,2\ldots ,$

where **A** is a *d* × *d* matrix. Every numerical implementation $\mathbf {y} \_{n}$ of the LL (or of a HOLL) $\mathbf {z} \_{n}$ of any order is generically called *Local Linearization scheme*.

#### Computing integrals involving matrix exponential

Among a number of algorithms to compute the integrals $\phi \_{j}$, those based on rational Padé and Krylov subspaces approximations for exponential matrix are preferred. For this, a central role is playing by the expression

$\sum \nolimits \_{i=1}^{l}\phi \_{i}(\mathbf {A} ,h)\mathbf {a} \_{i}=\mathbf {L} e^{h\mathbf {H} }\mathbf {r} ,$

where $\mathbf {a} \_{i}$ are *d*-dimensional vectors,

$\mathbf {H} ={\begin{bmatrix}\mathbf {A} &\mathbf {v} \_{l}&\mathbf {v} \_{l-1}&\cdots &\mathbf {v} \_{1}\\\mathbf {0} &\mathbf {0} &1&\cdots &0\\\mathbf {0} &\mathbf {0} &0&\ddots &0\\\vdots &\vdots &\vdots &\ddots &1\\\mathbf {0} &\mathbf {0} &0&\cdots &0\end{bmatrix}\in \mathbb {R} ^{(d+l)\times (d+l)},}$

$\mathbf {L} =\[\mathbf {I} \quad \mathbf {0} \_{d\times l}\]$, $\mathbf {r} =\[\mathbf {0} \_{1\times (d+l-1)}\quad 1\]^{\intercal },$ $\mathbf {v} \_{i}=\mathbf {a} \_{i}(i-1)!$, being $\mathbf {I} $ the *d*-dimensional identity matrix.

If $\mathbf {P} \_{p,q}(2^{-k}\mathbf {H} h)$ denotes the (*p*; *q*)-Padé approximation of $e^{2^{-k}\mathbf {H} h}$ and *k* is the smallest natural number such that $|2^{-k}\mathbf {H} h|\leq {\frac {1}{2},then}$

$\left\vert \sum \nolimits \_{i=1}^{l}\phi \_{i}(\mathbf {A} ,h)\mathbf {a} \_{i}-\mathbf {L} \left(\mathbf {\mathbf {P} } \_{p,q}(2^{-k}\mathbf {H} h)\right)^{2^{k}\mathbf {r} \right\vert \varpropto h^{p+q+1}.}$

If $\mathbf {\mathbf {k} } \_{m,k}^{p,q}(h,\mathbf {H} ,\mathbf {r} )$ denotes the *(m; p; q; k)* Krylov-Padé approximation of $e^{h\mathbf {H} }\mathbf {r} $, then

$\left\vert \sum \nolimits \_{i=1}^{l}\phi \_{i}(\mathbf {A} ,h)\mathbf {a} \_{i}-\mathbf {L\mathbf {k} } \_{m,k}^{p,q}(h,\mathbf {H} ,\mathbf {r} )\right\vert \varpropto h^{\min({m,p+q+1})},$

where $m\leq d$ is the dimension of the Krylov subspace.

#### Order-2 LL schemes

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {L} (\mathbf {P} \_{p,q}(2^{-k\_{n}\mathbf {M} \_{n}h\_{n}))^{2^{k\_{n}\mathbf {r,} }}}$ $\qquad \qquad (4.6)$

where the matrices $\mathbf {M} \_{n}$, **L** and **r** are defined as

$\mathbf {M} \_{n}={\begin{bmatrix}\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {y} \_{n})&\mathbf {f} \_{t}(t\_{n},\mathbf {y} \_{n})&\mathbf {f} (t\_{n},\mathbf {y} \_{n})\\0&0&1\\0&0&0\end{bmatrix}\in \mathbb {R} ^{(d+2)\times (d+2)},}$

$\mathbf {L} =\left\[{\begin{array}{ll}\mathbf {I} &\mathbf {0} \_{d\times 2}\end{array}\right\]}$ and $\mathbf {r} ^{\intercal }=\left\[{\begin{array}{ll}\mathbf {0} \_{1\times (d+1)}&1\end{array}\right\]}$ with $p+q\>1$ . For large systems of ODEs

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {L\mathbf {k} } \_{n},k\_{n}^{p,q}(h\_{n},\mathbf {M} \_{n},\mathbf {r} )\mathbf {,} \qquad {\text{ with }\qquad m\_{n}\>2.}$

#### Order-3 LL-Taylor schemes

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {L} \_{1}(\mathbf {P} \_{p,q}(2^{-k\_{n}\mathbf {T} \_{n}h\_{n}))^{2^{k\_{n}\mathbf {r} \_{1}\mathbf {,} }}}$ $\qquad \qquad (4.7)$

where for autonomous ODEs the matrices $\mathbf {T} \_{n},\mathbf {L} \_{1}$ and $\mathbf {r} \_{1}$ are defined as

$\mathbf {T} \_{n}=\left\[{\begin{array}{cccc}\mathbf {f} \_{\mathbf {x} }(\mathbf {y} \_{n})&(\mathbf {I} \otimes \mathbf {f} ^{\intercal }(\mathbf {y} \_{n}))\mathbf {f} \_{\mathbf {xx} }(\mathbf {y} \_{n})\mathbf {f} (\mathbf {y} \_{n})&\mathbf {0} &\mathbf {f} (\mathbf {y} \_{n})\\0&0&0&0\\0&0&0&1\\0&0&0&0\end{array}\right\]\in \mathbb {R} ^{(d+3)\times (d+3)},}$

$\mathbf {L} \_{1}=\left\[{\begin{array}{ll}\mathbf {I} &\mathbf {0} \_{d\times 3}\end{array}\right\]\quad and\quad \mathbf {r} \_{1}^{\intercal }=\left\[{\begin{array}{ll}\mathbf {0} \_{1\times (d+2)}&1\end{array}\right\]}}$. Here, $\mathbf {f} \_{\mathbf {xx} }$ denotes the second derivative of **f** with respect to **x**, and *p + q \> 2*. For large systems of ODEs

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {L\mathbf {k} } \_{n},k\_{n}^{p,q}(h\_{n},\mathbf {T} \_{n},\mathbf {r} )\mathbf {,} \qquad {\text{ with }\qquad m\_{n}\>3.}$

#### Order-4 LL-RK schemes

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {u} \_{4}+{\frac {h\_{n}{6}(2\mathbf {k} \_{2}+2\mathbf {k} \_{3}+\mathbf {k} \_{4}),\quad }}$ $\qquad \qquad (4.8)$

where

$\mathbf {u} \_{j}=\mathbf {L} (\mathbf {P} \_{p,q}(2^{-\kappa \_{j}\mathbf {M} \_{n}c\_{j}h\_{n}))^{2^{\kappa \_{j}\mathbf {r} }}}$

and

$\mathbf {k} \_{j}=\mathbf {f} \left(t\_{n}+c\_{j}h\_{n},\mathbf {y} \_{n}+\mathbf {u} \_{j}+c\_{j}h\_{n}\mathbf {k} \_{j-1}\right)-\mathbf {f} \left(t\_{n},\mathbf {y} \_{n}\right)-\mathbf {f} \_{\mathbf {x} }\left(t\_{n},\mathbf {y} \_{n}\right)\mathbf {u} \_{j}\ -\mathbf {f} \_{t}\left(t\_{n},\mathbf {y} \_{n}\right)c\_{j}h\_{n},$

with $\mathbf {k} \_{1}\equiv \mathbf {0} ,c=\left\[{\begin{array}{cccc}0&{\frac {1}{2}&{\frac {1}{2}&1\end{array}\right\],}}}$ and *p + q \> 3*. For large systems of ODEs, the vector $\mathbf {u} \_{j}$ in the above scheme is replaced by $\mathbf {u} \_{j}=\mathbf {L\mathbf {k} } \_{j},k\_{j}^{p,q}(c\_{j}h\_{n},\mathbf {M} \_{n},\mathbf {r} )$ with $m\_{j}\>4.$

#### Locally linearized Runge–Kutta scheme of Dormand and Prince

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {u} \_{s}+h\_{n}\sum \_{j=1}^{s}b\_{j}\mathbf {k} \_{j}\qquad {\text{ and }\qquad {\widehat {\mathbf {y} }\_{n+1}=\mathbf {y} \_{n}+\mathbf {u} \_{s}+h\_{n}\sum \_{j=1}^{s}{\widehat {b}\_{j}\mathbf {k} \_{j},\quad }}}$ $\qquad \qquad (4.9)$

where *s* = 7 is the number of stages,

$\mathbf {k} \_{j}=\mathbf {f(} t\_{n}+c\_{j}h\_{n},\mathbf {y} \_{n}+\mathbf {u} \_{j}+h\_{n}\sum \_{i=1}^{s-1}a\_{j,i}\mathbf {k} \_{i})-\mathbf {f} \left(t\_{n},\mathbf {y} \_{n}\right)-\mathbf {f} \_{\mathbf {x} }\left(t\_{n},\mathbf {y} \_{n}\right)\mathbf {u} \_{j}\ -\mathbf {f} \_{t}\left(t\_{n},\mathbf {y} \_{n}\right)c\_{j}h\_{n},$

with $\mathbf {k} \_{1}\equiv \mathbf {0} $, and $a\_{j,i},b\_{j},{\widehat {b}\_{j}\quad and\quad c\_{j}}$ are the Runge–Kutta coefficients of Dormand and Prince and *p* \+ *q* \> 4. The vector $\mathbf {u} \_{j}$ in the above scheme is computed by a Padé or Krylor–Padé approximation for small or large systems of ODE, respectively.

#### Stability and dynamics

By construction, the LL and HOLL discretizations inherit the stability and dynamics of the linear ODEs, but it is not the case of the LL schemes in general. With $p\leq q\leq p+2$, the LL schemes (4.6)-(4.9) are *A*-stable. With *q* = *p* \+ 1 or *q* = *p* \+ 2, the LL schemes (4.6)–(4.9) are also *L*-stable. For linear ODEs, the LL schemes (4.6)-(4.9) converge with order *p* \+ *q*. In addition, with *p = q = 6* and $m\_{n}$ = *d*, all the above described LL schemes yield to the ″exact computation″ (up to the precision of the floating-point arithmetic) of linear ODEs on the current personal computers. This includes stiff and highly oscillatory linear equations. Moreover, the LL schemes (4.6)-(4.9) are regular for linear ODEs and inherit the symplectic structure of Hamiltonian harmonic oscillators. These LL schemes are also linearization preserving, and display a better reproduction of the stable and unstable manifolds around hyperbolic equilibrium points and periodic orbits that other numerical schemes with the same stepsize. For instance, Figure 1 shows the phase portrait of the ODEs

$$\begin{aligned}&{\frac {dx\_{1}{dt}=-2x\_{1}+x\_{2}+1-\mu f(x\_{1},\lambda )\qquad \qquad (4.10)\\\\[6pt\]&{\frac {dx\_{2}{dt}=x\_{1}-2x\_{2}+1-\mu f(x\_{2},\lambda )\qquad \qquad \quad (4.11)\end{aligned}}}}}$$

with $f(u,\lambda )=u(1+u+\lambda u^{2})^{-1}$, $\mu =15$ and $\lambda =57$, and its approximation by various schemes. This system has two stable stationary points and one unstable stationary point in the region $0\leq x\_{1},x\_{2}\leq 1$.

## LL methods for DDEs

Consider the *d*-dimensional Delay Differential Equation (DDE)

$\frac {d\mathbf {x} (t)}{dt}=\mathbf {f} (t,\mathbf {x} (t),\mathbf {x} \_{t}(-\tau \_{1}),\ldots ,\mathbf {x} \_{t}(-\tau \_{m})),\qquad t\in \[t\_{0},T\],\qquad \qquad (5.1)$

with *m* constant delays $\tau \_{i}\>0$ and initial condition $\mathbf {x} \_{0}(s)=\mathbf {\varphi } (s)$ for all $s\in \[-\tau ,0\],$ where **f** is a differentiable function, $\mathbf {x} \_{t}:\[-\tau ,0\]\longrightarrow \mathbb {R} ^{d}$ is the segment function defined as

$\mathbf {x} \_{t}(s):=\mathbf {x} (t+s),{\text{ }s\in \[-\tau ,0\],}$

for all $t\in \[t\_{0},T\],\mathbf {\varphi } :\[-\tau ,0\]\longrightarrow \mathbb {R} ^{d}$ is a given function, and $\tau =\max \left\\{\tau \_{1},\ldots ,\tau \_{m}\right\\}.$

### Local linear discretization

For a time discretization $(t)\_{h}$ , the *Local Linear discretization* of the DDE (5.1) at each point $t\_{n+1}\in (t)\_{h}$ is defined by the recursive expression

$\mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\Phi (t\_{n},\mathbf {z} \_{n},h\_{n};{\widetilde {\mathbf {z} }\_{n}^{1},\ldots ,{\widetilde {\mathbf {z} }\_{n}^{m}),\qquad \qquad (5.2)}}$

where

$\Phi (t\_{n},\mathbf {z} \_{n},h\_{n};{\widetilde {\mathbf {z} }\_{n}^{1},\ldots ,{\widetilde {\mathbf {z} }\_{n}^{m})=\int \limits \_{0}^{h\_{n}e^{\mathbf {A} \_{n}(h\_{n}-u)}\left\[\sum \limits \_{i=1}^{m}\mathbf {B} \_{n}^{i}({\widetilde {\mathbf {z} }\_{n}^{i}(u-\tau \_{i})-{\widetilde {\mathbf {z} }\_{n}^{i}(-\tau \_{i}))+\mathbf {d} \_{n}\right\]\\,du+\int \limits \_{0}^{h\_{n}\int \limits \_{0}^{u}e^{\mathbf {A} \_{n}(h\_{n}-u)}\mathbf {c} \_{n}\\,dr\\,du}}}}}}$

$\widetilde {\mathbf {z} }\_{n}^{i}:\left\[-\tau \_{i},0\right\]\longrightarrow \mathbb {R} ^{d}$ is the segment function defined as

$\widetilde {\mathbf {z} }\_{n}^{i}(s):={\widetilde {\mathbf {z} }^{i}(t\_{n}+s),{\text{ }s\in \[-\tau \_{i},0\],}}$

and $\widetilde {\mathbf {z} }^{i}:\left\[t\_{n}-\tau \_{i},t\_{n}\right\]\longrightarrow \mathbb {R} ^{d}$ is a suitable approximation to $\mathbf {x} (t)$ for all $t\in \lbrack t\_{n}-\tau \_{i},t\_{n}\]$ such that $\widetilde {\mathbf {z} }^{i}(t\_{n})=\mathbf {z} \_{n}.$ Here,

$\mathbf {A} \_{n}=\mathbf {f} \_{x}(t\_{n},\mathbf {z} \_{n},{\widetilde {\mathbf {z} }\_{n}^{1}(-\tau \_{1}),\ldots ,{\widetilde {\mathbf {z} }\_{n}^{m}(-\tau \_{d})),{\text{ }\mathbf {B} \_{n}^{i}=\mathbf {f} \_{t}(-\tau \_{i})}(t\_{n},\mathbf {z} \_{n},{\widetilde {\mathbf {z} }\_{n}^{1}(-\tau \_{1}),\ldots ,{\widetilde {\mathbf {z} }\_{n}^{m}(-\tau \_{d}))}}}}$

are constant matrices and

$\mathbf {c} \_{n}=\mathbf {f} \_{t}(t\_{n},\mathbf {z} \_{n},{\widetilde {\mathbf {z} }\_{n}^{1}(-\tau \_{1}),\ldots ,{\widetilde {\mathbf {z} }\_{n}^{m}(-\tau \_{d})){\text{ and }\mathbf {d} \_{n}=\mathbf {f(} t\_{n},\mathbf {z} \_{n},{\widetilde {\mathbf {z} }\_{n}^{1}(-\tau \_{1}),\ldots ,{\widetilde {\mathbf {z} }\_{n}^{m}(-\tau \_{d}))}}}}}$

are constant vectors. $\mathbf {f} \_{t},\mathbf {f} \_{x}\quad and\quad \mathbf {f} \_{t}(-\tau \_{i})$ denote, respectively, the partial derivatives of **f** with respect to the variables *t* and ***x***, and $\mathbf {x} \_{t}(-\tau \_{i})$. The Local Linear discretization (5.2) converges to the solution of (5.1) with order $\alpha =\min\\{2,r\\},$ if $\widetilde {\mathbf {z} }\_{n}^{i}$ approximates $\mathbf {z} \_{n}^{i}$ with order $r\quad (i.e.,\left\vert \mathbf {z} \_{n}^{i}\mathbf {(} u-\tau \_{i}\mathbf {)} -{\widetilde {\mathbf {z} }\_{n}^{i}\mathbf {(} u-\tau \_{i}\mathbf {)} \right\vert \propto h\_{n}^{r}}$ for all $u\in \lbrack 0,h\_{n}\])$.

### Local linearization schemes

Depending on the approximations $\widetilde {\mathbf {z} }\_{n}^{i}$ and on the algorithm to compute $\mathbf {\phi } $ different Local Linearizations schemes can be defined. Every numerical implementation $\mathbf {y} \_{n}$ of a Local Linear discretization $\mathbf {z} \_{n}$ is generically called *local linearization scheme*.

#### Order-2 polynomial LL schemes

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {L} (\mathbf {P} \_{p,q}(2^{-k\_{n}\mathbf {M} \_{n}h\_{n}))^{2^{k\_{n}\mathbf {r,} \quad }}}$ $\qquad (5.3)$

where the matrices $\mathbf {M} \_{n},\mathbf {L} $ and $\mathbf {r} $ are defined as

$\mathbf {M} \_{n}={\begin{bmatrix}\mathbf {A} \_{n}&\mathbf {c} \_{n}+\sum \limits \_{i=1}^{m}\mathbf {B} \_{n}^{i}\mathbf {\alpha } \_{n}^{i}&\mathbf {d} \_{n}\\0&0&1\\0&0&0\end{bmatrix}\in \mathbb {R} ^{(d+2)\times (d+2)},}$

$\mathbf {L} =\left\[{\begin{array}{ll}\mathbf {I} &\mathbf {0} \_{d\times 2}\end{array}\right\]}$ and $\mathbf {r} ^{\intercal }=\left\[{\begin{array}{ll}\mathbf {0} \_{1\times (d+1)}&1\end{array}\right\],h\_{n}\leq \tau }$, and $p+q\>1$. Here, the matrices $\mathbf {A} \_{n}$, $\mathbf {B} \_{n}^{i}$, $\mathbf {c} \_{n}$ and $\mathbf {d} \_{n}$ are defined as in (5.2), but replacing $\mathbf {z} $ by $\mathbf {y} $ and $\mathbf {\alpha } \_{n}^{i}=(\mathbf {y} (t\_{n+1}-\tau \_{i})-\mathbf {y} (t\_{n}-\tau \_{i}))/h\_{n},$ where

$\mathbf {y} \left(t\right)=\mathbf {y} \_{t}+\mathbf {L} (\mathbf {P} \_{p,q}(2^{-k\_{n}\mathbf {M} \_{t}(t-t\_{t})))^{2^{k\_{n}\mathbf {r} ,}}}$

with $n\_{t}=\max\\{n=0,1,2,...,:t\_{n}\leq t{\text{ and }t\_{n}\in \left(t\right)\_{h}\\}}$, is the *Local Linear Approximation* to the solution of (5.1) defined through the LL scheme (5.3) for all $t\in \lbrack t\_{0},t\_{n}\]$ and by $\mathbf {y} \left(t\right)=\mathbf {\varphi } \left(t\right)$ for $t\in \left\[t\_{0}-\tau ,t\_{0}\right\]$. For large systems of DDEs

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {L\mathbf {k} } \_{n},k\_{n}^{p,q}(h\_{n},\mathbf {M} \_{n},\mathbf {r} )\quad and\quad \mathbf {y} \left(t\right)=\mathbf {y} \_{t}+\mathbf {L\mathbf {k} } \_{n\_{t},k\_{t}^{p,q}(t-t\_{t},\mathbf {M} \_{t},\mathbf {r} ),}$

with $p+q\>1$ and $m\_{n}\>2$. Fig. 2 Illustrates the stability of the LL scheme (5.3) and of that of an explicit scheme of similar order in the integration of a stiff system of DDEs.

## LL methods for RDEs

Consider the *d-*dimensional Random Differential Equation (RDE)

$\frac {d\mathbf {x} \left(t\right)}{dt}=\mathbf {f} (\mathbf {x} (t),\mathbf {\xi } (t)),\quad t\in \left\[t\_{0},T\right\],\qquad \qquad \qquad (6.1)$

with initial condition $\mathbf {x} (t\_{0})=\mathbf {x} \_{0},$ where $\mathbf {\xi } $ is a *k*-dimensional separable finite continuous stochastic process, and **f** is a differentiable function. Suppose that a realization (path) of $\mathbf {\xi } $ is given.

### Local Linear discretization

For a time discretization $\left(t\right)\_{h}$, the *Local Linear discretization* of the RDE (6.1) at each point $t\_{n+1}\in \left(t\right)\_{h}$ is defined by the recursive expression

$\mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\mathbf {\phi } (t\_{n},\mathbf {z} \_{n};h\_{n}),\qquad {\text{ with }\qquad \mathbf {z} \_{0}=\mathbf {x} \_{0},}$

where

$\mathbf {\phi } (t\_{n},\mathbf {z} \_{n};h\_{n})=\int \limits \_{0}^{h\_{n}e^{\mathbf {f} \_{\mathbf {x} }(\mathbf {z} \_{n},\mathbf {\xi } (t\_{n}))(h\_{n}-u)}(\mathbf {f(z} \_{n},\mathbf {\xi } (t\_{n}))+\mathbf {f} \_{\mathbf {\xi } }(\mathbf {z} \_{n},\mathbf {\xi } (t\_{n}))({\widetilde {\mathbf {\xi } }(t\_{n}+u)-{\widetilde {\mathbf {\xi } }(t\_{n})))\\,du}}}$

and $\widetilde {\mathbf {\xi } }$ is an approximation to the process $\mathbf {\xi } $ for all $t\in \left\[t\_{0},T\right\].$ Here, $\mathbf {f} \_{x}$ and $\mathbf {f} \_{\xi }$ denote the partial derivatives of $\mathbf {f} $ with respect to $\mathbf {x} $ and $\xi $, respectively.

### Local linearization schemes

Depending on the approximations $\widetilde {\mathbf {\xi } }$ to the process $\mathbf {\xi } $ and of the algorithm to compute $\mathbf {\phi } $, different Local Linearizations schemes can be defined. Every numerical implementation $\mathbf {y} \_{n}$ of the local linear discretization $\mathbf {z} \_{n}$ is generically called *local linearization scheme.*

#### LL schemes

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {L} (\mathbf {P} \_{p,q}(2^{-k\_{n}\mathbf {M} \_{n}h\_{n}))^{2^{k\_{n}\mathbf {r,} \quad }}}$

where the matrices $\mathbf {M} \_{n},\quad \mathbf {L} \quad and\quad \mathbf {r} $ are defined as

$\mathbf {M} \_{n}=\left\[{\begin{array}{ccc}\mathbf {f} \_{\mathbf {x} }\left(\mathbf {y} \_{n},\mathbf {\xi } (t\_{n})\right)&\mathbf {f} \_{\mathbf {\xi } }(\mathbf {y} \_{n},\mathbf {\xi } (t\_{n})(\mathbf {\xi } (t\_{n+1})-\mathbf {\xi } (t\_{n}))/h\_{n}&\mathbf {f} \left(\mathbf {y} \_{n},\mathbf {\xi } (t\_{n})\right)\\0&0&1\\0&0&0\end{array}\right\]}$

$\mathbf {L} =\left\[{\begin{array}{ll}\mathbf {I} &\mathbf {0} \_{d\times 2}\end{array}\right\]}$, $\mathbf {r} ^{\intercal }=\left\[{\begin{array}{ll}\mathbf {0} \_{1\times (d+1)}&1\end{array}\right\]}$, and *p+q\>1*. For large systems of RDEs,

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {L\mathbf {k} } \_{n},k\_{n}^{p,q}(h\_{n},\mathbf {M} \_{n},\mathbf {r} ),\quad p+q\>1\quad and\quad m\_{n}\>2.$

The convergence rate of both schemes is $min\\{2,2\gamma \\}$, where is $\gamma $ the exponent of the Holder condition of $\mathbf {\xi } $.

Figure 3 presents the phase portrait of the RDE

$\frac {dx\_{1}{dt}=-x\_{2}+\left(1-x\_{1}^{2}-x\_{2}^{2}\right)x\_{1}\sin(w^{H}(t))^{2},\quad \qquad x\_{1}(0)=0.8\qquad (6.2)}$

$\frac {dx\_{2}{dt}=x\_{1}+(1-x\_{1}^{2}-x\_{2}^{2})x\_{2}\sin(w^{H}(t))^{2},\qquad \qquad x\_{2}(0)=0.1,\qquad (6.3)}$

and its approximation by two numerical schemes, where $w^{H}$ denotes a fractional Brownian process with Hurst exponent *H=0.45*.

## Strong LL methods for SDEs

Consider the *d*-dimensional Stochastic Differential Equation (SDE)

$d\mathbf {x} (t)=\mathbf {f} (t,\mathbf {x} (t))dt+\sum \limits \_{i=1}^{m}\mathbf {g} \_{i}(t)d\mathbf {w} ^{i}(t),\quad t\in \left\[t\_{0},T\right\],\qquad \qquad \qquad (7.1)$

with initial condition $\mathbf {x} (t\_{0})=\mathbf {x} \_{0}$, where the drift coefficient $\mathbf {f} $ and the diffusion coefficient $\mathbf {g} \_{i}$ are differentiable functions, and $\mathbf {w=(\mathbf {w} } ^{1},\ldots ,\mathbf {w} ^{m}\mathbf {)} $ is an *m*-dimensional standard Wiener process.

### Local linear discretization

For a time discretization $\left(t\right)\_{h}$ , the order- $\mathbb {\gamma } $ (=1,1.5) *Strong Local Linear discretization* of the solution of the SDE (7.1) is defined by the recursive relation

$\mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\mathbf {\phi } \_{\mathbb {\gamma } }(t\_{n},\mathbf {z} \_{n};h\_{n})+\mathbf {\xi } (t\_{n},\mathbf {z} \_{n};h\_{n}),\quad with\quad \mathbf {z} \_{0}=\mathbf {x} \_{0},$

where

$\mathbf {\phi } \_{\mathbb {\gamma } }(t\_{n},\mathbf {z} \_{n};\delta )=\int \_{0}^{\delta }e^{\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {y} \_{n})(\delta -u)}(\mathbf {f(} t\_{n},\mathbf {z} \_{n})+\mathbf {a} ^{\mathbb {\gamma } }(t\_{n},\mathbf {z} \_{n})u)du$

and

$\mathbf {\xi } \left(t\_{n},\mathbf {z} \_{n};\delta \right)=\sum \limits \_{i=1}^{m}\int \nolimits \_{n}^{t\_{n}+\delta }e^{\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n})(t\_{n}+\delta -u)}\mathbf {g} \_{i}(u)\\,d\mathbf {w} ^{i}(u).$

Here,

$\mathbf {a} ^{\mathbb {\gamma } }(t\_{n},\mathbf {z} \_{n})=\left\\{{\begin{array}{cl}\mathbf {f} \_{t}(t\_{n},\mathbf {z} \_{n})&{\text{for }\qquad \mathbb {\gamma } =1\\\mathbf {f} \_{t}(t\_{n},\mathbf {z} \_{n})+{\frac {1}{2}\sum \limits \_{j=1}^{m}(\mathbf {I} \otimes \mathbf {g} \_{j}^{\intercal }(t\_{n}))\mathbf {f} \_{\mathbf {xx} }(t\_{n},\mathbf {z} \_{n})\mathbf {g} \_{j}(t\_{n})&{\text{for }\quad \mathbb {\gamma } =1.5,\end{array}\right.}}}}}$

$\mathbf {f} \_{\mathbf {x} },\mathbf {f} \_{t}$ denote the partial derivatives of $\mathbf {f} $ with respect to the variables $\mathbf {x} $ and ***t***, respectively, and $\mathbf {f} \_{\mathbf {xx} }$ the Hessian matrix of $\mathbf {f} $ with respect to $\mathbf {x} $. The strong Local Linear discretization $\mathbf {z} \_{n+1}$ converges with order $\mathbb {\gamma } $ (= 1, 1.5) to the solution of (7.1).

### High-order local linear discretizations

After the local linearization of the drift term of (7.1) at $(t\_{n},\mathbf {z} \_{n})$, the equation for the residual $\mathbf {r} $ is given by

$d\mathbf {r} (t)=\mathbf {q} \_{\gamma }(t\_{n},\mathbf {z} \_{n};t\mathbf {,\mathbf {r} } (t))\\,dt+\sum \limits \_{i=1}^{m}\mathbf {g} \_{i}(t)\\,d\mathbf {w} ^{i}(t)\mathbf {,} \qquad \mathbf {r} (t\_{n})=\mathbf {0} $

for all $t\in \lbrack t\_{n},t\_{n+1}\]$, where

$\mathbf {q} \_{\gamma }(t\_{n},\mathbf {z} \_{n};s\mathbf {,\xi } )=\mathbf {f} (s,\mathbf {z} \_{n}+\mathbf {\phi } \_{\gamma }(t\_{n},\mathbf {z} \_{n};s-t\_{n})+\mathbf {\xi } )-\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n})\mathbf {\phi } \_{\gamma }(t\_{n},\mathbf {z} \_{n};s-t\_{n})-\mathbf {a} ^{\gamma }(t\_{n},\mathbf {z} \_{n})(s-t\_{n})-\mathbf {f} (t\_{n},\mathbf {z} \_{n}).$

A *high-order local linear discretization* of the SDE *(7.1)* at each point $t\_{n+1}\in (t)\_{h}$ is then defined by the recursive expression

$\mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\mathbf {\phi } \_{\gamma }(t\_{n},\mathbf {z} \_{n};h\_{n})+{\widetilde {\mathbf {r} }(t\_{n},\mathbf {z} \_{n};h\_{n}),\qquad {\text{ with }\qquad \mathbf {z} \_{0}=\mathbf {x} \_{0},}}$

where $\widetilde {\mathbf {r} }$ is a strong approximation to the residual $\mathbf {r} $ of order $\alpha $ higher than **1.5**. The strong HOLL discretization $\mathbf {z} \_{n+1}$ converges with order $\alpha $ to the solution of (7.1).

### Local linearization schemes

Depending on the way of computing $\mathbf {\phi } \_{\mathbb {\gamma } }$ , $\mathbf {\xi } $ and $\widetilde {\mathbf {r} }$ different numerical schemes can be obtained. Every numerical implementation $\mathbf {y} \_{n}$ of a strong Local Linear discretization $\mathbf {z} \_{n}$ of any order is generically called *Strong Local Linearization (SLL) scheme*.

#### Order 1 SLL schemes

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {L} (\mathbf {P} \_{p,q}(2^{-k\_{n}\mathbf {M} \_{n}h\_{n}))^{2^{k\_{n}\mathbf {r+} \sum \limits \_{i=1}^{m}\mathbf {g} \_{i}(t\_{n})\Delta \mathbf {w} \_{n}^{i},\quad }}}$ $\qquad \qquad (7.2)$

where the matrices $\mathbf {M} \_{n}$, $\mathbf {L} $ and $\mathbf {r} $ are defined as in (4.6), $\Delta \mathbf {w} \_{n}^{i}$ is an i.i.d. zero mean Gaussian random variable with variance $h\_{n}$, and *p* + *q* \> 1. For large systems of SDEs, in the above scheme $(\mathbf {P} \_{p,q}(2^{-k\_{n}\mathbf {M} \_{n}h\_{n}))^{2^{k\_{n}\mathbf {r} }}}$ is replaced by $\mathbf {\mathbf {k} } \_{n},k\_{n}^{p,q}(h\_{n},\mathbf {M} \_{n},\mathbf {r} )$.

#### Order 1.5 SLL schemes

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {L} (\mathbf {P} \_{p,q}(2^{-k\_{n}\mathbf {M} \_{n}h\_{n}))^{2^{k\_{n}\mathbf {r} +\sum \limits \_{i=1}^{m}\left(\mathbf {g} \_{i}(t\_{n})\Delta \mathbf {w} \_{n}^{i}\mathbf {f} \_{\mathbf {x} }(t\_{n},{\widetilde {\mathbf {y} }\_{n})\mathbf {g} \_{i}(t\_{n})\Delta \mathbf {z} \_{n}^{i}+{\frac {d\mathbf {g} \_{i}(t\_{n})}{dt}(\Delta \mathbf {w} \_{n}^{i}h\_{n}-\Delta \mathbf {z} \_{n}^{i})\right),\qquad \qquad (7.3)}}}}}$

where the matrices $\mathbf {M} \_{n}$, $\mathbf {L} $ and $\mathbf {r} $ are defined as

$\mathbf {M} \_{n}={\begin{bmatrix}\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {y} \_{n})&\mathbf {f} \_{t}(t\_{n},\mathbf {y} \_{n})+{\frac {1}{2}\sum \limits \_{j=1}^{m}\left(\mathbf {I} \otimes \mathbf {g} \_{j}^{\intercal }(t\_{n})\right)\mathbf {f} \_{\mathbf {xx} }(t\_{n},\mathbf {y} \_{n})\mathbf {g} \_{j}(t\_{n})&\mathbf {f} (t\_{n},\mathbf {y} \_{n})\\0&0&1\\0&0&0\end{bmatrix}\in \mathbb {R} ^{(d+2)\times (d+2)},}}$

$\mathbf {L} =\left\[{\begin{array}{ll}\mathbf {I} &\mathbf {0} \_{d\times 2}\end{array}\right\],\mathbf {r} ^{\intercal }=\left\[{\begin{array}{ll}\mathbf {0} \_{1\times (d+1)}&1\end{array}\right\]}}$, $\Delta \mathbf {z} \_{n}^{i}$ is a i.i.d. zero mean Gaussian random variable with variance $E\left((\Delta \mathbf {z} \_{n}^{i})^{2}\right)={\frac {1}{3}h\_{n}^{3}}$ and covariance $E(\Delta \mathbf {w} \_{n}^{i}\Delta \mathbf {z} \_{n}^{i})={\frac {1}{2}h\_{n}^{2}}$ and *p+q\>1* . For large systems of SDEs, in the above scheme $(\mathbf {P} \_{p,q}(2^{-k\_{n}\mathbf {M} \_{n}h\_{n}))^{2^{k\_{n}\mathbf {r} }}}$ is replaced by $\mathbf {\mathbf {k} } \_{n},k\_{n}^{p,q}(h\_{n},\mathbf {M} \_{n},\mathbf {r} )$.

#### Order 2 SLL-Taylor schemes

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {L} (\mathbf {P} \_{p,q}(2^{-k\_{n}\mathbf {M} \_{n}h\_{n}))^{2^{k\_{n}\mathbf {r} +\sum \limits \_{j=1}^{m}\mathbf {g} \_{j}\left(t\_{n}\right)\Delta \mathbf {w} \_{n}^{j}+\sum \limits \_{j=1}^{m}\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {y} \_{n})\mathbf {g} \_{j}\left(t\_{n}\right){\widetilde {J}\_{\left(j,0\right)}+\sum \limits \_{j=1}^{m}{\frac {d\mathbf {g} \_{j}{dt}\left(t\_{n}\right){\widetilde {J}\_{\left(0,j\right)}}}}}}}}$

$\qquad \qquad +\sum \limits \_{1},j\_{2}=1}^{m}\left(\mathbf {I} \otimes \mathbf {g} \_{2}^{\intercal }\left(t\_{n}\right)\right)\mathbf {f} \_{\mathbf {xx} }(t\_{n},\mathbf {y} \_{n})\mathbf {g} \_{1}\left(t\_{n}\right){\widetilde {J}\_{1},j\_{2},0\right),}\qquad \qquad (7.4)$

where $\mathbf {M} \_{n}$, $\mathbf {L} $, $\mathbf {r} $ and $\Delta \mathbf {w} \_{n}^{i}$ are defined as in the order-1 SLL schemes, and $\widetilde {J}\_{\alpha }$ is order **2** approximation to the multiple Stratonovish integral $J\_{\alpha }$.

#### Order 2 SLL-RK schemes

For SDEs with a single Wiener noise *(m=1***)**

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+{\widetilde {\mathbf {\phi } }(t\_{n},\mathbf {y} \_{n};h\_{n})+{\frac {h\_{n}{2}\left(\mathbf {k} \_{1}+\mathbf {k} \_{2}\right)+\mathbf {g} \left(t\_{n}\right)\Delta w\_{n}+{\frac {\left(\mathbf {g} \left(t\_{n+1}\right)-\mathbf {g} \left(t\_{n}\right)\right)}{h\_{n}J\_{\left(0,1\right)}\quad (7.5)}}}}}$

$\quad \quad \quad $

where

$$\mathbf {k} \_{1}=\mathbf {f} (t\_{n}+{\frac {h\_{n}{2},\mathbf {y} \_{n}+{\widetilde {\mathbf {\phi } }(t\_{n},\mathbf {y} \_{n};{\frac {h\_{n}{2})+\gamma \_{+})-\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {y} \_{n}){\widetilde {\mathbf {\phi } }(t\_{n},\mathbf {y} \_{n};{\frac {h\_{n}{2})-\mathbf {f} \left(t\_{n},\mathbf {y} \_{n}\right)-\mathbf {f} \_{t}\left(t\_{n},\mathbf {y} \_{n}\right){\frac {h\_{n}{2},}}}}}}}}}}$$

$$\mathbf {k} \_{2}=\mathbf {f} (t\_{n}+{\frac {h\_{n}{2},\mathbf {y} \_{n}+{\widetilde {\mathbf {\phi } }(t\_{n},\mathbf {y} \_{n};{\frac {h\_{n}{2})+\gamma \_{-})-\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {y} \_{n}){\widetilde {\mathbf {\phi } }(t\_{n},\mathbf {y} \_{n};{\frac {h\_{n}{2})-\mathbf {f} \left(t\_{n},\mathbf {y} \_{n}\right)-\mathbf {f} \_{t}\left(t\_{n},\mathbf {y} \_{n}\right){\frac {h\_{n}{2},}}}}}}}}}}$$

with $\gamma \_{\pm }={\frac {1}{h\_{n}\mathbf {g} \left(t\_{n}\right){\Bigl (}{\widetilde {J}\_{\left(1,0\right)}\pm {\sqrt {2{\widetilde {J}\_{\left(1,1,0\right)}h\_{n}-{\widetilde {J}\_{\left(1,0\right)}^{2}{\Bigr )}}}}}}}}$.

Here, $\widetilde {\mathbf {\phi } }(t\_{n},\mathbf {y} \_{n};h\_{n})=\mathbf {L} (\mathbf {P} \_{p,q}(2^{-k\_{n}\mathbf {M} \_{n}h\_{n}))^{2^{k\_{n}\mathbf {r} }}}$ for low dimensional SDEs, and $\widetilde {\mathbf {\phi } }(t\_{n},\mathbf {y} \_{n};h\_{n})=\mathbf {L\mathbf {k} } \_{n},k\_{n}^{p,q}(h\_{n},\mathbf {M} \_{n},\mathbf {r} )$ for large systems of SDEs, where $\mathbf {M} \_{n}$, $\mathbf {L} $, $\mathbf {r} $, $\Delta \mathbf {w} \_{n}^{i}$ and $\widetilde {J}\_{\alpha }$ are defined as in the order-**2** SLL-Taylor schemes, *p+q\>1* and $m\_{n}\>2$.

#### Stability and dynamics

By construction, the strong LL and HOLL discretizations inherit the stability and dynamics of the linear SDEs, but it is not the case of the strong LL schemes in general. LL schemes (7.2)-(7.5) with $p\leq q\leq p+2$ are *A*-stable, including stiff and highly oscillatory linear equations. Moreover, for linear SDEs with random attractors, these schemes also have a random attractor that converges in probability to the exact one as the stepsize decreases and preserve the ergodicity of these equations for any stepsize. These schemes also reproduce essential dynamical properties of simple and coupled harmonic oscillators such as the linear growth of energy along the paths, the oscillatory behavior around 0, the symplectic structure of Hamiltonian oscillators, and the mean of the paths. For nonlinear SDEs with small noise (i.e., (7.1) with $\mathbf {g} \_{i}(t)\approx 0$), the paths of these SLL schemes are basically the nonrandom paths of the LL scheme (4.6) for ODEs plus a small disturbance related to the small noise. In this situation, the dynamical properties of that deterministic scheme, such as the linearization preserving and the preservation of the exact solution dynamics around hyperbolic equilibrium points and periodic orbits, become relevant for the paths of the SLL scheme. For instance, Fig 4 shows the evolution of domains in the phase plane and the energy of the stochastic oscillator

$\begin{array}{ll}dx(t)=y(t)dt,&x\_{1}(0)=0.01\\dy(t)=-(\omega ^{2}x(t)+\epsilon x^{4}(t))dt+\sigma dw\_{t},&x\_{1}(0)=0.1,\end{array}\qquad \qquad (7.6)$

and their approximations by two numerical schemes.

## Weak LL methods for SDEs

Consider the *d*-dimensional stochastic differential equation

$d\mathbf {x} (t)=\mathbf {f} (t,\mathbf {x} (t))dt+\sum \limits \_{i=1}^{m}\mathbf {g} \_{i}(t)d\mathbf {w} ^{i}(t),\qquad t\in \left\[t\_{0},T\right\],\qquad \qquad (8.1)$

with initial condition $\mathbf {x} (t\_{0})=\mathbf {x} \_{0}$, where the drift coefficient $\mathbf {f} $ and the diffusion coefficient $\mathbf {g} \_{i}$ are differentiable functions, and $\mathbf {w=(\mathbf {w} } ^{1},\ldots ,\mathbf {w} ^{m}\mathbf {)} $ is an *m*-dimensional standard Wiener process.

### Local Linear discretization

For a time discretization $\left(t\right)\_{h}$, the order- $\mathbb {\beta } $ $(=1,2)$ *Weak Local Linear discretization* of the solution of the SDE (8.1) is defined by the recursive relation

$\mathbf {z} \_{n+1}=\mathbf {z} \_{n}+\mathbf {\phi } \_{\mathbb {\beta } }(t\_{n},\mathbf {z} \_{n};h\_{n})+\mathbf {\eta } (t\_{n},\mathbf {z} \_{n};h\_{n}),\quad with\quad \mathbf {z} \_{0}=\mathbf {x} \_{0},$

where

$\mathbf {\phi } \_{\mathbb {\beta } }(t\_{n},\mathbf {z} \_{n};\delta )=\int \_{0}^{\delta }e^{\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n})(\delta -u)}(\mathbf {f(} t\_{n},\mathbf {z} \_{n})+\mathbf {b} ^{\mathbb {\beta } }(t\_{n},\mathbf {z} \_{n})u)du$

with

$\mathbf {b} ^{\mathbb {\beta } }(t\_{n},\mathbf {z} \_{n})={\begin{cases}\mathbf {f} \_{t}(t\_{n},\mathbf {z} \_{n})&{\text{for }\mathbb {\beta } =1\\\mathbf {f} \_{t}(t\_{n},\mathbf {z} \_{n})+{\frac {1}{2}\sum \limits \_{j=1}^{m}\left(\mathbf {I} \otimes \mathbf {g} \_{j}^{\intercal }\left(t\_{n}\right)\right)\mathbf {f} \_{\mathbf {xx} }(t\_{n},\mathbf {z} \_{n})\mathbf {g} \_{j}\left(t\_{n}\right)&{\text{for }\mathbb {\beta } =2,\end{cases}}}}}$

and $\mathbf {\eta } (t\_{n},\mathbf {z} \_{n};\delta )$ is a zero mean stochastic process with variance matrix

$\mathbf {\Sigma } (t\_{n},\mathbf {z} \_{n};\delta )=\int \limits \_{0}^{\delta }e^{\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {z} \_{n})(\delta -s)}\mathbf {G} (t\_{n}+s)\mathbf {G} ^{\intercal }(t\_{n}+s)e^{\mathbf {f} \_{\mathbf {x} }^{\intercal }(t\_{n},\mathbf {z} \_{n})(\delta -s)}ds.$

Here, $\mathbf {f} \_{\mathbf {x} }$, $\mathbf {f} \_{t}$ denote the partial derivatives of $\mathbf {f} $ with respect to the variables $\mathbf {x} $ and *t*, respectively, $\mathbf {f} \_{\mathbf {xx} }$ the Hessian matrix of $\mathbf {f} $ with respect to $\mathbf {x} $, and $\mathbf {G} (t)=\[\mathbf {g} \_{1}(t),\ldots ,\mathbf {g} \_{m}(t)\]$. The weak Local Linear discretization $\mathbf {z} \_{n+1}$ converges with order $\mathbb {\beta } $ (=1,2) to the solution of (8.1).

### Local Linearization schemes

Depending on the way of computing $\mathbf {\phi } \_{\mathbb {\beta } }$ and $\mathbf {\Sigma } $ different numerical schemes can be obtained. Every numerical implementation $\mathbf {y} \_{n}$ of the Weak Local Linear discretization $\mathbf {z} \_{n}$ is generically called *Weak Local Linearization (WLL) scheme*.

#### Order 1 WLL scheme

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {B} \_{14}+(\mathbf {B} \_{12}\mathbf {B} \_{11}^{\intercal })^{1/2}\mathbf {\xi } \_{n}$

where, for SDEs with autonomous diffusion coefficients, $\mathbf {B} \_{11}$, $\mathbf {B} \_{12}$ and $\mathbf {B} \_{14}$ are the submatrices defined by the partitioned matrix $\mathbf {B} =\mathbf {P} \_{p,q}(2^{-k\_{n}{\mathcal {M}\_{n}h\_{n}))^{2^{k\_{n}}}}}$, with

$\mathcal {M}\_{n}=\left\[{\begin{array}{cccc}\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {y} \_{n})&\mathbf {GG} ^{\intercal }&\mathbf {f} \_{t}(t\_{n},\mathbf {y} \_{n})&\mathbf {f} (t\_{n},\mathbf {y} \_{n})\\\mathbf {0} &-\mathbf {f} \_{\mathbf {x} }^{\intercal }(t\_{n},\mathbf {y} \_{n})&\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &0&1\\\mathbf {0} &\mathbf {0} &0&0\end{array}\right\]\in \mathbb {R} ^{(2d+2)\times (2d+2)},}$

and $\\{\mathbf {\xi } \_{n}\\}$ is a sequence of *d*-dimensional independent two-points distributed random vectors satisfying $P(\xi \_{n}^{k}=\pm 1)={\frac {1}{2}}$.

#### Order 2 WLL scheme

$\mathbf {y} \_{n+1}=\mathbf {y} \_{n}+\mathbf {B} \_{16}+(\mathbf {B} \_{14}\mathbf {B} \_{11}^{\intercal })^{1/2}\mathbf {\xi } \_{n},$

where $\mathbf {B} \_{11}$, $\mathbf {B} \_{14}$ and $\mathbf {B} \_{16}$ are the submatrices defined by the partitioned matrix $\mathbf {B} =\mathbf {P} \_{p,q}(2^{-k\_{n}{\mathcal {M}\_{n}h\_{n}))^{2^{k\_{n}}}}}$ with

$\mathcal {M}\_{n}=\left\[{\begin{array}{cccccc}\mathbf {J} &\mathbf {H} \_{2}&\mathbf {H} \_{1}&\mathbf {H} \_{0}&\mathbf {a} \_{2}&\mathbf {a} \_{1}\\\mathbf {0} &-\mathbf {J} ^{\intercal }&\mathbf {I} &\mathbf {0} &\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &-\mathbf {J} ^{\intercal }&\mathbf {I} &\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &\mathbf {0} &-\mathbf {J} ^{\intercal }&\mathbf {0} &\mathbf {0} \\\mathbf {0} &\mathbf {0} &\mathbf {0} &\mathbf {0} &0&1\\\mathbf {0} &\mathbf {0} &\mathbf {0} &\mathbf {0} &0&0\end{array}\right\]\in \mathbb {R} ^{(4d+2)\times (4d+2)},}$

$\mathbf {J} =\mathbf {f} \_{\mathbf {x} }(t\_{n},\mathbf {y} \_{n})\qquad \mathbf {a} \_{1}=\mathbf {f} (t\_{n},\mathbf {y} \_{n})\qquad \mathbf {a} \_{2}=\mathbf {f} \_{t}(t\_{n},\mathbf {y} \_{n})+{\frac {1}{2}\sum \limits \_{i=1}^{m}(\mathbf {I} \otimes (\mathbf {g} ^{i}(t\_{n}))^{\intercal })\mathbf {f} \_{\mathbf {xx} }(t\_{n},\mathbf {y} \_{n})\mathbf {g} ^{i}(t\_{n})}$

and

$\mathbf {H} \_{0}=\mathbf {G} (t\_{n})\mathbf {G} ^{\intercal }(t\_{n})\qquad \mathbf {H} \_{1}=\mathbf {G} (t\_{n}){\frac {d\mathbf {G} ^{\intercal }(t\_{n})}{dt}+{\frac {d\mathbf {G} (t\_{n})}{dt}\mathbf {G} ^{\intercal }(t\_{n})\qquad \mathbf {H} \_{2}={\frac {d\mathbf {G} (t\_{n})}{dt}{\frac {d\mathbf {G} ^{\intercal }(t\_{n})}{dt}{\text{.}}}}}}$

#### Stability and dynamics

By construction, the weak LL discretizations inherit the stability and dynamics of the linear SDEs, but it is not the case of the weak LL schemes in general. WLL schemes, with $p\leq q\leq p+2,$ preserve the first two moments of the linear SDEs, and inherits the mean-square stability or instability that such solution may have. This includes, for instance, the equations of coupled harmonic oscillators driven by random force, and large systems of stiff linear SDEs that result from the method of lines for linear stochastic partial differential equations. Moreover, these WLL schemes preserve the ergodicity of the linear equations, and are geometrically ergodic for some classes of nonlinear SDEs. For nonlinear SDEs with small noise (i.e., (8.1) with $\mathbf {g} \_{i}(t)\approx 0$), the solutions of these WLL schemes are basically the nonrandom paths of the LL scheme (4.6) for ODEs plus a small disturbance related to the small noise. In this situation, the dynamical properties of that deterministic scheme, such as the linearization preserving and the preservation of the exact solution dynamics around hyperbolic equilibrium points and periodic orbits, become relevant for the mean of the WLL scheme. For instance, Fig. 5 shows the approximate mean of the SDE

$dx=-t^{2}x{\text{ }dt+{\frac {3}{2(t+1)}e^{-t^{3}/3}{\text{ }dw\_{t},\qquad \qquad x(0)=1,\qquad \quad (8.2)}}}$

computed by various schemes.

## Historical notes

Below is a time line of the main developments of the Local Linearization (LL) method.

* Pope D.A. (1963) introduces the LL discretization for ODEs and the LL scheme based on Taylor expansion.
* Ozaki T. (1985) introduces the LL method for the integration and estimation of SDEs. The term "Local Linearization" is used for first time.
* Biscay R. et al. (1996) reformulate the strong LL method for SDEs.
* Shoji I. and Ozaki T. (1997) reformulate the weak LL method for SDEs.
* Hochbruck M. et al. (1998) introduce the LL scheme for ODEs based on Krylov subspace approximation.
* Jimenez J.C. (2002) introduces the LL scheme for ODEs and SDEs based on rational Padé approximation.
* Carbonell F.M. et al. (2005) introduce the LL method for RDEs.
* Jimenez J.C. et al. (2006) introduce the LL method for DDEs.
* De la Cruz H. et al. (2006, 2007) and Tokman M. (2006) introduce the two classes of HOLL integrators for ODEs: the integrator-based and the quadrature-based.
* De la Cruz H. et al. (2010) introduce strong HOLL method for SDEs.


title: Yampa River

abstract: The Yampa River flows 250 miles (400 km) through northwestern Colorado, United States. Rising in the Rocky Mountains, it is a tributary of the Green River and a major part of the Colorado River system. The Yampa is one of the few free-flowing rivers in the western United States, with only a few small dams and diversions. The name is derived from the Snake Indians word for the Perideridia plant, which has an edible root. John C. Frémont was among the first to record the name 'Yampah' in entries of his journal from 1843, as he found the plant was particularly abundant in the watershed.

text:
# Yampa River

The **Yampa River** flows 250 miles (400 km) through northwestern Colorado, United States. Rising in the Rocky Mountains, it is a tributary of the Green River and a major part of the Colorado River system. The Yampa is one of the few free-flowing rivers in the western United States, with only a few small dams and diversions.

The name is derived from the Snake Indians word for the Perideridia plant, which has an edible root. John C. Frémont was among the first to record the name 'Yampah' in entries of his journal from 1843, as he found the plant was particularly abundant in the watershed.

## Course

The headwaters of the Yampa are in the Park Range in Routt County, Colorado as the confluence of the Bear River and Phillips Creek, near the town of Yampa. The Bear River, larger of the two, flows from a source of 11,600-foot (3,500 m) at Derby Peak in the Flat Tops Wilderness. The Yampa River then flows north through a high mountain valley, through Stagecoach Reservoir and Lake Catamount, before reaching Steamboat Springs, where it turns sharply west. Below Steamboat Springs, the Yampa flows through a wider valley in the western foothills of the Rockies. It receives the Elk River from the north, then passes the towns of Milner and Hayden.

After entering Moffat County the Yampa passes Craig and is joined by the Williams Fork. West of Craig, the Yampa crosses arid, sparsely populated sagebrush country for about 50 miles (80 km) before reaching Cross Mountain Canyon, where the river slices a 1,000 ft (300 m) deep gap through the namesake mountain. Below Cross Mountain the Yampa enters the open valley of Lily Park, where it is joined by its largest tributary, the Little Snake River. Further west it enters Dinosaur National Monument, where it traverses more than 40 miles (64 km) of rugged canyons and rapids. The Yampa joins the Green in Echo Park at Steamboat Rock deep within the national monument, about 5 miles (8.0 km) from the Colorado–Utah border.

## Watershed

The Yampa drains 7,660 square miles (19,800 km2) of mostly semi-arid plateau country in northwestern Colorado and a small portion of southern Wyoming. The bulk of the watershed is located between the Park Range, to the east; the Flat Tops, to the south; and the Uinta Mountains, to the west. The Great Divide Basin, an area of closed drainage, borders the Yampa River basin to the north. The Continental Divide runs along the north and east sides of the Yampa basin, separating it from the headwaters of the North Platte River, which flows into the Mississippi River system. On the south, the Yampa River basin is bordered by that of the White River, which like the Yampa flows in a westerly direction to join the Green.

## Volume

The Yampa River is a typical Western snow-fed stream, but unlike most other rivers in the western United States its seasonal discharge patterns are not affected by large dams and water projects. The river forms a noticeably wide, shallow braided stream throughout much of its course. The lower three fourths of the Yampa, from the Elk River down, are navigable by small craft. However the meandering, shallow nature of the river can render the river unnavigable during late summer in low water years.

The average flow at the confluence of the Green is 2,154 cubic feet per second (61 m3/s), averaging 5,400 cu ft/s (153 m3/s) during the spring runoff of April–July, and falling below 500 cu ft/s (14 m3/s) during late summer and autumn. The upper section of the river freezes over between December and March.

According to a U.S. Geological Survey stream gage at Deerlodge Park, about 50 miles (80 km) above the mouth, the average river flow was 2,082 cu ft/s (59.0 m3/s) between 1983 and 2013. The highest annual mean was 4,431 cu ft/s (125.5 m3/s) in 2011, and the lowest 678 cu ft/s (19.2 m3/s) in 2002. Monthly recorded average flows were highest in May at 8,200 cu ft/s (230 m3/s) and lowest in September at 342 cu ft/s (9.7 m3/s). The highest recorded peak flow was 33,200 cu ft/s (940 m3/s) during the record snowmelt of May 18, 1984.

## Ecology

Because the Yampa River maintains a relatively natural flow pattern, it supports a productive riparian zone environment along much of its course. In addition, much of the river is unconstrained by levees allowing it to maintain its natural floodplain. This is also true of its main tributary, the Little Snake River. The Yampa River Preserve 17 miles (27 km) west of Steamboat Springs protects a rare riparian forest type consisting of narrowleaf cottonwood, box elder and red-osier dogwood that was once more common in the Upper Colorado River Basin. The Yampa's warm, silty waters are an ideal spawning ground for native fish such as the Colorado pikeminnow and humpback chub which have largely disappeared from dammed waterways in other parts of the Colorado River system.

## History

### Native Americans

Archaeological studies conducted in the Dinosaur National Monument have revealed evidence of human habitation up to 7000 BC. The Fremont culture or Desert Archaic people inhabited the Yampa River basin starting about 800 AD, but disappeared for reasons uncertain during the 1400s. The Fremont created petroglyphs along the Yampa River Canyon, of which more than 300 are still visible today. After the collapse of the Fremont culture, a branch of the Utes moved into the Yampa River basin. The White River Utes inhabited the valleys of the Yampa and White Rivers and the Rockies of northwestern Colorado. The band living in the Yampa River valley was known as the Yamparika or Yapudttka Utes. At times the river also served as a boundary between the Utes and Shoshone peoples to the north; there was also a band of Shoshone called the Yamparika.

The name *Yamparika* is a Snake Indians word meaning "yampa eaters", "yampa" referring to the edible roots of the *Perideridia* plant. "Yampa" itself probably meant "water-plant" or "common plant". In 1843 explorer John C. Frémont was among the first to record the name "Yampah", finding this plant to be of particular abundance in the watershed. Some fur traders in the 1800s thought Yampa was the Ute word for "bear", and the Yampa was often called the Bear River on early maps.

## Dam and diversion proposals

During the 1960s, the Yampa and Green River canyons were slated to be flooded under a reservoir created by Echo Park Dam, a U.S. Bureau of Reclamation water project. Due to environmentalists' opposition, this dam was never built, and the free-flowing character of the Yampa was retained. However, as part of the compromise to preserve Echo Park, the controversial Glen Canyon and Flaming Gorge dam projects were allowed to move forward, damming the Colorado and Green Rivers, respectively. Today the Yampa remains the only major tributary in the Colorado River system without a single large dam along its course (the moderately sized Stagecoach Dam, built 1989 and several others on the headwaters have only a limited impact on its flow).

In December, 2006, a report proposed to pump water from the Yampa River 200 miles east, under the Continental Divide, to the state's major cities along the Front Range. The diversion was proposed to start near Maybell, 20 miles (32 km) downstream of Craig, Colorado . The proposal faces widespread opposition because it could lower river flows in late summer due to the diversion.

The Yampa River, aerial view from the south, with the Craig Station coal-fired power plant and the municipality of Craig, Colorado, at the far right, and the town of Maybell, Colorado, at the far left; November 2017

## InfoBox

| Yampa River | |
| --- | --- |
| The Yampa River, c. 1871–1878 | |
| Map of the Yampa River watershed | |
| Location | |
| Country | United States |
| State | Colorado |
| Cities | Steamboat Springs, Craig, Hayden |
| Physical characteristics | |
|  | |
| Source | Rocky Mountains |
| coordinates | 40°09′30″N 106°53′59″W / 40.15833°N 106.89972°W / 40.15833; -106.89972 |
| elevation | 7,833 ft (2,387 m) |
|  | |
| Mouth | Green River |
| location | Dinosaur National Monument |
| coordinates | 40°31′39″N 108°59′04″W / 40.5276°N 108.9845°W / 40.5276; -108.9845 |
| elevation | 5,080 ft (1,550 m) |
| Length | 250 mi (400 km) |
| Basin size | 7,660 sq mi (19,800 km2) |
|  | |
| Discharge |  |
| location | Deerlodge Park |
| average | 2,069 cu ft/s (58.6 m3/s) |
| minimum | 1.9 cu ft/s (0.054 m3/s) |
| maximum | 33,200 cu ft/s (940 m3/s) |
|  | |
| Basin features | |
| Tributaries |  |
| left | Bear River, Williams Fork |
| right | Elk River, Little Snake River |


title: Mart O'Malley

abstract: Martin Joseph "Mart" O'Malley was a justice of the Indiana Supreme Court from January 4, 1943, to January 3, 1949. Born in Pittston, Pennsylvania, O'Malley was one of eleven siblings, and as a child had to drop out of school to work in a coal mine. Nevertheless, and without a high school education, he was able to attend Saint Thomas College in Scranton from 1910 to 1912, and received an LL.B. from Valparaiso University School of Law in 1915. He practiced law intermittently in Huntington, Indiana, from 1922 to 1965, at one point defending humorist H. Allen Smith from an obscenity charge arising from Smith's authorship of a risque article. O'Malley served as Huntington County Attorney from 1930 to 1933, and as Huntington City Attorney from 1939 to 1943, during which time he became heavily involved in the Republican Party, supporting the candidacy of Wendell Willkie in the 1940 United States presidential election. O'Malley was elected to the Indiana Supreme Court as a Republican in 1942, a strong year for Republicans in Indiana, in his first effort to obtain political office. He was defeated for re-election by Democratic challenger Paul G. Jasper in 1948, a strong year for Democrats. O'Malley's term ended in 1949. Following his defeat, O'Malley returned to the private practice of law, forming the law firm of O'Malley and Brown. O'Malley served on various boards and committees, and again served as Huntington City Attorney from 1955 to 1956, and retired to St. Petersburg, Florida. He died in Gainesville, Florida, after falling ill while visiting his son there.

text:
# Mart O'Malley

**Martin Joseph "Mart" O'Malley** (September 17, 1890 – September 9, 1972) was a justice of the Indiana Supreme Court from January 4, 1943, to January 3, 1949.

Born in Pittston, Pennsylvania, O'Malley was one of eleven siblings, and as a child had to drop out of school to work in a coal mine. Nevertheless, and without a high school education, he was able to attend Saint Thomas College in Scranton from 1910 to 1912, and received an LL.B. from Valparaiso University School of Law in 1915.

He practiced law intermittently in Huntington, Indiana, from 1922 to 1965, at one point defending humorist H. Allen Smith from an obscenity charge arising from Smith's authorship of a risque article. O'Malley served as Huntington County Attorney from 1930 to 1933, and as Huntington City Attorney from 1939 to 1943, during which time he became heavily involved in the Republican Party, supporting the candidacy of Wendell Willkie in the 1940 United States presidential election.

O'Malley was elected to the Indiana Supreme Court as a Republican in 1942, a strong year for Republicans in Indiana, in his first effort to obtain political office. He was defeated for re-election by Democratic challenger Paul G. Jasper in 1948, a strong year for Democrats. O'Malley's term ended in 1949.

Following his defeat, O'Malley returned to the private practice of law, forming the law firm of O'Malley and Brown. O'Malley served on various boards and committees, and again served as Huntington City Attorney from 1955 to 1956, and retired to St. Petersburg, Florida. He died in Gainesville, Florida, after falling ill while visiting his son there.


title: TRIO (gene)

abstract: Triple functional domain protein is a protein that in humans is encoded by the TRIO gene.

text:
# TRIO (gene)

**Triple functional domain protein** is a protein that in humans is encoded by the *TRIO* gene.

## Interactions

TRIO (gene) has been shown to interact with Filamin and RHOA.

## Further reading

* Overview of all the structural information available in the PDB for UniProt: *O75962* (Triple functional domain protein) at the PDBe-KB.

## InfoBox

| TRIO | | | |
| --- | --- | --- | --- |
|  | | | |
| Available structures  PDB Ortholog search: PDBe RCSB | | | |
| Identifiers | | | |
| Aliases | TRIO, ARHGEF23, tgat, trio Rho guanine nucleotide exchange factor, MEBAS, MRD44, MRD63, TRIO gene | | |
| External IDs | OMIM: 601893; MGI: 1927230; HomoloGene: 20847; GeneCards: TRIO; OMA:TRIO - orthologs | | |
|  | | | |
|  | | | |
|  | | | |
|  | | | |
| Orthologs  Species **Human** **Mouse**  Entrez   Ensembl   UniProt   RefSeq (mRNA)     NM\_007118   NM\_001081302   RefSeq (protein)     NP\_009049   NP\_001074771   Location (UCSC) Chr 5: 14.14 – 14.53 Mb Chr 15: 27.73 – 28.03 Mb  PubMed search | | | |
| Wikidata | | | |
| View/Edit Human View/Edit Mouse | | | |