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Fuzzy control system

A fuzzy control system is a control system based on fuzzy logic—a mathematical system that analyzes analog input values in terms of logical variables that take on continuous values between 0 and 1, in contrast to classical or digital logic, which operates on discrete values of either 1 or 0 (true or false, respectively).

Fuzzy logic is widely used in machine control. The term "fuzzy" refers to the fact that the logic involved can deal with concepts that cannot be expressed as the "true" or "false" but rather as "partially true". Although alternative approaches such as genetic algorithms and neural networks can perform just as well as fuzzy logic in many cases, fuzzy logic has the advantage that the solution to the problem can be cast in terms that human operators can understand, so that their experience can be used in the design of the controller. This makes it easier to mechanize tasks that are already successfully performed by humans.

History and applications
Fuzzy logic was proposed by Lotfi A. Zadeh of the University of California at Berkeley in a 1965 paper. He elaborated on his ideas in a 1973 paper that introduced the concept of "linguistic variables", which in this article equates to a variable defined as a fuzzy set. Other research followed, with the first industrial application, a cement kiln built in Denmark, coming on line in 1976.

Fuzzy systems were initially implemented in Japan.

 * Interest in fuzzy systems was sparked by Seiji Yasunobu and Soji Miyamoto of Hitachi, who in 1985 provided simulations that demonstrated the feasibility of fuzzy control systems for the Sendai Subway. Their ideas were adopted, and fuzzy systems were used to control accelerating, braking, and stopping when the Namboku Line opened in 1987.
 * In 1987, Takeshi Yamakawa demonstrated the use of fuzzy control, through a set of simple dedicated fuzzy logic chips, in an "inverted pendulum" experiment. This is a classic control problem, in which a vehicle tries to keep a pole mounted on its top by a hinge upright by moving back and forth. Yamakawa subsequently made the demonstration more sophisticated by mounting a wine glass containing water and even a live mouse to the top of the pendulum: the system maintained stability in both cases. Yamakawa eventually went on to organize his own fuzzy-systems research lab to help exploit his patents in the field.
 * Japanese engineers subsequently developed a wide range of fuzzy systems for both industrial and consumer applications. In 1988 Japan established the Laboratory for International Fuzzy Engineering (LIFE), a cooperative arrangement between 48 companies to pursue fuzzy research. The automotive company Volkswagen was the only foreign corporate member of LIFE, dispatching a researcher for a duration of three years.
 * Japanese consumer goods often incorporate fuzzy systems. Matsushita vacuum cleaners use microcontrollers running fuzzy algorithms to interrogate dust sensors and adjust suction power accordingly. Hitachi washing machines use fuzzy controllers to load-weight, fabric-mix, and dirt sensors and automatically set the wash cycle for the best use of power, water, and detergent.
 * Canon developed an autofocusing camera that uses a charge-coupled device (CCD) to measure the clarity of the image in six regions of its field of view and use the information provided to determine if the image is in focus. It also tracks the rate of change of lens movement during focusing, and controls its speed to prevent overshoot. The camera's fuzzy control system uses 12 inputs: 6 to obtain the current clarity data provided by the CCD and 6 to measure the rate of change of lens movement. The output is the position of the lens. The fuzzy control system uses 13 rules and requires 1.1 kilobytes of memory.
 * An industrial air conditioner designed by Mitsubishi uses 25 heating rules and 25 cooling rules. A temperature sensor provides input, with control outputs fed to an inverter, a compressor valve, and a fan motor. Compared to the previous design, the fuzzy controller heats and cools five times faster, reduces power consumption by 24%, increases temperature stability by a factor of two, and uses fewer sensors.
 * Other applications investigated or implemented include: character and handwriting recognition; optical fuzzy systems; robots, including one for making Japanese flower arrangements; voice-controlled robot helicopters (hovering is a "balancing act" rather similar to the inverted pendulum problem); rehabilitation robotics to provide patient-specific solutions (e.g. to control heart rate and blood pressure ); control of flow of powders in film manufacture; elevator systems; and so on.

Work on fuzzy systems is also proceeding in North America and Europe, although on a less extensive scale than in Japan.

 * The US Environmental Protection Agency has investigated fuzzy control for energy-efficient motors, and NASA has studied fuzzy control for automated space docking: simulations show that a fuzzy control system can greatly reduce fuel consumption.
 * Firms such as Boeing, General Motors, Allen-Bradley, Chrysler, Eaton, and Whirlpool have worked on fuzzy logic for use in low-power refrigerators, improved automotive transmissions, and energy-efficient electric motors.
 * In 1995 Maytag introduced an "intelligent" dishwasher based on a fuzzy controller and a "one-stop sensing module" that combines a thermistor, for temperature measurement; a conductivity sensor, to measure detergent level from the ions present in the wash; a turbidity sensor that measures scattered and transmitted light to measure the soiling of the wash; and a magnetostrictive sensor to read spin rate. The system determines the optimum wash cycle for any load to obtain the best results with the least amount of energy, detergent, and water. It even adjusts for dried-on foods by tracking the last time the door was opened, and estimates the number of dishes by the number of times the door was opened.
 * Xiera Technologies Inc. has developed the first auto-tuner for the fuzzy logic controller's knowledge base known as edeX. This technology was tested by Mohawk College and was able to solve non-linear 2x2 and 3x3 multi-input multi-output problems.

Research and development is also continuing on fuzzy applications in software, as opposed to firmware, design, including fuzzy expert systems and integration of fuzzy logic with neural-network and so-called adaptive "genetic" software systems, with the ultimate goal of building "self-learning" fuzzy-control systems. These systems can be employed to control complex, nonlinear dynamic plants, for example, human body.

Fuzzy sets
The input variables in a fuzzy control system are in general mapped by sets of membership functions similar to this, known as "fuzzy sets". The process of converting a crisp input value to a fuzzy value is called "fuzzification". The fuzzy logic based approach had been considered by designing two fuzzy systems, one for error heading angle and the other for velocity control.

A control system may also have various types of switch, or "ON-OFF", inputs along with its analog inputs, and such switch inputs of course will always have a truth value equal to either 1 or 0, but the scheme can deal with them as simplified fuzzy functions that happen to be either one value or another.

Given "mappings" of input variables into membership functions and truth values, the microcontroller then makes decisions for what action to take, based on a set of "rules", each of the form:

IF brake temperature IS warm AND speed IS not very fast THEN brake pressure IS slightly decreased.

In this example, the two input variables are "brake temperature" and "speed" that have values defined as fuzzy sets. The output variable, "brake pressure" is also defined by a fuzzy set that can have values like "static" or "slightly increased" or "slightly decreased" etc.

Fuzzy control in detail
Fuzzy controllers are very simple conceptually. They consist of an input stage, a processing stage, and an output stage. The input stage maps sensor or other inputs, such as switches, thumbwheels, and so on, to the appropriate membership functions and truth values. The processing stage invokes each appropriate rule and generates a result for each, then combines the results of the rules. Finally, the output stage converts the combined result back into a specific control output value.

The most common shape of membership functions is triangular, although trapezoidal and bell curves are also used, but the shape is generally less important than the number of curves and their placement. From three to seven curves are generally appropriate to cover the required range of an input value, or the "universe of discourse" in fuzzy jargon.

As discussed earlier, the processing stage is based on a collection of logic rules in the form of IF-THEN statements, where the IF part is called the "antecedent" and the THEN part is called the "consequent". Typical fuzzy control systems have dozens of rules.

Consider a rule for a thermostat:

IF (temperature is "cold") THEN turn (heater is "high")

This rule uses the truth value of the "temperature" input, which is some truth value of "cold", to generate a result in the fuzzy set for the "heater" output, which is some value of "high". This result is used with the results of other rules to finally generate the crisp composite output. Obviously, the greater the truth value of "cold", the higher the truth value of "high", though this does not necessarily mean that the output itself will be set to "high" since this is only one rule among many. In some cases, the membership functions can be modified by "hedges" that are equivalent to adverbs. Common hedges include "about", "near", "close to", "approximately", "very", "slightly", "too", "extremely", and "somewhat". These operations may have precise definitions, though the definitions can vary considerably between different implementations. "Very", for one example, squares membership functions; since the membership values are always less than 1, this narrows the membership function. "Extremely" cubes the values to give greater narrowing, while "somewhat" broadens the function by taking the square root.

In practice, the fuzzy rule sets usually have several antecedents that are combined using fuzzy operators, such as AND, OR, and NOT, though again the definitions tend to vary: AND, in one popular definition, simply uses the minimum weight of all the antecedents, while OR uses the maximum value. There is also a NOT operator that subtracts a membership function from 1 to give the "complementary" function.

There are several ways to define the result of a rule, but one of the most common and simplest is the "max-min" inference method, in which the output membership function is given the truth value generated by the premise.

Rules can be solved in parallel in hardware, or sequentially in software. The results of all the rules that have fired are "defuzzified" to a crisp value by one of several methods. There are dozens, in theory, each with various advantages or drawbacks.

The "centroid" method is very popular, in which the "center of mass" of the result provides the crisp value. Another approach is the "height" method, which takes the value of the biggest contributor. The centroid method favors the rule with the output of greatest area, while the height method obviously favors the rule with the greatest output value.

The diagram below demonstrates max-min inferencing and centroid defuzzification for a system with input variables "x", "y", and "z" and an output variable "n". Note that "mu" is standard fuzzy-logic nomenclature for "truth value":

Notice how each rule provides a result as a truth value of a particular membership function for the output variable. In centroid defuzzification the values are OR'd, that is, the maximum value is used and values are not added, and the results are then combined using a centroid calculation.

Fuzzy control system design is based on empirical methods, basically a methodical approach to trial-and-error. The general process is as follows:

 * Document the system's operational specifications and inputs and outputs.
 * Document the fuzzy sets for the inputs.
 * Document the rule set.
 * Determine the defuzzification method.
 * Run through test suite to validate system, adjust details as required.
 * Complete document and release to production.

As a general example, consider the design of a fuzzy controller for a steam turbine. The block diagram of this control system appears as follows:

The input and output variables map into the following fuzzy set:

—where:

N3:  Large negative. N2:  Medium negative. N1:  Small negative. Z:   Zero. P1:  Small positive. P2:  Medium positive. P3:  Large positive.

The rule set includes such rules as:

rule 1: IF temperature IS cool AND pressure IS weak, THEN throttle is P3.

rule 2: IF temperature IS cool AND pressure IS low, THEN throttle is P2.

rule 3: IF temperature IS cool AND pressure IS ok, THEN throttle is Z.

rule 4: IF temperature IS cool AND pressure IS strong, THEN throttle is N2.

In practice, the controller accepts the inputs and maps them into their membership functions and truth values. These mappings are then fed into the rules. If the rule specifies an AND relationship between the mappings of the two input variables, as the examples above do, the minimum of the two is used as the combined truth value; if an OR is specified, the maximum is used. The appropriate output state is selected and assigned a membership value at the truth level of the premise. The truth values are then defuzzified. For example, assume the temperature is in the "cool" state, and the pressure is in the "low" and "ok" states. The pressure values ensure that only rules 2 and 3 fire:

The two outputs are then defuzzified through centroid defuzzification: __________________________________________________________________

|         Z      P2                                    1 -+          *       * |        * *     * *                                       |        *   *   *   *                                       |       *     * *     *                                       |      *       222222222                                       |     *       22222222222                                       |    333333332222222222222                                       +---33333333222222222222222--> ^                                                      +150    __________________________________________________________________

The output value will adjust the throttle and then the control cycle will begin again to generate the next value.

Building a fuzzy controller
Consider implementing with a microcontroller chip a simple feedback controller:

A fuzzy set is defined for the input error variable "e", and the derived change in error, "delta", as well as the "output", as follows:

LP: large positive SP: small positive ZE: zero SN: small negative LN: large negative

If the error ranges from -1 to +1, with the analog-to-digital converter used having a resolution of 0.25, then the input variable's fuzzy set (which, in this case, also applies to the output variable) can be described very simply as a table, with the error / delta / output values in the top row and the truth values for each membership function arranged in rows beneath:

_______________________________________________________________________              -1    -0.75  -0.5   -0.25    0     0.25   0.5    0.75    1   _______________________________________________________________________    mu(LP)      0      0      0      0      0      0     0.3    0.7     1 mu(SP)     0      0      0      0     0.3    0.7     1     0.7    0.3 mu(ZE)     0      0     0.3    0.7     1     0.7    0.3     0      0 mu(SN)    0.3    0.7     1     0.7    0.3     0      0      0      0 mu(LN)     1     0.7    0.3     0      0      0      0      0      0 _______________________________________________________________________           —or, in graphical form (where each "X" has a value of 0.1):

LN          SN           ZE           SP           LP       +--+ |                                                                 | -1.0  |  XXXXXXXXXX   XXX          :            :            :           | -0.75 | XXXXXXX      XXXXXXX      :            :            :           | -0.5 |  XXX          XXXXXXXXXX   XXX          :            :           | -0.25 | :            XXXXXXX      XXXXXXX      :            :           | 0.0 |  :            XXX          XXXXXXXXXX   XXX          :           | 0.25 | :            :            XXXXXXX      XXXXXXX      :           | 0.5 |  :            :            XXX          XXXXXXXXXX   XXX         | 0.75 | :            :            :            XXXXXXX      XXXXXXX     | 1.0 |  :            :            :            XXX          XXXXXXXXXX  | |                                                                 |       +--+

Suppose this fuzzy system has the following rule base:

rule 1: IF e = ZE AND delta = ZE THEN output = ZE   rule 2:  IF e = ZE AND delta = SP THEN output = SN   rule 3:  IF e = SN AND delta = SN THEN output = LP   rule 4:  IF e = LP OR  delta = LP THEN output = LN

These rules are typical for control applications in that the antecedents consist of the logical combination of the error and error-delta signals, while the consequent is a control command output. The rule outputs can be defuzzified using a discrete centroid computation:

SUM( I = 1 TO 4 OF ( mu(I) * output(I) ) ) / SUM( I = 1 TO 4 OF mu(I) )

Now, suppose that at a given time: e    = 0.25 delta = 0.5

Then this gives:

________________________              e     delta ________________________  mu(LP)      0      0.3 mu(SP)    0.7      1 mu(ZE)    0.7     0.3 mu(SN)     0       0 mu(LN)     0       0 ________________________

Plugging this into rule 1 gives:

rule 1: IF e = ZE AND delta = ZE THEN output = ZE      mu(1)     = MIN( 0.7, 0.3 ) = 0.3 output(1) = 0

-- where:

 * mu(1): Truth value of the result membership function for rule 1. In terms of a centroid calculation, this is the "mass" of this result for this discrete case.
 * output(1): Value (for rule 1) where the result membership function (ZE) is maximum over the output variable fuzzy set range. That is, in terms of a centroid calculation, the location of the "center of mass" for this individual result. This value is independent of the value of "mu". It simply identifies the location of ZE along the output range.

The other rules give:

rule 2: IF e = ZE AND delta = SP THEN output = SN      mu(2)     = MIN( 0.7, 1 ) = 0.7 output(2) = -0.5

rule 3: IF e = SN AND delta = SN THEN output = LP     mu(3)     = MIN( 0.0, 0.0 ) = 0 output(3) = 1

rule 4: IF e = LP OR delta = LP THEN output = LN      mu(4)     = MAX( 0.0, 0.3 ) = 0.3 output(4) = -1

The centroid computation yields:

$$ \frac{mu(1) \cdot output(1)+mu(2) \cdot output(2)+mu(3) \cdot output(3)+mu(4) \cdot output(4)}{mu(1)+mu(2)+mu(3)+mu(4)} $$ $$=\frac{(0.3 \cdot 0)+(0.7 \cdot-0.5)+(0 \cdot 1) +(0.3 \cdot-1)}{0.3+0.7+0+0.3} $$ $$=-0.5$$—for the final control output. Simple. Of course the hard part is figuring out what rules actually work correctly in practice.

If you have problems figuring out the centroid equation, remember that a centroid is defined by summing all the moments (location times mass) around the center of gravity and equating the sum to zero. So if $$X_0$$ is the center of gravity, $$X_i$$ is the location of each mass, and $$M_i$$ is each mass, this gives:

$$0 = ( X_1 - X_0 ) \cdot M_1 + ( X_2 - X_0 ) \cdot M_2 + \ldots + ( X_n - X_0 ) \cdot M_n$$ $$0 = ( X_1 \cdot M_1 + X_2 \cdot M_2 + \ldots + X_n \cdot M_n ) - X_0 \cdot ( M_1 + M_2 + \ldots + M_n ) $$ $$ X_0 \cdot ( M_1 + M_2 + \ldots + M_n ) = X_1 \cdot M_1 + X_2 \cdot M_2 + \ldots + X_n \cdot M_n $$ $$ X_0 = \frac{ X_1 \cdot M_1 + X_2 \cdot M_2 + \ldots + X_n \cdot M_n }{ M_1 + M_2 + \ldots + M_n }$$

In our example, the values of mu correspond to the masses, and the values of X to location of the masses (mu, however, only 'corresponds to the masses' if the initial 'mass' of the output functions are all the same/equivalent. If they are not the same, i.e. some are narrow triangles, while others maybe wide trapezoids or shouldered triangles, then the mass or area of the output function must be known or calculated. It is this mass that is then scaled by mu and multiplied by its location X_i).

This system can be implemented on a standard microprocessor, but dedicated fuzzy chips are now available. For example, Adaptive Logic INC of San Jose, California, sells a "fuzzy chip", the AL220, that can accept four analog inputs and generate four analog outputs. A block diagram of the chip is shown below:

+-+                             +---+ analog --4-->| analog  |                              | mux / +--4--> analog in        |   mux   |                              |  SH   |        out +++                             +---+                   |                                       ^                   V                                       | +-+                            +--+--+            | ADC / latch |                             | DAC | +--+--+                            +-+                   |                                       ^                   |                                       |                   8         +-+                   |         |                             |                   |         V                             | |  +---+      +-+  |                   +-->| fuzzifier |      | defuzzifier +--+ +-+-+     +-+                             |                   ^                             |  +-+  |                             |  |    rule     |  | +->| processor  +--+ | (50 rules) | +--+--+                                      |                                +--+--+                                |  parameter  | |   memory   | |  256 x 8   | +-+

ADC: analog-to-digital converter DAC: digital-to-analog converter SH:  sample/hold

Antilock brakes
As an example, consider an anti-lock braking system, directed by a microcontroller chip. The microcontroller has to make decisions based on brake temperature, speed, and other variables in the system.

The variable "temperature" in this system can be subdivided into a range of "states": "cold", "cool", "moderate", "warm", "hot", "very hot". The transition from one state to the next is hard to define.

An arbitrary static threshold might be set to divide "warm" from "hot". For example, at exactly 90 degrees, warm ends and hot begins. But this would result in a discontinuous change when the input value passed over that threshold. The transition wouldn't be smooth, as would be required in braking situations.

The way around this is to make the states fuzzy. That is, allow them to change gradually from one state to the next. In order to do this, there must be a dynamic relationship established between different factors.

Start by defining the input temperature states using "membership functions":

With this scheme, the input variable's state no longer jumps abruptly from one state to the next. Instead, as the temperature changes, it loses value in one membership function while gaining value in the next. In other words, its ranking in the category of cold decreases as it becomes more highly ranked in the warmer category.

At any sampled timeframe, the "truth value" of the brake temperature will almost always be in some degree part of two membership functions: i.e.: '0.6 nominal and 0.4 warm', or '0.7 nominal and 0.3 cool', and so on.

The above example demonstrates a simple application, using the abstraction of values from multiple values. This only represents one kind of data, however, in this case, temperature.

Adding additional sophistication to this braking system, could be done by additional factors such as traction, speed, inertia, set up in dynamic functions, according to the designed fuzzy system.

Logical interpretation of fuzzy control
In spite of the appearance there are several difficulties to give a rigorous logical interpretation of the IF-THEN rules. As an example, interpret a rule as IF (temperature is "cold") THEN (heater is "high") by the first order formula Cold(x)→High(y) and assume that r is an input such that Cold(r) is false. Then the formula Cold(r)→High(t) is true for any t and therefore any t gives a correct control given r. A rigorous logical justification of fuzzy control is given in Hájek's book (see Chapter 7) where fuzzy control is represented as a theory of Hájek's basic logic.

In Gerla 2005 another logical approach to fuzzy control is proposed based on fuzzy logic programming: Denote by f the fuzzy function arising of an IF-THEN systems of rules. Then this system can be translated into a fuzzy program P containing a series of rules whose head is "Good(x,y)". The interpretation of this predicate in the least fuzzy Herbrand model of P coincides with f. This gives further useful tools to fuzzy control.

Fuzzy qualitative simulation
Before an Artificial Intelligence system is able to plan the action sequence, some kind of model is needed. For video games, the model is equal to the game rules. From the programming perspective, the game rules are implemented as a Physics engine which accepts an action from a player and calculates, if the action is valid. After the action was executed, the game is in follow up state. If the aim isn't only to play mathematical games but determine the actions for real world applications, the most obvious bottleneck is, that no game rules are available. The first step is to model the domain. System identification can be realized with precise mathematical equations or with Fuzzy rules.

Using Fuzzy logic and ANFIS systems (Adaptive network based fuzzy inference system) for creating the forward model for a domain has many disadvantages. A qualitative simulation isn't able to determine the correct follow up state, but the system will only guess what will happen if the action was taken. The Fuzzy qualitative simulation can't predict the exact numerical values, but it's using imprecise natural language to speculate about the future. It takes the current situation plus the actions from the past and generates the expected follow up state of the game.

The output of the ANFIS system isn't providing correct information, but only a Fuzzy set notation, for example [0,0.2,0.4,0]. After converting the set notation back into numerical values the accuracy get worse. This makes Fuzzy qualitative simulation a bad choice for practical applications.

Applications
Fuzzy control systems are suitable when the process complexity is high including uncertainty and nonlinear behavior, and there are no precise mathematical models available. Successful applications of fuzzy control systems have been reported worldwide mainly in Japan with pioneering solutions since 80s.

Some applications reported in the literature are:

 * Air conditioners
 * Automatic focus systems in cameras
 * Domestic appliances (refrigerators, washing machines...)
 * Control and optimization of industrial processes and system
 * Writing systems
 * Fuel efficiency in engines
 * Environment
 * Expert systems
 * Decision trees
 * Robotics
 * Autonomous vehicles

title: Fuzzy control system


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Statistician

A statistician is a person who works with theoretical or applied statistics. The profession exists in both the private and public sectors.

It is common to combine statistical knowledge with expertise in other subjects, and statisticians may work as employees or as statistical consultants.

Nature of the work
According to the United States Bureau of Labor Statistics, as of 2014, 26,970 jobs were classified as statistician in the United States. Of these people, approximately 30 percent worked for governments (federal, state, or local). As of October 2021, the median pay for statisticians in the United States was $92,270.

Additionally, there is a substantial number of people who use statistics and data analysis in their work but have job titles other than statistician, such as actuaries, applied mathematicians, economists, data scientists, data analysts (predictive analytics), financial analysts, psychometricians, sociologists, epidemiologists, and quantitative psychologists. Statisticians are included with the professions in various national and international occupational classifications.

In many countries, including the United States, employment in the field requires either a master's degree in statistics or a related field or a PhD.

According to one industry professional, "Typical work includes collaborating with scientists, providing mathematical modeling, simulations, designing randomized experiments and randomized sampling plans, analyzing experimental or survey results, and forecasting future events (such as sales of a product)."

According to the BLS, "Overall employment is projected to grow 33% from 2016 to 2026, much faster than average for all occupations. Businesses will need these workers to analyze the increasing volume of digital and electronic data." In October 2021, the CNBC rated it the fastest growing job in science and technology of the next decade, with a projected growth rate of 35.40%.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Statistician


text:
Computer number format

A computer number format is the internal representation of numeric values in digital device hardware and software, such as in programmable computers and calculators. Numerical values are stored as groupings of bits, such as bytes and words. The encoding between numerical values and bit patterns is chosen for convenience of the operation of the computer; the encoding used by the computer's instruction set generally requires conversion for external use, such as for printing and display. Different types of processors may have different internal representations of numerical values and different conventions are used for integer and real numbers. Most calculations are carried out with number formats that fit into a processor register, but some software systems allow representation of arbitrarily large numbers using multiple words of memory.

Binary number representation
Computers represent data in sets of binary digits. The representation is composed of bits, which in turn are grouped into larger sets such as bytes.

A bit is a binary digit that represents one of two states. The concept of a bit can be understood as a value of either 1 or 0, on or off, yes or no, true or false, or encoded by a switch or toggle of some kind.

While a single bit, on its own, is able to represent only two values, a string of bits may be used to represent larger values. For example, a string of three bits can represent up to eight distinct values as illustrated in Table 1.

As the number of bits composing a string increases, the number of possible 0 and 1 combinations increases exponentially. A single bit allows only two value-combinations, two bits combined can make four separate values, three bits for eight, and so on, increasing with the formula 2n. The amount of possible combinations doubles with each binary digit added as illustrated in Table 2.

Groupings with a specific number of bits are used to represent varying things and have specific names.

A byte is a bit string containing the number of bits needed to represent a character. On most modern computers, this is an eight bit string. Because the definition of a byte is related to the number of bits composing a character, some older computers have used a different bit length for their byte. In many computer architectures, the byte is the smallest addressable unit, the atom of addressability, say. For example, even though 64-bit processors may address memory sixty-four bits at a time, they may still split that memory into eight-bit pieces. This is called byte-addressable memory. Historically, many CPUs read data in some multiple of eight bits. Because the byte size of eight bits is so common, but the definition is not standardized, the term octet is sometimes used to explicitly describe an eight bit sequence.

A nibble (sometimes nybble), is a number composed of four bits. Being a half-byte, the nibble was named as a play on words. A person may need several nibbles for one bite from something; similarly, a nybble is a part of a byte. Because four bits allow for sixteen values, a nibble is sometimes known as a hexadecimal digit.

Octal and hexadecimal number display
Octal and hexadecimal encoding are convenient ways to represent binary numbers, as used by computers. Computer engineers often need to write out binary quantities, but in practice writing out a binary number such as 1001001101010001 is tedious and prone to errors. Therefore, binary quantities are written in a base-8, or "octal", or, much more commonly, a base-16, "hexadecimal" (hex), number format. In the decimal system, there are 10 digits, 0 through 9, which combine to form numbers. In an octal system, there are only 8 digits, 0 through 7. That is, the value of an octal "10" is the same as a decimal "8", an octal "20" is a decimal "16", and so on. In a hexadecimal system, there are 16 digits, 0 through 9 followed, by convention, with A through F. That is, a hexadecimal "10" is the same as a decimal "16" and a hexadecimal "20" is the same as a decimal "32". An example and comparison of numbers in different bases is described in the chart below.

When typing numbers, formatting characters are used to describe the number system, for example 000_0000B or 0b000_00000 for binary and 0F8H or 0xf8 for hexadecimal numbers.

Converting between bases
Each of these number systems is a positional system, but while decimal weights are powers of 10, the octal weights are powers of 8 and the hexadecimal weights are powers of 16. To convert from hexadecimal or octal to decimal, for each digit one multiplies the value of the digit by the value of its position and then adds the results. For example:

\begin{align} & \text{octal } 756 \\[5pt] = {} & (7 \times 8^2) + (5 \times 8^1) + (6 \times 8^0) \\[5pt] = {} & (7 \times 64) + (5 \times 8) + (6 \times 1) \\[5pt] = {} & 448 + 40 + 6 \\[5pt] = {} & \text{decimal } 494 \end{align} \qquad \begin{align} & \text{hex } \mathrm{3b2} \\[5pt] = {} & (3 \times 16^2) + (11 \times 16^1) + (2 \times 16^0) \\[5pt] = {} & (3 \times 256) + (11 \times 16) + (2 \times 1) \\[5pt] = {} & 768 + 176 + 2 \\[5pt] = {} & \text{decimal } 946 \end{align} $$

Fixed-point numbers
Fixed-point formatting can be useful to represent fractions in binary.

The number of bits needed for the precision and range desired must be chosen to store the fractional and integer parts of a number. For instance, using a 32-bit format, 16 bits may be used for the integer and 16 for the fraction.

The eight's bit is followed by the four's bit, then the two's bit, then the one's bit. The fractional bits continue the pattern set by the integer bits. The next bit is the half's bit, then the quarter's bit, then the ⅛'s bit, and so on. For example: This form of encoding cannot represent some values in binary. For example, the fraction $1⁄2$, 0.2 in decimal, the closest approximations would be as follows:

Even if more digits are used, an exact representation is impossible. The number $1 1⁄4$, written in decimal as 0.333333333..., continues indefinitely. If prematurely terminated, the value would not represent $7 3⁄8$ precisely.

Floating-point numbers
While both unsigned and signed integers are used in digital systems, even a 32-bit integer is not enough to handle all the range of numbers a calculator can handle, and that's not even including fractions. To approximate the greater range and precision of real numbers, we have to abandon signed integers and fixed-point numbers and go to a "floating-point" format.

In the decimal system, we are familiar with floating-point numbers of the form (scientific notation):

 * 1.1030402 × 105 = 1.1030402 × 100000 = 110304.02

or, more compactly:

 * 1.1030402E5

which means "1.1030402 times 1 followed by 5 zeroes". We have a certain numeric value (1.1030402) known as a "significand", multiplied by a power of 10 (E5, meaning 105 or 100,000), known as an "exponent". If we have a negative exponent, that means the number is multiplied by a 1 that many places to the right of the decimal point. For example:

 * 2.3434E−6 = 2.3434 × 10−6 = 2.3434 × 0.000001 = 0.0000023434

The advantage of this scheme is that by using the exponent we can get a much wider range of numbers, even if the number of digits in the significand, or the "numeric precision", is much smaller than the range. Similar binary floating-point formats can be defined for computers. There is a number of such schemes, the most popular has been defined by Institute of Electrical and Electronics Engineers (IEEE). The IEEE 754-2008 standard specification defines a 64 bit floating-point format with:

 * an 11-bit binary exponent, using "excess-1023" format. Excess-1023 means the exponent appears as an unsigned binary integer from 0 to 2047; subtracting 1023 gives the actual signed value
 * a 52-bit significand, also an unsigned binary number, defining a fractional value with a leading implied "1"
 * a sign bit, giving the sign of the number.

With the bits stored in 8 bytes of memory:

where "S" denotes the sign bit, "x" denotes an exponent bit, and "m" denotes a significand bit. Once the bits here have been extracted, they are converted with the computation:

 * <sign> × (1 + <fractional significand>) × 2<exponent> − 1023

This scheme provides numbers valid out to about 15 decimal digits, with the following range of numbers:

The specification also defines several special values that are not defined numbers, and are known as NaNs, for "Not A Number". These are used by programs to designate invalid operations and the like. Some programs also use 32-bit floating-point numbers. The most common scheme uses a 23-bit significand with a sign bit, plus an 8-bit exponent in "excess-127" format, giving seven valid decimal digits.

The bits are converted to a numeric value with the computation:

 * <sign> × (1 + <fractional significand>) × 2<exponent> − 127

leading to the following range of numbers:

Such floating-point numbers are known as "reals" or "floats" in general, but with a number of variations:

A 32-bit float value is sometimes called a "real32" or a "single", meaning "single-precision floating-point value".

A 64-bit float is sometimes called a "real64" or a "double", meaning "double-precision floating-point value".

The relation between numbers and bit patterns is chosen for convenience in computer manipulation; eight bytes stored in computer memory may represent a 64-bit real, two 32-bit reals, or four signed or unsigned integers, or some other kind of data that fits into eight bytes. The only difference is how the computer interprets them. If the computer stored four unsigned integers and then read them back from memory as a 64-bit real, it almost always would be a perfectly valid real number, though it would be junk data.

Only a finite range of real numbers can be represented with a given number of bits. Arithmetic operations can overflow or underflow, producing a value too large or too small to be represented.

The representation has a limited precision. For example, only 15 decimal digits can be represented with a 64-bit real. If a very small floating-point number is added to a large one, the result is just the large one. The small number was too small to even show up in 15 or 16 digits of resolution, and the computer effectively discards it. Analyzing the effect of limited precision is a well-studied problem. Estimates of the magnitude of round-off errors and methods to limit their effect on large calculations are part of any large computation project. The precision limit is different from the range limit, as it affects the significand, not the exponent.

The significand is a binary fraction that doesn't necessarily perfectly match a decimal fraction. In many cases a sum of reciprocal powers of 2 does not match a specific decimal fraction, and the results of computations will be slightly off. For example, the decimal fraction "0.1" is equivalent to an infinitely repeating binary fraction: 0.000110011 ...

Numbers in programming languages
Programming in assembly language requires the programmer to keep track of the representation of numbers. Where the processor does not support a required mathematical operation, the programmer must work out a suitable algorithm and instruction sequence to carry out the operation; on some microprocessors, even integer multiplication must be done in software.

High-level programming languages such as Ruby and Python offer an abstract number that may be an expanded type such as rational, bignum, or complex. Mathematical operations are carried out by library routines provided by the implementation of the language. A given mathematical symbol in the source code, by operator overloading, will invoke different object code appropriate to the representation of the numerical type; mathematical operations on any number—whether signed, unsigned, rational, floating-point, fixed-point, integral, or complex—are written exactly the same way.

Some languages, such as REXX and Java, provide decimal floating-points operations, which provide rounding errors of a different form.

title: Computer number format


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Yosemite National Park

Yosemite National Park is a national park in California. It is bordered on the southeast by Sierra National Forest and on the northwest by Stanislaus National Forest. The park is managed by the National Park Service and covers 759620 acre in four counties – centered in Tuolumne and Mariposa, extending north and east to Mono and south to Madera. Designated a World Heritage Site in 1984, Yosemite is internationally recognized for its granite cliffs, waterfalls, clear streams, groves of giant sequoia, lakes, mountains, meadows, glaciers, and biological diversity. Almost 95 percent of the park is designated wilderness. Yosemite is one of the largest and least fragmented habitat blocks in the Sierra Nevada.

Its geology is characterized by granite and remnants of older rock. About 10 million years ago, the Sierra Nevada was uplifted and tilted to form its unique slopes, which increased the steepness of stream and river beds, forming deep, narrow canyons. About one million years ago glaciers formed at higher elevations. They moved downslope, cutting and sculpting the U-shaped Yosemite Valley.

European American settlers first entered the valley in 1851. Other travelers entered earlier, but James D. Savage is credited with discovering the area that became Yosemite National Park. Native Americans had inhabited the region for nearly 4,000 years, although humans may have first visited as long as 8,000 to 10,000 years ago.

Yosemite was critical to the development of the concept of national parks. Galen Clark and others lobbied to protect Yosemite Valley from development, ultimately leading to President Abraham Lincoln's signing of the Yosemite Grant of 1864 that declared Yosemite as federally preserved land. In 1890, John Muir led a successful movement to motivate Congress to establish Yosemite Valley and its surrounding areas as a National Park. This helped pave the way for the National Park System. Yosemite draws about four million visitors annually. Most visitors spend the majority of their time in the valley's 7 sqmi. The park set a visitation record in 2016, surpassing five million visitors for the first time. In 2023, the park saw nearly four million visitors.

Toponym
The word Yosemite (derived from yohhe'meti, "they are killers" in Miwok) historically referred to the name that the Miwok gave to the Ahwahneechee People, the resident indigenous tribe. Previously, the region had been called "Ahwahnee" ("big mouth") by its only indigenous inhabitants, the Ahwahneechee. The term Yosemite in Miwok is easily confused with a similar term for "grizzly bear", and is still a common misconception.

Ahwahneechee and the Mariposa Wars
The indigenous natives of Yosemite called themselves the Ahwahneechee, meaning "dwellers" in Ahwahnee. The Ahwahneechee People were the only tribe that lived within the park boundaries, but other tribes lived in surrounding areas. Together they formed a larger Indigenous population in California, called the Southern Sierra Miwok. They are related to the Northern Paiute and Mono tribes. Other tribes like the Central Sierra Miwoks and the Yokuts, who both lived in the San Joaquin Valley and central California, visited Yosemite to trade and intermarry. This resulted in a blending of culture that helped preserve their presence in Yosemite after early American settlements and urban development threatened their survival. Vegetation and game in the region were similar to modern times; acorns were a dietary staple, as well as other seeds and plants, salmon and deer.

The 1848–1855 California Gold Rush was a major event impacting the native population. It drew more than 90,000 European Americans to the area in 1849, causing competition for resources between gold miners and residents. About 70 years before the Gold Rush, the indigenous population was estimated to be 300,000, quickly dropping to 150,000, and just ten years later, only about 50,000 remained. The reasons for such a decline included disease, birth rate decreases, starvation, and conflict. The conflict in Yosemite, which is known as the Mariposa War, was part of the California genocide, which was the systemic killing of indigenous peoples throughout the State between the 1840s and 1870s. It started in December 1850 when California funded a state militia to drive Native people from contested territory to suppress Native American resistance to the European American influx.

Yosemite tribes often stole from settlers and miners, sometimes killing them, in retribution for the extermination/domestication of their people, and loss of their lands and resources. The War and formation of the Mariposa Battalion was partially the result of a single incident involving James Savage, a Fresno trader whose trading post was attacked in December, 1850. After the incident, Savage rallied other miners and gained the support of local officials to pursue a war against the Natives. He was appointed United States Army Major and leader of the Mariposa Battalion in the beginning of 1851. He and Captain John Boling were responsible for pursuing the Ahwahneechee people, led by Chief Tenaya and driving them west, and out of Yosemite. In March 1851 under Savage's command, the Mariposa Battalion captured about 70 Ahwahneechee and planned to take them to a reservation in Fresno, but they escaped. Later in May, under the command of Boling, the battalion captured 35 Ahwahneechee, including Chief Tenaya, and marched them to the reservation. Most were allowed to eventually leave and the rest escaped. Tenaya and others fled across the Sierra Nevada and settled with the Mono Lake Paiutes. Tenaya and some of his companions were ultimately killed in 1853 either over stealing horses or a gambling conflict. The survivors of Tenaya's group and other Ahwahneechee were absorbed into the Mono Lake Paiute tribe. Accounts from this battalion were the first well-documented reports of European Americans entering Yosemite Valley. Attached to Savage's unit was Doctor Lafayette Bunnell, who later wrote about his awestruck impressions of the valley in The Discovery of the Yosemite. Bunnell is credited with naming Yosemite Valley, based on his interviews with Chief Tenaya. Bunnell wrote that Chief Tenaya was the founder of the Ahwahnee colony. Bunnell falsely believed that the word "Yosemite" meant "full-grown grizzly bear".

Indigenous peoples' continuing presence
After the Mariposa War, Native Americans continued to live in the Yosemite area in reduced numbers. The remaining Yosemite Ahwahneechee tribe members there were forced to relocate to a village constructed in 1851 by the state government. They learned to live within this camp and their limited rights, adapting to the changed environment by entering the tourism industry through employment and small businesses, manufacturing and selling goods and providing services. In 1953, the National Park Service banned all non-employee Natives from residing in the Park and evicted the non-employees who had residence. In 1969, with only a few families left in the Park, the National Park Service evicted the remaining Native people living within in the Park (all Park employees and their families) to a government housing area for park employees and destroyed the village as part of a fire-fighting exercise. A reconstructed "Indian Village of Ahwahnee" sits behind the Yosemite Museum, located next to the Yosemite Valley Visitor Center.

By the late 19th century, the population of all native inhabitants in Yosemite was difficult to determine, estimates ranged from thirty to several hundred. The Ahwahneechee people and their descendants were hard to identify. The last full-blooded Ahwahneechee died in 1931. Her name was Totuya, or Maria Lebrado. She was the granddaughter of Chief Tenaya and one of many forced out of her ancestral homelands. The Ahwahneechee live through the memory of their descendants, their fellow Yosemite Natives, and through the Yosemite exhibit in the Smithsonian and the Yosemite Museum. As a method of self-preservation and resilience, the Indigenous people of California proposed treaties in 1851 and 1852 that would have established land reservations for them, but Congress refused to ratify them. The Southern Sierra Miwuk Nation is seeking tribal sovereignty and federal recognition. The National Park Service created policies to protect sacred sites and allow Native People to return to their homelands and use National Park resources.

Early tourists
In 1855, entrepreneur James Mason Hutchings, artist Thomas Ayres and two others were the first tourists to visit. Hutchings and Ayres were responsible for much of Yosemite's earliest publicity, writing articles and special issues about the valley. Ayres' style was detailed with exaggerated angularity. His works and written accounts were distributed nationally, and an exhibition of his drawings was held in New York City. Hutchings' publicity efforts between 1855 and 1860 increased tourism to Yosemite. Natives supported the growing tourism industry by working as laborers or maids. Later, they performed dances for tourists, acted as guides, and sold handcrafted goods, notably woven baskets. The Indian village and its peoples fascinated visitors, especially James Hutchings who advocated for Yosemite tourism. He and others considered the indigenous presence to be one of Yosemite's greatest attractions.

Wawona was an early Indian encampment for Nuchu and Ahwahneechee people who were captured and relocated to a reservation on the Fresno River by Savage and the Mariposa Battalion in March 1851. Galen Clark discovered the Mariposa Grove of giant sequoia in Wawona in 1857. He had simple lodgings and roads built. In 1879, the Wawona Hotel was built to serve tourists visiting Mariposa Grove. As tourism increased, so did the number of trails and hotels to build on it.

The Wawona Tree, also known as the Tunnel Tree, was a giant sequoia that grew in the Mariposa Grove. It was 234 ft tall, and was 90 ft in circumference. When a carriage-wide tunnel was cut through the tree in 1881, it became even more popular as a tourist photo attraction. Carriages and automobiles, traversed the road that passed through the tree. The tree was permanently weakened by the tunnel, and it fell in 1969 under a heavy load of snow. It was estimated to have been 2,100 years old.

Yosemite's first concession was established in 1884 when John Degnan and his wife established a bakery and store. In 1916, the National Park Service granted a 20-year concession to the Desmond Park Service Company. It bought out or built hotels, stores, camps, a dairy, a garage, and other park facilities. The Hotel Del Portal was completed in 1908 by a subsidiary of the Yosemite Valley Railroad. It was located at El Portal, California just outside of Yosemite.

The Curry Company started in 1899, led by David and Jennie Curry to provide concessions. They founded Camp Curry, now Curry Village.

Park service administrators felt that limiting the number of concessionaires in the park would be more financially sound. The Curry Company and its rival, the Yosemite National Park Company, were forced to merge in 1925 to form the Yosemite Park & Curry Company (YP&CC). The company built the Ahwahnee Hotel in 1926–27.

Yosemite Grant
Concerned by the impact of commercial interests, citizens including Galen Clark and Senator John Conness advocated protection for the area. The 38th United States Congress passed legislation that was signed by President Abraham Lincoln on June 30, 1864, creating the Yosemite Grant. This is the first time land was set aside specifically for preservation and public use by the U.S. government, and set a precedent for the 1872 creation of Yellowstone national park, the nation's first. Yosemite Valley and the Mariposa Grove were ceded to California as a state park, and a board of commissioners was established two years later.

Galen Clark was appointed by the commission as the Grant's first guardian, but neither Clark nor the commissioners had the authority to evict homesteaders (which included Hutchings). The issue was not settled until 1872 when the homesteader land holdings were invalidated by the U.S. Supreme Court. Clark and the commissioners were ousted in a dispute that reached the Supreme Court in 1880. The two Supreme Court decisions affecting management of the Yosemite Grant are considered precedents in land management law. Hutchings became the new park guardian.

Tourist access to the park improved, and conditions in the Valley became more hospitable. Tourism significantly increased after the first transcontinental railroad was completed in 1869, while the long horseback ride to reach the area was a deterrent. Three stagecoach roads were built in the mid-1870s to provide better access for the growing number of visitors.

John Muir was a Scottish-born American naturalist and explorer. Muir's leadership ensured that many National Parks were left untouched, including Yosemite.

Muir wrote articles popularizing the area and increasing scientific interest in it. Muir was one of the first to theorize that the major landforms in Yosemite Valley were created by alpine glaciers, bucking established scientists such as Josiah Whitney. Muir wrote scientific papers on the area's biology. Landscape architect Frederick Law Olmsted emphasized the importance of conservation of Yosemite Valley.

Increased protection efforts
Overgrazing of meadows (especially by sheep), logging of giant sequoia, and other damage led Muir to become an advocate for further protection. Muir convinced prominent guests of the importance of putting the area under federal protection. One such guest was Robert Underwood Johnson, editor of Century Magazine. Muir and Johnson lobbied Congress for the Act that created Yosemite National Park on October 1, 1890. The State of California, however, retained control of Yosemite Valley and Mariposa Grove. Muir's writings raised awareness about the damage caused by sheep grazing, and he actively campaigned to virtually eliminate grazing from the Yosemite's high-country.

The newly created national park came under the jurisdiction of the United States Army's Troop I of the 4th Cavalry on May 19, 1891, which set up camp in Wawona with Captain Abram Epperson Wood as acting superintendent. By the late 1890s, sheep grazing was no longer a problem, and the Army made other improvements. However, the cavalry could not intervene to ease the worsening conditions. From 1899 to 1913, cavalry regiments of the Western Department, including the all Black 9th Cavalry (known as the "Buffalo Soldiers") and the 1st Cavalry, stationed two troops at Yosemite.

Muir and his Sierra Club continued to lobby the government and influential people for the creation of a unified Yosemite National Park. In May 1903, President Theodore Roosevelt camped with Muir near Glacier Point for three days. On that trip, Muir convinced Roosevelt to take control of Yosemite Valley and Mariposa Grove away from California and return it to the federal government. In 1906, Roosevelt signed a bill that shifted control.

National Park Service
The National Park Service (NPS) was formed in 1916, and Yosemite was transferred to that agency's jurisdiction. Tuolumne Meadows Lodge, Tioga Pass Road, and campgrounds at Tenaya and Merced lakes were also completed in 1916. Automobiles started to enter the park in ever-increasing numbers following the opening of all-weather highways to the park. The Yosemite Museum was founded in 1926 through the efforts of Ansel Franklin Hall. In the 1920s, the museum featured Native Americans practicing traditional crafts, and many Southern Sierra Miwok continued to live in Yosemite Valley until they were evicted from the park in the 1960s. Although the NPS helped create a museum that included Native American culture, its early actions and organizational values were dismissive of Yosemite Natives and the Ahwahneechee. NPS in the early 20th century criticized and restricted the expression of indigenous culture and behavior. For example, park officials penalized Natives for playing games and drinking during the Indian Field Days of 1924. In 1929, Park Superintendent Charles G. Thomson concluded that the Indian village was aesthetically unpleasant and was limiting white settler development and ordered the camp to be burned down. In 1969, many Native residents left in search of work as a result of the decline in tourism. NPS demolished their empty houses, evicted the remaining residents, and destroyed the entire village. This was the last Indigenous settlement within the park.

In 1903, a dam in Hetch Hetchy Valley in the northwestern region of the park was proposed. Its purpose was to provide water and hydroelectric power to San Francisco. Muir and the Sierra Club opposed the project, while others, including Gifford Pinchot, supported it. In 1913, the O'Shaughnessy Dam was approved via passage of the Raker Act.

In 1918, Clare Marie Hodges was hired as the first female Park Ranger in Yosemite. Following Hodges in 1921, Enid Michael was hired as a seasonal Park Ranger and continued to serve in that position for 20 years.

In 1937, conservationist Rosalie Edge, head of the Emergency Conservation Committee (ECC), successfully lobbied Congress to purchase about 8,000 acres of old-growth sugar pines on the perimeter of Yosemite National Park that were to be logged.

By 1968, traffic congestion and parking in Yosemite Valley during the summer months has become a concern. NPS reduced artificial inducements to visit the park, such as the Firefall, in which red-hot embers were pushed off a cliff near Glacier Point at night.

In 1984, preservationists persuaded Congress to designate 677600 acre, or about 89 percent of the park, as the Yosemite Wilderness. As a wilderness area, it would be preserved in its natural state with humans being only temporary visitors.

In 2016, The Trust for Public Land (TPL) purchased Ackerson Meadow, a 400 acre on the western edge of the park for $2.3 million. Ackerson Meadow was originally included in the proposed 1890 park boundary, but never acquired by the federal government. The purchase and donation of the meadow was made possible through a cooperative effort by TPL, NPS, and Yosemite Conservancy. On September 7, 2016, NPS accepted the donation of the land, making the meadow the largest addition to Yosemite since 1949.

Geography
Yosemite National Park is located in the central Sierra Nevada. Three wilderness areas are adjacent to Yosemite: the Ansel Adams Wilderness to the southeast, the Hoover Wilderness to the northeast, and the Emigrant Wilderness to the north.

The 1189 mi2 park contains thousands of lakes and ponds, 1600 mi of streams, 800 mi of hiking trails, and 350 mi of roads. Two federally designated Wild and Scenic Rivers, the Merced and the Tuolumne, begin within Yosemite's borders and flow westward through the Sierra foothills into the Central Valley of California.

Rocks and erosion
Almost all of the landforms are cut from the granitic rock of the Sierra Nevada Batholith (a batholith is a large mass of intrusive igneous rock that formed deep below the surface). About five percent of the park's landforms (mostly in its eastern margin near Mount Dana) are metamorphosed volcanic and sedimentary rocks. These roof pendants once formed the roof over the underlying granitic magma.

Erosion acting upon different types of uplift-generated joint and fracture systems is responsible for producing the valleys, canyons, domes, and other features. These joints and fracture systems do not move, and are therefore not faults. Spacing between joints is controlled by the amount of silica in the granite and granodiorite rocks; more silica tends to form a more resistant rock, resulting in larger spaces between joints and fractures.

Pillars and columns, such as Washington Column and Lost Arrow, are generated by cross joints. Erosion acting on master joints is responsible for shaping valleys and later canyons. The single most erosive force over the last few million years has been large alpine glaciers, which turned the previously V-shaped river-cut valleys into U-shaped glacial-cut canyons (such as Yosemite Valley and Hetch Hetchy Valley). Exfoliation (caused by the tendency of crystals in plutonic rocks to expand at the surface) acting on granitic rock with widely spaced joints is responsible for producing domes such as Half Dome and North Dome and inset arches like Royal Arches.

Popular features
Yosemite Valley represents only one percent of the park area. The Tunnel View gives a view of the valley. El Capitan is a prominent granite cliff that looms over the valley, and is a rock climbing favorite because of its sheer size, diverse climbing routes, and year-round accessibility. Granite domes such as Sentinel Dome and Half Dome rise 3000 and 4800 ft, respectively, above the valley floor. The park contains dozens of other granite domes.

The high country of Yosemite contains other important features such as Tuolumne Meadows, Dana Meadows, the Clark Range, the Cathedral Range, and the Kuna Crest. The Sierra crest and the Pacific Crest Trail run through Yosemite. Mount Dana and Mount Gibbs are peaks of red metamorphic rock. Granite peaks include Mount Conness, Cathedral Peak, and Matterhorn Peak. Mount Lyell is the highest point in the park, standing at 13,120 ft. The Lyell Glacier is the largest glacier in the park and one of the few remaining in the Sierra.

The park has three groves of ancient giant sequoia (Sequoiadendron giganteum) trees; the Mariposa Grove (200 trees), the Tuolumne Grove (25 trees), and the Merced Grove (20 trees). This species grows larger in volume than any other and is one of the tallest and longest-lived.

Water and ice
The Tuolumne and Merced River systems originate along the crest of the Sierra in the park and have carved river canyons 3000 to 4000 ft deep. The Tuolumne River drains the entire northern portion of the park, an area of approximately 680 mi2. The Merced River begins in the park's southern peaks, primarily the Cathedral and Clark Ranges, and drains an area of approximately 511 mi2.

Hydrologic processes, including glaciation, flooding, and fluvial geomorphic response, have been fundamental in creating park landforms. The park contains approximately 3,200 lakes (greater than 100 m2), two reservoirs, and 1700 mi of streams. Wetlands flourish in valley bottoms throughout the park, and are often hydrologically linked to nearby lakes and rivers through seasonal flooding and groundwater. Meadow habitats, distributed at elevations from 3000 to 11000 ft in the park, are generally wetlands, as are the riparian habitats found on the banks of Yosemite's watercourses.

Yosemite is famous for its high concentration of waterfalls in a small area. Numerous sheer drops, glacial steps and hanging valleys in the park feature spectacular cascades, especially during April, May, and June (as the snow melts). Located in Yosemite Valley, Yosemite Falls is the fourth tallest waterfall in North America at 2425 ft according to the World Waterfall Database. Also in the valley is the much lower volume Ribbon Falls, which has the highest single vertical drop, 1612 ft. Perhaps the most prominent of the valley waterfalls is Bridalveil Fall. Wapama Falls in Hetch Hetchy Valley is another notable waterfall. Hundreds of ephemeral waterfalls become active in the park after heavy rains or melting snowpack.

Park glaciers are relatively small and occupy areas that are in almost permanent shade, such as north- and northeast-facing cirques. Lyell Glacier is the largest glacier in Yosemite (the Palisades Glaciers are the largest in the Sierra Nevada) and covers 160 acre. None of the Yosemite glaciers are a remnant of the Ice Age alpine glaciers responsible for sculpting the Yosemite landscape. Instead, they were formed during one of the neoglacial episodes that have occurred since the thawing of the Ice Age (such as the Little Ice Age). Many Yosemite glaciers have disappeared, such as the Black Mountain Glacier that was marked in 1871 and had gone by the mid-1980s. Yosemite's final two glaciers – the Lyell and Maclure glaciers – have receded over the last 100 years and are expected to disappear as a result of climate change.

Climate
Yosemite has a Mediterranean climate (Köppen climate classification Csa), meaning most precipitation falls during the mild winter, and the other seasons are nearly dry (less than three percent of precipitation falls during the long, hot summers). Because of orographic lift, precipitation increases with elevation up to 8000 ft where it slowly decreases to the crest. Precipitation amounts vary from 36 in at 4000 ft elevation to 50 in at 8600 ft. Snow does not typically accumulate until November in the high country. It deepens into March or early April.

Mean daily temperatures range from 25 °F to 53 °F at Tuolumne Meadows at 8600 ft. At the Wawona Entrance (elevation 5130 ft), mean daily temperature ranges from 36 to 67 °F. At the lower elevations below 5000 ft, temperatures are hotter; the mean daily high temperature at Yosemite Valley (elevation 3966 ft) varies from 46 to 90 °F. At elevations above 8000 ft, the hot, dry summer temperatures are moderated by frequent summer thunderstorms, along with snow that can persist into July. The combination of dry vegetation, low relative humidity, and thunderstorms results in frequent lightning-caused fires as well.

At park headquarters (elevation 4018 ft), January averages 38.0 °F, while July averages 73.3 °F. In summer the nights are much cooler than the days. An average of 45.5 days have highs of 90 °F or higher and an average of 105.6 nights with freezing temperatures. Freezing temperatures have been recorded in every month of the year. The record high temperature was 112 °F on July 22 and July 24, 1915, while the record low temperature was -7 °F on January 1, 2009. Average annual precipitation is nearly 37 in, falling on 67 days. The wettest year was 1983 with 66.06 in and the driest year was 1976 with 14.84 in. The most precipitation in one month was 29.61 in in December 1955 and the most in one day was 6.92 in on December 23, 1955. Average annual snowfall is 39.4 in. The snowiest winter was 1948–1949 with 176.5 in. The most snow in one month was 175.0 in in January 1993.

Tectonic and volcanic activity
The location of the park was a passive continental margin during the Precambrian and early Paleozoic. Sediment was derived from continental sources and was deposited in shallow water. These rocks became deformed and metamorphosed.

Heat generated from the Farallon Plate subducting below the North American Plate led to the creation of an island arc of volcanoes on the west coast of proto-North America between the late Devonian and Permian periods. Material accreted onto the western edge of North America, and mountains were raised to the east in Nevada.

The first phase of regional plutonism started 210 million years ago in the late Triassic and continued throughout the Jurassic to about 150 million years before present (BP), which led to the creation of the Sierra Nevada Batholith. The resulting rocks were mostly granitic in composition and lay about 6 mi below the surface. Around the same time, the Nevadan orogeny built the Nevadan mountain range (also called the Ancestral Sierra Nevada) to a height of 15000 ft.

The second major pluton emplacement phase lasted from about 120 million to 80 million years ago during the Cretaceous. This was part of the Sevier orogeny.

Starting 20 million years ago (in the Cenozoic) and lasting until 5 million years ago, a now-extinct extension of Cascade Range volcanoes erupted, bringing large amounts of igneous material in the area. These igneous deposits blanketed the region north of the Yosemite region. Volcanic activity persisted past 5 million years BP east of the current park borders in the Mono Lake and Long Valley areas.

Uplift and erosion
Starting 10 million years ago, vertical movement along the Sierra fault started to uplift the Sierra Nevada. Subsequent tilting of the Sierra block and the resulting accelerated uplift of the Sierra Nevada increased the gradient of western-flowing streams. The streams consequently ran faster and thus cut their valleys more quickly. Additional uplift occurred when major faults developed to the east, forming Owens Valley from Basin and Range-associated extensional forces. Sierra uplift accelerated again about two million years ago during the Pleistocene.

The uplifting and increased erosion exposed granitic rocks to surface pressures, resulting in exfoliation (responsible for the rounded shape of the many domes in the park) and mass wasting following the numerous fracture joint planes (cracks; especially vertical ones) in the now solidified plutons. Pleistocene glaciers further accelerated this process, while glacial meltwater transported the resulting talus and till from valley floors.

Numerous vertical joint planes controlled where and how fast erosion took place. Most of these long, linear and very deep cracks trend northeast or northwest and form parallel, often regularly spaced sets.

Glacial sculpting
A series of glaciations further modified the region starting about 2 to 3 million years ago and ending sometime around 10,000 BP. At least four major glaciations occurred in the Sierra, locally called the Sherwin (also called the pre-Tahoe), Tahoe, Tenaya, and Tioga. The Sherwin glaciers were the largest, filling Yosemite and other valleys, while later stages produced much smaller glaciers. A Sherwin-age glacier was almost surely responsible for the major excavation and shaping of Yosemite Valley and other canyons in the area.

Glacial systems reached depths of up to 4000 ft and left their marks. The longest glacier ran down the Grand Canyon of the Tuolumne River for 60 mi, passing well beyond Hetch Hetchy Valley. Merced Glacier flowed out of Yosemite Valley and into the Merced River Gorge. Lee Vining Glacier carved Lee Vining Canyon and emptied into Lake Russel (the much-enlarged ice age version of Mono Lake). Only the highest peaks, such as Mount Dana and Mount Conness, were not covered by glaciers. Retreating glaciers often left recessional moraines that impounded lakes such as the 5.5 mi long Lake Yosemite (a shallow lake that periodically covered much of the floor of Yosemite Valley).

Habitats
The park has an elevation range from 2127 to 13114 ft and contains five major vegetation zones: chaparral and oak woodland, lower montane forest, upper montane forest, subalpine zone, and alpine. Of California's 7,000 plant species, approximately 50 percent occur in the Sierra Nevada and more than 20 percent within the park. The park contains suitable habitat for more than 160 rare plants, with rare local geologic formations and unique soils characterizing the restricted ranges many of these plants occupy.

With its scrubby sun-baked chaparral, stately groves of pine, fir, and sequoia, and expanses of alpine woodlands and meadows, Yosemite National Park preserves a Sierra landscape as it prevailed before Euro-American settlement. In contrast to surrounding lands, which have been significantly altered by logging, the park contains some 225510 acre of old-growth forest. Taken together, the park's varied habitats support over 250 species of vertebrates, which include fish, amphibians, reptiles, birds, and mammals.

Yosemite's western boundary has habitats dominated by mixed coniferous forests of ponderosa pine, sugar pine, incense cedar, white fir, Douglas fir, and a few stands of giant sequoia, interspersed by areas of black oak and canyon live oak. These habitats support relatively high wildlife diversity. Wildlife include black bear, coyote, raccoon, mountain kingsnake, Gilbert's skink, white-headed woodpecker, bobcat, river otter, gray fox, red fox, brown creeper, two species of skunk, cougar, spotted owl, and bats.

At higher elevation, the coniferous forests become purer stands of red fir, western white pine, Jeffrey pine, lodgepole pine, and the occasional foxtail pine. Fewer wildlife species tend to be found in these habitats, because of their higher elevation and lower complexity. Animals include golden-mantled ground squirrel, chickaree, fisher, Steller's jay, hermit thrush, and American goshawk. Reptiles are not common, but include rubber boa, western fence lizard, and northern alligator lizard.

As the landscape rises, trees become smaller and more sparse, with stands broken by areas of exposed granite. These include lodgepole pine, whitebark pine, and mountain hemlock that, at highest elevations, give way to vast expanses of granite as treeline is reached. The climate in these habitats is harsh and the growing season is short, but species such as pika, yellow-bellied marmot, white-tailed jackrabbit, Clark's nutcracker, and black rosy finch are adapted to these conditions. Treeless alpine habitats are favored by Sierra Nevada bighorn sheep. This species is found in the Yosemite area only around Tioga Pass, where a small, reintroduced population exists.

At a variety of elevations, meadows provide important habitat. Animals come to feed on the green grasses and use the flowing and standing water found in many meadows. Predators follow these animals. The interface between meadow and forest is favored by many animal species because of the proximity of open areas for foraging and cover for protection. Species that are highly dependent upon meadow habitat include great grey owl, willow flycatcher, Yosemite toad, and mountain beaver.

Management issues
The black bears of Yosemite were once famous for breaking into parked cars to steal food. They were an encouraged tourist sight for many years at the park's garbage dumps, where they congregated to eat garbage and tourists gathered to photograph them. Increasing bear/human encounters and increasing property damage led to an aggressive campaign to discourage bears from interacting with people and their stuff. The open-air dumps were closed; trash receptacles were replaced with bear-proof receptacles; campgrounds were equipped with bear-proof food lockers so that people would not leave food in their vehicles. Because bears who show aggression towards people usually are destroyed, park personnel have come up with innovative ways to lead bears to associate humans and their property with experiences such as getting hit with a rubber bullet. , about 30 bears a year were captured and ear-tagged and their DNA sampled so that, when bear damage occurs, rangers can ascertain which bear was causing the problem.

Despite the richness of high-quality habitats in Yosemite, the brown bear, California condor, and least Bell's vireo have become extinct in the park within historical time, and another 37 species currently have special status under either California or federal endangered species legislation. The most serious current threats include loss of a natural fire regime, exotic species, air pollution, habitat fragmentation, and climate change. On a more local basis, factors such as road kills and the availability of human food have affected some wildlife species.

Yosemite National Park has documented the presence of more than 130 non-native plant species within park boundaries. They were introduced into Yosemite following the migration of early Euro-American settlers in the late 1850s. Natural and human-caused disturbances, such as wildland fires and construction activities, have contributed to a rapid increase in the spread of non-native plants. Some of these species invade and displace the native plant communities, impacting park resources. Non-native plants can bring about significant ecosystem changes by altering native plant communities and the processes that support them. Some non-native species may cause an increase in fire frequency or increase the available soil nitrogen that allow other non-native plants to establish. Many non-native species, such as yellow star thistle (Centaurea solstitialis), are able to produce a long tap root that allows them to out-compete the native plants for available water.

Bull thistle (Cirsium vulgare), common mullein (Verbascum thapsus), and Klamath weed (Hypericum perforatum) have been identified as noxious pests in Yosemite since the 1940s. More recently recognized species are yellow star thistle (Centaurea solstitialis), sweet clover (Melilot spp.), Himalayan blackberry (Rubus armeniacus), cut-leaved blackberry (Rubus laciniatus) and large periwinkle (Vinca major).

Increasing ozone pollution causes tissue damage to sequoia trees, making them more vulnerable to insect infestation and disease. Since the cones of these trees require fire-touched soil to germinate, historic fire suppression has reduced their ability to reproduce. Planned prescribed fires may help the germination issue.

Wildfires
Indigenous residents intentionally set small fires in the early 1860s and before to clear the ground of brush as part as their farming practices. These fires are comparable to contemporary practices such as controlled burns that are done by the U.S. Forest Service and others. Although it was not their primary reason, Yosemite Natives helped preserve biodiversity and resilience by lighting these small fires. Native Americans used fire as an early wildlife management tool to keep certain lands clear, resulting in more food for large animals and decreasing the chance of large forest fires which that now devastate forest ecosystems. Some early uncontrolled forest fires were set accidentally by the militia group led by Major John Savage when the group burned down the Ahwahneechee camp in an attempt to expel them. The house fires eventually spread to a large section of the forest and the militia group ended up having to abandon their raid to save their own camp from the conflagration.

Forest fires clear the park of dead vegetation, making way for new growth. Small fires damage the income generated by tourism. During late July and early August, 2018, the Valley and other sections of the park, temporarily closed due to the Ferguson Fire at its western boundary. The closing was the largest in almost thirty years.

Public access
Yosemite National Park is open year-round, though certain roads close during snowy months, usually from November through May or June. Certain trails also close during winter, including The 4-Mile Trail and part of The Mist Trail.

Traffic congestion in the valley is heavy during peak summer season (June to August) and a free shuttle bus system operates in the valley. Parking in the valley during the summer is often full. Transit options are available from Fresno and Merced.

The natural and cultural history of Yosemite Valley is presented at the Yosemite Valley Visitor Center, the adjoining Yosemite Museum, and the Nature Center at Happy Isles. The parks' two National Historic Landmarks are the Sierra Club's LeConte Memorial Lodge (Yosemite's first public visitor center), and the Ahwahnee Hotel. Camp 4 is on the National Register of Historic Places.

Hiking
Over 800 mi of trails are available to hikers —everything from an easy stroll to a challenging mountain hike, or backpacking trips. The popular Half Dome hike to the summit of Half Dome requires a permit whenever the cables are up (usually from Memorial Day weekend to Columbus Day). A maximum of 300 hikers, selected by lottery, are permitted to advance beyond the base of the subdome each day, including 225 day hikers and 75 backpackers.

The park can be divided into five sections—Yosemite Valley, Wawona/Mariposa Grove/Glacier Point, Tuolumne Meadows, Hetch Hetchy, and Crane Flat/White Wolf. Numerous books describe park trails, while the National Park Service provides free information.

Between late spring and early fall, much of the park can be accessed for backpacking trips. All overnight trips into the back country require a wilderness permit and most require approved bear-resistant food storage.

Driving
While some locations in Yosemite are accessible only on foot, other locations can be reached via road. The most famous road is Tioga Road.

Bicycles are allowed on the roads, but only 12 mi of paved off-road trails are available in Yosemite Valley itself; mountain biking is not allowed.

Climbing
Rock climbing is an important part of Yosemite. In particular the valley is surrounded by summits such as Half Dome and El Capitan. Camp 4 is a walk-in campground in the Valley that was instrumental in the development of rock climbing as a sport, and is listed on the National Register of Historic Places. Climbers can generally be spotted in the snow-free months on anything from ten-foot-high (3 m) boulders to the 3300 ft face of El Capitan. Classes on rock climbing are offered there.

Tuolumne Meadows is well known for rock and mountain climbing.

Winter activities
Away from the Valley, the park is snowed in during the winter months, with many roads closed. Downhill skiing is available at the Badger Pass Ski Area—the oldest downhill skiing area in California, operating from mid-December through early April. Much of the park is open to cross-country skiing and snowshoeing, and backcountry ski huts are available. Wilderness permits are required for backcountry overnight ski trips.

The Bracebridge dinner is an annual holiday event, held since 1927 at the Ahwahnee Hotel, inspired by Washington Irving's descriptions of Squire Bracebridge and English Christmas traditions of the 18th century in his Sketch Book. Between 1929 and 1973, the show was organized by Ansel Adams.

Other
Yosemite has 13 official campgrounds.

Bicycle rentals are available from spring through fall. Over 12 mi of paved bike paths are available in Yosemite Valley. In addition, bicyclists can ride on roads. Helmets are required for children under 18 years of age. Off-trail riding and mountain biking are not permitted in the park.

Water activities are plentiful during warmer months. Rafting can be done through the valley on the Merced River from late May to July. Swimming pools are available at Yosemite Lodge and Curry Village.

Horsetail Fall
Horsetail Fall flows over the eastern edge of El Capitan. This small waterfall usually flows only during winter and is easy to miss. On rare occasions during mid- to late February, it can glow orange when backlit by sunset. This unique lighting effect happens only on evenings with a clear sky. Minor haze or cloudiness can spoil the effect. Although entirely natural, the phenomenon is reminiscent of the human-caused Firefall that historically occurred from Glacier Point.

In popular culture
The opening scenes of Star Trek V: The Final Frontier (1989) were filmed in Yosemite National Park. Films such as The Last of the Mohicans (1920) and Maverick (1994) have also been shot there. The 2014 documentary Valley Uprising is centered around Yosemite Valley and its history with an emphasis on climbing culture. The Academy Award-winning 2018 documentary Free Solo was filmed in Yosemite. The Dawn Wall, a 2017 documentary, was filmed there.

General references

 * Muir, John. "Features of the Proposed Yosemite National Park " The Century; a popular quarterly (Sept. 1890) 40#5
 * Muir, John. "Features of the Proposed Yosemite National Park " The Century; a popular quarterly (Sept. 1890) 40#5
 * Muir, John. "Features of the Proposed Yosemite National Park " The Century; a popular quarterly (Sept. 1890) 40#5
 * Muir, John. "Features of the Proposed Yosemite National Park " The Century; a popular quarterly (Sept. 1890) 40#5
 * Muir, John. "Features of the Proposed Yosemite National Park " The Century; a popular quarterly (Sept. 1890) 40#5

title: Yosemite National Park


text:
Tocantins River

The Tocantins River (Rio Tocantins, Parkatêjê: Pyti [pɨˈti]) is a river in Brazil, the central fluvial artery of the country. In the Tupi language, its name means "toucan's beak" (Tukã for "toucan" and Ti for "beak"). It runs from south to north for about 2450 km. It is not really a branch of the Amazon River, since its waters flow into the Atlantic Ocean alongside those of the Amazon. It flows through four Brazilian states (Goiás, Tocantins, Maranhão, and Pará) and gives its name to one of Brazil's newest states, formed in 1988 from what was until then the northern portion of Goiás.

The Tocantins is one of the largest clearwater rivers in South America.

Course
It rises in the mountainous district known as the Pireneus, west of the Federal District, but its western tributary, the Araguaia River, has its extreme southern headwaters on the slopes of the Serra dos Caiapós. The Araguaia flows 1670 km before its confluence with the Tocantins, to which it is almost equal in volume. Besides its main tributary, the Rio das Mortes, the Araguaia has twenty smaller branches, offering many miles of canoe navigation. In finding its way to the lowlands, it breaks frequently into waterfalls and rapids, or winds violently through rocky gorges, until, at a point about 160 km above its junction with the Tocantins, it saws its way across a rocky dyke for 20 km in roaring cataracts. Two other tributaries, called the Maranhão and Paranatinga, collect an immense volume of water from the highlands which surround them, especially on the south and south-east. Between the latter and the confluence with the Araguaia, the Tocantins is occasionally obstructed by rocky barriers which cross it almost at a right angle.

Fauna
The Tocantins basin (which include the Araguaia River) is the home of several large aquatic mammals such as Amazonian manatee, Araguaian river dolphin and tucuxi, and larger reptiles such as black caiman, spectacled caiman and yellow-spotted river turtle.

The Tocantins River Basin has a high richness of fish species, although it is relatively low by Amazon basin standards. More than 350 fish species have been registered, including more than 175 endemics. The most species rich families are Characidae (tetras and allies), Loricariidae (pleco catfish and allies) and Rivulidae (South American killifish). While most species essentially are of Amazonian origin, there are also some showing a connection with the Paraná and São Francisco rivers. The Tocantins and these two rivers flow in different directions, but all have their source in the Brazilian Plateau in a region where a low watershed allows some exchange between them. There are several fish species that migrate along the Tocantins to spawn, but this has been restricted by the dams. Following the construction of the massive Tucuruí Dam, the flow of the river changed. Some species have been adversely affected and there has been a substantial reduction in species richness in parts of the river.

The São Domingos karst in the upper Tocantins basin is home to an unusually high number of cavefish species (more than any other region in the Americas): Ancistrus cryptophthalmus, several Ituglanis species, Pimelodella spelaea, Aspidoras mephisto, an undescribed Cetopsorhamdia species and Eigenmannia vicentespelaea. The last is the only known cave-adapted knifefish and one of only two known non-catfish in caves of the South American mainland (the other is the characid Stygichthys typhlops).

In its lower reaches, the Tocantins separates the Tocantins–Araguaia–Maranhão moist forests ecoregion to the east from the Xingu–Tocantins–Araguaia moist forests ecoregion to the west. It acts as a barrier that prevents dispersal of flora and fauna between these ecoregions.

Dams
Downstream from the Araguaia confluence, in the state of Pará, the river used to have many cataracts and rapids, but they were flooded in the early 1980s by the artificial lake created by the Tucuruí Dam, one of the world's largest. When the second phase of the Tucuruí project was completed on November 30, 2010, a system of locks called Eclusas do Tucuruí was established with the goal of making a long extension of the river navigable.

In total there are seven dams on the river (Serra da Mesa dam, Cana Brava dam, São Salvador dam, Peixe Angical dam, Luiz Eduardo Magalhães (Lajeado) dam, Estreito dam, and Tucuruí dam), of which the largest are the Tucuruí and the Serra da Mesa dam.

Geology
The flat, broad valleys, composed of sand and clay, of both the Tocantins and its Araguaia branch are overlooked by steep bluffs. They are the margins of the great sandstone plateaus, from 300 to 600 m elevation above sea-level, through which the rivers have eroded their deep beds. Around the estuary of the Tocantins the great plateau has disappeared, to give place to a part of the forest-covered, half submerged alluvial plain, which extends far to the north-east and west. The Pará River, generally called one of the mouths of the Amazon, is only the lower reach of the Tocantins. If any portion of the waters of the Amazon runs round the southern side of the large island of Marajó into the river Para, it is only through tortuous, natural canals, which are in no sense outflow channels of the Amazon.

Discharge
The Tocantins River records a mean discharge rate of 13598 m3/s and a specific discharge rate of 14.4 L/s/km2. The sub-basins have the following specific discharge rates: Tocantins (11 L/s/km2), Araguaia (16 L/s/km2), Pará (17 L/s/km2) and Guamá (21 L/s/km2).

title: Tocantins River


text:
Perlite

Perlite is an amorphous volcanic glass that has a relatively high water content, typically formed by the hydration of obsidian. It occurs naturally and has the unusual property of greatly expanding when heated sufficiently. It is an industrial mineral, suitable "as ceramic flux to lower the sintering temperature", and a commercial product useful for its low density after processing.

Properties
Perlite softens when it reaches temperatures of 850 –. Water trapped in the structure of the material vaporises and escapes, and this causes the expansion of the material to 7–16 times its original volume. The expanded material is a brilliant white, due to the reflectivity of the trapped bubbles. Unexpanded ("raw") perlite has a bulk density around 1100 kg/m3 (1.1 g/cm3), while typical expanded perlite has a bulk density of about 30–150 kg/m3 (0.03–0.150 g/cm3).

Typical analysis

 * 70–75% silicon dioxide: SiO2
 * 12–15% aluminium oxide: Al2O3
 * 3–4% sodium oxide: Na2O
 * 3–5% potassium oxide: K2O
 * 0.5-2% iron oxide: Fe2O3
 * 0.2–0.7% magnesium oxide: MgO
 * 0.5–1.5% calcium oxide: CaO
 * 3–5% loss on ignition (chemical / combined water)

Sources and production
Perlite is a non-renewable resource. The world reserves of perlite are estimated at 700 million tonnes.

The confirmed resources of perlite existing in Armenia amount to 150 million m3, whereas the total amount of projected resources reaches up to 3 billion m3. Considering specific density of 1.1 ton/m3 confirmed reserves in Armenia amount to 165 million tons.

Other reported reserves are: Greece - 120 million tonnes, Turkey, USA and Hungary - about 49-57 million tonnes.

Perlite world production, led by China, Turkey, Greece, USA, Armenia and Hungary, summed up to 4.6 million tonnes in 2018.

Uses
Because of its low density and relatively low price (about US$150 per tonne of unexpanded perlite), many commercial applications for perlite have been developed.

Construction and manufacturing
In the construction and manufacturing fields, it is used in lightweight plasters, concrete and mortar, insulation and ceiling tiles. It may also be used to build composite materials that are sandwich-structured or to create syntactic foam.

Perlite filters are fairly common in filtering beer before it is bottled.

Small quantities of perlite are also used in foundries, cryogenic insulation, and ceramics (as a clay additive). It is also used by the explosives industry.

Aquatic filtration
Perlite is currently used in commercial pool filtration technology, as a replacement to diatomaceous earth filters. Perlite is an excellent filtration aid and is used extensively as an alternative to diatomaceous earth. The popularity of perlite usage as a filter medium is growing considerably worldwide. Several products exist in the market to provide perlite based filtration. Several perlite filters and perlite media have met NSF-50 approval (Aquify PMF Series and AquaPerl), which standardizes water quality and technology safety and performance. Perlite can be safely disposed of through existing sewage systems, although some pool operators choose to separate the perlite using settling tanks or screening systems to be disposed of separately.

Biotechnology
Due to thermal and mechanical stability, non-toxicity, and high resistance against microbial attacks and organic solvents, perlite is widely used in biotechnological applications. Perlite was found to be an excellent support for immobilization of biocatalysts such as enzymes for bioremediation and sensing applications.

Agriculture
In horticulture, perlite can be used as a soil amendment or alone as a medium for hydroponics or for starting cuttings. When used as an amendment, it has high permeability and low water retention and helps prevent soil compaction.

Cosmetics
Perlite is used in cosmetics as an absorbent and mechanical exfoliant.

Substitutes
Perlite can be replaced for all of its uses. Substitutes include:
 * Diatomite, used for filter-aids
 * Expanded clay, an alternative lightweight filler for building materials
 * Shale
 * Pumice
 * Slag
 * Vermiculite - many expanders of perlite are also exfoliating vermiculite and belong to both trade associations

Occupational safety
As perlite contains silicon dioxide, goggles and silica filtering masks are recommended when handling large quantities.

United States
The Occupational Safety and Health Administration (OSHA) has set the legal limit (permissible exposure limit) for perlite exposure in the workplace as 15 mg/m3 total exposure and 5 mg/m3 respiratory exposure over an 8-hour workday. The National Institute for Occupational Safety and Health (NIOSH) has set a recommended exposure limit (REL) of 10 mg/m3 total exposure and 5 mg/m3 respiratory exposure over an 8-hour workday.

title: Perlite