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In chemical kinetics , the overall rate of a reaction is often approximately determined by the slowest step, known as the rate-determining step ( RDS or RD-step [ 1 ] or r/d step [ 2 ] [ 3 ] ) or rate-limiting step . For a given reaction mechanism, the prediction of the corresponding rate equation (for comparison with the experimental rate law) is often simplified by using this approximation of the rate-determining step.
In principle, the time evolution of the reactant and product concentrations can be determined from the set of simultaneous rate equations for the individual steps of the mechanism, one for each step. However, the analytical solution of these differential equations is not always easy, and in some cases numerical integration may even be required. [ 4 ] The hypothesis of a single rate-determining step can greatly simplify the mathematics. In the simplest case the initial step is the slowest, and the overall rate is just the rate of the first step.
Also, the rate equations for mechanisms with a single rate-determining step are usually in a simple mathematical form, whose relation to the mechanism and choice of rate-determining step is clear. The correct rate-determining step can be identified by predicting the rate law for each possible choice and comparing the different predictions with the experimental law, as for the example of NO 2 and CO below.
The concept of the rate-determining step is very important to the optimization and understanding of many chemical processes such as catalysis and combustion .
As an example, consider the gas-phase reaction NO 2 + CO → NO + CO 2 . If this reaction occurred in a single step, its reaction rate ( r ) would be proportional to the rate of collisions between NO 2 and CO molecules: r = k [ NO 2 ][CO], where k is the reaction rate constant , and square brackets indicate a molar concentration . Another typical example is the Zel'dovich mechanism .
In fact, however, the observed reaction rate is second-order in NO 2 and zero-order in CO, [ 5 ] with rate equation r = k [ NO 2 ] 2 . This suggests that the rate is determined by a step in which two NO 2 molecules react, with the CO molecule entering at another, faster, step. A possible mechanism in two elementary steps that explains the rate equation is:
In this mechanism the reactive intermediate species NO 3 is formed in the first step with rate r 1 and reacts with CO in the second step with rate r 2 . However, NO 3 can also react with NO if the first step occurs in the reverse direction (NO + NO 3 → 2 NO 2 ) with rate r −1 , where the minus sign indicates the rate of a reverse reaction.
The concentration of a reactive intermediate such as [ NO 3 ] remains low and almost constant. It may therefore be estimated by the steady-state approximation, which specifies that the rate at which it is formed equals the (total) rate at which it is consumed. In this example NO 3 is formed in one step and reacts in two, so that
The statement that the first step is the slow step actually means that the first step in the reverse direction is slower than the second step in the forward direction, so that almost all NO 3 is consumed by reaction with CO and not with NO. That is, r −1 ≪ r 2 , so that r 1 − r 2 ≈ 0. But the overall rate of reaction is the rate of formation of final product (here CO 2 ), so that r = r 2 ≈ r 1 . That is, the overall rate is determined by the rate of the first step, and (almost) all molecules that react at the first step continue to the fast second step.
The other possible case would be that the second step is slow and rate-determining, meaning that it is slower than the first step in the reverse direction: r 2 ≪ r −1 . In this hypothesis, r 1 − r −1 ≈ 0, so that the first step is (almost) at equilibrium . The overall rate is determined by the second step: r = r 2 ≪ r 1 , as very few molecules that react at the first step continue to the second step, which is much slower. Such a situation in which an intermediate (here NO 3 ) forms an equilibrium with reactants prior to the rate-determining step is described as a pre-equilibrium [ 6 ] For the reaction of NO 2 and CO, this hypothesis can be rejected, since it implies a rate equation that disagrees with experiment.
If the first step were at equilibrium, then its equilibrium constant expression permits calculation of the concentration of the intermediate NO 3 in terms of more stable (and more easily measured) reactant and product species:
The overall reaction rate would then be
which disagrees with the experimental rate law given above, and so disproves the hypothesis that the second step is rate-determining for this reaction. However, some other reactions are believed to involve rapid pre-equilibria prior to the rate-determining step, as shown below .
Another example is the unimolecular nucleophilic substitution (S N 1) reaction in organic chemistry, where it is the first, rate-determining step that is unimolecular . A specific case is the basic hydrolysis of tert-butyl bromide ( t-C 4 H 9 Br ) by aqueous sodium hydroxide . The mechanism has two steps (where R denotes the tert-butyl radical t-C 4 H 9 ):
This reaction is found to be first-order with r = k [R−Br], which indicates that the first step is slow and determines the rate. The second step with OH − is much faster, so the overall rate is independent of the concentration of OH − .
In contrast, the alkaline hydrolysis of methyl bromide ( CH 3 Br ) is a bimolecular nucleophilic substitution (S N 2) reaction in a single bimolecular step. Its rate law is second-order : r = k [R−Br][ OH − ].
A useful rule in the determination of mechanism is that the concentration factors in the rate law indicate the composition and charge of the activated complex or transition state . [ 7 ] For the NO 2 –CO reaction above, the rate depends on [ NO 2 ] 2 , so that the activated complex has composition N 2 O 4 , with 2 NO 2 entering the reaction before the transition state, and CO reacting after the transition state.
A multistep example is the reaction between oxalic acid and chlorine in aqueous solution: H 2 C 2 O 4 + Cl 2 → 2 CO 2 + 2 H + + 2 Cl − . [ 7 ] The observed rate law is
which implies an activated complex in which the reactants lose 2 H + + Cl − before the rate-determining step. The formula of the activated complex is Cl 2 + H 2 C 2 O 4 − 2 H + − Cl − + x H 2 O , or C 2 O 4 Cl(H 2 O) – x (an unknown number of water molecules are added because the possible dependence of the reaction rate on H 2 O was not studied, since the data were obtained in water solvent at a large and essentially unvarying concentration).
One possible mechanism in which the preliminary steps are assumed to be rapid pre-equilibria occurring prior to the transition state is [ 7 ]
In a multistep reaction, the rate-determining step does not necessarily correspond to the highest Gibbs energy on the reaction coordinate diagram. [ 8 ] [ 6 ] If there is a reaction intermediate whose energy is lower than the initial reactants, then the activation energy needed to pass through any subsequent transition state depends on the Gibbs energy of that state relative to the lower-energy intermediate. The rate-determining step is then the step with the largest Gibbs energy difference relative either to the starting material or to any previous intermediate on the diagram. [ 8 ] [ 9 ]
Also, for reaction steps that are not first-order, concentration terms must be considered in choosing the rate-determining step. [ 8 ] [ 6 ]
Not all reactions have a single rate-determining step. In particular, the rate of a chain reaction is usually not controlled by any single step. [ 8 ]
In the previous examples the rate determining step was one of the sequential chemical reactions leading to a product. The rate-determining step can also be the transport of reactants to where they can interact and form the product. This case is referred to as diffusion control and, in general, occurs when the formation of product from the activated complex is very rapid and thus the provision of the supply of reactants is rate-determining. | https://en.wikipedia.org/wiki/Rate-determining_step |
In biochemistry , a rate-limiting step is a reaction step that controls the rate of a series of biochemical reactions. [ 1 ] [ 2 ] The statement is, however, a misunderstanding of how a sequence of enzyme - catalyzed reaction steps operate. Rather than a single step controlling the rate, it has been discovered that multiple steps control the rate. Moreover, each controlling step controls the rate to varying degrees.
Blackman (1905) [ 3 ] stated as an axiom: "when a process is conditioned as to its rapidity by a number of separate factors, the rate of the process is limited by the pace of the slowest factor." This implies that it should be possible, by studying the behavior of a complicated system such as a metabolic pathway , to characterize a single factor or reaction (namely the slowest), which plays the role of a master or rate-limiting step. In other words, the study of flux control can be simplified to the study of a single enzyme since, by definition, there can only be one 'rate-limiting' step. Since its conception, the 'rate-limiting' step has played a significant role in suggesting how metabolic pathways are controlled. Unfortunately, the notion of a 'rate-limiting' step is erroneous, at least under steady-state conditions. Modern biochemistry textbooks have begun to play down the concept. For example, the seventh edition of Lehninger Principles of Biochemistry [ 4 ] explicitly states: "It has now become clear that, in most pathways, the control of flux is distributed among several enzymes, and the extent to which each contributes to the control varies with metabolic circumstances". However, the concept is still incorrectly used in research articles. [ 5 ] [ 6 ]
From the 1920s to the 1950s, there were a number of authors who discussed the concept of rate-limiting steps, also known as master reactions. Several authors have stated that the concept of the 'rate-limiting' step is incorrect. Burton (1936) [ 7 ] was one of the first to point out that: "In the steady state of reaction chains, the principle of the master reaction has no application". Hearon (1952) [ 8 ] made a more general mathematical analysis and developed strict rules for the prediction of mastery in a linear sequence of enzyme-catalysed reactions. Webb (1963) [ 9 ] was highly critical of the concept of the rate-limiting step and of its blind application to solving problems of regulation in metabolism. Waley (1964) [ 10 ] made a simple but illuminating analysis of simple linear chains. He showed that provided the intermediate concentrations were low compared to the K m {\displaystyle K_{\mathrm {m} }} values of the enzymes, the following expression was valid:
1 F = 1 Q ( R e 1 + … X e i + … + Z e n ) {\displaystyle {\frac {1}{F}}={\frac {1}{Q}}\left({\frac {R}{e_{1}}}+\ldots {\frac {X}{e_{i}}}+\ldots +{\frac {Z}{e_{n}}}\right)}
where F {\displaystyle F} equals the pathway flux, and Q , R , … , X , … {\displaystyle Q,R,\ldots ,X,\ldots } and Z {\displaystyle Z} are functions of the rate constants and intermediate metabolite concentrations. The e i {\displaystyle e_{i}} terms are proportional to the limiting rate V {\displaystyle V} values of the enzymes. The first point to note from the above equation is that the pathway flux is a function of all the enzymes; there is no need for there to be a 'rate-limiting' step. If, however, all the terms X / e i {\displaystyle X/e_{i}} from S / e 2 {\displaystyle S/e_{2}} to Z / e n {\displaystyle Z/e_{n}} , are small relative to R / e 1 {\displaystyle R/e_{1}} then the first enzyme will contribute the most to determining the flux and therefore, could be termed the 'rate-limiting' step.
The modern perspective is that rate-limitingness should be quantitative and that it is distributed through a pathway to varying degrees. This idea was first considered by Higgins [ 11 ] in the late 1950s as part of his PhD thesis [ 12 ] where he introduced the quantitative measure he called the ‘reflection coefficient.’ This described the relative change of one variable to another for small perturbations. In his Ph.D. thesis, Higgins describes many properties of the reflection coefficients, and in later work, three groups, Savageau, [ 13 ] [ 14 ] Heinrich and Rapoport [ 15 ] [ 16 ] and Jim Burns in his thesis (1971) and subsequent publications [ 17 ] [ 18 ] independently and simultaneously developed this work into what is now called metabolic control analysis or, in the specific form developed by Savageau, biochemical systems theory . These developments extended Higgins’ original ideas significantly, and the formalism is now the primary theoretical approach to describing deterministic, continuous models of biochemical networks.
The variations in terminology between the different papers on metabolic control analysis [ 15 ] [ 17 ] were later harmonized by general agreement. [ 19 ] | https://en.wikipedia.org/wiki/Rate-limiting_step_(biochemistry) |
The rate of living theory postulates that the faster an organism's metabolism , the shorter its lifespan . First proposed by Max Rubner in 1908, the theory was based on his observation that smaller animals had faster metabolisms and shorter lifespans compared to larger animals with slower metabolisms. [ 1 ] The theory gained further credibility through the work of Raymond Pearl , who conducted experiments on drosophila and cantaloupe seeds, which supported Rubner's initial observation. Pearl's findings were later published in his book, The Rate of Living , in 1928, in which he expounded upon Rubner's theory and demonstrated a causal relationship between the slowing of metabolism and an increase in lifespan. [ 2 ]
The theory gained additional credibility with the discovery of Max Kleiber's law in 1932. Kleiber found that an organism's basal metabolic rate could be predicted by taking 3/4 the power of the organism's body weight. This finding was noteworthy because the inversion of the scaling exponent, between 0.2 and 0.33, also demonstrated the scaling for both lifespan and metabolic rate, and was colloquially called the "mouse-to-elephant" curve. [ 3 ]
Mechanistic evidence was provided by Denham Harman 's free radical theory of aging , created in the 1950s. This theory stated that organisms age over time due to the accumulation of damage from free radicals in the body. [ 4 ] It also showed that metabolic processes, specifically the mitochondria , are prominent producers of free radicals. [ 4 ] This provided a mechanistic link between Rubner's initial observations of decreased lifespan in conjunction with increased metabolism. [ citation needed ]
Support for this theory has been bolstered by studies linking a lower basal metabolic rate (evident with a lowered heartbeat) to increased life expectancy. [ 5 ] [ 6 ] [ 7 ] This has been proposed by some to be the key to why animals like the giant tortoise can live over 150 years. [ 8 ]
However, the ratio of resting metabolic rate to total daily energy expenditure can vary between 1.6 and 8.0 between species of mammals . Animals also vary in the degree of coupling between oxidative phosphorylation and ATP production , the amount of saturated fat in mitochondrial membranes , the amount of DNA repair , and many other factors that affect maximum life span. [ 9 ] Furthermore, a number of species with high metabolic rate, like bats and birds, are long-lived. [ 10 ] [ 11 ] In a 2007 analysis it was shown that, when modern statistical methods for correcting for the effects of body size and phylogeny are employed, metabolic rate does not correlate with longevity in mammals or birds. [ 12 ] | https://en.wikipedia.org/wiki/Rate-of-living_theory |
Rate-zonal centrifugation is a centrifugation technique employed to effectively separate particles of different sizes. [ 1 ] The tube is first filled with different concentrations of sucrose or another solute establishing layers with different densities and viscosities , forming a density gradient , within which the particles to be separated are added. The larger particles will be able to travel to the bottom layer because they are more massive. The greater mass allows the particles to travel through layers with a greater viscosity, while the smaller particles will remain at the top, as they lack the mass to travel through the more viscous layers. Once the centrifugation is over, fractions are collected.
This microbiology -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rate-zonal_centrifugation |
In mathematics , a rate is the quotient of two quantities , often represented as a fraction . [ 1 ] If the divisor (or fraction denominator) in the rate is equal to one expressed as a single unit, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable ), then the dividend (the fraction numerator) of the rate expresses the corresponding rate of change in the other ( dependent ) variable. In some cases, it may be regarded as a change to a value, which is caused by a change of a value in respect to another value. For example, acceleration is a change in velocity with respect to time
Temporal rate is a common type of rate ("per unit of time"), such as speed , heart rate , and flux . [ 2 ] In fact, often rate is a synonym of rhythm or frequency , a count per second (i.e., hertz ); e.g., radio frequencies or sample rates .
In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate; for example, a heart rate is expressed as "beats per minute".
Rates that have a non-time divisor or denominator include exchange rates , literacy rates , and electric field (in volts per meter).
A rate defined using two numbers of the same units will result in a dimensionless quantity , also known as ratio or simply as a rate (such as tax rates ) or counts (such as literacy rate ). Dimensionless rates can be expressed as a percentage (for example, the global literacy rate in 1998 was 80%), fraction , or multiple .
Rates and ratios often vary with time, location, particular element (or subset) of a set of objects, etc. Thus they are often mathematical functions .
A rate (or ratio) may often be thought of as an output-input ratio, benefit-cost ratio , all considered in the broad sense. For example, miles per hour in transportation is the output (or benefit) in terms of miles of travel, which one gets from spending an hour (a cost in time) of traveling (at this velocity).
A set of sequential indices may be used to enumerate elements (or subsets) of a set of ratios under study. For example, in finance, one could define I by assigning consecutive integers to companies, to political subdivisions (such as states), to different investments, etc. The reason for using indices I is so a set of ratios (i=0, N) can be used in an equation to calculate a function of the rates such as an average of a set of ratios. For example, the average velocity found from the set of v I 's mentioned above. Finding averages may involve using weighted averages and possibly using the harmonic mean .
A ratio r=a/b has both a numerator "a" and a denominator "b". The value of a and b may be a real number or integer . The inverse of a ratio r is 1/r = b/a. A rate may be equivalently expressed as an inverse of its value if the ratio of its units is also inverse. For example, 5 miles (mi) per kilowatt-hour (kWh) corresponds to 1/5 kWh/mi (or 200 Wh /mi).
Rates are relevant to many aspects of everyday life. For example: How fast are you driving? The speed of the car (often expressed in miles per hour) is a rate. What interest does your savings account pay you? The amount of interest paid per year is a rate.
Consider the case where the numerator f {\displaystyle f} of a rate is a function f ( a ) {\displaystyle f(a)} where a {\displaystyle a} happens to be the denominator of the rate δ f / δ a {\displaystyle \delta f/\delta a} . A rate of change of f {\displaystyle f} with respect to a {\displaystyle a} (where a {\displaystyle a} is incremented by h {\displaystyle h} ) can be formally defined in two ways: [ 3 ]
where f ( x ) is the function with respect to x over the interval from a to a + h . An instantaneous rate of change is equivalent to a derivative .
For example, the average speed of a car can be calculated using the total distance traveled between two points, divided by the travel time. In contrast, the instantaneous velocity can be determined by viewing a speedometer .
In chemistry and physics:
In computing:
Miscellaneous definitions: | https://en.wikipedia.org/wiki/Rate_(mathematics) |
In computer networking , Rate Based Satellite Control Protocol (RBSCP) is a tunneling method proposed by Cisco to improve the performance of satellite network links with high latency and error rates.
The problem RBSCP addresses is that the long RTT on the link keeps TCP virtual circuits in slow start for a long time. This, in addition to the high loss give a very low amount of bandwidth on the channel. Since satellite links may be high-throughput, the overall link utilized may be below what is optimal from a technical and economic view.
RBSCP works by tunneling the usual IP packets within IP packets. The transport protocol
identifier is 199. On each end of the tunnel, routers buffer packets to utilize the link
better. In addition to this, RBSCP tunnel routers:
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rate_Based_Satellite_Control_Protocol |
Rate analysis for construction works is the process of accessing rates for unit of work or supply. It breaks down the construction activity in its basic components such as labor, overheads, taxes, contractor profit and basic rate of individual material. [ 1 ] [ 2 ]
The aim to is determine project costs, preparation of estimates for the necessary invitation of bids. The process can be used by governmental and non governmental agencies for accessing the project costs in advance and compare the availability of budgets requiring for completion of projects. [ 3 ] The rate of analysis for the particular item in construction can vary over the years depending upon the inflation in basic rate of the item or the tax slabs in the particular country and other factors which form its core. It ensures project is transparent and accurately budgeted. [ 4 ] [ 5 ]
The factors that affect rate analysis are labor costs, material costs, tax slabs in countries, contractors profit, location of site/carriages, overhead costs like electricity charges and water charges. The material costs are verified from the local market or the maximum retail prices allowed by the government to sell the item. The labour costs for the unit item is taken as some percentage of basic material cost, wastages are added ranging from 1-5% depending upon nature of item, water charges and electricity charges are usually 1% each, all these quantities are added and suitable sales and income taxes are added to them, atlast contractors profit usually ranging from 10-20% is added and labour taxes are also incurred. The unit rate of item thus formed can be used for the preparation of bill of quantities. The necessary costing and estimation of projects can be accurately calculated by knowing the unit rates of items to be used in the project. Rate analysis thus have the impact on choices of materials to be used in the particular project by keeping in view the cost considerations of the project. [ 6 ] [ 7 ] | https://en.wikipedia.org/wiki/Rate_analysis |
In chemistry , the rate equation (also known as the rate law or empirical differential rate equation ) is an empirical differential mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and constant parameters (normally rate coefficients and partial orders of reaction) only. [ 1 ] For many reactions, the initial rate is given by a power law such as
where [ A ] {\displaystyle [\mathrm {A} ]} and [ B ] {\displaystyle [\mathrm {B} ]} are the molar concentrations of the species A {\displaystyle \mathrm {A} } and B , {\displaystyle \mathrm {B} ,} usually in moles per liter ( molarity , M {\displaystyle M} ). The exponents x {\displaystyle x} and y {\displaystyle y} are the partial orders of reaction for A {\displaystyle \mathrm {A} } and B {\displaystyle \mathrm {B} } , respectively, and the overall reaction order is the sum of the exponents. These are often positive integers, but they may also be zero, fractional, or negative. The order of reaction is a number which quantifies the degree to which the rate of a chemical reaction depends on concentrations of the reactants. [ 2 ] In other words, the order of reaction is the exponent to which the concentration of a particular reactant is raised. [ 2 ] The constant k {\displaystyle k} is the reaction rate constant or rate coefficient and at very few places velocity constant or specific rate of reaction . Its value may depend on conditions such as temperature, ionic strength, surface area of an adsorbent , or light irradiation . If the reaction goes to completion, the rate equation for the reaction rate v = k [ A ] x [ B ] y {\displaystyle v\;=\;k[{\ce {A}}]^{x}[{\ce {B}}]^{y}} applies throughout the course of the reaction.
Elementary (single-step) reactions and reaction steps have reaction orders equal to the stoichiometric coefficients for each reactant. The overall reaction order, i.e. the sum of stoichiometric coefficients of reactants, is always equal to the molecularity of the elementary reaction. However, complex (multi-step) reactions may or may not have reaction orders equal to their stoichiometric coefficients. This implies that the order and the rate equation of a given reaction cannot be reliably deduced from the stoichiometry and must be determined experimentally, since an unknown reaction mechanism could be either elementary or complex. When the experimental rate equation has been determined, it is often of use for deduction of the reaction mechanism .
The rate equation of a reaction with an assumed multi-step mechanism can often be derived theoretically using quasi-steady state assumptions from the underlying elementary reactions, and compared with the experimental rate equation as a test of the assumed mechanism. The equation may involve a fractional order , and may depend on the concentration of an intermediate species.
A reaction can also have an undefined reaction order with respect to a reactant if the rate is not simply proportional to some power of the concentration of that reactant; for example, one cannot talk about reaction order in the rate equation for a bimolecular reaction between adsorbed molecules :
Consider a typical chemical reaction in which two reactants A and B combine to form a product C:
This can also be written
The prefactors −1, −2 and 3 (with negative signs for reactants because they are consumed) are known as stoichiometric coefficients . One molecule of A combines with two of B to form 3 of C, so if we use the symbol [X] for the molar concentration of chemical X, [ 3 ]
If the reaction takes place in a closed system at constant temperature and volume, without a build-up of reaction intermediates , the reaction rate v {\displaystyle v} is defined as
where ν i is the stoichiometric coefficient for chemical X i , with a negative sign for a reactant. [ 4 ]
The initial reaction rate v 0 = v t = 0 {\displaystyle v_{0}=v_{t=0}} has some functional dependence on the concentrations of the reactants,
and this dependence is known as the rate equation or rate law . [ 5 ] This law generally cannot be deduced from the chemical equation and must be determined by experiment. [ 6 ]
A common form for the rate equation is a power law: [ 6 ]
The constant k {\displaystyle k} is called the rate constant . The exponents, which can be fractional, [ 6 ] are called partial orders of reaction and their sum is the overall order of reaction. [ 7 ]
In a dilute solution, an elementary reaction (one having a single step with a single transition state ) is empirically found to obey the law of mass action . This predicts that the rate depends only on the concentrations of the reactants, raised to the powers of their stoichiometric coefficients. [ 8 ]
The differential rate equation for an elementary reaction using mathematical product notation is:
Where:
The natural logarithm of the power-law rate equation is
This can be used to estimate the order of reaction of each reactant. For example, the initial rate can be measured in a series of experiments at different initial concentrations of reactant A {\displaystyle {\rm {A}}} with all other concentrations [ B ] , [ C ] , … {\displaystyle [{\rm {B],[{\rm {C],\dots }}}}} kept constant, so that
The slope of a graph of ln v {\displaystyle \ln v} as a function of ln [ A ] {\displaystyle \ln[{\ce {A}}]} then corresponds to the order x {\displaystyle x} with respect to reactant A {\displaystyle {\rm {A}}} . [ 9 ] [ 10 ]
However, this method is not always reliable because
The tentative rate equation determined by the method of initial rates is therefore normally verified by comparing the concentrations measured over a longer time (several half-lives) with the integrated form of the rate equation; this assumes that the reaction goes to completion.
For example, the integrated rate law for a first-order reaction is
where [ A ] {\displaystyle [{\rm {A]}}} is the concentration at time t {\displaystyle t} and [ A ] 0 {\displaystyle [{\rm {A]_{0}}}} is the initial concentration at zero time. The first-order rate law is confirmed if ln [ A ] {\displaystyle \ln {[{\ce {A}}]}} is in fact a linear function of time. In this case the rate constant k {\displaystyle k} is equal to the slope with sign reversed. [ 11 ] [ 12 ]
The partial order with respect to a given reactant can be evaluated by the method of flooding (or of isolation) of Ostwald . In this method, the concentration of one reactant is measured with all other reactants in large excess so that their concentration remains essentially constant. For a reaction a ·A + b ·B → c ·C with rate law v 0 = k ⋅ [ A ] x ⋅ [ B ] y , {\displaystyle v_{0}=k\cdot [{\rm {A}}]^{x}\cdot [{\rm {B}}]^{y},} the partial order x {\displaystyle x} with respect to A {\displaystyle {\rm {A}}} is determined using a large excess of B {\displaystyle {\rm {B}}} . In this case
v 0 = k ′ ⋅ [ A ] x {\displaystyle v_{0}=k'\cdot [{\rm {A}}]^{x}} with k ′ = k ⋅ [ B ] y , {\displaystyle k'=k\cdot [{\rm {B}}]^{y},}
and x {\displaystyle x} may be determined by the integral method. The order y {\displaystyle y} with respect to B {\displaystyle {\rm {B}}} under the same conditions (with B {\displaystyle {\rm {B}}} in excess) is determined by a series of similar experiments with a range of initial concentration [ B ] 0 {\displaystyle [{\rm {B]_{0}}}} so that the variation of k ′ {\displaystyle k'} can be measured. [ 13 ]
For zero-order reactions, the reaction rate is independent of the concentration of a reactant, so that changing its concentration has no effect on the rate of the reaction. Thus, the concentration changes linearly with time. The rate law for zero order reaction is
− d [ A ] d t = k [ A ] 0 = k , {\displaystyle -{d[A] \over dt}=k[A]^{0}=k,}
The unit of k is mol dm −3 s −1 . [ 14 ] This may occur when there is a bottleneck which limits the number of reactant molecules that can react at the same time, for example if the reaction requires contact with an enzyme or a catalytic surface. [ 15 ]
Many enzyme-catalyzed reactions are zero order, provided that the reactant concentration is much greater than the enzyme concentration which controls the rate, so that the enzyme is saturated . For example, the biological oxidation of ethanol to acetaldehyde by the enzyme liver alcohol dehydrogenase (LADH) is zero order in ethanol. [ 16 ]
Similarly, reactions with heterogeneous catalysis can be zero order if the catalytic surface is saturated. For example, the decomposition of phosphine ( PH 3 ) on a hot tungsten surface at high pressure is zero order in phosphine, which decomposes at a constant rate. [ 15 ]
In homogeneous catalysis zero order behavior can come about from reversible inhibition. For example, ring-opening metathesis polymerization using third-generation Grubbs catalyst exhibits zero order behavior in catalyst due to the reversible inhibition that occurs between pyridine and the ruthenium center. [ 17 ]
A first order reaction depends on the concentration of only one reactant (a unimolecular reaction ). Other reactants can be present, but their concentration has no effect on the rate. The rate law for a first order reaction is
The unit of k is s −1 . [ 14 ] Although not affecting the above math, the majority of first order reactions proceed via intermolecular collisions. Such collisions, which contribute the energy to the reactant, are necessarily second order. However according to the Lindemann mechanism the reaction consists of two steps: the bimolecular collision which is second order and the reaction of the energized molecule which is unimolecular and first order. The rate of the overall reaction depends on the slowest step, so the overall reaction will be first order when the reaction of the energized reactant is slower than the collision step.
The half-life is independent of the starting concentration and is given by t 1 / 2 = ln ( 2 ) k {\textstyle t_{1/2}={\frac {\ln {(2)}}{k}}} . The mean lifetime is τ = 1/ k . [ 18 ]
Examples of such reactions are:
In organic chemistry, the class of S N 1 (nucleophilic substitution unimolecular) reactions consists of first-order reactions. For example, in the reaction of aryldiazonium ions with nucleophiles in aqueous solution, ArN + 2 + X − → ArX + N 2 , the rate equation is v 0 = k [ ArN 2 + ] , {\displaystyle v_{0}=k[{\ce {ArN2+}}],} where Ar indicates an aryl group. [ 22 ]
A reaction is said to be second order when the overall order is two. The rate of a second-order reaction may be proportional to one concentration squared, v 0 = k [ A ] 2 , {\displaystyle v_{0}=k[{\ce {A}}]^{2},} or (more commonly) to the product of two concentrations, v 0 = k [ A ] [ B ] . {\displaystyle v_{0}=k[{\ce {A}}][{\ce {B}}].} As an example of the first type, the reaction NO 2 + CO → NO + CO 2 is second-order in the reactant NO 2 and zero order in the reactant CO. The observed rate is given by v 0 = k [ NO 2 ] 2 , {\displaystyle v_{0}=k[{\ce {NO2}}]^{2},} and is independent of the concentration of CO. [ 23 ]
For the rate proportional to a single concentration squared, the time dependence of the concentration is given by
The unit of k is mol −1 dm 3 s −1 . [ 14 ]
The time dependence for a rate proportional to two unequal concentrations is
if the concentrations are equal, they satisfy the previous equation.
The second type includes nucleophilic addition-elimination reactions , such as the alkaline hydrolysis of ethyl acetate : [ 22 ]
This reaction is first-order in each reactant and second-order overall:
If the same hydrolysis reaction is catalyzed by imidazole , the rate equation becomes [ 22 ]
The rate is first-order in one reactant (ethyl acetate), and also first-order in imidazole, which as a catalyst does not appear in the overall chemical equation.
Another well-known class of second-order reactions are the S N 2 (bimolecular nucleophilic substitution) reactions, such as the reaction of n-butyl bromide with sodium iodide in acetone :
This same compound can be made to undergo a bimolecular (E2) elimination reaction , another common type of second-order reaction, if the sodium iodide and acetone are replaced with sodium tert-butoxide as the salt and tert-butanol as the solvent:
If the concentration of a reactant remains constant (because it is a catalyst , or because it is in great excess with respect to the other reactants), its concentration can be included in the rate constant, leading to a pseudo–first-order (or occasionally pseudo–second-order) rate equation. For a typical second-order reaction with rate equation v 0 = k [ A ] [ B ] , {\displaystyle v_{0}=k[{\ce {A}}][{\ce {B}}],} if the concentration of reactant B is constant then v 0 = k [ A ] [ B ] = k ′ [ A ] , {\displaystyle v_{0}=k[{\ce {A}}][{\ce {B}}]=k'[{\ce {A}}],} where the pseudo–first-order rate constant k ′ = k [ B ] . {\displaystyle k'=k[{\ce {B}}].} The second-order rate equation has been reduced to a pseudo–first-order rate equation, which makes the treatment to obtain an integrated rate equation much easier.
One way to obtain a pseudo-first order reaction is to use a large excess of one reactant (say, [B]≫[A]) so that, as the reaction progresses, only a small fraction of the reactant in excess (B) is consumed, and its concentration can be considered to stay constant. For example, the hydrolysis of esters by dilute mineral acids follows pseudo- first order kinetics , where the concentration of water is constant because it is present in large excess:
The hydrolysis of sucrose ( C 12 H 22 O 11 ) in acid solution is often cited as a first-order reaction with rate v 0 = k [ C 12 H 22 O 11 ] . {\displaystyle v_{0}=k[{\ce {C12H22O11}}].} The true rate equation is third-order, v 0 = k [ C 12 H 22 O 11 ] [ H + ] [ H 2 O ] ; {\displaystyle v_{0}=k[{\ce {C12H22O11}}][{\ce {H+}}][{\ce {H2O}}];} however, the concentrations of both the catalyst H + and the solvent H 2 O are normally constant, so that the reaction is pseudo–first-order. [ 24 ]
Elementary reaction steps with order 3 (called ternary reactions ) are rare and unlikely to occur. However, overall reactions composed of several elementary steps can, of course, be of any (including non-integer) order.
[Except first order]
[Except first order]
[Limit is necessary for first order]
Here M {\displaystyle {\rm {M}}} stands for concentration in molarity (mol · L −1 ), t {\displaystyle t} for time, and k {\displaystyle k} for the reaction rate constant. The half-life of a first-order reaction is often expressed as t 1/2 = 0.693/ k (as ln(2)≈0.693).
In fractional order reactions, the order is a non-integer, which often indicates a chemical chain reaction or other complex reaction mechanism . For example, the pyrolysis of acetaldehyde ( CH 3 CHO ) into methane and carbon monoxide proceeds with an order of 1.5 with respect to acetaldehyde: v 0 = k [ CH 3 CHO ] 3 / 2 . {\displaystyle v_{0}=k[{\ce {CH3CHO}}]^{3/2}.} [ 26 ] The decomposition of phosgene ( COCl 2 ) to carbon monoxide and chlorine has order 1 with respect to phosgene itself and order 0.5 with respect to chlorine: v 0 = k [ COCl 2 ] [ Cl 2 ] 1 / 2 . {\displaystyle v_{0}=k{\ce {[COCl2] [Cl2]}}^{1/2}.} [ 27 ]
The order of a chain reaction can be rationalized using the steady state approximation for the concentration of reactive intermediates such as free radicals . For the pyrolysis of acetaldehyde, the Rice- Herzfeld mechanism is
where • denotes a free radical. [ 26 ] [ 28 ] To simplify the theory, the reactions of the *CHO to form a second *CH 3 are ignored.
In the steady state, the rates of formation and destruction of methyl radicals are equal, so that
so that the concentration of methyl radical satisfies
The reaction rate equals the rate of the propagation steps which form the main reaction products CH 4 and CO:
in agreement with the experimental order of 3/2. [ 26 ] [ 28 ]
More complex rate laws have been described as being mixed order if they approximate to the laws for more than one order at different concentrations of the chemical species involved. For example, a rate law of the form v 0 = k 1 [ A ] + k 2 [ A ] 2 {\displaystyle v_{0}=k_{1}[A]+k_{2}[A]^{2}} represents concurrent first order and second order reactions (or more often concurrent pseudo-first order and second order) reactions, and can be described as mixed first and second order. [ 29 ] For sufficiently large values of [A] such a reaction will approximate second order kinetics, but for smaller [A] the kinetics will approximate first order (or pseudo-first order). As the reaction progresses, the reaction can change from second order to first order as reactant is consumed.
Another type of mixed-order rate law has a denominator of two or more terms, often because the identity of the rate-determining step depends on the values of the concentrations. An example is the oxidation of an alcohol to a ketone by hexacyanoferrate (III) ion [Fe(CN) 6 3− ] with ruthenate (VI) ion (RuO 4 2− ) as catalyst . [ 30 ] For this reaction, the rate of disappearance of hexacyanoferrate (III) is v 0 = [ Fe ( CN ) 6 ] 2 − k α + k β [ Fe ( CN ) 6 ] 2 − {\displaystyle v_{0}={\frac {{\ce {[Fe(CN)6]^2-}}}{k_{\alpha }+k_{\beta }{\ce {[Fe(CN)6]^2-}}}}}
This is zero-order with respect to hexacyanoferrate (III) at the onset of the reaction (when its concentration is high and the ruthenium catalyst is quickly regenerated), but changes to first-order when its concentration decreases and the regeneration of catalyst becomes rate-determining.
Notable mechanisms with mixed-order rate laws with two-term denominators include:
A reaction rate can have a negative partial order with respect to a substance. For example, the conversion of ozone (O 3 ) to oxygen follows the rate equation v 0 = k [ O 3 ] 2 [ O 2 ] − 1 {\displaystyle v_{0}=k{\ce {[O_3]^2}}{\ce {[O_2]^{-1}}}} in an excess of oxygen. This corresponds to second order in ozone and order (−1) with respect to oxygen. [ 31 ]
When a partial order is negative, the overall order is usually considered as undefined. In the above example, for instance, the reaction is not described as first order even though the sum of the partial orders is 2 + ( − 1 ) = 1 {\displaystyle 2+(-1)=1} , because the rate equation is more complex than that of a simple first-order reaction.
A pair of forward and reverse reactions may occur simultaneously with comparable speeds. For example, A and B react into products P and Q and vice versa ( a, b, p , and q are the stoichiometric coefficients ):
The reaction rate expression for the above reactions (assuming each one is elementary) can be written as:
where: k 1 is the rate coefficient for the reaction that consumes A and B; k −1 is the rate coefficient for the backwards reaction, which consumes P and Q and produces A and B.
The constants k 1 and k −1 are related to the equilibrium coefficient for the reaction (K) by the following relationship (set v =0 in balance):
In a simple equilibrium between two species:
where the reaction starts with an initial concentration of reactant A, [ A ] 0 {\displaystyle {\ce {[A]0}}} , and an initial concentration of 0 for product P at time t =0.
Then the equilibrium constant K is expressed as:
where [ A ] e {\displaystyle [{\ce {A}}]_{e}} and [ P ] e {\displaystyle [{\ce {P}}]_{e}} are the concentrations of A and P at equilibrium, respectively.
The concentration of A at time t , [ A ] t {\displaystyle [{\ce {A}}]_{t}} , is related to the concentration of P at time t , [ P ] t {\displaystyle [{\ce {P}}]_{t}} , by the equilibrium reaction equation:
The term [ P ] 0 {\displaystyle {\ce {[P]0}}} is not present because, in this simple example, the initial concentration of P is 0.
This applies even when time t is at infinity; i.e., equilibrium has been reached:
then it follows, by the definition of K , that
and, therefore,
These equations allow us to uncouple the system of differential equations , and allow us to solve for the concentration of A alone.
The reaction equation was given previously as:
For A ↽ − − ⇀ P {\displaystyle {\ce {A <=> P}}} this is simply
The derivative is negative because this is the rate of the reaction going from A to P, and therefore the concentration of A is decreasing. To simplify notation, let x be [ A ] t {\displaystyle [{\ce {A}}]_{t}} , the concentration of A at time t . Let x e {\displaystyle x_{e}} be the concentration of A at equilibrium. Then:
Since:
the reaction rate becomes:
which results in:
A plot of the negative natural logarithm of the concentration of A in time minus the concentration at equilibrium versus time t gives a straight line with slope k 1 + k −1 . By measurement of [A] e and [P] e the values of K and the two reaction rate constants will be known. [ 32 ]
If the concentration at the time t = 0 is different from above, the simplifications above are invalid, and a system of differential equations must be solved. However, this system can also be solved exactly to yield the following generalized expressions:
When the equilibrium constant is close to unity and the reaction rates very fast for instance in conformational analysis of molecules, other methods are required for the determination of rate constants for instance by complete lineshape analysis in NMR spectroscopy .
If the rate constants for the following reaction are k 1 {\displaystyle k_{1}} and k 2 {\displaystyle k_{2}} ; A ⟶ B ⟶ C {\displaystyle {\ce {A -> B -> C}}} , then the rate equation is:
With the individual concentrations scaled by the total population of reactants to become probabilities, linear systems of differential equations such as these can be formulated as a master equation . The differential equations can be solved analytically and the integrated rate equations are
The steady state approximation leads to very similar results in an easier way.
When a substance reacts simultaneously to give two different products, a parallel or competitive reaction is said to take place.
A ⟶ B {\displaystyle {\ce {A -> B}}} and A ⟶ C {\displaystyle {\ce {A -> C}}} , with constants k 1 {\displaystyle k_{1}} and k 2 {\displaystyle k_{2}} and rate equations − d [ A ] d t = ( k 1 + k 2 ) [ A ] {\displaystyle -{\frac {d[{\ce {A}}]}{dt}}=(k_{1}+k_{2})[{\ce {A}}]} ; d [ B ] d t = k 1 [ A ] {\displaystyle {\frac {d[{\ce {B}}]}{dt}}=k_{1}[{\ce {A}}]} and d [ C ] d t = k 2 [ A ] {\displaystyle {\frac {d[{\ce {C}}]}{dt}}=k_{2}[{\ce {A}}]}
The integrated rate equations are then [ A ] = [ A ] 0 e − ( k 1 + k 2 ) t {\displaystyle [{\ce {A}}]={\ce {[A]0}}e^{-(k_{1}+k_{2})t}} ; [ B ] = k 1 k 1 + k 2 [ A ] 0 ( 1 − e − ( k 1 + k 2 ) t ) {\displaystyle [{\ce {B}}]={\frac {k_{1}}{k_{1}+k_{2}}}{\ce {[A]0}}\left(1-e^{-(k_{1}+k_{2})t}\right)} and [ C ] = k 2 k 1 + k 2 [ A ] 0 ( 1 − e − ( k 1 + k 2 ) t ) {\displaystyle [{\ce {C}}]={\frac {k_{2}}{k_{1}+k_{2}}}{\ce {[A]0}}\left(1-e^{-(k_{1}+k_{2})t}\right)} .
One important relationship in this case is [ B ] [ C ] = k 1 k 2 {\displaystyle {\frac {{\ce {[B]}}}{{\ce {[C]}}}}={\frac {k_{1}}{k_{2}}}}
This can be the case when studying a bimolecular reaction and a simultaneous hydrolysis (which can be treated as pseudo order one) takes place: the hydrolysis complicates the study of the reaction kinetics, because some reactant is being "spent" in a parallel reaction. For example, A reacts with R to give our product C, but meanwhile the hydrolysis reaction takes away an amount of A to give B, a byproduct: A + H 2 O ⟶ B {\displaystyle {\ce {A + H2O -> B}}} and A + R ⟶ C {\displaystyle {\ce {A + R -> C}}} . The rate equations are: d [ B ] d t = k 1 [ A ] [ H 2 O ] = k 1 ′ [ A ] {\displaystyle {\frac {d[{\ce {B}}]}{dt}}=k_{1}{\ce {[A][H2O]}}=k_{1}'[{\ce {A}}]} and d [ C ] d t = k 2 [ A ] [ R ] {\displaystyle {\frac {d[{\ce {C}}]}{dt}}=k_{2}{\ce {[A][R]}}} , where k 1 ′ {\displaystyle k_{1}'} is the pseudo first order constant. [ 33 ]
The integrated rate equation for the main product [C] is [ C ] = [ R ] 0 [ 1 − e − k 2 k 1 ′ [ A ] 0 ( 1 − e − k 1 ′ t ) ] {\displaystyle {\ce {[C]=[R]0}}\left[1-e^{-{\frac {k_{2}}{k_{1}'}}{\ce {[A]0}}\left(1-e^{-k_{1}'t}\right)}\right]} , which is equivalent to ln [ R ] 0 [ R ] 0 − [ C ] = k 2 [ A ] 0 k 1 ′ ( 1 − e − k 1 ′ t ) {\displaystyle \ln {\frac {{\ce {[R]0}}}{{\ce {[R]0-[C]}}}}={\frac {k_{2}{\ce {[A]0}}}{k_{1}'}}\left(1-e^{-k_{1}'t}\right)} . Concentration of B is related to that of C through [ B ] = − k 1 ′ k 2 ln ( 1 − [ C ] [ R ] 0 ) {\displaystyle [{\ce {B}}]=-{\frac {k_{1}'}{k_{2}}}\ln \left(1-{\frac {\ce {[C]}}{\ce {[R]0}}}\right)}
The integrated equations were analytically obtained but during the process it was assumed that [ A ] 0 − [ C ] ≈ [ A ] 0 {\displaystyle {\ce {[A]0}}-{\ce {[C]}}\approx {\ce {[A]0}}} . Therefore, previous equation for [C] can only be used for low concentrations of [C] compared to [A] 0
The most general description of a chemical reaction network considers a number N {\displaystyle N} of distinct chemical species reacting via R {\displaystyle R} reactions. [ 34 ] [ 35 ] The chemical equation of the j {\displaystyle j} -th reaction can then be written in the generic form
which is often written in the equivalent form
Here
The rate of such a reaction can be inferred by the law of mass action
which denotes the flux of molecules per unit time and unit volume. Here ( [ X ] ) = ( [ X 1 ] , [ X 2 ] , … , [ X N ] ) {\displaystyle {\ce {([\mathbf {X} ])=([X1],[X2],\ldots ,[X_{\mathit {N}}])}}} is the vector of concentrations. This definition includes the elementary reactions :
Each of these is discussed in detail below. One can define the stoichiometric matrix
denoting the net extent of molecules of i {\displaystyle i} in reaction j {\displaystyle j} . The reaction rate equations can then be written in the general form
This is the product of the stoichiometric matrix and the vector of reaction rate functions.
Particular simple solutions exist in equilibrium, d [ X i ] d t = 0 {\displaystyle {\frac {d[{\ce {X}}_{i}]}{dt}}=0} , for systems composed of merely reversible reactions. In this case, the rate of the forward and backward reactions are equal, a principle called detailed balance . Detailed balance is a property of the stoichiometric matrix N i j {\displaystyle N_{ij}} alone and does not depend on the particular form of the rate functions f j {\displaystyle f_{j}} . All other cases where detailed balance is violated are commonly studied by flux balance analysis , which has been developed to understand metabolic pathways . [ 36 ] [ 37 ]
For a general unimolecular reaction involving interconversion of N {\displaystyle N} different species, whose concentrations at time t {\displaystyle t} are denoted by X 1 ( t ) {\displaystyle X_{1}(t)} through X N ( t ) {\displaystyle X_{N}(t)} , an analytic form for the time-evolution of the species can be found. Let the rate constant of conversion from species X i {\displaystyle X_{i}} to species X j {\displaystyle X_{j}} be denoted as k i j {\displaystyle k_{ij}} , and construct a rate-constant matrix K {\displaystyle K} whose entries are the k i j {\displaystyle k_{ij}} .
Also, let X ( t ) = ( X 1 ( t ) , X 2 ( t ) , … , X N ( t ) ) T {\displaystyle X(t)=(X_{1}(t),X_{2}(t),\ldots ,X_{N}(t))^{T}} be the vector of concentrations as a function of time.
Let J = ( 1 , 1 , 1 , … , 1 ) T {\displaystyle J=(1,1,1,\ldots ,1)^{T}} be the vector of ones.
Let I {\displaystyle I} be the N × N {\displaystyle N\times N} identity matrix.
Let diag {\displaystyle \operatorname {diag} } be the function that takes a vector and constructs a diagonal matrix whose on-diagonal entries are those of the vector.
Let L − 1 {\displaystyle {\mathcal {L}}^{-1}} be the inverse Laplace transform from s {\displaystyle s} to t {\displaystyle t} .
Then the time-evolved state X ( t ) {\displaystyle X(t)} is given by
thus providing the relation between the initial conditions of the system and its state at time t {\displaystyle t} . | https://en.wikipedia.org/wiki/Rate_equation |
In computer networks , rate limiting is used to control the rate of requests sent or received by a network interface controller . It can be used to prevent DoS attacks [ 1 ] and limit web scraping . [ 2 ]
Research indicates flooding rates for one zombie machine are in excess of 20 HTTP GET requests per second, [ 3 ] legitimate rates much less.
Rate limiting should be used along with throttling pattern to minimize the number of throttling errors. [ 4 ]
Hardware appliances can limit the rate of requests on layer 4 or 5 of the OSI model .
Rate limiting can be induced by the network protocol stack of the sender due to a received ECN -marked packet and also by the network scheduler of any router along the way.
While a hardware appliance can limit the rate for a given range of IP-addresses on layer 4, it risks blocking a network with many users which are masked by NAT with a single IP address of an ISP .
Deep packet inspection can be used to filter on the session layer but will effectively disarm encryption protocols like TLS and SSL between the appliance and the protocol server (i.e. web server).
Protocol servers using a request / response model, such as FTP servers or typically Web servers may use a central in-memory key-value database , like Redis or Aerospike , for session management. A rate limiting algorithm is used to check if the user session (or IP address) has to be limited based on the information in the session cache.
In case a client made too many requests within a given time frame, HTTP servers can respond with status code 429: Too Many Requests .
However, in some cases (i.e. web servers) the session management and rate limiting algorithm should be built into the application (used for dynamic content) running on the web server, rather than the web server itself.
When a protocol server or a network device notice that the configured request limit is reached, then it will offload new requests and not respond to them. Sometimes they may be added to a queue to be processed once the input rate reaches an acceptable level, but at peak times the request rate can even exceed the capacities of such queues and requests have to be thrown away.
Data centers widely use rate limiting to control the share of resources given to different tenants and applications according to their service level agreement. [ 5 ] A variety of rate limiting techniques are applied in data centers using software and hardware. Virtualized data centers may also apply rate limiting at the hypervisor layer. Two important performance metrics of rate limiters in data centers are resource footprint (memory and CPU usage) which determines scalability, and precision. There usually exists a trade-off, that is, higher precision can be achieved by dedicating more resources to the rate limiters. A considerable body of research with focus on improving performance of rate limiting in data centers. [ 5 ] | https://en.wikipedia.org/wiki/Rate_limiting |
In aeronautics , the rate of climb ( RoC ) is an aircraft's vertical speed, that is the positive or negative rate of altitude change with respect to time. [ 1 ] In most ICAO member countries, even in otherwise metric countries, this is usually expressed in feet per minute (ft/min); elsewhere, it is commonly expressed in metres per second (m/s). The RoC in an aircraft is indicated with a vertical speed indicator (VSI) or instantaneous vertical speed indicator (IVSI).
The temporal rate of decrease in altitude is referred to as the rate of descent ( RoD ) or sink rate .
A negative rate of climb corresponds to a positive rate of descent: RoD = −RoC.
There are a number of designated airspeeds relating to optimum rates of ascent, the two most important of these are V X and V Y .
V X is the indicated forward airspeed for best angle of climb . This is the speed at which an aircraft gains the most altitude in a given horizontal distance , typically used to avoid a collision with an object a short distance away. By contrast, V Y is the indicated airspeed for best rate of climb, [ 2 ] a rate which allows the aircraft to climb to a specified altitude in the minimum amount of time regardless of the horizontal distance required. Except at the aircraft's ceiling, where they are equal, V X is always lower than V Y .
Climbing at V X allows pilots to maximize altitude gain per horizontal distance. This occurs at the speed for which the difference between thrust and drag is the greatest (maximum excess thrust ). In a jet airplane, this is approximately minimum drag speed, occurring at the bottom of the drag vs. speed curve.
Climbing at V Y allows pilots to maximize altitude gain per time. This occurs at the speed where the difference between engine power and the power required to overcome the aircraft's drag is greatest (maximum excess power). [ 3 ]
V x increases with altitude and V Y decreases with altitude until they converge at the airplane's absolute ceiling , the altitude above which the airplane cannot climb in steady flight.
The Cessna 172 is a four-seat aircraft. At maximum weight it has a V Y of 75 kn (139 km/h) indicated airspeed [ 4 ] providing a rate of climb of 721 ft/min (3.66 m/s).
Rate of climb at maximum power for a small aircraft is typically specified in its normal operating procedures but for large jet airliners it is usually mentioned in emergency operating procedures.
In aviation , a variometer – also known as a rate of climb and descent indicator (RCDI), rate-of-climb indicator, vertical speed indicator (VSI), or vertical velocity indicator (VVI) – is one of the flight instruments in an aircraft used to inform the pilot of the rate of descent or climb . [ 5 ] It can be calibrated in metres per second , feet per minute (1 ft/min = 0.00508 m/s) or knots (1 kn ≈ 0.514 m/s), depending on country and type of aircraft. It is typically connected to the aircraft's external static pressure source.
In powered flight , the pilot makes frequent use of the VSI to ascertain that level flight is being maintained, especially during turning maneuvers. In gliding , the instrument is used almost continuously during normal flight, often with an audible output, to inform the pilot of rising or sinking air . It is usual for gliders to be equipped with more than one type of variometer. The simpler type does not need an external source of power and can therefore be relied upon to function regardless of whether a battery or power source has been fitted. The electronic type with audio needs a power source to be operative during the flight. The instrument is of little interest during launching and landing, with the exception of aerotow , where the pilot will usually want to avoid releasing in sink. | https://en.wikipedia.org/wiki/Rate_of_climb |
The rate of heat flow is the amount of heat that is transferred per unit of time in some material, usually measured in watts ( joules per second). Heat is the flow of thermal energy driven by thermal non-equilibrium, so the term 'heat flow' is a redundancy (i.e. a pleonasm ). Heat must not be confused with stored thermal energy, and moving a hot object from one place to another must not be called heat transfer. However, it is common to say ‘heat flow’ to mean ‘ heat content ’. [ 1 ]
The equation of heat flow is given by Fourier's law of heat conduction.
Rate of heat flow = - (heat transfer coefficient) * (area of the body) * (variation of the temperature) / (length of the material)
The formula for the rate of heat flow is:
where
If a piece of material whose cross-sectional area is A {\displaystyle A} and thickness is Δ x {\displaystyle \Delta x} with a temperature difference Δ T {\displaystyle \Delta T} between its faces is observed, heat flows between the two faces in a direction perpendicular to the faces. The time rate of heat flow , Q Δ t {\displaystyle {\frac {Q}{\Delta t}}} , for small Q {\displaystyle Q} and small Δ t {\displaystyle \Delta t} , is proportional to A × Δ T Δ x {\displaystyle A\times {\frac {\Delta T}{\Delta x}}} . In the limit of infinitesimal thickness Δ x {\displaystyle \Delta x} , with temperature difference Δ T {\displaystyle \Delta T} , this becomes H = − k A ( Δ T Δ x ) {\displaystyle H=-kA({\frac {\Delta T}{\Delta x}})} , where H = ( Q Δ t ) {\displaystyle H=({\frac {Q}{\Delta t}})} is the time rate of heat flow through the area A {\displaystyle A} , Δ T Δ x {\displaystyle {\frac {\Delta T}{\Delta x}}} is the temperature gradient across the material, and k {\displaystyle k} , the proportionality constant, is the thermal conductivity of the material. [ 2 ] People often use k {\displaystyle k} , λ {\displaystyle \lambda } , or the Greek letter κ {\displaystyle \kappa } to represent this constant. [ citation needed ] The minus sign is there because the rate of heat flow is always negative—heat flows from the side at higher temperature to the one at lower temperature, not the other way around. [ 3 ]
This thermodynamics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rate_of_heat_flow |
In the drilling industry , the rate of penetration ( ROP ), [ 1 ] also known as penetration rate or drill rate , is the speed at which a drill bit breaks the rock under it to deepen the borehole . It is normally measured in feet per minute or meters per hour, but sometimes it is expressed in minutes per foot.
Generally, ROP increases in fast drilling formation such as sandstone (positive drill break) and decreases in slow drilling formations such as shale (reverse break). ROP decreases in shale due to diagenesis and overburden stresses. Over pressured zones can give twice of ROP as expected which is an indicative of a well kick . Drillers need to stop and do the bottoms up.
This industry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rate_of_penetration |
In epidemiology , a rate ratio , sometimes called an incidence density ratio or incidence rate ratio , is a relative difference measure used to compare the incidence rates of events occurring at any given point in time.
It is defined as:
where incidence rate is the occurrence of an event over person-time (for example person-years ):
The same time intervals must be used for both incidence rates. [ 1 ]
A common application for this measure in analytic epidemiologic studies is in the search for a causal association between a certain risk factor and an outcome. [ 2 ]
This statistics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rate_ratio |
Rate–distortion theory is a major branch of information theory which provides the theoretical foundations for lossy data compression ; it addresses the problem of determining the minimal number of bits per symbol, as measured by the rate R , that should be communicated over a channel, so that the source (input signal) can be approximately reconstructed at the receiver (output signal) without exceeding an expected distortion D .
Rate–distortion theory gives an analytical expression for how much compression can be achieved using lossy compression methods. Many of the existing audio, speech, image, and video compression techniques have transforms, quantization, and bit-rate allocation procedures that capitalize on the general shape of rate–distortion functions.
Rate–distortion theory was created by Claude Shannon in his foundational work on information theory.
In rate–distortion theory, the rate is usually understood as the number of bits per data sample to be stored or transmitted. The notion of distortion is a subject of on-going discussion. [ 1 ] In the most simple case (which is actually used in most cases), the distortion is defined as the expected value of the square of the difference between input and output signal (i.e., the mean squared error ). However, since we know that most lossy compression techniques operate on data that will be perceived by human consumers (listening to music , watching pictures and video) the distortion measure should preferably be modeled on human perception and perhaps aesthetics : much like the use of probability in lossless compression , distortion measures can ultimately be identified with loss functions as used in Bayesian estimation and decision theory . In audio compression, perceptual models (and therefore perceptual distortion measures) are relatively well developed and routinely used in compression techniques such as MP3 or Vorbis , but are often not easy to include in rate–distortion theory. In image and video compression, the human perception models are less well developed and inclusion is mostly limited to the JPEG and MPEG weighting ( quantization , normalization ) matrix.
Distortion functions measure the cost of representing a symbol x {\displaystyle x} by an approximated symbol x ^ {\displaystyle {\hat {x}}} . Typical distortion functions are the Hamming distortion and the Squared-error distortion.
The functions that relate the rate and distortion are found as the solution of the following minimization problem:
Here Q Y ∣ X ( y ∣ x ) {\displaystyle Q_{Y\mid X}(y\mid x)} , sometimes called a test channel, is the conditional probability density function (PDF) of the communication channel output (compressed signal) Y {\displaystyle Y} for a given input (original signal) X {\displaystyle X} , and I Q ( Y ; X ) {\displaystyle I_{Q}(Y;X)} is the mutual information between Y {\displaystyle Y} and X {\displaystyle X} defined as
where H ( Y ) {\displaystyle H(Y)} and H ( Y ∣ X ) {\displaystyle H(Y\mid X)} are the entropy of the output signal Y and the conditional entropy of the output signal given the input signal, respectively:
The problem can also be formulated as a distortion–rate function, where we find the infimum over achievable distortions for given rate constraint. The relevant expression is:
The two formulations lead to functions which are inverses of each other.
The mutual information can be understood as a measure for 'prior' uncertainty the receiver has about the sender's signal ( H ( Y )), diminished by the uncertainty that is left after receiving information about the sender's signal ( H ( Y ∣ X ) {\displaystyle H(Y\mid X)} ). Of course the decrease in uncertainty is due to the communicated amount of information, which is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle I \left(Y;X \right)} .
As an example, in case there is no communication at all, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle H(Y\mid X) = H (Y)} and I ( Y ; X ) = 0 {\displaystyle I(Y;X)=0} . Alternatively, if the communication channel is perfect and the received signal Y {\displaystyle Y} is identical to the signal X {\displaystyle X} at the sender, then H ( Y ∣ X ) = 0 {\displaystyle H(Y\mid X)=0} and I ( Y ; X ) = H ( X ) = H ( Y ) {\displaystyle I(Y;X)=H(X)=H(Y)} .
In the definition of the rate–distortion function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle D_Q} and D ∗ {\displaystyle D^{*}} are the distortion between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle Y} for a given Q Y ∣ X ( y ∣ x ) {\displaystyle Q_{Y\mid X}(y\mid x)} and the prescribed maximum distortion, respectively. When we use the mean squared error as distortion measure, we have (for amplitude - continuous signals ):
As the above equations show, calculating a rate–distortion function requires the stochastic description of the input X {\displaystyle X} in terms of the PDF P X ( x ) {\displaystyle P_{X}(x)} , and then aims at finding the conditional PDF Q Y ∣ X ( y ∣ x ) {\displaystyle Q_{Y\mid X}(y\mid x)} that minimize rate for a given distortion D ∗ {\displaystyle D^{*}} . These definitions can be formulated measure-theoretically to account for discrete and mixed random variables as well.
An analytical solution to this minimization problem is often difficult to obtain except in some instances for which we next offer two of the best known examples. The rate–distortion function of any source is known to obey several fundamental properties, the most important ones being that it is a continuous , monotonically decreasing convex (U) function and thus the shape for the function in the examples is typical (even measured rate–distortion functions in real life tend to have very similar forms).
Although analytical solutions to this problem are scarce, there are upper and lower bounds to these functions including the famous Shannon lower bound (SLB), which in the case of squared error and memoryless sources, states that for arbitrary sources with finite differential entropy,
where h ( D ) is the differential entropy of a Gaussian random variable with variance D. This lower bound is extensible to sources with memory and other distortion measures. One important feature of the SLB is that it is asymptotically tight in the low distortion regime for a wide class of sources and in some occasions, it actually coincides with the rate–distortion function. Shannon Lower Bounds can generally be found if the distortion between any two numbers can be expressed as a function of the difference between the value of these two numbers.
The Blahut–Arimoto algorithm , co-invented by Richard Blahut , is an elegant iterative technique for numerically obtaining rate–distortion functions of arbitrary finite input/output alphabet sources and much work has been done to extend it to more general problem instances.
The computation of the rate-distortion function requires knowledge of the underlying distribution, which is often unavailable in contemporary applications in data-science and machine learning. However, this challenge can be addressed using deep learning-based estimators of the rate-distortion function. [ 2 ] These estimators are typically referred to as 'neural estimators', involving the optimization of a parametrized variational form of the rate distortion objective.
When working with stationary sources with memory, it is necessary to modify the definition of the rate distortion function and it must be understood in the sense of a limit taken over sequences of increasing lengths.
where
and
where superscripts denote a complete sequence up to that time and the subscript 0 indicates initial state.
If we assume that X {\displaystyle X} is a Gaussian random variable with variance σ 2 {\displaystyle \sigma ^{2}} , and if we assume that successive samples of the signal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X} are stochastically independent (or equivalently, the source is memoryless , or the signal is uncorrelated ), we find the following analytical expression for the rate–distortion function:
The following figure shows what this function looks like:
Rate–distortion theory tell us that 'no compression system exists that performs outside the gray area'. The closer a practical compression system is to the red (lower) bound, the better it performs. As a general rule, this bound can only be attained by increasing the coding block length parameter. Nevertheless, even at unit blocklengths one can often find good (scalar) quantizers that operate at distances from the rate–distortion function that are practically relevant. [ 4 ]
This rate–distortion function holds only for Gaussian memoryless sources. It is known that the Gaussian source is the most "difficult" source to encode: for a given mean square error, it requires the greatest number of bits. The performance of a practical compression system working on—say—images, may well be below the R ( D ) {\displaystyle R\left(D\right)} lower bound shown.
The rate-distortion function of a Bernoulli random variable with Hamming distortion is given by:
where H b {\displaystyle H_{b}} denotes the binary entropy function .
Plot of the rate-distortion function for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle p=0.5} :
Suppose we want to transmit information about a source to the user with a distortion not exceeding D . Rate–distortion theory tells us that at least R ( D ) {\displaystyle R(D)} bits/symbol of information from the source must reach the user. We also know from Shannon's channel coding theorem that if the source entropy is H bits/symbol, and the channel capacity is C (where C < H {\displaystyle C<H} ), then H − C {\displaystyle H-C} bits/symbol will be lost when transmitting this information over the given channel. For the user to have any hope of reconstructing with a maximum distortion D , we must impose the requirement that the information lost in transmission does not exceed the maximum tolerable loss of H − R ( D ) {\displaystyle H-R(D)} bits/symbol. This means that the channel capacity must be at least as large as R ( D ) {\displaystyle R(D)} . [ 5 ] | https://en.wikipedia.org/wiki/Rate–distortion_theory |
Rathayibacter is a genus of bacteria of the order Actinomycetales which are gram-positive soil organisms. [ 1 ]
This Actinomycetota -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rathayibacter |
Rathayibacter tritici is a Gram-positive soil bacterium . It is a plant pathogen and causes spike blight in wheat .
This Actinomycetota -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rathayibacter_tritici |
In mathematics, Rathjen's ψ {\displaystyle \psi } psi function is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals M {\displaystyle M} to generate large countable ordinals . [ 1 ] A weakly Mahlo cardinal is a cardinal such that the set of regular cardinals below M {\displaystyle M} is closed under M {\displaystyle M} (i.e. all normal functions closed in M {\displaystyle M} are closed under some regular ordinal < M {\displaystyle <M} ). Rathjen uses this to diagonalise over the weakly inaccessible hierarchy.
It admits an associated ordinal notation T ( M ) {\displaystyle T(M)} whose limit (i.e. ordinal type) is ψ Ω ( χ ε M + 1 ( 0 ) ) {\displaystyle \psi _{\Omega }(\chi _{\varepsilon _{M}+1}(0))} , which is strictly greater than both | K P M | {\displaystyle \vert KPM\vert } and the limit of countable ordinals expressed by Rathjen's ψ {\displaystyle \psi } . | K P M | {\displaystyle \vert KPM\vert } , which is called the "Small Rathjen ordinal" is the proof-theoretic ordinal of K P M {\displaystyle {\mathsf {KPM}}} , Kripke–Platek set theory augmented by the axiom schema "for any Δ 0 {\displaystyle \Delta _{0}} -formula H ( x , y ) {\displaystyle H(x,y)} satisfying ∀ x ∃ y ( H ( x , y ) ) {\displaystyle \forall x\,\exists y\,(H(x,y))} , there exists an addmissible set z {\displaystyle z} satisfying ∀ x ∈ z ∃ y ( H ( x , y ) ) {\displaystyle \forall x\in z\,\exists y\,(H(x,y))} ". It is equal to ψ Ω ( ψ χ ε M + 1 ( 0 ) ( 0 ) ) {\displaystyle \psi _{\Omega }(\psi _{\chi _{\varepsilon _{M}+1}(0)}(0))} in Rathjen's ψ {\displaystyle \psi } function. [ 2 ]
Restrict π {\displaystyle \pi } and κ {\displaystyle \kappa } to uncountable regular cardinals < M {\displaystyle <M} ; for a function f {\displaystyle f} let dom ( f ) {\displaystyle \operatorname {dom} (f)} denote the domain of f {\displaystyle f} ; let cl M ( X ) {\displaystyle \operatorname {cl} _{M}(X)} denote X ∪ { α < M : α is a limit point of X } {\displaystyle X\cup \{\alpha <M:\alpha {\text{ is a limit point of }}X\}} , and let enum ( X ) {\displaystyle \operatorname {enum} (X)} denote the enumeration of X {\displaystyle X} . Lastly, an ordinal α {\displaystyle \alpha } is said to be to be strongly critical if φ α ( 0 ) = α {\displaystyle \varphi _{\alpha }(0)=\alpha } .
For α ∈ Γ M + 1 {\displaystyle \alpha \in \Gamma _{M+1}} and β ∈ M {\displaystyle \beta \in M} :
If κ = χ α ( β + 1 ) {\displaystyle \kappa =\chi _{\alpha }(\beta +1)} for some ( α , β ) ∈ Γ M + 1 × M {\displaystyle (\alpha ,\beta )\in \Gamma _{M+1}\times M} , define κ − := χ α ( β ) {\displaystyle \kappa ^{-}:=\chi _{\alpha }(\beta )} using the unique ( α , β ) {\displaystyle (\alpha ,\beta )} . Otherwise if κ = χ α ( 0 ) {\displaystyle \kappa =\chi _{\alpha }(0)} for some α ∈ Γ M + 1 {\displaystyle \alpha \in \Gamma _{M+1}} , then define κ − := sup ( SC M ( α ) ∪ { 0 } ) {\displaystyle \kappa ^{-}:=\sup(\operatorname {SC} _{M}(\alpha )\cup \{0\})} using the unique α {\displaystyle \alpha } , where SC M ( α ) {\displaystyle \operatorname {SC} _{M}(\alpha )} is a set of strongly critical ordinals < M {\displaystyle <M} explicitly defined in the original source.
For α ∈ Γ M + 1 {\displaystyle \alpha \in \Gamma _{M+1}} :
Rathjen originally defined the ψ {\displaystyle \psi } function in more complicated a way in order to create an ordinal notation associated to it. Therefore, it is not certain whether the simplified OCF above yields an ordinal notation or not. The original χ {\displaystyle \chi } functions used in Rathjen's original OCF are also not so easy to understand, and differ from the χ {\displaystyle \chi } functions defined above.
Rathjen's ψ {\displaystyle \psi } and the simplification provided above are not the same OCF. This is partially because the former is known to admit an ordinal notation, while the latter isn't known to admit an ordinal notation. [ citation needed ] Rathjen's ψ {\displaystyle \psi } is often confounded with another of his OCFs which also uses the symbol ψ {\displaystyle \psi } , but they are distinct notions. The former one is a published OCF, while the latter one is just a function symbol in an ordinal notation associated to an unpublished OCF. [ 3 ] | https://en.wikipedia.org/wiki/Rathjen's_psi_function |
In hydrology , a rating curve is a graph of discharge versus stage for a given point on a stream, usually at gauging stations , where the stream discharge is measured across the stream channel with a flow meter . [ 1 ] Numerous measurements of stream discharge are made over a range of stream stages. The rating curve is usually plotted as discharge on x-axis versus stage (surface elevation) on y-axis. [ 2 ]
The development of a rating curve involves two steps. In the first step the relationship between stage and discharge is established by measuring the stage and corresponding discharge in the river. And in the second part, stage of river is measured and discharge is calculated by using the relationship established in the first part. Stage is measured by reading a gauge installed in the river. If the stage-discharge relationship does not change with time, it is called permanent control. If the relationship does change, it is called shifting control. Shifting control is usually due to erosion or deposition of sediment at the stage measurement site. Bedrock-bottomed parts of rivers or concrete/metal weirs or structures are often, though not always, permanent controls.
If G represents stage for discharge Q, then the relationship between G and Q can possibly be approximated with an equation:
where C r {\displaystyle C_{r}} and β {\displaystyle \beta } are rating curve constants, and a {\displaystyle a} is a constant which represents the gauge reading corresponding to zero discharge. The constant a {\displaystyle a} can be measured when a stream is flowing under "section control" as the surveyed gauge height of the lowest point of the section control feature. When a stream is flowing under "channel control" conditions, the parameter a {\displaystyle a} does not have a physical analogue and must be estimated by following standard methods given in literature. The parameter β {\displaystyle \beta } is typically in the range of 2.0 to 3.0 when a stream is flowing under section control, and in the range of 1.0 to 2.0 when a stream is flowing under channel control.
A stream will typically transition from section control at lower gauge heights to channel control at higher gauge heights. The transition from section control to channel control can often be inferred by a change in the slope of a rating curve when plotted on log-log graph paper.
This hydrology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rating_curve |
In mathematics , a ratio ( / ˈ r eɪ ʃ ( i ) oʊ / ) shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7).
The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive .
A ratio may be specified either by giving both constituting numbers, written as " a to b " or " a : b ", or by giving just the value of their quotient a / b . [ 1 ] [ 2 ] [ 3 ] Equal quotients correspond to equal ratios.
A statement expressing the equality of two ratios is called a proportion .
Consequently, a ratio may be considered as an ordered pair of numbers, a fraction with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers , are rational numbers , and may sometimes be natural numbers.
A more specific definition adopted in physical sciences (especially in metrology ) for ratio is the dimensionless quotient between two physical quantities measured with the same unit . [ 4 ] A quotient of two quantities that are measured with different units may be called a rate . [ 5 ]
The ratio of numbers A and B can be expressed as: [ 6 ]
When a ratio is written in the form A : B , the two-dot character is sometimes the colon punctuation mark. [ 8 ] In Unicode , this is U+003A : COLON , although Unicode also provides a dedicated ratio character, U+2236 ∶ RATIO . [ 9 ]
The numbers A and B are sometimes called terms of the ratio , with A being the antecedent and B being the consequent . [ 10 ]
A statement expressing the equality of two ratios A : B and C : D is called a proportion , [ 11 ] written as A : B = C : D or A : B ∷ C : D . This latter form, when spoken or written in the English language, is often expressed as
A , B , C and D are called the terms of the proportion. A and D are called its extremes , and B and C are called its means . The equality of three or more ratios, like A : B = C : D = E : F , is called a continued proportion . [ 12 ]
Ratios are sometimes used with three or even more terms, e.g., the proportion for the edge lengths of a " two by four " that is ten inches long is therefore
a good concrete mix (in volume units) is sometimes quoted as
For a (rather dry) mixture of 4/1 parts in volume of cement to water, it could be said that the ratio of cement to water is 4:1, that there is 4 times as much cement as water, or that there is a quarter (1/4) as much water as cement.
The meaning of such a proportion of ratios with more than two terms is that the ratio of any two terms on the left-hand side is equal to the ratio of the corresponding two terms on the right-hand side.
It is possible to trace the origin of the word "ratio" to the Ancient Greek λόγος ( logos ). Early translators rendered this into Latin as ratio ("reason"; as in the word "rational"). A more modern interpretation of Euclid's meaning is more akin to computation or reckoning. [ 14 ] Medieval writers used the word proportio ("proportion") to indicate ratio and proportionalitas ("proportionality") for the equality of ratios. [ 15 ]
Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers. [ 16 ] The Pythagoreans' conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers ) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus . The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables. [ 17 ]
The existence of multiple theories seems unnecessarily complex since ratios are, to a large extent, identified with quotients and their prospective values. However, this is a comparatively recent development, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients. The reasons for this are twofold: first, there was the previously mentioned reluctance to accept irrational numbers as true numbers, and second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century. [ 18 ]
Book V of Euclid's Elements has 18 definitions, all of which relate to ratios. [ 19 ] In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them. The first two definitions say that a part of a quantity is another quantity that "measures" it and conversely, a multiple of a quantity is another quantity that it measures. In modern terminology, this means that a multiple of a quantity is that quantity multiplied by an integer greater than one—and a part of a quantity (meaning aliquot part ) is a part that, when multiplied by an integer greater than one, gives the quantity.
Euclid does not define the term "measure" as used here, However, one may infer that if a quantity is taken as a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity measures the second. These definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII.
Definition 3 describes what a ratio is in a general way. It is not rigorous in a mathematical sense and some have ascribed it to Euclid's editors rather than Euclid himself. [ 20 ] Euclid defines a ratio as between two quantities of the same type , so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists, when there is a multiple of each that exceeds the other. In modern notation, a ratio exists between quantities p and q , if there exist integers m and n such that mp > q and nq > p . This condition is known as the Archimedes property .
Definition 5 is the most complex and difficult. It defines what it means for two ratios to be equal. Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but such a definition would have been meaningless to Euclid. In modern notation, Euclid's definition of equality is that given quantities p , q , r and s , p : q ∷ r : s if and only if, for any positive integers m and n , np < mq , np = mq , or np > mq according as nr < ms , nr = ms , or nr > ms , respectively. [ 21 ] This definition has affinities with Dedekind cuts as, with n and q both positive, np stands to mq as p / q stands to the rational number m / n (dividing both terms by nq ). [ 22 ]
Definition 6 says that quantities that have the same ratio are proportional or in proportion . Euclid uses the Greek ἀναλόγον (analogon), this has the same root as λόγος and is related to the English word "analog".
Definition 7 defines what it means for one ratio to be less than or greater than another and is based on the ideas present in definition 5. In modern notation it says that given quantities p , q , r and s , p : q > r : s if there are positive integers m and n so that np > mq and nr ≤ ms .
As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid's editors. It defines three terms p , q and r to be in proportion when p : q ∷ q : r . This is extended to four terms p , q , r and s as p : q ∷ q : r ∷ r : s , and so on. Sequences that have the property that the ratios of consecutive terms are equal are called geometric progressions . Definitions 9 and 10 apply this, saying that if p , q and r are in proportion then p : r is the duplicate ratio of p : q and if p , q , r and s are in proportion then p : s is the triplicate ratio of p : q .
In general, a comparison of the quantities of a two-entity ratio can be expressed as a fraction derived from the ratio. For example, in a ratio of 2:3, the amount, size, volume, or quantity of the first entity is 2 3 {\displaystyle {\tfrac {2}{3}}} that of the second entity.
If there are 2 oranges and 3 apples, the ratio of oranges to apples is 2:3, and the ratio of oranges to the total number of pieces of fruit is 2:5. These ratios can also be expressed in fraction form: there are 2/3 as many oranges as apples, and 2/5 of the pieces of fruit are oranges. If orange juice concentrate is to be diluted with water in the ratio 1:4, then one part of concentrate is mixed with four parts of water, giving five parts total; the amount of orange juice concentrate is 1/4 the amount of water, while the amount of orange juice concentrate is 1/5 of the total liquid. In both ratios and fractions, it is important to be clear what is being compared to what, and beginners often make mistakes for this reason.
Fractions can also be inferred from ratios with more than two entities; however, a ratio with more than two entities cannot be completely converted into a single fraction, because a fraction can only compare two quantities. A separate fraction can be used to compare the quantities of any two of the entities covered by the ratio: for example, from a ratio of 2:3:7 we can infer that the quantity of the second entity is 3 7 {\displaystyle {\tfrac {3}{7}}} that of the third entity.
If we multiply all quantities involved in a ratio by the same number, the ratio remains valid. For example, a ratio of 3:2 is the same as 12:8. It is usual either to reduce terms to the lowest common denominator , or to express them in parts per hundred ( percent ).
If a mixture contains substances A , B , C and D in the ratio 5:9:4:2, then there are 5 parts of A for every 9 parts of B , 4 parts of C , and 2 parts of D . As 5 + 9 + 4 + 2 = 20, the total mixture contains 5/20 of A (5 parts out of 20), 9/20 of B , 4/20 of C , and 2/20 of D . If we divide all numbers by the total and multiply by 100, we have converted to percentages : 25% A , 45% B , 20% C , and 10% D (equivalent to writing the ratio as 25:45:20:10).
If the two or more ratio quantities encompass all of the quantities in a particular situation, it is said that "the whole" contains the sum of the parts: for example, a fruit basket containing two apples and three oranges and no other fruit is made up of two parts apples and three parts oranges. In this case, 2 5 {\displaystyle {\tfrac {2}{5}}} , or 40% of the whole is apples and 3 5 {\displaystyle {\tfrac {3}{5}}} , or 60% of the whole is oranges. This comparison of a specific quantity to "the whole" is called a proportion.
If the ratio consists of only two values, it can be represented as a fraction, in particular as a decimal fraction. For example, older televisions have a 4:3 aspect ratio , which means that the width is 4/3 of the height (this can also be expressed as 1.33:1 or just 1.33 rounded to two decimal places). More recent widescreen TVs have a 16:9 aspect ratio, or 1.78 rounded to two decimal places. One of the popular widescreen movie formats is 2.35:1 or simply 2.35. Representing ratios as decimal fractions simplifies their comparison. When comparing 1.33, 1.78 and 2.35, it is obvious which format offers wider image. Such a comparison works only when values being compared are consistent, like always expressing width in relation to height.
Ratios can be reduced (as fractions are) by dividing each quantity by the common factors of all the quantities. As for fractions, the simplest form is considered that in which the numbers in the ratio are the smallest possible integers.
Thus, the ratio 40:60 is equivalent in meaning to the ratio 2:3, the latter being obtained from the former by dividing both quantities by 20. Mathematically, we write 40:60 = 2:3, or equivalently 40:60∷2:3. The verbal equivalent is "40 is to 60 as 2 is to 3."
A ratio that has integers for both quantities and that cannot be reduced any further (using integers) is said to be in simplest form or lowest terms.
Sometimes it is useful to write a ratio in the form 1: x or x :1, where x is not necessarily an integer, to enable comparisons of different ratios. For example, the ratio 4:5 can be written as 1:1.25 (dividing both sides by 4) Alternatively, it can be written as 0.8:1 (dividing both sides by 5).
Where the context makes the meaning clear, a ratio in this form is sometimes written without the 1 and the ratio symbol (:), though, mathematically, this makes it a factor or multiplier .
Ratios may also be established between incommensurable quantities (quantities whose ratio, as value of a fraction, amounts to an irrational number ). The earliest discovered example, found by the Pythagoreans , is the ratio of the length of the diagonal d to the length of a side s of a square , which is the square root of 2 , formally a : d = 1 : 2 . {\displaystyle a:d=1:{\sqrt {2}}.} Another example is the ratio of a circle 's circumference to its diameter, which is called π , and is not just an irrational number , but a transcendental number .
Also well known is the golden ratio of two (mostly) lengths a and b , which is defined by the proportion
Taking the ratios as fractions and a : b {\displaystyle a:b} as having the value x , yields the equation
which has the positive, irrational solution x = a b = 1 + 5 2 . {\displaystyle x={\tfrac {a}{b}}={\tfrac {1+{\sqrt {5}}}{2}}.} Thus at least one of a and b has to be irrational for them to be in the golden ratio. An example of an occurrence of the golden ratio in math is as the limiting value of the ratio of two consecutive Fibonacci numbers : even though all these ratios are ratios of two integers and hence are rational, the limit of the sequence of these rational ratios is the irrational golden ratio.
Similarly, the silver ratio of a and b is defined by the proportion
This equation has the positive, irrational solution x = a b = 1 + 2 , {\displaystyle x={\tfrac {a}{b}}=1+{\sqrt {2}},} so again at least one of the two quantities a and b in the silver ratio must be irrational.
Odds (as in gambling) are expressed as a ratio. For example, odds of "7 to 3 against" (7:3) mean that there are seven chances that the event will not happen to every three chances that it will happen. The probability of success is 30%. In every ten trials, there are expected to be three wins and seven losses.
Ratios may be unitless , as in the case they relate quantities in units of the same dimension , even if their units of measurement are initially different.
For example, the ratio one minute : 40 seconds can be reduced by changing the first value to 60 seconds, so the ratio becomes 60 seconds : 40 seconds . Once the units are the same, they can be omitted, and the ratio can be reduced to 3:2.
On the other hand, there are non-dimensionless quotients, also known as rates (sometimes also as ratios). [ 23 ] [ 24 ] In chemistry, mass concentration ratios are usually expressed as weight/volume fractions.
For example, a concentration of 3% w/v usually means 3 g of substance in every 100 mL of solution. This cannot be converted to a dimensionless ratio, as in weight/weight or volume/volume fractions.
The locations of points relative to a triangle with vertices A , B , and C and sides AB , BC , and CA are often expressed in extended ratio form as triangular coordinates .
In barycentric coordinates , a point with coordinates α, β, γ is the point upon which a weightless sheet of metal in the shape and size of the triangle would exactly balance if weights were put on the vertices, with the ratio of the weights at A and B being α : β , the ratio of the weights at B and C being β : γ , and therefore the ratio of weights at A and C being α : γ .
In trilinear coordinates , a point with coordinates x : y : z has perpendicular distances to side BC (across from vertex A ) and side CA (across from vertex B ) in the ratio x : y , distances to side CA and side AB (across from C ) in the ratio y : z , and therefore distances to sides BC and AB in the ratio x : z .
Since all information is expressed in terms of ratios (the individual numbers denoted by α, β, γ, x, y, and z have no meaning by themselves), a triangle analysis using barycentric or trilinear coordinates applies regardless of the size of the triangle. | https://en.wikipedia.org/wiki/Ratio |
A Ratiometer type temperature indicating system consists of a sensing element and a moving-coil indicator, which unlike the conventional type has two coils moving together in a permanent-magnet field of non-uniform strength. The coil arrangements and the methods of obtaining the non-uniform field depends on the manufacturer's design. The main application behind in this ratiometer system is to find the unknown resistance, namely Rx. This plays a major role in the aircraft industry, in finding cylinder head temperatures exposed to turbine exhaust gases . [ 2 ] It can also be used to measure temperatures of systems such as engine oil and carburetor air.
Parallel coil is a method of obtaining non-uniform coil in ratiometer systems in which the hairsprings are wounded parallelly in order to allow the signal to pass through it.
The coil is wound from 99.99% pure copper on a wood which was treated with many oxidizers and chemicals .
The benefits from using foil inductors comes in the form of less distortion and higher dynamic headroom, when used on crossovers for modern high performance speakers.
In these methods two parallel resistance arms are formed; one containing the coil and a fixed calibrating resistance R1, and the other containing a coil in series with a calibrating resistance R2 and the temperature-sensing element Rx. Both arms are supplied with direct current from the aircraft's main power source, but the coils are so wound that current flows through them in opposite directions.
As in any moving-coil indicator, rotation of the measuring element is produced by forces which are proportional to product of the current and field strength, and the direction of rotation depends on the direction of current relative to the magnetic field. In a ratio meter, therefore, it follows that the force produced by one coil will always tend to rotate the measuring element in the opposite direction to the force produced by the second coil, and further more, as the magnetic field is of non-uniform strength, the coil carrying the greater current will always move towards the area of weaker field, and vice versa.
When the temperature at the sensing element Rx increases, then in accordance with the temperature/resistance relationship of the material used for the element, its resistance will increase and so cause a decrease in the force created by it.
When the measuring elements at the mid-position of its rotation, the currents in both windings can be in the same field strength simultaneously.
A ratiometer systems, however, do not require hair springs for exerting a controlling torque, this being provided solely by the appropriate coil winding and non-uniform field arrangements. Should variations occur in the power supply they will affect both coils equally so that the ratio of current flowing in the coils remains the same and tendencies for them to move to positions of differing field strength are counterbalanced. [ 2 ] | https://en.wikipedia.org/wiki/Ratio_meter_systems |
In mathematics , the ratio test is a test (or "criterion") for the convergence of a series
where each term is a real or complex number and a n is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test . [ 1 ]
The usual form of the test makes use of the limit
The ratio test states that:
It is possible to make the ratio test applicable to certain cases where the limit L fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when L = 1. More specifically, let
Then the ratio test states that: [ 2 ] [ 3 ]
If the limit L in ( 1 ) exists, we must have L = R = r . So the original ratio test is a weaker version of the refined one.
Consider the series
Applying the ratio test, one computes the limit
Since this limit is less than 1, the series converges.
Consider the series
Putting this into the ratio test:
Thus the series diverges.
Consider the three series
The first series ( 1 + 1 + 1 + 1 + ⋯ ) diverges, the second (the one central to the Basel problem ) converges absolutely and the third (the alternating harmonic series ) converges conditionally. However, the term-by-term magnitude ratios | a n + 1 a n | {\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|} of the three series are 1 , {\displaystyle 1,} n 2 ( n + 1 ) 2 {\displaystyle {\frac {n^{2}}{(n+1)^{2}}}} and n n + 1 {\displaystyle {\frac {n}{n+1}}} . So, in all three, the limit lim n → ∞ | a n + 1 a n | {\displaystyle \lim _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|} is equal to 1. This illustrates that when L = 1, the series may converge or diverge: the ratio test is inconclusive. In such cases, more refined tests are required to determine convergence or divergence.
Below is a proof of the validity of the generalized ratio test.
Suppose that r = lim inf n → ∞ | a n + 1 a n | > 1 {\displaystyle r=\liminf _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|>1} . We also suppose that ( a n ) {\displaystyle (a_{n})} has infinite non-zero members, otherwise the series is just a finite sum hence it converges. Then there exists some ℓ ∈ ( 1 ; r ) {\displaystyle \ell \in (1;r)} such that there exists a natural number n 0 ≥ 2 {\displaystyle n_{0}\geq 2} satisfying a n 0 ≠ 0 {\displaystyle a_{n_{0}}\neq 0} and | a n + 1 a n | > ℓ {\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|>\ell } for all n ≥ n 0 {\displaystyle n\geq n_{0}} , because if no such ℓ {\displaystyle \ell } exists then there exists arbitrarily large n {\displaystyle n} satisfying | a n + 1 a n | < ℓ {\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|<\ell } for every ℓ ∈ ( 1 ; r ) {\displaystyle \ell \in (1;r)} , then we can find a subsequence ( a n k ) k = 1 ∞ {\displaystyle \left(a_{n_{k}}\right)_{k=1}^{\infty }} satisfying lim sup n → ∞ | a n k + 1 a n k | ≤ ℓ < r {\displaystyle \limsup _{n\to \infty }\left|{\frac {a_{n_{k}+1}}{a_{n_{k}}}}\right|\leq \ell <r} , but this contradicts the fact that r {\displaystyle r} is the limit inferior of | a n + 1 a n | {\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|} as n → ∞ {\displaystyle n\to \infty } , implying the existence of ℓ {\displaystyle \ell } . Then we notice that for n ≥ n 0 + 1 {\displaystyle n\geq n_{0}+1} , | a n | > ℓ | a n − 1 | > ℓ 2 | a n − 2 | > . . . > ℓ n − n 0 | a n 0 | {\displaystyle |a_{n}|>\ell |a_{n-1}|>\ell ^{2}|a_{n-2}|>...>\ell ^{n-n_{0}}\left|a_{n_{0}}\right|} . Notice that ℓ > 1 {\displaystyle \ell >1} so ℓ n → ∞ {\displaystyle \ell ^{n}\to \infty } as n → ∞ {\displaystyle n\to \infty } and | a n 0 | > 0 {\displaystyle \left|a_{n_{0}}\right|>0} , this implies ( a n ) {\displaystyle (a_{n})} diverges so the series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} diverges by the n-th term test . Now suppose R = lim sup n → ∞ | a n + 1 a n | < 1 {\displaystyle R=\limsup _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|<1} . Similar to the above case, we may find a natural number n 1 {\displaystyle n_{1}} and a c ∈ ( R ; 1 ) {\displaystyle c\in (R;1)} such that | a n | ≤ c n − n 1 | a n 1 | {\displaystyle |a_{n}|\leq c^{n-n_{1}}\left|a_{n_{1}}\right|} for n ≥ n 1 {\displaystyle n\geq n_{1}} . Then ∑ n = 1 ∞ | a n | = ∑ k = 1 n 1 − 1 | a k | + ∑ n = n 1 ∞ | a n | ≤ ∑ k = 1 n 1 − 1 | a k | + ∑ n = n 1 ∞ c n − n 1 | a n 1 | = ∑ k = 1 n 1 − 1 | a k | + | a n 1 | ∑ n = 0 ∞ c n . {\displaystyle \sum _{n=1}^{\infty }|a_{n}|=\sum _{k=1}^{n_{1}-1}|a_{k}|+\sum _{n=n_{1}}^{\infty }|a_{n}|\leq \sum _{k=1}^{n_{1}-1}|a_{k}|+\sum _{n=n_{1}}^{\infty }c^{n-n_{1}}|a_{n_{1}}|=\sum _{k=1}^{n_{1}-1}|a_{k}|+\left|a_{n_{1}}\right|\sum _{n=0}^{\infty }c^{n}.} The series ∑ n = 0 ∞ c n {\displaystyle \sum _{n=0}^{\infty }c^{n}} is the geometric series with common ratio c ∈ ( 0 ; 1 ) {\displaystyle c\in (0;1)} , hence ∑ n = 0 ∞ c n = 1 1 − c {\displaystyle \sum _{n=0}^{\infty }c^{n}={\frac {1}{1-c}}} which is finite. The sum ∑ k = 1 n 1 − 1 | a k | {\displaystyle \sum _{k=1}^{n_{1}-1}|a_{k}|} is a finite sum and hence it is bounded, this implies the series ∑ n = 1 ∞ | a n | {\displaystyle \sum _{n=1}^{\infty }|a_{n}|} converges by the monotone convergence theorem and the series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges by the absolute convergence test. When the limit | a n + 1 a n | {\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|} exists and equals to L {\displaystyle L} then r = R = L {\displaystyle r=R=L} , this gives the original ratio test.
As seen in the previous example, the ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes allow one to deal with this case. [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] [ 11 ]
In all the tests below one assumes that Σ a n is a sum with positive a n . These tests also may be applied to any series with a finite number of negative terms. Any such series may be written as:
where a N is the highest-indexed negative term. The first expression on the right is a partial sum which will be finite, and so the convergence of the entire series will be determined by the convergence properties of the second expression on the right, which may be re-indexed to form a series of all positive terms beginning at n =1.
Each test defines a test parameter (ρ n ) which specifies the behavior of that parameter needed to establish convergence or divergence. For each test, a weaker form of the test exists which will instead place restrictions upon lim n->∞ ρ n .
All of the tests have regions in which they fail to describe the convergence properties of Σa n . In fact, no convergence test can fully describe the convergence properties of the series. [ 4 ] [ 10 ] This is because if Σa n is convergent, a second convergent series Σb n can be found which converges more slowly: i.e., it has the property that lim n->∞ (b n /a n ) = ∞. Furthermore, if Σa n is divergent, a second divergent series Σb n can be found which diverges more slowly: i.e., it has the property that lim n->∞ (b n /a n ) = 0. Convergence tests essentially use the comparison test on some particular family of a n , and fail for sequences which converge or diverge more slowly.
Augustus De Morgan proposed a hierarchy of ratio-type tests [ 4 ] [ 9 ]
The ratio test parameters ( ρ n {\displaystyle \rho _{n}} ) below all generally involve terms of the form D n a n / a n + 1 − D n + 1 {\displaystyle D_{n}a_{n}/a_{n+1}-D_{n+1}} . This term may be multiplied by a n + 1 / a n {\displaystyle a_{n+1}/a_{n}} to yield D n − D n + 1 a n + 1 / a n {\displaystyle D_{n}-D_{n+1}a_{n+1}/a_{n}} . This term can replace the former term in the definition of the test parameters and the conclusions drawn will remain the same. Accordingly, there will be no distinction drawn between references which use one or the other form of the test parameter.
The first test in the De Morgan hierarchy is the ratio test as described above.
This extension is due to Joseph Ludwig Raabe . Define:
(and some extra terms, see Ali, Blackburn, Feld, Duris (none), Duris2) [ clarification needed ]
The series will: [ 7 ] [ 10 ] [ 9 ]
For the limit version, [ 12 ] the series will:
When the above limit does not exist, it may be possible to use limits superior and inferior. [ 4 ] The series will:
Defining ρ n ≡ n ( a n a n + 1 − 1 ) {\displaystyle \rho _{n}\equiv n\left({\frac {a_{n}}{a_{n+1}}}-1\right)} , we need not assume the limit exists; if lim sup ρ n < 1 {\displaystyle \limsup \rho _{n}<1} , then ∑ a n {\displaystyle \sum a_{n}} diverges, while if lim inf ρ n > 1 {\displaystyle \liminf \rho _{n}>1} the sum converges.
The proof proceeds essentially by comparison with ∑ 1 / n R {\displaystyle \sum 1/n^{R}} . Suppose first that lim sup ρ n < 1 {\displaystyle \limsup \rho _{n}<1} . Of course
if lim sup ρ n < 0 {\displaystyle \limsup \rho _{n}<0} then a n + 1 ≥ a n {\displaystyle a_{n+1}\geq a_{n}} for large n {\displaystyle n} , so the sum diverges; assume then that 0 ≤ lim sup ρ n < 1 {\displaystyle 0\leq \limsup \rho _{n}<1} . There exists R < 1 {\displaystyle R<1} such that ρ n ≤ R {\displaystyle \rho _{n}\leq R} for all n ≥ N {\displaystyle n\geq N} , which is to say that a n / a n + 1 ≤ ( 1 + R n ) ≤ e R / n {\displaystyle a_{n}/a_{n+1}\leq \left(1+{\frac {R}{n}}\right)\leq e^{R/n}} . Thus a n + 1 ≥ a n e − R / n {\displaystyle a_{n+1}\geq a_{n}e^{-R/n}} , which implies that a n + 1 ≥ a N e − R ( 1 / N + ⋯ + 1 / n ) ≥ c a N e − R log ( n ) = c a N / n R {\displaystyle a_{n+1}\geq a_{N}e^{-R(1/N+\dots +1/n)}\geq ca_{N}e^{-R\log(n)}=ca_{N}/n^{R}} for n ≥ N {\displaystyle n\geq N} ; since R < 1 {\displaystyle R<1} this shows that ∑ a n {\displaystyle \sum a_{n}} diverges.
The proof of the other half is entirely analogous, with most of the inequalities simply reversed. We need a preliminary inequality to use
in place of the simple 1 + t < e t {\displaystyle 1+t<e^{t}} that was used above: Fix R {\displaystyle R} and N {\displaystyle N} . Note that log ( 1 + R n ) = R n + O ( 1 n 2 ) {\displaystyle \log \left(1+{\frac {R}{n}}\right)={\frac {R}{n}}+O\left({\frac {1}{n^{2}}}\right)} . So log ( ( 1 + R N ) … ( 1 + R n ) ) = R ( 1 N + ⋯ + 1 n ) + O ( 1 ) = R log ( n ) + O ( 1 ) {\displaystyle \log \left(\left(1+{\frac {R}{N}}\right)\dots \left(1+{\frac {R}{n}}\right)\right)=R\left({\frac {1}{N}}+\dots +{\frac {1}{n}}\right)+O(1)=R\log(n)+O(1)} ; hence ( 1 + R N ) … ( 1 + R n ) ≥ c n R {\displaystyle \left(1+{\frac {R}{N}}\right)\dots \left(1+{\frac {R}{n}}\right)\geq cn^{R}} .
Suppose now that lim inf ρ n > 1 {\displaystyle \liminf \rho _{n}>1} . Arguing as in the first paragraph, using the inequality established in the previous paragraph, we see that there exists R > 1 {\displaystyle R>1} such that a n + 1 ≤ c a N n − R {\displaystyle a_{n+1}\leq ca_{N}n^{-R}} for n ≥ N {\displaystyle n\geq N} ; since R > 1 {\displaystyle R>1} this shows that ∑ a n {\displaystyle \sum a_{n}} converges.
This extension is due to Joseph Bertrand and Augustus De Morgan .
Defining:
Bertrand's test [ 4 ] [ 10 ] asserts that the series will:
For the limit version, the series will:
When the above limit does not exist, it may be possible to use limits superior and inferior. [ 4 ] [ 9 ] [ 13 ] The series will:
This extension probably appeared at the first time by Margaret Martin in 1941. [ 14 ] A short proof based on Kummer's test and without technical assumptions (such as existence of the limits, for example) was provided by Vyacheslav Abramov in 2019. [ 15 ]
Let K ≥ 1 {\displaystyle K\geq 1} be an integer, and let ln ( K ) ( x ) {\displaystyle \ln _{(K)}(x)} denote the K {\displaystyle K} th iterate of natural logarithm , i.e. ln ( 1 ) ( x ) = ln ( x ) {\displaystyle \ln _{(1)}(x)=\ln(x)} and for any 2 ≤ k ≤ K {\displaystyle 2\leq k\leq K} , ln ( k ) ( x ) = ln ( k − 1 ) ( ln ( x ) ) {\displaystyle \ln _{(k)}(x)=\ln _{(k-1)}(\ln(x))} .
Suppose that the ratio a n / a n + 1 {\displaystyle a_{n}/a_{n+1}} , when n {\displaystyle n} is large, can be presented in the form
(The empty sum is assumed to be 0. With K = 1 {\displaystyle K=1} , the test reduces to Bertrand's test.)
The value ρ n {\displaystyle \rho _{n}} can be presented explicitly in the form
Extended Bertrand's test asserts that the series
For the limit version, the series
When the above limit does not exist, it may be possible to use limits superior and inferior. The series
For applications of Extended Bertrand's test see birth–death process .
This extension is due to Carl Friedrich Gauss .
Assuming a n > 0 and r > 1 , if a bounded sequence C n can be found such that for all n : [ 5 ] [ 7 ] [ 9 ] [ 10 ]
then the series will:
This extension is due to Ernst Kummer .
Let ζ n be an auxiliary sequence of positive constants. Define
Kummer's test states that the series will: [ 5 ] [ 6 ] [ 10 ] [ 11 ]
For the limit version, the series will: [ 16 ] [ 7 ] [ 9 ]
When the above limit does not exist, it may be possible to use limits superior and inferior. [ 4 ] The series will
All of the tests in De Morgan's hierarchy except Gauss's test can easily be seen as special cases of Kummer's test: [ 4 ]
where the empty product is assumed to be 1. Then,
Hence,
Note that for these four tests, the higher they are in the De Morgan hierarchy, the more slowly the 1 / ζ n {\displaystyle 1/\zeta _{n}} series diverges.
If ρ n > 0 {\displaystyle \rho _{n}>0} then fix a positive number 0 < δ < ρ n {\displaystyle 0<\delta <\rho _{n}} . There exists
a natural number N {\displaystyle N} such that for every n > N , {\displaystyle n>N,}
Since a n + 1 > 0 {\displaystyle a_{n+1}>0} , for every n > N , {\displaystyle n>N,}
In particular ζ n + 1 a n + 1 ≤ ζ n a n {\displaystyle \zeta _{n+1}a_{n+1}\leq \zeta _{n}a_{n}} for all n ≥ N {\displaystyle n\geq N} which means that starting from the index N {\displaystyle N} the sequence ζ n a n > 0 {\displaystyle \zeta _{n}a_{n}>0} is monotonically decreasing and
positive which in particular implies that it is bounded below by 0. Therefore, the limit
This implies that the positive telescoping series
and since for all n > N , {\displaystyle n>N,}
by the direct comparison test for positive series, the series ∑ n = 1 ∞ δ a n + 1 {\displaystyle \sum _{n=1}^{\infty }\delta a_{n+1}} is convergent.
On the other hand, if ρ < 0 {\displaystyle \rho <0} , then there is an N such that ζ n a n {\displaystyle \zeta _{n}a_{n}} is increasing for n > N {\displaystyle n>N} . In particular, there exists an ϵ > 0 {\displaystyle \epsilon >0} for which ζ n a n > ϵ {\displaystyle \zeta _{n}a_{n}>\epsilon } for all n > N {\displaystyle n>N} , and so ∑ n a n = ∑ n a n ζ n ζ n {\displaystyle \sum _{n}a_{n}=\sum _{n}{\frac {a_{n}\zeta _{n}}{\zeta _{n}}}} diverges by comparison with ∑ n ϵ ζ n {\displaystyle \sum _{n}{\frac {\epsilon }{\zeta _{n}}}} .
A new version of Kummer's test was established by Tong. [ 6 ] See also [ 8 ] [ 11 ] [ 17 ] for further discussions and new proofs. The provided modification of Kummer's theorem characterizes
all positive series, and the convergence or divergence can be formulated in the form of two necessary and sufficient conditions, one for convergence and another for divergence.
The first of these statements can be simplified as follows: [ 18 ]
The second statement can be simplified similarly:
However, it becomes useless, since the condition ∑ n = 1 ∞ 1 ζ n = ∞ {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{\zeta _{n}}}=\infty } in this case reduces to the original claim ∑ n = 1 ∞ a n = ∞ . {\displaystyle \sum _{n=1}^{\infty }a_{n}=\infty .}
Another ratio test that can be set in the framework of Kummer's theorem was presented by Orrin Frink [ 19 ] 1948.
Suppose a n {\displaystyle a_{n}} is a sequence in C ∖ { 0 } {\displaystyle \mathbb {C} \setminus \{0\}} ,
This result reduces to a comparison of ∑ n | a n | {\displaystyle \sum _{n}|a_{n}|} with a power series ∑ n n − p {\displaystyle \sum _{n}n^{-p}} , and can be seen to be related to Raabe's test. [ 20 ]
A more refined ratio test is the second ratio test: [ 7 ] [ 9 ] For a n > 0 {\displaystyle a_{n}>0} define:
By the second ratio test, the series will:
If the above limits do not exist, it may be possible to use the limits superior and inferior. Define:
Then the series will:
This test is a direct extension of the second ratio test. [ 7 ] [ 9 ] For 0 ≤ k ≤ m − 1 , {\displaystyle 0\leq k\leq m-1,} and positive a n {\displaystyle a_{n}} define:
By the m {\displaystyle m} th ratio test, the series will:
If the above limits do not exist, it may be possible to use the limits superior and inferior. For 0 ≤ k ≤ m − 1 {\displaystyle 0\leq k\leq m-1} define:
Then the series will:
This test is an extension of the m {\displaystyle m} th ratio test. [ 21 ]
Assume that the sequence a n {\displaystyle a_{n}} is a positive decreasing sequence.
Let φ : Z + → Z + {\displaystyle \varphi :\mathbb {Z} ^{+}\to \mathbb {Z} ^{+}} be such that lim n → ∞ n φ ( n ) {\displaystyle \lim _{n\to \infty }{\frac {n}{\varphi (n)}}} exists. Denote α = lim n → ∞ n φ ( n ) {\displaystyle \alpha =\lim _{n\to \infty }{\frac {n}{\varphi (n)}}} , and assume 0 < α < 1 {\displaystyle 0<\alpha <1} .
Assume also that lim n → ∞ a φ ( n ) a n = L . {\displaystyle \lim _{n\to \infty }{\frac {a_{\varphi (n)}}{a_{n}}}=L.}
Then the series will: | https://en.wikipedia.org/wiki/Ratio_test |
A rational agent or rational being is a person or entity that always aims to perform optimal actions based on given premises and information. A rational agent can be anything that makes decisions, typically a person , firm , machine , or software .
The concept of rational agents can be found in various disciplines such as artificial intelligence , cognitive science , decision theory , economics , ethics , game theory , and the study of practical reason .
In reference to economics, rational agent refers to hypothetical consumers and how they make decisions in a free market . This concept is one of the assumptions made in neoclassical economic theory . The concept of economic rationality arises from a tradition of marginal analysis used in neoclassical economics. The idea of a rational agent is important to the philosophy of utilitarianism , as detailed by philosopher Jeremy Bentham 's theory of the felicific calculus , also known as the hedonistic calculus.
The action a rational agent takes depends on:
In game theory and classical economics , it is often assumed that the actors , people, and firms are rational. However, the extent to which people and firms behave rationally is subject to debate. Economists often assume the models of rational choice theory and bounded rationality to formalize and predict the behavior of individuals and firms. Rational agents sometimes behave in manners that are counter-intuitive to many people, as in the traveler's dilemma .
Neuroeconomics is a concept that uses neuroscience , social psychology and other fields of science to better understand how people make decisions. Unlike rational agent theory, neuroeconomics does not attempt to predict large-scale human behavior but rather how individuals make decisions in case-by-case scenarios.
Artificial intelligence has borrowed the term "rational agents" from economics to describe autonomous programs that are capable of goal directed behavior. Today there is a considerable overlap between AI research, game theory and decision theory. Rational agents in AI are closely related to intelligent agents , autonomous software programs that display intelligence. [ 1 ] | https://en.wikipedia.org/wiki/Rational_agent |
In logic , a rational consequence relation is a non-monotonic consequence relation satisfying certain properties listed below.
A rational consequence relation is a logical framework that refines traditional deductive reasoning to better model real-world scenarios. It incorporates rules like reflexivity , left logical equivalence, right-hand weakening, cautious monotony, disjunction on the left-hand side, logical and on the right-hand side, and rational monotony. These rules enable the relation to handle everyday situations more effectively by allowing for non-monotonic reasoning, where conclusions can be drawn based on usual rather than absolute implications. This approach is particularly useful in cases where adding more information can change the outcome, providing a more nuanced understanding than monotone consequence relations.
A rational consequence relation ⊢ {\displaystyle \vdash } satisfies:
and the so-called Gabbay – Makinson rules: [ citation needed ]
The rational consequence relation is non-monotonic , and the relation θ ⊢ ϕ {\displaystyle \theta \vdash \phi } is intended to carry the meaning theta usually implies phi or phi usually follows from theta . In this sense it is more useful for modeling some everyday situations than a monotone consequence relation because the latter relation models facts in a more strict boolean fashion—something either follows under all circumstances or it does not.
The statement "If a cake contains sugar then it tastes good" implies under a monotone consequence relation the statement "If a cake contains sugar and soap then it tastes good." Clearly this doesn't match our own understanding of cakes. By asserting "If a cake contains sugar then it usually tastes good" a rational consequence relation allows for a more realistic model of the real world, and certainly it does not automatically follow that "If a cake contains sugar and soap then it usually tastes good."
Note that if we also have the information "If a cake contains sugar then it usually contains butter" then we may legally conclude (under CMO) that "If a cake contains sugar and butter then it usually tastes good." . Equally in the absence of a statement such as "If a cake contains sugar then usually it contains no soap " then we may legally conclude from RMO that "If the cake contains sugar and soap then it usually tastes good."
If this latter conclusion seems ridiculous to you then it is likely that you are subconsciously asserting your own preconceived knowledge about cakes when evaluating the validity of the statement. That is, from your experience you know that cakes that contain soap are likely to taste bad so you add to the system your own knowledge such as "Cakes that contain sugar do not usually contain soap." , even though this knowledge is absent from it. If the conclusion seems silly to you then you might consider replacing the word soap with the word eggs to see if it changes your feelings.
Consider the sentences:
We may consider it reasonable to conclude:
This would not be a valid conclusion under a monotonic deduction system (omitting of course the word 'usually'), since the third sentence would contradict the first two. In contrast the conclusion follows immediately using the Gabbay–Makinson rules: applying the rule CMO to the last two sentences yields the result.
The following consequences follow from the above rules:
Let L = { p 1 , … , p n } {\displaystyle L=\{p_{1},\ldots ,p_{n}\}} be a finite language . An atom is a formula of the form ⋀ i = 1 n p i ϵ {\displaystyle \bigwedge _{i=1}^{n}p_{i}^{\epsilon }} (where p 1 = p {\displaystyle p^{1}=p} and p − 1 = ¬ p {\displaystyle p^{-1}=\neg p} ). Notice that there is a unique valuation which makes any given atom true (and conversely each valuation satisfies precisely one atom). Thus an atom can be used to represent a preference about what we believe ought to be true.
Let A t L {\displaystyle At^{L}} be the set of all atoms in L. For θ ∈ {\displaystyle \theta \in } SL , define S θ = { α ∈ A t L | α ⊨ S C θ } {\displaystyle S_{\theta }=\{\alpha \in At^{L}|\alpha \models ^{SC}\theta \}} .
Let s → = s 1 , … , s m {\displaystyle {\vec {s}}=s_{1},\ldots ,s_{m}} be a sequence of subsets of A t L {\displaystyle At^{L}} . For θ {\displaystyle \theta } , ϕ {\displaystyle \phi } in SL, let the relation ⊢ s → {\displaystyle \vdash _{\vec {s}}} be such that θ ⊢ s → ϕ {\displaystyle \theta \vdash _{\vec {s}}\phi } if one of the following holds:
Then the relation ⊢ s → {\displaystyle \vdash _{\vec {s}}} is a rational consequence relation. This may easily be verified by checking directly that it satisfies the GM-conditions.
The idea behind the sequence of atom sets is that the earlier sets account for the most likely situations such as "young people are usually law abiding" whereas the later sets account for the less likely situations such as "young joyriders are usually not law abiding" .
It can be proven that any rational consequence relation on a finite language is representable via a sequence of atom preferences above. That is, for any such rational consequence relation ⊢ {\displaystyle \vdash } there is a sequence s → = s 1 , … , s m {\displaystyle {\vec {s}}=s_{1},\ldots ,s_{m}} of subsets of A t L {\displaystyle At^{L}} such that the associated rational consequence relation ⊢ s → {\displaystyle \vdash _{\vec {s}}} is the same relation: ⊢ s → = ⊢ {\displaystyle {\vdash _{\vec {s}}}={\vdash }} | https://en.wikipedia.org/wiki/Rational_consequence_relation |
In mathematics , a collection of real numbers is rationally independent if none of them can be written as a linear combination of the other numbers in the collection with rational coefficients. A collection of numbers which is not rationally independent is called rationally dependent . For instance we have the following example.
Because if we let x = 3 , y = 8 {\displaystyle x=3,y={\sqrt {8}}} , then 1 + 2 = 1 3 x + 1 2 y {\displaystyle 1+{\sqrt {2}}={\frac {1}{3}}x+{\frac {1}{2}}y} .
The real numbers ω 1 , ω 2 , ... , ω n are said to be rationally dependent if there exist integers k 1 , k 2 , ... , k n , not all of which are zero, such that
If such integers do not exist, then the vectors are said to be rationally independent . This condition can be reformulated as follows: ω 1 , ω 2 , ... , ω n are rationally independent if the only n -tuple of integers k 1 , k 2 , ... , k n such that
is the trivial solution in which every k i is zero.
The real numbers form a vector space over the rational numbers , and this is equivalent to the usual definition of linear independence in this vector space. | https://en.wikipedia.org/wiki/Rational_dependence |
In chemical biology and biomolecular engineering , rational design ( RD ) is an umbrella term which invites the strategy of creating new molecules with a certain functionality, based upon the ability to predict how the molecule's structure (specifically derived from motifs ) will affect its behavior through physical models. This can be done either from scratch or by making calculated variations on a known structure, and usually complements directed evolution .
As an example, rational design is used to decipher collagen stability, mapping ligand-receptor interactions, unveiling protein folding and dynamics, and creating extra-biological structures by using fluorinated amino acids . [ 1 ] To treat cancer , rational design is used for targeted therapies where proteins are engineered to modify the communication of cells with their environment. [ 2 ] There is also the rational design of alfa-alkyl auxin molecules, which are auxin analogs capable of binding and blocking the formation of the hormone receptor complex. [ 3 ]
Other applications of rational design include:
This nanotechnology-related article is a stub . You can help Wikipedia by expanding it .
This biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rational_design |
A rational difference equation is a nonlinear difference equation of the form [ 1 ] [ 2 ] [ 3 ] [ 4 ]
where the initial conditions x 0 , x − 1 , … , x − k {\displaystyle x_{0},x_{-1},\dots ,x_{-k}} are such that the denominator never vanishes for any n .
A first-order rational difference equation is a nonlinear difference equation of the form
When a , b , c , d {\displaystyle a,b,c,d} and the initial condition w 0 {\displaystyle w_{0}} are real numbers , this difference equation is called a Riccati difference equation . [ 3 ]
Such an equation can be solved by writing w t {\displaystyle w_{t}} as a nonlinear transformation of another variable x t {\displaystyle x_{t}} which itself evolves linearly. Then standard methods can be used to solve the linear difference equation in x t {\displaystyle x_{t}} .
Equations of this form arise from the infinite resistor ladder problem. [ 5 ] [ 6 ]
One approach [ 7 ] to developing the transformed variable x t {\displaystyle x_{t}} , when a d − b c ≠ 0 {\displaystyle ad-bc\neq 0} , is to write
where α = ( a + d ) / c {\displaystyle \alpha =(a+d)/c} and β = ( a d − b c ) / c 2 {\displaystyle \beta =(ad-bc)/c^{2}} and where w t = y t − d / c {\displaystyle w_{t}=y_{t}-d/c} .
Further writing y t = x t + 1 / x t {\displaystyle y_{t}=x_{t+1}/x_{t}} can be shown to yield
This approach [ 8 ] gives a first-order difference equation for x t {\displaystyle x_{t}} instead of a second-order one, for the case in which ( d − a ) 2 + 4 b c {\displaystyle (d-a)^{2}+4bc} is non-negative. Write x t = 1 / ( η + w t ) {\displaystyle x_{t}=1/(\eta +w_{t})} implying w t = ( 1 − η x t ) / x t {\displaystyle w_{t}=(1-\eta x_{t})/x_{t}} , where η {\displaystyle \eta } is given by η = ( d − a + r ) / 2 c {\displaystyle \eta =(d-a+r)/2c} and where r = ( d − a ) 2 + 4 b c {\displaystyle r={\sqrt {(d-a)^{2}+4bc}}} . Then it can be shown that x t {\displaystyle x_{t}} evolves according to
The equation
can also be solved by treating it as a special case of the more general matrix equation
where all of A, B, C, E, and X are n × n matrices (in this case n = 1); the solution of this is [ 9 ]
where
It was shown in [ 10 ] that a dynamic matrix Riccati equation of the form
which can arise in some discrete-time optimal control problems, can be solved using the second approach above if the matrix C has only one more row than column. | https://en.wikipedia.org/wiki/Rational_difference_equation |
Rational egoism (also called rational selfishness ) is the principle that an action is rational if and only if it maximizes one's self-interest . [ 1 ] [ 2 ] As such, it is considered a normative form of egoism , [ 3 ] though historically it has been associated with both positive and normative forms. [ 4 ] In its strong form, rational egoism holds that to not pursue one's own interest is unequivocally irrational . Its weaker form, however, holds that while it is rational to pursue self-interest, failing to pursue self-interest is not always irrational. [ 5 ]
Originally an element of nihilist philosophy in Russia, it was later popularised in English-speaking countries by Russian-American author Ayn Rand .
Rational egoism ( Russian : разумный эгоизм ) emerged as the dominant social philosophy of the Russian nihilist movement , having developed in the works of nihilist philosophers Nikolay Chernyshevsky and Dmitry Pisarev . However, their terminology was largely obfuscated to avoid government censorship and the name rational egoism explicitly is unmentioned in the writings of both philosophers. [ 4 ] [ 6 ] Rational egoism was further embodied in Chernyshevsky's 1863 novel What Is to Be Done? , [ 7 ] and was criticised in response by Fyodor Dostoyevsky in his 1864 work Notes from Underground . For Chernyshevsky, rational egoism served as the basis for the socialist development of human society. [ 4 ] [ 8 ]
English philosopher Henry Sidgwick discussed rational egoism in his book The Methods of Ethics , first published in 1872. [ 9 ] A method of ethics is "any rational procedure by which we determine what individual human beings 'ought'—or what it is 'right' for them—to do, or seek to realize by voluntary action". [ 10 ] Sidgwick considers three such procedures, namely, rational egoism, dogmatic intuitionism, and utilitarianism . Rational egoism is the view that, if rational, "an agent regards quantity of consequent pleasure and pain to himself alone important in choosing between alternatives of action; and seeks always the greatest attainable surplus of pleasure over pain". [ 11 ]
Sidgwick found it difficult to find any persuasive reason for preferring rational egoism over utilitarianism . Although utilitarianism can be provided with a rational basis and reconciled with the morality of common sense, rational egoism appears to be an equally plausible doctrine regarding what we have most reason to do. Thus we must "admit an ultimate and fundamental contradiction in our apparent intuitions of what is Reasonable in conduct; and from this admission it would seem to follow that the apparently intuitive operation of Practical Reason, manifested in these contradictory judgments, is after all illusory". [ 12 ]
The author and philosopher Ayn Rand also discusses a theory that she called rational egoism . She holds that it is both irrational and immoral to act against one's self-interest. [ 13 ] Thus, her view is a conjunction of both rational egoism (in the standard sense) and ethical egoism , because according to Objectivist philosophy , egoism cannot be properly justified without an epistemology based on reason .
Her book The Virtue of Selfishness (1964) explains the concept of rational egoism in depth. According to Rand, a rational man holds his own life as his highest value, rationality as his highest virtue , and his happiness as the final purpose of his life.
Conversely, Rand was sharply critical of the ethical doctrine of altruism :
Do not confuse altruism with kindness, good will or respect for the rights of others. These are not primaries, but consequences, which, in fact, altruism makes impossible. The irreducible primary of altruism, the basic absolute is self-sacrifice —which means self-immolation, self-abnegation, self-denial self-destruction—which means the self as a standard of evil, the selfless as a standard of the good.
Do not hide behind such superficialities as whether you should or should not give a dime to a beggar. This is not the issue. The issue is whether you do or do not have the right to exist without giving him that dime. The issue is whether you must keep buying your life, dime by dime, from any beggar who might choose to approach you. The issue is whether the need of others is the first mortgage on your life and the moral purpose of your existence. The issue is whether man is to be regarded as a sacrificial animal. Any man of self-esteem will answer: No. Altruism says: Yes. [ 14 ]
Two objections to rational egoism are given by the English philosopher Derek Parfit , who discusses the theory at length in Reasons and Persons (1984). First, from the rational egoist point of view, it is rational to contribute to a pension scheme now, even though this is detrimental to one's present interests (which are to spend the money now). But it seems equally reasonable to maximize one's interests now, given that one's reasons are not only relative to him, but to him as he is now (and not his future self, who is argued to be a "different" person). Parfit also argues that since the connections between the present mental state and the mental state of one's future self may decrease, it is not plausible to claim that one should be indifferent between one's present and future self. [ 15 ] | https://en.wikipedia.org/wiki/Rational_egoism |
Rational fideism is the philosophical view that considers faith to be precursor for any reliable knowledge . Every paradigmatic system, whether one considers rationalism or empiricism , is based on axioms that are neither self-founding nor self-evident (see the Münchhausen trilemma ), so it appeals to assumptions accepted as belief (in reason or experience respectively). Thus, faith is basic to knowability. On the other hand, such a conclusion is reached not with an act of faith but with reasoning, a rational argumentation.
"Rational fideism" has been defined variously. The following are some definitions.
For Joseph Glanvill rational fideism is the view that "Faith, and faith alone, is the basis for our belief in our reason. We believe in our reason because we believe in God's veracity. We do not try to prove that God is truthful; we believe this. Thus, faith in God gives us faith in reason, which in turn "justifies" our belief that God is no deceiver." [ 1 ]
Richard Popkin sees rational fideism as the opposite of "pure, blind, fideism". [ 2 ]
Similarly, Domenic Marbaniang sees rational fideism as "the view that the knowledge of God can be certified through faith alone that is based on a revelation that is rationally verified." [ 3 ] Observing that the way of both rationalism and empiricism towards the knowledge of ultimate or transcendent reality is bleak, he thinks that while fideism is the view that truth in religion rests solely on faith and not on a reasoning process, rational fideism "holds that truth in religion rests solely on faith; not blind faith, but faith that can give rational and cogent answers or reason to warrant the belief." [ 3 ]
According to C. Stephen Evans , rational fideism involves the possibility of reason becoming self-critical. Seeing it as the kind of responsible fideism, he states, "If human reason has limitations and also has some ability to recognise those limitations, then the possibility of responsible fideism emerges." [ 4 ] Evans states that not only does reason have limitations, it is also tainted by sin making one entitled to faith where reason fails. [ 5 ]
Patrick J. Clarke defines rational fideism as the approach that sees "reason as capable of providing the intellectual foundation of faith, not a priori but a posteriori , much as philosophy provides an intellectual foundation to theology." [ 6 ]
Brendan Sweetman notes a type of rational fideism as a view developed by some thinkers who hold that the pragmatic spiritual and moral success of believing in God on faith alone could be used as an "indirect argument for the truth of fideism." [ 7 ]
R.C. Sproul, John Gerstner, and Arthur Lindsley argue that "Rational fideism is a contradiction in terms" since requiring a criterion for belief is non-fideistic as fideism begins with belief; that is, fideistic belief proceeds from faith rather than reaching faith as a result of rational query. [ 8 ] | https://en.wikipedia.org/wiki/Rational_fideism |
Rational ignorance is refraining from acquiring knowledge when the supposed cost of educating oneself on an issue exceeds the expected potential benefit that the knowledge would provide.
Ignorance about an issue is said to be "rational" when the cost of educating oneself about the issue sufficiently to make an informed decision can outweigh any potential benefit one could reasonably expect to gain from that decision, and so it would be irrational to spend time doing so. This has consequences for the quality of decisions made by large numbers of people, such as in general elections , where the probability of any one vote changing the outcome is very small.
The term is most often found in economics , particularly public choice theory , but also used in other disciplines which study rationality and choice, including philosophy ( epistemology ) and game theory .
The term was coined by Anthony Downs in An Economic Theory of Democracy . [ 1 ]
Consider an employer attempting to choose between two candidates offering to complete a task at the cost of $10/hour. The length of time needed to complete the task may be longer or shorter depending on the skill of the person performing the task, so it is in the employer's best interests to find the fastest worker possible. Assume that the cost of another day of interviewing the candidates is $100. If the employer had deduced from the interviews so far that both candidates would complete the task in somewhere between 195 and 205 hours, it would be in the employer's best interests to choose one or the other by some easily applied metric (for example, flipping a coin) rather than spend the $100 on determining the better candidate, saving at most $100 in labor. In many cases, the decision may be made on the basis of heuristics ; a simple decision model which may not be completely accurate. For example, in deciding which brand of prepared food is most nutritious, a shopper might simply choose the one with (for example) the lowest amount of sugar, rather than conducting a research study of all the positive and negative factors in nutrition.
Marketers can take advantage of rational ignorance by increasing the complexity of a decision. If the difference in value between a quality product and a poor product is less than the cost to perform the research necessary to differentiate between them, then it is more rational for a consumer to just take his chances on whichever of the two is more convenient and available. Thus, it is in the interest of the producer of a lower value product to proliferate features, options, and package combinations which will tend to increase the number of shoppers who decide it's too much trouble to make an informed decision.
Politics and elections especially display the same dynamic. By increasing the number of issues that a person needs to consider to make a rational decision about candidates or policies, politicians and pundits encourage single-issue voting, party-line voting, jingoism , selling votes, or dart-throwing all of which may tip the playing field in favor of politicians who do not actually represent the electorate.
This does not mean that voters make poor and biased decisions: rather that in carrying out their everyday responsibilities (like working and taking care of a family), many people do not have the time to devote to researching every aspect of a candidate's policies. So many people find themselves making rational decisions meaning they let others who are more versed in the subject do the research and they form their opinion based on the evidence provided. They are being rationally ignorant not because they don't care but because they simply do not have the time.
Because the cost/benefit ratio increases with increasing costs or decreasing the benefit, the same effect can occur when politicians protect their policy decisions from the preferences of the public. To the degree that the electorate perceives their individual votes to count for less, they will have less incentive to spend any time actually learning any details about the candidate(s).
A more nuanced example occurs when a voter identifies with a particular political party, akin to the adoption of a favorite movie critic. Based on prior experience a responsible voter will seek politicians or a political party that draws conclusions about social policy that are similar to what their own conclusions would have been had they done a complete analysis. But when voters find themselves agreeing with the same party or politician across a number of election cycles, many voters simply trust that the same will continue to be true and "vote the ticket," also referred to as straight-ticket voting, instead of wasting time on a complete investigation.
Much of the empirical support for the idea of rational ignorance was drawn from studies of voter apathy , which reached particularly strong conclusions in the 1950s. [ 2 ] However, apathy appeared to decline sharply in the 1960s as concern about issues such as the Vietnam War mounted, and political polarization increased. [ 3 ] This is consistent with expectations from Public Choice Theory; as voters' interest in the results of policy decisions increase, the perceived benefit of the analysis (or the trip to the ballot box) increases, so more people will consider it rational to repair their ignorance.
There also may be situations when "the individual may perceive the situation as one that has carry over benefits to other situations, and treat the learning as a capital investment with payoff beyond the specific situation in which it is presented," and not a waste of time even though the time invested in the learning my not have immediate payoff. (Denzou and North, 1994). [ 4 ]
Additionally, rational ignorance is scrutinized for its broadening effect on the decisions that individuals make in different matters. The investment of time and energy on learning about the specified subject has ramifications on other decision areas. Individuals sometimes ignore this when unconsciously assessing the investment cost versus payout. The external benefits of acquiring knowledge in one area—those benefits occurring in other decision areas—are therefore subject to being overlooked. | https://en.wikipedia.org/wiki/Rational_ignorance |
The concept known as rational irrationality was popularized by economist Bryan Caplan in 2001 to reconcile the widespread existence of irrational behavior (particularly in the realms of religion and politics ) with the assumption of rationality made by mainstream economics and game theory . [ 1 ] [ 2 ] The theory, along with its implications for democracy, was expanded upon by Caplan in his book The Myth of the Rational Voter .
The original purpose of the concept was to explain how (allegedly) detrimental policies could be implemented in a democracy, and, unlike conventional public choice theory , Caplan posited that bad policies were selected by voters themselves. The theory has also been embraced by the ethical intuitionist philosopher Michael Huemer as an explanation for irrationality in politics. [ 3 ] [ 4 ] The theory has also been applied to explain religious belief . [ 5 ]
Caplan posits that there are two types of rationality:
Rational irrationality describes a situation in which it is instrumentally rational for an actor to be epistemically irrational.
Caplan argues that rational irrationality is more likely in situations in which:
In the framework of neoclassical economics , Caplan posits that there is a demand for irrationality . A person's demand curve describes the amount of irrationality that the person is willing to tolerate at any given cost of irrationality. By the law of demand , the lower the cost of irrationality, the higher the demand for it. When the cost of error is effectively zero, a person's demand for irrationality is high.
Rational irrationality is not doublethink and does not state that the individual deliberately chooses to believe something he or she knows to be false. Rather, the theory is that when the costs of having erroneous beliefs are low, people relax their intellectual standards and allow themselves to be more easily influenced by fallacious reasoning , cognitive biases , and emotional appeals . In other words, people do not deliberately seek to believe false things but stop putting in the intellectual effort to be open to evidence that may contradict their beliefs.
For rational irrationality to exist, people must have preferences over beliefs: certain beliefs must be appealing to people for reasons other than their truth value. In an essay on irrationality in politics Michael Huemer [ 3 ] identifies some possible sources of preferences over beliefs:
Many of the claims of religions are not easily verifiable in the day-to-day world. There are many competing religious theories about the origins of life , reincarnation , and paradise , but mistaken beliefs about these rarely impose real world costs upon the believers themselves. Thus, it may be instrumentally rational to be epistemically irrational about these matters. In other words, when forming or updating their religious beliefs, people may tend to relax their intellectual standards for the sake of driving popular support towards their beliefs. [ 5 ]
Politics is a situation where rational irrationality is expected to be common, according to Caplan's theory. In typical large democracies , each individual voter has a very low probability of influencing the outcome of an election or determining whether a particular policy will be implemented. Thus, the expected cost of supporting an erroneous policy (obtained by multiplying the cost of the policy by the probability that the individual voter will have a decisive role in influencing the policy) is very low. The psychological benefits of supporting policies that feel good but are in fact harmful may be greater than these small expected costs. This creates a situation where voters may be rationally irrational for practical morale reasons.
For rational irrationality at an individual level to have an effect on political outcomes, it is necessary that there be systemic ways in which people are irrational. In other words, people need to have systemic biases : there needs to be a systemic difference between people's preferences over beliefs and true beliefs. In the absence of systemic biases, different forms of irrationality would cancel out when aggregated using the voting process.
Caplan attempts to demonstrate empirically the existence of systemic biases in beliefs about economics in his book The Myth of the Rational Voter .
When a large number of individuals hold systematically biased beliefs, the total cost to the democracy of all these irrational beliefs could be significant. Thus, even though every individual voter may be behaving rationally, the voters as a whole are not acting in their collective self-interest. This is analogous to the tragedy of the commons . Another way of thinking about it is that each voter, by being rationally irrational, creates a small negative externality for other voters.
Caplan believes that the rational irrationality of voters is one of the reasons why democracies choose suboptimal economic policies, particularly in the area of free trade versus protectionism . Philosopher Michael Huemer , in a TEDx talk on rational irrationality in politics, cited the war on terror and protectionism as two examples of rational irrationality in politics. [ 4 ]
Any theory of democracy must take into account the empirical fact that most voters in a democracy have very little idea about the details of politics, including the names of their elected representatives, the terms of office, and the platforms of candidates of major political parties.
Like rational irrationality, some theories of democracy claim that democracies tend to choose bad policies. Other theories claim that despite the empirical observations about voter ignorance, democracies do in fact do fairly well. Below are listed some of these theories and their relation to rational irrationality.
The most famous theory of democratic failure is public choice theory . The theory, developed by James Buchanan , Gordon Tullock , and others, relies on rational ignorance . Voters have a very small probability of influencing policy outcomes, so they do not put much effort to stay up-to-date on politics. This allows special interests to manipulate the political process and engage in rent seeking . A key idea of public choice theory is that many harmful policies have concentrated benefits (experienced by special interests) and diffuse costs . The special interests experiencing the benefits are willing to lobby for the policies, while the costs are spread out very diffusely among a much larger group of people. Because these costs are diffuse, the people bearing the costs do not have enough at stake to lobby against the policies.
Rational irrationality and rational ignorance share some key similarities but are also different in a number of ways. The similarities are that both theories reject the claim that voters are rational and well-informed, and both theories claim that democracy does not function well. However, the theories differ in a number of ways:
There are two main objections to public choice theory and rational ignorance that do not apply to rational irrationality:
Brennan and Lomasky have an alternative theory of democratic failure that is quite similar to Caplan's theory of rational irrationality. Their theory, called expressive voting , states that people vote to express certain beliefs. The key difference between expressive voting and rational irrationality is that the former does not require people to actually hold systematically biased beliefs, while the latter does.
Loren Lomasky, one of the proponents of expressive voting, explained some of the key differences between the theories in a critical review of Caplan's book. [ 6 ]
Donald Wittman has argued that democracy works well. [ 7 ] Wittman's argument rests on raising a number of objections to public choice theory, such as those outlined above while contrasting public choice theory and rational irrationality. Caplan described his own work on rational irrationality as an attempt to rescue democratic failure from Wittman's attacks. After the publication of Caplan's book, Wittman and Caplan debated each other. [ 8 ] | https://en.wikipedia.org/wiki/Rational_irrationality |
Rational mysticism , which encompasses both rationalism and mysticism , is a term used by scholars , researchers , and other intellectuals , some of whom engage in studies of how altered states of consciousness or transcendence such as trance , visions , and prayer occur. Lines of investigation include historical and philosophical inquiry as well as scientific inquiry within such fields as neurophysiology and psychology .
The term "rational mysticism" was in use at least as early as 1911 when it was the subject of an article by Henry W. Clark in the Harvard Theological Review . [ 1 ] In a 1924 book, Rational Mysticism, theosophist William Kingsland correlated rational mysticism with scientific idealism . [ 2 ] [ 3 ] South African philosopher J. N. Findlay frequently used the term, developing the theme in Ascent to the Absolute and other works in the 1960s and 1970s. [ 4 ]
Columbia University pragmatist John Herman Randall, Jr. characterized both Plotinus and Baruch Spinoza as “rationalists with overtones of rational mysticism” in his 1970 book Hellenistic Ways of Deliverance and the Making of Christian Synthesis. [ 5 ] Rice University professor of religious studies Jeffrey J. Kripal , in his 2001 book Roads of Excess, Palaces of Wisdom, defined rational mysticism as “not a contradiction in terms” but “a mysticism whose limits are set by reason.” [ 6 ]
In response to criticism of his book The End of Faith , author Sam Harris used the term rational mysticism for the title of his rebuttal. [ 7 ] [ 8 ] [ 9 ] [ 10 ] University of Pennsylvania neurotheologist Andrew Newberg has been using nuclear medicine brain imaging in similar research since the early 1990s. [ 11 ] [ 12 ]
Executive editor of Discover magazine Corey S. Powell, in his 2002 book, God in the Equation, attributed the term to Albert Einstein : “In creating his radical cosmology, Einstein stitched together a rational mysticism, drawing on—but distinct from—the views that came before.” [ 13 ]
Science writer John Horgan interviewed and profiled James Austin , Terence McKenna , Michael Persinger , Christian Rätsch , Huston Smith , Ken Wilber , Alexander Shulgin and others for Rational Mysticism: Dispatches from the Border Between Science and Spirituality, [ 14 ] his 2003 study of “the scientific quest to explain the transcendent.” [ 15 ] | https://en.wikipedia.org/wiki/Rational_mysticism |
In mathematics , a rational number is a number that can be expressed as the quotient or fraction p q {\displaystyle {\tfrac {p}{q}}} of two integers , a numerator p and a non-zero denominator q . [ 1 ] For example, 3 7 {\displaystyle {\tfrac {3}{7}}} is a rational number, as is every integer (for example, − 5 = − 5 1 {\displaystyle -5={\tfrac {-5}{1}}} ). The set of all rational numbers, also referred to as " the rationals ", [ 2 ] the field of rationals [ 3 ] or the field of rational numbers is usually denoted by boldface Q , or blackboard bold Q . {\displaystyle \mathbb {Q} .}
A rational number is a real number . The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: 3/4 = 0.75 ), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545... ). [ 4 ] This statement is true not only in base 10 , but also in every other integer base , such as the binary and hexadecimal ones (see Repeating decimal § Extension to other bases ).
A real number that is not rational is called irrational . [ 5 ] Irrational numbers include the square root of 2 ( 2 {\displaystyle {\sqrt {2}}} ), π , e , and the golden ratio ( φ ). Since the set of rational numbers is countable , and the set of real numbers is uncountable , almost all real numbers are irrational. [ 1 ]
Rational numbers can be formally defined as equivalence classes of pairs of integers ( p, q ) with q ≠ 0 , using the equivalence relation defined as follows:
The fraction p q {\displaystyle {\tfrac {p}{q}}} then denotes the equivalence class of ( p, q ) . [ 6 ]
Rational numbers together with addition and multiplication form a field which contains the integers , and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field , and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of Q {\displaystyle \mathbb {Q} } are called algebraic number fields , and the algebraic closure of Q {\displaystyle \mathbb {Q} } is the field of algebraic numbers . [ 7 ]
In mathematical analysis , the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion , using Cauchy sequences , Dedekind cuts , or infinite decimals (see Construction of the real numbers ).
In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, to avoid confusion between " rational expression " and " rational function " (a polynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a rational curve is not a curve defined over the rationals, but a curve which can be parameterized by rational functions.
Although nowadays rational numbers are defined in terms of ratios , the term rational is not a derivation of ratio . On the contrary, it is ratio that is derived from rational : the first use of ratio with its modern meaning was attested in English about 1660, [ 8 ] while the use of rational for qualifying numbers appeared almost a century earlier, in 1570. [ 9 ] This meaning of rational came from the mathematical meaning of irrational , which was first used in 1551, and it was used in "translations of Euclid (following his peculiar use of ἄλογος )". [ 10 ] [ 11 ]
This unusual history originated in the fact that ancient Greeks "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers". [ 12 ] So such lengths were irrational , in the sense of illogical , that is "not to be spoken about" ( ἄλογος in Greek). [ 13 ]
Every rational number may be expressed in a unique way as an irreducible fraction a b , {\displaystyle {\tfrac {a}{b}},} where a and b are coprime integers and b > 0 . This is often called the canonical form of the rational number.
Starting from a rational number a b , {\displaystyle {\tfrac {a}{b}},} its canonical form may be obtained by dividing a and b by their greatest common divisor , and, if b < 0 , changing the sign of the resulting numerator and denominator.
Any integer n can be expressed as the rational number n 1 , {\displaystyle {\tfrac {n}{1}},} which is its canonical form as a rational number.
If both fractions are in canonical form, then:
If both denominators are positive (particularly if both fractions are in canonical form):
On the other hand, if either denominator is negative, then each fraction with a negative denominator must first be converted into an equivalent form with a positive denominator—by changing the signs of both its numerator and denominator. [ 6 ]
Two fractions are added as follows:
If both fractions are in canonical form, the result is in canonical form if and only if b, d are coprime integers . [ 6 ] [ 14 ]
If both fractions are in canonical form, the result is in canonical form if and only if b, d are coprime integers . [ 14 ]
The rule for multiplication is:
where the result may be a reducible fraction —even if both original fractions are in canonical form. [ 6 ] [ 14 ]
Every rational number a b {\displaystyle {\tfrac {a}{b}}} has an additive inverse , often called its opposite ,
If a b {\displaystyle {\tfrac {a}{b}}} is in canonical form, the same is true for its opposite.
A nonzero rational number a b {\displaystyle {\tfrac {a}{b}}} has a multiplicative inverse , also called its reciprocal ,
If a b {\displaystyle {\tfrac {a}{b}}} is in canonical form, then the canonical form of its reciprocal is either b a {\displaystyle {\tfrac {b}{a}}} or − b − a , {\displaystyle {\tfrac {-b}{-a}},} depending on the sign of a .
If b, c, d are nonzero, the division rule is
Thus, dividing a b {\displaystyle {\tfrac {a}{b}}} by c d {\displaystyle {\tfrac {c}{d}}} is equivalent to multiplying a b {\displaystyle {\tfrac {a}{b}}} by the reciprocal of c d : {\displaystyle {\tfrac {c}{d}}:} [ 14 ]
If n is a non-negative integer, then
The result is in canonical form if the same is true for a b . {\displaystyle {\tfrac {a}{b}}.} In particular,
If a ≠ 0 , then
If a b {\displaystyle {\tfrac {a}{b}}} is in canonical form, the canonical form of the result is b n a n {\displaystyle {\tfrac {b^{n}}{a^{n}}}} if a > 0 or n is even. Otherwise, the canonical form of the result is − b n − a n . {\displaystyle {\tfrac {-b^{n}}{-a^{n}}}.}
A finite continued fraction is an expression such as
where a n are integers. Every rational number a b {\displaystyle {\tfrac {a}{b}}} can be represented as a finite continued fraction, whose coefficients a n can be determined by applying the Euclidean algorithm to ( a, b ) .
are different ways to represent the same rational value.
The rational numbers may be built as equivalence classes of ordered pairs of integers . [ 6 ] [ 14 ]
More precisely, let ( Z × ( Z ∖ { 0 } ) ) {\displaystyle (\mathbb {Z} \times (\mathbb {Z} \setminus \{0\}))} be the set of the pairs ( m, n ) of integers such n ≠ 0 . An equivalence relation is defined on this set by
Addition and multiplication can be defined by the following rules:
This equivalence relation is a congruence relation , which means that it is compatible with the addition and multiplication defined above; the set of rational numbers Q {\displaystyle \mathbb {Q} } is the defined as the quotient set by this equivalence relation, ( Z × ( Z ∖ { 0 } ) ) / ∼ , {\displaystyle (\mathbb {Z} \times (\mathbb {Z} \backslash \{0\}))/\sim ,} equipped with the addition and the multiplication induced by the above operations. (This construction can be carried out with any integral domain and produces its field of fractions .) [ 6 ]
The equivalence class of a pair ( m, n ) is denoted m n . {\displaystyle {\tfrac {m}{n}}.} Two pairs ( m 1 , n 1 ) and ( m 2 , n 2 ) belong to the same equivalence class (that is are equivalent) if and only if
This means that
if and only if [ 6 ] [ 14 ]
Every equivalence class m n {\displaystyle {\tfrac {m}{n}}} may be represented by infinitely many pairs, since
Each equivalence class contains a unique canonical representative element . The canonical representative is the unique pair ( m, n ) in the equivalence class such that m and n are coprime , and n > 0 . It is called the representation in lowest terms of the rational number.
The integers may be considered to be rational numbers identifying the integer n with the rational number n 1 . {\displaystyle {\tfrac {n}{1}}.}
A total order may be defined on the rational numbers, that extends the natural order of the integers. One has
If
The set Q {\displaystyle \mathbb {Q} } of all rational numbers, together with the addition and multiplication operations shown above, forms a field . [ 6 ]
Q {\displaystyle \mathbb {Q} } has no field automorphism other than the identity. (A field automorphism must fix 0 and 1; as it must fix the sum and the difference of two fixed elements, it must fix every integer; as it must fix the quotient of two fixed elements, it must fix every rational number, and is thus the identity.)
Q {\displaystyle \mathbb {Q} } is a prime field , which is a field that has no subfield other than itself. [ 15 ] The rationals are the smallest field with characteristic zero. Every field of characteristic zero contains a unique subfield isomorphic to Q . {\displaystyle \mathbb {Q} .}
With the order defined above, Q {\displaystyle \mathbb {Q} } is an ordered field [ 14 ] that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique subfield isomorphic to Q . {\displaystyle \mathbb {Q} .}
Q {\displaystyle \mathbb {Q} } is the field of fractions of the integers Z . {\displaystyle \mathbb {Z} .} [ 16 ] The algebraic closure of Q , {\displaystyle \mathbb {Q} ,} i.e. the field of roots of rational polynomials, is the field of algebraic numbers .
The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. [ 6 ] For example, for any two fractions such that
(where b , d {\displaystyle b,d} are positive), we have
Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers. [ 17 ]
The set of positive rational numbers is countable , as is illustrated in the figure.
More precisely, one can sort the fractions by increasing values of the sum of the numerator and the denominator, and, for equal sums, by increasing numerator or denominator. This produces a sequence of fractions, from which one can remove the reducible fractions (in red on the figure), for getting a sequence that contains each rational number exactly once. This establishes a bijection between the rational numbers and the natural numbers, which maps each rational number to its rank in the sequence.
A similar method can be used for numbering all rational numbers (positive and negative).
As the set of all rational numbers is countable, and the set of all real numbers (as well as the set of irrational numbers) is uncountable, the set of rational numbers is a null set , that is, almost all real numbers are irrational, in the sense of Lebesgue measure . [ 18 ]
The rationals are a dense subset of the real numbers ; every real number has rational numbers arbitrarily close to it. [ 6 ] A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions . [ 19 ]
In the usual topology of the real numbers, the rationals are neither an open set nor a closed set . [ 20 ]
By virtue of their order, the rationals carry an order topology . The rational numbers, as a subspace of the real numbers, also carry a subspace topology . The rational numbers form a metric space by using the absolute difference metric d ( x , y ) = | x − y | , {\displaystyle d(x,y)=|x-y|,} and this yields a third topology on Q . {\displaystyle \mathbb {Q} .} All three topologies coincide and turn the rationals into a topological field . The rational numbers are an important example of a space which is not locally compact . The rationals are characterized topologically as the unique countable metrizable space without isolated points . The space is also totally disconnected . The rational numbers do not form a complete metric space , and the real numbers are the completion of Q {\displaystyle \mathbb {Q} } under the metric d ( x , y ) = | x − y | {\displaystyle d(x,y)=|x-y|} above. [ 14 ]
In addition to the absolute value metric mentioned above, there are other metrics which turn Q {\displaystyle \mathbb {Q} } into a topological field:
Let p be a prime number and for any non-zero integer a , let | a | p = p − n , {\displaystyle |a|_{p}=p^{-n},} where p n is the highest power of p dividing a .
In addition set | 0 | p = 0. {\displaystyle |0|_{p}=0.} For any rational number a b , {\displaystyle {\frac {a}{b}},} we set
Then
defines a metric on Q . {\displaystyle \mathbb {Q} .} [ 21 ]
The metric space ( Q , d p ) {\displaystyle (\mathbb {Q} ,d_{p})} is not complete, and its completion is the p -adic number field Q p . {\displaystyle \mathbb {Q} _{p}.} Ostrowski's theorem states that any non-trivial absolute value on the rational numbers Q {\displaystyle \mathbb {Q} } is equivalent to either the usual real absolute value or a p -adic absolute value. | https://en.wikipedia.org/wiki/Rational_number |
The rational planning model is a model of the planning process involving a number of rational actions or steps. Taylor (1998) outlines five steps, as follows: [ 1 ]
The rational planning model is used in planning and designing neighborhoods, cities, and regions. It has been central in the development of modern urban planning and transportation planning . The model has many limitations, particularly the lack of guidance on involving stakeholders and the community affected by planning, and other models of planning, such as collaborative planning , are now also widely used.
The very similar rational decision-making model , as it is called in organizational behavior , is a process for making logically sound decisions. [ 2 ] This multi-step model and aims to be logical and follow the orderly path from problem identification through solution. Rational decision making is a multi-step process for making logically sound decisions that aims to follow the orderly path from problem identification through solution.
Rational decision-making or planning follows a series of steps detailed below:
Verifying, defining & detailing the problem (problem definition, goal definition, information gathering). This step includes recognizing the problem, defining an initial solution, and starting primary analysis. Examples of this are creative devising, creative ideas, inspirations, breakthroughs, and brainstorms .
The very first step which is normally overlooked by the top level management is defining the exact problem. Though we think that the problem identification is obvious, many times it is not. When defining the problem situation, framing is essential part of the process. With correct framing, the situation is identified and possible previous experience with same kind of situation can be utilized. The rational decision making model is a group-based decision making process. If the problem is not identified properly then we may face a problem as each and every member of the group might have a different definition of the problem.
This step encloses two to three final solutions to the problem and preliminary implementation to the site. In planning, examples of this are Planned Units of Development and downtown revitalizations.
This activity is best done in groups, as different people may contribute different ideas or alternative solutions to the problem. Without alternative solutions, there is a chance of arriving at a non-optimal or a rational decision. For exploring the alternatives it is necessary to gather information. Technology may help with gathering this information.
Evaluative criteria are measurements to determine success and failure of alternatives. This step contains secondary and final analysis along with secondary solutions to the problem. Examples of this are site suitability and site sensitivity analysis. After going thoroughly through the process of defining the problem, exploring for all the possible alternatives for that problem and gathering information this step says evaluate the information and the possible options to anticipate the consequences of each and every possible alternative that is thought of. At this point optional criteria for measuring the success or failure of the decision taken needs to be considered.
The rational model of planning rest largely on objective assessment.
This step comprises a final solution and secondary implementation to the site. At this point the process has developed into different strategies of how to apply the solutions to the site.
Based on the criteria of assessment and the analysis done in previous steps, choose the best solution generated. These four steps form the core of the Rational Decision Making Model.
This step includes final implementation to the site and preliminary monitoring of the outcome and results of the site. This step is the building/renovations part of the process.
Modify future decisions and actions taken based on the above evaluation of outcomes. [ 3 ]
The rational model of decision-making is a process for making sound decisions in policy making in the public sector. Rationality is defined as “a style of behavior that is appropriate to the achievement of given goals, within the limits imposed by given conditions and constraints”. [ 4 ] It is important to note the model makes a series of assumptions in order for it to work, such as:
Indeed, some of the assumptions identified above are also pin pointed out in a study written by the historian H.A. Drake, as he states:
In its purest form, the Rational Actor approach presumes that such a figure [as Constantine] has complete freedom of action to achieve goals that he or she has articulated through a careful process of rational analysis involving full and objective study of all pertinent information and alternatives. At the same time, it presumes that this central actor is so fully in control of the apparatus of government that a decision once made is as good as implemented. There are no staffs on which to rely, no constituencies to placate, no generals or governors to cajole. By attributing all decision making to one central figure who is always fully in control and who acts only after carefully weighing all options, the Rational Actor method allows scholars to filter out extraneous details and focus attention on central issues . [ 5 ]
Furthermore, as we have seen, in the context of policy rational models are intended to achieve maximum social gain. For this purpose, Simon identifies an outline of a step by step mode of analysis to achieve rational decisions. Ian Thomas describes Simon's steps as follows:
In similar lines, Wiktorowicz and Deber describe through their study on ‘Regulating biotechnology: a rational-political model of policy development’ the rational approach to policy development. The main steps involved in making a rational decision for these authors are the following:
The approach of Wiktorowicz and Deber is similar to Simon and they assert that the rational model tends to deal with “the facts” (data, probabilities) in steps 1 to 3, leaving the issue of assessing values to the final step. According to Wiktorowicz and Deber values are introduced in the final step of the rational model, where the utility of each policy option is assessed.
Many authors have attempted to interpret the above-mentioned steps, amongst others, Patton and Sawicki [ 8 ] who summarize the model as presented in the following figure (missing):
The model of rational decision-making has also proven to be very useful to several decision making processes in industries outside the public sphere. Nonetheless, many criticisms of the model arise due to claim of the model being impractical and lying on unrealistic assumptions. For instance, it is a difficult model to apply in the public sector because social problems can be very complex, ill-defined and interdependent. The problem lies in the thinking procedure implied by the model which is linear and can face difficulties in extra ordinary problems or social problems which have no sequences of happenings. This latter argument can be best illustrated by the words of Thomas R. Dye, the president of the Lincoln Center for Public Service, who wrote in his book `Understanding Public Policy´ the following passage:
There is no better illustration of the dilemmas of rational policy making in America than in the field of health…the first obstacle to rationalism is defining the problem. Is our goal to have good health — that is, whether we live at all (infant mortality), how well we live (days lost to sickness), and how long we live (life spans and adult mortality)? Or is our goal to have good medical care — frequent visits to the doctor, wellequipped and accessible hospitals, and equal access to medical care by rich and poor alike? [ 9 ]
The problems faced when using the rational model arise in practice because social and environmental values can be difficult to quantify and forge consensus around. [ 10 ] Furthermore, the assumptions stated by Simon are never fully valid in a real world context.
However, as Thomas states the rational model provides a good perspective since in modern society rationality plays a central role and everything that is rational tends to be prized. Thus, it does not seem strange that “we ought to be trying for rational decision-making”. [ 6 ]
As illustrated in Figure 1, rational policy analysis can be broken into 6 distinct stages of analysis. Step 2 highlights the need to understand which factors should be considered as part of the decision making process. At this part of the process, all the economic, social, and environmental factors that are important to the policy decision need to be identified and then expressed as policy decision criteria. For example, the decision criteria used in the analysis of environmental policy is often a mix of —
Some criteria, such as economic benefit, will be more easily measurable or definable, while others such as environmental quality will be harder to measure or express quantitatively. Ultimately though, the set of decision criteria needs to embody all of the policy goals, and overemphasising the more easily definable or measurable criteria, will have the undesirable impact of biasing the analysis towards a subset of the policy goals. [ 12 ]
The process of identifying a suitably comprehensive decision criteria set is also vulnerable to being skewed by pressures arising at the political interface. For example, decision makers may tend to give " more weight to policy impacts that are concentrated, tangible, certain, and immediate than to impacts that are diffuse, intangible, uncertain, and delayed ."^8. For example, with a cap-and-trade system for carbon emissions the net financial cost in the first five years of policy implementation is a far easier impact to conceptualise than the more diffuse and uncertain impact of a country's improved position to influence global negotiations on climate change action .
Displaying the impacts of policy alternatives can be done using a policy analysis matrix (PAM) such that shown in Table 1. As shown, a PAM provides a summary of the policy impacts for the various alternatives and examination of the matrix can reveal the tradeoffs associated with the different alternatives.
Table 1. Policy analysis matrix (PAM) for SO2 emissions control.
Once policy alternatives have been evaluated, the next step is to decide which policy alternative should be implemented. This is shown as step 5 in Figure 1. At one extreme, comparing the policy alternatives can be relatively simple if all the policy goals can be measured using a single metric and given equal weighting. In this case, the decision method is an exercise in benefit cost analysis (BCA).
At the other extreme, the numerous goals will require the policy impacts to be expressed using a variety of metrics that are not readily comparable. In such cases, the policy analyst may draw on the concept of utility to aggregate the various goals into a single score. With the utility concept, each impact is given a weighting such that 1 unit of each weighted impact is considered to be equally valuable (or desirable) with regards to the collective well-being.
Weimer and Vining also suggest that the " go, no go " rule can be a useful method for deciding amongst policy alternatives^8. Under this decision making regime, some or all policy impacts can be assigned thresholds which are used to eliminate at least some of the policy alternatives. In their example, one criterion " is to minimize SO2 emissions " and so a threshold might be a reduction SO2 emissions "of at least 8.0 million tons per year". As such, any policy alternative that does not meet this threshold can be removed from consideration. If only a single policy alternative satisfies all the impact thresholds then it is the one that is considered a "go" for each impact. Otherwise it might be that all but a few policy alternatives are eliminated and those that remain need to be more closely examined in terms of their trade-offs so that a decision can be made.
To demonstrate the rational analysis process as described above, let’s examine the policy paper “Stimulating the use of biofuels in the European Union: Implications for climate change policy” by Lisa Ryan where the substitution of fossil fuels with biofuels has been proposed in the European Union (EU) between 2005–2010 as part of a strategy to mitigate greenhouse gas emissions from road transport, increase security of energy supply and support development of rural communities.
Considering the steps of Patton and Sawicki model as in Figure 1 above, this paper only follows components 1 to 5 of the rationalist policy analysis model:
However, there are a lot of assumptions, requirements without which the rational decision model is a failure. Therefore, they all have to be considered.
The model assumes that we have or should or can obtain adequate information, both in terms of quality, quantity and accuracy. This applies to the situation as well as the alternative technical situations. It further assumes that you have or should or can obtain substantive knowledge of the cause and effect relationships relevant to the evaluation of the alternatives. In other words, it assumes that you have a thorough knowledge of all the alternatives and the consequences of the alternatives chosen. It further assumes that you can rank the alternatives and choose the best of it.
The following are the limitations for the Rational Decision Making Model:
While the rational planning model was innovative at its conception, the concepts are controversial and questionable processes today. The rational planning model has fallen out of mass use as of the last decade. Rather than conceptualising human agents as rational planners, Lucy Suchman argues, agents can better be understood as engaging in situated action . [ 14 ] Going further, Guy Benveniste argued that the rational model could not be implemented without taking the political context into account. [ 15 ] | https://en.wikipedia.org/wiki/Rational_planning_model |
In mathematics , in the representation theory of algebraic groups , a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map of algebraic varieties.
Finite direct sums and products of rational representations are rational.
A rational G {\displaystyle G} module is a module that can be expressed as a sum (not necessarily direct) of rational representations.
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rational_representation |
In mathematics and computer science , a rational series is a generalisation of the concept of formal power series over a ring to the case when the basic algebraic structure is no longer a ring but a semiring , and the indeterminates adjoined are not assumed to commute . They can be regarded as algebraic expressions of a formal language over a finite alphabet .
Let R be a semiring and A a finite alphabet.
A non-commutative polynomial over A is a finite formal sum of words over A . They form a semiring R ⟨ A ⟩ {\displaystyle R\langle A\rangle } .
A formal series is a R -valued function c , on the free monoid A * , which may be written as
The set of formal series is denoted R ⟨ ⟨ A ⟩ ⟩ {\displaystyle R\langle \langle A\rangle \rangle } and becomes a semiring under the operations
A non-commutative polynomial thus corresponds to a function c on A * of finite support .
In the case when R is a ring, then this is the Magnus ring over R . [ 1 ]
If L is a language over A , regarded as a subset of A * we can form the characteristic series of L as the formal series
corresponding to the characteristic function of L .
In R ⟨ ⟨ A ⟩ ⟩ {\displaystyle R\langle \langle A\rangle \rangle } one can define an operation of iteration expressed as
and formalised as
The rational operations are the addition and multiplication of formal series, together with iteration.
A rational series is a formal series obtained by rational operations from R ⟨ A ⟩ . {\displaystyle R\langle A\rangle .}
This abstract algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rational_series |
Rational thermodynamics is a school of thought in statistical thermodynamics developed in the 1960s.
Its introduction is attributed to Clifford Truesdell , Bernard Coleman [ fr ] and Walter Noll . [ 1 ] [ 2 ] The aim was to develop a mathematical model of thermodynamics that would go beyond the traditional "thermodynamics of irreversible processes " or TIP developed in the late 19th to early 20th centuries.
Truesdell's "flamboyant style" and "satirical verve" caused controversy between "rational thermodynamics" and proponents of traditional thermodynamics. [ 3 ] | https://en.wikipedia.org/wiki/Rational_thermodynamics |
In elementary algebra , root rationalisation (or rationalization ) is a process by which radicals in the denominator of an algebraic fraction are eliminated.
If the denominator is a monomial in some radical, say a x n k , {\displaystyle a{\sqrt[{n}]{x}}^{k},} with k < n , rationalisation consists of multiplying the numerator and the denominator by x n n − k {\displaystyle {\sqrt[{n}]{x}}^{n-k}} , and replacing x n n {\displaystyle {\sqrt[{n}]{x}}^{n}} by x (this is allowed, as, by definition, a n th root of x is a number that has x as its n th power). If k ≥ n , one writes k = qn + r with 0 ≤ r < n ( Euclidean division ), and x n k = x q x n r ; {\displaystyle {\sqrt[{n}]{x}}^{k}=x^{q}{\sqrt[{n}]{x}}^{r};} then one proceeds as above by multiplying by x n n − r . {\displaystyle {\sqrt[{n}]{x}}^{n-r}.}
If the denominator is linear in some square root, say a + b x , {\displaystyle a+b{\sqrt {x}},} rationalisation consists of multiplying the numerator and the denominator by the conjugate a − b x , {\displaystyle a-b{\sqrt {x}},} and expanding the product in the denominator.
This technique may be extended to any algebraic denominator, by multiplying the numerator and the denominator by all algebraic conjugates of the denominator, and expanding the new denominator into the norm of the old denominator. However, except in special cases, the resulting fractions may have huge numerators and denominators, and, therefore, the technique is generally used only in the above elementary cases.
For the fundamental technique, the numerator and denominator must be multiplied by the same factor.
Example 1:
To rationalise this kind of expression , bring in the factor 5 {\displaystyle {\sqrt {5}}} :
The square root disappears from the denominator, because ( 5 ) 2 = 5 {\displaystyle \left({\sqrt {5}}\right)^{2}=5} by definition of a square root:
which is the result of the rationalisation.
Example 2:
To rationalise this radical, bring in the factor a 3 2 {\displaystyle {\sqrt[{3}]{a}}^{2}} :
The cube root disappears from the denominator, because it is cubed; so
which is the result of the rationalisation.
For a denominator that is:
Rationalisation can be achieved by multiplying by the conjugate :
and applying the difference of two squares identity, which here will yield −1. To get this result, the entire fraction should be multiplied by
This technique works much more generally. It can easily be adapted to remove one square root at a time, i.e. to rationalise
by multiplication by
Example:
The fraction must be multiplied by a quotient containing 3 ∓ 5 {\displaystyle {{\sqrt {3}}\mp {\sqrt {5}}}} .
Now, we can proceed to remove the square roots in the denominator:
Example 2:
This process also works with complex numbers with i = − 1 {\displaystyle i={\sqrt {-1}}}
The fraction must be multiplied by a quotient containing 1 ∓ − 5 {\displaystyle {1\mp {\sqrt {-5}}}} .
Rationalisation can be extended to all algebraic numbers and algebraic functions (as an application of norm forms ). For example, to rationalise a cube root , two linear factors involving cube roots of unity should be used, or equivalently a quadratic factor.
This material is carried in classic algebra texts. For example: | https://en.wikipedia.org/wiki/Rationalisation_(mathematics) |
In philosophy , rationalism is the epistemological view that "regards reason as the chief source and test of knowledge" [ 1 ] or “the position that reason has precedence over other ways of acquiring knowledge”, [ 2 ] often in contrast to other possible sources of knowledge such as faith , tradition, or sensory experience . More formally, rationalism is defined as a methodology or a theory "in which the criterion of truth is not sensory but intellectual and deductive ". [ 3 ]
In a major philosophical debate during the Enlightenment , [ 4 ] rationalism (sometimes here equated with innatism ) was opposed to empiricism . On the one hand, rationalists like René Descartes emphasized that knowledge is primarily innate and the intellect, the inner faculty of the human mind, can therefore directly grasp or derive logical truths ; on the other hand, empiricists like John Locke emphasized that knowledge is not primarily innate and is best gained by careful observation of the physical world outside the mind, namely through sensory experiences. Rationalists asserted that certain principles exist in logic , mathematics , ethics , and metaphysics that are so fundamentally true that denying them causes one to fall into contradiction. The rationalists had such a high confidence in reason that empirical proof and physical evidence were regarded as unnecessary to ascertain certain truths – in other words, "there are significant ways in which our concepts and knowledge are gained independently of sense experience". [ 5 ]
Different degrees of emphasis on this method or theory lead to a range of rationalist standpoints, from the moderate position "that reason has precedence over other ways of acquiring knowledge" to the more extreme position that reason is "the unique path to knowledge". [ 2 ] Given a pre-modern understanding of reason, rationalism is identical to philosophy , the Socratic life of inquiry, or the zetetic ( skeptical ) clear interpretation of authority (open to the underlying or essential cause of things as they appear to our sense of certainty).
Rationalism has a philosophical history dating from antiquity . The analytical nature of much of philosophical enquiry, the awareness of apparently a priori domains of knowledge such as mathematics, combined with the emphasis of obtaining knowledge through the use of rational faculties (commonly rejecting, for example, direct revelation ) have made rationalist themes very prevalent in the history of philosophy .
Since the Enlightenment, rationalism is usually associated with the introduction of mathematical methods into philosophy as seen in the works of Descartes , Leibniz , and Spinoza . [ 3 ] This is commonly called continental rationalism , because it was predominant in the continental schools of Europe, whereas in Britain empiricism dominated.
Even then, the distinction between rationalists and empiricists was drawn at a later period and would not have been recognized by the philosophers involved. Also, the distinction between the two philosophies is not as clear-cut as is sometimes suggested; for example, Descartes and Locke have similar views about the nature of human ideas. [ 5 ]
Proponents of some varieties of rationalism argue that, starting with foundational basic principles, like the axioms of geometry , one could deductively derive the rest of all possible knowledge. Notable philosophers who held this view most clearly were Baruch Spinoza and Gottfried Leibniz , whose attempts to grapple with the epistemological and metaphysical problems raised by Descartes led to a development of the fundamental approach of rationalism. Both Spinoza and Leibniz asserted that, in principle , all knowledge, including scientific knowledge, could be gained through the use of reason alone, though they both observed that this was not possible in practice for human beings except in specific areas such as mathematics . On the other hand, Leibniz admitted in his book Monadology that "we are all mere Empirics in three fourths of our actions." [ 6 ]
In politics , rationalism, since the Enlightenment , historically emphasized a "politics of reason" centered upon rationality , deontology , utilitarianism , secularism , and irreligion [ 7 ] – the latter aspect's antitheism was later softened by the adoption of pluralistic reasoning methods practicable regardless of religious or irreligious ideology. [ 8 ] [ 9 ] In this regard, the philosopher John Cottingham [ 10 ] noted how rationalism, a methodology , became socially conflated with atheism , a worldview :
In the past, particularly in the 17th and 18th centuries, the term 'rationalist' was often used to refer to free thinkers of an anti-clerical and anti-religious outlook, and for a time the word acquired a distinctly pejorative force (thus in 1670 Sanderson spoke disparagingly of 'a mere rationalist, that is to say in plain English an atheist of the late edition...'). The use of the label 'rationalist' to characterize a world outlook which has no place for the supernatural is becoming less popular today; terms like ' humanist ' or ' materialist ' seem largely to have taken its place. But the old usage still survives.
Rationalism is often contrasted with empiricism . Taken very broadly, these views are not mutually exclusive, since – on some definitions – a philosopher can be both rationalist and empiricist. [ 11 ] [ 2 ] Taken to extremes, the empiricist view holds that all ideas come to us a posteriori , that is to say, through experience; either through the external senses or through such inner sensations as pain and gratification. The empiricist essentially believes that knowledge is based on or derived directly from experience. The rationalist believes we come to knowledge a priori – through the use of logic – and is thus independent of sensory experience. In other words, as Galen Strawson once wrote, "you can see that it is true just lying on your couch. You don't have to get up off your couch and go outside and examine the way things are in the physical world. You don't have to do any science." [ 12 ]
Between both philosophies, the issue at hand is the fundamental source of human knowledge and the proper techniques for verifying what we think we know. Whereas both philosophies are under the umbrella of epistemology , their argument lies in the understanding of the warrant, which is under the wider epistemic umbrella of the theory of justification . Part of epistemology , this theory attempts to understand the justification of propositions and beliefs . Epistemologists are concerned with various epistemic features of belief, which include the ideas of justification , warrant, rationality , and probability . Of these four terms, the term that has been most widely used and discussed by the early 21st century is "warrant". Loosely speaking, justification is the reason that someone (probably) holds a belief.
If A makes a claim and then B casts doubt on it, A ' s next move would normally be to provide justification for the claim. The precise method one uses to provide justification is where the lines are drawn between rationalism and empiricism (among other philosophical views). Much of the debate in these fields are focused on analyzing the nature of knowledge and how it relates to connected notions such as truth , belief , and justification .
At its core, rationalism consists of three basic claims. For people to consider themselves rationalists, they must adopt at least one of these three claims: the intuition/deduction thesis, the innate knowledge thesis, or the innate concept thesis. In addition, a rationalist can choose to adopt the claim of Indispensability of Reason and or the claim of Superiority of Reason, although one can be a rationalist without adopting either thesis. [ citation needed ]
The indispensability of reason thesis : "The knowledge we gain in subject area, S , by intuition and deduction, as well as the ideas and instances of knowledge in S that are innate to us, could not have been gained by us through sense experience." [ 1 ] In short, this thesis claims that experience cannot provide what we gain from reason.
The superiority of reason thesis : '"The knowledge we gain in subject area S by intuition and deduction or have innately is superior to any knowledge gained by sense experience". [ 1 ] In other words, this thesis claims reason is superior to experience as a source for knowledge.
Rationalists often adopt similar stances on other aspects of philosophy. Most rationalists reject skepticism for the areas of knowledge they claim are knowable a priori . When you claim some truths are innately known to us, one must reject skepticism in relation to those truths. Especially for rationalists who adopt the Intuition/Deduction thesis, the idea of epistemic foundationalism tends to crop up. This is the view that we know some truths without basing our belief in them on any others and that we then use this foundational knowledge to know more truths. [ 1 ]
"Some propositions in a particular subject area, S, are knowable by us by intuition alone; still others are knowable by being deduced from intuited propositions." [ 13 ]
Generally speaking, intuition is a priori knowledge or experiential belief characterized by its immediacy; a form of rational insight. We simply "see" something in such a way as to give us a warranted belief. Beyond that, the nature of intuition is hotly debated. In the same way, generally speaking, deduction is the process of reasoning from one or more general premises to reach a logically certain conclusion. Using valid arguments , we can deduce from intuited premises.
For example, when we combine both concepts, we can intuit that the number three is prime and that it is greater than two. We then deduce from this knowledge that there is a prime number greater than two. Thus, it can be said that intuition and deduction combined to provide us with a priori knowledge – we gained this knowledge independently of sense experience.
To argue in favor of this thesis, Gottfried Wilhelm Leibniz , a prominent German philosopher, says,
The senses, although they are necessary for all our actual knowledge, are not sufficient to give us the whole of it, since the senses never give anything but instances, that is to say particular or individual truths. Now all the instances which confirm a general truth, however numerous they may be, are not sufficient to establish the universal necessity of this same truth, for it does not follow that what happened before will happen in the same way again. … From which it appears that necessary truths, such as we find in pure mathematics, and particularly in arithmetic and geometry, must have principles whose proof does not depend on instances, nor consequently on the testimony of the senses, although without the senses it would never have occurred to us to think of them… [ 14 ]
Empiricists such as David Hume have been willing to accept this thesis for describing the relationships among our own concepts. [ 13 ] In this sense, empiricists argue that we are allowed to intuit and deduce truths from knowledge that has been obtained a posteriori .
By injecting different subjects into the Intuition/Deduction thesis, we are able to generate different arguments. Most rationalists agree mathematics is knowable by applying the intuition and deduction. Some go further to include ethical truths into the category of things knowable by intuition and deduction. Furthermore, some rationalists also claim metaphysics is knowable in this thesis. Naturally, the more subjects the rationalists claim to be knowable by the Intuition/Deduction thesis, the more certain they are of their warranted beliefs, and the more strictly they adhere to the infallibility of intuition, the more controversial their truths or claims and the more radical their rationalism. [ 13 ]
In addition to different subjects, rationalists sometimes vary the strength of their claims by adjusting their understanding of the warrant. Some rationalists understand warranted beliefs to be beyond even the slightest doubt; others are more conservative and understand the warrant to be belief beyond a reasonable doubt.
Rationalists also have different understanding and claims involving the connection between intuition and truth. Some rationalists claim that intuition is infallible and that anything we intuit to be true is as such. More contemporary rationalists accept that intuition is not always a source of certain knowledge – thus allowing for the possibility of a deceiver who might cause the rationalist to intuit a false proposition in the same way a third party could cause the rationalist to have perceptions of nonexistent objects .
"We have knowledge of some truths in a particular subject area, S, as part of our rational nature." [ 15 ]
The Innate Knowledge thesis is similar to the Intuition/Deduction thesis in the regard that both theses claim knowledge is gained a priori . The two theses go their separate ways when describing how that knowledge is gained. As the name, and the rationale, suggests, the Innate Knowledge thesis claims knowledge is simply part of our rational nature. Experiences can trigger a process that allows this knowledge to come into our consciousness, but the experiences do not provide us with the knowledge itself. The knowledge has been with us since the beginning and the experience simply brought into focus, in the same way a photographer can bring the background of a picture into focus by changing the aperture of the lens. The background was always there, just not in focus.
This thesis targets a problem with the nature of inquiry originally postulated by Plato in Meno . Here, Plato asks about inquiry; how do we gain knowledge of a theorem in geometry? We inquire into the matter. Yet, knowledge by inquiry seems impossible. [ 16 ] In other words, "If we already have the knowledge, there is no place for inquiry. If we lack the knowledge, we don't know what we are seeking and cannot recognize it when we find it. Either way we cannot gain knowledge of the theorem by inquiry. Yet, we do know some theorems." [ 15 ] The Innate Knowledge thesis offers a solution to this paradox . By claiming that knowledge is already with us, either consciously or unconsciously , a rationalist claims we don't really learn things in the traditional usage of the word, but rather that we simply use words we know.
"We have some of the concepts we employ in a particular subject area, S, as part of our rational nature." [ 17 ]
Similar to the Innate Knowledge thesis, the Innate Concept thesis suggests that some concepts are simply part of our rational nature. These concepts are a priori in nature and sense experience is irrelevant to determining the nature of these concepts (though, sense experience can help bring the concepts to our conscious mind ).
In his book Meditations on First Philosophy , [ 18 ] René Descartes postulates three classifications for our ideas when he says, "Among my ideas, some appear to be innate, some to be adventitious, and others to have been invented by me. My understanding of what a thing is, what truth is, and what thought is, seems to derive simply from my own nature. But my hearing a noise, as I do now, or seeing the sun, or feeling the fire, comes from things which are located outside me, or so I have hitherto judged. Lastly, sirens , hippogriffs and the like are my own invention." [ 19 ]
Adventitious ideas are those concepts that we gain through sense experiences, ideas such as the sensation of heat, because they originate from outside sources; transmitting their own likeness rather than something else and something you simply cannot will away. Ideas invented by us, such as those found in mythology , legends and fairy tales , are created by us from other ideas we possess. Lastly, innate ideas, such as our ideas of perfection , are those ideas we have as a result of mental processes that are beyond what experience can directly or indirectly provide.
Gottfried Wilhelm Leibniz defends the idea of innate concepts by suggesting the mind plays a role in determining the nature of concepts, to explain this, he likens the mind to a block of marble in the New Essays on Human Understanding ,
This is why I have taken as an illustration a block of veined marble, rather than a wholly uniform block or blank tablets, that is to say what is called tabula rasa in the language of the philosophers. For if the soul were like those blank tablets, truths would be in us in the same way as the figure of Hercules is in a block of marble, when the marble is completely indifferent whether it receives this or some other figure. But if there were veins in the stone which marked out the figure of Hercules rather than other figures, this stone would be more determined thereto, and Hercules would be as it were in some manner innate in it, although labour would be needed to uncover the veins, and to clear them by polishing, and by cutting away what prevents them from appearing. It is in this way that ideas and truths are innate in us, like natural inclinations and dispositions, natural habits or potentialities, and not like activities, although these potentialities are always accompanied by some activities which correspond to them, though they are often imperceptible." [ 20 ]
Some philosophers, such as John Locke (who is considered one of the most influential thinkers of the Enlightenment and an empiricist ), argue that the Innate Knowledge thesis and the Innate Concept thesis are the same. [ 21 ] Other philosophers, such as Peter Carruthers , argue that the two theses are distinct from one another. As with the other theses covered under the umbrella of rationalism, the more types and greater number of concepts a philosopher claims to be innate, the more controversial and radical their position; "the more a concept seems removed from experience and the mental operations we can perform on experience the more plausibly it may be claimed to be innate. Since we do not experience perfect triangles but do experience pains, our concept of the former is a more promising candidate for being innate than our concept of the latter. [ 17 ]
Although rationalism in its modern form post-dates antiquity, philosophers from this time laid down the foundations of rationalism. In particular, the understanding that we may be aware of knowledge available only through the use of rational thought. [ citation needed ]
Pythagoras was one of the first Western philosophers to stress rationalist insight. [ 22 ] He is often revered as a great mathematician , mystic and scientist , but he is best known for the Pythagorean theorem , which bears his name, and for discovering the mathematical relationship between the length of strings on lute and the pitches of the notes. Pythagoras "believed these harmonies reflected the ultimate nature of reality. He summed up the implied metaphysical rationalism in the words 'All is number'. It is probable that he had caught the rationalist's vision, later seen by Galileo (1564–1642), of a world governed throughout by mathematically formulable laws". [ 23 ] It has been said that he was the first man to call himself a philosopher, or lover of wisdom. [ 24 ]
Plato held rational insight to a very high standard, as is seen in his works such as Meno and The Republic . He taught on the Theory of Forms (or the Theory of Ideas) [ 25 ] [ 26 ] [ 27 ] which asserts that the highest and most fundamental kind of reality is not the material world of change known to us through sensation , but rather the abstract, non-material (but substantial ) world of forms (or ideas). [ 28 ] For Plato, these forms were accessible only to reason and not to sense. [ 23 ] In fact, it is said that Plato admired reason, especially in geometry , so highly that he had the phrase "Let no one ignorant of geometry enter" inscribed over the door to his academy. [ 29 ]
Aristotle 's main contribution to rationalist thinking was the use of syllogistic logic and its use in argument. Aristotle defines syllogism as "a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so." [ 30 ] Despite this very general definition, Aristotle limits himself to categorical syllogisms which consist of three categorical propositions in his work Prior Analytics . [ 31 ] These included categorical modal syllogisms. [ 32 ]
Although the three great Greek philosophers disagreed with one another on specific points, they all agreed that rational thought could bring to light knowledge that was self-evident – information that humans otherwise could not know without the use of reason. After Aristotle's death, Western rationalistic thought was generally characterized by its application to theology, such as in the works of Augustine , the Islamic philosopher Avicenna (Ibn Sina) , Averroes (Ibn Rushd) , and Jewish philosopher and theologian Maimonides . The Waldensians sect also incorporated rationalism into their movement. [ 33 ] One notable event in the Western timeline was the philosophy of Thomas Aquinas who attempted to merge Greek rationalism and Christian revelation in the thirteenth-century. [ 23 ] [ 34 ] Generally, the Roman Catholic Church viewed Rationalists as a threat, labeling them as those who "while admitting revelation, reject from the word of God whatever, in their private judgment, is inconsistent with human reason." [ 35 ]
Descartes was the first of the modern rationalists and has been dubbed the 'Father of Modern Philosophy.' Much subsequent Western philosophy is a response to his writings, [ 36 ] [ 37 ] [ 38 ] which are studied closely to this day.
Descartes thought that only knowledge of eternal truths – including the truths of mathematics, and the epistemological and metaphysical foundations of the sciences – could be attained by reason alone; other knowledge, the knowledge of physics, required experience of the world, aided by the scientific method . He also argued that although dreams appear as real as sense experience , these dreams cannot provide persons with knowledge. Also, since conscious sense experience can be the cause of illusions, then sense experience itself can be doubtable. As a result, Descartes deduced that a rational pursuit of truth should doubt every belief about sensory reality. He elaborated these beliefs in such works as Discourse on the Method , Meditations on First Philosophy , and Principles of Philosophy . Descartes developed a method to attain truths according to which nothing that cannot be recognised by the intellect (or reason ) can be classified as knowledge. These truths are gained "without any sensory experience", according to Descartes. Truths that are attained by reason are broken down into elements that intuition can grasp, which, through a purely deductive process, will result in clear truths about reality.
Descartes therefore argued, as a result of his method, that reason alone determined knowledge, and that this could be done independently of the senses. For instance, his famous dictum, cogito ergo sum or "I think, therefore I am", is a conclusion reached a priori i.e., prior to any kind of experience on the matter. The simple meaning is that doubting one's existence, in and of itself, proves that an "I" exists to do the thinking. In other words, doubting one's own doubting is absurd. [ 22 ] This was, for Descartes, an irrefutable principle upon which to ground all forms of other knowledge. Descartes posited a metaphysical dualism , distinguishing between the substances of the human body (" res extensa ") and the mind or soul (" res cogitans "). This crucial distinction would be left unresolved and lead to what is known as the mind–body problem , since the two substances in the Cartesian system are independent of each other and irreducible.
The philosophy of Baruch Spinoza is a systematic, logical, rational philosophy developed in seventeenth-century Europe . [ 39 ] [ 40 ] [ 41 ] Spinoza's philosophy is a system of ideas constructed upon basic building blocks with an internal consistency with which he tried to answer life's major questions and in which he proposed that "God exists only philosophically." [ 41 ] [ 42 ] He was heavily influenced by Descartes, [ 43 ] Euclid [ 42 ] and Thomas Hobbes , [ 43 ] as well as theologians in the Jewish philosophical tradition such as Maimonides . [ 43 ] But his work was in many respects a departure from the Judeo-Christian-Islamic tradition. Many of Spinoza's ideas continue to vex thinkers today and many of his principles, particularly regarding the emotions , have implications for modern approaches to psychology . To this day, many important thinkers have found Spinoza's "geometrical method" [ 41 ] difficult to comprehend: Goethe admitted that he found this concept confusing. [ citation needed ] His magnum opus , Ethics , contains unresolved obscurities and has a forbidding mathematical structure modeled on Euclid's geometry. [ 42 ] Spinoza's philosophy attracted believers such as Albert Einstein [ 44 ] and much intellectual attention. [ 45 ] [ 46 ] [ 47 ] [ 48 ] [ 49 ]
Leibniz was the last major figure of seventeenth-century rationalism who contributed heavily to other fields such as metaphysics , epistemology , logic , mathematics , physics , jurisprudence , and the philosophy of religion ; he is also considered to be one of the last "universal geniuses". [ 50 ] He did not develop his system, however, independently of these advances. Leibniz rejected Cartesian dualism and denied the existence of a material world. In Leibniz's view there are infinitely many simple substances, which he called " monads " (which he derived directly from Proclus ).
Leibniz developed his theory of monads in response to both Descartes and Spinoza , because the rejection of their visions forced him to arrive at his own solution. Monads are the fundamental unit of reality, according to Leibniz, constituting both inanimate and animate objects. These units of reality represent the universe, though they are not subject to the laws of causality or space (which he called " well-founded phenomena "). Leibniz, therefore, introduced his principle of pre-established harmony to account for apparent causality in the world.
Kant is one of the central figures of modern philosophy , and set the terms by which all subsequent thinkers have had to grapple. He argued that human perception structures natural laws, and that reason is the source of morality. His thought continues to hold a major influence in contemporary thought, especially in fields such as metaphysics, epistemology, ethics, political philosophy, and aesthetics. [ 51 ]
Kant named his brand of epistemology " Transcendental Idealism ", and he first laid out these views in his famous work The Critique of Pure Reason . In it he argued that there were fundamental problems with both rationalist and empiricist dogma. To the rationalists he argued, broadly, that pure reason is flawed when it goes beyond its limits and claims to know those things that are necessarily beyond the realm of every possible experience: the existence of God , free will, and the immortality of the human soul. Kant referred to these objects as "The Thing in Itself" and goes on to argue that their status as objects beyond all possible experience by definition means we cannot know them. To the empiricist, he argued that while it is correct that experience is fundamentally necessary for human knowledge, reason is necessary for processing that experience into coherent thought. He therefore concludes that both reason and experience are necessary for human knowledge. In the same way, Kant also argued that it was wrong to regard thought as mere analysis. "In Kant's views, a priori concepts do exist, but if they are to lead to the amplification of knowledge, they must be brought into relation with empirical data". [ 52 ]
Rationalism has become a rarer label of philosophers today; rather many different kinds of specialised rationalisms are identified. For example, Robert Brandom has appropriated the terms "rationalist expressivism" and "rationalist pragmatism" as labels for aspects of his programme in Articulating Reasons , and identified "linguistic rationalism", the claim that the contents of propositions "are essentially what can serve as both premises and conclusions of inferences", as a key thesis of Wilfred Sellars . [ 53 ]
Outside of academic philosophy, some participants in the internet communities surrounding LessWrong and Slate Star Codex have described themselves as "rationalists" or the " rationalist community " in reference to rationality , rather than rationalism. [ 54 ] [ 55 ] [ 56 ] The term has also been used in this way by critics such as Timnit Gebru . [ 57 ]
Rationalism was criticized by American psychologist William James for being out of touch with reality. James also criticized rationalism for representing the universe as a closed system, which contrasts with his view that the universe is an open system. [ 58 ]
Proponents of emotional choice theory criticize rationalism by drawing on new findings from emotion research in psychology and neuroscience . They point out that the rationalist paradigm is generally based on the assumption that decision-making is a conscious and reflective process based on thoughts and beliefs. It presumes that people decide on the basis of calculation and deliberation. However, cumulative research in neuroscience suggests that only a small part of the brain's activities operate at the level of conscious reflection. The vast majority of its activities consist of unconscious appraisals and emotions. [ 59 ] The significance of emotions in decision-making has generally been ignored by rationalism, according to these critics. Moreover, emotional choice theorists contend that the rationalist paradigm has difficulty incorporating emotions into its models, because it cannot account for the social nature of emotions. Even though emotions are felt by individuals, psychologists and sociologists have shown that emotions cannot be isolated from the social environment in which they arise. Emotions are inextricably intertwined with people's social norms and identities, which are typically outside the scope of standard rationalist accounts. [ 60 ] Emotional choice theory seeks to capture not only the social but also the physiological and dynamic character of emotions. It represents a unitary action model to organize, explain, and predict the ways in which emotions shape decision-making. [ 61 ] | https://en.wikipedia.org/wiki/Rationalism |
Rational choice (also termed rationalism ) is a prominent framework in international relations scholarship. Rational choice is not a substantive theory of international politics , but rather a methodological approach that focuses on certain types of social explanation for phenomena. [ 1 ] In that sense, it is similar to constructivism , and differs from liberalism and realism , which are substantive theories of world politics. [ 1 ] [ 2 ] Rationalist analyses have been used to substantiate realist theories, [ 3 ] [ 4 ] [ 5 ] as well as liberal theories of international relations. [ 6 ] [ 7 ]
Rational choice research tends to explain conditions that bring about outcomes or patterns of behavior if relevant actors behave rationally. [ 1 ] Key concepts in rational choice research in international relations include incomplete information, credibility , signaling, transaction costs, trust, and audience costs .
According to James D. Fearon , a rational choice research project typically proceeds in the following fashion: [ 1 ]
Actors do not have to be fully rational. [ 1 ] There are varieties of rationality (e.g. thick and thin rationality). [ 1 ] Rational choice scholarship may emphasize materialist variables, but rational choice and materialism are not necessarily synonymous. [ 1 ]
Rational choice explanations for conflict and the lack of cooperation in international politics frequently point to factors such as incomplete information, and a lack of credibility. Chances of cooperation and peaceful resolution can be increased through costly signaling, [ 7 ] [ 8 ] [ 9 ] long shadows of the future, [ 10 ] [ 11 ] and tit-for-tat bargaining strategies. [ 12 ] [ 13 ] According to rationalist analyses, institutions may facilitate cooperation by increasing information, reducing transaction costs, and reducing collective action problems. [ 14 ]
Rational choice analyses tend to conceptualize norms as adhering to a "logic of consequence" rather than the constructivist “ logic of appropriateness ”. The “logic of consequences” entails that actors are assumed to choose the most efficient means to reach their goals on the basis of a cost-benefit analysis. [ 15 ] This stands in contrast to the logic of appropriateness whereby actors follow “internalized prescriptions of what is socially defined as normal, true, right, or good, without, or in spite of calculation of consequences and expected utility”. [ 16 ] Jeffrey Checkel writes that there are two common types of explanations for the efficacy of norms: [ 17 ]
According to Duncan Snidal , the advantages of rational choice research is that the formalization of arguments helps to clarify the underlying logic of authors' claims, the clarity of arguments makes rational choice arguments falsifiable , and rational choice arguments lend themselves to empirical validation through case studies . [ 18 ]
Constructivist scholars argue that while rational choice approaches may be useful to explain the interactions of actors with given interests, rationalist approaches are ultimately limited in explaining how those interests emerged in the first place. [ 1 ] In other words, rationalists use exogenously given interests, but struggle to account for endogenously given interests. [ 18 ] According to Duncan Snidal, rationalists are good at explaining continuity and stability (equilibrium solutions), but are less adept at explaining why change occurs. [ 18 ] He also argues that rationalists are ill-equipped to incorporate norms in their models. [ 18 ] According to Sidney Verba , a rational choice model of international relations depends on the quality of assumptions in the model; bad assumptions undercut the usefulness and adequacy of the model. [ 19 ]
International relations scholars who use methods and theories of psychology and cognitive science have criticized rational choice models of international relations. [ 20 ] [ 21 ]
In international relations theory , the bargaining model of war is a method of representing the potential gains and losses and ultimate outcome of war between two actors as a bargaining interaction. [ 22 ] A central puzzle that motivates research in this vein is the "inefficiency puzzle of war": why do wars occur when it would be better for all parties involved to reach an agreement that goes short of war? [ 23 ] [ 3 ]
Thomas Schelling was an early proponent of formalizing conflicts as bargaining situations. [ 23 ] [ 24 ] Stanford University political scientist James Fearon brought prominence to the bargaining model in the 1990s. [ 25 ] His 1995 article "Rationalist Explanations for War" is the most assigned journal article in International Relations graduate training at U.S. universities. [ 25 ] [ 3 ] The bargaining model of war has been described as "the dominant framework used in the study of war in the international relations field." [ 26 ]
According to James D. Fearon, there are three conditions where war is possible under the bargaining model: [ 3 ]
In short, Fearon argues that a lack of information and bargaining indivisibilities can lead rational states into war. [ 3 ] Robert Powell modified the model as presented by Fearon, arguing that three prominent kinds of commitment problems ( preventive war , preemptive war , and bargaining failure over rising powers) tended to be caused by large and rapid shifts in the distribution of power. [ 28 ] The fundamental cause for war in Powell's view is that actors cannot under those circumstances credibly commit to abide by any agreement. [ 28 ] Powell also argued that bargaining indivisibilities were a form of commitment problem, as opposed to something that intrinsically prevented actors from reaching a bargain (because actors could reach an agreement over side payments over an indivisible good). [ 28 ]
Applications of the bargaining model have indicated that third-party mediators can reduce the potential for war (by providing information). [ 29 ] Some scholars have argued that democratic states can more credibly reveal their resolve because of the domestic costs that stem from making empty threats towards other states. [ 30 ]
University of Pennsylvania political scientist Alex Weisiger has tackled the puzzle of prolonged wars, arguing that commitment problems can account for lengthy wars. Weisiger argues that "situational" commitment problems where one power is declining and preemptively attacks a rising power can be lengthy because the rising power believes that the declining power will not agree to any bargain. [ 31 ] He also argues that "dispositional" commitment problems, whereby states will not accept anything except unconditional surrender (because they believe the other state will never abide by any bargain), can be lengthy. [ 31 ]
Rochester University political scientist Hein Goemans argues that prolonged wars can be rational because actors in wars still have incentives to misrepresent their capabilities and resolve, both to be in a better position at the war settlement table and to affect interventions by third parties in the war. [ 32 ] Actors may also raise or reduce their war aims once it becomes clear that they have the upper hand. Goemans also argues that it can be rational for leaders to "gamble for resurrection", which means that leaders become reluctant to settle wars if they believe they will be punished severely in domestic politics (e.g. punished through exile, imprisonment or death ) if they do not outright win the war. [ 32 ]
Building on canonical work by James Fearon, there are two prominent signaling mechanisms in the rational choice literature: sinking costs and tying hands. [ 33 ] [ 34 ] The former refers to signals that involve sunk irrecoverable costs, whereas the latter refers to signals that will incur costs in the future if the signaler reneges. [ 35 ]
The applicability of the bargaining model is limited by numerous factors, including:
According to Robert Powell, the bargaining model has limitations in terms of explaining prolonged wars (because actors should quickly learn about the other side's commitment and capabilities). It can also give ahistorical readings of certain historical cases, as the implications of the model is that there would be no war between rational actors if the actors had perfect information. [ 41 ] Ahsan Butt argues that in some wars, one actor is insistent on war and there are no plausible concessions that can be made by the other state. [ 42 ]
Stephen Walt argues that while the bargaining model of war (as presented by Fearon) is an "insightful and intelligent" formalization of how a lack of information and commitment problems under anarchy can lead states into conflict, it is ultimately not a "new theoretical claim" but rather another way of expressing ideas that the likes of Robert Art, Robert Jervis and Kenneth Oye have previously presented. [ 24 ]
Jonathan Kirshner has criticized the assumption of the bargaining model that states will reach a bargain if they have identical information. [ 43 ] Kirshner notes that sports pundits have high-quality identical information available to them, yet they make different predictions about how sporting events will turn out. International politics is likely to be even more complicated to predict than sporting events. [ 40 ]
According to Erik Gartzke, the bargaining model is useful for thinking probabilistically about international conflict, but the onset of any specific war is theoretically indeterminate. [ 44 ]
In a prominent 1999 critique of rational choice scholarship in security studies , Stephen Walt argued that a lot of rational choice research in security had limited originality, produced a lot of trivial results, and failed to empirically verify the validity of its theoretical claims. While he praised the logical consistency and precision of rational choice scholarship, he argued that formal modeling was not a prerequisite for logical consistency and precision. He added that rationalist models were limited in their empirical applicability due to the presence of multiple equilibria (i.e. folk theorem ) and flaws in human updating. He criticized the shift in security studies research towards formal models, arguing that it added unnecessary complexity (which created an appearance of greater scientism) which forced scholars and student to invest time in reading rational choice scholarship and learning formal modeling skills when the time could be spent on more productive endeavors. [ 24 ]
Rational choice scholars warn against conflating analytical assumptions in rational choice scholarship with empirical assumptions. [ 1 ] [ 45 ]
In terms of rationalist models in IPE scholarship, Martha Finnemore and Henry Farrell have raised questions about the strong relationship between rational choice models and quantitative methods , pointing out that qualitative methods may be more or equally suitable in empirical tests of rational choice models due to problems in quantitatively assessing strategic interactions . [ 46 ]
According to Peter Katzenstein , Robert Keohane and Stephen Krasner , rational choice research is limited in the sense that it struggles to explain the sources of actors' preferences. [ 47 ]
Rational choice scholarship has provided potential explanations for democratic peace theory, which is the notion that democracies are hesitant to engage in armed conflict with other identified democracies.
One prominent mechanism for the democratic theory is audience costs. An audience cost is a term in international relations theory that describes the electoral penalty a leader incurs from his or her constituency if they escalate a foreign policy crisis and are then seen as backing down. [ 48 ] The term was popularized in a 1994 academic article by James Fearon where he argued that democracies carry greater audience costs than authoritarian states, which makes them better at signaling their intentions in interstate disputes. [ 49 ] [ 50 ] Branislav Slantchev has argued that the presence of a free media is a key component of audience costs. [ 51 ]
Fearon's argument regarding the credibility of democratic states in disputes has been subject to debate among international relations scholars. Two studies 2001, using the MID and ICB datasets, provided empirical support for the notion that democracies were more likely to issue effective threats. [ 52 ] [ 53 ] There is survey experiment data that substantiates that specified threats induce audience costs, [ 54 ] but also data with mixed findings. [ 55 ]
A 2012 study by Alexander B. Downes and Todd S. Sechser found that existing datasets were not suitable to draw any conclusions as to whether democratic states issued more effective threats. [ 56 ] They constructed their own dataset specifically for interstate military threats and outcomes, which found no relationship between regime type and effective threats. [ 56 ] A 2017 study which recoded flaws in the MID dataset ultimately conclude, " that there are no regime-based differences in dispute reciprocation, and prior findings may be based largely on poorly coded data." [ 57 ] A 2012 study by Marc Trachtenberg, which analyzed a dozen great power crises, found no evidence of the presence of audience costs in these crises. [ 58 ]
Other scholars have disputed the democratic credibility argument, questioning its causal logic and empirical validity. [ 59 ] Research by Jessica Weeks argued that some authoritarian regime types have similar audience costs as in democratic states. [ 60 ] [ 61 ] A 2014 study by Jessica Chen Weiss argued that the Chinese regime fomented or clamped down on nationalist (or anti-foreign) protests in China in order to signal resolve. Fomenting or permitting nationalist protests entail audience costs, as they make it harder for the Chinese regime to back down in a foreign policy crisis out of fear that the protestors turn against the regime. [ 62 ]
Other rational choice scholars argue that the democratic peace is in part explained by the greater transparency of democratic political systems, which reduces the likelihood that states miscalculate the resolve of democratic states. [ 63 ] [ 64 ]
Rational Choice Institutionalism ( RCI ) is a theoretical approach to the study of institutions arguing that actors use institutions to maximize their utility, and that institutions affect rational individual behavior. [ 65 ] [ 66 ] This approach has been applied to the study of domestic institutions, as well as international institutions. [ 67 ] In the institutionalist literature, RCI is one of the three prominent approaches, along with historical institutionalism and sociological institutionalism . [ 68 ]
According to Erik Voeten, rational choice scholarship on international institutions can be divided between (1) rational functionalism and (2) Distributive rationalism. [ 69 ] The former sees organizations as functional optimal solutions to collective problems, whereas the latter sees organizations as an outcome of actors' individual and collective goals. [ 69 ] A prominent example of rational functionalism is the "Rational Design of International Institutions" literature. [ 70 ] [ 71 ]
Barbara Koremenos defines international cooperation as "any explicit arrangement – negotiated among international actors – that prescribes, proscribes, and/or authorizes behavior." [ 72 ] She has provided a rationalist account for the design of international institutions, arguing, "because agreements matter, they are designed in rational ways, and the fact that people make efforts to design them in such ways corroborates their significance." [ 73 ]
Proponents of emotional choice theory criticize rationalism by drawing on new findings from emotion research in psychology and neuroscience . They point out that the rationalist paradigm is generally based on the assumption that decision-making is a conscious and reflective process based on thoughts and beliefs. It presumes that people decide on the basis of calculation and deliberation. However, cumulative research in neuroscience suggests that only a small part of the brain's activities operate at the level of conscious reflection. The vast majority of its activities consist of unconscious appraisals and emotions. [ 74 ] The significance of emotions in decision-making has generally been ignored by rationalism, according to these critics.
Moreover, emotional choice theorists contend that the rationalist paradigm has difficulty incorporating emotions into its models, because it cannot account for the social nature of emotions. Even though emotions are felt by individuals, psychologists and sociologists have shown that emotions cannot be isolated from the social environment in which they arise. Emotions are inextricably intertwined with people's social norms and identities, which are typically outside the scope of standard rationalist accounts. [ 75 ] Emotional choice theory seeks to capture not only the social but also the physiological and dynamic character of emotions. It represents a unitary action model to organize, explain, and predict the ways in which emotions shape decision-making. [ 76 ] | https://en.wikipedia.org/wiki/Rationalism_(international_relations) |
Rationalist ( Polish : Racjonalista ) was a Polish magazine published in Warsaw from October 1930 to December 1935 by the Warsaw Circle of Intellectuals, Polish Association of Free Thought .
Editor and publisher of "rationalist" was Józef Landau . The leading publicists were: Tadeusz Kotarbiński , Henryk Ułaszyn , and Józef Landau.
This Polish magazine or academic journal-related article is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about magazines . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Rationalist_(magazine) |
The Rationalist Association was a charity in the United Kingdom which published New Humanist magazine between 1885 and 2025. Since 2025, the Rationalist Press has been the publishing imprint of Humanists UK . [ 1 ] [ 2 ]
The original Rationalist Press Association (RPA) was founded in 1885 by a group of freethinkers who were unhappy with the increasingly political and decreasingly intellectual tenor of the British secularist movement , [ 3 ] which made its name publishing cheap reprints of classic literature – such as works by Charles Darwin and John Stuart Mill – through its Thinker's Library series, along with literature that was deemed too anti-religious to be handled by mainstream publishers and booksellers.
In 2002, the RPA became a wholly owned subsidiary of the Rationalist Association, a charity established to continue its work. [ 4 ] In 2025, the Rationalist Association merged with Humanists UK, which took over ownership of the RPA and publication of New Humanist . As the Rationalist Press, the original 1885 RPA became the publishing imprint of Humanists UK.
The impetus for the creation of the Rationalist Press Association can be traced back to Charles Albert Watts , the publisher who printed the National Reformer and a majority of Charles Bradlaugh 's books. [ 3 ] In 1890 Watts formed the Propagandist Press Committee, with George Jacob Holyoake as president, in order to circumvent the problem caused by booksellers who refused to handle secularist books. Holyoake remained president as the committee changed its name to the Rationalist Press Committee and finally settled on the Rationalist Press Association in 1899. [ 5 ] Members of the association paid a subscription fee and received books annually to the value of that fee. [ 3 ]
The Association became quite successful after 1902, when it started selling reprints of serious scientific works by authors such as Julian Huxley , Ernst Haeckel and Matthew Arnold . It achieved even greater success through the Thinker's Library series of books, published by Watts & Co. from 1929 until 1951 under the leadership of Charles Watts's son Fredrick. The Association's continued success in selling books of a heretical nature, mostly by agnostic or atheist authors, contributed to a growing rationalist zeal and a growing demand for this type of literature. By 1959 the Association had reached its highest membership, with more than 5,000 members. Yet its success also contributed to its demise: rationalist literature became so popular that the Association's readership was taken by larger, more established mainstream publishers. The result was a steady decline in membership. [ 3 ]
In 2002, the Association changed its name to The Rationalist Association.
In 2006, Jonathan Miller was chosen to be its president. He said in response to being chosen: "Not believing in religion is very widespread, but I think this community gets overlooked. I am flattered and honoured". [ 6 ]
In Jan 2025, the organisation merged with Humanists UK , which now publishes the quarterly magazine, New Humanist . | https://en.wikipedia.org/wiki/Rationalist_Association |
The Rationalist Society of Australia ( RSA ) promotes the interests of rationalists nationally in Australia. Originally formed as the Victorian Rationalist Association, the society originated in a meeting of freethinkers in the University of Melbourne in 1906. [ 1 ] It is the operational arm of the rationalist movement in Australia.
The society created a rationalist library in 1909, and grew its collection though donations. The society ran the 1910 and 1913 Australian tours of rationalist thinker, Joseph McCabe . [ 2 ] A number of trade unionists and social campaigners sought to advance political causes, including Robert Samuel Ross and Alfred Foster . John Samuel Langley became the secretary in 1919, and William Glanville Cook became the secretary in 1938. [ 3 ]
Its aims include:
The RSA publishes the Australian Rationalist journal. Issues are archived in the National Library of Australia , and previous issues of the journal can be found on their website. [ 4 ] Victoria University maintains a Rationalist Collection from the society. [ 2 ] Contributors have included Brian Fitzpatrick and Ian Robinson .
The Australian Bureau of Statistics in the national census categorises rationalists under "No Religion". In the 2016 census , 29.6% of respondents (or 6,933,708 people) selected "no religion" [ 5 ] or irreligious , a category that includes rationalists as well as Humanists , agnostics and atheists .
This article about an organisation in Australia is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rationalist_Society_of_Australia |
The rationalist community is a 21st century movement that formed around a group of internet blogs including LessWrong and Astral Codex Ten (formerly known as Slate Star Codex ). The movement gained prominence in the San Francisco Bay Area . Its adherents claim to use rationality to avoid cognitive biases . Common interests include transhumanism , statistics, effective altruism , and mitigating existential risk from artificial general intelligence .
Rationalists are concerned with applying Bayesian inference to understand the world as it really is, avoiding cognitive biases, emotionality, or political correctness. [ 1 ] [ 2 ] [ 3 ] [ 4 ] Writing for The New Atlantis , Tara Burton describes rationalist culture as having a " technocratic focus on ameliorating the human condition through hyper-utilitarian goals", [ 5 ] with the "distinctly liberal optimism... that defines so much of Silicon Valley ideology — that intelligent people, using the right epistemic tools, can think better, and save the world by doing so". [ 6 ]
Early rationalist blogs LessWrong and Slate Star Codex attracted a STEM-interested audience that cared about self-improvement, and was suspicious of the humanities and how human emotions inhibit rational thinking. [ 7 ] The movement connected to the founder culture of Silicon Valley and its faith in the power of intelligent capitalists and technocrats to create widespread prosperity. [ 8 ] [ 9 ]
Bloomberg Businessweek journalist Ellen Huet adds that the rationalist movement "valorizes extremes: seeking rational truth above all else, donating the most money and doing the utmost good for the most important reason. This way of thinking can lend an attractive clarity, but it can also provide cover for destructive or despicable behavior". [ 10 ] Huet also notes that the borders of the community are blurry, [ 11 ] and members who have drifted from core orthodoxies will self-describe as post-rationalist or EA-adjacent. [ 12 ]
One of the main interests of the rationalist community is combating existential risk posed by the emergence of an artificial superintelligence . [ 13 ] [ 14 ] Many members of the rationalist community believe it is critical in that sense and one of the only communities that have a chance at saving humanity from extinction. [ 15 ] [ 16 ] [ 17 ]
The rationalist community emerged in the 2000s on various blogs on the Internet, including Overcoming Bias , LessWrong , and Slate Star Codex . [ 18 ] [ 19 ] [ 20 ]
Eliezer Yudkowsky , who created LessWrong and is regarded as a major figure within the movement, serially published the Harry Potter fanfiction Harry Potter and the Methods of Rationality from 2010 to 2015, which led people towards LessWrong and the rationalist community. [ 21 ] [ 22 ] Harry Potter and the Methods of Rationality was a highly popular fanfiction and is well-regarded within the rationalist community. [ 23 ] [ 24 ] Yudkowsky has used the work to solicit donations for the Center for Applied Rationality , which teaches courses based on it, [ 25 ] [ 26 ] and a 2013 LessWrong survey revealed a quarter of its users had found the site due to the fanfiction. [ 27 ]
In the 2010s, the rationalist community emerged as a major force in Silicon Valley , with many rationalists working for large technology companies. [ 28 ] [ 29 ] Billionaires Elon Musk , Peter Thiel and Ethereum creator Vitalik Buterin have donated to rationalist-associated institutions. [ 30 ] [ 31 ]
Despite the online origins of the movement, the community is active and close-knit offline, especially in the San Francisco Bay Area , where many rationalists live in intentional communities and engage in polyamorous relationships with other rationalists. [ 32 ] [ 33 ] [ 34 ] Bay Area organizations associated with the rationalist community include the Center for Applied Rationality , which teaches the techniques of rationality espoused by rationalists, and the Machine Intelligence Research Institute , which conducts research on AI safety . [ 35 ] [ 36 ] [ 37 ]
According to Ellen Huet writing in Bloomberg Businessweek in 2023, "Several current and former members of the community say its dynamics can be "cult-like"". [ 38 ] Journalist Allegra Rosenberg describes adherents who became "disillusioned with that whole scene, because it's a little culty, it's a little dogmatic." [ 39 ] Émile Torres describes TESCREALism , which includes rationalists, as "operat[ing] like a cult." [ 40 ]
Huet also reports the stories of eight women with allegations of sexual misconduct, which they describe as pervasive in the rationalist community. [ 41 ]
Writing in The New Yorker , Gideon Lewis-Kraus argues that rationalists "have given safe harbor to some genuinely egregious ideas," such as scientific racism and neoreactionary views, and that "the rationalists' general willingness to pursue orderly exchanges on objectionable topics, often with monstrous people, remains not only a point of pride but a constitutive part of the subculture's self-understanding." [ 42 ]
The rationalist community has a large overlap with effective altruism [ 43 ] [ 44 ] and transhumanism . [ 45 ] Critics such as computer scientist Timnit Gebru and philosopher Émile P. Torres additionally link rationalists with other philosophies they collectively name TESCREAL : Transhumanism, extropianism , singularitarianism , cosmism , rationalism, effective altruism, and longtermism . [ 46 ]
The postrationalists are a loose group of one-time rationalists who became disillusioned with the rationalist community, which they came to perceive as cultlike [ 47 ] and unhumanistic. [ 5 ] The term is also used as a hedge by people in the community who have drifted from its orthodoxy. [ 12 ] This community also goes by the acronym TPOT, standing for This Part of Twitter . [ 48 ] [ 49 ]
The Zizians are a spin-off group from rationalism with an ideological emphasis on veganism and anarchism , which became well known in 2025 for being suspected of involvement in four murders. [ 50 ] The Zizians formed around the Bay Area rationalist community, but became disillusioned with rationalist organizations and leaders. Among the Zizians' accusations against them were anti-transgender discrimination, misuse of donor funds to pay off a sexual misconduct accuser, and not valuing animal welfare in plans for human- friendly AI . [ 51 ] | https://en.wikipedia.org/wiki/Rationalist_community |
Rationality is the quality of being guided by or based on reason . In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence . This quality can apply to an ability, as in a rational animal , to a psychological process , like reasoning , to mental states , such as beliefs and intentions , or to persons who possess these other forms of rationality. A thing that lacks rationality is either arational , if it is outside the domain of rational evaluation, or irrational , if it belongs to this domain but does not fulfill its standards.
There are many discussions about the essential features shared by all forms of rationality. According to reason-responsiveness accounts, to be rational is to be responsive to reasons. For example, dark clouds are a reason for taking an umbrella , which is why it is rational for an agent to do so in response. An important rival to this approach are coherence-based accounts, which define rationality as internal coherence among the agent's mental states. Many rules of coherence have been suggested in this regard, for example, that one should not hold contradictory beliefs or that one should intend to do something if one believes that one should do it. Goal-based accounts characterize rationality in relation to goals, such as acquiring truth in the case of theoretical rationality. Internalists believe that rationality depends only on the person's mind . Externalists contend that external factors may also be relevant. Debates about the normativity of rationality concern the question of whether one should always be rational. A further discussion is whether rationality requires that all beliefs be reviewed from scratch rather than trusting pre-existing beliefs.
Various types of rationality are discussed in the academic literature. The most influential distinction is between theoretical and practical rationality. Theoretical rationality concerns the rationality of beliefs. Rational beliefs are based on evidence that supports them. Practical rationality pertains primarily to actions. This includes certain mental states and events preceding actions, like intentions and decisions . In some cases, the two can conflict, as when practical rationality requires that one adopts an irrational belief. Another distinction is between ideal rationality, which demands that rational agents obey all the laws and implications of logic, and bounded rationality , which takes into account that this is not always possible since the computational power of the human mind is too limited. Most academic discussions focus on the rationality of individuals. This contrasts with social or collective rationality, which pertains to collectives and their group beliefs and decisions.
Rationality is important for solving all kinds of problems in order to efficiently reach one's goal. It is relevant to and discussed in many disciplines. In ethics , one question is whether one can be rational without being moral at the same time. Psychology is interested in how psychological processes implement rationality. This also includes the study of failures to do so, as in the case of cognitive biases . Cognitive and behavioral sciences usually assume that people are rational enough to predict how they think and act. Logic studies the laws of correct arguments . These laws are highly relevant to the rationality of beliefs. A very influential conception of practical rationality is given in decision theory , which states that a decision is rational if the chosen option has the highest expected utility . Other relevant fields include game theory , Bayesianism , economics , and artificial intelligence .
In its most common sense, rationality is the quality of being guided by reasons or being reasonable. [ 1 ] [ 2 ] [ 3 ] For example, a person who acts rationally has good reasons for what they do. This usually implies that they reflected on the possible consequences of their action and the goal it is supposed to realize. In the case of beliefs , it is rational to believe something if the agent has good evidence for it and it is coherent with the agent's other beliefs. [ 4 ] [ 5 ] While actions and beliefs are the most paradigmatic forms of rationality, the term is used both in ordinary language and in many academic disciplines to describe a wide variety of things, such as persons , desires , intentions , decisions , policies, and institutions. [ 6 ] [ 7 ] Because of this variety in different contexts, it has proven difficult to give a unified definition covering all these fields and usages. In this regard, different fields often focus their investigation on one specific conception, type, or aspect of rationality without trying to cover it in its most general sense. [ 8 ]
These different forms of rationality are sometimes divided into abilities , processes , mental states , and persons. [ 6 ] [ 2 ] [ 1 ] [ 8 ] [ 9 ] For example, when it is claimed that humans are rational animals , this usually refers to the ability to think and act in reasonable ways. It does not imply that all humans are rational all the time: this ability is exercised in some cases but not in others. [ 6 ] [ 8 ] [ 9 ] On the other hand, the term can also refer to the process of reasoning that results from exercising this ability. Often many additional activities of the higher cognitive faculties are included as well, such as acquiring concepts, judging , deliberating , planning, and deciding as well as the formation of desires and intentions. These processes usually affect some kind of change in the thinker's mental states. In this regard, one can also talk of the rationality of mental states, like beliefs and intentions. [ 6 ] A person who possesses these forms of rationality to a sufficiently high degree may themselves be called rational . [ 1 ] In some cases, also non-mental results of rational processes may qualify as rational. For example, the arrangement of products in a supermarket can be rational if it is based on a rational plan. [ 6 ] [ 2 ]
The term "rational" has two opposites: irrational and arational . Arational things are outside the domain of rational evaluation, like digestive processes or the weather. Things within the domain of rationality are either rational or irrational depending on whether they fulfill the standards of rationality. [ 10 ] [ 7 ] For example, beliefs, actions, or general policies are rational if there is a good reason for them and irrational otherwise. It is not clear in all cases what belongs to the domain of rational assessment. For example, there are disagreements about whether desires and emotions can be evaluated as rational and irrational rather than arational. [ 6 ] The term "irrational" is sometimes used in a wide sense to include cases of arationality. [ 11 ]
The meaning of the terms "rational" and "irrational" in academic discourse often differs from how they are used in everyday language. Examples of behaviors considered irrational in ordinary discourse are giving into temptations , going out late even though one has to get up early in the morning, smoking despite being aware of the health risks, or believing in astrology . [ 12 ] [ 13 ] In the academic discourse, on the other hand, rationality is usually identified with being guided by reasons or following norms of internal coherence. Some of the earlier examples may qualify as rational in the academic sense depending on the circumstances. Examples of irrationality in this sense include cognitive biases and violating the laws of probability theory when assessing the likelihood of future events. [ 12 ] This article focuses mainly on irrationality in the academic sense.
The terms "rationality", " reason ", and "reasoning" are frequently used as synonyms. But in technical contexts, their meanings are often distinguished. [ 7 ] [ 12 ] [ 1 ] Reason is usually understood as the faculty responsible for the process of reasoning. [ 7 ] [ 14 ] This process aims at improving mental states. Reasoning tries to ensure that the norms of rationality obtain. It differs from rationality nonetheless since other psychological processes besides reasoning may have the same effect. [ 7 ] Rationality derives etymologically from the Latin term rationalitas . [ 6 ]
There are many disputes about the essential characteristics of rationality. It is often understood in relational terms: something, like a belief or an intention, is rational because of how it is related to something else. [ 6 ] [ 1 ] But there are disagreements as to what it has to be related to and in what way. For reason-based accounts, the relation to a reason that justifies or explains the rational state is central. For coherence-based accounts, the relation of coherence between mental states matters. There is a lively discussion in the contemporary literature on whether reason-based accounts or coherence-based accounts are superior. [ 15 ] [ 5 ] Some theorists also try to understand rationality in relation to the goals it tries to realize. [ 1 ] [ 16 ]
Other disputes in this field concern whether rationality depends only on the agent's mind or also on external factors, whether rationality requires a review of all one's beliefs from scratch, and whether we should always be rational. [ 6 ] [ 1 ] [ 12 ]
A common idea of many theories of rationality is that it can be defined in terms of reasons. In this view, to be rational means to respond correctly to reasons. [ 2 ] [ 1 ] [ 15 ] For example, the fact that a food is healthy is a reason to eat it. So this reason makes it rational for the agent to eat the food. [ 15 ] An important aspect of this interpretation is that it is not sufficient to merely act accidentally in accordance with reasons. Instead, responding to reasons implies that one acts intentionally because of these reasons. [ 2 ]
Some theorists understand reasons as external facts. This view has been criticized based on the claim that, in order to respond to reasons, people have to be aware of them, i.e. they have some form of epistemic access. [ 15 ] [ 5 ] But lacking this access is not automatically irrational. In one example by John Broome , the agent eats a fish contaminated with salmonella , which is a strong reason against eating the fish. But since the agent could not have known this fact, eating the fish is rational for them. [ 17 ] [ 18 ] Because of such problems, many theorists have opted for an internalist version of this account. This means that the agent does not need to respond to reasons in general, but only to reasons they have or possess. [ 2 ] [ 15 ] [ 5 ] [ 19 ] The success of such approaches depends a lot on what it means to have a reason and there are various disagreements on this issue. [ 7 ] [ 15 ] A common approach is to hold that this access is given through the possession of evidence in the form of cognitive mental states , like perceptions and knowledge . A similar version states that "rationality consists in responding correctly to beliefs about reasons". So it is rational to bring an umbrella if the agent has strong evidence that it is going to rain. But without this evidence, it would be rational to leave the umbrella at home, even if, unbeknownst to the agent, it is going to rain. [ 2 ] [ 19 ] These versions avoid the previous objection since rationality no longer requires the agent to respond to external factors of which they could not have been aware. [ 2 ]
A problem faced by all forms of reason-responsiveness theories is that there are usually many reasons relevant and some of them may conflict with each other. So while salmonella contamination is a reason against eating the fish, its good taste and the desire not to offend the host are reasons in favor of eating it. This problem is usually approached by weighing all the different reasons. This way, one does not respond directly to each reason individually but instead to their weighted sum . Cases of conflict are thus solved since one side usually outweighs the other. So despite the reasons cited in favor of eating the fish, the balance of reasons stands against it, since avoiding a salmonella infection is a much weightier reason than the other reasons cited. [ 17 ] [ 18 ] This can be expressed by stating that rational agents pick the option favored by the balance of reasons. [ 7 ] [ 20 ]
However, other objections to the reason-responsiveness account are not so easily solved. They often focus on cases where reasons require the agent to be irrational, leading to a rational dilemma. For example, if terrorists threaten to blow up a city unless the agent forms an irrational belief, this is a very weighty reason to do all in one's power to violate the norms of rationality. [ 2 ] [ 21 ]
An influential rival to the reason-responsiveness account understands rationality as internal coherence. [ 15 ] [ 5 ] On this view, a person is rational to the extent that their mental states and actions are coherent with each other. [ 15 ] [ 5 ] Diverse versions of this approach exist that differ in how they understand coherence and what rules of coherence they propose. [ 7 ] [ 20 ] [ 2 ] A general distinction in this regard is between negative and positive coherence. [ 12 ] [ 22 ] Negative coherence is an uncontroversial aspect of most such theories: it requires the absence of contradictions and inconsistencies . This means that the agent's mental states do not clash with each other. In some cases, inconsistencies are rather obvious, as when a person believes that it will rain tomorrow and that it will not rain tomorrow. In complex cases, inconsistencies may be difficult to detect, for example, when a person believes in the axioms of Euclidean geometry and is nonetheless convinced that it is possible to square the circle . Positive coherence refers to the support that different mental states provide for each other. For example, there is positive coherence between the belief that there are eight planets in the Solar System and the belief that there are less than ten planets in the Solar System: the earlier belief implies the latter belief. Other types of support through positive coherence include explanatory and causal connections. [ 12 ] [ 22 ]
Coherence-based accounts are also referred to as rule-based accounts since the different aspects of coherence are often expressed in precise rules. In this regard, to be rational means to follow the rules of rationality in thought and action. According to the enkratic rule, for example, rational agents are required to intend what they believe they ought to do. This requires coherence between beliefs and intentions. The norm of persistence states that agents should retain their intentions over time. This way, earlier mental states cohere with later ones. [ 15 ] [ 12 ] [ 5 ] It is also possible to distinguish different types of rationality, such as theoretical or practical rationality, based on the different sets of rules they require. [ 7 ] [ 20 ]
One problem with such coherence-based accounts of rationality is that the norms can enter into conflict with each other, so-called rational dilemmas . For example, if the agent has a pre-existing intention that turns out to conflict with their beliefs, then the enkratic norm requires them to change it, which is disallowed by the norm of persistence. This suggests that, in cases of rational dilemmas, it is impossible to be rational, no matter which norm is privileged. [ 15 ] [ 23 ] [ 24 ] Some defenders of coherence theories of rationality have argued that, when formulated correctly, the norms of rationality cannot enter into conflict with each other. That means that rational dilemmas are impossible. This is sometimes tied to additional non-trivial assumptions, such that ethical dilemmas also do not exist. A different response is to bite the bullet and allow that rational dilemmas exist. This has the consequence that, in such cases, rationality is not possible for the agent and theories of rationality cannot offer guidance to them. [ 15 ] [ 23 ] [ 24 ] These problems are avoided by reason-responsiveness accounts of rationality since they "allow for rationality despite conflicting reasons but [coherence-based accounts] do not allow for rationality despite conflicting requirements". Some theorists suggest a weaker criterion of coherence to avoid cases of necessary irrationality: rationality requires not to obey all norms of coherence but to obey as many norms as possible. So in rational dilemmas, agents can still be rational if they violate the minimal number of rational requirements. [ 15 ]
Another criticism rests on the claim that coherence-based accounts are either redundant or false. On this view, either the rules recommend the same option as the balance of reasons or a different option. If they recommend the same option, they are redundant. If they recommend a different option, they are false since, according to its critics, there is no special value in sticking to rules against the balance of reasons. [ 7 ] [ 20 ]
A different approach characterizes rationality in relation to the goals it aims to achieve. [ 1 ] [ 16 ] In this regard, theoretical rationality aims at epistemic goals, like acquiring truth and avoiding falsehood. Practical rationality, on the other hand, aims at non-epistemic goals, like moral , prudential, political, economic, or aesthetic goals. This is usually understood in the sense that rationality follows these goals but does not set them. So rationality may be understood as a " minister without portfolio " since it serves goals external to itself. [ 1 ] This issue has been the source of an important historical discussion between David Hume and Immanuel Kant . The slogan of Hume's position is that "reason is the slave of the passions". This is often understood as the claim that rationality concerns only how to reach a goal but not whether the goal should be pursued at all. So people with perverse or weird goals may still be perfectly rational. This position is opposed by Kant, who argues that rationality requires having the right goals and motives . [ 7 ] [ 25 ] [ 26 ] [ 27 ] [ 1 ]
According to William Frankena there are four conceptions of rationality based on the goals it tries to achieve. They correspond to egoism , utilitarianism , perfectionism , and intuitionism . [ 1 ] [ 28 ] [ 29 ] According to the egoist perspective, rationality implies looking out for one's own happiness . This contrasts with the utilitarian point of view, which states that rationality entails trying to contribute to everyone's well-being or to the greatest general good. For perfectionism, a certain ideal of perfection, either moral or non-moral, is the goal of rationality. According to the intuitionist perspective, something is rational "if and only if [it] conforms to self-evident truths, intuited by reason". [ 1 ] [ 28 ] These different perspectives diverge a lot concerning the behavior they prescribe. One problem for all of them is that they ignore the role of the evidence or information possessed by the agent. In this regard, it matters for rationality not just whether the agent acts efficiently towards a certain goal but also what information they have and how their actions appear reasonable from this perspective. Richard Brandt responds to this idea by proposing a conception of rationality based on relevant information: "Rationality is a matter of what would survive scrutiny by all relevant information." [ 1 ] This implies that the subject repeatedly reflects on all the relevant facts, including formal facts like the laws of logic. [ 1 ]
An important contemporary discussion in the field of rationality is between internalists and externalists . [ 1 ] [ 30 ] [ 31 ] Both sides agree that rationality demands and depends in some sense on reasons. They disagree on what reasons are relevant or how to conceive those reasons. Internalists understand reasons as mental states, for example, as perceptions, beliefs, or desires. In this view, an action may be rational because it is in tune with the agent's beliefs and realizes their desires. Externalists, on the other hand, see reasons as external factors about what is good or right. They state that whether an action is rational also depends on its actual consequences. [ 1 ] [ 30 ] [ 31 ] The difference between the two positions is that internalists affirm and externalists reject the claim that rationality supervenes on the mind. This claim means that it only depends on the person's mind whether they are rational and not on external factors. So for internalism, two persons with the same mental states would both have the same degree of rationality independent of how different their external situation is. Because of this limitation, rationality can diverge from actuality. So if the agent has a lot of misleading evidence, it may be rational for them to turn left even though the actually correct path goes right. [ 2 ] [ 1 ]
Bernard Williams has criticized externalist conceptions of rationality based on the claim that rationality should help explain what motivates the agent to act. This is easy for internalism but difficult for externalism since external reasons can be independent of the agent's motivation. [ 1 ] [ 32 ] [ 33 ] Externalists have responded to this objection by distinguishing between motivational and normative reasons . [ 1 ] Motivational reasons explain why someone acts the way they do while normative reasons explain why someone ought to act in a certain way. Ideally, the two overlap, but they can come apart. For example, liking chocolate cake is a motivational reason for eating it while having high blood pressure is a normative reason for not eating it. [ 34 ] [ 35 ] The problem of rationality is primarily concerned with normative reasons. This is especially true for various contemporary philosophers who hold that rationality can be reduced to normative reasons. [ 2 ] [ 17 ] [ 18 ] The distinction between motivational and normative reasons is usually accepted, but many theorists have raised doubts that rationality can be identified with normativity. On this view, rationality may sometimes recommend suboptimal actions, for example, because the agent lacks important information or has false information. In this regard, discussions between internalism and externalism overlap with discussions of the normativity of rationality. [ 1 ]
An important implication of internalist conceptions is that rationality is relative to the person's perspective or mental states. Whether a belief or an action is rational usually depends on which mental states the person has. So carrying an umbrella for the walk to the supermarket is rational for a person believing that it will rain but irrational for another person who lacks this belief. [ 6 ] [ 36 ] [ 37 ] According to Robert Audi , this can be explained in terms of experience : what is rational depends on the agent's experience. Since different people make different experiences, there are differences in what is rational for them. [ 36 ]
Rationality is normative in the sense that it sets up certain rules or standards of correctness: to be rational is to comply with certain requirements. [ 2 ] [ 15 ] [ 16 ] For example, rationality requires that the agent does not have contradictory beliefs. Many discussions on this issue concern the question of what exactly these standards are. Some theorists characterize the normativity of rationality in the deontological terms of obligations and permissions . Others understand them from an evaluative perspective as good or valuable. A further approach is to talk of rationality based on what is praise- and blameworthy. [ 1 ] It is important to distinguish the norms of rationality from other types of norms. For example, some forms of fashion prescribe that men do not wear bell-bottom trousers . Understood in the strongest sense, a norm prescribes what an agent ought to do or what they have most reason to do. The norms of fashion are not norms in this strong sense: that it is unfashionable does not mean that men ought not to wear bell-bottom trousers. [ 2 ]
Most discussions of the normativity of rationality are interested in the strong sense, i.e. whether agents ought always to be rational. [ 2 ] [ 18 ] [ 17 ] [ 38 ] This is sometimes termed a substantive account of rationality in contrast to structural accounts. [ 2 ] [ 15 ] One important argument in favor of the normativity of rationality is based on considerations of praise- and blameworthiness. It states that we usually hold each other responsible for being rational and criticize each other when we fail to do so. This practice indicates that irrationality is some form of fault on the side of the subject that should not be the case. [ 39 ] [ 38 ] A strong counterexample to this position is due to John Broome , who considers the case of a fish an agent wants to eat. It contains salmonella, which is a decisive reason why the agent ought not to eat it. But the agent is unaware of this fact, which is why it is rational for them to eat the fish. [ 17 ] [ 18 ] So this would be a case where normativity and rationality come apart. This example can be generalized in the sense that rationality only depends on the reasons accessible to the agent or how things appear to them. What one ought to do, on the other hand, is determined by objectively existing reasons. [ 40 ] [ 38 ] In the ideal case, rationality and normativity may coincide but they come apart either if the agent lacks access to a reason or if he has a mistaken belief about the presence of a reason. These considerations are summed up in the statement that rationality supervenes only on the agent's mind but normativity does not. [ 41 ] [ 42 ]
But there are also thought experiments in favor of the normativity of rationality. One, due to Frank Jackson , involves a doctor who receives a patient with a mild condition and has to prescribe one out of three drugs: drug A resulting in a partial cure, drug B resulting in a complete cure, or drug C resulting in the patient's death. [ 43 ] The doctor's problem is that they cannot tell which of the drugs B and C results in a complete cure and which one in the patient's death. The objectively best case would be for the patient to get drug B, but it would be highly irresponsible for the doctor to prescribe it given the uncertainty about its effects. So the doctor ought to prescribe the less effective drug A, which is also the rational choice. This thought experiment indicates that rationality and normativity coincide since what is rational and what one ought to do depends on the agent's mind after all. [ 40 ] [ 38 ]
Some theorists have responded to these thought experiments by distinguishing between normativity and responsibility . [ 38 ] On this view, critique of irrational behavior, like the doctor prescribing drug B, involves a negative evaluation of the agent in terms of responsibility but remains silent on normative issues. On a competence-based account, which defines rationality in terms of the competence of responding to reasons, such behavior can be understood as a failure to execute one's competence. But sometimes we are lucky and we succeed in the normative dimension despite failing to perform competently, i.e. rationally, due to being irresponsible. [ 38 ] [ 44 ] The opposite can also be the case: bad luck may result in failure despite a responsible, competent performance. This explains how rationality and normativity can come apart despite our practice of criticizing irrationality. [ 38 ] [ 45 ]
The concept of normativity can also be used to distinguish different theories of rationality. Normative theories explore the normative nature of rationality. They are concerned with rules and ideals that govern how the mind should work. Descriptive theories, on the other hand, investigate how the mind actually works. This includes issues like under which circumstances the ideal rules are followed as well as studying the underlying psychological processes responsible for rational thought. Descriptive theories are often investigated in empirical psychology while philosophy tends to focus more on normative issues. This division also reflects how different these two types are investigated. [ 6 ] [ 46 ] [ 16 ] [ 47 ]
Descriptive and normative theorists usually employ different methodologies in their research. Descriptive issues are studied by empirical research . This can take the form of studies that present their participants with a cognitive problem. It is then observed how the participants solve the problem, possibly together with explanations of why they arrived at a specific solution. Normative issues, on the other hand, are usually investigated in similar ways to how the formal sciences conduct their inquiry. [ 6 ] [ 46 ] In the field of theoretical rationality, for example, it is accepted that deductive reasoning in the form of modus ponens leads to rational beliefs. This claim can be investigated using methods like rational intuition or careful deliberation toward a reflective equilibrium . These forms of investigation can arrive at conclusions about what forms of thought are rational and irrational without depending on empirical evidence . [ 6 ] [ 48 ] [ 49 ]
An important question in this field concerns the relation between descriptive and normative approaches to rationality. [ 6 ] [ 16 ] [ 47 ] One difficulty in this regard is that there is in many cases a huge gap between what the norms of ideal rationality prescribe and how people actually reason. Examples of normative systems of rationality are classical logic , probability theory , and decision theory . Actual reasoners often diverge from these standards because of cognitive biases , heuristics, or other mental limitations. [ 6 ]
Traditionally, it was often assumed that actual human reasoning should follow the rules described in normative theories. In this view, any discrepancy is a form of irrationality that should be avoided. However, this usually ignores the human limitations of the mind. Given these limitations, various discrepancies may be necessary (and in this sense rational ) to get the most useful results. [ 6 ] [ 12 ] [ 1 ] For example, the ideal rational norms of decision theory demand that the agent should always choose the option with the highest expected value. However, calculating the expected value of each option may take a very long time in complex situations and may not be worth the trouble. This is reflected in the fact that actual reasoners often settle for an option that is good enough without making certain that it is really the best option available. [ 1 ] [ 50 ] A further difficulty in this regard is Hume's law , which states that one cannot deduce what ought to be based on what is. [ 51 ] [ 52 ] So just because a certain heuristic or cognitive bias is present in a specific case, it should not be inferred that it should be present. One approach to these problems is to hold that descriptive and normative theories talk about different types of rationality. This way, there is no contradiction between the two and both can be correct in their own field. Similar problems are discussed in so-called naturalized epistemology . [ 6 ] [ 53 ]
Rationality is usually understood as conservative in the sense that rational agents do not start from zero but already possess many beliefs and intentions. Reasoning takes place on the background of these pre-existing mental states and tries to improve them. This way, the original beliefs and intentions are privileged: one keeps them unless a reason to doubt them is encountered. Some forms of epistemic foundationalism reject this approach. According to them, the whole system of beliefs is to be justified by self-evident beliefs. Examples of such self-evident beliefs may include immediate experiences as well as simple logical and mathematical axioms . [ 12 ] [ 54 ] [ 55 ]
An important difference between conservatism and foundationalism concerns their differing conceptions of the burden of proof . According to conservativism, the burden of proof is always in favor of already established belief: in the absence of new evidence, it is rational to keep the mental states one already has. According to foundationalism, the burden of proof is always in favor of suspending mental states. For example, the agent reflects on their pre-existing belief that the Taj Mahal is in Agra but is unable to access any reason for or against this belief. In this case, conservatives think it is rational to keep this belief while foundationalists reject it as irrational due to the lack of reasons. In this regard, conservatism is much closer to the ordinary conception of rationality. One problem for foundationalism is that very few beliefs, if any, would remain if this approach was carried out meticulously. Another is that enormous mental resources would be required to constantly keep track of all the justificatory relations connecting non-fundamental beliefs to fundamental ones. [ 12 ] [ 54 ] [ 55 ]
Rationality is discussed in a great variety of fields, often in very different terms. While some theorists try to provide a unifying conception expressing the features shared by all forms of rationality, the more common approach is to articulate the different aspects of the individual forms of rationality. The most common distinction is between theoretical and practical rationality. Other classifications include categories for ideal and bounded rationality as well as for individual and social rationality. [ 6 ] [ 56 ]
The most influential distinction contrasts theoretical or epistemic rationality with practical rationality. Its theoretical side concerns the rationality of beliefs : whether it is rational to hold a given belief and how certain one should be about it. Practical rationality, on the other hand, is about the rationality of actions , intentions , and decisions . [ 7 ] [ 12 ] [ 56 ] [ 27 ] This corresponds to the distinction between theoretical reasoning and practical reasoning: theoretical reasoning tries to assess whether the agent should change their beliefs while practical reasoning tries to assess whether the agent should change their plans and intentions. [ 12 ] [ 56 ] [ 27 ]
Theoretical rationality concerns the rationality of cognitive mental states, in particular, of beliefs. [ 7 ] [ 4 ] It is common to distinguish between two factors. The first factor is about the fact that good reasons are necessary for a belief to be rational. This is usually understood in terms of evidence provided by the so-called sources of knowledge , i.e. faculties like perception , introspection , and memory . In this regard, it is often argued that to be rational, the believer has to respond to the impressions or reasons presented by these sources. For example, the visual impression of the sunlight on a tree makes it rational to believe that the sun is shining. [ 27 ] [ 7 ] [ 4 ] In this regard, it may also be relevant whether the formed belief is involuntary and implicit
The second factor pertains to the norms and procedures of rationality that govern how agents should form beliefs based on this evidence. These norms include the rules of inference discussed in regular logic as well as other norms of coherence between mental states. [ 7 ] [ 4 ] In the case of rules of inference, the premises of a valid argument offer support to the conclusion and make therefore the belief in the conclusion rational. [ 27 ] The support offered by the premises can either be deductive or non-deductive . [ 57 ] [ 58 ] In both cases, believing in the premises of an argument makes it rational to also believe in its conclusion. The difference between the two is given by how the premises support the conclusion. For deductive reasoning, the premises offer the strongest possible support: it is impossible for the conclusion to be false if the premises are true. The premises of non-deductive arguments also offer support for their conclusion. But this support is not absolute: the truth of the premises does not guarantee the truth of the conclusion. Instead, the premises make it more likely that the conclusion is true. In this case, it is usually demanded that the non-deductive support is sufficiently strong if the belief in the conclusion is to be rational. [ 56 ] [ 27 ] [ 57 ]
An important form of theoretical irrationality is motivationally biased belief, sometimes referred to as wishful thinking . In this case, beliefs are formed based on one's desires or what is pleasing to imagine without proper evidential support. [ 7 ] [ 59 ] Faulty reasoning in the form of formal and informal fallacies is another cause of theoretical irrationality. [ 60 ]
All forms of practical rationality are concerned with how we act. It pertains both to actions directly as well as to mental states and events preceding actions, like intentions and decisions . There are various aspects of practical rationality, such as how to pick a goal to follow and how to choose the means for reaching this goal. Other issues include the coherence between different intentions as well as between beliefs and intentions. [ 61 ] [ 62 ] [ 1 ]
Some theorists define the rationality of actions in terms of beliefs and desires. In this view, an action to bring about a certain goal is rational if the agent has the desire to bring about this goal and the belief that their action will realize it. A stronger version of this view requires that the responsible beliefs and desires are rational themselves. [ 6 ] A very influential conception of the rationality of decisions comes from decision theory . In decisions, the agent is presented with a set of possible courses of action and has to choose one among them. Decision theory holds that the agent should choose the alternative that has the highest expected value . [ 61 ] Practical rationality includes the field of actions but not of behavior in general. The difference between the two is that actions are intentional behavior, i.e. they are performed for a purpose and guided by it. In this regard, intentional behavior like driving a car is either rational or irrational while non-intentional behavior like sneezing is outside the domain of rationality. [ 6 ] [ 63 ] [ 64 ]
For various other practical phenomena, there is no clear consensus on whether they belong to this domain or not. For example, concerning the rationality of desires, two important theories are proceduralism and substantivism. According to proceduralism, there is an important distinction between instrumental and noninstrumental desires . A desire is instrumental if its fulfillment serves as a means to the fulfillment of another desire. [ 65 ] [ 12 ] [ 6 ] For example, Jack is sick and wants to take medicine to get healthy again. In this case, the desire to take the medicine is instrumental since it only serves as a means to Jack's noninstrumental desire to get healthy. Both proceduralism and substantivism usually agree that a person can be irrational if they lack an instrumental desire despite having the corresponding noninstrumental desire and being aware that it acts as a means. Proceduralists hold that this is the only way a desire can be irrational. Substantivists, on the other hand, allow that noninstrumental desires may also be irrational. In this regard, a substantivist could claim that it would be irrational for Jack to lack his noninstrumental desire to be healthy. [ 7 ] [ 65 ] [ 6 ] Similar debates focus on the rationality of emotions . [ 6 ]
Theoretical and practical rationality are often discussed separately and there are many differences between them. In some cases, they even conflict with each other. However, there are also various ways in which they overlap and depend on each other. [ 61 ] [ 6 ]
It is sometimes claimed that theoretical rationality aims at truth while practical rationality aims at goodness . [ 61 ] According to John Searle , the difference can be expressed in terms of " direction of fit ". [ 6 ] [ 66 ] [ 67 ] On this view, theoretical rationality is about how the mind corresponds to the world by representing it. Practical rationality, on the other hand, is about how the world corresponds to the ideal set up by the mind and how it should be changed. [ 6 ] [ 7 ] [ 68 ] [ 1 ] Another difference is that arbitrary choices are sometimes needed for practical rationality. For example, there may be two equally good routes available to reach a goal. On the practical level, one has to choose one of them if one wants to reach the goal. It would even be practically irrational to resist this arbitrary choice, as exemplified by Buridan's ass . [ 12 ] [ 69 ] But on the theoretical level, one does not have to form a belief about which route was taken upon hearing that someone reached the goal. In this case, the arbitrary choice for one belief rather than the other would be theoretically irrational. Instead, the agent should suspend their belief either way if they lack sufficient reasons. Another difference is that practical rationality is guided by specific goals and desires, in contrast to theoretical rationality. So it is practically rational to take medicine if one has the desire to cure a sickness. But it is theoretically irrational to adopt the belief that one is healthy just because one desires this. This is a form of wishful thinking . [ 12 ]
In some cases, the demands of practical and theoretical rationality conflict with each other. For example, the practical reason of loyalty to one's child may demand the belief that they are innocent while the evidence linking them to the crime may demand a belief in their guilt on the theoretical level. [ 12 ] [ 68 ]
But the two domains also overlap in certain ways. For example, the norm of rationality known as enkrasia links beliefs and intentions. It states that "rationality requires of you that you intend to F if you believe your reasons require you to F". Failing to fulfill this requirement results in cases of irrationality known as akrasia or weakness of the will . [ 2 ] [ 1 ] [ 15 ] [ 7 ] [ 59 ] Another form of overlap is that the study of the rules governing practical rationality is a theoretical matter. [ 7 ] [ 70 ] And practical considerations may determine whether to pursue theoretical rationality on a certain issue as well as how much time and resources to invest in the inquiry. [ 68 ] [ 59 ] It is often held that practical rationality presupposes theoretical rationality. This is based on the idea that to decide what should be done, one needs to know what is the case. But one can assess what is the case independently of knowing what should be done. So in this regard, one can study theoretical rationality as a distinct discipline independent of practical rationality but not the other way round. [ 6 ] However, this independence is rejected by some forms of doxastic voluntarism. They hold that theoretical rationality can be understood as one type of practical rationality. This is based on the controversial claim that we can decide what to believe. It can take the form of epistemic decision theory , which states that people try to fulfill epistemic aims when deciding what to believe. [ 6 ] [ 71 ] [ 72 ] A similar idea is defended by Jesús Mosterín . He argues that the proper object of rationality is not belief but acceptance . He understands acceptance as a voluntary and context-dependent decision to affirm a proposition. [ 73 ]
Various theories of rationality assume some form of ideal rationality, for example, by demanding that rational agents obey all the laws and implications of logic . This can include the requirement that if the agent believes a proposition , they should also believe in everything that logically follows from this proposition. However, many theorists reject this form of logical omniscience as a requirement for rationality. They argue that, since the human mind is limited, rationality has to be defined accordingly to account for how actual finite humans possess some form of resource-limited rationality. [ 12 ] [ 6 ] [ 1 ]
According to the position of bounded rationality , theories of rationality should take into account cognitive limitations, such as incomplete knowledge, imperfect memory, and limited capacities of computation and representation. An important research question in this field is about how cognitive agents use heuristics rather than brute calculations to solve problems and make decisions. According to the satisficing heuristic, for example, agents usually stop their search for the best option once an option is found that meets their desired achievement level. In this regard, people often do not continue to search for the best possible option, even though this is what theories of ideal rationality commonly demand. [ 6 ] [ 1 ] [ 50 ] Using heuristics can be highly rational as a way to adapt to the limitations of the human mind, especially in complex cases where these limitations make brute calculations impossible or very time- and resource-intensive. [ 6 ] [ 1 ]
Most discussions and research in the academic literature focus on individual rationality. This concerns the rationality of individual persons, for example, whether their beliefs and actions are rational. But the question of rationality can also be applied to groups as a whole on the social level. This form of social or collective rationality concerns both theoretical and practical issues like group beliefs and group decisions. [ 6 ] [ 74 ] [ 75 ] And just like in the individual case, it is possible to study these phenomena as well as the processes and structures that are responsible for them. On the social level, there are various forms of cooperation to reach a shared goal. In theoretical cases, a group of jurors may first discuss and then vote to determine whether the defendant is guilty. Or in the practical case, politicians may cooperate to implement new regulations to combat climate change . These forms of cooperation can be judged on their social rationality depending on how they are implemented and on the quality of the results they bear. Some theorists try to reduce social rationality to individual rationality by holding that the group processes are rational to the extent that the individuals participating in them are rational. But such a reduction is frequently rejected. [ 6 ] [ 74 ]
Various studies indicate that group rationality often outperforms individual rationality. For example, groups of people working together on the Wason selection task usually perform better than individuals by themselves. This form of group superiority is sometimes termed "wisdom of crowds" and may be explained based on the claim that competent individuals have a stronger impact on the group decision than others. [ 6 ] [ 76 ] However, this is not always the case and sometimes groups perform worse due to conformity or unwillingness to bring up controversial issues. [ 6 ]
Many other classifications are discussed in the academic literature. One important distinction is between approaches to rationality based on the output or on the process. Process-oriented theories of rationality are common in cognitive psychology and study how cognitive systems process inputs to generate outputs. Output-oriented approaches are more common in philosophy and investigate the rationality of the resulting states. [ 6 ] [ 2 ] Another distinction is between relative and categorical judgments of rationality. In the relative case, rationality is judged based on limited information or evidence while categorical judgments take all the evidence into account and are thus judgments all things considered . [ 6 ] [ 1 ] For example, believing that one's investments will multiply can be rational in a relative sense because it is based on one's astrological horoscope . But this belief is irrational in a categorical sense if the belief in astrology is itself irrational. [ 6 ]
Rationality is central to solving many problems, both on the local and the global scale. This is often based on the idea that rationality is necessary to act efficiently and to reach all kinds of goals. [ 6 ] [ 16 ] This includes goals from diverse fields, such as ethical goals, humanist goals, scientific goals, and even religious goals. [ 6 ] The study of rationality is very old and has occupied many of the greatest minds since ancient Greek. This interest is often motivated by discovering the potentials and limitations of our minds. Various theorists even see rationality as the essence of being human, often in an attempt to distinguish humans from other animals. [ 6 ] [ 8 ] [ 9 ] However, this strong affirmation has been subjected to many criticisms, for example, that humans are not rational all the time and that non-human animals also show diverse forms of intelligence. [ 6 ]
The topic of rationality is relevant to a variety of disciplines. It plays a central role in philosophy, psychology, Bayesianism , decision theory , and game theory . [ 7 ] But it is also covered in other disciplines, such as artificial intelligence , behavioral economics , microeconomics , and neuroscience . Some forms of research restrict themselves to one specific domain while others investigate the topic in an interdisciplinary manner by drawing insights from different fields. [ 56 ]
The term paradox of rationality has a variety of meanings. It is often used for puzzles or unsolved problems of rationality. Some are just situations where it is not clear what the rational person should do. Others involve apparent faults within rationality itself, for example, where rationality seems to recommend a suboptimal course of action. [ 7 ] A special case are so-called rational dilemmas, in which it is impossible to be rational since two norms of rationality conflict with each other. [ 23 ] [ 24 ] Examples of paradoxes of rationality include Pascal's Wager , the Prisoner's dilemma , Buridan's ass , and the St. Petersburg paradox . [ 7 ] [ 77 ] [ 21 ]
The German scholar Max Weber proposed an interpretation of social action that distinguished between four different idealized types of rationality. [ 78 ]
The first, which he called Zweckrational or purposive/ instrumental rationality , is related to the expectations about the behavior of other human beings or objects in the environment. These expectations serve as means for a particular actor to attain ends, ends which Weber noted were "rationally pursued and calculated." [ This quote needs a citation ] The second type, Weber called Wertrational or value/belief-oriented. Here the action is undertaken for what one might call reasons intrinsic to the actor: some ethical, aesthetic, religious or other motives, independent of whether it will lead to success. The third type was affectual, determined by an actor's specific affect, feeling, or emotion—to which Weber himself said that this was a kind of rationality that was on the borderline of what he considered "meaningfully oriented." The fourth was traditional or conventional, determined by ingrained habituation. Weber emphasized that it was very unusual to find only one of these orientations: combinations were the norm. His usage also makes clear that he considered the first two as more significant than the others, and it is arguable that the third and fourth are subtypes of the first two.
The advantage in Weber's interpretation of rationality is that it avoids a value-laden assessment, say, that certain kinds of beliefs are irrational. Instead, Weber suggests that ground or motive can be given—for religious or affect reasons, for example—that may meet the criterion of explanation or justification even if it is not an explanation that fits the Zweckrational orientation of means and ends. The opposite is therefore also true: some means-ends explanations will not satisfy those whose grounds for action are Wertrational .
Weber's constructions of rationality have been critiqued both from a Habermasian (1984) perspective (as devoid of social context and under-theorised in terms of social power) [ 79 ] and also from a feminist perspective (Eagleton, 2003) whereby Weber's rationality constructs are viewed as imbued with masculine values and oriented toward the maintenance of male power. [ 80 ] An alternative position on rationality (which includes both bounded rationality , [ 81 ] as well as the affective and value-based arguments of Weber) can be found in the critique of Etzioni (1988), [ 82 ] who reframes thought on decision-making to argue for a reversal of the position put forward by Weber. Etzioni illustrates how purposive/instrumental reasoning is subordinated by normative considerations (ideas on how people 'ought' to behave) and affective considerations (as a support system for the development of human relationships).
Richard Brandt proposed a "reforming definition" of rationality, arguing someone is rational if their notions survive a form of cognitive-psychotherapy . [ 83 ]
Robert Audi developed a comprehensive account of rationality that covers both the theoretical and the practical side of rationality. [ 36 ] [ 84 ] This account centers on the notion of a ground : a mental state is rational if it is "well-grounded" in a source of justification . [ 84 ] : 19 Irrational mental states, on the other hand, lack a sufficient ground. For example, the perceptual experience of a tree when looking outside the window can ground the rationality of the belief that there is a tree outside.
Audi is committed to a form of foundationalism : the idea that justified beliefs, or in his case, rational states in general, can be divided into two groups: the foundation and the superstructure . [ 84 ] : 13, 29–31 The mental states in the superstructure receive their justification from other rational mental states while the foundational mental states receive their justification from a more basic source. [ 84 ] : 16–18 For example, the above-mentioned belief that there is a tree outside is foundational since it is based on a basic source: perception. Knowing that trees grow in soil, we may deduce that there is soil outside. This belief is equally rational, being supported by an adequate ground, but it belongs to the superstructure since its rationality is grounded in the rationality of another belief. Desires, like beliefs, form a hierarchy: intrinsic desires are at the foundation while instrumental desires belong to the superstructure. In order to link the instrumental desire to the intrinsic desire an extra element is needed: a belief that the fulfillment of the instrumental desire is a means to the fulfillment of the intrinsic desire. [ 85 ]
Audi asserts that all the basic sources providing justification for the foundational mental states come from experience . As for beliefs , there are four types of experience that act as sources: perception, memory, introspection, and rational intuition. [ 86 ] The main basic source of the rationality of desires , on the other hand, comes in the form of hedonic experience: the experience of pleasure and pain. [ 87 ] : 20 So, for example, a desire to eat ice-cream is rational if it is based on experiences in which the agent enjoyed the taste of ice-cream, and irrational if it lacks such a support. Because of its dependence on experience, rationality can be defined as a kind of responsiveness to experience. [ 87 ] : 21
Actions , in contrast to beliefs and desires, do not have a source of justification of their own. Their rationality is grounded in the rationality of other states instead: in the rationality of beliefs and desires. Desires motivate actions. Beliefs are needed here, as in the case of instrumental desires, to bridge a gap and link two elements. [ 84 ] : 62 Audi distinguishes the focal rationality of individual mental states from the global rationality of persons . Global rationality has a derivative status: it depends on the focal rationality. [ 36 ] Or more precisely: "Global rationality is reached when a person has a sufficiently integrated system of sufficiently well-grounded propositional attitudes, emotions, and actions". [ 84 ] : 232 Rationality is relative in the sense that it depends on the experience of the person in question. Since different people undergo different experiences, what is rational to believe for one person may be irrational to believe for another person. [ 36 ] That a belief is rational does not entail that it is true . [ 85 ]
The problem of rationality is relevant to various issues in ethics and morality . [ 7 ] Many debates center around the question of whether rationality implies morality or is possible without it. Some examples based on common sense suggest that the two can come apart. For example, some immoral psychopaths are highly intelligent in the pursuit of their schemes and may, therefore, be seen as rational. However, there are also considerations suggesting that the two are closely related to each other. For example, according to the principle of universality, "one's reasons for acting are acceptable only if it is acceptable that everyone acts on such reasons". [ 12 ] A similar formulation is given in Immanuel Kant 's categorical imperative : "act only according to that maxim whereby you can, at the same time, will that it should become a universal law". [ 88 ] The principle of universality has been suggested as a basic principle both for morality and for rationality. [ 12 ] This is closely related to the question of whether agents have a duty to be rational. Another issue concerns the value of rationality. In this regard, it is often held that human lives are more important than animal lives because humans are rational. [ 12 ] [ 8 ]
Many psychological theories have been proposed to describe how reasoning happens and what underlying psychological processes are responsible. One of their goals is to explain how the different types of irrationality happen and why some types are more prevalent than others. They include mental logic theories , mental model theories , and dual process theories . [ 56 ] [ 89 ] [ 90 ] An important psychological area of study focuses on cognitive biases . Cognitive biases are systematic tendencies to engage in erroneous or irrational forms of thinking , judging , and acting. Examples include the confirmation bias , the self-serving bias , the hindsight bias , and the Dunning–Kruger effect . [ 91 ] [ 92 ] [ 93 ] Some empirical findings suggest that metacognition is an important aspect of rationality. The idea behind this claim is that reasoning is carried out more efficiently and reliably if the responsible thought processes are properly controlled and monitored. [ 56 ]
The Wason selection task is an influential test for studying rationality and reasoning abilities. In it, four cards are placed before the participants. Each has a number on one side and a letter on the opposite side. In one case, the visible sides of the four cards are A, D, 4, and 7. The participant is then asked which cards need to be turned around in order to verify the conditional claim "if there is a vowel on one side of the card, then there is an even number on the other side of the card". The correct answer is A and 7. But this answer is only given by about 10%. Many choose card 4 instead even though there is no requirement on what letters may appear on its opposite side. [ 6 ] [ 89 ] [ 94 ] An important insight from using these and similar tests is that the rational ability of the participants is usually significantly better for concrete and realistic cases than for abstract or implausible cases. [ 89 ] [ 94 ] Various contemporary studies in this field use Bayesian probability theory to study subjective degrees of belief, for example, how the believer's certainty in the premises is carried over to the conclusion through reasoning. [ 6 ]
In the psychology of reasoning , psychologists and cognitive scientists have defended different positions on human rationality. One prominent view, due to Philip Johnson-Laird and Ruth M. J. Byrne among others is that humans are rational in principle but they err in practice, that is, humans have the competence to be rational but their performance is limited by various factors. [ 95 ] However, it has been argued that many standard tests of reasoning, such as those on the conjunction fallacy , on the Wason selection task , or the base rate fallacy suffer from methodological and conceptual problems. This has led to disputes in psychology over whether researchers should (only) use standard rules of logic, probability theory and statistics, or rational choice theory as norms of good reasoning. Opponents of this view, such as Gerd Gigerenzer , favor a conception of bounded rationality , especially for tasks under high uncertainty. [ 96 ] The concept of rationality continues to be debated by psychologists, economists and cognitive scientists. [ 97 ]
The psychologist Jean Piaget gave an influential account of how the stages in human development from childhood to adulthood can be understood in terms of the increase of rational and logical abilities. [ 6 ] [ 98 ] [ 99 ] [ 100 ] He identifies four stages associated with rough age groups: the sensorimotor stage below the age of two, the preoperational state until the age of seven, the concrete operational stage until the age of eleven, and the formal operational stage afterward. Rational or logical reasoning only takes place in the last stage and is related to abstract thinking , concept formation , reasoning, planning, and problem-solving . [ 6 ]
According to A. C. Grayling, rationality "must be independent of emotions, personal feelings or any kind of instincts". [ 101 ] Certain findings [ which? ] in cognitive science and neuroscience show that no human has ever satisfied this criterion, except perhaps a person with no affective feelings, for example, an individual with a massively damaged amygdala or severe psychopathy. Thus, such an idealized form of rationality is best exemplified by computers, and not people. However, scholars may productively appeal to the idealization as a point of reference. [ citation needed ] In his book, The Edge of Reason: A Rational Skeptic in an Irrational World, British philosopher Julian Baggini sets out to debunk myths about reason (e.g., that it is "purely objective and requires no subjective judgment"). [ 102 ]
Cognitive and behavioral sciences try to describe, explain, and predict how people think and act. Their models are often based on the assumption that people are rational. For example, classical economics is based on the assumption that people are rational agents that maximize expected utility. However, people often depart from the ideal standards of rationality in various ways. For example, they may only look for confirming evidence and ignore disconfirming evidence. Another factor studied in this regard are the limitations of human intellectual capacities. Many discrepancies from rationality are caused by limited time, memory, or attention. Often heuristics and rules of thumb are used to mitigate these limitations, but they may lead to new forms of irrationality. [ 12 ] [ 1 ] [ 50 ]
Theoretical rationality is closely related to logic , but not identical to it. [ 12 ] [ 6 ] Logic is often defined as the study of correct arguments . This concerns the relation between the propositions used in the argument: whether its premises offer support to its conclusion. Theoretical rationality, on the other hand, is about what to believe or how to change one's beliefs. The laws of logic are relevant to rationality since the agent should change their beliefs if they violate these laws. But logic is not directly about what to believe. Additionally, there are also other factors and norms besides logic that determine whether it is rational to hold or change a belief. [ 12 ] The study of rationality in logic is more concerned with epistemic rationality, that is, attaining beliefs in a rational manner, than instrumental rationality.
An influential account of practical rationality is given by decision theory . [ 12 ] [ 56 ] [ 6 ] Decisions are situations where a number of possible courses of action are available to the agent, who has to choose one of them. Decision theory investigates the rules governing which action should be chosen. It assumes that each action may lead to a variety of outcomes. Each outcome is associated with a conditional probability and a utility . The expected gain of an outcome can be calculated by multiplying its conditional probability with its utility. The expected utility of an act is equivalent to the sum of all expected gains of the outcomes associated with it. From these basic ingredients, it is possible to define the rationality of decisions: a decision is rational if it selects the act with the highest expected utility. [ 12 ] [ 6 ] While decision theory gives a very precise formal treatment of this issue, it leaves open the empirical problem of how to assign utilities and probabilities. So decision theory can still lead to bad empirical decisions if it is based on poor assignments. [ 12 ]
According to decision theorists, rationality is primarily a matter of internal consistency. This means that a person's mental states like beliefs and preferences are consistent with each other or do not go against each other. One consequence of this position is that people with obviously false beliefs or perverse preferences may still count as rational if these mental states are consistent with their other mental states. [ 7 ] Utility is often understood in terms of self-interest or personal preferences . However, this is not a necessary aspect of decisions theory and it can also be interpreted in terms of goodness or value in general . [ 7 ] [ 70 ]
Game theory is closely related to decision theory and the problem of rational choice. [ 7 ] [ 56 ] Rational choice is based on the idea that rational agents perform a cost-benefit analysis of all available options and choose the option that is most beneficial from their point of view. In the case of game theory, several agents are involved. This further complicates the situation since whether a given option is the best choice for one agent may depend on choices made by other agents. Game theory can be used to analyze various situations, like playing chess, firms competing for business, or animals fighting over prey. Rationality is a core assumption of game theory: it is assumed that each player chooses rationally based on what is most beneficial from their point of view. This way, the agent may be able to anticipate how others choose and what their best choice is relative to the behavior of the others. [ 7 ] [ 103 ] [ 104 ] [ 105 ] This often results in a Nash equilibrium , which constitutes a set of strategies, one for each player, where no player can improve their outcome by unilaterally changing their strategy. [ 7 ] [ 103 ] [ 104 ]
A popular contemporary approach to rationality is based on Bayesian epistemology . [ 7 ] [ 106 ] Bayesian epistemology sees belief as a continuous phenomenon that comes in degrees. For example, Daniel is relatively sure that the Boston Celtics will win their next match and absolutely certain that two plus two equals four. In this case, the degree of the first belief is weaker than the degree of the second belief. These degrees are usually referred to as credences and represented by numbers between 0 and 1, where 0 corresponds to full disbelief, 1 corresponds to full belief and 0.5 corresponds to suspension of belief. Bayesians understand this in terms of probability : the higher the credence, the higher the subjective probability that the believed proposition is true. As probabilities, they are subject to the laws of probability theory . These laws act as norms of rationality: beliefs are rational if they comply with them and irrational if they violate them. [ 107 ] [ 108 ] [ 109 ] For example, it would be irrational to have a credence of 0.9 that it will rain tomorrow together with another credence of 0.9 that it will not rain tomorrow. This account of rationality can also be extended to the practical domain by requiring that agents maximize their subjective expected utility. This way, Bayesianism can provide a unified account of both theoretical and practical rationality. [ 7 ] [ 106 ] [ 6 ]
Rationality plays a key role in economics and there are several strands to this. [ 110 ] Firstly, there is the concept of instrumentality—basically the idea that people and organisations are instrumentally rational—that is, adopt the best actions to achieve their goals. Secondly, there is an axiomatic concept that rationality is a matter of being logically consistent within your preferences and beliefs. Thirdly, people have focused on the accuracy of beliefs and full use of information—in this view, a person who is not rational has beliefs that do not fully use the information they have.
Debates within economic sociology also arise as to whether or not people or organizations are "really" rational, as well as whether it makes sense to model them as such in formal models. Some have argued that a kind of bounded rationality makes more sense for such models.
Others think that any kind of rationality along the lines of rational choice theory is a useless concept for understanding human behavior; the term homo economicus (economic man: the imaginary man being assumed in economic models who is logically consistent but amoral) was coined largely in honor of this view. Behavioral economics aims to account for economic actors as they actually are, allowing for psychological biases, rather than assuming idealized instrumental rationality.
The field of artificial intelligence is concerned, among other things, with how problems of rationality can be implemented and solved by computers. [ 56 ] Within artificial intelligence , a rational agent is typically one that maximizes its expected utility, given its current knowledge. Utility is the usefulness of the consequences of its actions. The utility function is arbitrarily defined by the designer, but should be a function of "performance", which is the directly measurable consequences, such as winning or losing money. In order to make a safe agent that plays defensively, a nonlinear function of performance is often desired, so that the reward for winning is lower than the punishment for losing. An agent might be rational within its own problem area, but finding the rational decision for arbitrarily complex problems is not practically possible. The rationality of human thought is a key problem in the psychology of reasoning . [ 111 ]
There is an ongoing debate over the merits of using "rationality" in the study of international relations (IR). Some scholars hold it indispensable. [ 112 ] Others are more critical. [ 113 ] Still, the pervasive and persistent usage of "rationality" in political science and IR is beyond dispute. "Rationality" remains ubiquitous in this field. Abulof finds that Some 40% of all scholarly references to "foreign policy" allude to "rationality"—and this ratio goes up to more than half of pertinent academic publications in the 2000s. He further argues that when it comes to concrete security and foreign policies, IR employment of rationality borders on "malpractice": rationality-based descriptions are largely either false or unfalsifiable; many observers fail to explicate the meaning of "rationality" they employ; and the concept is frequently used politically to distinguish between "us and them." [ 114 ]
The concept of rationality has been subject to criticism by various philosophers who question its universality and capacity to provide a comprehensive understanding of reality and human existence .
Friedrich Nietzsche , in his work " Beyond Good and Evil " (1886), criticized the overemphasis on rationality and argued that it neglects the irrational and instinctual aspects of human nature. Nietzsche advocated for a reevaluation of values based on individual perspectives and the will to power , stating, "There are no facts, only interpretations." [ 115 ]
Martin Heidegger , in " Being and Time " (1927), offered a critique of the instrumental and calculative view of reason , emphasizing the primacy of our everyday practical engagement with the world. Heidegger challenged the notion that rationality alone is the sole arbiter of truth and understanding. [ 116 ]
Max Horkheimer and Theodor Adorno , in their seminal work " Dialectic of Enlightenment " [ 117 ] (1947), questioned the Enlightenment 's rationality. They argued that the dominance of instrumental reason in modern society leads to the domination of nature and the dehumanization of individuals. Horkheimer and Adorno highlighted how rationality narrows the scope of human experience and hinders critical thinking.
Michel Foucault , in " Discipline and Punish " [ 118 ] (1975) and " The Birth of Biopolitics " [ 119 ] (1978), critiqued the notion of rationality as a neutral and objective force. Foucault emphasized the intertwining of rationality with power structures and its role in social control. He famously stated, "Power is not an institution, and not a structure; neither is it a certain strength we are endowed with; it is the name that one attributes to a complex strategic situation in a particular society." [ 120 ]
These philosophers' critiques of rationality shed light on its limitations, assumptions, and potential dangers. Their ideas challenge the universal application of rationality as the sole framework for understanding the complexities of human existence and the world. | https://en.wikipedia.org/wiki/Rationality |
Rationality: What It Is, Why It Seems Scarce, Why It Matters is a 2021 book written by Canadian-American cognitive scientist Steven Pinker . [ 1 ] The book was published on September 28, 2021, by the Viking imprint of Penguin Random House . [ 2 ]
It argues that rationality is a key driver of moral and social progress, and it attempts to resolve the apparent conflict between scientific progress and increasing levels of disinformation. Pinker explains several concepts underlying rationality, including from the fields of logic , probability theory , statistics , and social choice . [ 3 ]
The book debuted at number nine on The New York Times nonfiction best-seller list for the week ending October 2, 2021. [ 4 ]
In its starred review, Publishers Weekly wrote, "He manages to be scrupulously rigorous yet steadily accessible and entertaining." [ 5 ] To Andrew Anthony on The Guardian , Pinker, not a "dry and humourless slave to rational thought", "knows that what we find funny is often nothing more than clever inversions of logic". [ 6 ] Kirkus Reviews wrote, "The author can be heady and geeky, but seldom to the point that his discussions shade off into inaccessibility." [ 7 ]
On The New York Times , Jennifer Szalai commented that "The trouble arrives when he [Pinker] tries to gussy up his psychologist's hat with his more elaborate public intellectual's attire", [ 8 ] while Anthony Gottlieb noted Pinker's tendencies to "exaggerate the popularity of ill-founded beliefs" and to devote "plenty of space to advocating rationality, which the authors of similar works have not found necessary to do, perhaps because anybody who chooses to read about rationality is probably already in favor of it." [ 9 ]
This article about a non-fiction book is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rationality_(book) |
Rationalizability is a solution concept in game theory . It is the most permissive possible solution concept that still requires both players to be at least somewhat rational and know the other players are also somewhat rational, i.e. that they do not play dominated strategies . A strategy is rationalizable if there exists some possible set of beliefs both players could have about each other's actions, that would still result in the strategy being played.
Rationalizability is a broader concept than a Nash equilibrium . Both require players to respond optimally to some belief about their opponents' actions, but Nash equilibrium requires these beliefs to be correct, while rationalizability does not. Rationalizability was first defined, independently, by Bernheim (1984) and Pearce (1984).
Starting with a normal-form game , the rationalizable set of actions can be computed as follows:
In a game with finitely many actions, this process always terminates and leaves a non-empty set of actions for each player. These are the rationalizable actions.
The iterated elimination (or deletion, or removal) of dominated strategies (also denominated as IESDS, or IDSDS, or IRSDS) is one common technique for solving games that involves iteratively removing dominated strategies. In the first step, at most one dominated strategy is removed from the strategy space of each of the players since no rational player would ever play these strategies. This results in a new, smaller game. Some strategies—that were not dominated before—may be dominated in the smaller game. The first step is repeated, creating a new even smaller game, and so on. The process stops when no dominated strategy is found for any player. This process is valid since it is assumed that rationality among players is common knowledge , that is, each player knows that the rest of the players are rational, and each player knows that the rest of the players know that he knows that the rest of the players are rational, and so on ad infinitum (see Aumann, 1976).
There are two versions of this process. One version involves only eliminating strictly dominated strategies. If, after completing this process, there is only one strategy for each player remaining, that strategy set is the unique Nash equilibrium. [ 1 ] Moreover, iterated elimination of strictly dominated strategies is path independent. That is, if at any point in the process there are multiple strictly dominated strategies, then it doesn't matter for the end result which strategies we remove first. [ 2 ]
Strict Dominance Deletion Step-by-Step Example:
Another version involves eliminating both strictly and weakly dominated strategies. If, at the end of the process, there is a single strategy for each player, this strategy set is also a Nash equilibrium . However, unlike the first process, elimination of weakly dominated strategies may eliminate some Nash equilibria. As a result, the Nash equilibrium found by eliminating weakly dominated strategies may not be the only Nash equilibrium. (In some games, if we remove weakly dominated strategies in a different order, we may end up with a different Nash equilibrium.)
Weak Dominance Deletion Step-by-Step Example:
In any case, if by iterated elimination of dominated strategies there is only one strategy left for each player, the game is called a dominance-solvable game.
There are instances when there is no pure strategy that dominates another pure strategy, but a mixture of two or more pure strategies can dominate another strategy. This is called Strictly Dominant Mixed Strategies. Some authors allow for elimination of strategies dominated by a mixed strategy in this way.
Example 1:
In this scenario, for player 1, there is no pure strategy that dominates another pure strategy. Let's define the probability of player 1 playing up as p, and let p = 1 / 2 . We can set a mixed strategy where player 1 plays up and down with probabilities ( 1 / 2 , 1 / 2 ). When player 2 plays left, then the payoff for player 1 playing the mixed strategy of up and down is 1, when player 2 plays right, the payoff for player 1 playing the mixed strategy is 0.5. Thus regardless of whether player 2 chooses left or right, player 1 gets more from playing this mixed strategy between up and down than if the player were to play the middle strategy. In this case, we should eliminate the middle strategy for player 1 since it's been dominated by the mixed strategy of playing up and down with probability ( 1 / 2 , 1 / 2 ).
Example 2:
We can demonstrate the same methods on a more complex game and solve for the rational strategies. In this scenario, the blue coloring represents the dominating numbers in the particular strategy.
Step-by-step solving:
For Player 2, X is dominated by the mixed strategy 1 / 2 Y and 1 / 2 Z.
The expected payoff for playing strategy 1 / 2 Y + 1 / 2 Z must be greater than the expected payoff for playing pure strategy X, assigning 1 / 2 and 1 / 2 as tester values. The argument for mixed strategy dominance can be made if there is at least one mixed strategy that allows for dominance.
Testing with 1 / 2 and 1 / 2 gets the following:
Expected average payoff of 1 / 2 Strategy Y: 1 / 2 (4+0+4) = 4
Expected average payoff of 1 / 2 Strategy Z: 1 / 2 (0+5+5) = 5
Expected average payoff of pure strategy X: (1+1+3) = 5
Set up the inequality to determine whether the mixed strategy will dominate the pure strategy based on expected payoffs.
u 1 / 2 Y + u 1 / 2 Z ⩼ u X
4 + 5 > 5
Mixed strategy 1 / 2 Y and 1 / 2 Z will dominate pure strategy X for Player 2, and thus X can be eliminated from the rationalizable strategies for P2.
For Player 1, U is dominated by the pure strategy D.
For player 2, Y is dominated by the pure strategy Z.
This leaves M dominating D for Player 1.
The only rationalizable strategy for Players 1 and 2 is then (M,Z) or (3,5).
Consider a simple coordination game (the payoff matrix is to the right). The row player can play a if he can reasonably believe that the column player could play A , since a is a best response to A . He can reasonably believe that the column player can play A if it is reasonable for the column player to believe that the row player could play a . She can believe that he will play a if it is reasonable for her to believe that he could play a , etc.
This provides an infinite chain of consistent beliefs that result in the players playing ( a , A ). This makes ( a , A ) a rationalizable pair of actions. A similar process can be repeated for ( b , B ).
As an example where not all strategies are rationalizable, consider a prisoner's dilemma pictured to the left. Row player would never play c , since c is not a best response to any strategy by the column player. For this reason, c is not rationalizable.
Conversely, for two-player games, the set of all rationalizable strategies can be found by iterated elimination of strictly dominated strategies. For this method to hold however, one also needs to consider strict domination by mixed strategies . Consider the game on the right with payoffs of the column player omitted for simplicity. Notice that "b" is not strictly dominated by either "t" or "m" in the pure strategy sense, but it is still dominated by a strategy that would mix "t" and "m" with probability of each equal to 1/2. This is due to the fact that given any belief about the action of the column player, the mixed strategy will always yield higher expected payoff. [ 3 ] This implies that "b" is not rationalizable.
Moreover, "b" is not a best response to either "L" or "R" or any mix of the two. This is because an action that is not rationalizable can never be a best response to any opponent's strategy (pure or mixed). This would imply another version of the previous method of finding rationalizable strategies as those that survive the iterated elimination of strategies that are never a best response (in pure or mixed sense).
In games with more than two players, however, there may be strategies that are not strictly dominated, but which can never be the best response. By the iterated elimination of all such strategies one can find the rationalizable strategies for a multiplayer game.
It can be easily proved that a Nash equilibrium is a rationalizable equilibrium; however, the converse is not true. Some rationalizable equilibria are not Nash equilibria. This makes the rationalizability concept a generalization of Nash equilibrium concept.
As an example, consider the game matching pennies pictured to the right. In this game the only Nash equilibrium is row playing h and t with equal probability and column playing H and T with equal probability. However, all pure strategies in this game are rationalizable.
Consider the following reasoning: row can play h if it is reasonable for her to believe that column will play H . Column can play H if its reasonable for him to believe that row will play t . Row can play t if it is reasonable for her to believe that column will play T . Column can play T if it is reasonable for him to believe that row will play h (beginning the cycle again). This provides an infinite set of consistent beliefs that results in row playing h . A similar argument can be given for row playing t , and for column playing either H or T . | https://en.wikipedia.org/wiki/Rationalizable_strategy |
In mathematics , Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows . The study of the dynamics of unipotent flows played a decisive role in the proof of the Oppenheim conjecture by Grigory Margulis . Ratner's theorems have guided key advances in the understanding of the dynamics of unipotent flows. Their later generalizations provide ways to both sharpen the results and extend the theory to the setting of arbitrary semisimple algebraic groups over a local field .
The Ratner orbit closure theorem asserts that the closures of orbits of unipotent flows on the quotient of a Lie group by a lattice are nice, geometric subsets. The Ratner equidistribution theorem further asserts that each such orbit is equidistributed in its closure. The Ratner measure classification theorem is the weaker statement that every ergodic invariant probability measure is homogeneous, or algebraic : this turns out to be an important step towards proving the more general equidistribution property. There is no universal agreement on the names of these theorems: they are variously known as the "measure rigidity theorem", the "theorem on invariant measures" and its "topological version", and so on.
The formal statement of such a result is as follows. Let G {\displaystyle G} be a Lie group , Γ {\displaystyle {\mathit {\Gamma }}} a lattice in G {\displaystyle G} , and u t {\displaystyle u^{t}} a one-parameter subgroup of G {\displaystyle G} consisting of unipotent elements, with the associated flow ϕ t {\displaystyle \phi _{t}} on Γ ∖ G {\displaystyle {\mathit {\Gamma }}\setminus G} . Then the closure of every orbit { x u t } {\displaystyle \left\{xu^{t}\right\}} of ϕ t {\displaystyle \phi _{t}} is homogeneous. This means that there exists a connected , closed subgroup S {\displaystyle S} of G {\displaystyle G} such that the image of the orbit x S {\displaystyle \,xS\,} for the action of S {\displaystyle S} by right translations on G {\displaystyle G} under the canonical projection to Γ ∖ G {\displaystyle {\mathit {\Gamma }}\setminus G} is closed, has a finite S {\displaystyle S} -invariant measure, and contains the closure of the ϕ t {\displaystyle \phi _{t}} -orbit of x {\displaystyle x} as a dense subset .
The simplest case to which the statement above applies is G = S L 2 ( R ) {\displaystyle G=SL_{2}(\mathbb {R} )} . In this case it takes the following more explicit form; let Γ {\displaystyle \Gamma } be a lattice in S L 2 ( R ) {\displaystyle SL_{2}(\mathbb {R} )} and F ⊂ Γ ∖ G {\displaystyle F\subset \Gamma \backslash G} a closed subset which is invariant under all maps Γ g ↦ Γ ( g u t ) {\displaystyle \Gamma g\mapsto \Gamma (gu_{t})} where u t = ( 1 t 0 1 ) {\displaystyle u_{t}={\begin{pmatrix}1&t\\0&1\end{pmatrix}}} . Then either there exists an x ∈ Γ ∖ G {\displaystyle x\in \Gamma \backslash G} such that F = x U {\displaystyle F=xU} (where U = { u t , t ∈ R } {\displaystyle U=\{u_{t},t\in \mathbb {R} \}} ) or F = Γ ∖ G {\displaystyle F=\Gamma \backslash G} .
In geometric terms Γ {\displaystyle \Gamma } is a cofinite Fuchsian group , so the quotient M = Γ ∖ H 2 {\displaystyle M=\Gamma \backslash \mathbb {H} ^{2}} of the hyperbolic plane by Γ {\displaystyle \Gamma } is a hyperbolic orbifold of finite volume. The theorem above implies that every horocycle of H 2 {\displaystyle \mathbb {H} ^{2}} has an image in M {\displaystyle M} which is either a closed curve (a horocycle around a cusp of M {\displaystyle M} ) or dense in M {\displaystyle M} . | https://en.wikipedia.org/wiki/Ratner's_theorems |
A rattleback is a semi-ellipsoidal top which will rotate on its axis in a preferred direction. If spun in the opposite direction, it becomes unstable, "rattles" to a stop and reverses its spin to the preferred direction.
For most rattlebacks the motion will happen when the rattleback is spun in one direction, but not when spun in the other. Some exceptional rattlebacks will reverse when spun in either direction. [ 1 ] This counterintuitive behavior makes the rattleback a physical curiosity that has excited human imagination since prehistoric times. [ 2 ]
A rattleback may also be known as a "anagyre", "(rebellious) celt ", "Celtic stone", "druid stone", "rattlerock", "Robinson Reverser", "spin bar", "wobble stone" (or "wobblestone") and by product names including "ARK", "Bizzaro Swirl", "Space Pet" and "Space Toy".
Archeologists who investigated ancient Celtic and Egyptian sites in the 19th century found celts which exhibited the spin-reversal motion. [ citation needed ] The antiquarian word celt (the "c" is soft, pronounced as "s") describes lithic tools and weapons shaped like an adze , axe , chisel , or hoe .
The first modern descriptions of these celts were published in the 1890s when Gilbert Walker wrote his "On a curious dynamical property of celts" for the Proceedings of the Cambridge Philosophical Society in Cambridge, England, and "On a dynamical top" for the Quarterly Journal of Pure and Applied Mathematics in Somerville, Massachusetts, US. [ 3 ] [ 4 ]
Additional examinations of rattlebacks were published in 1909 and 1918, and by the 1950s and 1970s, several more examinations were made. But, the popular fascination with the objects has increased notably since the 1980s when no fewer than 28 examinations were published.
Rattleback artifacts are typically stone and come in various sizes. Modern ones sold as novelty puzzles and toys are generally made of plastic, wood, or glass, and come in sizes from a few inches up to 12 inches (300 mm) long. A rattleback can also be made by bending a spoon. [ 5 ] Two rattleback design types exist: they have either an asymmetrical base with a skewed rolling axis, or a symmetrical base with offset weighting at the ends.
The spin-reversal motion follows from the growth of instabilities on the other rotation axes, that are rolling (on the main axis) and pitching (on the crosswise axis). [ 6 ]
When there is an asymmetry in the mass distribution with respect to the plane formed by the pitching and the vertical axes, a coupling of these two instabilities arises; one can imagine how the asymmetry in mass will deviate the rattleback when pitching, which will create some rolling.
The amplified mode will differ depending on the spin direction, which explains the rattleback's asymmetrical behavior. Depending on whether it is rather a pitching or rolling instability that dominates, the growth rate will be very high or quite low.
This explains why, due to friction, most rattlebacks appear to exhibit spin-reversal motion only when spun in the pitching-unstable direction, also known as the strong reversal direction. When the rattleback is spun in the "stable direction", also known as the weak reversal direction, friction and damping often slow the rattleback to a stop before the rolling instability has time to fully build. Some rattlebacks, however, exhibit "unstable behavior" when spun in either direction, and incur several successive spin reversals per spin. [ 7 ]
Other ways to add motion to a rattleback include tapping by pressing down momentarily on either of its ends, and rocking by pressing down repeatedly on either of its ends.
For a comprehensive analysis of rattleback's motion, see V.Ph. Zhuravlev and D.M. Klimov (2008). [ 8 ] The previous papers were based on simplified assumptions and limited to studying local instability of its steady-state oscillation.
Realistic mathematical modelling of a rattleback is presented by G. Kudra and J. Awrejcewicz (2015). [ 9 ] They focused on modelling of the contact forces and tested different versions of models of friction and rolling resistance, obtaining good agreement with the experimental results.
Numerical simulations predict that a rattleback situated on a harmonically oscillating base can exhibit rich bifurcation dynamics, including different types of periodic, quasi-periodic and chaotic motions. [ 10 ] | https://en.wikipedia.org/wiki/Rattleback |
The Rauhut–Currier reaction, also called the vinylogous Morita–Baylis–Hillman reaction , [ 1 ] is an organic reaction describing (in its original scope) the dimerization or isomerization of electron-deficient alkenes such as enones by action of an organophosphine of the type R 3 P. [ 2 ] In a more general description the RC reaction is any coupling of one active alkene / latent enolate to a second Michael acceptor , creating a new C–C bond between the alpha-position of one activated alkene and the beta-position of a second alkene under the influence of a nucleophilic catalyst . [ 3 ] The reaction mechanism is essentially that of the related and better known Baylis–Hillman reaction (DABCO not phosphine, carbonyl not enone) but the Rauhut–Currier reaction actually predates it by several years. In comparison to the MBH reaction, the RC reaction lacks substrate reactivity and regioselectivity .
The original 1963 reaction described the dimerization of the ethyl acrylate to the ethyl diester of 2- methylene - glutaric acid with tributylphosphine in acetonitrile :
This reaction was also found to work for acrylonitrile .
RC cross-couplings are known but suffer from lack of selectivity. Amines such as DABCO can also act as catalyst. The reactivity is improved in intramolecular RC reactions, for example in the isomerization of di-enones to form cyclopentenes : [ 4 ]
A similar reaction by asymmetric synthesis organocatalyzed by a protected cysteine and potassium tert-butoxide afforded a cyclohexene with 95% enantiomeric excess : [ 5 ]
In this reaction the phosphine is replaced by the thiol group of cysteine but the reaction is the same. | https://en.wikipedia.org/wiki/Rauhut–Currier_reaction |
The Raunkiær system is a system for categorizing plants using life-form categories, devised by Danish botanist Christen C. Raunkiær and later extended by various authors.
It was first proposed in a talk to the Danish Botanical Society in 1904 as can be inferred from the printed discussion of that talk, but not the talk itself, nor its title. The journal, Botanisk Tidsskrift , published brief comments on the talk by M.P. Porsild, with replies by Raunkiær. A fuller account appeared in French the following year. [ 1 ] Raunkiær elaborated further on the system and published this in Danish in 1907. [ 2 ] [ 3 ]
The original note and the 1907 paper were much later translated to English and published with Raunkiær's collected works. [ 4 ] [ 3 ] [ 5 ]
Raunkiær's life-form scheme has subsequently been revised and modified by various authors, [ 6 ] [ 7 ] [ 8 ] but the main structure has survived. Raunkiær's life-form system may be useful in researching the transformations of biotas and the genesis of some groups of phytophagous animals. [ 9 ]
The subdivisions of the Raunkiær system are premised on the location of the bud of a plant during seasons with adverse conditions, i. e. cold seasons and dry seasons:
These plants, normally woody perennials , grow stems into the air, with their resting buds being more than 50 cm above the soil surface, [ 10 ] e.g. trees and shrubs , and also epiphytes , which Raunkiær later separated as a distinct class (see below).
Raunkiær further divided the phanerophytes according to height as
Further division was premised on the characters of duration of foliage, i. e. evergreen or deciduous, and presence of covering bracts on buds, for 12 classes. Three further divisions were made to increase the total of classes to 15: [ 3 ]
Epiphytes were originally included in the phanerophytes (see above) but then separated because they do not grow in soil, so the soil location is irrelevant in classifying them. They form characteristic communities of moist climatic conditions. [ 11 ]
These plants have buds on persistent shoots near the soil surface; woody plants with perennating buds borne close to the soil surface, a maximum of 25 cm above the soil surface, e.g., bilberry and periwinkle .
These plants have buds at or near the soil surface, e.g. common daisy and dandelion , and are divided into:
These plants have subterranean or under water resting buds, and are divided into:
These are annual plants that complete their lives rapidly in favorable conditions and survive the unfavorable cold or dry season in the form of seeds. About 6% of plants are therophytes but their proportion is much higher in region with hot-dry summer. [ 12 ]
Aerophytes were a later addition to the system. [ 13 ] These are plants that obtain moisture and nutrients from the air and rain. [ 14 ] They usually grow on other plants yet are not parasitic on them. These are perennial plants and are like epiphytes but whose root system have been reduced. [ 15 ] They occur in communities that inhabit exclusively hyper-arid areas with abundant fog. [ 13 ] Like epiphytes and hemicryptophytes, their buds are near the soil surface. Some Tillandsia species are classified as aerophytes.
Farley Mowat , in his book, Never Cry Wolf , described using a Raunkiær's Circle in making a "cover degree" study to determine the ratios of various plants one to the other. He spoke of it as "a device designed in hell." [ 16 ] | https://en.wikipedia.org/wiki/Raunkiær_plant_life-form |
Ravi Bhushan (born 12 April 1953, in Muzaffarnagar, India) was a Professor of Chemistry at Indian Institute of Technology Roorkee who worked in the areas of natural products chemistry, protein chemistry , and chiral analysis by liquid chromatography.
Bhushan began his education in his native India, completing his undergraduate and master's degrees from the University of Jodhpur . He received his Ph.D. in chemistry (working on structure elucidation of natural products isolated from certain desert plants) in 1978 at the University of Jodhpur . Bhushan joined as a lecturer at the University of Roorkee (now Indian Institute of Technology Roorkee , India) in 1979 and was later selected for the position of full professor of chemistry in 1996 and served there till retirement in 2018. [ citation needed ]
He started his research in the chemistry of natural products. At Washington State University he established early steps in the metabolism of d-neomethyl-α-D-glucoside in pipermint ( Mentha piperita ) rhizomes via in vivo studies. Bhushan developed a de novo method for direct resolution of certain racemates by liquid chromatography . Later, the approach was applied for direct enantioseparation of several active pharmaceutical ingredients ( APIs ). It is now an established approach in literature. 1994 onwards, the method was extended to such resolutions by ligand exchange principle. [ 1 ] The method is of significant importance to pharmaceutical industry and analytical laboratories associated with regulatory agencies for determination and control of enantiomeric purity (and isolation of native enantiomers) of a variety of APIs since many of them are marketed and administered as racemic mixture while only one enantiomer is therapeutically useful. [ 2 ]
Bhushan supervised the Ph.D. theses of > 30 scholars and has published more than 270 research papers.
Bhushan is a member of editorial board of | https://en.wikipedia.org/wiki/Ravi_Bhushan |
Ravi Naidu (also known as Ravendra Naidu; born in Nadi, Fiji ) is a Fijian-Australian scientist, working in soil contamination and sustainable soil management. In 2023, he was awarded the Glinka World Soil Prize ... [ 1 ]
He is a Distinguished Laureate Professor and Director of the Global Centre for Environmental Remediation in the University of Newcastle , Australia. Apart from his academic role, he holds leadership position in the Cooperative Research Centre for the Contamination Assessment and Remediation of the Environment.
According to his publicly available biography, it appears that Ravi takes a risk-based approach to environmental remediation, in which he helped bring industry, academia and government stakeholders together to work collaboratively. [ 2 ]
He was awarded the European Geosciences Union's Kabata-Pendias Medal in 2023, [ 3 ] and the Mahatma Gandhi Leadership in 2022 [ 4 ]
After graduating from University of the South Pacific (USP), Fiji, in Chemistry and Mathematics, Ravi Naidu obtained his M.Sc. (Mineralogy) jointly from University of Aberdeen , UK, in 1979. His career started as a lecturer at USP. Soon after, he completed his PhD (Soil chemistry) from Massey University, New Zealand in 1986. In 2015, he obtained Doctor of Science (Soil Chemistry) degree from Massey University , following the same ( Hon Causa ) by the Tamil Nadu Agricultural University in 2013.
Ravi Naidu's detailed analysis focuses on contaminants from agriculture and industry and their effects on the environment and humans. He explains that his research examines how contaminants are present in the environment, their interactions, and how they reach people or nature. [ 2 ]
In his Glinka Award description, [ 1 ] it is acknowledged that he had contributed research about contaminant and soil properties that helped sustainable soil management today. However, to do this, his investigations mostly target industrial contaminants, driven largely by the industries themselves. [ 7 ]
Ravi Naidu and a few of his collaborators published a prospective review article in 2021 titled "Chemical pollution: A growing peril and potentially catastrophic risk to humanity." [ 8 ] Accordingly to PlumX metrics, this article has captured the attention of various other articles, policy documents, social media, and news. [ 8 ]
The following articles authored by Ravi Naidu cover his multidisciplinary environmental science career: | https://en.wikipedia.org/wiki/Ravi_Naidu |
Raw intelligence is raw data gathered by an intelligence operation, such as espionage or signal interception . Such data commonly requires processing and analysis to make it useful and reliable. To turn the raw intelligence into a finished form, the steps required may include decryption , translation , collation , evaluation and confirmation . [ 2 ] [ 3 ] [ 4 ]
In the period after the First World War , British practise was to circulate raw intelligence with little analysis or context. [ 5 ] Such direct intelligence was a strong influence on policy-makers. [ 5 ] Churchill was especially keen to see raw intelligence and was supplied this by Desmond Morton during his period outside the government. [ 6 ] When Churchill became Prime Minister in 1940, he still insisted on receiving raw intelligence and wanted it all until it was explained that the volume was now too great. [ 6 ] A selection of daily intercepts was provided to him each day by Bletchley Park and he called these his "golden eggs". [ 6 ]
US intelligence has a different tradition from the British. The key event for the US was the failure to prevent the attack on Pearl Harbor and the inquiries which followed concluded that this was not due to the lack of raw intelligence so much as the failure to make effective use of it. The Central Intelligence Agency was created to collate, analyse and summarise the raw intelligence collected by the other departments. US agencies which focus on the collection of raw intelligence include the National Reconnaissance Office and the National Security Agency . [ 7 ]
This espionage -related article is a stub . You can help Wikipedia by expanding it .
This law enforcement –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Raw_intelligence |
The Rawalpindi experiments were experiments involving use of mustard gas carried out by British scientists from Porton Down on hundreds of soldiers from the British Indian Army . These experiments were carried out before and during the Second World War in a military installation at Rawalpindi , in modern-day Pakistan . [ 1 ] These experiments began in the early 1930s and lasted more than 10 years. [ 2 ] Since the publication of the story in The Guardian on 1 September 2007, the experiments have been referred to as the Rawalpindi experiments or Rawalpindi mustard gas experiments in the media and elsewhere.
The experiments in Rawalpindi were part of a much larger project intended to test the effects of chemical weapons on humans. More than 20,000 British servicemen were subjected to chemical warfare trials between 1916 and 1989 at the Defence Ministry's Porton Down research centre in southwest England. The Rawalpindi experiments focused on mustard gas , now known to be highly carcinogenic. According to documents at The National Archives in London , British scientists and doctors tested the effects of mustard gas on hundreds of Indian soldiers [ 3 ] over a ten-year period. Beginning in the early 1930s, scientists at Rawalpindi sent British Indian Army soldiers, wearing shorts and cotton shirts, into gas chambers to experience the effects of mustard gas . The scientists hoped to determine the appropriate dosage to use on battlefields. Many of the subjects suffered severe burns from their exposure to the gas. [ 4 ]
These tests caused large numbers of burns, some of which were so damaging that the subjects had to be hospitalized. According to the report severely burned patients were often very miserable and depressed and in considerable discomfort. [ 5 ] No long-term effects of exposure were documented or studied. [ citation needed ] The patients were treated at the Indian Military Hospital Rawalpindi (now known as the Military Hospital Rawalpindi ). The exact place where the British facility equipped with gas chambers was located in Rawalpindi is unknown. Porton Down officials have argued that trials took place in a different era, during a conflict, and so their conduct should not be judged by today's standards. [ 1 ] | https://en.wikipedia.org/wiki/Rawalpindi_experiments |
In computing , the term raw disk , [ 1 ] [ 2 ] often referred to as raw , is used to refer to hard disk access at a raw, binary level, beneath the file system level, and using partition data at the MBR .
A notable example is in the context of platform virtualization , and a feature of certain virtualization software is the ability to access a hard disk at the raw level. Virtualization software may typically function via the usage of a virtual drive format like OVF , but some users may want the virtualization software to be able to run an operating system that has been installed on another disk or disk partition. In order to do this, the virtualization software must allow raw disk access to that disk or disk partition, and then allow that entire operating system to boot within the virtualization window.
RAW file system indicates a state of your hard drive which has no or unknown file system. A disk or drive with a RAW file system is also known as RAW disk or RAW drive. When a hard drive or external storage device is shown as RAW, it could be:
This computing article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rawdisk |
A raw mill is the equipment used to grind raw materials into " rawmix " during the manufacture of cement . Rawmix is then fed to a cement kiln , which transforms it into clinker , which is then ground to make cement in the cement mill . The raw milling stage of the process effectively defines the chemistry (and therefore physical properties) of the finished cement, and has a large effect upon the efficiency of the whole manufacturing process.
The history of the development of the technology of raw material grinding defines the early history of cement technology. Other stages of cement manufacture used existing technology in the early days. Early hydraulic materials such as hydraulic limes , natural cements and Parker's Roman cement were all based on "natural" raw materials, burned "as-dug". Because these natural blends of minerals occur only rarely, manufacturers were interested in making a fine-grained artificial mixture of readily available minerals such as limestone and clay that could be used in the same way. A typical problem would be to make an intimate mixture of 75% chalk and 25% clay, and burn this to produce an ”artificial cement". The development of the "wet" method of producing fine-grained clay in the ceramics industry afforded a means of doing this. For this reason, the early cement industry used the "wet process", in which the raw materials are ground together with water, to produce a slurry, containing 20–50% water. Both Louis Vicat and James Frost used this technique in the early 19th century, and it remained the only way of making rawmix for Portland cement until 1890. A modification of the technique used by the early industry was "double-burning", in which a hard limestone would be burned and slaked before combining with clay slurry. This technique avoided the grinding of hard stone, and was employed by, among others, Joseph Aspdin . Early grinding technology was poor, and early slurries were made thin, with a high water content. The slurry was then allowed to stand in large reservoirs ("slurry-backs") for several weeks. Large, un-ground particles would drop to the bottom, and excess water rose to the top. The water was periodically decanted until a stiff cake, of the consistency of pottery clay, was left. This was sliced up, discarding the coarse material at the bottom, and burned in the kiln. Wet grinding is comparatively energy-efficient, and so when good dry-grinding equipment became available, the wet process continued in use throughout the 20th century, often employing equipment that Josiah Wedgwood would have recognized.
Rawmixes are formulated to contain a correctly balanced chemistry for the production of calcium silicates ( alite and belite ) and fluxes ( aluminate and ferrite ) in the kiln. Chemical analysis data in cement manufacture are expressed in terms of oxides, and the most important of these in rawmix design are SiO 2 , Al 2 O 3 , Fe 2 O 3 and CaO. In principle, any material that can contribute any of these oxides can be used as a rawmix component. Because the major oxide required is CaO, the most prevalent rawmix component is limestone , while the others are mostly contributed by clay or shale . Minor adjustments to the chemistry are made by smaller additions of materials such as those shown below.
Typical rawmix component chemical analyses:
Note: LoI 950 is the Loss on ignition at 950 °C, and represents (approximately) the components lost during kiln processing. It consists mainly of CO 2 from carbonates, H 2 O from clay hydrates, and organic carbon.
Using these materials, typical rawmixes could be composed:
The chemical analyses of these rawmixes would be:
The raw materials and mixes shown are only "typical": considerable variations are possible depending on the raw materials available.
Apart from the major oxides (CaO, SiO 2 , Al 2 O 3 and Fe 2 O 3 ) the minor oxides are, at best, diluents of the clinker, and may be deleterious. However, cement raw materials are for the most part dug from the Earth's crust and contain most of the elements in the periodic table in some amount. The manufacturer therefore selects materials so that the deleterious effects of minor elements are minimized or kept under control. Minor elements that are frequently encountered are as follows:
Wet grinding is more efficient than dry grinding because water coats the newly formed surfaces of broken particles and prevents re-agglomeration. The process of blending and homogenizing the rawmix is also much easier when it is in slurry form. The disadvantage is that the water in the resultant slurry has to be removed subsequently, and this usually requires a lot of energy. While energy was cheap, wet grinding was common, but since 1970 the situation has changed dramatically, and new wet process plant is now rarely installed. Wet grinding is performed by two distinct means: washmills and ballmills.
This represents the earliest rawmilling technology, and was used to grind soft materials such as chalk and clay. It is rather similar to a food processor. It consists of a large bowl (up to 15 m in diameter) into which the crushed (to less than 250 mm) raw materials are tipped along with a stream of water. The material is stirred by rotating sets of harrows . The outside walls of the bowl consist of gratings or perforated plates through which fine product can pass. Grinding is largely autogenous (i.e. it takes place by collision between lumps of raw material), and is very efficient, producing little waste heat, provided that the materials are soft. Typically two or three washmills are connected in series, these being provided with successively smaller outlet perforations. The entire system can produce slurry with the expenditure of as little as 5 kW·h of electricity per dry tonne. Relatively hard minerals (such as flint) in the mix, are more or less untouched by the grinding process, and settle out in the base of the mill, from where they are periodically dug out.
The ballmill allows grinding of the harder limestones that are more common than chalk. A ballmill consists of a horizontal cylinder that rotates on its axis. It holds spherical, cylindrical or rod-like grinding media of size 15–100 mm that may be steel or a variety of ceramic materials, and occupy 20–30% of the mill volume. The shell of the mill is lined with steel or rubber plates. Grinding is effected by impact and attrition between the grinding media. The various mineral components of the rawmix are fed to the mill at a constant rate along with water, and the slurry runs from the outlet end. The washdrum has a similar concept, but contains little or no grinding media, grinding being autogenous, by the cascading action of the larger raw material pieces. It is suitable for soft materials, and particularly for flinty chalk, where the unground flint acts as grinding media.
It is essential that large particles (> 150 μm for calcium carbonate and > 45 μm for quartz) should be eliminated from the rawmix, to facilitate chemical combination in the kiln. In the case of slurries, larger particles can be removed by hydrocyclones or sieving devices. These require a certain amount of energy, supplied by high pressure pumping. This process, and the moving and blending of the slurry, require careful control of the slurry viscosity. Clearly, a thinner slurry is easily obtained by adding more water, but at the expense of high energy consumption for its subsequent removal. In practice, the slurry is therefore made as thick as the plant equipment can handle. Cement rawmix slurries are Bingham plastics which can also exhibit thixotropic or rheopectic behaviour. The energy needed to pump slurry at a desired rate is controlled mainly by the slurry's yield stress , and this in turn varies more or less exponentially with the slurry solids/liquid ratio. In practice, deflocculants are often added in order to maintain pumpability at low moisture contents. Common deflocculants used (at typical dose rates of 0.005–0.03%) are sodium carbonate , sodium silicate , sodium polyphosphates and lignosulfonates . Under favourable circumstances, pumpable slurries with less than 25% water can be obtained.
Rawmixes frequently contain minerals of contrasting hardness, such as calcite and quartz. Simultaneous grinding of these in a rawmill is inefficient, because the grinding energy is preferentially used in grinding the softer material. This results in a large amount of excessively fine soft material, which "cushions" the grinding of the harder mineral. For this reason, sand is sometimes ground separately, then fed to the main rawmill as a fine slurry.
Dry rawmills are the normal technology installed today, allowing minimization of energy consumption and CO 2 emissions. In general, cement raw materials are mainly quarried, and so contain a certain amount of natural moisture. Attempting to grind a wet material is unsuccessful because an intractable "mud" forms. On the other hand, it is much easier to dry a fine material than a coarse one, because large particles hold moisture deep in their structure. It is therefore usual to simultaneously dry and grind the materials in the rawmill. A hot-air furnace may be used to supply this heat, but usually hot waste gases from the kiln are used. For this reason, the rawmill is usually placed close to the kiln preheater. Types of dry rawmill include ball mills, roller mills and hammer mills.
These are similar to cement mills , but often with a larger gas flow. The gas temperature is controlled by cold-air bleeds to ensure a dry product without overheating the mill. The product passes into an air separator, which returns oversized particles to the mill inlet. Occasionally, the mill is preceded by a hot-air-swept hammer mill which does most of the drying and produces millimetre-sized feed for the mill. Ball mills are rather inefficient, and typically require 10–20 kW·h of electric power to make a tonne of rawmix. The Aerofall mill is sometimes used for pre-grinding large wet feeds. It is a short, large-diameter semi-autogenous mill, typically containing 15% by volume of very large (130 mm) grinding balls. Feed can be up to 250 mm, and the larger chunks produce much of the grinding action. The mill is air-swept, and the fines are carried away in the gas stream. Crushing and drying are efficient, but the product is coarse (around 100 μm), and is usually re-ground in a separate ball mill.
These are the standard form in modern installations, occasionally called vertical spindle mills . In a typical arrangement, the material is fed onto a rotating table, onto which steel rollers press down. A high velocity of hot gas flow is maintained close to the dish so that fine particles are swept away as soon as they are produced. The gas flow carries the fines into an integral air separator, which returns larger particles to the grinding path. The fine material is swept out in the exhaust gas and is captured by a cyclone before being pumped to storage. The remaining dusty gas is usually returned to the main kiln dust control equipment for cleaning. Feed size can be up to 100 mm. Roller mills are efficient, using about half the energy of a ball mill, and there seems to be no limit to the size available. Roller mills with output in excess of 800 tonnes per hour have been installed. Unlike ball mills, feed to the mill must be regular and uninterrupted; otherwise damaging resonant vibration sets in.
Hammer mills (or "crusher driers") swept with hot kiln exhaust gases have limited application where a soft, wet raw material is being ground. The simple design means that it can be operated at a higher temperature than other mills, giving it high drying capacity. However, the grinding action is poor, and the product is often re-ground in a ball mill. | https://en.wikipedia.org/wiki/Rawmill |
In geometry , a straight line , usually abbreviated line , is an infinitely long object with no width, depth, or curvature , an idealization of such physical objects as a straightedge , a taut string, or a ray of light . Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment , which is a part of a line delimited by two points (its endpoints ).
Euclid's Elements defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean , projective , and affine geometry .
In the Greek deductive geometry of Euclid's Elements , a general line (now called a curve ) is defined as a "breadthless length", and a straight line (now called a line segment ) was defined as a line "which lies evenly with the points on itself". [ 1 ] : 291 These definitions appeal to readers' physical experience, relying on terms that are not themselves defined, and the definitions are never explicitly referenced in the remainder of the text. In modern geometry, a line is usually either taken as a primitive notion with properties given by axioms , [ 1 ] : 95 or else defined as a set of points obeying a linear relationship, for instance when real numbers are taken to be primitive and geometry is established analytically in terms of numerical coordinates .
In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (modern mathematicians added to Euclid's original axioms to fill perceived logical gaps), [ 1 ] : 108 a line is stated to have certain properties that relate it to other lines and points . For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect at most at one point. [ 1 ] : 300 In two dimensions (i.e., the Euclidean plane ), two lines that do not intersect are called parallel . In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane , or skew if they are not.
On a Euclidean plane , a line can be represented as a boundary between two regions. [ 2 ] : 104 Any collection of finitely many lines partitions the plane into convex polygons (possibly unbounded); this partition is known as an arrangement of lines .
In three-dimensional space , a first degree equation in the variables x , y , and z defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes. More generally, in n -dimensional space n −1 first-degree equations in the n coordinate variables define a line under suitable conditions.
In more general Euclidean space , R n (and analogously in every other affine space ), the line L passing through two different points a and b is the subset L = { ( 1 − t ) a + t b ∣ t ∈ R } . {\displaystyle L=\left\{(1-t)\,a+tb\mid t\in \mathbb {R} \right\}.} The direction of the line is from a reference point a ( t = 0) to another point b ( t = 1), or in other words, in the direction of the vector b − a . Different choices of a and b can yield the same line.
Three or more points are said to be collinear if they lie on the same line. If three points are not collinear, there is exactly one plane that contains them.
In affine coordinates , in n -dimensional space the points X = ( x 1 , x 2 , ..., x n ), Y = ( y 1 , y 2 , ..., y n ), and Z = ( z 1 , z 2 , ..., z n ) are collinear if the matrix [ 1 x 1 x 2 ⋯ x n 1 y 1 y 2 ⋯ y n 1 z 1 z 2 ⋯ z n ] {\displaystyle {\begin{bmatrix}1&x_{1}&x_{2}&\cdots &x_{n}\\1&y_{1}&y_{2}&\cdots &y_{n}\\1&z_{1}&z_{2}&\cdots &z_{n}\end{bmatrix}}} has a rank less than 3. In particular, for three points in the plane ( n = 2), the above matrix is square and the points are collinear if and only if its determinant is zero.
Equivalently for three points in a plane, the points are collinear if and only if the slope between one pair of points equals the slope between any other pair of points (in which case the slope between the remaining pair of points will equal the other slopes). By extension, k points in a plane are collinear if and only if any ( k –1) pairs of points have the same pairwise slopes.
In Euclidean geometry , the Euclidean distance d ( a , b ) between two points a and b may be used to express the collinearity between three points by: [ 3 ] [ 4 ]
However, there are other notions of distance (such as the Manhattan distance ) for which this property is not true.
In the geometries where the concept of a line is a primitive notion , as may be the case in some synthetic geometries , other methods of determining collinearity are needed.
In Euclidean geometry, all lines are congruent , meaning that every line can be obtained by moving a specific line. However, lines may play special roles with respect to other geometric objects and can be classified according to that relationship.
For instance, with respect to a conic (a circle , ellipse , parabola , or hyperbola ), lines can be:
In the context of determining parallelism in Euclidean geometry, a transversal is a line that intersects two other lines that may or not be parallel to each other.
For more general algebraic curves , lines could also be:
With respect to triangles we have:
For a convex quadrilateral with at most two parallel sides, the Newton line is the line that connects the midpoints of the two diagonals . [ 7 ]
For a hexagon with vertices lying on a conic we have the Pascal line and, in the special case where the conic is a pair of lines, we have the Pappus line .
Parallel lines are lines in the same plane that never cross. Intersecting lines share a single point in common. Coincidental lines coincide with each other—every point that is on either one of them is also on the other.
Perpendicular lines are lines that intersect at right angles . [ 8 ]
In three-dimensional space , skew lines are lines that are not in the same plane and thus do not intersect each other.
In synthetic geometry , the concept of a line is often considered as a primitive notion , [ 1 ] : 95 meaning it is not being defined by using other concepts, but it is defined by the properties, called axioms , that it must satisfy. [ 9 ]
However, the axiomatic definition of a line does not explain the relevance of the concept and is often too abstract for beginners. So, the definition is often replaced or completed by a mental image or intuitive description that allows understanding what is a line. Such descriptions are sometimes referred to as definitions, but are not true definitions since they cannot used in mathematical proofs . The "definition" of line in Euclid's Elements falls into this category; [ 1 ] : 95 and is never used in proofs of theorems.
Lines in a Cartesian plane or, more generally, in affine coordinates , are characterized by linear equations. More precisely, every line L {\displaystyle L} (including vertical lines) is the set of all points whose coordinates ( x , y ) satisfy a linear equation; that is, L = { ( x , y ) ∣ a x + b y = c } , {\displaystyle L=\{(x,y)\mid ax+by=c\},} where a , b and c are fixed real numbers (called coefficients ) such that a and b are not both zero. Using this form, vertical lines correspond to equations with b = 0.
One can further suppose either c = 1 or c = 0 , by dividing everything by c if it is not zero.
There are many variant ways to write the equation of a line which can all be converted from one to another by algebraic manipulation. The above form is sometimes called the standard form . If the constant term is put on the left, the equation becomes a x + b y − c = 0 , {\displaystyle ax+by-c=0,} and this is sometimes called the general form of the equation. However, this terminology is not universally accepted, and many authors do not distinguish these two forms.
These forms are generally named by the type of information (data) about the line that is needed to write down the form. Some of the important data of a line is its slope, x-intercept , known points on the line and y-intercept.
The equation of the line passing through two different points P 0 ( x 0 , y 0 ) {\displaystyle P_{0}(x_{0},y_{0})} and P 1 ( x 1 , y 1 ) {\displaystyle P_{1}(x_{1},y_{1})} may be written as ( y − y 0 ) ( x 1 − x 0 ) = ( y 1 − y 0 ) ( x − x 0 ) . {\displaystyle (y-y_{0})(x_{1}-x_{0})=(y_{1}-y_{0})(x-x_{0}).} If x 0 ≠ x 1 , this equation may be rewritten as y = ( x − x 0 ) y 1 − y 0 x 1 − x 0 + y 0 {\displaystyle y=(x-x_{0})\,{\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}+y_{0}} or y = x y 1 − y 0 x 1 − x 0 + x 1 y 0 − x 0 y 1 x 1 − x 0 . {\displaystyle y=x\,{\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}+{\frac {x_{1}y_{0}-x_{0}y_{1}}{x_{1}-x_{0}}}\,.} In two dimensions , the equation for non-vertical lines is often given in the slope–intercept form :
y = m x + b {\displaystyle y=mx+b} where:
The slope of the line through points A ( x a , y a ) {\displaystyle A(x_{a},y_{a})} and B ( x b , y b ) {\displaystyle B(x_{b},y_{b})} , when x a ≠ x b {\displaystyle x_{a}\neq x_{b}} , is given by m = ( y b − y a ) / ( x b − x a ) {\displaystyle m=(y_{b}-y_{a})/(x_{b}-x_{a})} and the equation of this line can be written y = m ( x − x a ) + y a {\displaystyle y=m(x-x_{a})+y_{a}} .
As a note, lines in three dimensions may also be described as the simultaneous solutions of two linear equations a 1 x + b 1 y + c 1 z − d 1 = 0 {\displaystyle a_{1}x+b_{1}y+c_{1}z-d_{1}=0} a 2 x + b 2 y + c 2 z − d 2 = 0 {\displaystyle a_{2}x+b_{2}y+c_{2}z-d_{2}=0} such that ( a 1 , b 1 , c 1 ) {\displaystyle (a_{1},b_{1},c_{1})} and ( a 2 , b 2 , c 2 ) {\displaystyle (a_{2},b_{2},c_{2})} are not proportional (the relations a 1 = t a 2 , b 1 = t b 2 , c 1 = t c 2 {\displaystyle a_{1}=ta_{2},b_{1}=tb_{2},c_{1}=tc_{2}} imply t = 0 {\displaystyle t=0} ). This follows since in three dimensions a single linear equation typically describes a plane and a line is what is common to two distinct intersecting planes.
Parametric equations are also used to specify lines, particularly in those in three dimensions or more because in more than two dimensions lines cannot be described by a single linear equation.
In three dimensions lines are frequently described by parametric equations: x = x 0 + a t y = y 0 + b t z = z 0 + c t {\displaystyle {\begin{aligned}x&=x_{0}+at\\y&=y_{0}+bt\\z&=z_{0}+ct\end{aligned}}} where:
Parametric equations for lines in higher dimensions are similar in that they are based on the specification of one point on the line and a direction vector.
The normal form (also called the Hesse normal form , [ 10 ] after the German mathematician Ludwig Otto Hesse ), is based on the normal segment for a given line, which is defined to be the line segment drawn from the origin perpendicular to the line. This segment joins the origin with the closest point on the line to the origin. The normal form of the equation of a straight line on the plane is given by: x cos φ + y sin φ − p = 0 , {\displaystyle x\cos \varphi +y\sin \varphi -p=0,} where φ {\displaystyle \varphi } is the angle of inclination of the normal segment (the oriented angle from the unit vector of the x -axis to this segment), and p is the (positive) length of the normal segment. The normal form can be derived from the standard form a x + b y = c {\displaystyle ax+by=c} by dividing all of the coefficients by a 2 + b 2 . {\displaystyle {\sqrt {a^{2}+b^{2}}}.} and also multiplying through by − 1 {\displaystyle -1} if c < 0. {\displaystyle c<0.}
Unlike the slope-intercept and intercept forms, this form can represent any line but also requires only two finite parameters, φ {\displaystyle \varphi } and p , to be specified. If p > 0 , then φ {\displaystyle \varphi } is uniquely defined modulo 2 π . On the other hand, if the line is through the origin ( c = p = 0 ), one drops the c /| c | term to compute sin φ {\displaystyle \sin \varphi } and cos φ {\displaystyle \cos \varphi } , and it follows that φ {\displaystyle \varphi } is only defined modulo π .
The vector equation of the line through points A and B is given by r = O A + λ A B {\displaystyle \mathbf {r} =\mathbf {OA} +\lambda \,\mathbf {AB} } (where λ is a scalar ).
If a is vector OA and b is vector OB , then the equation of the line can be written: r = a + λ ( b − a ) {\displaystyle \mathbf {r} =\mathbf {a} +\lambda (\mathbf {b} -\mathbf {a} )} .
A ray starting at point A is described by limiting λ. One ray is obtained if λ ≥ 0, and the opposite ray comes from λ ≤ 0.
In a Cartesian plane , polar coordinates ( r , θ ) are related to Cartesian coordinates by the parametric equations: [ 11 ] x = r cos θ , y = r sin θ . {\displaystyle x=r\cos \theta ,\quad y=r\sin \theta .}
In polar coordinates, the equation of a line not passing through the origin —the point with coordinates (0, 0) —can be written r = p cos ( θ − φ ) , {\displaystyle r={\frac {p}{\cos(\theta -\varphi )}},} with r > 0 and φ − π / 2 < θ < φ + π / 2. {\displaystyle \varphi -\pi /2<\theta <\varphi +\pi /2.} Here, p is the (positive) length of the line segment perpendicular to the line and delimited by the origin and the line, and φ {\displaystyle \varphi } is the (oriented) angle from the x -axis to this segment.
It may be useful to express the equation in terms of the angle α = φ + π / 2 {\displaystyle \alpha =\varphi +\pi /2} between the x -axis and the line. In this case, the equation becomes r = p sin ( θ − α ) , {\displaystyle r={\frac {p}{\sin(\theta -\alpha )}},} with r > 0 and 0 < θ < α + π . {\displaystyle 0<\theta <\alpha +\pi .}
These equations can be derived from the normal form of the line equation by setting x = r cos θ , {\displaystyle x=r\cos \theta ,} and y = r sin θ , {\displaystyle y=r\sin \theta ,} and then applying the angle difference identity for sine or cosine.
These equations can also be proven geometrically by applying right triangle definitions of sine and cosine to the right triangle that has a point of the line and the origin as vertices, and the line and its perpendicular through the origin as sides.
The previous forms do not apply for a line passing through the origin, but a simpler formula can be written: the polar coordinates ( r , θ ) {\displaystyle (r,\theta )} of the points of a line passing through the origin and making an angle of α {\displaystyle \alpha } with the x -axis, are the pairs ( r , θ ) {\displaystyle (r,\theta )} such that r ≥ 0 , and θ = α or θ = α + π . {\displaystyle r\geq 0,\qquad {\text{and}}\quad \theta =\alpha \quad {\text{or}}\quad \theta =\alpha +\pi .}
In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it.
When a geometry is described by a set of axioms , the notion of a line is usually left undefined (a so-called primitive object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in differential geometry , a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries , a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line.
In many models of projective geometry , the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry. In elliptic geometry we see a typical example of this. [ 1 ] : 108 In the spherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametrically opposite points identified. In a different model of elliptic geometry, lines are represented by Euclidean planes passing through the origin. Even though these representations are visually distinct, they satisfy all the properties (such as, two points determining a unique line) that make them suitable representations for lines in this geometry.
The "shortness" and "straightness" of a line, interpreted as the property that the distance along the line between any two of its points is minimized (see triangle inequality ), can be generalized and leads to the concept of geodesics in metric spaces .
Given a line and any point A on it, we may consider A as decomposing this line into two parts.
Each such part is called a ray and the point A is called its initial point . It is also known as half-line (sometimes, a half-axis if it plays a distinct role, e.g., as part of a coordinate axis ). It is a one-dimensional half-space . The point A is considered to be a member of the ray. [ a ] Intuitively, a ray consists of those points on a line passing through A and proceeding indefinitely, starting at A , in one direction only along the line. However, in order to use this concept of a ray in proofs a more precise definition is required.
Given distinct points A and B , they determine a unique ray with initial point A . As two points define a unique line, this ray consists of all the points between A and B (including A and B ) and all the points C on the line through A and B such that B is between A and C . [ 12 ] This is, at times, also expressed as the set of all points C on the line determined by A and B such that A is not between B and C . [ 13 ] A point D , on the line determined by A and B but not in the ray with initial point A determined by B , will determine another ray with initial point A . With respect to the AB ray, the AD ray is called the opposite ray .
Thus, we would say that two different points, A and B , define a line and a decomposition of this line into the disjoint union of an open segment ( A , B ) and two rays, BC and AD (the point D is not drawn in the diagram, but is to the left of A on the line AB ). These are not opposite rays since they have different initial points.
In Euclidean geometry two rays with a common endpoint form an angle . [ 14 ]
The definition of a ray depends upon the notion of betweenness for points on a line. It follows that rays exist only for geometries for which this notion exists, typically Euclidean geometry or affine geometry over an ordered field . On the other hand, rays do not exist in projective geometry nor in a geometry over a non-ordered field, like the complex numbers or any finite field .
A line segment is a part of a line that is bounded by two distinct end points and contains every point on the line between its end points. Depending on how the line segment is defined, either of the two end points may or may not be part of the line segment. Two or more line segments may have some of the same relationships as lines, such as being parallel, intersecting, or skew, but unlike lines they may be none of these, if they are coplanar and either do not intersect or are collinear .
A point on number line corresponds to a real number and vice versa. [ 15 ] Usually, integers are evenly spaced on the line, with positive numbers are on the right, negative numbers on the left. As an extension to the concept, an imaginary line representing imaginary numbers can be drawn perpendicular to the number line at zero. [ 16 ] The two lines forms the complex plane , a geometrical representation of the set of complex numbers . | https://en.wikipedia.org/wiki/Ray_(geometry) |
In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity , absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces, complicating analysis.
Historically, ray tracing involved analytic solutions to the ray's trajectories. In modern applied physics and engineering physics , the term also encompasses numerical solutions to the Eikonal equation . For example, ray-marching involves repeatedly advancing idealized narrow beams called rays through the medium by discrete amounts. Simple problems can be analyzed by propagating a few rays using simple mathematics. More detailed analysis can be performed by using a computer to propagate many rays.
When applied to problems of electromagnetic radiation , ray tracing often relies on approximate solutions to Maxwell's equations such as geometric optics , that are valid as long as the light waves propagate through and around objects whose dimensions are much greater than the light's wavelength . Ray theory can describe interference by accumulating the phase during ray tracing (e.g., complex-valued Fresnel coefficients and Jones calculus ). It can also be extended to describe edge diffraction , with modifications such as the geometric theory of diffraction , which enables tracing diffracted rays .
More complicated phenomena require methods such as physical optics or wave theory .
Ray tracing works by assuming that the particle or wave can be modeled as a large number of very narrow beams ( rays ), and that there exists some distance, possibly very small, over which such a ray is locally straight. The ray tracer will advance the ray over this distance, and then use a local derivative of the medium to calculate the ray's new direction. From this location, a new ray is sent out and the process is repeated until a complete path is generated. If the simulation includes solid objects, the ray may be tested for intersection with them at each step, making adjustments to the ray's direction if a collision is found. Other properties of the ray may be altered as the simulation advances as well, such as intensity , wavelength , or polarization . This process is repeated with as many rays as are necessary to understand the behavior of the system.
Ray tracing is being increasingly used in astronomy to simulate realistic images of the sky. Unlike conventional simulations, ray tracing does not use the expected or calculated point spread function (PSF) of a telescope and instead traces the journey of each photon from entrance in the upper atmosphere to collision with the detector. [ 1 ] Most of the dispersion and distortion, arising mainly from atmosphere, optics and detector are taken into account. While this method of simulating images is inherently slow, advances in CPU and GPU capabilities has somewhat mitigated this problem. It can also be used in designing telescopes. Notable examples include Large Synoptic Survey Telescope where this kind of ray tracing was first used with PhoSim [ 2 ] to create simulated images. [ 3 ]
One particular form of ray tracing is radio signal ray tracing, which traces radio signals, modeled as rays, through the ionosphere where they are refracted and/or reflected back to the Earth. This form of ray tracing involves the integration of differential equations that describe the propagation of electromagnetic waves through dispersive and anisotropic media such as the ionosphere. An example of physics-based radio signal ray tracing is shown to the right. Radio communicators use ray tracing to help determine the precise behavior of radio signals as they propagate through the ionosphere.
The image at the right illustrates the complexity of the situation. Unlike optical ray tracing where the medium between objects typically has a constant refractive index , signal ray tracing must deal with the complexities of a spatially varying refractive index, where changes in ionospheric electron densities influence the refractive index and hence, ray trajectories. Two sets of signals are broadcast at two different elevation angles. When the main signal penetrates into the ionosphere, the magnetic field splits the signal into two component waves which are separately ray traced through the ionosphere. The ordinary wave (red) component follows a path completely independent of the extraordinary wave (green) component.
Sound velocity in the ocean varies with depth due to changes in density and temperature , reaching a local minimum near a depth of 800–1000 meters. This local minimum, called the SOFAR channel , acts as a waveguide , as sound tends to bend towards it. Ray tracing may be used to calculate the path of sound through the ocean up to very large distances, incorporating the effects of the SOFAR channel, as well as reflections and refractions off the ocean surface and bottom. From this, locations of high and low signal intensity may be computed, which are useful in the fields of ocean acoustics , underwater acoustic communication , and acoustic thermometry .
Ray tracing may be used in the design of lenses and optical systems , such as in cameras , microscopes , telescopes , and binoculars , and its application in this field dates back to the 1900s. Geometric ray tracing is used to describe the propagation of light rays through a lens system or optical instrument, allowing the image-forming properties of the system to be modeled. The following effects can be integrated into a ray tracer in a straightforward fashion:
For the application of lens design, two special cases of wave interference are important to account for. In a focal point , rays from a point light source meet again and may constructively or destructively interfere with each other. Within a very small region near this point, incoming light may be approximated by plane waves which inherit their direction from the rays. The optical path length from the light source is used to compute the phase . The derivative of the position of the ray in the focal region on the source position is used to obtain the width of the ray, and from that the amplitude of the plane wave. The result is the point spread function , whose Fourier transform is the optical transfer function . From this, the Strehl ratio can also be calculated.
The other special case to consider is that of the interference of wavefronts, which are approximated as planes. However, when the rays come close together or even cross, the wavefront approximation breaks down. Interference of spherical waves is usually not combined with ray tracing, thus diffraction at an aperture cannot be calculated. However, these limitations can be resolved by an advanced modeling technique called Field Tracing . Field Tracing is a modelling technique, combining geometric optics with physical optics enabling to overcome the limitations of interference and diffraction in designing.
The ray tracing techniques are used to optimize the design of the instrument by minimizing aberrations , for photography, and for longer wavelength applications such as designing microwave or even radio systems, and for shorter wavelengths, such as ultraviolet and X-ray optics.
Before the advent of the computer , ray tracing calculations were performed by hand using trigonometry and logarithmic tables. The optical formulas of many classic photographic lenses were optimized by roomfuls of people, each of whom handled a small part of the large calculation. Now they are worked out in optical design software . A simple version of ray tracing known as ray transfer matrix analysis is often used in the design of optical resonators used in lasers . The basic principles of the most frequently used algorithm could be found in Spencer and Murty's fundamental paper: "General ray tracing Procedure". [ 4 ]
There is a ray tracing technique called focal-plane ray tracing where how an optical ray will be after a lens is determined based on a lens focal plane and how the ray crosses the plane. [ 5 ] This method utilizes the fact that rays from a point on the front focal plane of a positive lens will be parallel right after the lens and rays toward a point on the back or rear focal plane of a negative lens will also be parallel after the lens. In each case, the direction of the parallel rays after the lens is determined by a ray appearing to cross the lens nodal points (or the lens center for a thin lens).
In seismology , geophysicists use ray tracing to aid in earthquake location and tomographic reconstruction of the Earth's interior . [ 6 ] [ 7 ] Seismic wave velocity varies within and beneath Earth's crust , causing these waves to bend and reflect. Ray tracing may be used to compute paths through a geophysical model, following them back to their source, such as an earthquake, or deducing the properties of the intervening material. [ 8 ] In particular, the discovery of the seismic shadow zone (illustrated at right) allowed scientists to deduce the presence of Earth's molten core.
In general relativity , where gravitational lensing can occur, the geodesics of the light rays receiving at the observer are integrated backwards in time until they hit the region of interest. Image synthesis under this technique can be view as an extension of the usual ray tracing in computer graphics. [ 9 ] [ 10 ] An example of such synthesis is found in the 2014 film Interstellar . [ 11 ]
In laser-plasma physics ray-tracing can be used to simplify the calculations of laser propagation inside a plasma. Analytic solutions for ray trajectories in simple plasma density profiles are a well established, [ 12 ] however researchers in laser-plasma physics often rely on ray-marching techniques due to the complexity of plasma density, temperature, and flow profiles which are often solved for using computational fluid dynamics simulations. | https://en.wikipedia.org/wiki/Ray_tracing_(physics) |
Nebraska Wesleyan University
Raychelle Burks is an associate professor of analytical chemistry at American University in Washington, D.C. , and science communicator , who has regularly appeared on the Science Channel . In 2020, the American Chemical Society awarded her the Grady-Stack award for her public engagement excellence. [ 1 ]
Burks developed an interest in forensic chemistry when she was 12 after a field trip that presented students with a science interaction challenge, asking students to solve a real-world problem using science. [ 2 ] Burks earned her BS in chemistry at the University of Northern Iowa , her MSc in Forensic Science at Nebraska Wesleyan University , her PhD in chemistry from the University of Nebraska - Lincoln , and was a postdoctoral research associate at the Doane College . [ 3 ]
Burks became an assistant professor of chemistry at St. Edward's University in Austin, Texas , in 2016, where she taught and conducted research until 2020. She then moved to Washington, D.C. , to join the faculty at American University as an associate professor of chemistry. [ 4 ]
Her current research centers on developing low-cost colorimetric sensors for detecting chemicals of forensic interest including explosives and illicit drugs. [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] To maximize portability in the field, her group focuses on transforming smartphones into detection devices. [ 7 ] Her research interests lie in the applied science domain, which she believes is well-suited to capturing and holding students' attention because they are working to solve real-world problems. [ 10 ] She has spoken about her intersectional research approach to equipping students with the technical knowledge they need to work on these real-world challenges with the United States Department of Defense Science, Technology, and Innovation Exchange. [ 10 ] [ 11 ]
Burks is a popular science communicator, using pop culture as an anchor to explore chemistry. She appeared on the Science Channel 's Outrageous Acts of Science and Reactions , the video series for the American Chemical Society . [ 12 ] [ 13 ] She has appeared on Mother Jones ' Inquiring Minds podcast to share how chemistry can save you from a zombie apocalypse and on The Story Collider podcast with a story from her time working in a crime lab. [ 14 ] [ 15 ] In early 2020, she appeared on the NPR Short Wave podcast on the episode "A Short Wave Guide to Good - and Bad - TV Forensics". [ 16 ] Burks has also contributed to scientific interest pieces for St. Andrew University on using chemistry in every day life. [ 17 ] Her writing has been featured in Slate , The Washington Post , UNDARK , and Chemistry World . [ 18 ] [ 19 ] [ 20 ] [ 21 ]
Burks is also an advocate for women and underrepresented groups in science, speaking from her experiences as a black woman in STEM . [ 22 ] [ 23 ] In 2018, Burks was a co-principal investigator for a $1.5 million NSF STEM grant to fund the establishment of the St. Andrew's Institute for Interdisciplinary Science (I4), which would promote internships and research opportunities for underrepresented groups in STEM. [ 24 ] She founded the DIYSciZone at GeekGirlCon , bringing scientists and science educators together to give convention attendees hands-on experiences with science experiments. [ 25 ] The citation for her American Chemical Society Grady-Stack award read, “Raychelle is a public-scientist extraordinaire... She inspires a love of chemistry by bringing chemistry directly to where her audience is. This direct engagement — her commitment to finding chemistry that can entertain and enlighten people who wouldn’t normally think of science — is nothing short of phenomenal". [ 26 ] Burks is active on social media to promote her field and fellow scientists. [ 27 ] [ 28 ]
In 2020, Burks appeared in the Tribeca Film Festival in the film " Picture a Scientist ." [ 29 ] [ 30 ] [ 31 ]
Her awards and honors include; | https://en.wikipedia.org/wiki/Raychelle_Burks |
In fluid dynamics , Rayleigh's equation or Rayleigh stability equation is a linear ordinary differential equation to study the hydrodynamic stability of a parallel, incompressible and inviscid shear flow . The equation is: [ 1 ]
with U ( z ) {\displaystyle U(z)} the flow velocity of the steady base flow whose stability is to be studied and z {\displaystyle z} is the cross-stream direction (i.e. perpendicular to the flow direction). Further φ ( z ) {\displaystyle \varphi (z)} is the complex valued amplitude of the infinitesimal streamfunction perturbations applied to the base flow, k {\displaystyle k} is the wavenumber of the perturbations and c {\displaystyle c} is the phase speed with which the perturbations propagate in the flow direction. The prime denotes differentiation with respect to z . {\displaystyle z.}
The equation is named after Lord Rayleigh , who introduced it in 1880. [ 2 ] The Orr–Sommerfeld equation – introduced later, for the study of stability of parallel viscous flow – reduces to Rayleigh's equation when the viscosity is zero. [ 3 ]
Rayleigh's equation, together with appropriate boundary conditions , most often poses an eigenvalue problem . For given (real-valued) wavenumber k {\displaystyle k} and mean flow velocity U ( z ) , {\displaystyle U(z),} the eigenvalues are the phase speeds c , {\displaystyle c,} and the eigenfunctions are the associated streamfunction amplitudes φ ( z ) . {\displaystyle \varphi (z).} In general, the eigenvalues form a continuous spectrum . In certain cases there may further be a discrete spectrum of complex conjugate pairs of c . {\displaystyle c.} Since the wavenumber k {\displaystyle k} occurs only as a square k 2 {\displaystyle k^{2}} in Rayleigh's equation, a solution (i.e. φ ( z ) {\displaystyle \varphi (z)} and c {\displaystyle c} ) for wavenumber + k {\displaystyle +k} is also a solution for the wavenumber − k . {\displaystyle -k.} [ 3 ]
Rayleigh's equation only concerns two-dimensional perturbations to the flow. From Squire's theorem it follows that the two-dimensional perturbations are less stable than three-dimensional perturbations.
If a real-valued phase speed c {\displaystyle c} is in between the minimum and maximum of U ( z ) , {\displaystyle U(z),} the problem has so-called critical layers near z = z c r i t {\displaystyle z=z_{\mathrm {crit} }} where U ( z c r i t ) = c . {\displaystyle U(z_{\mathrm {crit} })=c.} At the critical layers Rayleigh's equation becomes singular . These were first being studied by Lord Kelvin , also in 1880. [ 4 ] His solution gives rise to a so-called cat's eye pattern of streamlines near the critical layer, when observed in a frame of reference moving with the phase speed c . {\displaystyle c.} [ 3 ]
Consider a parallel shear flow U ( z ) {\displaystyle U(z)} in the x {\displaystyle x} direction, which varies only in the cross-flow direction z . {\displaystyle z.} [ 1 ] The stability of the flow is studied by adding small perturbations to the flow velocity u ( x , z , t ) {\displaystyle u(x,z,t)} and w ( x , z , t ) {\displaystyle w(x,z,t)} in the x {\displaystyle x} and z {\displaystyle z} directions, respectively. The flow is described using the incompressible Euler equations , which become after linearization – using velocity components U ( z ) + u ( x , z , t ) {\displaystyle U(z)+u(x,z,t)} and w ( x , z , t ) : {\displaystyle w(x,z,t):}
with ∂ t {\displaystyle \partial _{t}} the partial derivative operator with respect to time, and similarly ∂ x {\displaystyle \partial _{x}} and ∂ z {\displaystyle \partial _{z}} with respect to x {\displaystyle x} and z . {\displaystyle z.} The pressure fluctuations p ( x , z , t ) {\displaystyle p(x,z,t)} ensure that the continuity equation ∂ x u + ∂ z w = 0 {\displaystyle \partial _{x}u+\partial _{z}w=0} is fulfilled. The fluid density is denoted as ρ {\displaystyle \rho } and is a constant in the present analysis. The prime U ′ {\displaystyle U'} denotes differentiation of U ( z ) {\displaystyle U(z)} with respect to its argument z . {\displaystyle z.}
The flow oscillations u {\displaystyle u} and w {\displaystyle w} are described using a streamfunction ψ ( x , z , t ) , {\displaystyle \psi (x,z,t),} ensuring that the continuity equation is satisfied:
Taking the z {\displaystyle z} - and x {\displaystyle x} -derivatives of the x {\displaystyle x} - and z {\displaystyle z} -momentum equation, and thereafter subtracting the two equations, the pressure p {\displaystyle p} can be eliminated:
which is essentially the vorticity transport equation, ∂ x 2 ψ + ∂ z 2 ψ {\displaystyle \partial _{x}^{2}\psi +\partial _{z}^{2}\psi } being (minus) the vorticity.
Next, sinusoidal fluctuations are considered:
with φ ( z ) {\displaystyle \varphi (z)} the complex-valued amplitude of the streamfunction oscillations, while i {\displaystyle i} is the imaginary unit ( i 2 = − 1 {\displaystyle i^{2}=-1} ) and ℜ { ⋅ } {\displaystyle \Re \left\{\cdot \right\}} denotes the real part of the expression between the brackets. Using this in the vorticity transport equation, Rayleigh's equation is obtained.
The boundary conditions for flat impermeable walls follow from the fact that the streamfunction is a constant at them. So at impermeable walls the streamfunction oscillations are zero, i.e. φ = 0. {\displaystyle \varphi =0.} For unbounded flows the common boundary conditions are that lim z → ± ∞ φ ( z ) = 0. {\displaystyle \lim _{z\to \pm \infty }\varphi (z)=0.} | https://en.wikipedia.org/wiki/Rayleigh's_equation_(fluid_dynamics) |
The Rayleigh quotient represents a quick method to estimate the natural frequency of both discrete and continuous oscillating systems.
where ω n {\displaystyle \omega _{n}} is the natural frequency of the nth mode, V {\displaystyle V} is the potential energy of the system and T ~ {\displaystyle {\tilde {T}}} is a property equivalent to the kinetic energy but with velocity replaced by position.
For multi degree-of-freedom vibration system, in which the mass and the stiffness matrices are known, the Rayleigh quotient can be derived starting from the equation of motion.
The eigenvalue problem for a general system of the form
in absence of damping and external forces reduces to:
The previous equation can be written also as the following:
where ω {\displaystyle \omega } represents the natural frequency and M and K are the real positive symmetric mass and stiffness matrices respectively.
For an N -degree-of-freedom system the equation has N solutions ω n 2 {\displaystyle \omega _{n}^{2}} , u n {\displaystyle {\textbf {u}}_{n}} for n = 1, 2, 3, ..., N . By multiplying both sides of the equation by u n T {\displaystyle {\textbf {u}}_{n}^{T}} and dividing by the scalar u n T M u n {\displaystyle {\textbf {u}}_{n}^{T}\,M\,{\textbf {u}}_{n}} , it is possible to express the eigenvalue problem as follows:
In the previous equation it is also possible to observe that the numerator is proportional to the potential energy while the denominator depicts a measure of the kinetic energy. Moreover, the equation allow us to calculate the natural frequency only if the eigenvector (as well as any other displacement vector) u n {\displaystyle {\textbf {u}}_{n}} is known. For academic interests, if the modal vectors are not known, we can repeat the foregoing process but with ω 2 {\displaystyle \omega ^{2}} and u {\displaystyle {\textbf {u}}} taking the place of ω n 2 {\displaystyle \omega _{n}^{2}} and u n {\displaystyle {\textbf {u}}_{n}} , respectively. By doing so we obtain the scalar R ( u ) {\displaystyle R({\textbf {u}})} , also known as Rayleigh's quotient: [ 1 ]
Therefore, the Rayleigh's quotient is a scalar whose value depends on the vector u {\displaystyle {\textbf {u}}} and it can be calculated with good approximation for any arbitrary vector u {\displaystyle {\textbf {u}}} as long as it lays reasonably far from the modal vectors u i {\displaystyle {\textbf {u}}_{i}} , i = 1,2,3,..., N .
Since, it is possible to state that the vector u {\displaystyle {\textbf {u}}} differs from the modal vector u n {\displaystyle {\textbf {u}}_{n}} by a small quantity of first order, the correct result of the Rayleigh's quotient will differ not sensitively from the estimated one and that's what makes this method very useful. A good way to estimate the lowest modal vector ( u 1 ) {\displaystyle (u_{1})} , that generally works well for most structures (even though is not guaranteed), is to assume ( u 1 ) {\displaystyle (u_{1})} equal to the static displacement from an applied force that has the same relative distribution of the diagonal mass matrix terms. The latter can be elucidated by the following 3-DOF example.
As an example, we can consider a 3-degree-of-freedom system in which the mass and the stiffness matrices of them are known as follows: M = [ 1 0 0 0 1 0 0 0 3 ] , K = [ 3 − 1 0 − 1 3 − 2 0 − 2 2 ] {\displaystyle M={\begin{bmatrix}1&0&0\\0&1&0\\0&0&3\end{bmatrix}}\;,\quad K={\begin{bmatrix}3&-1&0\\-1&3&-2\\0&-2&2\end{bmatrix}}}
To get an estimation of the lowest natural frequency we choose a trial vector of static displacement obtained by loading the system with a force proportional to the masses: F = k [ m 1 m 2 m 3 ] = 1 [ 1 1 3 ] {\displaystyle {\textbf {F}}=k{\begin{bmatrix}m_{1}\\m_{2}\\m_{3}\end{bmatrix}}=1{\begin{bmatrix}1\\1\\3\end{bmatrix}}}
Thus, the trial vector will become u = K − 1 F = [ 2.5 6.5 8 ] {\displaystyle {\textbf {u}}=K^{-1}{\textbf {F}}={\begin{bmatrix}2.5\\6.5\\8\end{bmatrix}}} that allow us to calculate the Rayleigh quotient: R = u T K u u T M u = ⋯ = 0.137214 {\displaystyle R={\frac {{\textbf {u}}^{T}\,K\,{\textbf {u}}}{{\textbf {u}}^{T}\,M\,{\textbf {u}}}}=\cdots =0.137214}
Thus, the lowest natural frequency, calculated by means of the Rayleigh quotient is: ω Ray = 0.370424 {\displaystyle \omega _{\text{Ray}}=0.370424}
Using a calculation tool is pretty fast to verify how much it differs from the "real" one. In this case, using MATLAB, it has been calculated that the lowest natural frequency is: ω real = 0.369308 {\displaystyle \omega _{\text{real}}=0.369308} that has led to an error of 0.302315 % {\displaystyle 0.302315\%} using the Rayleigh's approximation, a remarkable result.
The example shows how the Rayleigh quotient is capable of getting an accurate estimation of the lowest natural frequency. The practice of using the static displacement vector as a trial vector is valid as the static displacement vector tends to resemble the lowest vibration mode.
For continuous systems, the concept of mass and stiffness matrices does not apply, but it can be seen that the Rayleigh quotient is still the ratio of the potential energy to the "kinetic energy without the time derivatives":
The arises for the concept of the maximum kinetic energy being equal to the maximum potential energy for conservative systems [ 2 ] For the case of a string of mass per unit length m under tension P: | https://en.wikipedia.org/wiki/Rayleigh's_quotient_in_vibrations_analysis |
In distillation , a Rayleigh Still is a separation process where a feed stream is charged and withdrawn batch-wise with a separate stream fed and removed continuously . It is also known as a Rayleigh Distillation. It consists of a single stage distillation where the batch-charged phase is well mixed during operation. This was developed originally by Lord Rayleigh in 1902. [ 1 ]
This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rayleigh_Still |
In fluid dynamics , Rayleigh flow (after English physicist Lord Rayleigh ) refers to frictionless , non- adiabatic fluid flow through a constant-area duct where the effect of heat transfer is considered. Compressibility effects often come into consideration, although the Rayleigh flow model certainly also applies to incompressible flow . For this model, the duct area remains constant and no mass is added within the duct. Therefore, unlike Fanno flow , the stagnation temperature is a variable. The heat addition causes a decrease in stagnation pressure , which is known as the Rayleigh effect and is critical in the design of combustion systems. Heat addition will cause both supersonic and subsonic Mach numbers to approach Mach 1, resulting in choked flow . Conversely, heat rejection decreases a subsonic Mach number and increases a supersonic Mach number along the duct. It can be shown that for calorically perfect flows the maximum entropy occurs at M = 1 .
The Rayleigh flow model begins with a differential equation that relates the change in Mach number with the change in stagnation temperature , T 0 . The differential equation is shown below.
Solving the differential equation leads to the relation shown below, where T 0 * is the stagnation temperature at the throat location of the duct which is required for thermally choking the flow.
These values are significant in the design of combustion systems. For example, if a turbojet combustion chamber has a maximum temperature of T 0 * = 2000 K, T 0 and M at the entrance to the combustion chamber must be selected so thermal choking does not occur, which will limit the mass flow rate of air into the engine and decrease thrust.
For the Rayleigh flow model, the dimensionless change in entropy relation is shown below.
The above equation can be used to plot the Rayleigh line on a Mach number versus ΔS graph, but the dimensionless enthalpy, H, versus ΔS diagram, is more often used. The dimensionless enthalpy equation is shown below with an equation relating the static temperature with its value at the choke location for a calorically perfect gas where the heat capacity at constant pressure, c p , remains constant.
The above equation can be manipulated to solve for M as a function of H. However, due to the form of the T/T* equation, a complicated multi-root relation is formed for M = M(T/T*). Instead, M can be chosen as an independent variable where ΔS and H can be matched up in a chart as shown in Figure 1. Figure 1 shows that heating will increase an upstream, subsonic Mach number until M = 1.0 and the flow chokes . Conversely, adding heat to a duct with an upstream, supersonic Mach number will cause the Mach number to decrease until the flow chokes. Cooling produces the opposite result for each of those two cases. The Rayleigh flow model reaches maximum entropy at M = 1.0 For subsonic flow, the maximum value of H occurs at M = 0.845. This indicates that cooling, instead of heating, causes the Mach number to move from 0.845 to 1.0 This is not necessarily correct as the stagnation temperature always increases to move the flow from a subsonic Mach number to M = 1, but from M = 0.845 to M = 1.0 the flow accelerates faster than heat is added to it. Therefore, this is a situation where heat is added but T/T* decreases in that region.
The area and mass flow rate are held constant for Rayleigh flow. Unlike Fanno flow, the Fanning friction factor , f , remains constant. These relations are shown below with the * symbol representing the throat location where choking can occur.
Differential equations can also be developed and solved to describe Rayleigh flow property ratios with respect to the values at the choking location. The ratios for the pressure, density, static temperature, velocity and stagnation pressure are shown below, respectively. They are represented graphically along with the stagnation temperature ratio equation from the previous section. A stagnation property contains a '0' subscript.
The Rayleigh flow model has many analytical uses, most notably involving aircraft engines. For instance, the combustion chambers inside turbojet engines usually have a constant area and the fuel mass addition is negligible. These properties make the Rayleigh flow model applicable for heat addition to the flow through combustion, assuming the heat addition does not result in dissociation of the air-fuel mixture. Producing a shock wave inside the combustion chamber of an engine due to thermal choking is very undesirable due to the decrease in mass flow rate and thrust. Therefore, the Rayleigh flow model is critical for an initial design of the duct geometry and combustion temperature for an engine.
The Rayleigh flow model is also used extensively with the Fanno flow model. These two models intersect at points on the enthalpy-entropy and Mach number-entropy diagrams, which is meaningful for many applications. However, the entropy values for each model are not equal at the sonic state. The change in entropy is 0 at M = 1 for each model, but the previous statement means the change in entropy from the same arbitrary point to the sonic point is different for the Fanno and Rayleigh flow models. If initial values of s i and M i are defined, a new equation for dimensionless entropy versus Mach number can be defined for each model. These equations are shown below for Fanno and Rayleigh flow, respectively.
Figure 3 shows the Rayleigh and Fanno lines intersecting with each other for initial conditions of s i = 0 and M i = 3.0 The intersection points are calculated by equating the new dimensionless entropy equations with each other, resulting in the relation below.
The intersection points occur at the given initial Mach number and its post- normal shock value. For Figure 3, these values are M = 3.0 and 0.4752, which can be found the normal shock tables listed in most compressible flow textbooks. A given flow with a constant duct area can switch between the Rayleigh and Fanno models at these points. | https://en.wikipedia.org/wiki/Rayleigh_flow |
Rayleigh fractionation describes the evolution of a system with multiple phases in which one phase is continuously removed from the system through fractional distillation . It is used in particular to describe isotopic enrichment or depletion as material moves between reservoirs in an equilibrium process . Rayleigh fractionation holds particular importance in hydrology and meteorology as a model for the isotopic differentiation of meteoric water due to condensation.
The original Rayleigh equation was derived by Lord Rayleigh for the case of fractional distillation of mixed liquids. [ 1 ]
This is an exponential relation that describes the partitioning of isotopes between two reservoirs as one reservoir decreases in size. The equations can be used to describe an isotope fractionation process if: (1) material is continuously removed from a mixed system containing molecules of two or more isotopic species (e.g., water with 18 O and 16 O, or sulfate with 34 S and 32 S), (2) the fractionation accompanying the removal process at any instance is described by the fractionation factor a, and (3) a does not change during the process. Under these conditions, the evolution of the isotopic composition in the residual (reactant) material is described by:
R R 0 = ( X X 0 ) a − 1 {\displaystyle {\frac {R}{R^{0}}}=\left({\frac {X}{X^{0}}}\right)^{a-1}}
where R = ratio of the isotopes (e.g., 18 O/ 16 O) in the reactant, R 0 = initial ratio, X = the concentration or amount of the more abundant (lighter) isotope (e.g., 16 O), and X 0 = initial concentration. Because the concentration of X >> Xh (heavier isotope concentration), X is approximately equal to the amount of original material in the phase. Hence, if f = X / X 0 {\displaystyle f=X/X^{0}} = fraction of material remaining, then:
R = R 0 f a − 1 {\displaystyle R=R^{0}f^{a-1}}
For large changes in concentration, such as they occur during e.g. distillation of heavy water, these formulae need to be integrated over the distillation trajectory. For small changes such as occur during transport of water vapour through the atmosphere , the differentiated equation will usually be sufficient. | https://en.wikipedia.org/wiki/Rayleigh_fractionation |
In fluid mechanics , the Rayleigh number ( Ra , after Lord Rayleigh [ 1 ] ) for a fluid is a dimensionless number associated with buoyancy -driven flow, also known as free (or natural) convection . [ 2 ] [ 3 ] [ 4 ] It characterises the fluid's flow regime: [ 5 ] a value in a certain lower range denotes laminar flow ; a value in a higher range, turbulent flow . Below a certain critical value, there is no fluid motion and heat transfer is by conduction rather than convection. For most engineering purposes, the Rayleigh number is large, somewhere around 10 6 to 10 8 .
The Rayleigh number is defined as the product of the Grashof number ( Gr ), which describes the relationship between buoyancy and viscosity within a fluid, and the Prandtl number ( Pr ), which describes the relationship between momentum diffusivity and thermal diffusivity : Ra = Gr × Pr . [ 4 ] [ 3 ] Hence it may also be viewed as the ratio of buoyancy and viscosity forces multiplied by the ratio of momentum and thermal diffusivities: Ra = B/ μ × ν / α . It is closely related to the Nusselt number ( Nu ). [ 5 ]
The Rayleigh number describes the behaviour of fluids (such as water or air) when the mass density of the fluid is non-uniform. The mass density differences are usually caused by temperature differences. Typically a fluid expands and becomes less dense as it is heated. Gravity causes denser parts of the fluid to sink, which is called convection . Lord Rayleigh studied [ 2 ] the case of Rayleigh-Bénard convection . [ 6 ] When the Rayleigh number, Ra, is below a critical value for a fluid, there is no flow and heat transfer is purely by conduction ; when it exceeds that value, heat is transferred by natural convection. [ 3 ]
When the mass density difference is caused by temperature difference, Ra is, by definition, the ratio of the time scale for diffusive thermal transport to the time scale for convective thermal transport at speed u {\displaystyle u} : [ 4 ]
R a = time scale for thermal transport via diffusion time scale for thermal transport via convection at speed u . {\displaystyle \mathrm {Ra} ={\frac {\text{time scale for thermal transport via diffusion}}{{\text{time scale for thermal transport via convection at speed}}~u}}.}
This means the Rayleigh number is a type [ 4 ] of Péclet number . For a volume of fluid of size l {\displaystyle l} in all three dimensions [ clarification needed ] and mass density difference Δ ρ {\displaystyle \Delta \rho } , the force due to gravity is of the order Δ ρ l 3 g {\displaystyle \Delta \rho l^{3}g} , where g {\displaystyle g} is acceleration due to gravity. From the Stokes equation , when the volume of fluid is sinking, viscous drag is of the order η l u {\displaystyle \eta lu} , where η {\displaystyle \eta } is the dynamic viscosity of the fluid. When these two forces are equated, the speed u ∼ Δ ρ l 2 g / η {\displaystyle u\sim \Delta \rho l^{2}g/\eta } . Thus the time scale for transport via flow is l / u ∼ η / Δ ρ l g {\displaystyle l/u\sim \eta /\Delta \rho lg} . The time scale for thermal diffusion across a distance l {\displaystyle l} is l 2 / α {\displaystyle l^{2}/\alpha } , where α {\displaystyle \alpha } is the thermal diffusivity . Thus the Rayleigh number Ra is
R a = l 2 / α η / Δ ρ l g = Δ ρ l 3 g η α = ρ β Δ T l 3 g η α {\displaystyle \mathrm {Ra} ={\frac {l^{2}/\alpha }{\eta /\Delta \rho lg}}={\frac {\Delta \rho l^{3}g}{\eta \alpha }}={\frac {\rho \beta \Delta Tl^{3}g}{\eta \alpha }}}
where we approximated the density difference Δ ρ = ρ β Δ T {\displaystyle \Delta \rho =\rho \beta \Delta T} for a fluid of average mass density ρ {\displaystyle \rho } , thermal expansion coefficient β {\displaystyle \beta } and a temperature difference Δ T {\displaystyle \Delta T} across distance l {\displaystyle l} .
The Rayleigh number can be written as the product of the Grashof number and the Prandtl number : [ 4 ] [ 3 ] R a = G r P r . {\displaystyle \mathrm {Ra} =\mathrm {Gr} \mathrm {Pr} .}
For free convection near a vertical wall, the Rayleigh number is defined as:
R a x = g β ν α ( T s − T ∞ ) x 3 = G r x P r {\displaystyle \mathrm {Ra} _{x}={\frac {g\beta }{\nu \alpha }}(T_{s}-T_{\infty })x^{3}=\mathrm {Gr} _{x}\mathrm {Pr} }
where:
In the above, the fluid properties Pr, ν , α and β are evaluated at the film temperature , which is defined as:
T f = T s + T ∞ 2 . {\displaystyle T_{f}={\frac {T_{s}+T_{\infty }}{2}}.}
For a uniform wall heating flux, the modified Rayleigh number is defined as:
R a x ∗ = g β q o ″ ν α k x 4 {\displaystyle \mathrm {Ra} _{x}^{*}={\frac {g\beta q''_{o}}{\nu \alpha k}}x^{4}}
where:
The Rayleigh number can also be used as a criterion to predict convectional instabilities, such as A-segregates , in the mushy zone of a solidifying alloy. The mushy zone Rayleigh number is defined as:
R a = Δ ρ ρ 0 g K ¯ L α ν = Δ ρ ρ 0 g K ¯ R ν {\displaystyle \mathrm {Ra} ={\frac {{\frac {\Delta \rho }{\rho _{0}}}g{\bar {K}}L}{\alpha \nu }}={\frac {{\frac {\Delta \rho }{\rho _{0}}}g{\bar {K}}}{R\nu }}}
where:
A-segregates are predicted to form when the Rayleigh number exceeds a certain critical value. This critical value is independent of the composition of the alloy, and this is the main advantage of the Rayleigh number criterion over other criteria for prediction of convectional instabilities, such as Suzuki criterion.
Torabi Rad et al. showed that for steel alloys the critical Rayleigh number is 17. [ 8 ] Pickering et al. explored Torabi Rad's criterion, and further verified its effectiveness. Critical Rayleigh numbers for lead–tin and nickel-based super-alloys were also developed. [ 9 ]
The Rayleigh number above is for convection in a bulk fluid such as air or water, but convection can also occur when the fluid is inside and fills a porous medium, such as porous rock saturated with water. [ 10 ] Then the Rayleigh number, sometimes called the Rayleigh-Darcy number , is different. In a bulk fluid, i.e., not in a porous medium, from the Stokes equation , the falling speed of a domain of size l {\displaystyle l} of liquid u ∼ Δ ρ l 2 g / η {\displaystyle u\sim \Delta \rho l^{2}g/\eta } . In porous medium, this expression is replaced by that from Darcy's law u ∼ Δ ρ k g / η {\displaystyle u\sim \Delta \rho kg/\eta } , with k {\displaystyle k} the permeability of the porous medium. The Rayleigh or Rayleigh-Darcy number is then
R a = ρ β Δ T k l g η α {\displaystyle \mathrm {Ra} ={\frac {\rho \beta \Delta Tklg}{\eta \alpha }}}
This also applies to A-segregates , in the mushy zone of a solidifying alloy. [ 8 ]
In geophysics , the Rayleigh number is of fundamental importance: it indicates the presence and strength of convection within a fluid body such as the Earth's mantle . The mantle is a solid that behaves as a fluid over geological time scales. The Rayleigh number for the Earth's mantle due to internal heating alone, Ra H , is given by:
R a H = g ρ 0 2 β H D 5 η α k {\displaystyle \mathrm {Ra} _{H}={\frac {g\rho _{0}^{2}\beta HD^{5}}{\eta \alpha k}}}
where:
A Rayleigh number for bottom heating of the mantle from the core, Ra T , can also be defined as:
R a T = ρ 0 2 g β Δ T sa D 3 C P η k {\displaystyle \mathrm {Ra} _{T}={\frac {\rho _{0}^{2}g\beta \Delta T_{\text{sa}}D^{3}C_{P}}{\eta k}}}
where:
High values for the Earth's mantle indicates that convection within the Earth is vigorous and time-varying, and that convection is responsible for almost all the heat transported from the deep interior to the surface. | https://en.wikipedia.org/wiki/Rayleigh_number |
In fluid dynamics, Rayleigh problem also known as Stokes first problem is a problem of determining the flow created by a sudden movement of an infinitely long plate from rest, named after Lord Rayleigh and Sir George Stokes . This is considered as one of the simplest unsteady problems that have an exact solution for the Navier-Stokes equations . The impulse movement of semi-infinite plate was studied by Keith Stewartson . [ 1 ]
Consider an infinitely long plate which is suddenly made to move with constant velocity U {\displaystyle U} in the x {\displaystyle x} direction, which is located at y = 0 {\displaystyle y=0} in an infinite domain of fluid, which is at rest initially everywhere. The incompressible Navier-Stokes equations reduce to [ 2 ] [ 3 ]
where ν {\displaystyle \nu } is the kinematic viscosity . The initial and the no-slip condition on the wall are
the last condition is due to the fact that the motion at y = 0 {\displaystyle y=0} is not felt at infinity. The flow is only due to the motion of the plate, there is no imposed pressure gradient.
The problem on the whole is similar to the one dimensional heat conduction problem. Hence a self-similar variable can be introduced [ 4 ]
Substituting this into the partial differential equation reduces it to an ordinary differential equation
with boundary conditions
The solution to the above problem can be written in terms of complementary error function
The force per unit area exerted on the plate is
Instead of using a step boundary condition for the wall movement, the velocity of the wall can be prescribed as an arbitrary function of time, i.e., U = f ( t ) {\displaystyle U=f(t)} . Then the solution is given by [ 5 ]
Consider an infinitely long cylinder of radius a {\displaystyle a} starts rotating suddenly at time t = 0 {\displaystyle t=0} with an angular velocity Ω {\displaystyle \Omega } . Then the velocity in the θ {\displaystyle \theta } direction is given by
where K 1 {\displaystyle K_{1}} is the modified Bessel function of the second kind. As t → ∞ {\displaystyle t\rightarrow \infty } , the solution approaches that of a rigid vortex. The force per unit area exerted on the cylinder is
where I 0 {\displaystyle I_{0}} is the modified Bessel function of the first kind.
Exact solution is also available when the cylinder starts to slide in the axial direction with constant velocity U {\displaystyle U} . If we consider the cylinder axis to be in x {\displaystyle x} direction, then the solution is given by | https://en.wikipedia.org/wiki/Rayleigh_problem |
In mathematics , the Rayleigh quotient [ 1 ] ( / ˈ r eɪ . l i / ) for a given complex Hermitian matrix M {\displaystyle M} and nonzero vector x {\displaystyle x} is defined as: [ 2 ] [ 3 ] R ( M , x ) = x ∗ M x x ∗ x . {\displaystyle R(M,x)={x^{*}Mx \over x^{*}x}.} For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric , and the conjugate transpose x ∗ {\displaystyle x^{*}} to the usual transpose x ′ {\displaystyle x'} . Note that R ( M , c x ) = R ( M , x ) {\displaystyle R(M,cx)=R(M,x)} for any non-zero scalar c {\displaystyle c} . Recall that a Hermitian (or real symmetric) matrix is diagonalizable with only real eigenvalues . It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value λ min {\displaystyle \lambda _{\min }} (the smallest eigenvalue of M {\displaystyle M} ) when x {\displaystyle x} is v min {\displaystyle v_{\min }} (the corresponding eigenvector ). [ 4 ] Similarly, R ( M , x ) ≤ λ max {\displaystyle R(M,x)\leq \lambda _{\max }} and R ( M , v max ) = λ max {\displaystyle R(M,v_{\max })=\lambda _{\max }} .
The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms (such as Rayleigh quotient iteration ) to obtain an eigenvalue approximation from an eigenvector approximation.
The range of the Rayleigh quotient (for any matrix, not necessarily Hermitian) is called a numerical range and contains its spectrum . When the matrix is Hermitian, the numerical radius is equal to the spectral norm. Still in functional analysis, λ max {\displaystyle \lambda _{\max }} is known as the spectral radius . In the context of C ⋆ {\displaystyle C^{\star }} -algebras or algebraic quantum mechanics, the function that to M {\displaystyle M} associates the Rayleigh–Ritz quotient R ( M , x ) {\displaystyle R(M,x)} for a fixed x {\displaystyle x} and M {\displaystyle M} varying through the algebra would be referred to as vector state of the algebra.
In quantum mechanics , the Rayleigh quotient gives the expectation value of the observable corresponding to the operator M {\displaystyle M} for a system whose state is given by x {\displaystyle x} .
If we fix the complex matrix M {\displaystyle M} , then the resulting Rayleigh quotient map (considered as a function of x {\displaystyle x} ) completely determines M {\displaystyle M} via the polarization identity ; indeed, this remains true even if we allow M {\displaystyle M} to be non-Hermitian. However, if we restrict the field of scalars to the real numbers, then the Rayleigh quotient only determines the symmetric part of M {\displaystyle M} .
As stated in the introduction, for any vector x , one has R ( M , x ) ∈ [ λ min , λ max ] {\displaystyle R(M,x)\in \left[\lambda _{\min },\lambda _{\max }\right]} , where λ min , λ max {\displaystyle \lambda _{\min },\lambda _{\max }} are respectively the smallest and largest eigenvalues of M {\displaystyle M} . This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of M : R ( M , x ) = x ∗ M x x ∗ x = ∑ i = 1 n λ i y i 2 ∑ i = 1 n y i 2 {\displaystyle R(M,x)={x^{*}Mx \over x^{*}x}={\frac {\sum _{i=1}^{n}\lambda _{i}y_{i}^{2}}{\sum _{i=1}^{n}y_{i}^{2}}}} where ( λ i , v i ) {\displaystyle (\lambda _{i},v_{i})} is the i {\displaystyle i} -th eigenpair after orthonormalization and y i = v i ∗ x {\displaystyle y_{i}=v_{i}^{*}x} is the i {\displaystyle i} th coordinate of x in the eigenbasis. It is then easy to verify that the bounds are attained at the corresponding eigenvectors v min , v max {\displaystyle v_{\min },v_{\max }} .
The fact that the quotient is a weighted average of the eigenvalues can be used to identify the second, the third, ... largest eigenvalues. Let λ max = λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n = λ min {\displaystyle \lambda _{\max }=\lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}=\lambda _{\min }} be the eigenvalues in decreasing order. If n = 2 {\displaystyle n=2} and x {\displaystyle x} is constrained to be orthogonal to v 1 {\displaystyle v_{1}} , in which case y 1 = v 1 ∗ x = 0 {\displaystyle y_{1}=v_{1}^{*}x=0} , then R ( M , x ) {\displaystyle R(M,x)} has maximum value λ 2 {\displaystyle \lambda _{2}} , which is achieved when x = v 2 {\displaystyle x=v_{2}} .
An empirical covariance matrix M {\displaystyle M} can be represented as the product A ′ A {\displaystyle A'A} of the data matrix A {\displaystyle A} pre-multiplied by its transpose A ′ {\displaystyle A'} . Being a positive semi-definite matrix, M {\displaystyle M} has non-negative eigenvalues, and orthogonal (or orthogonalisable) eigenvectors, which can be demonstrated as follows.
Firstly, that the eigenvalues λ i {\displaystyle \lambda _{i}} are non-negative: M v i = A ′ A v i = λ i v i ⇒ v i ′ A ′ A v i = v i ′ λ i v i ⇒ ‖ A v i ‖ 2 = λ i ‖ v i ‖ 2 ⇒ λ i = ‖ A v i ‖ 2 ‖ v i ‖ 2 ≥ 0. {\displaystyle {\begin{aligned}&Mv_{i}=A'Av_{i}=\lambda _{i}v_{i}\\\Rightarrow {}&v_{i}'A'Av_{i}=v_{i}'\lambda _{i}v_{i}\\\Rightarrow {}&\left\|Av_{i}\right\|^{2}=\lambda _{i}\left\|v_{i}\right\|^{2}\\\Rightarrow {}&\lambda _{i}={\frac {\left\|Av_{i}\right\|^{2}}{\left\|v_{i}\right\|^{2}}}\geq 0.\end{aligned}}}
Secondly, that the eigenvectors v i {\displaystyle v_{i}} are orthogonal to one another: M v i = λ i v i ⇒ v j ′ M v i = v j ′ λ i v i ⇒ ( M v j ) ′ v i = λ j v j ′ v i ⇒ λ j v j ′ v i = λ i v j ′ v i ⇒ ( λ j − λ i ) v j ′ v i = 0 ⇒ v j ′ v i = 0 {\displaystyle {\begin{aligned}&Mv_{i}=\lambda _{i}v_{i}\\\Rightarrow {}&v_{j}'Mv_{i}=v_{j}'\lambda _{i}v_{i}\\\Rightarrow {}&\left(Mv_{j}\right)'v_{i}=\lambda _{j}v_{j}'v_{i}\\\Rightarrow {}&\lambda _{j}v_{j}'v_{i}=\lambda _{i}v_{j}'v_{i}\\\Rightarrow {}&\left(\lambda _{j}-\lambda _{i}\right)v_{j}'v_{i}=0\\\Rightarrow {}&v_{j}'v_{i}=0\end{aligned}}} if the eigenvalues are different – in the case of multiplicity, the basis can be orthogonalized.
To now establish that the Rayleigh quotient is maximized by the eigenvector with the largest eigenvalue, consider decomposing an arbitrary vector x {\displaystyle x} on the basis of the eigenvectors v i {\displaystyle v_{i}} : x = ∑ i = 1 n α i v i , {\displaystyle x=\sum _{i=1}^{n}\alpha _{i}v_{i},} where α i = x ′ v i v i ′ v i = ⟨ x , v i ⟩ ‖ v i ‖ 2 {\displaystyle \alpha _{i}={\frac {x'v_{i}}{v_{i}'v_{i}}}={\frac {\langle x,v_{i}\rangle }{\left\|v_{i}\right\|^{2}}}} is the coordinate of x {\displaystyle x} orthogonally projected onto v i {\displaystyle v_{i}} . Therefore, we have: R ( M , x ) = x ′ A ′ A x x ′ x = ( ∑ j = 1 n α j v j ) ′ ( A ′ A ) ( ∑ i = 1 n α i v i ) ( ∑ j = 1 n α j v j ) ′ ( ∑ i = 1 n α i v i ) = ( ∑ j = 1 n α j v j ) ′ ( ∑ i = 1 n α i ( A ′ A ) v i ) ( ∑ i = 1 n α i 2 v i ′ v i ) = ( ∑ j = 1 n α j v j ) ′ ( ∑ i = 1 n α i λ i v i ) ( ∑ i = 1 n α i 2 ‖ v i ‖ 2 ) {\displaystyle {\begin{aligned}R(M,x)&={\frac {x'A'Ax}{x'x}}\\&={\frac {{\Bigl (}\sum _{j=1}^{n}\alpha _{j}v_{j}{\Bigr )}'\left(A'A\right){\Bigl (}\sum _{i=1}^{n}\alpha _{i}v_{i}{\Bigr )}}{{\Bigl (}\sum _{j=1}^{n}\alpha _{j}v_{j}{\Bigr )}'{\Bigl (}\sum _{i=1}^{n}\alpha _{i}v_{i}{\Bigr )}}}\\&={\frac {{\Bigl (}\sum _{j=1}^{n}\alpha _{j}v_{j}{\Bigr )}'{\Bigl (}\sum _{i=1}^{n}\alpha _{i}(A'A)v_{i}{\Bigr )}}{{\Bigl (}\sum _{i=1}^{n}\alpha _{i}^{2}{v_{i}}'{v_{i}}{\Bigr )}}}\\&={\frac {{\Bigl (}\sum _{j=1}^{n}\alpha _{j}v_{j}{\Bigr )}'{\Bigl (}\sum _{i=1}^{n}\alpha _{i}\lambda _{i}v_{i}{\Bigr )}}{{\Bigl (}\sum _{i=1}^{n}\alpha _{i}^{2}\|{v_{i}}\|^{2}{\Bigr )}}}\end{aligned}}} which, by orthonormality of the eigenvectors, becomes: R ( M , x ) = ∑ i = 1 n α i 2 λ i ∑ i = 1 n α i 2 = ∑ i = 1 n λ i ( x ′ v i ) 2 ( x ′ x ) ( v i ′ v i ) 2 = ∑ i = 1 n λ i ( x ′ v i ) 2 ( x ′ x ) {\displaystyle {\begin{aligned}R(M,x)&={\frac {\sum _{i=1}^{n}\alpha _{i}^{2}\lambda _{i}}{\sum _{i=1}^{n}\alpha _{i}^{2}}}\\&=\sum _{i=1}^{n}\lambda _{i}{\frac {(x'v_{i})^{2}}{(x'x)(v_{i}'v_{i})^{2}}}\\&=\sum _{i=1}^{n}\lambda _{i}{\frac {(x'v_{i})^{2}}{(x'x)}}\end{aligned}}}
The last representation establishes that the Rayleigh quotient is the sum of the squared cosines of the angles formed by the vector x {\displaystyle x} and each eigenvector v i {\displaystyle v_{i}} , weighted by corresponding eigenvalues.
If a vector x {\displaystyle x} maximizes R ( M , x ) {\displaystyle R(M,x)} , then any non-zero scalar multiple k x {\displaystyle kx} also maximizes R {\displaystyle R} , so the problem can be reduced to the Lagrange problem of maximizing ∑ i = 1 n α i 2 λ i {\textstyle \sum _{i=1}^{n}\alpha _{i}^{2}\lambda _{i}} under the constraint that ∑ i = 1 n α i 2 = 1 {\textstyle \sum _{i=1}^{n}\alpha _{i}^{2}=1} .
Define: β i = α i 2 {\displaystyle \beta _{i}=\alpha _{i}^{2}} . This then becomes a linear program , which always attains its maximum at one of the corners of the domain. A maximum point will have α 1 = ± 1 {\displaystyle \alpha _{1}=\pm 1} and α i = 0 {\displaystyle \alpha _{i}=0} for all i > 1 {\displaystyle i>1} (when the eigenvalues are ordered by decreasing magnitude).
Thus, the Rayleigh quotient is maximized by the eigenvector with the largest eigenvalue.
Alternatively, this result can be arrived at by the method of Lagrange multipliers . The first part is to show that the quotient is constant under scaling x → c x {\displaystyle x\to cx} , where c {\displaystyle c} is a scalar R ( M , c x ) = ( c x ) ∗ M c x ( c x ) ∗ c x = c ∗ c c ∗ c x ∗ M x x ∗ x = R ( M , x ) . {\displaystyle R(M,cx)={\frac {(cx)^{*}Mcx}{(cx)^{*}cx}}={\frac {c^{*}c}{c^{*}c}}{\frac {x^{*}Mx}{x^{*}x}}=R(M,x).}
Because of this invariance, it is sufficient to study the special case ‖ x ‖ 2 = x T x = 1 {\displaystyle \|x\|^{2}=x^{T}x=1} . The problem is then to find the critical points of the function R ( M , x ) = x T M x , {\displaystyle R(M,x)=x^{\mathsf {T}}Mx,} subject to the constraint ‖ x ‖ 2 = x T x = 1. {\displaystyle \|x\|^{2}=x^{T}x=1.} In other words, it is to find the critical points of L ( x ) = x T M x − λ ( x T x − 1 ) , {\displaystyle {\mathcal {L}}(x)=x^{\mathsf {T}}Mx-\lambda \left(x^{\mathsf {T}}x-1\right),} where λ {\displaystyle \lambda } is a Lagrange multiplier. The stationary points of L ( x ) {\displaystyle {\mathcal {L}}(x)} occur at d L ( x ) d x = 0 ⇒ 2 x T M − 2 λ x T = 0 ⇒ 2 M x − 2 λ x = 0 (taking the transpose of both sides and noting that M is Hermitian) ⇒ M x = λ x {\displaystyle {\begin{aligned}&{\frac {d{\mathcal {L}}(x)}{dx}}=0\\\Rightarrow {}&2x^{\mathsf {T}}M-2\lambda x^{\mathsf {T}}=0\\\Rightarrow {}&2Mx-2\lambda x=0{\text{ (taking the transpose of both sides and noting that }}M{\text{ is Hermitian)}}\\\Rightarrow {}&Mx=\lambda x\end{aligned}}} and ∴ R ( M , x ) = x T M x x T x = λ x T x x T x = λ . {\displaystyle \therefore R(M,x)={\frac {x^{\mathsf {T}}Mx}{x^{\mathsf {T}}x}}=\lambda {\frac {x^{\mathsf {T}}x}{x^{\mathsf {T}}x}}=\lambda .}
Therefore, the eigenvectors x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} of M {\displaystyle M} are the critical points of the Rayleigh quotient and their corresponding eigenvalues λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are the stationary values of L {\displaystyle {\mathcal {L}}} . This property is the basis for principal components analysis and canonical correlation .
Sturm–Liouville theory concerns the action of the linear operator L ( y ) = 1 w ( x ) ( − d d x [ p ( x ) d y d x ] + q ( x ) y ) {\displaystyle L(y)={\frac {1}{w(x)}}\left(-{\frac {d}{dx}}\left[p(x){\frac {dy}{dx}}\right]+q(x)y\right)} on the inner product space defined by ⟨ y 1 , y 2 ⟩ = ∫ a b w ( x ) y 1 ( x ) y 2 ( x ) d x {\displaystyle \langle {y_{1},y_{2}}\rangle =\int _{a}^{b}w(x)y_{1}(x)y_{2}(x)\,dx} of functions satisfying some specified boundary conditions at a and b . In this case the Rayleigh quotient is ⟨ y , L y ⟩ ⟨ y , y ⟩ = ∫ a b y ( x ) ( − d d x [ p ( x ) d y d x ] + q ( x ) y ( x ) ) d x ∫ a b w ( x ) y ( x ) 2 d x . {\displaystyle {\frac {\langle {y,Ly}\rangle }{\langle {y,y}\rangle }}={\frac {\int _{a}^{b}y(x)\left(-{\frac {d}{dx}}\left[p(x){\frac {dy}{dx}}\right]+q(x)y(x)\right)dx}{\int _{a}^{b}{w(x)y(x)^{2}}dx}}.}
This is sometimes presented in an equivalent form, obtained by separating the integral in the numerator and using integration by parts : ⟨ y , L y ⟩ ⟨ y , y ⟩ = { ∫ a b y ( x ) ( − d d x [ p ( x ) y ′ ( x ) ] ) d x } + { ∫ a b q ( x ) y ( x ) 2 d x } ∫ a b w ( x ) y ( x ) 2 d x = { − y ( x ) [ p ( x ) y ′ ( x ) ] | a b } + { ∫ a b y ′ ( x ) [ p ( x ) y ′ ( x ) ] d x } + { ∫ a b q ( x ) y ( x ) 2 d x } ∫ a b w ( x ) y ( x ) 2 d x = { − p ( x ) y ( x ) y ′ ( x ) | a b } + { ∫ a b [ p ( x ) y ′ ( x ) 2 + q ( x ) y ( x ) 2 ] d x } ∫ a b w ( x ) y ( x ) 2 d x . {\displaystyle {\begin{aligned}{\frac {\langle {y,Ly}\rangle }{\langle {y,y}\rangle }}&={\frac {\left\{\int _{a}^{b}y(x)\left(-{\frac {d}{dx}}\left[p(x)y'(x)\right]\right)dx\right\}+\left\{\int _{a}^{b}{q(x)y(x)^{2}}\,dx\right\}}{\int _{a}^{b}{w(x)y(x)^{2}}\,dx}}\\&={\frac {\left\{\left.-y(x)\left[p(x)y'(x)\right]\right|_{a}^{b}\right\}+\left\{\int _{a}^{b}y'(x)\left[p(x)y'(x)\right]\,dx\right\}+\left\{\int _{a}^{b}{q(x)y(x)^{2}}\,dx\right\}}{\int _{a}^{b}w(x)y(x)^{2}\,dx}}\\&={\frac {\left\{\left.-p(x)y(x)y'(x)\right|_{a}^{b}\right\}+\left\{\int _{a}^{b}\left[p(x)y'(x)^{2}+q(x)y(x)^{2}\right]\,dx\right\}}{\int _{a}^{b}{w(x)y(x)^{2}}\,dx}}.\end{aligned}}} | https://en.wikipedia.org/wiki/Rayleigh_quotient |
Rayleigh scattering ( / ˈ r eɪ l i / RAY -lee ) is the scattering or deflection of light , or other electromagnetic radiation , by particles with a size much smaller than the wavelength of the radiation. For light frequencies well below the resonance frequency of the scattering medium (normal dispersion regime), the amount of scattering is inversely proportional to the fourth power of the wavelength (e.g., a blue color is scattered much more than a red color as light propagates through air). The phenomenon is named after the 19th-century British physicist Lord Rayleigh (John William Strutt). [ 1 ]
Rayleigh scattering results from the electric polarizability of the particles. The oscillating electric field of a light wave acts on the charges within a particle, causing them to move at the same frequency. The particle, therefore, becomes a small radiating dipole whose radiation we see as scattered light. The particles may be individual atoms or molecules; it can occur when light travels through transparent solids and liquids, but is most prominently seen in gases .
Rayleigh scattering of sunlight in Earth's atmosphere causes diffuse sky radiation , which is the reason for the blue color of the daytime and twilight sky , as well as the yellowish to reddish hue of the low Sun . Sunlight is also subject to Raman scattering , which changes the rotational state of the molecules and gives rise to polarization effects. [ 2 ]
Scattering by particles with a size comparable to, or larger than, the wavelength of the light is typically treated by the Mie theory , the discrete dipole approximation and other computational techniques. Rayleigh scattering applies to particles that are small with respect to wavelengths of light, and that are optically "soft" (i.e., with a refractive index close to 1). Anomalous diffraction theory applies to optically soft but larger particles.
In 1869, while attempting to determine whether any contaminants remained in the purified air he used for infrared experiments, John Tyndall discovered that bright light scattering off nanoscopic particulates was faintly blue-tinted. [ 3 ] He conjectured that a similar scattering of sunlight gave the sky its blue hue , but he could not explain the preference for blue light, nor could atmospheric dust explain the intensity of the sky's color.
In 1871, Lord Rayleigh published two papers on the color and polarization of skylight to quantify Tyndall's effect in water droplets in terms of the tiny particulates' volumes and refractive indices . [ 4 ] [ 5 ] [ 6 ] In 1881, with the benefit of James Clerk Maxwell 's 1865 proof of the electromagnetic nature of light , he showed that his equations followed from electromagnetism . [ 7 ] In 1899, he showed that they applied to individual molecules, with terms containing particulate volumes and refractive indices replaced with terms for molecular polarizability . [ 8 ]
The size of a scattering particle is often parameterized by the ratio
x = 2 π r λ {\displaystyle x={\frac {2\pi r}{\lambda }}}
where r is the particle's radius, λ is the wavelength of the light and x is a dimensionless parameter that characterizes the particle's interaction with the incident radiation such that: Objects with x ≫ 1 act as geometric shapes, scattering light according to their projected area. At the intermediate x ≃ 1 of Mie scattering , interference effects develop through phase variations over the object's surface. Rayleigh scattering applies to the case when the scattering particle is very small (x ≪ 1, with a particle size < 1/10 of wavelength [ 9 ] ) and the whole surface re-radiates with the same phase. Because the particles are randomly positioned, the scattered light arrives at a particular point with a random collection of phases; it is incoherent and the resulting intensity is just the sum of the squares of the amplitudes from each particle and therefore proportional to the inverse fourth power of the wavelength and the sixth power of its size. [ 10 ] [ 11 ] The wavelength dependence is characteristic of dipole scattering [ 10 ] and the volume dependence will apply to any scattering mechanism. In detail, the intensity of light scattered by any one of the small spheres of radius r and refractive index n from a beam of unpolarized light of wavelength λ and intensity I 0 is given by [ 12 ] I s = I 0 1 + cos 2 θ 2 R 2 ( 2 π λ ) 4 ( n 2 − 1 n 2 + 2 ) 2 r 6 {\displaystyle I_{s}=I_{0}{\frac {1+\cos ^{2}\theta }{2R^{2}}}\left({\frac {2\pi }{\lambda }}\right)^{4}\left({\frac {n^{2}-1}{n^{2}+2}}\right)^{2}r^{6}} where R is the observer's distance to the particle and θ is the scattering angle. Averaging this over all angles gives the Rayleigh scattering cross-section of the particles in air: [ 13 ] σ s = 8 π 3 ( 2 π λ ) 4 ( n 2 − 1 n 2 + 2 ) 2 r 6 . {\displaystyle \sigma _{\text{s}}={\frac {8\pi }{3}}\left({\frac {2\pi }{\lambda }}\right)^{4}\left({\frac {n^{2}-1}{n^{2}+2}}\right)^{2}r^{6}.} Here n is the refractive index of the spheres that approximate the molecules of the gas; the index of the gas surrounding the spheres is neglected, an approximation that introduces an error of less than 0.05%. [ 14 ]
The fraction of light scattered by scattering particles over the unit travel length (e.g., meter) is the number of particles per unit volume N times the cross-section. For example, air has a refractive index of 1.0002793 at atmospheric pressure, where there are about 2 × 10 25 molecules per cubic meter, and therefore the major constituent of the atmosphere, nitrogen, has a Rayleigh cross section of 5.1 × 10 −31 m 2 at a wavelength of 532 nm (green light). [ 14 ] This means that about a fraction 10 −5 of the light will be scattered for every meter of travel.
The strong wavelength dependence of the scattering (~ λ −4 ) means that shorter (blue) wavelengths are scattered more strongly than longer (red) wavelengths.
The expression above can also be written in terms of individual molecules by expressing the dependence on refractive index in terms of the molecular polarizability α , proportional to the dipole moment induced by the electric field of the light. In this case, the Rayleigh scattering intensity for a single particle is given in CGS-units by [ 15 ] I s = I 0 8 π 4 α 2 λ 4 R 2 ( 1 + cos 2 θ ) {\displaystyle I_{s}=I_{0}{\frac {8\pi ^{4}\alpha ^{2}}{\lambda ^{4}R^{2}}}(1+\cos ^{2}\theta )} and in SI-units by I s = I 0 π 2 α 2 ε 0 2 λ 4 R 2 1 + cos 2 ( θ ) 2 . {\displaystyle I_{s}=I_{0}{\frac {\pi ^{2}\alpha ^{2}}{{\varepsilon _{0}}^{2}\lambda ^{4}R^{2}}}{\frac {1+\cos ^{2}(\theta )}{2}}.}
When the dielectric constant ϵ {\displaystyle \epsilon } of a certain region of volume V {\displaystyle V} is different from the average dielectric constant of the medium ϵ ¯ {\displaystyle {\bar {\epsilon }}} , then any incident light will be scattered according to the following equation [ 16 ]
I = I 0 π 2 V 2 σ ϵ 2 2 λ 4 R 2 ( 1 + cos 2 θ ) {\displaystyle I=I_{0}{\frac {\pi ^{2}V^{2}\sigma _{\epsilon }^{2}}{2\lambda ^{4}R^{2}}}{\left(1+\cos ^{2}\theta \right)}} where σ ϵ 2 {\displaystyle \sigma _{\epsilon }^{2}} represents the variance of the fluctuation in the dielectric constant ϵ {\displaystyle \epsilon } .
The blue color of the sky is a consequence of three factors: [ 17 ]
The strong wavelength dependence of the Rayleigh scattering (~ λ −4 ) means that shorter ( blue ) wavelengths are scattered more strongly than longer ( red ) wavelengths. This results in the indirect blue and violet light coming from all regions of the sky. The human eye responds to this wavelength combination as if it were a combination of blue and white light. [ 17 ]
Some of the scattering can also be from sulfate particles. For years after large Plinian eruptions , the blue cast of the sky is notably brightened by the persistent sulfate load of the stratospheric gases. Some works of the artist J. M. W. Turner may owe their vivid red colours to the eruption of Mount Tambora in his lifetime. [ 18 ]
In locations with little light pollution , the moonlit night sky is also blue, because moonlight is reflected sunlight, with a slightly lower color temperature due to the brownish color of the Moon. The moonlit sky is not perceived as blue, however, because at low light levels human vision comes mainly from rod cells that do not produce any color perception ( Purkinje effect ). [ 19 ]
Rayleigh scattering is also an important mechanism of wave scattering in amorphous solids such as glass, and is responsible for acoustic wave damping and phonon damping in glasses and granular matter at low or not too high temperatures. [ 20 ] This is because in glasses at higher temperatures the Rayleigh-type scattering regime is obscured by the anharmonic damping (typically with a ~ λ −2 dependence on wavelength), which becomes increasingly more important as the temperature rises.
Rayleigh scattering is an important component of the scattering of optical signals in optical fibers . Silica fibers are glasses, disordered materials with microscopic variations of density and refractive index. These give rise to energy losses due to the scattered light, with the following coefficient: [ 21 ] α scat = 8 π 3 3 λ 4 n 8 p 2 k T f β {\displaystyle \alpha _{\text{scat}}={\frac {8\pi ^{3}}{3\lambda ^{4}}}n^{8}p^{2}kT_{\text{f}}\beta }
where n is the refraction index, p is the photoelastic coefficient of the glass, k is the Boltzmann constant , and β is the isothermal compressibility. T f is a fictive temperature , representing the temperature at which the density fluctuations are "frozen" in the material.
Rayleigh-type λ −4 scattering can also be exhibited by porous materials. An example is the strong optical scattering by nanoporous materials. [ 23 ] The strong contrast in refractive index between pores and solid parts of sintered alumina results in very strong scattering, with light completely changing direction each five micrometers on average. The λ −4 -type scattering is caused by the nanoporous structure (a narrow pore size distribution around ~70 nm) obtained by sintering monodispersive alumina powder. | https://en.wikipedia.org/wiki/Rayleigh_scattering |
In mathematics, the Rayleigh theorem for eigenvalues pertains to the behavior of the solutions of an eigenvalue equation as the number of basis functions employed in its resolution increases. Rayleigh, Lord Rayleigh , and 3rd Baron Rayleigh are the titles of John William Strutt , after the death of his father, the 2nd Baron Rayleigh. Lord Rayleigh made contributions not just to both theoretical and experimental physics, but also to applied mathematics. The Rayleigh theorem for eigenvalues, as discussed below, enables the energy minimization that is required in many self-consistent calculations of electronic and related properties of materials, from atoms , molecules , and nanostructures to semiconductors , insulators , and metals . Except for metals, most of these other materials have an energy or a band gap , i.e., the difference between the lowest, unoccupied energy and the highest, occupied energy. For crystals , the energy spectrum is in bands and there is a band gap, if any, as opposed to energy gap . Given the diverse contributions of Lord Rayleigh, his name is associated with other theorems, including Parseval's theorem . For this reason, keeping the full name of "Rayleigh Theorem for Eigenvalues" avoids confusions.
The theorem, as indicated above, applies to the resolution of equations called eigenvalue equations. i.e., the ones of the form HѰ = λѰ , where H is an operator, Ѱ is a function and λ is number called the eigenvalue . To solve problems of this type, we expand the unknown function Ѱ in terms of known functions. The number of these known functions is the size of the basis set. The expansion coefficients are also numbers. The number of known functions included in the expansion, the same as that of coefficients, is the dimension of the Hamiltonian matrix that will be generated. The statement of the theorem follows. [ 1 ] [ 2 ]
Let an eigenvalue equation be solved by linearly expanding the unknown function in terms of N known functions. Let the resulting eigenvalues be ordered from the smallest (lowest), λ 1 , to the largest (highest), λ N . Let the same eigenvalue equation be solved using a basis set of dimension N + 1 that comprises the previous N functions plus an additional one. Let the resulting eigenvalues be ordered from the smallest, λ ′ 1 , to the largest, λ ′ N +1 . Then, the Rayleigh theorem for eigenvalues states that λ ′ i ≤ λ i for i = 1 to N .
A subtle point about the above statement is that the smaller of the two sets of functions must be a subset of the larger one. The above inequality does not hold otherwise.
In quantum mechanics , [ 3 ] where the operator H is the Hamiltonian , the lowest eigenvalues are occupied (by electrons) up to the applicable number of electrons; the remaining eigenvalues, not occupied by electrons, are empty energy levels. The energy content of the Hamiltonian is the sum of the occupied eigenvalues. The Rayleigh theorem for eigenvalues is extensively utilized in calculations of electronic and related properties of materials. The electronic energies of materials are obtained through calculations said to be self-consistent , as explained below.
In density functional theory (DFT) calculations of electronic energies of materials, the eigenvalue equation, HѰ = λѰ , has a companion equation that gives the electronic charge density of the material in terms of the wave functions of the occupied energies. To be reliable, these calculations have to be self-consistent , as explained below.
The process of obtaining the electronic energies of material begins with the selection of an initial set of known functions (and related coefficients) in terms of which one expands the unknown function Ѱ . Using the known functions for the occupied states, one constructs an initial charge density for the material. For density functional theory calculations, once the charge density is known, the potential, the Hamiltonian, and the eigenvalue equation are generated. Solving this equation leads to eigenvalues (occupied or unoccupied) and their corresponding wave functions (in terms of the known functions and new coefficients of expansion). Using only the new wave functions of the occupied energies, one repeats the cycle of constructing the charge density and of generating the potential and the Hamiltonian. Then, using all the new wave functions (for occupied and empty states), one regenerates the eigenvalue equation and solves it. Each one of these cycles is called an iteration. The calculations are complete when the difference between the potentials generated in Iteration n + 1 and the one immediately preceding it (i.e., n ) is 10 −5 or less. The iterations are then said to have converged and the outcomes of the last iteration are the self-consistent results that are reliable.
The characteristics and number [ 1 ] [ 2 ] of the known functions utilized in the expansion of Ѱ naturally have a bearing on the quality of the final, self-consistent results. The selection of atomic orbitals that include exponential or Gaussian functions, in additional to polynomial and angular features that apply, practically ensures the high quality of self-consistent results, except for the effects of the size [ 1 ] [ 2 ] and of attendant characteristics (features) of the basis set. These characteristics include the polynomial and angular functions that are inherent to the description of s, p, d, and f states for an atom. While the s functions [ 4 ] are spherically symmetric, the others are not; they are often called polarization orbitals or functions.
The conundrum is the following. Density functional theory is for the description of the ground state of materials, i.e., the state of lowest energy. The second theorem [ 5 ] [ 6 ] of DFT states that the energy functional for the Hamiltonian [i.e., the energy content of the Hamiltonian] reaches its minimum value (i.e., the ground state) if the charge density employed in the calculation is that of the ground state. We described above the selection of an initial basis set in order to perform self-consistent calculations. A priori, there is no known mechanism for selecting a single basis set so that, after self consistency, the charge density it generates is that of the ground state. Self consistency with a given basis set leads to the reliable energy content of the Hamiltonian for that basis set. As per the Rayleigh theorem for eigenvalues, upon augmenting that initial basis set, the ensuing self consistent calculations lead to an energy content of the Hamiltonian that is lower than or equal to that obtained with the initial basis set. We recall that the reliable, self-consistent energy content of the Hamiltonian obtained with a basis set, after self consistency, is relative to that basis set. A larger basis set that contains the first one generally leads self consistent eigenvalues that are lower than or equal to their corresponding values from the previous calculation. One may paraphrase the issue as follows. Several basis sets of different sizes, upon the attainment of self-consistency, lead to stationary (converged) solutions. There exists an infinite number of such stationary solutions. The conundrum stems from the fact that, a priori , one has no means to determine the basis set, if any, after self consistency, leads to the ground state charge density of the material, and, according to the second DFT theorem, to the ground state energy of the material under study.
Let us first recall that a self-consistent density functional theory calculation, with a single basis set, produces a stationary solution which cannot be claimed to be that of the ground state. To find the DFT ground state of a material, one has to vary [ 5 ] [ 6 ] the basis set (in size and attendant features) in order to minimize the energy content of the Hamiltonian, while keeping the number of particles constant. Hohenberg and Kohn , [ 5 ] specifically stated that the energy content of the Hamiltonian "has a minimum at the 'correct' ground state Ψ, relative to arbitrary variations of Ψ ′ in which the total number of particles is kept constant." Hence, the trial basis set is to be varied in order to minimize the energy. The Rayleigh theorem for eigenvalues shows how to perform such a minimization with successive augmentation of the basis set. The first trial basis set has to be a small one that accounts for all the electrons in the system. After performing a self consistent calculation (following many iterations) with this initial basis set, one augments it with one atomic orbital . Depending on the s , p , d , or f character of this orbital, the size of the new basis set (and the dimension of the Hamiltonian matrix) will be larger than that of the initial one by 2, 6, 10, or 14, respectively, taking the spin into account. Given that the initial, trial basis set was deliberately selected to be small, the resulting self consistent results cannot be assumed to describe the ground state of the material. Upon performing self-consistent calculations with the augmented basis set, one compares the occupied energies from Calculations I and II, after setting the Fermi level to zero. Invariably, [ 7 ] [ 8 ] the occupied energies from Calculation II are lower than or equal to their corresponding values from Calculation I. Naturally, one cannot affirm that the results from Calculation II describe the ground state of the material, given the absence of any proof that the occupied energies cannot be lowered further. Hence, one continues the process of augmenting the basis set with one orbital and of performing the next self-consistent calculation. The process is complete when three consecutive calculations yield the same occupied energies. One can affirm that the occupied energies from these three calculations represent the ground state of the material. Indeed, while two consecutive calculations can produce the same occupied energies, these energies may be for a local minimum energy content of the Hamiltonian as opposed to the absolute minimum. To have three consecutive calculations produce the same occupied energies is the robust criterion [ 9 ] [ 10 ] for the attainment of the ground state of a material (i.e., the state where the occupied energies have their absolute minimal values). This paragraph described how successive augmentation of the basis set solves one aspect of the conundrum, i.e., a generalized minimization of the energy content of the Hamiltonian to reach the ground state of the system under study.
Even though the paragraph above shows how the Rayleigh theorem enables the generalized minimization of the energy content of the Hamiltonian, to reach the ground state, we are still left with the fact that three different calculations produced this ground state. Let the respective numbers of these calculations be N, (N+1), and (N+2). While the occupied energies from these calculations are the same (i.e., the ground state), the unoccupied energies are not identical. Indeed, the general trend is that the unoccupied energies from the calculations [ 1 ] [ 2 ] are in the reverse order of the sizes of the basis sets for these calculations. In other words, for a given unoccupied eigenvalue (say the lowest one of the unoccupied energies), the result from Calculation (N+2) is smaller than or equal to that from Calculation (N+1). The latter, in turn, is smaller than or equal to the result from Calculation N. In the case of semiconductors , the lowest-laying unoccupied energies from the three calculations are generally the same, up to 6 to 10 eV or above, depending on the material, if the sizes of the basis sets of the three calculations are not vastly different. Still, for higher, unoccupied energies, the Rayleigh theorem for eigenvalues applies. This paragraph poses the question as to which one of the three, consecutive, self-consistent calculations leading to the ground state energy provides the true DFT description of the material – given the differences between some of their unoccupied energies. There are two distinct ways of determining the calculation providing the DFT description of the material.
The value of the above determination of the physically meaningful calculation is that it avoids the consideration of basis sets that are larger than that of Calculation N and are heretofore over-complete for the description of the ground state of the material. In the current literature, the only calculations that have reproduced [ 8 ] [ 9 ] [ 10 ] or predicted [ 11 ] [ 12 ] [ 13 ] the correct, electronic properties of semiconductors have been the ones that (1) searched for and reached the true ground state of materials and (2) avoided the utilization of over complete basis sets as described above. These accurate DFT calculations did not invoke the self-interaction correction (SIC) [ 14 ] or the derivative discontinuity [ 15 ] [ 16 ] [ 17 ] employed extensively in the literature to explain the woeful underestimation of the band gaps of semiconductors [ 16 ] and insulators . [ 16 ] [ 17 ] In light of the content of the two bullets above, an alternative, plausible explanation of the energy and band gap underestimation in the literature is the use of over-complete basis sets that lead to an unphysical lowering of some unoccupied energies, including some of the lowest-laying ones. [ 8 ] | https://en.wikipedia.org/wiki/Rayleigh_theorem_for_eigenvalues |
The Rayleigh–Kuo criterion (sometimes called the Kuo criterion ) is a stability condition for a fluid . This criterion determines whether or not a barotropic instability can occur, leading to the presence of vortices (like eddies and storms ). The Kuo criterion states that for barotropic instability to occur, the gradient of the absolute vorticity must change its sign at some point within the boundaries of the current. [ 1 ] [ 2 ] Note that this criterion is a necessary condition, so if it does not hold it is not possible for a barotropic instability to form. But it is not a sufficient condition, meaning that if the criterion is met, this does not automatically mean that the fluid is unstable. If the criterion is not met, it is certain that the flow is stable. [ 3 ]
This criterion was formulated by Hsiao-Lan Kuo and is based on Rayleigh's equation named after the Lord Rayleigh who first introduced this equation in fluid dynamics .
Vortices like eddies are created by instabilities in a flow. When there are instabilities within the mean flow, energy can be transferred from the mean flow to the small perturbations which can then grow. In a barotropic fluid the density is a function of only the pressure and not the temperature (in contrast to a baroclinic fluid, where the density is a function of both the pressure and temperature [ 3 ] ). This means that surfaces of constant density ( isopycnals ) are also surfaces of constant pressure ( isobars ). [ 4 ] Barotropic instability can form in different ways. Two examples are; when there is an interaction between the fluid flow and the bathymetry or topography of the domain; when there are frontal instabilities (may also lead to baroclinic instabilities). These instabilities are not dependent on the density and might even occur when the density of the fluid is constant. Instead, most of the instabilities are caused by a shear on the flow as can be seen in Figure 1. This shear in the velocity field induces a vertical and horizontal vorticity within the flow. As a result, there is upwelling on the right of the flow and downwelling on the left. This situation might lead to a barotropic unstable flow. The eddies that form alternatingly on both sides of the flow are part of this instability.
Another way to achieve this instability is to displace the Rossby waves in the horizontal direction (see Figure 2). This leads to a transfer of kinetic energy (not potential energy ) from the mean flow towards the small perturbations (the eddies). [ 5 ] The Rayleigh–Kuo criterion states that the gradient of the absolute vorticity should change sign within the domain. In the example of the shear induced eddies on the right, this means that the second derivative of the flow in the cross-flow direction, should be zero somewhere. This happens in the centre of the eddies, where the acceleration of the flow perpendicular to the flow changes direction.
The presence of these instabilities in a rotating fluid have been observed in laboratory experiments. The settings of the experiment were based on the conditions in the Gulf Stream and showed that within the ocean currents such as the Gulf Stream, it is possible for barotropic instabilities to occur. [ 6 ] But barotropic instabilities were also observed in other Western Boundary Currents (WBC). In the Agulhas current , the barotropic instability leads to ring shedding . The Agulhas current retroflects (turns back) near the coast of South Africa. At this same location, some anti-cyclonic rings of warm water escape from the mean current and travel along the coast of Africa. The formation of these rings is a manifestation of a barotropic instability. [ 7 ]
The derivation of the Rayleigh–Kuo criterion was first written down by Hsiao-Lan Kuo in his paper called ' dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere' from 1949 . [ 1 ] This derivation is repeated and simplified below. [ 2 ]
First, the assumptions made by Hsiao-Lan Kuo are discussed. Second, the Rayleigh equation is derived in order continue to derive the Rayleigh–Kuo criterion. By integrating this equation and filling in the boundary conditions, the Kuo criterion can be obtained.
In order to derive the Rayleigh–Kuo criterion, some assumptions are made on the fluids properties. We consider a nondivergent, two-dimensional barotropic fluid . The fluid has a mean zonal flow direction which can vary in the meridional direction. On this mean flow, some small perturbations are imposed in both the zonal and meridional direction: u ( y , t ) = U ( y ) + u ∗ ( y , t ) {\displaystyle u(y,t)=U(y)+u^{*}(y,t)} and v = v ∗ {\displaystyle v=v^{*}} . The perturbations need to be small in order to linearize the vorticity equation. Vertical motion and divergence and convergence of the fluid are neglected. When taking into account these factors, a similar result would have been obtained with only a small shift in the position of the criterion within the velocity profile. [ 1 ]
The derivation of the Kuo criterion will be done within the domain L = [ 0 , y ] {\displaystyle L=[0,y]} . On the northern and southern boundary of this domain, the meridional fluid is zero.
To derive the Rayleigh equation for a barotropic fluid, the barotropic vorticity equation is used. This equation assumes that the absolute vorticity is conserved: d ζ a d t = 0 {\displaystyle {\frac {d\zeta _{a}}{dt}}=0} here, d d t {\displaystyle {\frac {d}{dt}}} is the material derivative . The absolute vorticity is the relative vorticity plus the planetary vorticity: ζ a = ζ + f {\displaystyle \zeta _{a}=\zeta +f} . The relative vorticity, ζ {\displaystyle \zeta } , is the rotation of the fluid with respect to the Earth. The planetary vorticity (also called Coriolis frequency ), f {\displaystyle f} , is the vorticity of a parcel induced by the rotation of the Earth. When applying the beta-plane approximation for the planetary vorticity, the conservation of absolute vorticity looks like:
d ζ a d t = d d t ( ζ + β y ) = 0 {\displaystyle {\frac {d\zeta _{a}}{dt}}={\frac {d}{dt}}\left(\zeta +\beta y\right)=0} The relative vorticity is defined as ζ = ∂ v ∂ x − ∂ u ∂ y . {\displaystyle \zeta ={\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}.} Since the flow field consist of a mean flow with small perturbations, it can be written as ζ = ζ ¯ + ζ ∗ {\displaystyle \zeta ={\overline {\zeta }}+\zeta ^{*}} with ζ ¯ = − ∂ U ∂ y {\displaystyle {\overline {\zeta }}=-{\frac {\partial U}{\partial y}}} and ζ ∗ = ∂ v ∗ ∂ x − ∂ u ∗ ∂ y . {\displaystyle \zeta ^{*}={\frac {\partial v^{*}}{\partial x}}-{\frac {\partial u^{*}}{\partial y}}.} This formulation is used in the vorticity equation:
0 = d d t ( ζ + β y ) 0 = d d t ( ζ ′ + ζ ¯ + β y ) 0 = ( ∂ ∂ t + u ∂ ∂ x + v ∂ ∂ y ) ( ζ ∗ − ∂ U ∂ y + β y ) {\displaystyle {\begin{aligned}0&={\frac {d}{dt}}\left(\zeta +\beta y\right)\\0&={\frac {d}{dt}}\left(\zeta '+{\overline {\zeta }}+\beta y\right)\\0&=\left({\frac {\partial }{\partial t}}+u{\frac {\partial }{\partial x}}+v{\frac {\partial }{\partial y}}\right)\left(\zeta ^{*}-{\frac {\partial U}{\partial y}}+\beta y\right)\end{aligned}}} Here, u {\displaystyle u} and v {\displaystyle v} are the zonal and meridional components of the flow and ζ ′ {\displaystyle \zeta '} is the relative vorticity induced by the perturbations on the flow ( u ′ {\displaystyle u'} and v ′ {\displaystyle v'} ). U {\displaystyle U} is the mean zonal flow and β {\displaystyle \beta } is derivative of the planetary vorticity f {\displaystyle f} with respect to y {\displaystyle y} .
A zonal mean flow with small perturbations was assumed, u = U + u ∗ {\displaystyle u=U+u^{*}} , and a meridional flow with a zero mean, v = v ∗ {\displaystyle v=v^{*}} . Since it was assumed that the perturbations are small, a linearization can be performed on the barotropic vorticity equation above, ignoring all the non-linear terms (terms where two or more small variables, i.e. u ∗ , v ∗ , ζ ∗ {\displaystyle u^{*},v^{*},\zeta ^{*}} , are multiplied with one another). Also the derivative of u {\displaystyle u} in the zonal direction, the time derivative of the mean flow U {\displaystyle U} and the time derivative of β y {\displaystyle \beta y} are zero. This results in a simplified equation:
0 = ( ∂ ∂ t + U ∂ ∂ x ) ζ ′ − v ′ ∂ ∂ y ∂ U ∂ y + v ′ ∂ ∂ y ( β y ) 0 = ( ∂ ∂ t + U ∂ ∂ x ) ζ ′ + v ′ ( β − ∂ 2 U ∂ y 2 ) . {\displaystyle {\begin{aligned}0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\zeta '-v'{\frac {\partial }{\partial y}}{\frac {\partial U}{\partial y}}+v'{\frac {\partial }{\partial y}}\left(\beta y\right)\\0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\zeta '+v'\left(\beta -{\frac {\partial ^{2}U}{\partial y^{2}}}\right).\end{aligned}}}
With ζ ′ {\displaystyle \zeta '} as defined above ( ζ ∗ = ∂ v ∗ ∂ x − ∂ u ∗ ∂ y {\displaystyle \zeta ^{*}={\frac {\partial v^{*}}{\partial x}}-{\frac {\partial u^{*}}{\partial y}}} ) and u ∗ {\displaystyle u^{*}} and v ∗ {\displaystyle v^{*}} the small perturbations in the zonal and meridional components of the flow.
To find the solution to the linearized equation, a stream function was introduced by Lord Rayleigh for the perturbations of the flow velocity:
u ∗ = ∂ ψ ∂ y , v ∗ = − ∂ ψ ∂ x . {\displaystyle u^{*}={\frac {\partial \psi }{\partial y}},\;\;\;\;v^{*}={\frac {-\partial \psi }{\partial x}}.} These new definitions of the stream function are used to rewrite the linearized barotropic vorticity equation. 0 = ( ∂ ∂ t + U ∂ ∂ x ) ( ∂ v ∗ ∂ x − ∂ u ∗ ∂ y ) + v ∗ ( β − ∂ 2 U ∂ y 2 ) 0 = ( ∂ ∂ t + U ∂ ∂ x ) ( − ∂ 2 ψ ∂ x 2 − ∂ 2 ψ ∂ y 2 ) − ∂ ψ ∂ x ( β − U ″ ) 0 = ( ∂ ∂ t + U ∂ ∂ x ) ∇ 2 ψ + ∂ ψ ∂ x ( β − U ″ ) {\displaystyle {\begin{aligned}0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\left({\frac {\partial v^{*}}{\partial x}}-{\frac {\partial u^{*}}{\partial y}}\right)+v^{*}\left(\beta -{\frac {\partial ^{2}U}{\partial y^{2}}}\right)\\0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\left(-{\frac {\partial ^{2}\psi }{\partial x^{2}}}-{\frac {\partial ^{2}\psi }{\partial y^{2}}}\right)-{\frac {\partial \psi }{\partial x}}\left(\beta -U''\right)\\0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\nabla ^{2}\psi +{\frac {\partial \psi }{\partial x}}\left(\beta -U''\right)\end{aligned}}} Here, U ″ {\displaystyle U''} is the second derivative of U {\displaystyle U} with respect to y {\displaystyle y} ( U ″ = ∂ 2 U ∂ y 2 ) {\displaystyle (U''={\frac {\partial ^{2}U}{\partial y^{2}}})} . To solve this equation for the stream function, a wave-like solution was proposed by Rayleigh which reads ψ ( x , y , t ) = Ψ ( y ) e i α ( x − c t ) {\displaystyle \psi (x,y,t)=\Psi (y)e^{i\alpha \left(x-ct\right)}} . The amplitude Ψ ( y ) {\displaystyle \Psi (y)} may be complex number , α {\displaystyle \alpha } is the wave number which is a real number and c {\displaystyle c} is the phase velocity which may be complex as well. Inserting this proposed solution leads us to the equation which is known as Rayleigh's equation .
0 = ( ∂ ∂ t + U ∂ ∂ x ) ∇ 2 ψ + ∂ ψ ∂ x ( β − U ″ ) 0 = ( ∂ ∂ t + U ∂ ∂ x ) ( ( i α ) 2 Ψ e i α ( x − c t ) + Ψ ″ e i α ( x − c t ) ) + i α Ψ e i α ( x − c t ) ( β − U ″ ) 0 = ( i α ) 2 Ψ e i α ( x − c t ) ( − i α c + i α U ) + Ψ ″ e i α ( x − c t ) ( − i α c + i α U ) + i α Ψ e i α ( x − c t ) ( β − U ″ ) 0 = i α e i α ( x − c t ) [ ( α 2 c − α 2 U ) Ψ + Ψ ″ ( − c + U ) + Ψ ( β − U ″ ) ] 0 = ( U − c ) ( Ψ ″ − α 2 Ψ ) + ( β − U ″ ) Ψ ) {\displaystyle {\begin{aligned}0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\nabla ^{2}\psi +{\frac {\partial \psi }{\partial x}}\left(\beta -U''\right)\\[14pt]0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\left((i\alpha )^{2}\Psi e^{i\alpha \left(x-ct\right)}+\Psi ''e^{i\alpha \left(x-ct\right)}\right)+i\alpha \Psi e^{i\alpha (x-ct)}(\beta -U'')\\[14pt]0&=(i\alpha )^{2}\Psi e^{i\alpha (x-ct)}(-i\alpha c+i\alpha U)+\Psi ''e^{i\alpha (x-ct)}(-i\alpha c+i\alpha U)+i\alpha \Psi e^{i\alpha (x-ct)}(\beta -U'')\\[14pt]0&=i\alpha e^{i\alpha (x-ct)}[(\alpha ^{2}c-\alpha ^{2}U)\Psi +\Psi ''(-c+U)+\Psi (\beta -U'')]\\[14pt]0&=(U-c)(\Psi ''-\alpha ^{2}\Psi )+(\beta -U'')\Psi )\\[14pt]\end{aligned}}} To get to this equation, in the last step it was used that α {\displaystyle \alpha } can't be zero and neither can the exponential. This means that the terms in the square brackets needs to be zero. The symbol Ψ ″ {\displaystyle \Psi ''} denotes the second derivative of the amplitude of the stream function, Ψ {\displaystyle \Psi } with respect to y {\displaystyle y} ( Ψ ″ = ∂ 2 Ψ ∂ y 2 ) {\displaystyle (\Psi ''={\frac {\partial ^{2}\Psi }{\partial y^{2}}})} . This last equation that was derived, is known as Rayleigh's equation which is a linear ordinary differential equation . It is very difficult to explicitly solve this equation. It is therefore that Hsiao-Lan Kuo came up with a stability criterion for this problem without actually solving it.
Instead of solving Rayleigh's equation, Hsiao-Lan Kuo came up with a necessary stability condition which had to be met in order for the fluid to be able to get unstable. To get to this criterion, Rayleigh's equation was rewritten and the boundary conditions of the flow field are used.
The first step is to divide Rayleigh's equation by ( U − c ) {\displaystyle (U-c)} and multiplying the equation by the complex conjugate of Ψ ( Ψ ∗ = Ψ r − i Ψ i ) {\displaystyle \Psi \;\;(\Psi ^{*}=\Psi _{r}-i\Psi _{i})} .
0 = ( Ψ ″ − α 2 Ψ ) + ( β − U ″ U − c ) Ψ 0 = Ψ ∗ ( Ψ r ″ + i Ψ i ″ − α 2 Ψ r − α 2 i Ψ i ) + Ψ ∗ ( β − U ″ U − c ) ( Ψ r − i Ψ i ) 0 = Ψ r Ψ r ″ + Ψ i Ψ i ″ − α 2 ( Ψ r 2 − Ψ i 2 ) + ( β − U ″ U − c ) ( Ψ r 2 − Ψ i 2 ) + i ( − Ψ i Ψ r ″ + Ψ r Ψ i ″ ) 0 = Ψ r ″ Ψ r + Ψ i ″ Ψ i + ( − α 2 + U − c r | U − c | 2 ( β − U ″ ) ) | Ψ | 2 + i ( c i | U − c | 2 ( β − U ″ ) | Ψ | 2 − Ψ r ″ Ψ i + Ψ i ″ Ψ r ) {\displaystyle {\begin{aligned}0&=(\Psi ''-\alpha ^{2}\Psi )+\left({\frac {\beta -U''}{U-c}}\right)\Psi \\0&=\Psi ^{*}(\Psi _{r}''+i\Psi _{i}''-\alpha ^{2}\Psi _{r}-\alpha ^{2}i\Psi _{i})+\Psi ^{*}\left({\frac {\beta -U''}{U-c}}\right)(\Psi _{r}-i\Psi _{i})\\0&=\Psi _{r}\Psi _{r}''+\Psi _{i}\Psi _{i}''-\alpha ^{2}(\Psi _{r}^{2}-\Psi _{i}^{2})+\left({\frac {\beta -U''}{U-c}}\right)(\Psi _{r}^{2}-\Psi _{i}^{2})+i(-\Psi _{i}\Psi _{r}''+\Psi _{r}\Psi _{i}'')\\0&=\Psi _{r}''\Psi _{r}+\Psi _{i}''\Psi _{i}+\left(-\alpha ^{2}+{\frac {U-c_{r}}{|U-c|^{2}}}(\beta -U'')\right)|\Psi |^{2}+i\left({\frac {c_{i}}{|U-c|^{2}}}(\beta -U'')|\Psi |^{2}-\Psi _{r}''\Psi _{i}+\Psi _{i}''\Psi _{r}\right)\end{aligned}}} In the last step, ( U − c ) {\displaystyle (U-c)} is multiplied with its complex conjugate leading to the following equality is used: 1 U − c = 1 U − c r − i c i = U − c r + i c i | U − c | 2 {\displaystyle {\frac {1}{U-c}}={\frac {1}{U-c_{r}-ic_{i}}}={\frac {U-c_{r}+ic_{i}}{|U-c|^{2}}}} . For the solution of Rayleigh's equation to exist, both the real and imaginary part of the equation above need to be equal to zero.
To get to the Kuo criterion, the imaginary part is integrated over the domain ( y = [ 0 , L ] {\displaystyle y=[0,L]} ) . The stream function at the boundaries of the domain is zero, Ψ ( 0 ) = Ψ ( L ) = 0 {\displaystyle \Psi (0)=\Psi (L)=0} , as already stated in the assumptions. The zonal flow must vanish at the boundaries of the domain. This leads to a constant stream function which is set to zero for convenience.
∫ 0 L ( Ψ r Ψ i ″ − Ψ i Ψ r ″ ) d y + ∫ 0 L ( c i | Ψ | 2 | U − c | ( β − U ″ ) ) = 0 {\displaystyle \int _{0}^{L}(\Psi _{r}\Psi _{i}''-\Psi _{i}\Psi _{r}'')dy+\int _{0}^{L}\left(c_{i}{\frac {|\Psi |^{2}}{|U-c|}}(\beta -U'')\right)=0}
The first integral can be solved:
∫ 0 L ( Ψ r Ψ i ″ − Ψ i Ψ r ″ ) d y = ∫ 0 L ∂ ∂ y ( Ψ r Ψ i ′ − Ψ i Ψ r ′ ) d y = ( Ψ r Ψ i ′ − Ψ i Ψ r ′ ) | 0 L = 0 {\displaystyle {\begin{aligned}\int _{0}^{L}(\Psi _{r}\Psi _{i}''-\Psi _{i}\Psi _{r}'')dy&=\int _{0}^{L}{\frac {\partial }{\partial y}}(\Psi _{r}\Psi _{i}'-\Psi _{i}\Psi _{r}')dy\\[8pt]&=(\Psi _{r}\Psi _{i}'-\Psi _{i}\Psi _{r}')|_{0}^{L}\\[8pt]&=0\\[12pt]\end{aligned}}}
So the first integral is equal to zero. This means that the second integral should also be zero, making it possible to solve this integral numerically.
∫ 0 L ( c i | Ψ | 2 | U − c | ( β − U ″ ) ) d y = 0 {\displaystyle {\begin{aligned}\int _{0}^{L}\left(c_{i}{\frac {|\Psi |^{2}}{|U-c|}}(\beta -U'')\right)dy&=0\\\end{aligned}}}
When c i {\displaystyle c_{i}} is zero, we are dealing with a stable amplitude of the solution, this means that the solution is stable. We are looking for un unstable situation, so then | Ψ | 2 | U − c | ( β − U ″ ) {\displaystyle {\frac {|\Psi |^{2}}{|U-c|}}(\beta -U'')} should be zero. Since the fraction in front of ( β − U ″ ) {\displaystyle (\beta -U'')} is non-zero and positive, this leads to the conclusion that ( β − U ″ ) {\displaystyle (\beta -U'')} should be zero. This leads to the final formulation, the Kuo criterion:
β − U ″ = 0 β = U ″ ∂ ( β y ) ∂ y = ∂ 2 U ∂ y 2 {\displaystyle {\begin{aligned}\beta -U''&=0\\[10pt]\beta &=U''\\[10pt]{\frac {\partial (\beta y)}{\partial y}}&={\frac {\partial ^{2}U}{\partial y^{2}}}\end{aligned}}} Here, U {\displaystyle U} is the mean zonal flow and β {\displaystyle \beta } is the derivative of the planetary vorticity f {\displaystyle f} with respect to y {\displaystyle y} . | https://en.wikipedia.org/wiki/Rayleigh–Kuo_criterion |
Rayleigh–Lorentz pendulum (or Lorentz pendulum ) is a simple pendulum , but subjected to a slowly varying frequency due to an external action (frequency is varied by varying the pendulum length), named after Lord Rayleigh and Hendrik Lorentz . [ 1 ] This problem formed the basis for the concept of adiabatic invariants in mechanics. On account of the slow variation of frequency, it is shown that the ratio of average energy to frequency is constant.
The pendulum problem was first formulated by Lord Rayleigh in 1902, although some mathematical aspects have been discussed before by Léon Lecornu in 1895 and Charles Bossut in 1778. [ 2 ] [ 3 ] [ 4 ] Unaware of Rayleigh's work, at the first Solvay conference in 1911, Hendrik Lorentz proposed a question, How does a simple pendulum behave when the length of the suspending thread is gradually shortened? , in order to clarify the quantum theory at that time. To that Albert Einstein responded the next day by saying that both energy and frequency of the quantum pendulum changes such that their ratio is constant, so that the pendulum is in the same quantum state as the initial state. These two separate works formed the basis for the concept of adiabatic invariant , which found applications in various fields and old quantum theory . In 1958, Subrahmanyan Chandrasekhar took interest in the problem and studied it so that a renewed interest in the problem was set, subsequently to be studied by many other researchers like John Edensor Littlewood etc. [ 5 ] [ 6 ] [ 7 ]
The equation of the simple harmonic motion with frequency ω {\displaystyle \omega } for the displacement x ( t ) {\displaystyle x(t)} is given by x ¨ + ω 2 x = 0. {\displaystyle {\ddot {x}}+\omega ^{2}x=0.}
If the frequency is constant, the solution is simply given by x = A cos ( ω t + ϕ ) {\displaystyle x=A\cos(\omega t+\phi )} . But if the frequency is allowed to vary slowly with time ω = ω ( t ) {\displaystyle \omega =\omega (t)} , or precisely, if the characteristic time scale for the frequency variation is much smaller than the time period of oscillation, i.e., | 1 ω d ω d t | ≪ ω , {\displaystyle \left|{\frac {1}{\omega }}{\frac {d\omega }{dt}}\right|\ll \omega ,} then it can be shown that E ¯ ω = constant , {\displaystyle {\frac {\bar {E}}{\omega }}={\text{constant}},} where E ¯ {\displaystyle {\bar {E}}} is the average energy averaged over an oscillation. Since the frequency is changing with time due to external action, conservation of energy no longer holds and the energy over a single oscillation is not constant. During an oscillation, the frequency changes (however slowly), so does its energy. Therefore, to describe the system, one defines the average energy per unit mass for a given potential V ( x ; ω ) {\displaystyle V(x;\omega )} as follows E ¯ = ∮ d t [ 1 2 ( x ˙ ) 2 + V ( x ( t ) ; ω ( t ) ) ] ∮ d t {\displaystyle {\bar {E}}={\frac {\displaystyle \oint dt\left[{\tfrac {1}{2}}\left({\dot {x}}\right)^{2}+V(x(t);\omega (t))\right]}{\displaystyle \oint dt}}} where the closed integral denotes that it is taken over a complete oscillation. Defined this way, it can be seen that the averaging is done, weighting each element of the orbit by the fraction of time that the pendulum spends in that element. For simple harmonic oscillator, it reduces to E ¯ = 1 2 A 2 ω 2 {\displaystyle {\bar {E}}={\tfrac {1}{2}}A^{2}\omega ^{2}} where both the amplitude and frequency are now functions of time. | https://en.wikipedia.org/wiki/Rayleigh–Lorentz_pendulum |
In fluid mechanics , the Rayleigh–Plesset equation or Besant–Rayleigh–Plesset equation is a nonlinear ordinary differential equation which governs the dynamics of a spherical bubble in an infinite body of incompressible fluid. [ 1 ] [ 2 ] [ 3 ] [ 4 ] Its general form is usually written as
R d 2 R d t 2 + 3 2 ( d R d t ) 2 + 4 ν L R d R d t + 2 σ ρ L R + Δ P ( t ) ρ L = 0 {\displaystyle R{\frac {d^{2}R}{dt^{2}}}+{\frac {3}{2}}\left({\frac {dR}{dt}}\right)^{2}+{\frac {4\nu _{L}}{R}}{\frac {dR}{dt}}+{\frac {2\sigma }{\rho _{L}R}}+{\frac {\Delta P(t)}{\rho _{L}}}=0}
where
Provided that P B ( t ) {\displaystyle P_{B}(t)} is known and P ∞ ( t ) {\displaystyle P_{\infty }(t)} is given, the Rayleigh–Plesset equation can be used to solve for the time-varying bubble radius R ( t ) {\displaystyle R(t)} .
The Rayleigh–Plesset equation can be derived from the Navier–Stokes equations under the assumption of spherical symmetry . [ 4 ] It can also be derived using an energy balance. [ 5 ]
Neglecting surface tension and viscosity, the equation was first derived by W. H. Besant in his 1859 book with the problem statement stated as An infinite mass of homogeneous incompressible fluid acted upon by no forces is at rest, and a spherical portion of the fluid is suddenly annihilated; it is required to find the instantaneous alteration of pressure at any point of the mass, and the time in which the cavity will be filled up, the pressure at an infinite distance being supposed to remain constant (in fact, Besant attributes the problem to Cambridge Senate-House problems of 1847). [ 6 ] Besant predicted the time required to fill an empty cavity of initial radius R 0 {\displaystyle R_{0}} to be
Lord Rayleigh found a simpler derivation of the same result, based on conservation of energy . The kinetic energy of the inrushing fluid is 2 π ρ U 2 R 3 {\displaystyle 2\pi \rho U^{2}R^{3}} where R {\displaystyle R} is the time-dependent radius of the void, and U {\displaystyle U} the radial velocity of the fluid there. The work done by the fluid pressing in at infinity is 4 π P ∞ ( R 0 3 − R 3 ) / 3 {\displaystyle 4\pi P_{\infty }(R_{0}^{3}-R^{3})/3} , and equating these two energies gives a relation between R {\displaystyle R} and U {\displaystyle U} . Then, noting that U = ∂ R / ∂ t {\displaystyle U=\partial R/\partial t} , separation of variables gives Besant's result. Rayleigh went further than Besant, in evaluating the integral (Euler's beta function ) in terms of gamma functions . Rayleigh adapted this approach to the case of a cavity filled with an ideal gas (a bubble) by including a term for the work done compressing the gas.
For the case of the perfectly empty void, Rayleigh determined that the pressure P {\displaystyle P} in the fluid at a radius r {\displaystyle r} is given by:
When the void is at least one quarter of its initial volume, then the pressure decreases monotonically from P ∞ {\displaystyle P_{\infty }} at infinity to zero at R {\displaystyle R} . As the void shrinks further a pressure maximum, greater than P ∞ {\displaystyle P_{\infty }} appears at
very rapidly growing and converging on the void.
The equation was first applied to traveling cavitation bubbles by Milton S. Plesset in 1949 by including effects of surface tension. [ 7 ]
The Rayleigh–Plesset equation can be derived entirely from first principles using the bubble radius as the dynamic parameter. [ 3 ] Consider a spherical bubble with time-dependent radius R ( t ) {\displaystyle R(t)} , where t {\displaystyle t} is time. Assume that the bubble contains a homogeneously distributed vapor/gas with a uniform temperature T B ( t ) {\displaystyle T_{B}(t)} and pressure P B ( t ) {\displaystyle P_{B}(t)} . Outside the bubble is an infinite domain of liquid with constant density ρ L {\displaystyle \rho _{L}} and dynamic viscosity μ L {\displaystyle \mu _{L}} . Let the temperature and pressure far from the bubble be T ∞ {\displaystyle T_{\infty }} and P ∞ ( t ) {\displaystyle P_{\infty }(t)} . The temperature T ∞ {\displaystyle T_{\infty }} is assumed to be constant. At a radial distance r {\displaystyle r} from the center of the bubble, the varying liquid properties are pressure P ( r , t ) {\displaystyle P(r,t)} , temperature T ( r , t ) {\displaystyle T(r,t)} , and radially outward velocity u ( r , t ) {\displaystyle u(r,t)} . Note that these liquid properties are only defined outside the bubble, for r ≥ R ( t ) {\displaystyle r\geq R(t)} .
By conservation of mass , the inverse-square law requires that the radially outward velocity u ( r , t ) {\displaystyle u(r,t)} must be inversely proportional to the square of the distance from the origin (the center of the bubble). [ 7 ] Therefore, letting F ( t ) {\displaystyle F(t)} be some function of time,
In the case of zero mass transport across the bubble surface, the velocity at the interface must be
which gives that
In the case where mass transport occurs and assuming the bubble contents are at constant density, the rate of mass increase inside the bubble is given by
with V {\displaystyle V} being the volume of the bubble. If u L {\displaystyle u_{L}} is the velocity of the liquid relative to the bubble at r = R {\displaystyle r=R} , then the mass entering the bubble is given by
with A {\displaystyle A} being the surface area of the bubble. Now by conservation of mass d m v / d t = d m L / d t {\displaystyle dm_{v}/dt=dm_{L}/dt} , therefore u L = ( ρ V / ρ L ) d R / d t {\displaystyle u_{L}=(\rho _{V}/\rho _{L})dR/dt} . Hence
Therefore
In many cases, the liquid density is much greater than the vapor density, ρ L ≫ ρ V {\displaystyle \rho _{L}\gg \rho _{V}} , so that F ( t ) {\displaystyle F(t)} can be approximated by the original zero mass transfer form F ( t ) = R 2 d R / d t {\displaystyle F(t)=R^{2}dR/dt} , so that [ 7 ]
Assuming that the liquid is a Newtonian fluid , the incompressible Navier–Stokes equation in spherical coordinates for motion in the radial direction gives
Substituting kinematic viscosity ν L = μ L / ρ L {\displaystyle \nu _{L}=\mu _{L}/\rho _{L}} and rearranging gives
whereby substituting u ( r , t ) {\displaystyle u(r,t)} from mass conservation yields
Note that the viscous terms cancel during substitution. [ 7 ] Separating variables and integrating from the bubble boundary r = R {\displaystyle r=R} to r → ∞ {\displaystyle r\rightarrow \infty } gives
Let σ r r {\displaystyle \sigma _{rr}} be the normal stress in the liquid that points radially outward from the center of the bubble. In spherical coordinates, for a fluid with constant density and constant viscosity,
Therefore at some small portion of the bubble surface, the net force per unit area acting on the lamina is
where σ {\displaystyle \sigma } is the surface tension . [ 7 ] If there is no mass transfer across the boundary, then this force per unit area must be zero, therefore
P ( R ) = P B − 4 μ L R d R d t − 2 σ R {\displaystyle P(R)=P_{B}-{\frac {4\mu _{L}}{R}}{\frac {dR}{dt}}-{\frac {2\sigma }{R}}}
and so the result from momentum conservation becomes
whereby rearranging and letting ν L = μ L / ρ L {\displaystyle \nu _{L}=\mu _{L}/\rho _{L}} gives the Rayleigh–Plesset equation [ 7 ]
Using dot notation to represent derivatives with respect to time, the Rayleigh–Plesset equation can be more succinctly written as
More recently, analytical closed-form solutions were found for the Rayleigh–Plesset equation for both an empty and gas-filled bubble [ 8 ] and were generalized to the N-dimensional case. [ 9 ] The case when the surface tension is present due to the effects of capillarity were also studied. [ 9 ] [ 10 ]
Also, for the special case where surface tension and viscosity are neglected, high-order analytical approximations are also known. [ 11 ]
In the static case, the Rayleigh–Plesset equation simplifies, yielding the Young–Laplace equation :
When only infinitesimal periodic variations in the bubble radius and pressure are considered, the RP equation also yields the expression of the natural frequency of the bubble oscillation . | https://en.wikipedia.org/wiki/Rayleigh–Plesset_equation |
The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues , originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz .
In this method, an infinite-dimensional linear operator is approximated by a finite-dimensional compression , on which we can use an eigenvalue algorithm .
It is used in all applications that involve approximating eigenvalues and eigenvectors , often under different names. In quantum mechanics , where a system of particles is described using a Hamiltonian , the Ritz method uses trial wave functions to approximate the ground state eigenfunction with the lowest energy. In the finite element method context, mathematically the same algorithm is commonly called the Ritz-Galerkin method . The Rayleigh–Ritz method or Ritz method terminology is typical in mechanical and structural engineering to approximate the eigenmodes and resonant frequencies of a structure.
The name of the method and its origin story have been debated by historians. [ 1 ] [ 2 ] It has been called Ritz method after Walther Ritz , since the numerical procedure has been published by Walther Ritz in 1908-1909. According to A. W. Leissa, [ 1 ] Lord Rayleigh wrote a paper congratulating Ritz on his work in 1911, but stating that he himself had used Ritz's method in many places in his book and in another publication. This statement, although later disputed, and the fact that the method in the trivial case of a single vector results in the Rayleigh quotient make the case for the name Rayleigh–Ritz method. According to S. Ilanko, [ 2 ] citing Richard Courant , both Lord Rayleigh and Walther Ritz independently conceived the idea of utilizing the equivalence between boundary value problems of partial differential equations on the one hand and problems of the calculus of variations on the other hand for numerical calculation of the solutions, by substituting for the variational problems simpler approximating extremum problems in which a finite number of parameters need to be determined. Ironically for the debate, the modern justification of the algorithm drops the calculus of variations in favor of the simpler and more general approach of orthogonal projection as in Galerkin method named after Boris Galerkin , thus leading also to the Ritz-Galerkin method naming. [ citation needed ]
Let T {\displaystyle T} be a linear operator on a Hilbert space H {\displaystyle {\mathcal {H}}} , with inner product ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} . Now consider a finite set of functions L = { φ 1 , . . . , φ n } {\displaystyle {\mathcal {L}}=\{\varphi _{1},...,\varphi _{n}\}} . Depending on the application these functions may be:
One could use the orthonormal basis generated from the eigenfunctions of the operator, which will produce diagonal approximating matrices, but in this case we would have already had to calculate the spectrum.
We now approximate T {\displaystyle T} by T L {\displaystyle T_{\mathcal {L}}} , which is defined as the matrix with entries [ 3 ]
( T L ) i , j = ( T φ i , φ j ) . {\displaystyle (T_{\mathcal {L}})_{i,j}=(T\varphi _{i},\varphi _{j}).}
and solve the eigenvalue problem T L u = λ u {\displaystyle T_{\mathcal {L}}u=\lambda u} . It can be shown that the matrix T L {\displaystyle T_{\mathcal {L}}} is the compression of T {\displaystyle T} to L {\displaystyle {\mathcal {L}}} . [ 3 ]
For differential operators (such as Sturm-Liouville operators ), the inner product ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} can be replaced by the weak formulation A ( ⋅ , ⋅ ) {\displaystyle {\mathcal {A}}(\cdot ,\cdot )} . [ 4 ] [ 6 ]
If a subset of the orthonormal basis was used to find the matrix, the eigenvectors of T L {\displaystyle T_{\mathcal {L}}} will be linear combinations of orthonormal basis functions, and as a result they will be approximations of the eigenvectors of T {\displaystyle T} . [ 7 ]
It is possible for the Rayleigh–Ritz method to produce values which do not converge to actual values in the spectrum of the operator as the truncation gets large. These values are known as spectral pollution. [ 3 ] [ 5 ] [ 8 ] In some cases (such as for the Schrödinger equation ), there is no approximation which both includes all eigenvalues of the equation, and contains no pollution. [ 9 ]
The spectrum of the compression (and thus pollution) is bounded by the numerical range of the operator; in many cases it is bounded by a subset of the numerical range known as the essential numerical range . [ 10 ] [ 11 ]
In numerical linear algebra, the Rayleigh–Ritz method is commonly [ 12 ] applied to approximate an eigenvalue problem A x = λ x {\displaystyle A\mathbf {x} =\lambda \mathbf {x} } for the matrix A ∈ C N × N {\displaystyle A\in \mathbb {C} ^{N\times N}} of size N {\displaystyle N} using a projected matrix of a smaller size m < N {\displaystyle m<N} , generated from a given matrix V ∈ C N × m {\displaystyle V\in \mathbb {C} ^{N\times m}} with orthonormal columns. The matrix version of the algorithm is the most simple:
If the subspace with the orthonormal basis given by the columns of the matrix V ∈ C N × m {\displaystyle V\in \mathbb {C} ^{N\times m}} contains k ≤ m {\displaystyle k\leq m} vectors that are close to eigenvectors of the matrix A {\displaystyle A} , the Rayleigh–Ritz method above finds k {\displaystyle k} Ritz vectors that well approximate these eigenvectors. The easily computable quantity ‖ A x ~ i − λ ~ i x ~ i ‖ {\displaystyle \|A{\tilde {\mathbf {x} }}_{i}-{\tilde {\lambda }}_{i}{\tilde {\mathbf {x} }}_{i}\|} determines the accuracy of such an approximation for every Ritz pair.
In the easiest case m = 1 {\displaystyle m=1} , the N × m {\displaystyle N\times m} matrix V {\displaystyle V} turns into a unit column-vector v {\displaystyle v} , the m × m {\displaystyle m\times m} matrix V ∗ A V {\displaystyle V^{*}AV} is a scalar that is equal to the Rayleigh quotient ρ ( v ) = v ∗ A v / v ∗ v {\displaystyle \rho (v)=v^{*}Av/v^{*}v} , the only i = 1 {\displaystyle i=1} solution to the eigenvalue problem is y i = 1 {\displaystyle y_{i}=1} and μ i = ρ ( v ) {\displaystyle \mu _{i}=\rho (v)} , and the only one Ritz vector is v {\displaystyle v} itself. Thus, the Rayleigh–Ritz method turns into computing of the Rayleigh quotient if m = 1 {\displaystyle m=1} .
Another useful connection to the Rayleigh quotient is that μ i = ρ ( v i ) {\displaystyle \mu _{i}=\rho (v_{i})} for every Ritz pair ( λ ~ i , x ~ i ) {\displaystyle ({\tilde {\lambda }}_{i},{\tilde {\mathbf {x} }}_{i})} , allowing to derive some properties of Ritz values μ i {\displaystyle \mu _{i}} from the corresponding theory for the Rayleigh quotient . For example, if A {\displaystyle A} is a Hermitian matrix , its Rayleigh quotient (and thus its every Ritz value) is real and takes values within the closed interval of the smallest and largest eigenvalues of A {\displaystyle A} .
The matrix A = [ 2 0 0 0 2 1 0 1 2 ] {\displaystyle A={\begin{bmatrix}2&0&0\\0&2&1\\0&1&2\end{bmatrix}}} has eigenvalues 1 , 2 , 3 {\displaystyle 1,2,3} and the corresponding eigenvectors x λ = 1 = [ 0 1 − 1 ] , x λ = 2 = [ 1 0 0 ] , x λ = 3 = [ 0 1 1 ] . {\displaystyle \mathbf {x} _{\lambda =1}={\begin{bmatrix}0\\1\\-1\end{bmatrix}},\quad \mathbf {x} _{\lambda =2}={\begin{bmatrix}1\\0\\0\end{bmatrix}},\quad \mathbf {x} _{\lambda =3}={\begin{bmatrix}0\\1\\1\end{bmatrix}}.} Let us take V = [ 0 0 1 0 0 1 ] , {\displaystyle V={\begin{bmatrix}0&0\\1&0\\0&1\end{bmatrix}},} then V ∗ A V = [ 2 1 1 2 ] {\displaystyle V^{*}AV={\begin{bmatrix}2&1\\1&2\end{bmatrix}}} with eigenvalues 1 , 3 {\displaystyle 1,3} and the corresponding eigenvectors y μ = 1 = [ 1 − 1 ] , y μ = 3 = [ 1 1 ] , {\displaystyle \mathbf {y} _{\mu =1}={\begin{bmatrix}1\\-1\end{bmatrix}},\quad \mathbf {y} _{\mu =3}={\begin{bmatrix}1\\1\end{bmatrix}},} so that the Ritz values are 1 , 3 {\displaystyle 1,3} and the Ritz vectors are x ~ λ ~ = 1 = [ 0 1 − 1 ] , x ~ λ ~ = 3 = [ 0 1 1 ] . {\displaystyle \mathbf {\tilde {x}} _{{\tilde {\lambda }}=1}={\begin{bmatrix}0\\1\\-1\end{bmatrix}},\quad \mathbf {\tilde {x}} _{{\tilde {\lambda }}=3}={\begin{bmatrix}0\\1\\1\end{bmatrix}}.} We observe that each one of the Ritz vectors is exactly one of the eigenvectors of A {\displaystyle A} for the given V {\displaystyle V} as well as the Ritz values give exactly two of the three eigenvalues of A {\displaystyle A} . A mathematical explanation for the exact approximation is based on the fact that the column space of the matrix V {\displaystyle V} happens to be exactly the same as the subspace spanned by the two eigenvectors x λ = 1 {\displaystyle \mathbf {x} _{\lambda =1}} and x λ = 3 {\displaystyle \mathbf {x} _{\lambda =3}} in this example.
Truncated singular value decomposition (SVD) in numerical linear algebra can also use the Rayleigh–Ritz method to find approximations to left and right singular vectors of the matrix M ∈ C M × N {\displaystyle M\in \mathbb {C} ^{M\times N}} of size M × N {\displaystyle M\times N} in given subspaces by turning the singular value problem into an eigenvalue problem.
The definition of the singular value σ {\displaystyle \sigma } and the corresponding left and right singular vectors is M v = σ u {\displaystyle Mv=\sigma u} and M ∗ u = σ v {\displaystyle M^{*}u=\sigma v} . Having found one set (left of right) of approximate singular vectors and singular values by applying naively the Rayleigh–Ritz method to the Hermitian normal matrix M ∗ M ∈ C N × N {\displaystyle M^{*}M\in \mathbb {C} ^{N\times N}} or M M ∗ ∈ C M × M {\displaystyle MM^{*}\in \mathbb {C} ^{M\times M}} , whichever one is smaller size, one could determine the other set of left of right singular vectors simply by dividing by the singular values, i.e., u = M v / σ {\displaystyle u=Mv/\sigma } and v = M ∗ u / σ {\displaystyle v=M^{*}u/\sigma } . However, the division is unstable or fails for small or zero singular values.
An alternative approach, e.g., defining the normal matrix as A = M ∗ M ∈ C N × N {\displaystyle A=M^{*}M\in \mathbb {C} ^{N\times N}} of size N × N {\displaystyle N\times N} , takes advantage of the fact that for a given N × m {\displaystyle N\times m} matrix W ∈ C N × m {\displaystyle W\in \mathbb {C} ^{N\times m}} with orthonormal columns the eigenvalue problem of the Rayleigh–Ritz method for the m × m {\displaystyle m\times m} matrix W ∗ A W = W ∗ M ∗ M W = ( M W ) ∗ M W {\displaystyle W^{*}AW=W^{*}M^{*}MW=(MW)^{*}MW} can be interpreted as a singular value problem for the N × m {\displaystyle N\times m} matrix M W {\displaystyle MW} . This interpretation allows simple simultaneous calculation of both left and right approximate singular vectors as follows.
The algorithm can be used as a post-processing step where the matrix W {\displaystyle W} is an output of an eigenvalue solver, e.g., such as LOBPCG , approximating numerically selected eigenvectors of the normal matrix A = M ∗ M {\displaystyle A=M^{*}M} .
The matrix M = [ 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 ] {\displaystyle M={\begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&3&0\\0&0&0&4\\0&0&0&0\end{bmatrix}}} has its normal matrix A = M ∗ M = [ 1 0 0 0 0 4 0 0 0 0 9 0 0 0 0 16 ] , {\displaystyle A=M^{*}M={\begin{bmatrix}1&0&0&0\\0&4&0&0\\0&0&9&0\\0&0&0&16\\\end{bmatrix}},} singular values 1 , 2 , 3 , 4 {\displaystyle 1,2,3,4} and the corresponding thin SVD A = [ 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 ] [ 4 0 0 0 0 3 0 0 0 0 2 0 0 0 0 1 ] [ 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 ] , {\displaystyle A={\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\\0&0&0&0\end{bmatrix}}{\begin{bmatrix}4&0&0&0\\0&3&0&0\\0&0&2&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}},} where the columns of the first multiplier from the complete set of the left singular vectors of the matrix A {\displaystyle A} , the diagonal entries of the middle term are the singular values, and the columns of the last multiplier transposed (although the transposition does not change it) [ 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 ] ∗ = [ 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 ] {\displaystyle {\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}}^{*}\quad =\quad {\begin{bmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\end{bmatrix}}} are the corresponding right singular vectors.
Let us take W = [ 1 / 2 1 / 2 1 / 2 − 1 / 2 0 0 0 0 ] {\displaystyle W={\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}\\1/{\sqrt {2}}&-1/{\sqrt {2}}\\0&0\\0&0\end{bmatrix}}} with the column-space that is spanned by the two exact right singular vectors [ 0 1 1 0 0 0 0 0 ] {\displaystyle {\begin{bmatrix}0&1\\1&0\\0&0\\0&0\end{bmatrix}}} corresponding to the singular values 1 and 2.
Following the algorithm step 1, we compute M W = [ 1 / 2 1 / 2 2 − 2 0 0 0 0 ] , {\displaystyle MW={\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}\\{\sqrt {2}}&-{\sqrt {2}}\\0&0\\0&0\end{bmatrix}},} and on step 2 its thin SVD M W = U Σ V h {\displaystyle MW=\mathbf {U} {\Sigma }\mathbf {V} _{h}} with U = [ 0 1 1 0 0 0 0 0 0 0 ] , Σ = [ 2 0 0 1 ] , V h = [ 1 / 2 − 1 / 2 1 / 2 1 / 2 ] . {\displaystyle \mathbf {U} ={\begin{bmatrix}0&1\\1&0\\0&0\\0&0\\0&0\end{bmatrix}},\quad \Sigma ={\begin{bmatrix}2&0\\0&1\end{bmatrix}},\quad \mathbf {V} _{h}={\begin{bmatrix}1/{\sqrt {2}}&-1/{\sqrt {2}}\\1/{\sqrt {2}}&1/{\sqrt {2}}\end{bmatrix}}.} Thus we already obtain the singular values 2 and 1 from Σ {\displaystyle \Sigma } and from U {\displaystyle \mathbf {U} } the corresponding two left singular vectors u {\displaystyle u} as [ 0 , 1 , 0 , 0 , 0 ] ∗ {\displaystyle [0,1,0,0,0]^{*}} and [ 1 , 0 , 0 , 0 , 0 ] ∗ {\displaystyle [1,0,0,0,0]^{*}} , which span the column-space of the matrix W {\displaystyle W} , explaining why the approximations are exact for the given W {\displaystyle W} .
Finally, step 3 computes the matrix V h = V h W ∗ {\displaystyle V_{h}=\mathbf {V} _{h}W^{*}} V h = [ 1 / 2 − 1 / 2 1 / 2 1 / 2 ] [ 1 / 2 1 / 2 0 0 1 / 2 − 1 / 2 0 0 ] = [ 0 1 0 0 1 0 0 0 ] {\displaystyle \mathbf {V} _{h}={\begin{bmatrix}1/{\sqrt {2}}&-1/{\sqrt {2}}\\1/{\sqrt {2}}&1/{\sqrt {2}}\end{bmatrix}}\,{\begin{bmatrix}1/{\sqrt {2}}&1/{\sqrt {2}}&0&0\\1/{\sqrt {2}}&-1/{\sqrt {2}}&0&0\end{bmatrix}}={\begin{bmatrix}0&1&0&0\\1&0&0&0\end{bmatrix}}} recovering from its rows the two right singular vectors v {\displaystyle v} as [ 0 , 1 , 0 , 0 ] ∗ {\displaystyle [0,1,0,0]^{*}} and [ 1 , 0 , 0 , 0 ] ∗ {\displaystyle [1,0,0,0]^{*}} .
We validate the first vector: M v = σ u {\displaystyle Mv=\sigma u} [ 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 ] [ 0 1 0 0 ] = 2 [ 0 1 0 0 0 ] {\displaystyle {\begin{bmatrix}1&0&0&0\\0&2&0&0\\0&0&3&0\\0&0&0&4\\0&0&0&0\end{bmatrix}}\,{\begin{bmatrix}0\\1\\0\\0\end{bmatrix}}=\,2\,{\begin{bmatrix}0\\1\\0\\0\\0\end{bmatrix}}} and M ∗ u = σ v {\displaystyle M^{*}u=\sigma v} [ 1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 4 0 ] [ 0 1 0 0 0 ] = 2 [ 0 1 0 0 ] . {\displaystyle {\begin{bmatrix}1&0&0&0&0\\0&2&0&0&0\\0&0&3&0&0\\0&0&0&4&0\end{bmatrix}}\,{\begin{bmatrix}0\\1\\0\\0\\0\end{bmatrix}}=\,2\,{\begin{bmatrix}0\\1\\0\\0\end{bmatrix}}.} Thus, for the given matrix W {\displaystyle W} with its column-space that is spanned by two exact right singular vectors, we determine these right singular vectors, as well as the corresponding left singular vectors and the singular values, all exactly. For an arbitrary matrix W {\displaystyle W} , we obtain approximate singular triplets which are optimal given W {\displaystyle W} in the sense of optimality of the Rayleigh–Ritz method.
In quantum physics, where the spectrum of the Hamiltonian is the set of discrete energy levels allowed by a quantum mechanical system, the Rayleigh–Ritz method is used to approximate the energy states and wavefunctions of a complicated atomic or nuclear system. [ 7 ] In fact, for any system more complicated than a single hydrogen atom, there is no known exact solution for the spectrum of the Hamiltonian. [ 6 ]
In this case, a trial wave function , Ψ {\displaystyle \Psi } , is tested on the system. This trial function is selected to meet boundary conditions (and any other physical constraints). The exact function is not known; the trial function contains one or more adjustable parameters, which are varied to find a lowest energy configuration.
It can be shown that the ground state energy, E 0 {\displaystyle E_{0}} , satisfies an inequality: E 0 ≤ ⟨ Ψ | H ^ | Ψ ⟩ ⟨ Ψ | Ψ ⟩ . {\displaystyle E_{0}\leq {\frac {\langle \Psi |{\hat {H}}|\Psi \rangle }{\langle \Psi |\Psi \rangle }}.}
That is, the ground-state energy is less than this value.
The trial wave-function will always give an expectation value larger than or equal to the ground-energy.
If the trial wave function is known to be orthogonal to the ground state, then it will provide a boundary for the energy of some excited state.
The Ritz ansatz function is a linear combination of N known basis functions { Ψ i } {\displaystyle \left\lbrace \Psi _{i}\right\rbrace } , parametrized by unknown coefficients: Ψ = ∑ i = 1 N c i Ψ i . {\displaystyle \Psi =\sum _{i=1}^{N}c_{i}\Psi _{i}.}
With a known Hamiltonian, we can write its expected value as ε = ⟨ ∑ i = 1 N c i Ψ i | H ^ | ∑ i = 1 N c i Ψ i ⟩ ⟨ ∑ i = 1 N c i Ψ i | ∑ i = 1 N c i Ψ i ⟩ = ∑ i = 1 N ∑ j = 1 N c i ∗ c j H i j ∑ i = 1 N ∑ j = 1 N c i ∗ c j S i j ≡ A B . {\displaystyle \varepsilon ={\frac {\left\langle \displaystyle \sum _{i=1}^{N}c_{i}\Psi _{i}\right|{\hat {H}}\left|\displaystyle \sum _{i=1}^{N}c_{i}\Psi _{i}\right\rangle }{\left\langle \left.\displaystyle \sum _{i=1}^{N}c_{i}\Psi _{i}\right|\displaystyle \sum _{i=1}^{N}c_{i}\Psi _{i}\right\rangle }}={\frac {\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{N}c_{i}^{*}c_{j}H_{ij}}{\displaystyle \sum _{i=1}^{N}\displaystyle \sum _{j=1}^{N}c_{i}^{*}c_{j}S_{ij}}}\equiv {\frac {A}{B}}.}
The basis functions are usually not orthogonal, so that the overlap matrix S has nonzero nondiagonal elements. Either { c i } {\displaystyle \left\lbrace c_{i}\right\rbrace } or { c i ∗ } {\displaystyle \left\lbrace c_{i}^{*}\right\rbrace } (the conjugation of the first) can be used to minimize the expectation value. For instance, by making the partial derivatives of ε {\displaystyle \varepsilon } over { c i ∗ } {\displaystyle \left\lbrace c_{i}^{*}\right\rbrace } zero, the following equality is obtained for every k = 1, 2, ..., N : ∂ ε ∂ c k ∗ = ∑ j = 1 N c j ( H k j − ε S k j ) B = 0 , {\displaystyle {\frac {\partial \varepsilon }{\partial c_{k}^{*}}}={\frac {\displaystyle \sum _{j=1}^{N}c_{j}(H_{kj}-\varepsilon S_{kj})}{B}}=0,} which leads to a set of N secular equations : ∑ j = 1 N c j ( H k j − ε S k j ) = 0 for k = 1 , 2 , … , N . {\displaystyle \sum _{j=1}^{N}c_{j}\left(H_{kj}-\varepsilon S_{kj}\right)=0\quad {\text{for}}\quad k=1,2,\dots ,N.}
In the above equations, energy ε {\displaystyle \varepsilon } and the coefficients { c j } {\displaystyle \left\lbrace c_{j}\right\rbrace } are unknown. With respect to c , this is a homogeneous set of linear equations, which has a solution when the determinant of the coefficients to these unknowns is zero: det ( H − ε S ) = 0 , {\displaystyle \det \left(H-\varepsilon S\right)=0,} which in turn is true only for N values of ε {\displaystyle \varepsilon } . Furthermore, since the Hamiltonian is a hermitian operator , the H matrix is also hermitian and the values of ε i {\displaystyle \varepsilon _{i}} will be real. The lowest value among ε i {\displaystyle \varepsilon _{i}} (i=1,2,..,N), ε 0 {\displaystyle \varepsilon _{0}} , will be the best approximation to the ground state for the basis functions used. The remaining N-1 energies are estimates of excited state energies. An approximation for the wave function of state i can be obtained by finding the coefficients { c j } {\displaystyle \left\lbrace c_{j}\right\rbrace } from the corresponding secular equation.
The Rayleigh–Ritz method is often used in mechanical engineering for finding the approximate real resonant frequencies of multi degree of freedom systems, such as spring mass systems or flywheels on a shaft with varying cross section . It is an extension of Rayleigh's method. It can also be used for finding buckling loads and post-buckling behaviour for columns.
Consider the case whereby we want to find the resonant frequency of oscillation of a system. First, write the oscillation in the form, y ( x , t ) = Y ( x ) cos ω t {\displaystyle y(x,t)=Y(x)\cos \omega t} with an unknown mode shape Y ( x ) {\displaystyle Y(x)} . Next, find the total energy of the system, consisting of a kinetic energy term and a potential energy term. The kinetic energy term involves the square of the time derivative of y ( x , t ) {\displaystyle y(x,t)} and thus gains a factor of ω 2 {\displaystyle \omega ^{2}} . Thus, we can calculate the total energy of the system and express it in the following form: E = T + V ≡ A [ Y ( x ) ] ω 2 sin 2 ω t + B [ Y ( x ) ] cos 2 ω t {\displaystyle E=T+V\equiv A[Y(x)]\omega ^{2}\sin ^{2}\omega t+B[Y(x)]\cos ^{2}\omega t}
By conservation of energy, the average kinetic energy must be equal to the average potential energy. Thus, ω 2 = B [ Y ( x ) ] A [ Y ( x ) ] = R [ Y ( x ) ] {\displaystyle \omega ^{2}={\frac {B[Y(x)]}{A[Y(x)]}}=R[Y(x)]} which is also known as the Rayleigh quotient . Thus, if we knew the mode shape Y ( x ) {\displaystyle Y(x)} , we would be able to calculate A [ Y ( x ) ] {\displaystyle A[Y(x)]} and B [ Y ( x ) ] {\displaystyle B[Y(x)]} , and in turn get the eigenfrequency. However, we do not yet know the mode shape. In order to find this, we can approximate Y ( x ) {\displaystyle Y(x)} as a combination of a few approximating functions Y i ( x ) {\displaystyle Y_{i}(x)} Y ( x ) = ∑ i = 1 N c i Y i ( x ) {\displaystyle Y(x)=\sum _{i=1}^{N}c_{i}Y_{i}(x)} where c 1 , c 2 , ⋯ , c N {\displaystyle c_{1},c_{2},\cdots ,c_{N}} are constants to be determined. In general, if we choose a random set of c 1 , c 2 , ⋯ , c N {\displaystyle c_{1},c_{2},\cdots ,c_{N}} , it will describe a superposition of the actual eigenmodes of the system. However, if we seek c 1 , c 2 , ⋯ , c N {\displaystyle c_{1},c_{2},\cdots ,c_{N}} such that the eigenfrequency ω 2 {\displaystyle \omega ^{2}} is minimised, then the mode described by this set of c 1 , c 2 , ⋯ , c N {\displaystyle c_{1},c_{2},\cdots ,c_{N}} will be close to the lowest possible actual eigenmode of the system. Thus, this finds the lowest eigenfrequency. If we find eigenmodes orthogonal to this approximated lowest eigenmode, we can approximately find the next few eigenfrequencies as well.
In general, we can express A [ Y ( x ) ] {\displaystyle A[Y(x)]} and B [ Y ( x ) ] {\displaystyle B[Y(x)]} as a collection of terms quadratic in the coefficients c i {\displaystyle c_{i}} : B [ Y ( x ) ] = ∑ i ∑ j c i c j K i j = c T K c {\displaystyle B[Y(x)]=\sum _{i}\sum _{j}c_{i}c_{j}K_{ij}=\mathbf {c} ^{\mathsf {T}}K\mathbf {c} } A [ Y ( x ) ] = ∑ i ∑ j c i c j M i j = c T M c {\displaystyle A[Y(x)]=\sum _{i}\sum _{j}c_{i}c_{j}M_{ij}=\mathbf {c} ^{\mathsf {T}}M\mathbf {c} } where K {\displaystyle K} and M {\displaystyle M} are the stiffness matrix and mass matrix of a discrete system respectively.
The minimization of ω 2 {\displaystyle \omega ^{2}} becomes: ∂ ω 2 ∂ c i = ∂ ∂ c i c T K c c T M c = 0 {\displaystyle {\frac {\partial \omega ^{2}}{\partial c_{i}}}={\frac {\partial }{\partial c_{i}}}{\frac {\mathbf {c} ^{\mathsf {T}}K\mathbf {c} }{\mathbf {c} ^{\mathsf {T}}M\mathbf {c} }}=0}
Solving this, c T M c ∂ c T K c ∂ c − c T K c ∂ c T M c ∂ c = 0 {\displaystyle \mathbf {c} ^{\mathsf {T}}M\mathbf {c} {\frac {\partial \mathbf {c} ^{\mathsf {T}}K\mathbf {c} }{\partial \mathbf {c} }}-\mathbf {c} ^{\mathsf {T}}K\mathbf {c} {\frac {\partial \mathbf {c} ^{\mathsf {T}}M\mathbf {c} }{\partial \mathbf {c} }}=0} K c − c T K c c T M c M c = 0 {\displaystyle K\mathbf {c} -{\frac {\mathbf {c} ^{\mathsf {T}}K\mathbf {c} }{\mathbf {c} ^{\mathsf {T}}M\mathbf {c} }}M\mathbf {c} =\mathbf {0} } K c − ω 2 M c = 0 {\displaystyle K\mathbf {c} -\omega ^{2}M\mathbf {c} =\mathbf {0} }
For a non-trivial solution of c, we require determinant of the matrix coefficient of c to be zero. det ( K − ω 2 M ) = 0 {\displaystyle \det(K-\omega ^{2}M)=0}
This gives a solution for the first N eigenfrequencies and eigenmodes of the system, with N being the number of approximating functions.
The following discussion uses the simplest case, where the system has two lumped springs and two lumped masses, and only two mode shapes are assumed. Hence M = [ m 1 , m 2 ] and K = [ k 1 , k 2 ] .
A mode shape is assumed for the system, with two terms, one of which is weighted by a factor B , e.g. Y = [1, 1] + B [1, −1]. Simple harmonic motion theory says that the velocity at the time when deflection is zero, is the angular frequency ω {\displaystyle \omega } times the deflection (y) at time of maximum deflection. In this example the kinetic energy (KE) for each mass is 1 2 ω 2 Y 1 2 m 1 {\textstyle {\frac {1}{2}}\omega ^{2}Y_{1}^{2}m_{1}} etc., and the potential energy (PE) for each spring is 1 2 k 1 Y 1 2 {\textstyle {\frac {1}{2}}k_{1}Y_{1}^{2}} etc.
We also know that without damping, the maximal KE equals the maximal PE. Thus, ∑ i = 1 2 ( 1 2 ω 2 Y i 2 M i ) = ∑ i = 1 2 ( 1 2 K i Y i 2 ) {\displaystyle \sum _{i=1}^{2}\left({\frac {1}{2}}\omega ^{2}Y_{i}^{2}M_{i}\right)=\sum _{i=1}^{2}\left({\frac {1}{2}}K_{i}Y_{i}^{2}\right)}
The overall amplitude of the mode shape cancels out from each side, always. That is, the actual size of the assumed deflection does not matter, just the mode shape .
Mathematical manipulations then obtain an expression for ω {\displaystyle \omega } , in terms of B, which can be differentiated with respect to B, to find the minimum, i.e. when d ω / d B = 0 {\displaystyle d\omega /dB=0} . This gives the value of B for which ω {\displaystyle \omega } is lowest. This is an upper bound solution for ω {\displaystyle \omega } if ω {\displaystyle \omega } is hoped to be the predicted fundamental frequency of the system because the mode shape is assumed , but we have found the lowest value of that upper bound, given our assumptions, because B is used to find the optimal 'mix' of the two assumed mode shape functions.
There are many tricks with this method, the most important is to try and choose realistic assumed mode shapes. For example, in the case of beam deflection problems it is wise to use a deformed shape that is analytically similar to the expected solution. A quartic may fit most of the easy problems of simply linked beams even if the order of the deformed solution may be lower. The springs and masses do not have to be discrete, they can be continuous (or a mixture), and this method can be easily used in a spreadsheet to find the natural frequencies of quite complex distributed systems, if you can describe the distributed KE and PE terms easily, or else break the continuous elements up into discrete parts.
This method could be used iteratively, adding additional mode shapes to the previous best solution, or you can build up a long expression with many Bs and many mode shapes, and then differentiate them partially .
The Koopman operator allows a finite-dimensional nonlinear system to be encoded as an infinite-dimensional linear system . In general, both of these problems are difficult to solve, but for the latter we can use the Ritz-Galerkin method to approximate a solution. [ 13 ]
In the language of the finite element method, the matrix H k j {\displaystyle H_{kj}} is precisely the stiffness matrix of the Hamiltonian in the piecewise linear element space, and the matrix S k j {\displaystyle S_{kj}} is the mass matrix . In the language of linear algebra, the value ϵ {\displaystyle \epsilon } is an eigenvalue of the discretized Hamiltonian, and the vector c {\displaystyle c} is a discretized eigenvector. | https://en.wikipedia.org/wiki/Rayleigh–Ritz_method |
The Rayleigh–Taylor instability , or RT instability (after Lord Rayleigh and G. I. Taylor ), is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. [ 2 ] [ 3 ] [ 4 ] Examples include the behavior of water suspended above oil in the gravity of Earth , [ 3 ] mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions , [ 5 ] supernova explosions in which expanding core gas is accelerated into denser shell gas, [ 6 ] [ 7 ] merging binary quantum fluids in metastable configuration, [ 8 ] instabilities in plasma fusion reactors and [ 9 ] inertial confinement fusion. [ 10 ]
Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, and it may be modeled by two completely plane-parallel layers of immiscible fluid, the denser fluid on top of the less dense one and both subject to the Earth's gravity. The equilibrium here is unstable to any perturbations or disturbances of the interface: if a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state. Thus the disturbance will grow and lead to a further release of potential energy , as the denser material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards. This was the set-up as studied by Lord Rayleigh. [ 3 ] The important insight by G. I. Taylor was his realisation that this situation is equivalent to the situation when the fluids are accelerated , with the less dense fluid accelerating into the denser fluid. [ 3 ] This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion. [ 11 ]
As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear growth phase, eventually developing "plumes" flowing upwards (in the gravitational buoyancy sense) and "spikes" falling downwards. In the linear phase, the fluid movement can be closely approximated by linear equations , and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for a linear approximation, and non-linear equations are required to describe fluid motions. In general, the density disparity between the fluids determines the structure of the subsequent non-linear RT instability flows (assuming other variables such as surface tension and viscosity are negligible here). The difference in the fluid densities divided by their sum is defined as the Atwood number , A. For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid; for A close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes. [ 2 ]
This process is evident not only in many terrestrial examples, from salt domes to weather inversions , but also in astrophysics and electrohydrodynamics . For example, RT instability structure is evident in the Crab Nebula , in which the expanding pulsar wind nebula powered by the Crab pulsar is sweeping up ejected material from the supernova explosion 1000 years ago. [ 12 ] The RT instability has also recently been discovered in the Sun's outer atmosphere, or solar corona , when a relatively dense solar prominence overlies a less dense plasma bubble. [ 13 ] This latter case resembles magnetically modulated RT instabilities. [ 14 ] [ 15 ] [ 16 ]
Note that the RT instability is not to be confused with the Plateau–Rayleigh instability (also known as Rayleigh instability) of a liquid jet. This instability, sometimes called the hosepipe (or firehose) instability, occurs due to surface tension, which acts to break a cylindrical jet into a stream of droplets having the same total volume but higher surface area.
Many people have witnessed the RT instability by looking at a lava lamp , although some might claim this is more accurately described as an example of Rayleigh–Bénard convection due to the active heating of the fluid layer at the bottom of the lamp.
The evolution of the RTI follows four main stages. [ 2 ] In the first stage, the perturbation amplitudes are small when compared to their wavelengths, the equations of motion can be linearized, resulting in exponential instability growth. In the early portion of this stage, a sinusoidal initial perturbation retains its sinusoidal shape. However, after the end of this first stage, when non-linear effects begin to appear, one observes the beginnings of the formation of the ubiquitous mushroom-shaped spikes (fluid structures of heavy fluid growing into light fluid) and bubbles (fluid structures of light fluid growing into heavy fluid). The growth of the mushroom structures continues in the second stage and can be modeled using buoyancy drag models, resulting in a growth rate that is approximately constant in time. At this point, nonlinear terms in the equations of motion can no longer be ignored. The spikes and bubbles then begin to interact with one another in the third stage. Bubble merging takes place, where the nonlinear interaction of mode coupling acts to combine smaller spikes and bubbles to produce larger ones. Also, bubble competition takes places, where spikes and bubbles of smaller wavelength that have become saturated are enveloped by larger ones that have not yet saturated. This eventually develops into a region of turbulent mixing, which is the fourth and final stage in the evolution. It is generally assumed that the mixing region that finally develops is self-similar and turbulent, provided that the Reynolds number is sufficiently large. [ 17 ]
The inviscid two-dimensional Rayleigh–Taylor (RT) instability provides an excellent springboard into the mathematical study of stability because of the simple nature of the base state. [ 18 ] Consider a base state in which there is an interface, located at z = 0 {\displaystyle z=0} that separates fluid media with different densities, ρ 1 {\displaystyle \rho _{1}} for z < 0 {\displaystyle z<0} and ρ 2 {\displaystyle \rho _{2}} for z > 0 {\displaystyle z>0} . The gravitational acceleration is described by the vector g = − g e z {\displaystyle \mathbf {g} =-g\,\mathbf {e} _{z}} . The velocity field and pressure field in this equilibrium state, denoted with an overbar, are given by
where the reference location for the pressure is taken to be at z = 0 {\displaystyle z=0} . Let this interface be slightly perturbed, so that it assumes the position z = f ( x , t ) {\displaystyle z=f(x,t)} . Correspondingly, the base state is also slightly perturbed. In the linear theory, we can write
where k {\displaystyle k} is the real wavenumber in the x {\displaystyle x} -direction and σ {\displaystyle \sigma } is the growth rate of the perturbation. Then the linear stability analysis based on the inviscid governing equations shows that
Thus, if ρ 2 < ρ 1 {\displaystyle \rho _{2}<\rho _{1}} , the base state is stable and while if ρ 2 > ρ 1 {\displaystyle \rho _{2}>\rho _{1}} , it is unstable for all wavenumbers. If the interface has a surface tension γ {\displaystyle \gamma } , then the dispersion relation becomes
which indicates that the instability occurs only for a range of wavenumbers 0 < k < k c {\displaystyle 0<k<k_{c}} where k c 2 = ( ρ 2 − ρ 1 ) g / γ {\displaystyle k_{c}^{2}=(\rho _{2}-\rho _{1})g/\gamma } ; that is to say, surface tension stabilises large wavenumbers or small length scales. Then the maximum growth rate occurs at the wavenumber k m = k c / 3 {\displaystyle k_{m}=k_{c}/{\sqrt {3}}} and its value is
The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude, ( u ′ ( x , z , t ) , w ′ ( x , z , t ) ) . {\displaystyle (u'(x,z,t),w'(x,z,t)).\,} Because the fluid is assumed incompressible, this velocity field has the streamfunction representation
u ′ = ( u ′ ( x , z , t ) , w ′ ( x , z , t ) ) = ( ψ z , − ψ x ) , {\displaystyle {\textbf {u}}'=(u'(x,z,t),w'(x,z,t))=(\psi _{z},-\psi _{x}),\,}
where the subscripts indicate partial derivatives . Moreover, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays irrotational , hence ∇ × u ′ = 0 {\displaystyle \nabla \times {\textbf {u}}'=0\,} . In the streamfunction representation, ∇ 2 ψ = 0. {\displaystyle \nabla ^{2}\psi =0.\,} Next, because of the translational invariance of the system in the x -direction, it is possible to make the ansatz
ψ ( x , z , t ) = e i α ( x − c t ) Ψ ( z ) , {\displaystyle \psi \left(x,z,t\right)=e^{i\alpha \left(x-ct\right)}\Psi \left(z\right),\,}
where α {\displaystyle \alpha \,} is a spatial wavenumber. Thus, the problem reduces to solving the equation
( D 2 − α 2 ) Ψ j = 0 , D = d d z , j = L , G . {\displaystyle \left(D^{2}-\alpha ^{2}\right)\Psi _{j}=0,\,\,\,\ D={\frac {d}{dz}},\,\,\,\ j=L,G.\,}
The domain of the problem is the following: the fluid with label 'L' lives in the region − ∞ < z ≤ 0 {\displaystyle -\infty <z\leq 0\,} , while the fluid with the label 'G' lives in the upper half-plane 0 ≤ z < ∞ {\displaystyle 0\leq z<\infty \,} . To specify the solution fully, it is necessary to fix conditions at the boundaries and interface. This determines the wave speed c , which in turn determines the stability properties of the system.
The first of these conditions is provided by details at the boundary. The perturbation velocities w i ′ {\displaystyle w'_{i}\,} should satisfy a no-flux condition, so that fluid does not leak out at the boundaries z = ± ∞ . {\displaystyle z=\pm \infty .\,} Thus, w L ′ = 0 {\displaystyle w_{L}'=0\,} on z = − ∞ {\displaystyle z=-\infty \,} , and w G ′ = 0 {\displaystyle w_{G}'=0\,} on z = ∞ {\displaystyle z=\infty \,} . In terms of the streamfunction, this is
Ψ L ( − ∞ ) = 0 , Ψ G ( ∞ ) = 0. {\displaystyle \Psi _{L}\left(-\infty \right)=0,\qquad \Psi _{G}\left(\infty \right)=0.\,}
The other three conditions are provided by details at the interface z = η ( x , t ) {\displaystyle z=\eta \left(x,t\right)\,} .
Continuity of vertical velocity: At z = η {\displaystyle z=\eta } , the vertical velocities match, w L ′ = w G ′ {\displaystyle w'_{L}=w'_{G}\,} . Using the stream function representation, this gives
Ψ L ( η ) = Ψ G ( η ) . {\displaystyle \Psi _{L}\left(\eta \right)=\Psi _{G}\left(\eta \right).\,}
Expanding about z = 0 {\displaystyle z=0\,} gives
Ψ L ( 0 ) = Ψ G ( 0 ) + H.O.T. , {\displaystyle \Psi _{L}\left(0\right)=\Psi _{G}\left(0\right)+{\text{H.O.T.}},\,}
where H.O.T. means 'higher-order terms'. This equation is the required interfacial condition.
The free-surface condition: At the free surface z = η ( x , t ) {\displaystyle z=\eta \left(x,t\right)\,} , the kinematic condition holds:
∂ η ∂ t + u ′ ∂ η ∂ x = w ′ ( η ) . {\displaystyle {\frac {\partial \eta }{\partial t}}+u'{\frac {\partial \eta }{\partial x}}=w'\left(\eta \right).\,}
Linearizing, this is simply
∂ η ∂ t = w ′ ( 0 ) , {\displaystyle {\frac {\partial \eta }{\partial t}}=w'\left(0\right),\,}
where the velocity w ′ ( η ) {\displaystyle w'\left(\eta \right)\,} is linearized on to the surface z = 0 {\displaystyle z=0\,} . Using the normal-mode and streamfunction representations, this condition is c η = Ψ {\displaystyle c\eta =\Psi \,} , the second interfacial condition.
Pressure relation across the interface: For the case with surface tension , the pressure difference over the interface at z = η {\displaystyle z=\eta } is given by the Young–Laplace equation:
p G ( z = η ) − p L ( z = η ) = σ κ , {\displaystyle p_{G}\left(z=\eta \right)-p_{L}\left(z=\eta \right)=\sigma \kappa ,\,}
where σ is the surface tension and κ is the curvature of the interface, which in a linear approximation is
κ = ∇ 2 η = η x x . {\displaystyle \kappa =\nabla ^{2}\eta =\eta _{xx}.\,}
Thus,
p G ( z = η ) − p L ( z = η ) = σ η x x . {\displaystyle p_{G}\left(z=\eta \right)-p_{L}\left(z=\eta \right)=\sigma \eta _{xx}.\,}
However, this condition refers to the total pressure (base+perturbed), thus
[ P G ( η ) + p G ′ ( 0 ) ] − [ P L ( η ) + p L ′ ( 0 ) ] = σ η x x . {\displaystyle \left[P_{G}\left(\eta \right)+p'_{G}\left(0\right)\right]-\left[P_{L}\left(\eta \right)+p'_{L}\left(0\right)\right]=\sigma \eta _{xx}.\,}
(As usual, The perturbed quantities can be linearized onto the surface z=0 .) Using hydrostatic balance , in the form
P L = − ρ L g z + p 0 , P G = − ρ G g z + p 0 , {\displaystyle P_{L}=-\rho _{L}gz+p_{0},\qquad P_{G}=-\rho _{G}gz+p_{0},\,}
this becomes
p G ′ − p L ′ = g η ( ρ G − ρ L ) + σ η x x , on z = 0. {\displaystyle p'_{G}-p'_{L}=g\eta \left(\rho _{G}-\rho _{L}\right)+\sigma \eta _{xx},\qquad {\text{on }}z=0.\,}
The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised Euler equations for the perturbations, ∂ u i ′ ∂ t = − 1 ρ i ∂ p i ′ ∂ x {\displaystyle {\frac {\partial u_{i}'}{\partial t}}=-{\frac {1}{\rho _{i}}}{\frac {\partial p_{i}'}{\partial x}}\,} with i = L , G , {\displaystyle i=L,G,\,} to yield
p i ′ = ρ i c D Ψ i , i = L , G . {\displaystyle p_{i}'=\rho _{i}cD\Psi _{i},\qquad i=L,G.\,}
Putting this last equation and the jump condition on p G ′ − p L ′ {\displaystyle p'_{G}-p'_{L}} together,
c ( ρ G D Ψ G − ρ L D Ψ L ) = g η ( ρ G − ρ L ) + σ η x x . {\displaystyle c\left(\rho _{G}D\Psi _{G}-\rho _{L}D\Psi _{L}\right)=g\eta \left(\rho _{G}-\rho _{L}\right)+\sigma \eta _{xx}.\,}
Substituting the second interfacial condition c η = Ψ {\displaystyle c\eta =\Psi \,} and using the normal-mode representation, this relation becomes
c 2 ( ρ G D Ψ G − ρ L D Ψ L ) = g Ψ ( ρ G − ρ L ) − σ α 2 Ψ , {\displaystyle c^{2}\left(\rho _{G}D\Psi _{G}-\rho _{L}D\Psi _{L}\right)=g\Psi \left(\rho _{G}-\rho _{L}\right)-\sigma \alpha ^{2}\Psi ,\,}
where there is no need to label Ψ {\displaystyle \Psi \,} (only its derivatives) because Ψ L = Ψ G {\displaystyle \Psi _{L}=\Psi _{G}\,} at z = 0. {\displaystyle z=0.\,}
Now that the model of stratified flow has been set up, the solution is at hand. The streamfunction equation ( D 2 − α 2 ) Ψ i = 0 , {\displaystyle \left(D^{2}-\alpha ^{2}\right)\Psi _{i}=0,\,} with the boundary conditions Ψ ( ± ∞ ) {\displaystyle \Psi \left(\pm \infty \right)\,} has the solution
Ψ L = A L e α z , Ψ G = A G e − α z . {\displaystyle \Psi _{L}=A_{L}e^{\alpha z},\qquad \Psi _{G}=A_{G}e^{-\alpha z}.\,}
The first interfacial condition states that Ψ L = Ψ G {\displaystyle \Psi _{L}=\Psi _{G}\,} at z = 0 {\displaystyle z=0\,} , which forces A L = A G = A . {\displaystyle A_{L}=A_{G}=A.\,} The third interfacial condition states that
c 2 ( ρ G D Ψ G − ρ L D Ψ L ) = g Ψ ( ρ G − ρ L ) − σ α 2 Ψ . {\displaystyle c^{2}\left(\rho _{G}D\Psi _{G}-\rho _{L}D\Psi _{L}\right)=g\Psi \left(\rho _{G}-\rho _{L}\right)-\sigma \alpha ^{2}\Psi .\,}
Plugging the solution into this equation gives the relation
A c 2 α ( − ρ G − ρ L ) = A g ( ρ G − ρ L ) − σ α 2 A . {\displaystyle Ac^{2}\alpha \left(-\rho _{G}-\rho _{L}\right)=Ag\left(\rho _{G}-\rho _{L}\right)-\sigma \alpha ^{2}A.\,}
The A cancels from both sides and we are left with
c 2 = g α ρ L − ρ G ρ L + ρ G + σ α ρ L + ρ G . {\displaystyle c^{2}={\frac {g}{\alpha }}{\frac {\rho _{L}-\rho _{G}}{\rho _{L}+\rho _{G}}}+{\frac {\sigma \alpha }{\rho _{L}+\rho _{G}}}.\,}
To understand the implications of this result in full, it is helpful to consider the case of zero surface tension. Then,
c 2 = g α ρ L − ρ G ρ L + ρ G , σ = 0 , {\displaystyle c^{2}={\frac {g}{\alpha }}{\frac {\rho _{L}-\rho _{G}}{\rho _{L}+\rho _{G}}},\qquad \sigma =0,\,}
and clearly
Now, when the heavier fluid sits on top, c 2 < 0 {\displaystyle c^{2}<0\,} , and
c = ± i g A α , A = ρ G − ρ L ρ G + ρ L , {\displaystyle c=\pm i{\sqrt {\frac {g{\mathcal {A}}}{\alpha }}},\qquad {\mathcal {A}}={\frac {\rho _{G}-\rho _{L}}{\rho _{G}+\rho _{L}}},\,}
where A {\displaystyle {\mathcal {A}}\,} is the Atwood number . By taking the positive solution, we see that the solution has the form
Ψ ( x , z , t ) = A e − α | z | exp [ i α ( x − c t ) ] = A exp ( α g A ~ α t ) exp ( i α x − α | z | ) {\displaystyle \Psi \left(x,z,t\right)=Ae^{-\alpha |z|}\exp \left[i\alpha \left(x-ct\right)\right]=A\exp \left(\alpha {\sqrt {\frac {g{\tilde {\mathcal {A}}}}{\alpha }}}t\right)\exp \left(i\alpha x-\alpha |z|\right)\,}
and this is associated to the interface position η by: c η = Ψ . {\displaystyle c\eta =\Psi .\,} Now define B = A / c . {\displaystyle B=A/c.\,}
When the two layers of the fluid are allowed to have a relative velocity, the instability is generalized to the Kelvin–Helmholtz–Rayleigh–Taylor instability, which includes both the Kelvin–Helmholtz instability and the Rayleigh–Taylor instability as special cases. It was recently discovered that the fluid equations governing the linear dynamics of the system admit a parity-time symmetry , and the Kelvin–Helmholtz–Rayleigh–Taylor instability occurs when and only when the parity-time symmetry breaks spontaneously. [ 19 ]
The RT instability can be seen as the result of baroclinic torque created by the misalignment of the pressure and density gradients at the perturbed interface, as described by the two-dimensional inviscid vorticity equation, D ω D t = 1 ρ 2 ∇ ρ × ∇ p {\displaystyle {\frac {D\omega }{Dt}}={\frac {1}{\rho ^{2}}}\nabla \rho \times \nabla p} , where ω is vorticity, ρ density and p is the pressure. In this case the dominant pressure gradient is hydrostatic , resulting from the acceleration.
When in the unstable configuration, for a particular harmonic component of the initial perturbation, the torque on the interface creates vorticity that will tend to increase the misalignment of the gradient vectors . This in turn creates additional vorticity, leading to further misalignment. This concept is depicted in the figure, where it is observed that the two counter-rotating vortices have velocity fields that sum at the peak and trough of the perturbed interface. In the stable configuration, the vorticity, and thus the induced velocity field, will be in a direction that decreases the misalignment and therefore stabilizes the system. [ 17 ] [ 20 ]
A much simpler explanation of the basic physics of the Rayleigh–Taylor instability was published in 2006. [ 21 ]
The analysis in the previous section breaks down when the amplitude of the perturbation is large. The growth then becomes non-linear as the spikes and bubbles of the instability tangle and roll up into vortices. Then, as in the figure, numerical simulation of the full problem is required to describe the system. | https://en.wikipedia.org/wiki/Rayleigh–Taylor_instability |
Raymond Clare Archibald (7 October 1875 – 26 July 1955) was a prominent Canadian-American mathematician. He is known for his work as a historian of mathematics, his editorships of mathematical journals and his contributions to the teaching of mathematics. [ 1 ]
Raymond Clare Archibald was born in South Branch, Stewiacke, Nova Scotia on 7 October 1875. He was the son of Abram Newcomb Archibald (1849–1883) and Mary Mellish Archibald (1849–1901). He was the fourth cousin twice removed of the famous Canadian-American astronomer and mathematician Simon Newcomb . [ 2 ]
Archibald graduated in 1894 from Mount Allison College with B.A. degree in mathematics and teacher's certificate in violin. After teaching mathematics and violin for a year at the Mount Allison Ladies' College he went to Harvard where he received a B.A. 1896 and a M.A. in 1897. He then traveled to Europe where he attended the Humboldt University of Berlin during 1898 and received a Ph.D. cum laude from the University of Strasbourg in 1900. His advisor was Karl Theodor Reye and title of his dissertation was The Cardioide and Some of its Related Curves.
He returned to Canada in 1900 and taught mathematics and violin at the Mount Allison Ladies' College until 1907. After a one-year appointment at Acadia University he accepted an invitation of join the mathematics department at Brown University . He stayed at Brown for the rest of his career becoming a Professor Emeritus in 1943. While at Brown he created one of the finest mathematical libraries in the western hemisphere. [ 3 ]
Archibald returned to Mount Allison in 1954 to curate the Mary Mellish Archibald Memorial Library, the library he had founded in 1905 to honor his mother. At his death the library contained 23,000 volumes, 2,700 records, and 70,000 songs in American and English poetry and drama. [ 4 ]
Raymond Clare Archibald was a world-renowned historian of mathematics with a lifelong concern for the teaching of mathematics in secondary schools. At the presentation of his portrait to Brown University the head of the mathematics department, Professor Clarence Raymond Adams said of him:
"The instincts of the bibliophile were also his from early years. Possessing a passion for accurate detail, systematic by nature and blessed with a memory that was the marvel of his friends, he gradually acquired a knowledge of mathematical books and their values which has scarcely been equalled. This knowledge and an untiring energy he dedicated to the upbuilding of the mathematical library at Brown University . From modest beginnings he has developed this essential equipment of the mathematical investigator to a point where it has no superior, in completeness and in convenience for the user."
Archibald received honorary degrees from the University of Padua (LL.D., 1922), Mount Allison University (LL.D., 1923) and from Brown University (M.A. ad eundem, 1943). [ 5 ]
Archibald's bibliography contains over 1,000 entries. [ 7 ] He contributed to over 20 different journals, mathematical, scientific, educational and literary. The following are the books of which he is an author: | https://en.wikipedia.org/wiki/Raymond_C._Archibald |
Raymond Daudel (2 February 1920 [ 1 ] [ 2 ] [ 3 ] – 20 June 2006 [ 3 ] ) was a French theoretical and quantum chemist .
Trained as a physicist, he was an assistant to Irène Joliot-Curie at the Radium Institute . [ 1 ] Daudel spent almost the entirety of his career as professor at the Sorbonne and director of a laboratory of the Centre National de la Recherche Scientifique (CNRS). He is quoted as saying that the latter "was much better because the CNRS was very rich". [ 1 ] This allowed Daudel to attract many co-workers from elsewhere in France and internationally.
Raymond Daudel was Officier de la Légion d'honneur and Officier de l' Ordre National du Mérite . [ 2 ] He served as President of the European Academy of Arts Sciences and Humanities , [ 2 ] in Paris, France. Daudel was a founding member and Honorary President of the International Academy of Quantum Molecular Science . [ 3 ]
An author as well as an academic, Raymond Daudel authored several books, including Quantum chemistry , originally with R. Lefebyre and C. Moser in 1959 (Interscience Publishers, Inc., New York) and later with G. Leroy, D. Peeters, and M. Sana, published by Wiley in 1983. [ 2 ] He was responsible for the organization of the first International Congress in Quantum Chemistry , held in Menton , France in 1973. | https://en.wikipedia.org/wiki/Raymond_Daudel |
Raymond K. Sheline (March 31, 1922 – February 10, 2016) was a member of the Manhattan Project and spent much of his career as a professor in chemistry and physics at Florida State University . [ 1 ] Sheline's research focused on spectroscopic studies of atomic nuclei and molecular structures.
Sheline was born in Port Clinton, Ohio and a graduate of Woodward High School . [ 2 ] He studied at Bethany College in West Virginia , where he graduated in 1943. [ 3 ] From 1943 till 1945, he worked on the Manhattan Project as a chemist at Columbia University . Sheline went to graduate school at University of California, Berkeley after World War II and obtained his PhD in chemistry there in 1949 under the supervision of Kenneth Pitzer . His PhD thesis dealt with vibrational spectroscopy of polyatomic molecules .
Sheline taught at the University of Chicago for two years after his PhD. From 1951 to 1999, Sheline was a professor in chemistry and physics at Florida State University. Between 1966 and 1967, he was named Robert O. Lawton Distinguished Professor. [ 4 ]
Sheline was a three-time Guggenheim fellow [ 5 ] and a Fulbright scholar. [ 6 ]
Sheline married Yvonne Sheline in 1951, they have seven children. [ 7 ] | https://en.wikipedia.org/wiki/Raymond_Sheline |
The Ray–Dutt twist is a mechanism proposed for the racemization of octahedral complexes containing three bidentate chelate rings. Such complexes typically adopt an octahedral molecular geometry in their ground states , in which case they possess helical chirality . The pathway entails formation of an intermediate of C 2v point group symmetry . [ 1 ] An alternative pathway that also does not break any metal-ligand bonds is called the Bailar twist . Both of these mechanism product complexes wherein the ligating atoms (X in the scheme) are arranged in an approximate trigonal prism.
This pathway is called the Ray–Dutt twist in honor of Priyadaranjan Ray (not Prafulla Chandra Ray ) and N. K. Dutt, inorganic chemists at the Indian Association for the Cultivation of Science abbr. IACS [ 2 ] who proposed this process. [ 3 ] [ 4 ] [ 5 ] [ 6 ]
This stereochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ray–Dutt_twist |
Rubidium oxide is the chemical compound with the formula Rb 2 O . Rubidium oxide is highly reactive towards water, and therefore it would not be expected to occur naturally. The rubidium content in minerals is often calculated and quoted in terms of Rb 2 O . In reality, the rubidium is typically present as a component of (actually, an impurity in) silicate or aluminosilicate . A major source of rubidium is lepidolite , KLi 2 Al(Al,Si) 3 O 10 (F,OH) 2 , wherein Rb sometimes replaces K.
Rb 2 O is a yellow colored solid. The related species Na 2 O, K 2 O , and Cs 2 O are colorless, pale-yellow, and orange, respectively.
The alkali metal oxides M 2 O (M = Li, Na, K, Rb) crystallise in the antifluorite structure. In the antifluorite motif, the positions of the anions and cations are reversed relative to their positions in CaF 2 , with rubidium ions 4-coordinate (tetrahedral) and oxide ions 8-coordinate (cubic). [ 1 ]
Like other alkali metal oxides, Rb 2 O is a strong base . Thus, Rb 2 O reacts exothermically with water to form rubidium hydroxide .
So reactive is Rb 2 O toward water that it is considered hygroscopic . Upon heating, Rb 2 O reacts with hydrogen to rubidium hydroxide and rubidium hydride : [ 2 ]
For laboratory use, RbOH is usually used in place of the oxide. RbOH can be purchased for ca. US$ 5/g (2006). The hydroxide is more useful, less reactive toward atmospheric moisture, and less expensive than the oxide.
As for most alkali metal oxides, [ 3 ] the best synthesis of Rb 2 O does not entail oxidation of the metal but reduction of the anhydrous nitrate:
Typical for alkali metal hydroxides, RbOH cannot be dehydrated to the oxide. Instead, the hydroxide can be decomposed to the oxide (by reduction of the hydrogen ion) using Rb metal:
Metallic Rb reacts with O 2 , as indicated by its tendency to rapidly tarnish in air. The tarnishing process is relatively colorful as it proceeds via bronze-colored Rb 6 O and copper-colored Rb 9 O 2 . [ 4 ] The suboxides of rubidium that have been characterized by X-ray crystallography include Rb 9 O 2 and Rb 6 O, as well as the mixed Cs -Rb suboxides Cs 11 O 3 Rb n ( n = 1, 2, 3). [ 5 ]
The final product of oxygenation of Rb is principally RbO 2 , rubidium superoxide :
This superoxide can then be reduced to Rb 2 O using excess rubidium metal: | https://en.wikipedia.org/wiki/Rb2O |
Rubidium sesquioxide is a chemical compound with the formula Rb 2 O 3 or more accurately Rb 4 O 6 . In terms of oxidation states , Rubidium in this compound has a nominal charge of +1, and the oxygen is a mixed peroxide ( O 2− 2 ) and superoxide ( O − 2 ) for a structural formula of (Rb + ) 4 (O − 2 ) 2 (O 2− 2 ) . [ 4 ] It has been studied theoretically as an example of a strongly correlated material . [ 5 ]
The compound was predicted to be a rare example of a ferromagnetic compound that is magnetic due to a p-block element , [ 6 ] and a half-metal that was conducting in the minority spin band. [ 7 ] However, while the material does have exotic magnetic behavior, experimental results instead showed an electrically insulating magnetically frustrated system . [ 1 ] [ 5 ] Rb 4 O 6 also displays a Verwey transition where charge ordering appears at 290 K. [ 8 ]
Rubidium sesquioxide can be prepared by reacting the peroxide Rb 2 O 2 and the superoxide RbO 2 : [ 2 ]
It is initially discovered in 1907, [ 9 ] [ 3 ] and more thoroughly characterized in 1939. [ 10 ] The compound crystallizes in a body-centered cubic form with the same crystal structure as Pu 2 C 3 and Cs 4 O 6 . [ 10 ] [ 3 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rb2O3 |
Rubidium sulfide is an inorganic compound and a salt with the chemical formula Rb 2 S. It is a white solid with similar properties to other alkali metal sulfides .
By dissolving hydrogen sulfide into rubidium hydroxide solution, it will produce rubidium bisulfide , followed by rubidium sulfide. [ 3 ] [ 4 ]
Rubidium sulfide has a cubic crystal similar to lithium sulfide , sodium sulfide and potassium sulfide , known as the anti-fluorite structure. Their space groups are F m 3 ¯ m {\displaystyle Fm{\bar {3}}m} . Rubidium sulfide has a crystal lattice unit cell dimension of = 765.0 pm . [ 1 ]
Rubidium sulfide reacts with sulfur in hydrogen gas to form rubidium pentasulfide , Rb 2 S 5 . [ 4 ] [ 5 ] | https://en.wikipedia.org/wiki/Rb2S |
Rubidium selenide is an inorganic compound composed of selenium and rubidium . It is a selenide with a chemical formula of Rb 2 Se. Rubidium selenide is used together with caesium selenide in photovoltaic cells . [ 5 ]
Rubidium selenide can be prepared by reacting mercury selenide and metallic rubidium. [ 6 ] The elements can be synthesized in liquid ammonia . [ 7 ]
Hydrogen selenide can also be dissolved in an aqueous solution of rubidium hydroxide to eventually form rubidium selenide. [ 8 ] This method is similar to the method for preparing rubidium sulfide , because they are both chalcogenide compounds.
Rubidium selenide has cubic crystal structure , which belongs to the antifluorite structure, and the space group is F m 3 ¯ m {\displaystyle Fm{\bar {3}}m} and the lattice parameters are a=801.0 pm, per unit. The unit cell has 4 units. [ 1 ] | https://en.wikipedia.org/wiki/Rb2Se |
Rubidium telluride is the inorganic compound with the formula Rb 2 Te . It is a yellow-green powder that melts at either 775 °C or 880 °C (two different values have been reported). It is an obscure material of minor academic interest. [ 1 ]
Like other alkali metal chalcogenides, Rb 2 Te is prepared from the elements in liquid ammonia . [ 2 ]
Rubidium telluride is used in some space-based UV detectors. [ citation needed ]
The compound has several polymorphs . At room temperature, ω-Rb 2 Te is a metastable antiflourite type structure , and transforms to α-Rb 2 Te upon heating, which is a PbCl 2 type structure. [ 3 ] | https://en.wikipedia.org/wiki/Rb2Te |
Rubidium sesquioxide is a chemical compound with the formula Rb 2 O 3 or more accurately Rb 4 O 6 . In terms of oxidation states , Rubidium in this compound has a nominal charge of +1, and the oxygen is a mixed peroxide ( O 2− 2 ) and superoxide ( O − 2 ) for a structural formula of (Rb + ) 4 (O − 2 ) 2 (O 2− 2 ) . [ 4 ] It has been studied theoretically as an example of a strongly correlated material . [ 5 ]
The compound was predicted to be a rare example of a ferromagnetic compound that is magnetic due to a p-block element , [ 6 ] and a half-metal that was conducting in the minority spin band. [ 7 ] However, while the material does have exotic magnetic behavior, experimental results instead showed an electrically insulating magnetically frustrated system . [ 1 ] [ 5 ] Rb 4 O 6 also displays a Verwey transition where charge ordering appears at 290 K. [ 8 ]
Rubidium sesquioxide can be prepared by reacting the peroxide Rb 2 O 2 and the superoxide RbO 2 : [ 2 ]
It is initially discovered in 1907, [ 9 ] [ 3 ] and more thoroughly characterized in 1939. [ 10 ] The compound crystallizes in a body-centered cubic form with the same crystal structure as Pu 2 C 3 and Cs 4 O 6 . [ 10 ] [ 3 ]
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Rb4O6 |
Rubidium oxide is the chemical compound with the formula Rb 2 O . Rubidium oxide is highly reactive towards water, and therefore it would not be expected to occur naturally. The rubidium content in minerals is often calculated and quoted in terms of Rb 2 O . In reality, the rubidium is typically present as a component of (actually, an impurity in) silicate or aluminosilicate . A major source of rubidium is lepidolite , KLi 2 Al(Al,Si) 3 O 10 (F,OH) 2 , wherein Rb sometimes replaces K.
Rb 2 O is a yellow colored solid. The related species Na 2 O, K 2 O , and Cs 2 O are colorless, pale-yellow, and orange, respectively.
The alkali metal oxides M 2 O (M = Li, Na, K, Rb) crystallise in the antifluorite structure. In the antifluorite motif, the positions of the anions and cations are reversed relative to their positions in CaF 2 , with rubidium ions 4-coordinate (tetrahedral) and oxide ions 8-coordinate (cubic). [ 1 ]
Like other alkali metal oxides, Rb 2 O is a strong base . Thus, Rb 2 O reacts exothermically with water to form rubidium hydroxide .
So reactive is Rb 2 O toward water that it is considered hygroscopic . Upon heating, Rb 2 O reacts with hydrogen to rubidium hydroxide and rubidium hydride : [ 2 ]
For laboratory use, RbOH is usually used in place of the oxide. RbOH can be purchased for ca. US$ 5/g (2006). The hydroxide is more useful, less reactive toward atmospheric moisture, and less expensive than the oxide.
As for most alkali metal oxides, [ 3 ] the best synthesis of Rb 2 O does not entail oxidation of the metal but reduction of the anhydrous nitrate:
Typical for alkali metal hydroxides, RbOH cannot be dehydrated to the oxide. Instead, the hydroxide can be decomposed to the oxide (by reduction of the hydrogen ion) using Rb metal:
Metallic Rb reacts with O 2 , as indicated by its tendency to rapidly tarnish in air. The tarnishing process is relatively colorful as it proceeds via bronze-colored Rb 6 O and copper-colored Rb 9 O 2 . [ 4 ] The suboxides of rubidium that have been characterized by X-ray crystallography include Rb 9 O 2 and Rb 6 O, as well as the mixed Cs -Rb suboxides Cs 11 O 3 Rb n ( n = 1, 2, 3). [ 5 ]
The final product of oxygenation of Rb is principally RbO 2 , rubidium superoxide :
This superoxide can then be reduced to Rb 2 O using excess rubidium metal: | https://en.wikipedia.org/wiki/Rb9O2 |
Rubidium bromide is an inorganic compound with the chemical formula Rb Br . It is a salt of hydrogen bromide . It consists of bromide anions Br − and rubidium cations Rb + . It has a NaCl crystal structure, with a lattice constant of 685 picometres. [ 1 ]
There are several methods for synthesising rubidium bromide. One involves reacting rubidium hydroxide with hydrobromic acid :
Another method is to neutralize rubidium carbonate with hydrobromic acid:
Rubidium metal would react directly with bromine to form RbBr, but this is not a sensible production method, since rubidium metal is substantially more expensive than the carbonate or hydroxide; moreover, the reaction would be explosive.
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/RbBr |
Rubidium chloride is the chemical compound with the formula RbCl. This alkali metal halide salt is composed of rubidium and chlorine , and finds diverse uses ranging from electrochemistry to molecular biology .
In its gas phase, RbCl is diatomic with a bond length estimated at 2.7868 Å. [ 1 ] This distance increases to 3.285 Å for cubic RbCl, reflecting the higher coordination number of the ions in the solid phase. [ 2 ]
Depending on conditions, solid RbCl exists in one of three arrangements or polymorphs as determined with holographic imaging: [ 3 ]
The sodium chloride (NaCl) polymorph is most common. A cubic close-packed arrangement of chloride anions with rubidium cations filling the octahedral holes describes this polymorph. [ 4 ] Both ions are six-coordinate in this arrangement. The lattice energy of this polymorph is only 3.2 kJ/mol less than the following structure's. [ 5 ]
At high temperature and pressure, RbCl adopts the caesium chloride (CsCl) structure (NaCl and KCl undergo the same structural change at high pressures). Here, the chloride ions form a simple cubic arrangement with chloride anions occupying the vertices of a cube surrounding a central Rb + . This is RbCl's densest packing motif. [ 2 ] Because a cube has eight vertices, both ions' coordination numbers equal eight. This is RbCl's highest possible coordination number. Therefore, according to the radius ratio rule, cations in this polymorph will reach their largest apparent radius because the anion-cation distances are greatest. [ 4 ]
The sphalerite polymorph of rubidium chloride has not been observed experimentally. This is consistent with the theory; the lattice energy is predicted to be nearly 40.0 kJ/mol smaller in magnitude than those of the preceding structures. [ 5 ]
The most common preparation of pure rubidium chloride involves the reaction of its hydroxide with hydrochloric acid , followed by recrystallization : [ 6 ]
Because RbCl is hygroscopic , it must be protected from atmospheric moisture, e.g. using a desiccator . RbCl is primarily used in laboratories. Therefore, numerous suppliers (see below) produce it in smaller quantities as needed. It is offered in a variety of forms for chemical and biomedical research.
Rubidium chloride reacts with sulfuric acid to give rubidium hydrogen sulfate .
Every 18 mg of rubidium chloride is equivalent to approximately one banana equivalent dose due to the large fraction (27.8%) of naturally-occurring radioactive isotope rubidium-87 . | https://en.wikipedia.org/wiki/RbCl |
Rubidium perchlorate , RbClO 4 , is the perchlorate of rubidium . It is an oxidizing agent , as are all perchlorates .
Rubidium perchlorate can be obtained through the careful heating of a rubidium chlorate solution, leading to a disproportionation reaction with the release of oxygen gas: [ 3 ]
When heated, it decomposes into the chloride and oxygen: [ 4 ]
It has two polymorphs . Below 279 °C, it crystallizes in orthorhombic crystal system with lattice constants a = 0.927 nm, b = 0.581 nm, c = 0.753 nm. Over 279 °C, it has a cubic structure with lattice constant a = 0.770 nm. [ 1 ]
Table of solubility in water: [ 1 ] | https://en.wikipedia.org/wiki/RbClO4 |
Rubidium fluoride (RbF) is the fluoride salt of rubidium . It is a cubic crystal with rock-salt structure.
There are several methods for synthesising rubidium fluoride. One involves reacting rubidium hydroxide with hydrofluoric acid : [ 1 ]
Another method is to neutralize rubidium carbonate with hydrofluoric acid: [ 1 ]
Another possible method is to react rubidium hydroxide with ammonium fluoride :
The least used method due to expense of rubidium metal is to react it directly with fluorine gas, as rubidium reacts violently with halogens : [ 1 ]
Rubidium fluoride is a white crystalline substance with a cubic crystal structure that looks very similar to common salt (NaCl). The crystals belong to the space group Fm3m (space group no. 225) with the lattice parameter a = 565 pm and four formula units per unit cell . [ 2 ] The refractive index of the crystals is nD = 1.398. [ 2 ] Rubidium fluoride colors a flame (Bunsen burner flame) purple or magenta red (spectral analysis).
Rubidium fluoride forms two different hydrates, a sesquihydrate with the stoichiometric composition 2RbF·3H 2 O and a third hydrate with the composition 3RbF·H 2 O. [ 3 ]
In addition to simple rubidium fluoride, an acidic rubidium fluoride with the molecular formula HRbF 2 is also known, [ 4 ] which can be produced by reacting rubidium fluoride and hydrogen fluoride. [ 4 ] The compounds H 2 RbF 3 and H 3 RbF 4 were also synthesized. [ 5 ] [ 4 ]
The solubility in acetone is 0.0036 g/kg at 18 °C and 0.0039 g/kg at 37 °C. [ 6 ]
The standard enthalpy of formation of rubidium fluoride is Δ f H 0 298 = −552.2 kJ mol−1, [ 7 ] the standard free enthalpy of formation ΔG 0 298 = −520.4 kJ mol−1, [ 7 ] and the standard molar entropy S 0 298 = 113.9 J K −1 ·mol−1. [ 7 ] The enthalpy of solution of rubidium fluoride was determined to be −24.28 kJ/mol. [ 8 ] | https://en.wikipedia.org/wiki/RbF |
Rubidium nitrate is an inorganic compound with the formula Rb NO 3 . This alkali metal nitrate salt is white and highly soluble in water.
Rubidium nitrate is a white crystalline powder that is highly soluble in water and very slightly soluble in acetone . In a flame test , RbNO 3 gives a mauve/light purple colour.
Rubidium compounds have very few applications. [ 1 ] Like caesium nitrate , it is used in infrared radiation optics, in pyrotechnic compositions as a pyrotechnic colorant and as an oxidizer , e.g. in decoys and illumination flares although it is rarely used in fireworks to produce a red-violet colour. It is also used as a raw material for preparation of other rubidium compounds and rubidium metal, for manufacture of catalysts and in scintillation counters .
RbNO 3 can be prepared either by dissolving rubidium metal, its hydroxide or carbonate in nitric acid. | https://en.wikipedia.org/wiki/RbNO3 |
Rubidium superoxide or rubidium hyperoxide is a chemical compound with the chemical formula RbO 2 . In terms of oxidation states , the negatively charged superoxide and positively charged rubidium give it a structural formula of Rb + [O 2 ] − . [ 2 ]
It can be created by slowly exposing elemental rubidium to oxygen gas: [ 3 ]
Like other alkali metal hyperoxides, crystals can also be grown in liquid ammonia . [ 4 ]
Between 280 and 360 °C, Rubidium superoxide will decompose , leaving not rubidium sesquioxide ( Rb 2 O 3 ), but rather rubidium peroxide ( Rb 2 O 2 ). [ 3 ]
An even more oxygen rich compound, that of rubidium ozonide ( RbO 3 ) can be created using RbO 2 . [ 5 ]
Roughly speaking, RbO 2 has a crystal structure similar to tetragonal calcium carbide , but is rather distorted due to the Jahn–Teller effect , which makes the crystal structure less symmetrical. [ 2 ]
RbO 2 is stable in dry air, but is extremely hygroscopic . [ 3 ]
The compound has been studied as an example of magnetism arising intrinsically from the p-shell . [ 6 ] RbO 2 has been predicted to be a paramagnetic Mott insulator . [ 7 ] At low temperatures, it transitions to antiferromagnetic order, with a Neel temperature of 15 K. [ 2 ] | https://en.wikipedia.org/wiki/RbO2 |
Re-Pair (short for recursive pairing ) is a grammar-based compression algorithm that, given an input text, builds a straight-line program , i.e. a context-free grammar generating a single string: the input text. In order to perform the compression in linear time, it consumes the amount of memory that is approximately five times the size of its input.
The grammar is built by recursively replacing the most frequent pair of characters occurring in the text.
Once there is no pair of characters occurring twice, the resulting string is used as the axiom of the grammar.
Therefore, the output grammar is such that all rules but the axiom have two symbols on the right-hand side.
Re-Pair was first introduced by NJ. Larsson and A. Moffat [ 1 ] in 1999.
In their paper the algorithm is presented together with a detailed description of the data structures required to implement it with linear time and space complexity. The experiments showed that Re-Pair achieves high compression ratios and offers good performance for decompression. However, the major drawback of the algorithm is its memory consumption, which is approximately 5 times the size of the input. Such memory usage is required in order to perform the compression in linear time but makes the algorithm impractical for compressing large files.
The image on the right shows how the algorithm works compresses the string w = x a b c a b c y 123123 z a b c {\displaystyle w=xabcabcy123123zabc} .
During the first iteration, the pair a b {\displaystyle ab} , which occurs three times in w {\displaystyle w} , is replaced by a new symbol R 1 {\displaystyle R_{1}} .
On the second iteration, the most frequent pair in the string w = x R 1 c R 1 c y 123123 z R 1 c {\displaystyle w=xR_{1}cR_{1}cy123123zR_{1}c} , which is R 1 c {\displaystyle R_{1}c} , is replaced by a new symbol R 2 {\displaystyle R_{2}} .
Thus, at the end of the second iteration, the remaining string is w = x R 2 R 2 y 123123 z R 2 {\displaystyle w=xR_{2}R_{2}y123123zR_{2}} .
In the next two iterations, the pairs 12 {\displaystyle 12} and R 3 3 {\displaystyle R_{3}3} are replaced by symbols R 3 {\displaystyle R_{3}} and R 4 {\displaystyle R_{4}} respectively.
Finally, the string w = x R 2 R 2 y R 4 R 4 z R 2 {\displaystyle w=xR_{2}R_{2}yR_{4}R_{4}zR_{2}} contains no repeated pair and therefore it is used as the axiom of the output grammar.
In order to achieve linear time complexity, Re-Pair requires the following data structures
Since the hash table and the priority queue refer to the same elements (pairs), they can be implemented by a common data structure called PAIR with pointers for the hash table (h_next) and the priority queue (p_next and p_prev). Furthermore, each PAIR points to the beginning of the first (f_pos) and the last (b_pos) occurrences of the string represented by the PAIR in the sequence. The following picture shows an overview of this data structure.
The following two pictures show an example of how these data structures look after the initialization and after applying one step of the pairing process (pointers to NULL are not displayed):
Once the grammar has been built for a given input string, in order to achieve effective compression, this grammar has to be encoded efficiently.
One of the simplest methods for encoding the grammar is the implicit encoding , which consists on invoking function encodeCFG(X) , described below, sequentially on all the axiom's symbols.
Intuitively, rules are encoded as they are visited in a depth-first traversal of the grammar. The first time a rule is visited, its right hand side is encoded recursively and a new code is assigned to the rule. From that point, whenever the rule is reached, the assigned value is written.
Another possibility is to separate the rules of the grammar into generations such that a rule X → Y Z {\displaystyle X\to YZ} belongs to generation i {\displaystyle i} iff at least one of Y {\displaystyle Y} or Z {\displaystyle Z} belongs to generation i − 1 {\displaystyle i-1} and the other belongs to generation j {\displaystyle j} with j ≤ i − 1 {\displaystyle j\leq i-1} . Then these generations are encoded subsequently starting from generation 0 {\displaystyle 0} . This was the method proposed originally when Re-Pair was first introduced. However, most implementations of Re-Pair use the implicit encoding method due to its simplicity and good performance. Furthermore, it allows on-the-fly decompression.
There exists a number of different implementations of Re-Pair. Each of these versions aims at improving one specific aspect of the algorithm, such as reducing the runtime, reducing the space consumption or increasing the compression ratio. | https://en.wikipedia.org/wiki/Re-Pair |
Re-amping is a process often used in multitrack recording in which a recorded signal is routed back out of the editing environment and run through external processing using effects units and then into a guitar amplifier and a guitar speaker cabinet or a reverb chamber. Originally, the technique was used mostly for electric guitars : it facilitates a separation of guitar playing from guitar amplifier processing—a previously recorded audio program is played back and re-recorded at a later time for the purpose of adding effects, ambiance such as reverb or echo, and the tone shaping imbued by certain amps and cabinets. The technique has since evolved over the 2000s to include many other applications. Re-amping can also be applied to other instruments and program, such as recorded drums, synthesizers, and virtual instruments .
Examples of common re-amping objectives include taking a pre-recorded electric guitar track and adding musically pleasing amplifier distortion/overdrive , room tone such as reverb , audio compression , EQ/filters, envelope followers , resonance, and gating. Re-amping is often used to "warm up" dry tracks, which often means adding complex, musically interesting effects. By playing a dry signal through a studio's main monitors and then using room mics to capture the ambiance, engineers are able to create realistic reverbs and blend the "wet" (modified) signal with the original dry recorded sound to achieve the desired amount of depth.
The technique is especially useful for "softening" stereo drum tracks. By pointing the monitors away from each other and miking each speaker individually, the stereo image can be well preserved and a new sense of "depth" can be added to the track. It is important for audio engineers to check that the microphones being used are in phase to avoid problems with the mix.
An electric guitarist records a dry, unprocessed, unaffected track in a recording studio. This is often achieved by connecting the guitar into a DI unit (a Direct Input or Direct Inject buffer box) that is fed to a recording console or, alternatively, bypassing the console by using an outboard preamplifier. Often, the guitarist's signal is sent to both recorder and guitar amp simultaneously, providing the guitarist with a proper amplifier "feel" while they are playing while also tracking (recording) a dry (un-effects processed) signal.
Later, the dry, direct, unprocessed guitar recording is fed to a bridging device (a Reamp unit or reverse DI box) to "re-record" the guitarist's unprocessed performance through a dedicated guitar amplifier cabinet and/or external effects units. The guitar amplifier is placed in the live room or isolation booth of the recording studio and is set up to produce the desired tonal quality, including distortion character and room reverberation. A microphone is placed near the guitar speaker and a new track is recorded, producing the re-amplified, processed track. The microphone cable is connected to the mixing console or mic preamp using a cable, as usual, without using a bridging device.
External effects such as stomp boxes and guitar multi-fx processors can also be included in the re-amping process. As well as physical devices that require an impedance-matched guitar pickup signal, software-based virtual guitar effects and amps can be included in the re-amping process.
Re-amping allows guitarists and other electronic musicians to record their tracks and go home, leaving the engineer and producer to spend more time dialing in "just right" settings and effects on prerecorded tracks. When re-amping electric guitar tracks, the guitarist need not be present for the engineer to experiment with a range of effects, mic positions, speaker cabinets, amplifiers, effects pedals, and overall tonality – continuously replaying the prerecorded tracks while experimenting with new settings and tones. When a desired tone is finally achieved, the guitarist's dry performance is re-recorded, or "re-amped," with all added effects.
Manufacturers of instrument processing gear such as guitar effects, or equipment reviewers, can gather a library of dry performance tracks, performed and edited well, and then run these ideal tracks through the processing gear to demonstrate the sounds that the processing gear can produce. An unlimited number of performance playback passes, including looping, enables trying out many combinations of settings quickly, including microphone techniques. When guitar amp or amp simulator designers try various circuit component values or settings, they can use the dry tracks as prepared, always available input test signals, and consistent reference signals.
Another advantage of re-amping is that it enables producers and band members to have more options for remixing and redoing a recording a long time after the original recording. If the original recording of a song with electric guitar is done in 1985, and the electric guitarist's sound was only recorded with a mic in front of their combo amp or speaker cabinet, the recording will lock in the specific type of distortion, reverb, flanger processing and other effects that were used, which might sound "dated" several decades later. A producer who is tasked with doing a remix of this song 30 years later cannot "undo" or remove the distortion, reverb, chorus or flanger effects, and so there are limited options for remixing the guitar sound. On the other hand, if the engineering and production team in 1985 had simultaneously tracked (recorded) a "dry", DI out signal from the electric guitar's pickups, a producer remixing the song 30 years later could take the dry guitar signal and re-amp it through 2000s-era digital effects and speaker systems, giving a new sound to this 1985 track.
Direct inject (DI) is a device for connecting an unbalanced, high-impedance, low-level signal (commonly a guitar pickup's signal) into audio equipment designed for a low-impedance balanced signal (such as a DAW ) or audio consoles . Reverse-DI means running this same device or technique in reverse – connecting a high-level (typically balanced, low-impedance) signal into audio equipment that was designed for low-level, unbalanced, high-impedance signals, such as a guitar amplifier.
Playing back a signal from recording studio equipment directly into a guitar amplifier can cause unwanted side-effects such as input-stage distortion, treble loss or overemphasis, and ground-loop hum; thus there is sometimes a need for impedance conversion , level-matching, and ground alteration. Like running a guitar signal through a guitar effects pedal that is set to Bypass, re-amping introduces some degree of sonic degradation compared to playing a guitar live directly into a guitar amp rig.
A re-amping device commonly employs a reversed Direct Inject (DI) transformer with some resistors added for level and impedance shift. Level and impedance adjustment can be achieved by adding a potentiometer or adjustable resistor. A proper re-amping device converts a balanced signal to an unbalanced signal, reduces a high studio-level (line-level) signal down to a low guitar-level signal, and shifts the output to a high instrument-level impedance (typically a guitar pickup impedance).
In conventional re-amplification, a dry recorded signal is sent into a balanced XLR input. An unbalanced ¼" (Tip-Sleeve) phone connector is typically used for the output, which is connected to the guitar amp rig. Some re-amping devices offer a pad (attenuator) switch to reduce a too-hot output level. Sometimes a guitar volume pedal or buffered effects pedal can work adequately for re-amping, depending on grounding, levels, and impedance. Another approach to simulating the high impedance of a guitar pickup is to use a passive DI and add a 10 K-ohm resistor in series with the signal connection inside a 1/4" plug.
While "reverse DI" re-recording techniques have been used for decades, the process was popularized in part by the introduction of the Reamp device in 1993. The registered trademark "Reamp" describes a patented invention ( U.S. patent 6,005,950 ) filed in 1994 by audio engineer John Cuniberti , perhaps best known for his lifelong engineering work with guitarist Joe Satriani . The Reamp inductively couples balanced line-level sources into unbalanced guitar-level destinations (e.g., DAW output to guitar amp input) and includes a potentiometer that alters both signal level and source impedance. Derivations of the Reamp trademark, such as "reamping" and "re-amplification," have become common terminology in professional audio to describe the process of amplified re-recording – much like the word "Band-aid" is often used to describe adhesive bandages (see Genericized trademark ).
The process of re-recording has been used throughout the history of recording studios. Pierre Schaeffer in the 1930s and 1940s used recorded sounds, such as trains, and played them back with ambient alteration, re-recording the net result. Karlheinz Stockhausen and Edgard Varèse later used similar techniques. [ 1 ] Les Paul and Mary Ford recorded layered vocal harmonies and guitar parts, modifying prior tracks with effects such as ambient reverb while recording the net result together on a new track. Les Paul placed a loudspeaker at one end of a tunnel and a microphone at the other end. The loudspeaker played back previously recorded material - the microphone recorded the resulting altered sound.
Roger Nichols claims to have used a guitar re-recording process (not reverse DI) in 1968, partly to spread the stress on cranked tube amps across multiple amps, one at a time. A sound would be dialed in for several hours on one cranked guitar amplifier, and if this stress audibly wore down the amplifier components, another amplifier would be used to record the remaining work. It has been noted that Phil Spector , re-mixing the original Beatles’ Let It Be master tapes in 1970, may have re-recorded dry electric guitar program through a guitar amplifier.
Film sound re-recording is a time-honored practice. Sound designer Walter Murch is known for a technique called "worldizing" in which "real world" ambiance is added, via re-recording, to dry recorded program. Sound designer Nick Peck describes the worldizing process: "Place a speaker in a room or location with the desired aural fingerprint and position a microphone some distance from the speaker. Next, play back your original sounds through the speaker and re-record them on another recorder, capturing the sound with all the reverberant characteristics of the space. This requires much time and effort, but when only the most authentic reproduction will do, worldizing can get you there." [ 2 ]
Radial and Millennia Media products use the Reamp patent, or a variant of the patent, under license. The Reamp Company acknowledges that words such as "reamping" have become generic/colloquial audio expressions, but asserts that the word Reamp remains their legally registered trademark. In late 2010 Recording engineer John Cuniberti announced the sale of his Reamp patent, trademark and all business assets to Vancouver, Canada-based Radial Engineering Ltd., a leading manufacturer of products used by audio professionals and musicians around the world. [ 3 ] | https://en.wikipedia.org/wiki/Re-amp |
A re-recording mixer in North America, also known as a dubbing mixer in Europe, is a post-production audio engineer who mixes recorded dialogue, sound effects and music to create the final version of a soundtrack for a feature film , television program , or television advertisement . The final mix must achieve a desired sonic balance between its various elements, and must match the director 's or sound designer 's original vision for the project. For material intended for broadcast, the final mix must also comply with all applicable laws governing sound mixing (e.g., the CALM Act in the United States and the EBU R 128 loudness protocol in Europe).
The different names of this profession are both based on the fact that the mixer is not mixing a live performance to a live audience nor recording live on a set . That is, the mixer is re-recording sound already recorded elsewhere (the basis of the North American name) after passing it through mixing equipment such as a digital audio workstation and may dub in additional sounds in the process (the basis of the European name). While mixing can be performed in a recording studio or home office, a full-size mixing stage or dubbing stage is used for feature films intended for release to movie theaters in order to help the mixer envision how the final mix will be heard in such large spaces.
During production or earlier parts of post-production, sound editors, sound designers, sound engineers, production sound mixers and/or music editors assemble the tracks that become raw materials for the re-recording mixer to work with. Those tracks in turn originate with sounds created by professional musicians, singers, actors, or Foley artists.
The first part of the traditional re-recording process is called the "premix." In the dialog premix the re-recording mixer does preliminary processing, including making initial loudness adjustments, cross-fading , and reducing environmental noise or spill that the on-set microphone picked up. In most instances, audio restoration software may be employed. For film or television productions, they may add a temporary/permanent music soundtrack that will have been prepared by the music editor, then the resulting work will be previewed by test audiences, and then the film or television program is re-cut and the soundtrack must be mixed again. Re-recording mixer may also augment or minimize audience reactions for television programs recorded in front of a studio audience. In some cases, a laugh track may augment these reactions.
During the "final mix" the re-recording/dubbing mixers, guided by the director or producer, must make creative decisions from moment to moment in each scene about how loud each major sound element (dialog, sound effects, laugh track and music) should be relative to each other. They also modify individual sounds when desired by adjusting their loudness and spectral content and by adding artificial reverberation. They can insert sounds into a three-dimensional space of the listening environment for a variety of venues and release formats: movie theaters, home theater systems, etc. that have stereo and multi-channel ( 5.1 , 7.1 , etc.) surround sound systems. Today, films may be mixed in 'object-based' audio formats such as Dolby Atmos , which adds height channels and metadata to allow for real-time rendering of audio objects in a three-dimensional coordinate space. | https://en.wikipedia.org/wiki/Re-recording_mixer |
Rhenium(VII) oxide is the inorganic compound with the formula Re 2 O 7 . This yellowish solid is the anhydride of HOReO 3 . Perrhenic acid , Re 2 O 7 ·2H 2 O, is closely related to Re 2 O 7 . Re 2 O 7 is the raw material for all rhenium compounds, being the volatile fraction obtained upon roasting the host ore. [ 2 ]
Solid Re 2 O 7 consists of alternating octahedral and tetrahedral Re centres. Upon heating, the polymer cracks to give molecular (nonpolymeric) Re 2 O 7 . This molecular species closely resembles manganese heptoxide , consisting of a pair of ReO 4 tetrahedra that share a vertex, i.e., O 3 Re–O–ReO 3 . [ 3 ]
Rhenium(VII) oxide is formed when metallic rhenium or its oxides or sulfides are oxidized at 500–700 °C (900–1,300 °F) in air. [ 4 ]
Re 2 O 7 dissolves in water to give perrhenic acid .
Heating Re 2 O 7 gives rhenium dioxide , a reaction signalled by the appearance of the dark blue coloration: [ 5 ]
Using tetramethyltin , it converts to methylrhenium trioxide ("MTO"), a catalyst for oxidations: [ 6 ]
In a related reaction, it reacts with hexamethyldisiloxane to give the siloxide : [ 4 ]
Rhenium(VII) oxide finds some use in organic synthesis as a catalyst for ethenolysis , [ 7 ] carbonyl reduction and amide reduction . [ 8 ] | https://en.wikipedia.org/wiki/Re2O7 |
Rhenium(VII) sulfide is a chemical compound with the formula Re 2 S 7 . It has a complex structure, but can be synthesized from direct combination of the elements: [ 1 ] 2 Re + 7 S → Δ Re 2 S 7 {\displaystyle {\ce {2Re{}+7S->[{} \atop \Delta ]Re2S7}}} Alternatively, rhenium(VII) oxide reacts with hydrogen sulfide in 4N HCl to the same end: [ 2 ] Re 2 O 7 + 7 H 2 S → Δ Re 2 S 7 + 7 H 2 O {\displaystyle {\ce {Re2O7{}+7H2S->[{} \atop \Delta ]Re2S7{}+7H2O}}}
The compound catalyses the reduction of nitric oxide to nitrous oxide and hydrogenation of double bonds. In this regard, it unusually tolerates sulfur compounds, which poison noble metal catalysts. [ 1 ]
Rhenium(VII) sulfide decomposes when heated. In vacuum , it generates rhenium(IV) sulfide :
In air, the sulfide oxidizes to sulfur dioxide :
This inorganic compound –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Re2S7 |
Trirhenium nonachloride is a compound with the formula ReCl 3 , sometimes also written Re 3 Cl 9 . It is a dark red hygroscopic solid that is insoluble in ordinary solvents. The compound is important in the history of inorganic chemistry as an early example of a cluster compound with metal-metal bonds. [ 1 ] It is used as a starting material for synthesis of other rhenium complexes.
As shown by X-ray crystallography trirhenium nonachloride consists of Re 3 Cl 12 subunits that share three chloride bridges with adjacent clusters. The interconnected network of clusters forms sheets. Around each Re center are seven ligands, four bridging chlorides, one terminal chloride, and two Re-Re bonds. [ 2 ]
The hydrate is molecular with the formula Re 3 Cl 9 (H 2 O) 3 . [ 3 ]
The heat of oxidation is evaluated according to the equation:
The enthalpy for this process is 190.7 ± 0.2 kcal/mol. [ 2 ]
The compound was discovered in 1932. [ 4 ] Trirhenium nonachloride is efficiently prepared by thermal decomposition of rhenium pentachloride or hexachlororhenic(IV) acid: [ 5 ]
If the sample is vacuum sublimed at 500 °C, the resulting material is comparatively unreactive. The partially hydrated material such as Re 3 Cl 9 (H 2 O) 4 [ 6 ] can be more useful synthetically. Other synthetic methods include treating rhenium with sulfuryl chloride . This process is sometimes conducted with the addition of aluminium chloride . [ 2 ] It is also obtained by heating Re 2 (O 2 CCH 3 ) 4 Cl 2 under HCl:
Reaction of the tri- and pentachlorides gives rhenium tetrachloride : | https://en.wikipedia.org/wiki/Re3Cl9 |
Rhenium diboride (ReB 2 ) is a synthetic high- hardness material that was first synthesized in 1962. [ 3 ] [ 4 ] The compound is formed from a mixture of rhenium , noted for its resistance to high pressure, and boron , which forms short, strong covalent bonds with rhenium. It has regained popularity in recent times in hopes of finding a material that possesses hardness comparable to that of diamond . [ 5 ]
Unlike other high-hardness synthetic materials, such as the c-BN, rhenium diboride can be synthesized at ambient pressure , [ 4 ] potentially simplifying a mass production . However, the high cost of rhenium and commercial availability of alternatives such as polycrystalline c-BN , make a prospect of large-scale applications less likely. [ 4 ]
ReB 2 can be synthesized by at least three different methods at standard atmospheric pressure: solid-state metathesis , melting in an electric arc , and direct heating of the elements. [ 5 ]
In the metathesis reaction, rhenium trichloride and magnesium diboride are mixed and heated in an inert atmosphere and the magnesium chloride byproduct is washed away. Excess boron is needed to prevent the formation of other phases such as Re 7 B 3 and Re 3 B.
In the arc-melting method, rhenium and boron powders are mixed and a large electric current is passed through the mixture, also in an inert atmosphere.
In the direct reaction method, the rhenium-boron mixture is sealed in a vacuum and held at a high temperature over a longer period (1,000 °C for five days).
At least the last two methods are capable of producing pure ReB 2 without any other phases, as confirmed by X-ray crystallography .
Rhenium diboride is occasionally, and controversially, [ 4 ] [ 6 ] cited as a " superhard material " due to its high hardness level. However, tested in the asymptotic-hardness region, as recommended for hard and superhard materials, [ 4 ] rhenium diboride demonstrates a Vickers hardness of only 30.1 ± 1.3 GPa at 4.9 N, well below the generally-accepted threshold of 40 GPa or more needed to classify it as "superhard". [ 4 ] Another research has estimated the H v of full-dense ReB 2 at about 22 GPa under an applied load of 2.94 N, [ 6 ] comparable to that of tungsten carbide , silicon carbide , titanium diboride or zirconium diboride . [ 6 ]
Values greater than 40 GPa have been observed only in tests with very low loads, which is not a suitable testing method for this type of solids. [ 4 ] In one test, the lowest tested load of 0.49 N yielded the average hardness of 48 ± 5.6 GPa and a maximum hardness of 55.5 GPa, which is comparable to the hardness of cubic boron nitride (c-BN) under an equivalent load. [ 5 ] Such phenomenon of inverse relationship between the applied load and hardness is known as the indentation size effect . [ 5 ]
In recent times, there has been a significant amount of research into improving the hardness and other properties of the ReB 2 . In one study, the hardness for the ReB 2 (R-3m) polymorph was estimated at 41.7 GPa, while for the ReB 2 (P6 3 /mmc) it was placed at c.a. 40.6 GPa. [ 7 ] In another study, a fully dense B 4 C - 27 wt.% ReB 2 ceramic composite nanopowder was fabricated by spark plasma sintering . It has exhibited a microhardness of 50 ± 3 GPa under a 49 N load in the asymptotic-hardness region and had a 3.2 g/cm 3 density, comparable with the hardness and density of the c-BN. [ 8 ]
The hardness of ReB 2 exhibits considerable anisotropy because of its hexagonal layered structure , being greatest along the c axis. Two factors contribute to the high hardness of ReB 2 : a high density of valence electrons , and an abundance of short covalent bonds . [ 5 ] [ 9 ] Rhenium has one of the highest valence electron densities of any transition metal (476 electrons/nm 3 , compare to 572 electrons/nm 3 for osmium and 705 electrons/nm 3 for diamond [ 10 ] ). The addition of boron requires only a 5% expansion of the rhenium lattice because the small boron atoms fill the existing spaces between the rhenium atoms. Furthermore, the electronegativities of rhenium and boron are close enough (1.9 and 2.04 on the Pauling scale ) that they form covalent bonds in which the electrons are shared almost equally. | https://en.wikipedia.org/wiki/ReB2 |
Trirhenium nonachloride is a compound with the formula ReCl 3 , sometimes also written Re 3 Cl 9 . It is a dark red hygroscopic solid that is insoluble in ordinary solvents. The compound is important in the history of inorganic chemistry as an early example of a cluster compound with metal-metal bonds. [ 1 ] It is used as a starting material for synthesis of other rhenium complexes.
As shown by X-ray crystallography trirhenium nonachloride consists of Re 3 Cl 12 subunits that share three chloride bridges with adjacent clusters. The interconnected network of clusters forms sheets. Around each Re center are seven ligands, four bridging chlorides, one terminal chloride, and two Re-Re bonds. [ 2 ]
The hydrate is molecular with the formula Re 3 Cl 9 (H 2 O) 3 . [ 3 ]
The heat of oxidation is evaluated according to the equation:
The enthalpy for this process is 190.7 ± 0.2 kcal/mol. [ 2 ]
The compound was discovered in 1932. [ 4 ] Trirhenium nonachloride is efficiently prepared by thermal decomposition of rhenium pentachloride or hexachlororhenic(IV) acid: [ 5 ]
If the sample is vacuum sublimed at 500 °C, the resulting material is comparatively unreactive. The partially hydrated material such as Re 3 Cl 9 (H 2 O) 4 [ 6 ] can be more useful synthetically. Other synthetic methods include treating rhenium with sulfuryl chloride . This process is sometimes conducted with the addition of aluminium chloride . [ 2 ] It is also obtained by heating Re 2 (O 2 CCH 3 ) 4 Cl 2 under HCl:
Reaction of the tri- and pentachlorides gives rhenium tetrachloride : | https://en.wikipedia.org/wiki/ReCl3 |
Rhenium pentachloride is an inorganic compound with the formula Re 2 Cl 10 . This red-brown solid [ 3 ] is paramagnetic . [ 4 ]
Rhenium pentachloride has a bioctahedral structure and can be described as Cl 4 Re(μ-Cl) 2 ReCl 4 . The (μ-Cl) 2 part of this formula indicates that two chloride ligands are bridging ligands , i.e. they connect to two Re atoms. The Re-Re distance is 3.74 Å. [ 1 ] The motif is similar to that seen for tantalum pentachloride .
This compound was first prepared in 1933, [ 5 ] a few years after the discovery of rhenium. The preparation involves chlorination of rhenium at temperatures up to 900 °C. [ 3 ] The material can be purified by sublimation.
ReCl 5 is one of the most oxidized binary chlorides of Re. It does not undergo further chlorination. ReCl 6 has been prepared from rhenium hexafluoride . [ 6 ] Rhenium heptafluoride is known but not the heptachloride. [ 7 ]
It degrades in air to a brown liquid. [ 8 ]
Although rhenium pentachloride has no commercial applications, it is of historic significance as one of the early catalysts for olefin metathesis . [ 9 ] Reduction gives trirhenium nonachloride .
Oxygenation affords the Re(VII) oxychloride: [ 10 ]
Comproportionation of the penta- and trichloride gives rhenium tetrachloride . | https://en.wikipedia.org/wiki/ReCl5 |
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