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There are, however, some potential drawbacks to using stratified sampling. First, identifying strata and implementing such an approach can increase the cost and complexity of sample selection, as well as leading to increased complexity of population estimates. Second, when examining multiple criteria, stratifying variables may be related to some, but not to others, further complicating the design, and potentially reducing the utility of the strata. Finally, in some cases (such as designs with a large number of strata, or those with a specified minimum sample size per group), stratified sampling can potentially require a larger sample than would other methods (although in most cases, the required sample size would be no larger than would be required for simple random sampling).
A stratified sampling approach is most effective when three conditions are met
Variability within strata are minimized
Variability between strata are maximized
The variables upon which the population is stratified are strongly correlated with the desired dependent variable.
Advantages over other sampling methods
Focuses on important subpopulations and ignores irrelevant ones.
Allows use of different sampling techniques for different subpopulations.
Improves the accuracy/efficiency of estimation.
Permits greater balancing of statistical power of tests of differences between strata by sampling equal numbers from strata varying widely in size.
Disadvantages
Requires selection of relevant stratification variables which can be difficult.
Is not useful when there are no homogeneous subgroups.
Can be expensive to implement.
Poststratification
Stratification is sometimes introduced after the sampling phase in a process called "poststratification". This approach is typically implemented due to a lack of prior knowledge of an appropriate stratifying variable or when the experimenter lacks the necessary information to create a stratifying variable during the sampling phase. Although the method is susceptible to the pitfalls of post hoc approaches, it can provide several benefits in the right situation. Implementation usually follows a simple random sample. In addition to allowing for stratification on an ancillary variable, poststratification can be used to implement weighting, which can improve the precision of a sample's estimates. | Sampling (statistics) | Wikipedia | 425 | 160361 | https://en.wikipedia.org/wiki/Sampling%20%28statistics%29 | Mathematics | Statistics and probability | null |
Oversampling
Choice-based sampling or oversampling is one of the stratified sampling strategies. In choice-based sampling, the data are stratified on the target and a sample is taken from each stratum so that rarer target classes will be more represented in the sample. The model is then built on this biased sample. The effects of the input variables on the target are often estimated with more precision with the choice-based sample even when a smaller overall sample size is taken, compared to a random sample. The results usually must be adjusted to correct for the oversampling.
Probability-proportional-to-size sampling
In some cases the sample designer has access to an "auxiliary variable" or "size measure", believed to be correlated to the variable of interest, for each element in the population. These data can be used to improve accuracy in sample design. One option is to use the auxiliary variable as a basis for stratification, as discussed above.
Another option is probability proportional to size ('PPS') sampling, in which the selection probability for each element is set to be proportional to its size measure, up to a maximum of 1. In a simple PPS design, these selection probabilities can then be used as the basis for Poisson sampling. However, this has the drawback of variable sample size, and different portions of the population may still be over- or under-represented due to chance variation in selections.
Systematic sampling theory can be used to create a probability proportionate to size sample. This is done by treating each count within the size variable as a single sampling unit. Samples are then identified by selecting at even intervals among these counts within the size variable. This method is sometimes called PPS-sequential or monetary unit sampling in the case of audits or forensic sampling. | Sampling (statistics) | Wikipedia | 369 | 160361 | https://en.wikipedia.org/wiki/Sampling%20%28statistics%29 | Mathematics | Statistics and probability | null |
Example: Suppose we have six schools with populations of 150, 180, 200, 220, 260, and 490 students respectively (total 1500 students), and we want to use student population as the basis for a PPS sample of size three. To do this, we could allocate the first school numbers 1 to 150, the second school 151 to 330 (= 150 + 180), the third school 331 to 530, and so on to the last school (1011 to 1500). We then generate a random start between 1 and 500 (equal to 1500/3) and count through the school populations by multiples of 500. If our random start was 137, we would select the schools which have been allocated numbers 137, 637, and 1137, i.e. the first, fourth, and sixth schools.
The PPS approach can improve accuracy for a given sample size by concentrating sample on large elements that have the greatest impact on population estimates. PPS sampling is commonly used for surveys of businesses, where element size varies greatly and auxiliary information is often available – for instance, a survey attempting to measure the number of guest-nights spent in hotels might use each hotel's number of rooms as an auxiliary variable. In some cases, an older measurement of the variable of interest can be used as an auxiliary variable when attempting to produce more current estimates.
Cluster sampling
Sometimes it is more cost-effective to select respondents in groups ('clusters'). Sampling is often clustered by geography, or by time periods. (Nearly all samples are in some sense 'clustered' in time – although this is rarely taken into account in the analysis.) For instance, if surveying households within a city, we might choose to select 100 city blocks and then interview every household within the selected blocks.
Clustering can reduce travel and administrative costs. In the example above, an interviewer can make a single trip to visit several households in one block, rather than having to drive to a different block for each household.
It also means that one does not need a sampling frame listing all elements in the target population. Instead, clusters can be chosen from a cluster-level frame, with an element-level frame created only for the selected clusters. In the example above, the sample only requires a block-level city map for initial selections, and then a household-level map of the 100 selected blocks, rather than a household-level map of the whole city. | Sampling (statistics) | Wikipedia | 494 | 160361 | https://en.wikipedia.org/wiki/Sampling%20%28statistics%29 | Mathematics | Statistics and probability | null |
Cluster sampling (also known as clustered sampling) generally increases the variability of sample estimates above that of simple random sampling, depending on how the clusters differ between one another as compared to the within-cluster variation. For this reason, cluster sampling requires a larger sample than SRS to achieve the same level of accuracy – but cost savings from clustering might still make this a cheaper option.
Cluster sampling is commonly implemented as multistage sampling. This is a complex form of cluster sampling in which two or more levels of units are embedded one in the other. The first stage consists of constructing the clusters that will be used to sample from. In the second stage, a sample of primary units is randomly selected from each cluster (rather than using all units contained in all selected clusters). In following stages, in each of those selected clusters, additional samples of units are selected, and so on. All ultimate units (individuals, for instance) selected at the last step of this procedure are then surveyed. This technique, thus, is essentially the process of taking random subsamples of preceding random samples.
Multistage sampling can substantially reduce sampling costs, where the complete population list would need to be constructed (before other sampling methods could be applied). By eliminating the work involved in describing clusters that are not selected, multistage sampling can reduce the large costs associated with traditional cluster sampling. However, each sample may not be a full representative of the whole population.
Quota sampling
In quota sampling, the population is first segmented into mutually exclusive sub-groups, just as in stratified sampling. Then judgement is used to select the subjects or units from each segment based on a specified proportion. For example, an interviewer may be told to sample 200 females and 300 males between the age of 45 and 60.
It is this second step which makes the technique one of non-probability sampling. In quota sampling the selection of the sample is non-random. For example, interviewers might be tempted to interview those who look most helpful. The problem is that these samples may be biased because not everyone gets a chance of selection. This random element is its greatest weakness and quota versus probability has been a matter of controversy for several years.
Minimax sampling | Sampling (statistics) | Wikipedia | 448 | 160361 | https://en.wikipedia.org/wiki/Sampling%20%28statistics%29 | Mathematics | Statistics and probability | null |
In imbalanced datasets, where the sampling ratio does not follow the population statistics, one can resample the dataset in a conservative manner called minimax sampling. The minimax sampling has its origin in Anderson minimax ratio whose value is proved to be 0.5: in a binary classification, the class-sample sizes should be chosen equally. This ratio can be proved to be minimax ratio only under the assumption of LDA classifier with Gaussian distributions. The notion of minimax sampling is recently developed for a general class of classification rules, called class-wise smart classifiers. In this case, the sampling ratio of classes is selected so that the worst case classifier error over all the possible population statistics for class prior probabilities, would be the best.
Accidental sampling
Accidental sampling (sometimes known as grab, convenience or opportunity sampling) is a type of nonprobability sampling which involves the sample being drawn from that part of the population which is close to hand. That is, a population is selected because it is readily available and convenient. It may be through meeting the person or including a person in the sample when one meets them or chosen by finding them through technological means such as the internet or through phone. The researcher using such a sample cannot scientifically make generalizations about the total population from this sample because it would not be representative enough. For example, if the interviewer were to conduct such a survey at a shopping center early in the morning on a given day, the people that they could interview would be limited to those given there at that given time, which would not represent the views of other members of society in such an area, if the survey were to be conducted at different times of day and several times per week. This type of sampling is most useful for pilot testing. Several important considerations for researchers using convenience samples include:
Are there controls within the research design or experiment which can serve to lessen the impact of a non-random convenience sample, thereby ensuring the results will be more representative of the population?
Is there good reason to believe that a particular convenience sample would or should respond or behave differently than a random sample from the same population?
Is the question being asked by the research one that can adequately be answered using a convenience sample? | Sampling (statistics) | Wikipedia | 460 | 160361 | https://en.wikipedia.org/wiki/Sampling%20%28statistics%29 | Mathematics | Statistics and probability | null |
In social science research, snowball sampling is a similar technique, where existing study subjects are used to recruit more subjects into the sample. Some variants of snowball sampling, such as respondent driven sampling, allow calculation of selection probabilities and are probability sampling methods under certain conditions.
Voluntary sampling
The voluntary sampling method is a type of non-probability sampling. Volunteers choose to complete a survey.
Volunteers may be invited through advertisements in social media. The target population for advertisements can be selected by characteristics like location, age, sex, income, occupation, education, or interests using tools provided by the social medium. The advertisement may include a message about the research and link to a survey. After following the link and completing the survey, the volunteer submits the data to be included in the sample population. This method can reach a global population but is limited by the campaign budget. Volunteers outside the invited population may also be included in the sample.
It is difficult to make generalizations from this sample because it may not represent the total population. Often, volunteers have a strong interest in the main topic of the survey.
Line-intercept sampling
Line-intercept sampling is a method of sampling elements in a region whereby an element is sampled if a chosen line segment, called a "transect", intersects the element.
Panel sampling
Panel sampling is the method of first selecting a group of participants through a random sampling method and then asking that group for (potentially the same) information several times over a period of time. Therefore, each participant is interviewed at two or more time points; each period of data collection is called a "wave". The method was developed by sociologist Paul Lazarsfeld in 1938 as a means of studying political campaigns. This longitudinal sampling-method allows estimates of changes in the population, for example with regard to chronic illness to job stress to weekly food expenditures. Panel sampling can also be used to inform researchers about within-person health changes due to age or to help explain changes in continuous dependent variables such as spousal interaction. There have been several proposed methods of analyzing panel data, including MANOVA, growth curves, and structural equation modeling with lagged effects.
Snowball sampling
Snowball sampling involves finding a small group of initial respondents and using them to recruit more respondents. It is particularly useful in cases where the population is hidden or difficult to enumerate.
Theoretical sampling | Sampling (statistics) | Wikipedia | 477 | 160361 | https://en.wikipedia.org/wiki/Sampling%20%28statistics%29 | Mathematics | Statistics and probability | null |
Theoretical sampling occurs when samples are selected on the basis of the results of the data collected so far with a goal of developing a deeper understanding of the area or develop theories. An initial, general sample is first collected with the goal of investigating general trends, where further sampling may consist of extreme or very specific cases might be selected in order to maximize the likelihood a phenomenon will actually be observable.
Active sampling
In active sampling, the samples which are used for training a machine learning algorithm are actively selected, also compare active learning (machine learning).
Judgmental selection
Judgement sampling is a type non-random sampling where samples are selected based on the opinion of an expert, who can select participants based on how valuable the information they provide is.
Haphazard sampling
Haphazard sampling refers to the idea of using human judgement to simulate randomness. Despite samples being hand-picked, the goal is to ensure that no conscious bias exists within the choice of samples, but often fails due to selection bias. Haphazard sampling is generally opted for due to its convenience, when the tools or capacity to perform other sampling methods may not exist.
Replacement of selected units
Sampling schemes may be without replacement ('WOR' – no element can be selected more than once in the same sample) or with replacement ('WR' – an element may appear multiple times in the one sample). For example, if we catch fish, measure them, and immediately return them to the water before continuing with the sample, this is a WR design, because we might end up catching and measuring the same fish more than once. However, if we do not return the fish to the water or tag and release each fish after catching it, this becomes a WOR design.
Sample size determination
Formulas, tables, and power function charts are well known approaches to determine sample size.
Steps for using sample size tables:
Postulate the effect size of interest, α, and β.
Check sample size table
Select the table corresponding to the selected α
Locate the row corresponding to the desired power
Locate the column corresponding to the estimated effect size.
The intersection of the column and row is the minimum sample size required.
Sampling and data collection
Good data collection involves:
Following the defined sampling process
Keeping the data in time order
Noting comments and other contextual events
Recording non-responses | Sampling (statistics) | Wikipedia | 464 | 160361 | https://en.wikipedia.org/wiki/Sampling%20%28statistics%29 | Mathematics | Statistics and probability | null |
Applications of sampling
Sampling enables the selection of right data points from within the larger data set to estimate the characteristics of the whole population. For example, there are about 600 million tweets produced every day. It is not necessary to look at all of them to determine the topics that are discussed during the day, nor is it necessary to look at all the tweets to determine the sentiment on each of the topics. A theoretical formulation for sampling Twitter data has been developed.
In manufacturing different types of sensory data such as acoustics, vibration, pressure, current, voltage, and controller data are available at short time intervals. To predict down-time it may not be necessary to look at all the data but a sample may be sufficient.
Errors in sample surveys
Survey results are typically subject to some error. Total errors can be classified into sampling errors and non-sampling errors. The term "error" here includes systematic biases as well as random errors.
Sampling errors and biases
Sampling errors and biases are induced by the sample design. They include:
Selection bias: When the true selection probabilities differ from those assumed in calculating the results.
Random sampling error: Random variation in the results due to the elements in the sample being selected at random.
Non-sampling error
Non-sampling errors are other errors which can impact final survey estimates, caused by problems in data collection, processing, or sample design. Such errors may include:
Over-coverage: inclusion of data from outside of the population
Under-coverage: sampling frame does not include elements in the population.
Measurement error: e.g. when respondents misunderstand a question, or find it difficult to answer
Processing error: mistakes in data coding
Non-response or Participation bias: failure to obtain complete data from all selected individuals
After sampling, a review is held of the exact process followed in sampling, rather than that intended, in order to study any effects that any divergences might have on subsequent analysis.
A particular problem involves non-response. Two major types of non-response exist:
unit nonresponse (lack of completion of any part of the survey)
item non-response (submission or participation in survey but failing to complete one or more components/questions of the survey) | Sampling (statistics) | Wikipedia | 451 | 160361 | https://en.wikipedia.org/wiki/Sampling%20%28statistics%29 | Mathematics | Statistics and probability | null |
In survey sampling, many of the individuals identified as part of the sample may be unwilling to participate, not have the time to participate (opportunity cost), or survey administrators may not have been able to contact them. In this case, there is a risk of differences between respondents and nonrespondents, leading to biased estimates of population parameters. This is often addressed by improving survey design, offering incentives, and conducting follow-up studies which make a repeated attempt to contact the unresponsive and to characterize their similarities and differences with the rest of the frame. The effects can also be mitigated by weighting the data (when population benchmarks are available) or by imputing data based on answers to other questions. Nonresponse is particularly a problem in internet sampling. Reasons for this problem may include improperly designed surveys, over-surveying (or survey fatigue),
and the fact that potential participants may have multiple e-mail addresses, which they do not use anymore or do not check regularly.
Survey weights
In many situations, the sample fraction may be varied by stratum and data will have to be weighted to correctly represent the population. Thus for example, a simple random sample of individuals in the United Kingdom might not include some in remote Scottish islands who would be inordinately expensive to sample. A cheaper method would be to use a stratified sample with urban and rural strata. The rural sample could be under-represented in the sample, but weighted up appropriately in the analysis to compensate.
More generally, data should usually be weighted if the sample design does not give each individual an equal chance of being selected. For instance, when households have equal selection probabilities but one person is interviewed from within each household, this gives people from large households a smaller chance of being interviewed. This can be accounted for using survey weights. Similarly, households with more than one telephone line have a greater chance of being selected in a random digit dialing sample, and weights can adjust for this.
Weights can also serve other purposes, such as helping to correct for non-response.
Methods of producing random samples
Random number table
Mathematical algorithms for pseudo-random number generators
Physical randomization devices such as coins, playing cards or sophisticated devices such as ERNIE | Sampling (statistics) | Wikipedia | 454 | 160361 | https://en.wikipedia.org/wiki/Sampling%20%28statistics%29 | Mathematics | Statistics and probability | null |
In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in dimensions is called a hyperball or -ball and is bounded by a hypersphere or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment.
In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In the field of topology the closed -dimensional ball is often denoted as or while the open -dimensional ball is or .
In Euclidean space
In Euclidean -space, an (open) -ball of radius and center is the set of all points of distance less than from . A closed -ball of radius is the set of all points of distance less than or equal to away from .
In Euclidean -space, every ball is bounded by a hypersphere. The ball is a bounded interval when , is a disk bounded by a circle when , and is bounded by a sphere when .
Volume
The -dimensional volume of a Euclidean ball of radius in -dimensional Euclidean space is:
where is Leonhard Euler's gamma function (which can be thought of as an extension of the factorial function to fractional arguments). Using explicit formulas for particular values of the gamma function at the integers and half integers gives formulas for the volume of a Euclidean ball that do not require an evaluation of the gamma function. These are:
In the formula for odd-dimensional volumes, the double factorial is defined for odd integers as .
In general metric spaces
Let be a metric space, namely a set with a metric (distance function) , and let be a positive real number. The open (metric) ball of radius centered at a point in , usually denoted by or , is defined the same way as a Euclidean ball, as the set of points in of distance less than away from ,
The closed (metric) ball, sometimes denoted or , is likewise defined as the set of points of distance less than or equal to away from , | Ball (mathematics) | Wikipedia | 504 | 160556 | https://en.wikipedia.org/wiki/Ball%20%28mathematics%29 | Mathematics | Measurement | null |
In particular, a ball (open or closed) always includes itself, since the definition requires . A unit ball (open or closed) is a ball of radius 1.
A ball in a general metric space need not be round. For example, a ball in real coordinate space under the Chebyshev distance is a hypercube, and a ball under the taxicab distance is a cross-polytope. A closed ball also need not be compact. For example, a closed ball in any infinite-dimensional normed vector space is never compact. However, a ball in a vector space will always be convex as a consequence of the triangle inequality.
A subset of a metric space is bounded if it is contained in some ball. A set is totally bounded if, given any positive radius, it is covered by finitely many balls of that radius.
The open balls of a metric space can serve as a base, giving this space a topology, the open sets of which are all possible unions of open balls. This topology on a metric space is called the topology induced by the metric .
Let denote the closure of the open ball in this topology. While it is always the case that it is always the case that For example, in a metric space with the discrete metric, one has but for any
In normed vector spaces
Any normed vector space with norm is also a metric space with the metric In such spaces, an arbitrary ball of points around a point with a distance of less than may be viewed as a scaled (by ) and translated (by ) copy of a unit ball Such "centered" balls with are denoted with
The Euclidean balls discussed earlier are an example of balls in a normed vector space.
-norm
In a Cartesian space with the -norm , that is one chooses some and definesThen an open ball around the origin with radius is given by the set
For , in a 2-dimensional plane , "balls" according to the -norm (often called the taxicab or Manhattan metric) are bounded by squares with their diagonals parallel to the coordinate axes; those according to the -norm, also called the Chebyshev metric, have squares with their sides parallel to the coordinate axes as their boundaries. The -norm, known as the Euclidean metric, generates the well known disks within circles, and for other values of , the corresponding balls are areas bounded by Lamé curves (hypoellipses or hyperellipses). | Ball (mathematics) | Wikipedia | 498 | 160556 | https://en.wikipedia.org/wiki/Ball%20%28mathematics%29 | Mathematics | Measurement | null |
For , the - balls are within octahedra with axes-aligned body diagonals, the -balls are within cubes with axes-aligned edges, and the boundaries of balls for with are superellipsoids. generates the inner of usual spheres.
Often can also consider the case of in which case we define
General convex norm
More generally, given any centrally symmetric, bounded, open, and convex subset of , one can define a norm on where the balls are all translated and uniformly scaled copies of . Note this theorem does not hold if "open" subset is replaced by "closed" subset, because the origin point qualifies but does not define a norm on .
In topological spaces
One may talk about balls in any topological space , not necessarily induced by a metric. An (open or closed) -dimensional topological ball of is any subset of which is homeomorphic to an (open or closed) Euclidean -ball. Topological -balls are important in combinatorial topology, as the building blocks of cell complexes.
Any open topological -ball is homeomorphic to the Cartesian space and to the open unit -cube (hypercube) . Any closed topological -ball is homeomorphic to the closed -cube .
An -ball is homeomorphic to an -ball if and only if . The homeomorphisms between an open -ball and can be classified in two classes, that can be identified with the two possible topological orientations of .
A topological -ball need not be smooth; if it is smooth, it need not be diffeomorphic to a Euclidean -ball.
Regions
A number of special regions can be defined for a ball:
cap, bounded by one plane
sector, bounded by a conical boundary with apex at the center of the sphere
segment, bounded by a pair of parallel planes
shell, bounded by two concentric spheres of differing radii
wedge, bounded by two planes passing through a sphere center and the surface of the sphere | Ball (mathematics) | Wikipedia | 392 | 160556 | https://en.wikipedia.org/wiki/Ball%20%28mathematics%29 | Mathematics | Measurement | null |
An axle or axletree is a central shaft for a rotating wheel or gear. On wheeled vehicles, the axle may be fixed to the wheels, rotating with them, or fixed to the vehicle, with the wheels rotating around the axle. In the former case, bearings or bushings are provided at the mounting points where the axle is supported. In the latter case, a bearing or bushing sits inside a central hole in the wheel to allow the wheel or gear to rotate around the axle. Sometimes, especially on bicycles, the latter type of axle is referred to as a spindle.
Terminology
On cars and trucks, several senses of the word axle occur in casual usage, referring to the shaft itself, its housing, or simply any transverse pair of wheels. Strictly speaking, a shaft that rotates with the wheel, being either bolted or splined in fixed relation to it, is called an axle or axle shaft. However, in looser usage, an entire assembly including the surrounding axle housing (typically a casting) is also called an axle.
An even broader (somewhat figurative) sense of the word refers to every pair of parallel wheels on opposite sides of a vehicle, regardless of their mechanical connection to each other and to the vehicle frame or body. Thus, transverse pairs of wheels in an independent suspension may be called an axle in some contexts. This very loose definition of "axle" is often used in assessing toll roads or vehicle taxes, and is taken as a rough proxy for the overall weight-bearing capacity of a vehicle, and its potential for causing wear or damage to roadway surfaces.
Vehicle axles
Axles are an integral component of most practical wheeled vehicles. In a solid, "live-axle" suspension system, the rotating inner axle cores (or half-shafts) serve to transmit driving torque to the wheels at each end, while the rigid outer tube maintains the position of the wheels at fixed angles relative to the axle, and controls the angle of the axle and wheels assembly to the vehicle body. The solid axles (housings) in this system must also bear the weight of the vehicle plus any cargo. A non-driving axle, such as the front beam axle in heavy-duty trucks and some two-wheel drive light trucks and vans, will have no shaft, and serves only as a suspension and steering component. Conversely, many front-wheel drive cars have a one-piece rear beam axle. | Axle | Wikipedia | 485 | 160573 | https://en.wikipedia.org/wiki/Axle | Technology | Components_2 | null |
In other types of suspension systems, the axles serve only to transmit driving torque to the wheels: the position and angle of the wheel hubs is made independent from the axles by the function of the suspension system. This is typical of the independent suspensions found on most newer cars, and even SUVs, and on the front of many light trucks. An exception to this rule is the independent (rear) swing axle suspension, wherein the half-axles are also load-bearing suspension arms.
Independent drive-trains still need differentials (or diffs), but without fixed axle-housing tubes attached. The diff may be attached to the vehicle frame or body, and/or be integrated with the transmission (or gearbox) in a combined transaxle unit. The axle (half-)shafts then transmit driving torque to the wheels, usually via constant-velocity joints. Like a full floating axle system, the drive shafts in a front-wheel-drive independent suspension system do not support any vehicle weight.
Structural features and design
A straight axle is a single rigid shaft connecting a wheel on the left side of the vehicle to a wheel on the right side. The axis of rotation fixed by the axle is common to both wheels. Such a design can keep the wheel positions steady under heavy stress, and can therefore support heavy loads. Straight axles are used on trains (that is, locomotives and railway wagons), for the rear axles of commercial trucks, and on heavy-duty off-road vehicles. The axle can optionally be protected and further reinforced by enclosing the length of the axle in a housing.
In split-axle designs, the wheel on each side is attached to a separate shaft. Modern passenger cars have split-drive axles. In some designs, this allows independent suspension of the left and right wheels, and therefore a smoother ride. Even when the suspension is not independent, split axles permit the use of a differential, allowing the left and right drive wheels to be driven at different speeds as the automobile turns, improving traction and extending tire life.
A tandem axle is a group of two or more axles situated close together. Truck designs use such a configuration to provide a greater weight capacity than a single axle. Semi-trailers usually have a tandem axle at the rear. | Axle | Wikipedia | 467 | 160573 | https://en.wikipedia.org/wiki/Axle | Technology | Components_2 | null |
Axles are typically made from SAE grade 41xx steel or SAE grade 10xx steel. SAE grade 41xx steel is commonly known as "chrome-molybdenum steel" (or "chrome-moly") while SAE grade 10xx steel is known as "carbon steel". The primary differences between the two are that chrome-moly steel is significantly more resistant to bending or breaking, and is very difficult to weld with tools normally found outside a professional welding shop.
Drive axle
An axle that is driven by the engine or prime mover is called a drive axle.
Modern front-wheel drive cars typically combine the transmission (gearbox and differential) and front axle into a single unit called a transaxle. The drive axle is a split axle with a differential and universal joints between the two half axles. Each half axle connects to the wheel by use of a constant velocity (CV) joint which allows the wheel assembly to move freely vertically as well as to pivot when making turns.
In rear-wheel drive cars and trucks, the engine turns a driveshaft (also called a propeller shaft or tailshaft) which transmits the rotational force to a drive axle at the rear of the vehicle. The drive axle may be a live axle, but modern rear-wheel drive automobiles generally use a split axle with a differential. In this case, one half-axle or half-shaft connects the differential with the left rear wheel, a second half-shaft does the same with the right rear wheel; thus the two half-axles and the differential constitute the rear axle. The front drive axle is providing the force to drive the truck. In fact, only one wheel of that axle is actually moving the truck and trailer down the road.
Some simple vehicle designs, such as leisure go-karts, may have a single driven wheel where the drive axle is a split axle with only one of the two shafts driven by the engine, or else have both wheels connected to one shaft without a differential (kart racing). However, other go-karts have two rear drive wheels too.
Lift axle | Axle | Wikipedia | 432 | 160573 | https://en.wikipedia.org/wiki/Axle | Technology | Components_2 | null |
Some dump trucks and trailers may be configured with a lift axle (also known as an airlift axle or drop axle), which may be mechanically raised or lowered. The axle is lowered to increase the weight capacity, or to distribute the weight of the cargo over more wheels, for example, to cross a weight-restricted bridge. When not needed, the axle is lifted off the ground to save wear on the tires and axle, and to increase traction in the remaining wheels, and to decrease fuel consumption. Lifting an axle also alleviates lateral scrubbing of the additional axle in very tight turns, allowing the vehicle to turn more readily. In some situations, the removal of pressure from the additional axle is necessary for the vehicle to complete a turn at all.
Several manufacturers offer computer-controlled airlifts so that the dead axles are automatically lowered when the main axle reaches its weight limit. The dead axles can still be lifted by the press of a button if needed, for better maneuverability.
Lift axles were in use in the early 1940s. Initially, the axle was lifted by a mechanical device. Soon hydraulics replaced the mechanical lift system. One of the early manufacturers was Zetterbergs, located in Östervåla, Sweden. Their brand was Zeta-lyften.
The liftable tandem drive axle was invented in 1957 by the Finnish truck manufacturer Vanajan Autotehdas, a company sharing history with Sisu Auto.
Full-floating vs semi-floating
A full-floating axle carries the vehicle's weight on the axle casing, not the half-shafts; they serve only to transmit torque from the differential to the wheels. They "float" inside an assembly that carries the vehicle's weight. Thus the only stress it must endure is torque (not lateral bending force). Full-floating axle shafts are retained by a flange bolted to the hub, while the hub and bearings are retained on the spindle by a large nut. | Axle | Wikipedia | 396 | 160573 | https://en.wikipedia.org/wiki/Axle | Technology | Components_2 | null |
In contrast, a semi-floating design carries the weight of the vehicle on the axle shaft itself; there is a single bearing at the end of the axle housing that carries the load from the axle and that the axle rotates through. To be "semi-floating" the axle shafts must be able to "float" in the housing, bearings and seals, and not subject to axial "thrust" and/or bearing preload. Needle bearings and separate lip seals are used in semi-floating axles with axle retained in the housing at their inner ends typically with circlips which are 3¾-round hardened washers that slide into grooves machined at the inner end of the shafts and retained in/by recesses in the differential carrier side gears which are themselves retained by the differential pinion gear (or "spider gear") shaft. A true semi-floating axle assembly places no side loads on the axle housing tubes or axle shafts.
Axles that are pressed into ball or tapered roller bearings, which are in turn retained in the axle housings with flanges, bolts, and nuts do not "float" and place axial loads on the bearings, housings, and only a short section of the shaft itself, that also carries all radial loads.
The full-floating design is typically used in most ¾- and 1-ton light trucks, medium-duty trucks, and heavy-duty trucks. The overall assembly can carry more weight than a semi-floating or non-floating axle assembly because the hubs have two bearings riding on a fixed spindle. A full-floating axle can be identified by a protruding hub to which the axle shaft flange is bolted.
The semi-floating axle setup is commonly used on half-ton and lighter 4×4 trucks in the rear. This setup allows the axle shaft to be the means of propulsion, and also support the weight of the vehicle. The main difference between the full- and semi-floating axle setups is the number of bearings. The semi-floating axle features only one bearing, while the full-floating assembly has bearings on both the inside and outside of the wheel hub. The other difference is axle removal. To remove the semi-floating axle, the wheel must be removed first; if such an axle breaks, the wheel is most likely to come off the vehicle. The semi-floating design is found under most ½-ton and lighter trucks, as well as in SUVs and rear-wheel-drive passenger cars, usually being smaller or less expensive models. | Axle | Wikipedia | 512 | 160573 | https://en.wikipedia.org/wiki/Axle | Technology | Components_2 | null |
A benefit of a full-floating axle is that even if an axle shaft (used to transmit torque or power) breaks, the wheel will not come off, preventing serious accidents. | Axle | Wikipedia | 36 | 160573 | https://en.wikipedia.org/wiki/Axle | Technology | Components_2 | null |
Tachycardia, also called tachyarrhythmia, is a heart rate that exceeds the normal resting rate. In general, a resting heart rate over 100 beats per minute is accepted as tachycardia in adults. Heart rates above the resting rate may be normal (such as with exercise) or abnormal (such as with electrical problems within the heart).
Complications
Tachycardia can lead to fainting.
When the rate of blood flow becomes too rapid, or fast blood flow passes on damaged endothelium, it increases the friction within vessels resulting in turbulence and other disturbances. According to the Virchow's triad, this is one of the three conditions (along with hypercoagulability and endothelial injury/dysfunction) that can lead to thrombosis (i.e., blood clots within vessels).
Causes
Some causes of tachycardia include:
Adrenergic storm
Anaemia
Anxiety
Atrial fibrillation
Atrial flutter
Atrial tachycardia
Atrioventricular reentrant tachycardia
AV nodal reentrant tachycardia
Brugada syndrome
Circulatory shock and its various causes (obstructive shock, cardiogenic shock, hypovolemic shock, distributive shock)
Dehydration
Dysautonomia
Exercise
Fear
Hypoglycemia
Hypovolemia
Hyperthyroidism
Hyperventilation
Inappropriate sinus tachycardia
Junctional tachycardia
Metabolic myopathy
Multifocal atrial tachycardia
Pacemaker mediated
Pain
Panic attack
Pheochromocytoma
Sinus tachycardia
Sleep deprivation
Supraventricular tachycardia
Ventricular tachycardia
Wolff–Parkinson–White syndrome
Drug related:
Alcohol (Ethanol) intoxication
Stimulants
Cannabis
Drug withdrawal
Tricyclic antidepressants
Nefopam
Opioids (rare) | Tachycardia | Wikipedia | 404 | 160581 | https://en.wikipedia.org/wiki/Tachycardia | Biology and health sciences | Symptoms and signs | Health |
Diagnosis
The upper threshold of a normal human resting heart rate is based on age. Cutoff values for tachycardia in different age groups are fairly well standardized; typical cutoffs are listed below:
1–2 days: Tachycardia >159 beats per minute (bpm)
3–6 days: Tachycardia >166 bpm
1–3 weeks: Tachycardia >182 bpm
1–2 months: Tachycardia >179 bpm
3–5 months: Tachycardia >186 bpm
6–11 months: Tachycardia >169 bpm
1–2 years: Tachycardia >151 bpm
3–4 years: Tachycardia >137 bpm
5–7 years: Tachycardia >133 bpm
8–11 years: Tachycardia >130 bpm
12–15 years: Tachycardia >119 bpm
>15 years – adult: Tachycardia >100 bpm
Heart rate is considered in the context of the prevailing clinical picture. When the heart beats excessively or rapidly, the heart pumps less efficiently and provides less blood flow to the rest of the body, including the heart itself. The increased heart rate also leads to increased work and oxygen demand by the heart, which can lead to rate related ischemia.
Differential diagnosis | Tachycardia | Wikipedia | 274 | 160581 | https://en.wikipedia.org/wiki/Tachycardia | Biology and health sciences | Symptoms and signs | Health |
An electrocardiogram (ECG) is used to classify the type of tachycardia. They may be classified into narrow and wide complex based on the QRS complex. Equal or less than 0.1s for narrow complex. Presented in order of most to least common, they are:
Narrow complex
Sinus tachycardia, which originates from the sino-atrial (SA) node, near the base of the superior vena cava
Atrial fibrillation
Atrial flutter
AV nodal reentrant tachycardia
Accessory pathway mediated tachycardia
Atrial tachycardia
Multifocal atrial tachycardia
Cardiac Tamponade
Junctional tachycardia (rare in adults)
Wide complex
Ventricular tachycardia, any tachycardia that originates in the ventricles
Any narrow complex tachycardia combined with a problem with the conduction system of the heart, often termed "supraventricular tachycardia with aberrancy"
A narrow complex tachycardia with an accessory conduction pathway, often termed "supraventricular tachycardia with pre-excitation" (e.g. Wolff–Parkinson–White syndrome)
Pacemaker-tracked or pacemaker-mediated tachycardia
Tachycardias may be classified as either narrow complex tachycardias (supraventricular tachycardias) or wide complex tachycardias. Narrow and wide refer to the width of the QRS complex on the ECG. Narrow complex tachycardias tend to originate in the atria, while wide complex tachycardias tend to originate in the ventricles. Tachycardias can be further classified as either regular or irregular.
Sinus
The body has several feedback mechanisms to maintain adequate blood flow and blood pressure. If blood pressure decreases, the heart beats faster in an attempt to raise it. This is called reflex tachycardia. This can happen in response to a decrease in blood volume (through dehydration or bleeding), or an unexpected change in blood flow. The most common cause of the latter is orthostatic hypotension (also called postural hypotension). Fever, hyperventilation, diarrhea and severe infections can also cause tachycardia, primarily due to increase in metabolic demands. | Tachycardia | Wikipedia | 491 | 160581 | https://en.wikipedia.org/wiki/Tachycardia | Biology and health sciences | Symptoms and signs | Health |
Upon exertion, sinus tachycardia can also be seen in some inborn errors of metabolism that result in metabolic myopathies, such as McArdle's disease (GSD-V). Metabolic myopathies interfere with the muscle's ability to create energy. This energy shortage in muscle cells causes an inappropriate rapid heart rate in response to exercise. The heart tries to compensate for the energy shortage by increasing heart rate to maximize delivery of oxygen and other blood borne fuels to the muscle cells.
"In McArdle's, our heart rate tends to increase in what is called an 'inappropriate' response. That is, after the start of exercise it increases much more quickly than would be expected in someone unaffected by McArdle's." As skeletal muscle relies predominantly on glycogenolysis for the first few minutes as it transitions from rest to activity, as well as throughout high-intensity aerobic activity and all anaerobic activity, individuals with GSD-V experience during exercise: sinus tachycardia, tachypnea, muscle fatigue and pain, during the aforementioned activities and time frames. Those with GSD-V also experience "second wind", after approximately 6–10 minutes of light-moderate aerobic activity, such as walking without an incline, where the heart rate drops and symptoms of exercise intolerance improve.
An increase in sympathetic nervous system stimulation causes the heart rate to increase, both by the direct action of sympathetic nerve fibers on the heart and by causing the endocrine system to release hormones such as epinephrine (adrenaline), which have a similar effect. Increased sympathetic stimulation is usually due to physical or psychological stress. This is the basis for the so-called fight-or-flight response, but such stimulation can also be induced by stimulants such as ephedrine, amphetamines or cocaine. Certain endocrine disorders such as pheochromocytoma can also cause epinephrine release and can result in tachycardia independent of nervous system stimulation. Hyperthyroidism can also cause tachycardia. The upper limit of normal rate for sinus tachycardia is thought to be 220 bpm minus age.
Inappropriate sinus tachycardia | Tachycardia | Wikipedia | 468 | 160581 | https://en.wikipedia.org/wiki/Tachycardia | Biology and health sciences | Symptoms and signs | Health |
Inappropriate sinus tachycardia (IST) is a diagnosis of exclusion, a rare but benign type of cardiac arrhythmia that may be caused by a structural abnormality in the sinus node. It can occur in seemingly healthy individuals with no history of cardiovascular disease. Other causes may include autonomic nervous system deficits, autoimmune response, or drug interactions. Although symptoms might be distressing, treatment is not generally needed.
Ventricular
Ventricular tachycardia (VT or V-tach) is a potentially life-threatening cardiac arrhythmia that originates in the ventricles. It is usually a regular, wide complex tachycardia with a rate between 120 and 250 beats per minute. A medically significant subvariant of ventricular tachycardia is called torsades de pointes (literally meaning "twisting of the points", due to its appearance on an EKG), which tends to result from a long QT interval.
Both of these rhythms normally last for only a few seconds to minutes (paroxysmal tachycardia), but if VT persists it is extremely dangerous, often leading to ventricular fibrillation.
Supraventricular
This is a type of tachycardia that originates from above the ventricles, such as the atria. It is sometimes known as paroxysmal atrial tachycardia (PAT). Several types of supraventricular tachycardia are known to exist.
Atrial fibrillation
Atrial fibrillation is one of the most common cardiac arrhythmias. In general, it is an irregular, narrow complex rhythm. However, it may show wide QRS complexes on the ECG if a bundle branch block is present. At high rates, the QRS complex may also become wide due to the Ashman phenomenon. It may be difficult to determine the rhythm's regularity when the rate exceeds 150 beats per minute. Depending on the patient's health and other variables such as medications taken for rate control, atrial fibrillation may cause heart rates that span from 50 to 250 beats per minute (or even higher if an accessory pathway is present). However, new-onset atrial fibrillation tends to present with rates between 100 and 150 beats per minute. | Tachycardia | Wikipedia | 486 | 160581 | https://en.wikipedia.org/wiki/Tachycardia | Biology and health sciences | Symptoms and signs | Health |
AV nodal reentrant tachycardia
AV nodal reentrant tachycardia (AVNRT) is the most common reentrant tachycardia. It is a regular narrow complex tachycardia that usually responds well to the Valsalva maneuver or the drug adenosine. However, unstable patients sometimes require synchronized cardioversion. Definitive care may include catheter ablation.
AV reentrant tachycardia
AV reentrant tachycardia (AVRT) requires an accessory pathway for its maintenance. AVRT may involve orthodromic conduction (where the impulse travels down the AV node to the ventricles and back up to the atria through the accessory pathway) or antidromic conduction (which the impulse travels down the accessory pathway and back up to the atria through the AV node). Orthodromic conduction usually results in a narrow complex tachycardia, and antidromic conduction usually results in a wide complex tachycardia that often mimics ventricular tachycardia. Most antiarrhythmics are contraindicated in the emergency treatment of AVRT, because they may paradoxically increase conduction across the accessory pathway.
Junctional tachycardia
Junctional tachycardia is an automatic tachycardia originating in the AV junction. It tends to be a regular, narrow complex tachycardia and may be a sign of digitalis toxicity.
Management
The management of tachycardia depends on its type (wide complex versus narrow complex), whether or not the person is stable or unstable, and whether the instability is due to the tachycardia. Unstable means that either important organ functions are affected or cardiac arrest is about to occur. Stable means that there is a tachycardia, but it does not seem an immediate threat for the patient's health, but only a symptom of an unknown disease, or a reaction that is not very dangerous in that moment.
Unstable
In those that are unstable with a narrow complex tachycardia, intravenous adenosine may be attempted. In all others, immediate cardioversion is recommended. | Tachycardia | Wikipedia | 449 | 160581 | https://en.wikipedia.org/wiki/Tachycardia | Biology and health sciences | Symptoms and signs | Health |
Stable
If the problem is a simple acceleration of the heart rate that worries the patient, but the heart and the general patient's health remain stable enough, it is possible to correct it by a simple deceleration using some physical maneuvers called vagal maneuvers. But, if the cause of the tachycardia is chronic (permanent), it would return after some time, unless that cause is corrected.
Besides, the patient should avoid receiving external effects that cause or increase tachycardia.
The same measures than in unstable tachycardia can also be taken, with medications and the type of cardioversion that is appropriate for the patient's tachycardia. | Tachycardia | Wikipedia | 137 | 160581 | https://en.wikipedia.org/wiki/Tachycardia | Biology and health sciences | Symptoms and signs | Health |
The word tachycardia came to English from Neo-Latin as a neoclassical compound built from the combining forms tachy- + -cardia, which are from the Greek ταχύς tachys, "quick, rapid" and καρδία, kardia, "heart". As a matter both of usage choices in the medical literature and of idiom in natural language, the words tachycardia and tachyarrhythmia are usually used interchangeably, or loosely enough that precise differentiation is not explicit. Some careful writers have tried to maintain a logical differentiation between them, which is reflected in major medical dictionaries and major general dictionaries. The distinction is that tachycardia be reserved for the rapid heart rate itself, regardless of cause, physiologic or pathologic (that is, from healthy response to exercise or from cardiac arrhythmia), and that tachyarrhythmia be reserved for the pathologic form (that is, an arrhythmia of the rapid rate type). This is why five of the previously referenced dictionaries do not enter cross-references indicating synonymy between their entries for the two words (as they do elsewhere whenever synonymy is meant), and it is why one of them explicitly specifies that the two words not be confused. But the prescription will probably never be successfully imposed on general usage, not only because much of the existing medical literature ignores it even when the words stand alone but also because the terms for specific types of arrhythmia (standard collocations of adjectives and noun) are deeply established idiomatically with the tachycardia version as the more commonly used version. Thus SVT is called supraventricular tachycardia more than twice as often as it is called supraventricular tachyarrhythmia; moreover, those two terms are always completely synonymous—in natural language there is no such term as "healthy/physiologic supraventricular tachycardia". The same themes are also true of AVRT and AVNRT. Thus this pair is an example of when a particular prescription (which may have been tenable 50 or 100 years earlier) can no longer be invariably enforced without violating idiom | Tachycardia | Wikipedia | 483 | 160581 | https://en.wikipedia.org/wiki/Tachycardia | Biology and health sciences | Symptoms and signs | Health |
But the power to differentiate in an idiomatic way is not lost, regardless, because when the specification of physiologic tachycardia is needed, that phrase aptly conveys it | Tachycardia | Wikipedia | 41 | 160581 | https://en.wikipedia.org/wiki/Tachycardia | Biology and health sciences | Symptoms and signs | Health |
Plovers ( , ) are members of a widely distributed group of wading birds of subfamily Charadriinae. The term "plover" applies to all the members of the subfamily, though only about half of them include it in their name.
Species list in taxonomic sequence
The taxonomy of family Charadriidae is unsettled. At various times the plovers, dotterels, and lapwings of family Charadriidae have been distributed among several subfamilies, with Charadriinae including most of the species. The International Ornithological Congress (IOC) and the Clements taxonomy do not assign species to subfamilies. The South American Classification Committee of the American Ornithological Society (AOS) includes all of the species in Charadriinae. The North American Classification Committee of the AOS and BirdLife International's Handbook of the Birds of the World separate the four members of genus Pluvialis as subfamily Pluvialinae.
The IOC recognizes these 69 species of plovers, dotterels, and lapwings in family Charadriidae. They are distributed among 11 genera, some of which have only one species. This list is presented according to the IOC taxonomic sequence and can also be sorted alphabetically by common name and binomial.
Description
Plovers are found throughout the world, with the exception of the Sahara and the polar regions, and are characterised by relatively short bills. They hunt by sight, rather than by feel as longer-billed waders like snipes do. They feed mainly on insects, worms or other invertebrates, depending on the habitat, which are obtained by a run-and-pause technique, rather than the steady probing of some other wader groups. Plovers engage in false brooding, a type of distraction display. Examples include pretending to change position or to sit on an imaginary nest site.
In folklore
The European golden plover spends summers in Iceland, and in Icelandic folklore, the appearance of the first plover in the country means that spring has arrived. The Icelandic media always covers the first plover sighting. | Plover | Wikipedia | 427 | 160710 | https://en.wikipedia.org/wiki/Plover | Biology and health sciences | Charadriiformes | Animals |
A tunnel is an underground or undersea passageway. It is dug through surrounding soil, earth or rock, or laid under water, and is usually completely enclosed except for the two portals common at each end, though there may be access and ventilation openings at various points along the length. A pipeline differs significantly from a tunnel, though some recent tunnels have used immersed tube construction techniques rather than traditional tunnel boring methods.
A tunnel may be for foot or vehicular road traffic, for rail traffic, or for a canal. The central portions of a rapid transit network are usually in the tunnel. Some tunnels are used as sewers or aqueducts to supply water for consumption or for hydroelectric stations. Utility tunnels are used for routing steam, chilled water, electrical power or telecommunication cables, as well as connecting buildings for convenient passage of people and equipment.
Secret tunnels are built for military purposes, or by civilians for smuggling of weapons, contraband, or people. Special tunnels, such as wildlife crossings, are built to allow wildlife to cross human-made barriers safely. Tunnels can be connected together in tunnel networks.
A tunnel is relatively long and narrow; the length is often much greater than twice the diameter, although similar shorter excavations can be constructed, such as cross passages between tunnels. The definition of what constitutes a tunnel can vary widely from source to source. For example, in the United Kingdom, a road tunnel is defined as "a subsurface highway structure enclosed for a length of or more." In the United States, the NFPA definition of a tunnel is "An underground structure with a design length greater than and a diameter greater than ."
Etymology
The word "tunnel" comes from the Middle English tonnelle, meaning "a net", derived from Old French tonnel, a diminutive of tonne ("cask"). The modern meaning, referring to an underground passageway, evolved in the 16th century as a metaphor for a narrow, confined space like the inside of a cask.
History
In Babylon, about 2200 B.C., it is believed that the first artificial tunnel was constructed. To join the temple of Belos with the palace, this was built with the aid of the cut and cover technique. | Tunnel | Wikipedia | 450 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
In the Mahabharata, the Pandavas built a secret tunnel within their new home, called "Lakshagriha" (House of Lac), which was constructed by Purochana under the orders of Duryodhana by the intention of burning them alive inside, allowing them to escape when the palace was set on fire; this act of foresight by the Pandavas saved their lives
Some of the earliest tunnels used by humans were paleoburrows excavated by prehistoric mammals.
Much of the early technology of tunnelling evolved from mining and military engineering. The etymology of the terms "mining" (for mineral extraction or for siege attacks), "military engineering", and "civil engineering" reveals these deep historic connections.
Antiquity and early middle ages
Predecessors of modern tunnels were adits that transported water for irrigation, drinking, or sewerage. The first qanats are known from before 2000 BC.
The earliest tunnel known to have been excavated from both ends is the Siloam Tunnel, built in Jerusalem by the kings of Judah around the 8th century BC. Another tunnel excavated from both ends, maybe the second known, is the Tunnel of Eupalinos, which is a tunnel aqueduct long running through Mount Kastro in Samos, Greece. It was built in the 6th century BC to serve as an aqueduct.
In Pakistan, the mughal era tunnel has been restored in the Lahore.
In Ethiopia, the Siqurto foot tunnel, hand-hewn in the Middle Ages, crosses a mountain ridge.
In the Gaza Strip, the network of tunnels was used by Jewish strategists as rock-cut shelters, in first links to Judean resistance against Roman rule in the Bar Kokhba revolt during the 2nd century AD.
Geotechnical investigation and design
A major tunnel project must start with a comprehensive investigation of ground conditions by collecting samples from boreholes and by other geophysical techniques. An informed choice can then be made of machinery and methods for excavation and ground support, which will reduce the risk of encountering unforeseen ground conditions. In planning the route, the horizontal and vertical alignments can be selected to make use of the best ground and water conditions. It is common practice to locate a tunnel deeper than otherwise would be required, in order to excavate through solid rock or other material that is easier to support during construction. | Tunnel | Wikipedia | 479 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Conventional desk and preliminary site studies may yield insufficient information to assess such factors as the blocky nature of rocks, the exact location of fault zones, or the stand-up times of softer ground. This may be a particular concern in large-diameter tunnels. To give more information, a pilot tunnel (or "drift tunnel") may be driven ahead of the main excavation. This smaller tunnel is less likely to collapse catastrophically should unexpected conditions be met, and it can be incorporated into the final tunnel or used as a backup or emergency escape passage. Alternatively, horizontal boreholes may sometimes be drilled ahead of the advancing tunnel face.
Other key geotechnical factors:
Stand-up time is the amount of time a newly excavated cavity can support itself without any added structures. Knowing this parameter allows the engineers to determine how far an excavation can proceed before support is needed, which in turn affects the speed, efficiency, and cost of construction. Generally, certain configurations of rock and clay will have the greatest stand-up time, while sand and fine soils will have a much lower stand-up time.
Groundwater control is very important in tunnel construction. Water leaking into a tunnel or vertical shaft will greatly decrease stand-up time, causing the excavation to become unstable and risking collapse. The most common way to control groundwater is to install dewatering pipes into the ground and to simply pump the water out. A very effective but expensive technology is ground freezing, using pipes which are inserted into the ground surrounding the excavation, which are then cooled with special refrigerant fluids. This freezes the ground around each pipe until the whole space is surrounded with frozen soil, keeping water out until a permanent structure can be built.
Tunnel cross-sectional shape is also very important in determining stand-up time. If a tunnel excavation is wider than it is high, it will have a harder time supporting itself, decreasing its stand-up time. A square or rectangular excavation is more difficult to make self-supporting, because of a concentration of stress at the corners.
Choice of tunnels versus bridges
For water crossings, a tunnel is generally more costly to construct than a bridge. However, both navigational and traffic considerations may limit the use of high bridges or drawbridges intersecting with shipping channels, necessitating a tunnel. | Tunnel | Wikipedia | 460 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Bridges usually require a larger footprint on each shore than tunnels. In areas with expensive real estate, such as Manhattan and urban Hong Kong, this is a strong factor in favor of a tunnel. Boston's Big Dig project replaced elevated roadways with a tunnel system to increase traffic capacity, hide traffic, reclaim land, redecorate, and reunite the city with the waterfront.
The 1934 Queensway Tunnel under the River Mersey at Liverpool was chosen over a massively high bridge partly for defence reasons; it was feared that aircraft could destroy a bridge in times of war, not merely impairing road traffic but blocking the river to navigation. Maintenance costs of a massive bridge to allow the world's largest ships to navigate under were considered higher than for a tunnel. Similar conclusions were reached for the 1971 Kingsway Tunnel under the Mersey. In Hampton Roads, Virginia, tunnels were chosen over bridges for strategic considerations; in the event of damage, bridges might prevent US Navy vessels from leaving Naval Station Norfolk.
Water-crossing tunnels built instead of bridges include the Seikan Tunnel in Japan; the Holland Tunnel and Lincoln Tunnel between New Jersey and Manhattan in New York City; the Queens-Midtown Tunnel between Manhattan and the borough of Queens on Long Island; the Detroit-Windsor Tunnel between Michigan and Ontario; and the Elizabeth River tunnels between Norfolk and Portsmouth, Virginia; the 1934 River Mersey road Queensway Tunnel; the Western Scheldt Tunnel, Zeeland, Netherlands; and the North Shore Connector tunnel in Pittsburgh, Pennsylvania. The Sydney Harbour Tunnel was constructed to provide a second harbour crossing and to alleviate traffic congestion on the Sydney Harbour Bridge, without spoiling the iconic view.
Other reasons for choosing a tunnel instead of a bridge include avoiding difficulties with tides, weather, and shipping during construction (as in the Channel Tunnel), aesthetic reasons (preserving the above-ground view, landscape, and scenery), and also for weight capacity reasons (it may be more feasible to build a tunnel than a sufficiently strong bridge).
Some water crossings are a mixture of bridges and tunnels, such as the Denmark to Sweden link and the Chesapeake Bay Bridge-Tunnel in Virginia. | Tunnel | Wikipedia | 433 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
There are particular hazards with tunnels, especially from vehicle fires when combustion gases can asphyxiate users, as happened at the Gotthard Road Tunnel in Switzerland in 2001. One of the worst railway disasters ever, the Balvano train disaster, was caused by a train stalling in the Armi tunnel in Italy in 1944, killing 426 passengers. Designers try to reduce these risks by installing emergency ventilation systems or isolated emergency escape tunnels parallel to the main passage.
Project planning and cost estimates
Government funds are often required for the creation of tunnels. When a tunnel is being planned or constructed, economics and politics play a large factor in the decision making process. Civil engineers usually use project management techniques for developing a major structure. Understanding the amount of time the project requires, and the amount of labor and materials needed is a crucial part of project planning. The project duration must be identified using a work breakdown structure and critical path method. Also, the land needed for excavation and construction staging, and the proper machinery must be selected. Large infrastructure projects require millions or even billions of dollars, involving long-term financing, usually through issuance of bonds.
The costs and benefits for an infrastructure such as a tunnel must be identified. Political disputes can occur, as in 2005 when the US House of Representatives approved a $100 million federal grant to build a tunnel under New York Harbor. However, the Port Authority of New York and New Jersey was not aware of this bill and had not asked for a grant for such a project. Increased taxes to finance a large project may cause opposition.
Construction
Tunnels are dug in types of materials varying from soft clay to hard rock. The method of tunnel construction depends on such factors as the ground conditions, the groundwater conditions, the length and diameter of the tunnel drive, the depth of the tunnel, the logistics of supporting the tunnel excavation, the final use and the shape of the tunnel and appropriate risk management.
There are three basic types of tunnel construction in common use. Cut-and-cover tunnels are constructed in a shallow trench and then covered over. Bored tunnels are constructed in situ, without removing the ground above. Finally, a tube can be sunk into a body of water, which is called an immersed tunnel.
Cut-and-cover
Cut-and-cover is a simple method of construction for shallow tunnels where a trench is excavated and roofed over with an overhead support system strong enough to carry the load of what is to be built above the tunnel. | Tunnel | Wikipedia | 495 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
There are two basic forms of cut-and-cover tunnelling:
Bottom-up method: A trench is excavated, with ground support as necessary, and the tunnel is constructed in it. The tunnel may be of in situ concrete, precast concrete, precast arches, or corrugated steel arches; in early days brickwork was used. The trench is then carefully back-filled and the surface is reinstated.
Top-down method: Side support walls and capping beams are constructed from ground level by such methods as slurry walling or contiguous bored piling. Only a shallow excavation is needed to construct the tunnel roof using precast beams or in situ concrete sitting on the walls. The surface is then reinstated except for access openings. This allows early reinstatement of roadways, services, and other surface features. Excavation then takes place under the permanent tunnel roof, and the base slab is constructed.
Shallow tunnels are often of the cut-and-cover type (if under water, of the immersed-tube type), while deep tunnels are excavated, often using a tunnelling shield. For intermediate levels, both methods are possible.
Large cut-and-cover boxes are often used for underground metro stations, such as Canary Wharf tube station in London. This construction form generally has two levels, which allows economical arrangements for ticket hall, station platforms, passenger access and emergency egress, ventilation and smoke control, staff rooms, and equipment rooms. The interior of Canary Wharf station has been likened to an underground cathedral, owing to the sheer size of the excavation. This contrasts with many traditional stations on London Underground, where bored tunnels were used for stations and passenger access. Nevertheless, the original parts of the London Underground network, the Metropolitan and District Railways, were constructed using cut-and-cover. These lines pre-dated electric traction and the proximity to the surface was useful to ventilate the inevitable smoke and steam.
A major disadvantage of cut-and-cover is the widespread disruption generated at the surface level during construction. This, and the availability of electric traction, brought about London Underground's switch to bored tunnels at a deeper level towards the end of the 19th century. | Tunnel | Wikipedia | 439 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Prior to the replacement of manual excavation by the use of boring machines, Victorian tunnel excavators developed a specialized method called clay-kicking for digging tunnels in clay-based soils. The clay-kicker lies on a plank at a 45-degree angle away from the working face and rather than a mattock with his hands, inserts with his feet a tool with a cup-like rounded end, then turns the tool with his hands to extract a section of soil, which is then placed on the waste extract.
Clay-kicking is a specialized method developed in the United Kingdom of digging tunnels in strong clay-based soil structures. This method of cut and cover construction required relatively little disturbance of property during the renewal of the United Kingdom's then ancient sewerage systems. It was also used during the First World War by Royal Engineer tunnelling companies placing mines beneath German lines, because it was almost silent and so not susceptible to listening methods of detection.
Boring machines
Tunnel boring machines (TBMs) and associated back-up systems are used to highly automate the entire tunnelling process, reducing tunnelling costs. In certain predominantly urban applications, tunnel boring is viewed as a quick and cost-effective alternative to laying surface rails and roads. Expensive compulsory purchase of buildings and land, with potentially lengthy planning inquiries, is eliminated. Disadvantages of TBMs arise from their usually large size – the difficulty of transporting the large TBM to the site of tunnel construction, or (alternatively) the high cost of assembling the TBM on-site, often within the confines of the tunnel being constructed. | Tunnel | Wikipedia | 320 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
There are a variety of TBM designs that can operate in a variety of conditions, from hard rock to soft water-bearing ground. Some TBMs, the bentonite slurry and earth-pressure balance types, have pressurized compartments at the front end, allowing them to be used in difficult conditions below the water table. This pressurizes the ground ahead of the TBM cutter head to balance the water pressure. The operators work in normal air pressure behind the pressurized compartment, but may occasionally have to enter that compartment to renew or repair the cutters. This requires special precautions, such as local ground treatment or halting the TBM at a position free from water. Despite these difficulties, TBMs are now preferred over the older method of tunnelling in compressed air, with an airlock/decompression chamber some way back from the TBM, which required operators to work in high pressure and go through decompression procedures at the end of their shifts, much like deep-sea divers.
In February 2010, Aker Wirth delivered a TBM to Switzerland, for the expansion of the Linth–Limmern Power Stations located south of Linthal in the canton of Glarus. The borehole has a diameter of . The four TBMs used for excavating the Gotthard Base Tunnel, in Switzerland, had a diameter of about . A larger TBM was built to bore the Green Heart Tunnel (Dutch: Tunnel Groene Hart) as part of the HSL-Zuid in the Netherlands, with a diameter of . This in turn was superseded by the Madrid M30 ringroad, Spain, and the Chong Ming tunnels in Shanghai, China. All of these machines were built at least partly by Herrenknecht. , the world's largest TBM was "Big Bertha", a diameter machine built by Hitachi Zosen Corporation, which dug the Alaskan Way Viaduct replacement tunnel in Seattle, Washington (US).
Shafts | Tunnel | Wikipedia | 404 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
A temporary access shaft is sometimes necessary during the excavation of a tunnel. They are usually circular and go straight down until they reach the level at which the tunnel is going to be built. A shaft normally has concrete walls and is usually built to be permanent. Once the access shafts are complete, TBMs are lowered to the bottom and excavation can start. Shafts are the main entrance in and out of the tunnel until the project is completed. If a tunnel is going to be long, multiple shafts at various locations may be bored so that entrance to the tunnel is closer to the unexcavated area.
Once construction is complete, construction access shafts are often used as ventilation shafts, and may also be used as emergency exits.
Sprayed concrete techniques
The new Austrian tunnelling method (NATM)—also referred to as the Sequential Excavation Method (SEM)—was developed in the 1960s.
The main idea of this method is to use the geological stress of the surrounding rock mass to stabilize the tunnel, by allowing a measured relaxation and stress reassignment into the surrounding rock to prevent full loads becoming imposed on the supports. Based on geotechnical measurements, an optimal cross section is computed. The excavation is protected by a layer of sprayed concrete, commonly referred to as shotcrete. Other support measures can include steel arches, rock bolts, and mesh. Technological developments in sprayed concrete technology have resulted in steel and polypropylene fibers being added to the concrete mix to improve lining strength. This creates a natural load-bearing ring, which minimizes the rock's deformation.
By special monitoring the NATM method is flexible, even at surprising changes of the geomechanical rock consistency during the tunneling work. The measured rock properties lead to appropriate tools for tunnel strengthening.
Pipe jacking
In pipe jacking, hydraulic jacks are used to push specially made pipes through the ground behind a TBM or shield. This method is commonly used to create tunnels under existing structures, such as roads or railways. Tunnels constructed by pipe jacking are normally small diameter bores with a maximum size of around . | Tunnel | Wikipedia | 421 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Box jacking
Box jacking is similar to pipe jacking, but instead of jacking tubes, a box-shaped tunnel is used. Jacked boxes can be a much larger span than a pipe jack, with the span of some box jacks in excess of . A cutting head is normally used at the front of the box being jacked, and spoil removal is normally by excavator from within the box. Recent developments of the Jacked Arch and Jacked deck have enabled longer and larger structures to be installed to close accuracy.
Underwater tunnels
There are also several approaches to underwater tunnels, the two most common being bored tunnels or immersed tubes, examples are Bjørvika Tunnel and Marmaray. Submerged floating tunnels are a novel approach under consideration; however, no such tunnels have been constructed to date.
Temporary way
During construction of a tunnel it is often convenient to install a temporary railway, particularly to remove excavated spoil, often narrow gauge so that it can be double track to allow the operation of empty and loaded trains at the same time. The temporary way is replaced by the permanent way at completion, thus explaining the term "Perway".
Enlargement
The vehicles or traffic using a tunnel can outgrow it, requiring replacement or enlargement:
The original single line Gib Tunnel near Mittagong was replaced with a double-track tunnel, with the original tunnel used for growing mushrooms.
The 1832 double-track -long tunnel from Edge Hill to Lime Street in Liverpool was near totally removed, apart from a section at Edge Hill and a section nearer to Lime Street, as four tracks were required. The tunnel was dug out into a very deep four-track cutting, with short tunnels in places along the cutting. Train services were not interrupted as the work progressed. There are other occurrences of tunnels being replaced by open cuts, for example, the Auburn Tunnel.
The Farnworth Tunnel in England was enlarged using a tunnel boring machine (TBM) in 2015. The Rhyndaston Tunnel was enlarged using a borrowed TBM so as to be able to take ISO containers.
Tunnels can also be enlarged by lowering the floor.
Open building pit
An open building pit consists of a horizontal and a vertical boundary that keeps groundwater and soil out of the pit. There are several potential alternatives and combinations for (horizontal and vertical) building pit boundaries. The most important difference with cut-and-cover is that the open building pit is muted after tunnel construction; no roof is placed.
Other construction methods | Tunnel | Wikipedia | 506 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Drilling and blasting
Hydraulic splitter
Slurry-shield machine
Wall-cover construction method.
Variant tunnel types
Double-deck and multipurpose tunnels
Some tunnels are double-deck, for example, the two major segments of the San Francisco–Oakland Bay Bridge (completed in 1936) are linked by a double-deck tunnel section through Yerba Buena Island, the largest-diameter bored tunnel in the world. At construction this was a combination bidirectional rail and truck pathway on the lower deck with automobiles above, now converted to one-way road vehicle traffic on each deck.
In Turkey, the Eurasia Tunnel under the Bosphorus, opened in 2016, has at its core a two-deck road tunnel with two lanes on each deck.
Additionally, in 2015 the Turkish government announced that it will build three-level tunnel, also under the Bosporus. The tunnel is intended to carry both the Istanbul metro and a two-level highway, over a length of .
The French in west Paris consists of two bored tunnel tubes, the eastern one of which has two levels for light motorized vehicles, over a length of . Although each level offers a physical height of , only traffic up to tall is allowed in this tunnel tube, and motorcyclists are directed to the other tube. Each level was built with a three-lane roadway, but only two lanes per level are used – the third serves as a hard shoulder within the tunnel. The A86 Duplex is Europe's longest double-deck tunnel.
In Shanghai, China, a two-tube double-deck tunnel was built starting in 2002. In each tube of the both decks are for motor vehicles. In each direction, only cars and taxis travel on the high two-lane upper deck, and heavier vehicles, like trucks and buses, as well as cars, may use the high single-lane lower level.
In the Netherlands, a two-storey, eight-lane, cut-and-cover road tunnel under the city of Maastricht was opened in 2016. Each level accommodates a full height, two by two-lane highway. The two lower tubes of the tunnel carry the A2 motorway, which originates in Amsterdam, through the city; and the two upper tubes take the N2 regional highway for local traffic. | Tunnel | Wikipedia | 463 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
The Alaskan Way Viaduct replacement tunnel, is a $3.3 billion , double-decker bored highway tunnel under Downtown Seattle. Construction began in July 2013 using "Bertha", at the time the world's largest earth pressure balance tunnel boring machine, with a cutterhead diameter. After several delays, tunnel boring was completed in April 2017, and the tunnel opened to traffic on 4 February 2019.
New York City's 63rd Street Tunnel under the East River, between the boroughs of Manhattan and Queens, was intended to carry subway trains on the upper level and Long Island Rail Road commuter trains on the lower level. Construction started in 1969, and the two sides of the tunnel were bored through in 1972. The upper level, used by the IND 63rd Street Line () of the New York City Subway, was not opened for passenger service until 1989. The lower level, intended for commuter rail, saw passenger service after completion of the East Side Access project, in late 2022.
In the UK, the 1934 Queensway Tunnel under the River Mersey between Liverpool and Birkenhead was originally to have road vehicles running on the upper deck and trams on the lower. During construction the tram usage was cancelled. The lower section is only used for cables, pipes and emergency accident refuge enclosures.
Hong Kong's Lion Rock Tunnel, built in the mid 1960s, connecting New Kowloon and Sha Tin, carries a motorway but also serves as an aqueduct, featuring a gallery containing five water mains lines with diameters between below the road section of the tunnel.
Wuhan's Yangtze River Highway and Railway Tunnel is a two-tube double-deck tunnel under the Yangtze River completed in 2018. Each tube carries three lanes of local traffic on the top deck with one track Wuhan Metro Line 7 on the lower deck. | Tunnel | Wikipedia | 369 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Mount Baker Tunnel has three levels. The bottom level is to be used by Sound Transit light rail. The middle level is used by car traffic, and the top layer is for bicycle and pedestrian access.
Some tunnels have more than one purpose. The SMART Tunnel in Malaysia is the first multipurpose "Stormwater Management And Road Tunnel" in the world, created to convey both traffic and occasional flood waters in Kuala Lumpur. When necessary, floodwater is first diverted into a separate bypass tunnel located underneath the double-deck roadway tunnel. In this scenario, traffic continues normally. Only during heavy, prolonged rains when the threat of extreme flooding is high, the upper tunnel tube is closed off to vehicles and automated flood control gates are opened so that water can be diverted through both tunnels.
Covered passageways
Over-bridges can sometimes be built by covering a road or river or railway with brick or steel arches, and then levelling the surface with earth. In railway parlance, a surface-level track which has been built or covered over is normally called a "covered way".
Snow sheds are a kind of artificial tunnel built to protect a railway from avalanches of snow. Similarly the Stanwell Park, New South Wales "steel tunnel", on the Illawarra railway line, protects the line from rockfalls.
Underpass
An underpass is a road or railway or other passageway passing under another road or railway, under an overpass. This is not strictly a tunnel.
Utility Tunnel
A Utility Tunnel is built for the purpose of carrying one or more utilities in the same space, for this reason they are also referred to as Multi-Utility Tunnels or MUTs. Through co-location of different utilities in one tunnel, organizations are able to reduce the financial and environmental costs of building and maintaining utilities. These tunnels can be used for many types of utilities, routing steam, chilled water, electrical power or telecommunication cables, as well as connecting buildings for convenient passage of people and equipment.
Safety and security | Tunnel | Wikipedia | 398 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Owing to the enclosed space of a tunnel, fires can have very serious effects on users. The main dangers are gas and smoke production, with even low concentrations of carbon monoxide being highly toxic. Fires killed 11 people in the Gotthard tunnel fire of 2001 for example, all of the victims succumbing to smoke and gas inhalation. Over 400 passengers died in the Balvano train disaster in Italy in 1944, when the locomotive halted in a long tunnel. Carbon monoxide poisoning was the main cause of death. In the Caldecott Tunnel fire of 1982, the majority of fatalities were caused by toxic smoke, rather than by the initial crash. Likewise 84 people were killed in the Paris Métro train fire of 1904.
Motor vehicle tunnels usually require ventilation shafts and powered fans to remove toxic exhaust gases during routine operation.
Rail tunnels usually require fewer air changes per hour, but still may require forced-air ventilation. Both types of tunnels often have provisions to increase ventilation under emergency conditions, such as a fire. Although there is a risk of increasing the rate of combustion through increased airflow, the primary focus is on providing breathable air to persons trapped in the tunnel, as well as firefighters.
The aerodynamic pressure wave produced by high speed trains entering a tunnel reflect at its open ends and change sign (compression wavefront changes to rarefaction wavefront and vice versa). When two wavefronts of the same sign meet the train, significant and rapid air pressure may cause ear discomfort for passengers and crew. When a high-speed train exits a tunnel, a loud "Tunnel boom" may occur, which can disturb residents near the mouth of the tunnel, and it is exacerbated in mountain valleys where the sound can echo.
When there is a parallel, separate tunnel available, airtight but unlocked emergency doors are usually provided which allow trapped personnel to escape from a smoke-filled tunnel to the parallel tube.
Larger, heavily used tunnels, such as the Big Dig tunnel in Boston, Massachusetts, may have a dedicated 24-hour staffed operations center which monitors and reports on traffic conditions, and responds to emergencies. Video surveillance equipment is often used, and real-time pictures of traffic conditions for some highways may be viewable by the general public via the Internet.
A database of seismic damage to underground structures using 217 case histories shows the following general observations can be made regarding the seismic performance of underground structures: | Tunnel | Wikipedia | 485 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Underground structures suffer appreciably less damage than surface structures.
Reported damage decreases with increasing over burden depth. Deep tunnels seem to be safer and less vulnerable to earthquake shaking than are shallow tunnels.
Underground facilities constructed in soils can be expected to suffer more damage compared to openings constructed in competent rock.
Lined and grouted tunnels are safer than unlined tunnels in rock. Shaking damage can be reduced by stabilizing the ground around the tunnel and by improving the contact between the lining and the surrounding ground through grouting.
Tunnels are more stable under a symmetric load, which improves ground-lining interaction. Improving the tunnel lining by placing thicker and stiffer sections without stabilizing surrounding poor ground may result in excess seismic forces in the lining. Backfilling with non-cyclically mobile material and rock-stabilizing measures may improve the safety and stability of shallow tunnels.
Damage may be related to peak ground acceleration and velocity based on the magnitude and epicentral distance of the affected earthquake.
Duration of strong-motion shaking during earthquakes is of utmost importance because it may cause fatigue failure and therefore, large deformations.
High frequency motions may explain the local spalling of rock or concrete along planes of weakness. These frequencies, which rapidly attenuate with distance, may be expected mainly at small distances from the causative fault.
Ground motion may be amplified upon incidence with a tunnel if wavelengths are between one and four times the tunnel diameter.
Damage at and near tunnel portals may be significant due to slope instability.
Earthquakes are one of nature's most formidable threats. A magnitude 6.7 earthquake shook the San Fernando valley in Los Angeles in 1994. The earthquake caused extensive damage to various structures, including buildings, freeway overpasses and road systems throughout the area. The National Center for Environmental Information estimates total damages to be 40 billion dollars. According to an article issued by Steve Hymon of TheSource – Transportation News and Views, there was no serious damage sustained by the LA subway system. Metro, the owner of the LA subway system, issued a statement through their engineering staff about the design and consideration that goes into a tunnel system. Engineers and architects perform extensive analysis as to how hard they expect earthquakes to hit that area. All of this goes into the overall design and flexibility of the tunnel. | Tunnel | Wikipedia | 464 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
This same trend of limited subway damage following an earthquake can be seen in many other places. In 1985 a magnitude 8.1 earthquake shook Mexico City; there was no damage to the subway system, and in fact the subway systems served as a lifeline for emergency personnel and evacuations. A magnitude 7.2 ripped through Kobe Japan in 1995, leaving no damage to the tunnels themselves. Entry portals sustained minor damages, however these damages were attributed to inadequate earthquake design that originated from the original construction date of 1965. In 2010 a magnitude 8.8, massive by any scale, afflicted Chile. Entrance stations to subway systems suffered minor damages, and the subway system was down for the rest of the day. By the next afternoon, the subway system was operational again.
Examples
In history
The history of ancient tunnels and tunneling in the world is reviewed in various sources which include many examples of these structures that were built for different purposes. Some well known ancient and modern tunnels are briefly introduced below: | Tunnel | Wikipedia | 199 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
The qanat or kareez of Persia are water management systems used to provide a reliable supply of water to human settlements or for irrigation in hot, arid and semi-arid climates. The deepest known qanat is in the Iranian city of Gonabad, which after 2700 years, still provides drinking and agricultural water to nearly 40,000 people. Its main well depth is more than , and its length is .
The Siloam Tunnel was built before 701 BC for a reliable supply of water, to withstand siege attacks.
The Eupalinian aqueduct on the island of Samos (North Aegean, Greece) was built in 520 BC by the ancient Greek engineer Eupalinos of Megara under a contract with the local community. Eupalinos organised the work so that the tunnel was begun from both sides of Mount Kastro. The two teams advanced simultaneously and met in the middle with excellent accuracy, something that was extremely difficult in that time. The aqueduct was of utmost defensive importance, since it ran underground, and it was not easily found by an enemy who could otherwise cut off the water supply to Pythagoreion, the ancient capital of Samos. The tunnel's existence was recorded by Herodotus (as was the mole and harbour, and the third wonder of the island, the great temple to Hera, thought by many to be the largest in the Greek world). The precise location of the tunnel was only re-established in the 19th century by German archaeologists. The tunnel proper is and visitors can still enter it.
One of the first known drainage and sewage networks in form of tunnels was constructed at Persepolis in Iran at the same time as the construction of its foundation in 518 BC. In most places the network was dug in the sound rock of the mountain and then covered by large pieces of rock and stone followed by earth and piles of rubble to level the ground. During investigations and surveys, long sections of similar rock tunnels extending beneath the palace area were traced by Herzfeld and later by Schmidt and their archaeological teams.
The Via Flaminia, an important Roman road, penetrated the Furlo pass in the Apennines through a tunnel which emperor Vespasian had ordered built in 76–77 AD. A modern road, the SS 3 Flaminia, still uses this tunnel, which had a precursor dating back to the 3rd century BC, remnants of this earlier tunnel (one of the first road tunnels) are also still visible. | Tunnel | Wikipedia | 505 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
The world's oldest tunnel traversing under a water body is claimed to be the Terelek kaya tüneli under Kızıl River, a little south of the towns of Boyabat and Durağan in Turkey, just downstream from where Kizil River joins its tributary Gökırmak. The tunnel is presently under a narrow part of a lake formed by a dam some kilometres further downstream. Estimated to have been built more than 2000 years ago, possibly by the same civilization that also built the royal tombs in a rock face nearby, it is assumed to have had a defensive purpose.
Sapperton Canal Tunnel on the Thames and Severn Canal in England, dug through hills, which opened in 1789, was long and allowed boat transport of coal and other goods. Above it the Sapperton Long Tunnel was constructed which carries the "Golden Valley" railway line between Swindon and Gloucester.
The 1791 Dudley canal tunnel is on the Dudley Canal, in Dudley, England. The tunnel is long. Closed in 1962 the tunnel was reopened in 1973. The series of tunnels was extended in 1984 and 1989.
Fritchley Tunnel, constructed in 1793 in Derbyshire by the Butterley Company to transport limestone to its ironworks factory. The Butterley company engineered and built its own railway. A victim of the depression the company closed after 219 years in 2009. The tunnel is the world's oldest railway tunnel traversed by rail wagons. Gravity and horse haulage was utilised. The railway was converted to steam locomotion in 1813 using a Steam Horse locomotive engineered and built by the Butterley company, however reverted to horses. Steam trains used the tunnel continuously from the 1840s when the railway was converted to a narrow gauge. The line closed in 1933. In the Second World War, the tunnel was used as an air raid shelter. Sealed up in 1977 it was rediscovered in 2013 and inspected. The tunnel was resealed to preserved the construction as it was designated an ancient monument.
The 1794 Butterley canal tunnel canal tunnel is in length on the Cromford Canal in Ripley, Derbyshire, England. The tunnel was built simultaneously with the 1793 Fritchley railway tunnel. The tunnel partially collapsed in 1900 splitting the Cromford Canal, and has not been used since. The Friends of Cromford Canal, a group of volunteers, are working at fully restoring the Cromford Canal and the Butterley Tunnel. | Tunnel | Wikipedia | 485 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
The 1796 Stoddart Tunnel in Chapel-en-le-Frith in Derbyshire is reputed to be the oldest rail tunnel in the world. The rail wagons were originally horse-drawn.
Derby Tunnels in Salem, Massachusetts, were built in 1801 to smuggle imports affected by President Thomas Jefferson's new customs duties. Jefferson had ordered local militias to help the Custom House in each port collect these dues, but the smugglers, led by Elias Derby, hired the Salem militia to dig the tunnels and hide the spoil.
A tunnel was created for the first true steam locomotive, from Penydarren to Abercynon. The Penydarren locomotive was built by Richard Trevithick. The locomotive made the historic journey from Penydarren to Abercynon in 1804. Part of this tunnel can still be seen at Pentrebach, Merthyr Tydfil, Wales. This is arguably the oldest railway tunnel in the world, dedicated only to self-propelled steam engines on rails.
The Montgomery Bell Tunnel in Tennessee, an water diversion tunnel, , to power a water wheel, was built by slave labour in 1819, being the first full-scale tunnel in North America.
Bourne's Tunnel, Rainhill, near Liverpool, England. It is long. Built in the late 1820s, the exact date is unknown, however probably built in 1828 or 1829. This is the first tunnel in the world constructed under a railway line. The construction of the Liverpool to Manchester Railway ran over a horse-drawn tramway that ran from the Sutton collieries to the Liverpool-Warrington turnpike road. A tunnel was bored under the railway for the tramway. As the railway was being constructed the tunnel was made operational, opening prior to the Liverpool tunnels on the Liverpool to Manchester line. The tunnel was made redundant in 1844 when the tramway was dismantled.
Crown Street station, Liverpool, England, 1829. Built by George Stephenson, a single track railway tunnel , was bored from Edge Hill to Crown Street to serve the world's first intercity passenger railway terminus station. The station was abandoned in 1836 being too far from Liverpool city centre, with the area converted for freight use. Closed down in 1972, the tunnel is disused. However it is the oldest passenger rail tunnel running under streets in the world. | Tunnel | Wikipedia | 463 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
The 1829 Wapping Tunnel in Liverpool, England, at long on a twin track railway, was the first rail tunnel bored under a metropolis. The tunnel's path is from Edge Hill in the east of the city to Wapping Dock in the south end Liverpool docks. The tunnel was used only for freight terminating at the Park Lane goods terminal. Currently disused since 1972, the tunnel was to be a part of the Merseyrail metro network, with work started and abandoned because of costs. The tunnel is in excellent condition and is still being considered for reuse by Merseyrail, maybe with an underground station cut into the tunnel for Liverpool university. The river portal is opposite the new King's Dock Liverpool Arena being an ideal location for a serving station. If reused the tunnel will be the oldest used underground rail tunnel in the world and oldest section of any underground metro system.
1832, Lime Street railway station tunnel, Liverpool. A two track rail tunnel, long was bored under the metropolis from Edge Hill in the east of the city to Lime Street in Liverpool's city centre. The tunnel was in use from 1832 being used to transport building materials to the new Lime St station while under construction. The station and tunnel was opened to passengers in 1836. In the 1880s the tunnel was converted to a deep cutting, open to the atmosphere, being four tracks wide. This is the only occurrence of a major tunnel being removed. Two short sections of the original tunnel still exist at Edge Hill station and further towards Lime Street, giving the two tunnels the distinction of being the oldest rail tunnels in the world still in use, and the oldest in use under streets. Over time a section of the deep cutting has been converted back into tunnel due to sections having buildings built over.
Box Tunnel in England, which opened in 1841, was the longest railway tunnel in the world at the time of construction. It was dug by hand, and has a length of .
The 1842 Prince of Wales Tunnel, in Shildon near Darlington, England, is the oldest sizeable tunnel in the world still in use under a settlement. | Tunnel | Wikipedia | 420 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
The Victoria Tunnel Newcastle opened in 1842, is a subterranean wagonway with a maximum depth of that drops from entrance to exit. The tunnel runs under Newcastle upon Tyne, England, and originally exited at the River Tyne. It remains largely intact. Originally designed to carry coal from Spital Tongues to the river, in WW2 part of the tunnel was used as a shelter. Under the management of a charitable foundation called the Ouseburn Trust it is currently used for heritage tours.
The Thames Tunnel, built by Marc Isambard Brunel and his son Isambard Kingdom Brunel opened in 1843, was the first tunnel (after Terelek) traversing under a water body, and the first to be built using a tunnelling shield. Originally used as a foot-tunnel, the tunnel was converted to a railway tunnel in 1869 and was a part of the East London Line of the London Underground until 2007. It was the oldest section of the network, although not the oldest purpose built rail section. From 2010 the tunnel became a part of the London Overground network.
The Victoria Tunnel/Waterloo Tunnel in Liverpool, England, was bored under a metropolis opening in 1848. The tunnel was initially used only for rail freight serving the Waterloo Freight terminal, and later freight and passengers serving the Liverpool ship liner terminal. The tunnel's path is from Edge Hill in the east of the city to the north end Liverpool docks at Waterloo Dock. The tunnel is split into two tunnels with a short open air cutting linking the two. The cutting is where the cable hauled trains from Edge Hill were hitched and unhitched. The two tunnels are effectively one on the same centre line and are regarded as one. However, as initially the long Victoria section was originally cable hauled and the shorter Waterloo section was locomotive hauled, two separate names were given, the short section was named the Waterloo Tunnel. In 1895 the two tunnels were converted to locomotive haulage. Used until 1972, the tunnel is still in excellent condition. A short section of the Victoria tunnel at Edge Hill is still used for shunting trains. The tunnel is being considered for reuse by the Merseyrail network. Stations cut into the tunnel are being considered and also reuse by a monorail system from the proposed Liverpool Waters redevelopment of Liverpool's Central Docks has been proposed. | Tunnel | Wikipedia | 469 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
The summit tunnel of the Semmering railway, the first Alpine tunnel, was opened in 1848 and was long. It connected rail traffic between Vienna, the capital of Austro-Hungarian Empire, and Trieste, its port.
The Giovi Rail Tunnel through the Appennini Mounts opened in 1854, linking the capital city of the Kingdom of Sardinia, Turin, to its port, Genoa. The tunnel was long.
The oldest underground sections of the London Underground were built using the cut-and-cover method in the 1860s, and opened in January 1863. What are now the Metropolitan, Hammersmith & City and Circle lines were the first to prove the success of a metro or subway system.
On 18 June 1868, the Central Pacific Railroad's Summit Tunnel (Tunnel #6) at Donner Pass in the California Sierra Nevada mountains was opened, permitting the establishment of the commercial mass transportation of passengers and freight over the Sierras for the first time. It remained in daily use until 1993, when the Southern Pacific Railroad closed it and transferred all rail traffic through the long Tunnel #41 (a.k.a. "The Big Hole") built a mile to the south in 1925.
In 1870, after fourteen years of works, the Fréjus Rail Tunnel was completed between France and Italy, being the second-oldest Alpine tunnel, long. At that time it was the longest in the world.
The third Alpine tunnel, the Gotthard Rail Tunnel, between northern and southern Switzerland, opened in 1882 and was the longest rail tunnel in the world, measuring .
The 1882 Col de Tende Road Tunnel, at long, was one of the first long road tunnels under a pass, running between France and Italy.
The Mersey Railway tunnel opened in 1886, running from Liverpool to Birkenhead under the River Mersey. The Mersey Railway was the world's first deep-level underground railway. By 1892 the extensions on land from Birkenhead Park station to Liverpool Central Low level station gave a tunnel in length. The under river section is in length, and was the longest underwater tunnel in world in January 1886.
The rail Severn Tunnel was opened in late 1886, at long, although only of the tunnel is actually under the River Severn. The tunnel replaced the Mersey Railway tunnel's longest under water record, which was held for less than a year.
James Greathead, in constructing the City & South London Railway tunnel beneath the Thames, opened in 1890, brought together three key elements of tunnel construction under water: | Tunnel | Wikipedia | 509 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
shield method of excavation;
permanent cast iron tunnel lining;
construction in a compressed air environment to inhibit water flowing through soft ground material into the tunnel heading.
Built in sections between 1890 and 1939, the section of London Underground's Northern line from Morden to East Finchley via Bank was the longest railway tunnel in the world at in length.
St. Clair Tunnel, also opened later in 1890, linked the elements of the Greathead tunnels on a larger scale.
In 1906 the fourth Alpine tunnel opened, the Simplon Tunnel, between Switzerland and Italy. It is long, and was the longest tunnel in the world until 1982. It was also the deepest tunnel in the world, with a maximum rock overlay of approximately .
The 1927 Holland Tunnel was the first underwater tunnel designed for automobiles. The construction required a novel ventilation system.
In 1945 the Delaware Aqueduct tunnel was completed, supplying water to New York City. At it is the longest tunnel in the world.
In 1988 the long Seikan Tunnel in Japan was completed under the Tsugaru Strait, linking the islands of Honshu and Hokkaido. It was the longest railway tunnel in the world at that time.
Ryfast is the longest undersea road tunnel. It is in length. The tunnel opened for use in 2020. | Tunnel | Wikipedia | 256 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Longest
The Thirlmere Aqueduct in North West England, United Kingdom is sometimes considered the longest tunnel, of any type, in the world at , though the aqueduct's tunnel section is not continuous.
The Dahuofang Water Tunnel in China, opened in 2009, is the third longest water tunnel in the world at length.
The Gotthard Base Tunnel in Switzerland, opened in 2016, is the longest and deepest railway tunnel in the world at length and maximum depth below the Gotthard Massif. It provides a flat transit route between the North and South of Europe under the Swiss Alps, at a maximum elevation of .
The Seikan Tunnel in Japan connects the main island of Honshu with the northern island of Hokkaido by rail. It is long, of which are crossing the Tsugaru Strait undersea.
The Channel Tunnel crosses the English Channel between France and the United Kingdom. It has a total length of , of which are the world's longest undersea tunnel section.
The Lötschberg Base Tunnel in Switzerland was the longest land rail tunnel, with a length of , from its inauguration in 2007 until the completion of the Gotthard Base Tunnel in 2016.
The Lærdal Tunnel in Norway from Lærdal to Aurland is the world's longest road tunnel, intended for cars and similar vehicles, at .
The Zhongnanshan Tunnel in People's Republic of China opened in January 2007 is the world's second longest highway tunnel and the longest mountain road tunnel in Asia, at .
The longest canal tunnel is the Rove Tunnel in France, over long.
Notable | Tunnel | Wikipedia | 324 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
The Moffat Tunnel, opened in 1928, passes under the Continental Divide of the Americas in Colorado. The tunnel is long and at an elevation of is the highest active railroad tunnel in the U.S. (The inactive Tennessee Pass Line and the historic Alpine Tunnel are higher.)
Williamson's tunnels in Liverpool, from 1804 and completed around 1840 by a wealthy eccentric, are probably the largest underground folly in the world. The tunnels were built with no functional purpose.
The Chicago freight tunnel network is the largest urban street tunnel network, comprising of tunnels beneath the majority of downtown Chicago streets. It operated between 1906 and 1956 as a freight network, connecting building basements and railway stations. Following a 1992 flood the network was sealed, although some parts still carry utility and communications infrastructure.
The Pennsylvania Turnpike opened in 1940 with seven tunnels, most of which were bored as part of the stillborn South Pennsylvania Railroad and giving the highway the nickname "Tunnel Highway". Four of the tunnels (Allegheny Mountain, Tuscarora Mountain, Kittatinny Mountain, and Blue Mountain) remain in active use, while the other three (Laurel Hill, Rays Hill, and Sideling Hill) were bypassed in the 1960s; the latter two tunnels are on a bypassed section of the Turnpike now commonly known as the Abandoned Pennsylvania Turnpike.
The Fredhälls road tunnel was opened in 1966, in Stockholm, Sweden, and the New Elbe road tunnel opened in 1975 in Hamburg, Germany. Both tunnels handle around 150,000 vehicles a day, making them two of the most trafficked tunnels in the world.
The Honningsvåg Tunnel ( long) opened in 1999 on European route E69 in Norway as the world's northernmost road tunnel, except for mines (which exist on Svalbard).
The Central Artery road tunnel in Boston, Massachusetts, is a part of the larger Big Dig completed around 2007, and carries approximately 200,000 vehicles/day under the city along Interstate 93, US Route 1, and Massachusetts Route 3, which share a concurrency through the tunnels. The Big Dig replaced Boston's old badly deteriorated I-93 elevated highway.
The Stormwater Management And Road Tunnel or SMART Tunnel, is a combined storm drainage and road structure opened in 2007 in Kuala Lumpur, Malaysia. The tunnel is the longest stormwater drainage tunnel in South East Asia and second longest in Asia. The facility can be operated as a simultaneous traffic and stormwater passage, or dedicated exclusively to stormwater when necessary. | Tunnel | Wikipedia | 499 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
The Eiksund Tunnel on national road Rv 653 in Norway is the world's deepest subsea road tunnel, measuring long, with deepest point at below the sea level, opened in February 2008.
Gerrards Cross railway tunnel, in England, opened in 2010, is notable in that it converted an existing railway cutting into a tunnel to create ground to build a supermarket over the tunnel. The railway in the cutting was first opened around 1906, stretching over 104 years to complete a railway tunnel. The tunnel was built using the cover method with craned in prefabricated forms in order to keep the busy railway operating. A branch of the Tesco supermarket chain occupies the newly created ground above the railway tunnel, with an adjacent existing railway station at the end of the tunnel. During construction, a portion of the tunnel collapsed when soil cover was added. The prefabricated forms were covered with a layer of reinforced concrete after the collapse.
The Fenghuoshan tunnel, completed in 2005 on the Qinghai-Tibet railway is the world's highest railway tunnel, about above sea level and long.
The La Linea Tunnel in Colombia, 2016, is the longest, , mountain tunnel in South America. It crosses beneath a mountain at above sea level with six traffic lanes, and it has a parallel emergency tunnel. The tunnel is subject to serious groundwater pressure. The tunnel will link Bogotá and its urban area with the coffee-growing region, and with the main port on the Colombian Pacific coast.
The Chicago Deep Tunnel Project is a network of of drainage tunnels designed to reduce flooding in the Chicago area. Started in the mid-1970s, the project is due to be completed in 2029.
New York City Water Tunnel No. 3, started in 1970, has an expected completion beyond 2026, and will measure more than . | Tunnel | Wikipedia | 368 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Mining
The use of tunnels for mining is called drift mining. Drift mining can help find coal, goal, iron, and other minerals, just like normal mining.
Sub-surface mining consists of digging tunnels or shafts into the earth to reach buried ore deposits.
Military use
Some tunnels are not for transport at all but rather, are fortifications, for example Mittelwerk and Cheyenne Mountain Complex. Excavation techniques, as well as the construction of underground bunkers and other habitable areas, are often associated with military use during armed conflict, or civilian responses to threat of attack. Another use for tunnels was for the storage of chemical weapons .
Secret tunnels
Secret tunnels have given entrance to or escape from an area, such as the Cu Chi Tunnels or the smuggling tunnels in the Gaza Strip which connect it to Egypt. Although the Underground Railroad network used to transport escaped slaves was "underground" mostly in the sense of secrecy, hidden tunnels were occasionally used. Secret tunnels were also used during the Cold War, under the Berlin Wall and elsewhere, to smuggle refugees, and for espionage.
Smugglers use secret tunnels to transport or store contraband, such as illegal drugs and weapons. Elaborately engineered tunnels built to smuggle drugs across the Mexico-US border were estimated to require up to 9 months to complete, and an expenditure of up to $1 million. Some of these tunnels were equipped with lighting, ventilation, telephones, drainage pumps, hydraulic elevators, and in at least one instance, an electrified rail transport system. Secret tunnels have also been used by thieves to break into bank vaults and retail stores after hours. Several tunnels have been discovered by the Border Security Forces across the Line of Control along the India-Pakistan border, mainly to allow terrorists access to the Indian territory of Jammu and Kashmir.
The actual usage of erdstall tunnels is unknown but theories connect it to a rebirth ritual.
Natural tunnels | Tunnel | Wikipedia | 377 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Lava tubes are emptied lava conduits, formed during volcanic eruptions by flowing and cooling lava.
Natural Tunnel State Park (Virginia, US) features an natural tunnel, really a limestone cave, that has been used as a railroad tunnel since 1890.
Punarjani Guha in Kerala, India. Hindus believe that crawling through the tunnel (which they believe was created by a Hindu god) from one end to the other will wash away all of one's sins and thus allow one to attain rebirth. Only men are permitted to crawl through the tunnel.
Torghatten, a Norwegian island with a hat-shaped silhouette, has a natural tunnel in the middle of the hat, letting light come through. The long, high, and wide tunnel is said to be the hole made by an arrow of the angry troll Hestmannen, the hill being the hat of the troll-king of Sømna trying to save the beautiful Lekamøya. The tunnel is thought actually to be the work of ice. The sun shines through the tunnel during two few minutes long periods every year.
Major accidents
Clayton Tunnel rail crash (1861) – confusion about block signals leading to collision, 23 killed.
Welwyn Tunnel rail crash (1866) – train failed in tunnel, guard did not protect train.
Paris Métro train fire (1904) – train fire in Couronnes underground station, 84 killed by smoke and gases.
Church Hill Tunnel collapse (1925) – tunnel collapse on a work train during renovation, killing four men and trapping a steam locomotive and ten flat cars.
Balvano train disaster (1944) – asphyxiation of about 500 "unofficial" passengers on freight train.
Caldecott Tunnel fire (1982) – major motor vehicle tunnel crash and fire.
Channel Tunnel fire (1996) – Train carrying Heavy Good Vehicles (HGV) caught on fire.
Princess Diana's death (1997) – Car crash in Pont de l'Alma tunnel, Paris, which killed Princess Diana.
Mont Blanc Tunnel fire (1999) – Transport truck caught on fire and combusted inside tunnel.
Big Dig Ceiling collapse (2006) – Concrete ceiling panel falls in Fort Point tunnel, Boston, which causes the Big Dig project to be closed for a year. | Tunnel | Wikipedia | 457 | 160832 | https://en.wikipedia.org/wiki/Tunnel | Technology | Transport infrastructure | null |
Felt is a textile that is produced by matting, condensing, and pressing fibers together. Felt can be made of natural fibers such as wool or animal fur, or from synthetic fibers such as petroleum-based acrylic or acrylonitrile or wood pulp–based rayon. Blended fibers are also common. Natural fiber felt has special properties that allow it to be used for a wide variety of purposes. It is "fire-retardant and self-extinguishing; it dampens vibration and absorbs sound; and it can hold large amounts of fluid without feeling wet..."
History
Felt from wool is one of the oldest known textiles. Many cultures have legends about the origins of felt-making. Sumerian legend claims that the secret of feltmaking was discovered by Urnamman of Lagash. The story of Saint Clement and Saint Christopher relates that the men packed their sandals with wool to prevent blisters while fleeing from persecution. At the end of their journey the movement and sweat had turned the wool into felt socks.
Most likely felt's origins can be found in central Asia, where there is evidence of feltmaking in Siberia (Altai mountains) in Northern Mongolia and more recently evidence dating back to the first century CE in Mongolia. Siberian tombs (7th to 2nd century BCE) show the broad uses of felt in that culture, including clothing, jewelry, wall hangings, and elaborate horse blankets. Employing careful color use, stitching, and other techniques, these feltmakers were able to use felt as an illustrative and decorative medium on which they could depict abstract designs and realistic scenes with great skill. Over time these makers became known for the beautiful abstract patterns they used that were derived from plant, animal, and other symbolic designs.
From Siberia and Mongolia feltmaking spread across the areas held by the Turkic-Mongolian tribes. Sheep and camel herds were central to the wealth and lifestyle of these tribes, both of which animals were critical to producing the fibers needed for felting. For nomads traveling frequently and living on fairly treeless plains felt provided housing (yurts, tents etc.), insulation, floor coverings, and inside walling, as well as many household necessities from bedding and coverings to clothing. In the case of nomadic peoples, an area where feltmaking was particularly visible was in trappings for their animals and for travel. Felt was often featured in the blankets that went under saddles. | Felt | Wikipedia | 499 | 160853 | https://en.wikipedia.org/wiki/Felt | Technology | Fabrics and fibers | null |
Dyes provided rich coloring, and colored slices of pre-felts (semi-felted sheets that could be cut in decorative ways) along with dyed yarns and threads were combined to create beautiful designs on the wool backgrounds. Felt was even used to create totems and amulets with protective functions. In traditional societies the patterns embedded in the felt were also imbued with significant religious and symbolic meaning.
Feltmaking is still practised by nomadic peoples (such as Mongols and Turkic people) in Central Asia, where rugs, tents and clothing are regularly made. Some of these are traditional items, such as the classic yurt, or ger, while others are designed for the tourist market, such as decorated slippers. In the Western world, felt is widely used as a medium for expression in both textile art and contemporary art and design, where it has significance as an ecologically responsible textile and building material.
In addition to Central Asian traditions of felting, Scandinavian countries have also supported feltmaking, particularly for clothing.
Manufacturing methods
Wet felting
In the wet felting process, hot water is applied to layers of animal hairs, while repeated agitation and compression causes the fibers to hook together or weave together into a single piece of fabric. Wrapping the properly arranged fiber in a sturdy, textured material, such as a bamboo mat or burlap, will speed up the felting process. The felted material may be finished by fulling.
Only certain types of fiber can be wet felted successfully. Most types of fleece, such as those taken from the alpaca or the Merino sheep, can be put through the wet felting process. One may also use mohair (goat), angora (rabbit), or hair from rodents such as beavers and muskrats. These types of fiber are covered in tiny scales, similar to the scales found on a strand of human hair. Heat, motion, and moisture of the fleece causes the scales to open, while agitating them causes them to latch onto each other, creating felt. There is an alternative theory that the fibers wind around each other during felting. Plant fibers and synthetic fibers will not wet felt. | Felt | Wikipedia | 444 | 160853 | https://en.wikipedia.org/wiki/Felt | Technology | Fabrics and fibers | null |
In order to make multi-colored designs, felters conduct a two-step process in which they create pre-felts of specialized colors—these semi-completed sheets of colored felt can then be cut with a sharp implement (knife or scissors) and the distinctive colors placed next to each other as in making a mosaic. The felting process is then resumed and the edges of the fabric attach to each other as the felting process is completed. Shyrdak carpets (Turkmenistan) use a form of this method wherein two pieces of contrasting color are cut out with the same pattern, the cut-outs are then switched, fitting one into the other, which makes a sharply defined and colorful patterned piece. In order to strengthen the joints of a mosaic style felt, feltmakers often add a backing layer of fleece that is felted along with the other components. Feltmakers can differ in their orientation to this added layer—where some will lay it on top of the design before felting and others will place the design on top of the strengthening layer.
The process of felting was adapted to the lifestyles of the different cultures in which it flourished. In Central Asia, it is common to conduct the rolling/friction process with the aid of a horse, donkey, or camel, which will pull the rolled felt until the process is complete. Alternately, a group of people in a line might roll the felt along, kicking it regularly with their feet. Further fulling can include throwing or slamming and working the edges with careful rolling. In Turkey, some baths had areas dedicated to feltmaking, making use of the steam and hot water that were already present for bathing.
Development of felting as a profession
As felting grew in importance to a society, so, too, did the knowledge about techniques and approaches. Amateur or community felting obviously continued in many communities at the same time that felting specialists and felting centers began to develop. However, the importance of felting to community life can be seen in the fact that, in many Central Asian communities, felt production is directed by a leader who oversees the process as a ritual that includes prayers—words and actions to bring good luck to the process. Successfully completing the creation of felt (certainly large felt pieces) is reason for celebration, feasting, and the sharing of traditional stories. | Felt | Wikipedia | 469 | 160853 | https://en.wikipedia.org/wiki/Felt | Technology | Fabrics and fibers | null |
In Turkey, craft guilds called "ahi" came into being, and these groups were responsible for registering members and protecting the knowledge of felting. In Istanbul at one time, there were 1,000 felters working in 400 workshops registered in this ahi.
Needle felting
Needle felting is a method of creating felt that uses specially designed needles instead of water. Felting needles have angled notches along the shaft that catch fibers and tangle them together to produce felt. These notches are sometimes erroneously called "barbs", but barbs are protrusions (like barbed wire) and would be too difficult to thrust into the wool and nearly impossible to pull out. Felting needles are thin and sharp, with shafts of a variety of different gauges and shapes. Needle felting is used in industrial felt making as well as for individual art and craft applications.
Felting needles are sometimes fitted in holders that allow the use of 2 or more needles at one time to sculpt wool objects and shapes. Individual needles are often used for detail while multiple needles that are paired together are used for larger areas or to form the base of the project. At any point in time a variety of fibers and fiber colors may be added, using needles to incorporate them into the project.
Needle felting can be used to create both 2 dimensional and 3 dimensional artwork, including soft sculpture, dolls, figurines, jewelry, and 2 dimensional wool paintings. Needle felting is popular with artists and craftspeople worldwide. One example is Ikuyo Fujita(藤田育代 Fujita Ikuyo), a Japanese artist who works primarily in needle felt painting and mogol (pipe cleaner) art.
Recently, needle-felting machines have become popular for art or craft felters. Similar to a sewing machine, these tools have several needles that punch fibers together. These machines can be used to create felted products more efficiently. The embellishment machine allows the user to create unique combinations of fibers and designs. | Felt | Wikipedia | 407 | 160853 | https://en.wikipedia.org/wiki/Felt | Technology | Fabrics and fibers | null |
Carroting
Invented in the mid 17th century and used until the mid-20th centuries, a process called "carroting" was used in the manufacture of good quality felt for making men's hats. Beaver, rabbit or hare skins were treated with a dilute solution of the mercury compound mercuric nitrate. The skins were dried in an oven where the thin fur at the sides turned orange, the color of carrots. Pelts were stretched over a bar in a cutting machine, and the skin was sliced off in thin shreds, with the fleece coming away entirely. The fur was blown onto a cone-shaped colander and then treated with hot water to consolidate it. The cone then peeled off and passed through wet rollers to cause the fur to felt. These 'hoods' were then dyed and blocked to make hats. The toxic solutions from the carrot and the vapours it produced resulted in widespread cases of mercury poisoning among hatters. This may be the origin of the phrase "mad as a hatter" which was used to humorous effect by Lewis Carroll in the chapter "A Mad Tea Party" of the novel Alice in Wonderland.
Uses
Felt is used in a wide range of industries and manufacturing processes, from the automotive industry and casinos to musical instruments and home construction, as well as in gun wadding, either inside cartridges or pushed down the barrel of a muzzleloader. Felt had many uses in ancient times and continues to be widely used today.
Industrial uses
Felt is frequently used in industry as a sound or vibration damper, as a non-woven fabric for air filtration, and in machinery for cushioning and padding moving parts.
Home Decor
Felt can be used in home furnishings like table runners, placemats, coasters, and even as backing for area rugs. It can add a touch of warmth and texture to a space.
Clothing
During the 18th and 19th centuries gentlemen's headwear made from beaver felt were popular. In the early part of the 20th century, cloth felt hats, such as fedoras, trilbies and homburgs, were worn by many men in the western world. Felt is often used in footwear as boot liners, with the Russian valenki being an example. | Felt | Wikipedia | 459 | 160853 | https://en.wikipedia.org/wiki/Felt | Technology | Fabrics and fibers | null |
Musical instruments
Many musical instruments use felt. It is often used as a damper. On drum cymbal stands, it protects the cymbal from cracking and ensures a clean sound. It is used to wrap bass drum strikers and timpani mallets. Felt is used extensively in pianos; for example, piano hammers are made of wool felt around a wooden core. The density and springiness of the felt is a major part of what creates a piano's tone. As the felt becomes grooved and "packed" with use and age, the tone suffers. Felt is placed under the piano keys on accordions to control touch and key noise; it is also used on the pallets to silence notes not sounded by preventing air flow. Felt is used with other instruments, particularly stringed instruments, as a damper to reduce volume or eliminate unwanted sounds.
Arts and crafts
Felt is used for framing paintings. It is laid between the slip mount and picture as a protective measure to avoid damage from rubbing to the edge of the painting. This is commonly found as a preventive measure on paintings which have already been restored or professionally framed. It is widely used to protect paintings executed on various surfaces including canvas, wood panel and copper plate.
A felt-covered board can be used in storytelling to small children. Small felt cutouts or figures of animals, people, or other objects will adhere to a felt board, and in the process of telling the story, the storyteller also acts it out on the board with the animals or people. Puppets can also be made with felt. The best known example of felt puppets are Jim Henson's Muppets. Felt pressed dolls, such as Lenci dolls, were very popular in the nineteenth century and just after World War I.
As part of the overall renewal of interest in textile and fiber arts, beginning in the 1970s and continuing through today, felt has experienced a strong revival in interest, including its historical roots. Polly Stirling, a fiber artist from New South Wales, Australia, is commonly associated with the development of nuno felting, a key technique for contemporary art felting. German artist Joseph Beuys prominently integrates felt within his works. English artist Jenny Cowern shifted from traditional drawing and painting media into using felt as her primary media. | Felt | Wikipedia | 463 | 160853 | https://en.wikipedia.org/wiki/Felt | Technology | Fabrics and fibers | null |
Modern day felters with access to a broad range of sheep and other animal fibers have exploited knowledge of these different breeds to produce special effects in their felt. Fleece locks are classified by the Bradford or Micron count, both which designate the fineness to coarseness of the material. Fine wools range from 64 to 80 (Bradford); medium 40–60 (Bradford); and coarse 36–60 (Bradford). Merino, the finest and most delicate sheep fleece, will be employed for clothing that goes next to the body. Claudy Jongstra raises traditional and rare breeds of sheep with much hardier coats (Drenthe, Heath, Gotland, Schoonbeek, and Wensleydale) on her property in Friesland and these are used in her interior design projects. Exploitation of these characteristics of the fleece in tandem with the use of other techniques, such as stitching and incorporation of other fibers, provides felters with a broad range of possibilities | Felt | Wikipedia | 200 | 160853 | https://en.wikipedia.org/wiki/Felt | Technology | Fabrics and fibers | null |
A cockpit or flight deck is the area, on the front part of an aircraft, spacecraft, or submersible, from which a pilot controls the vehicle.
The cockpit of an aircraft contains flight instruments on an instrument panel, and the controls that enable the pilot to fly the aircraft. In most airliners, a door separates the cockpit from the aircraft cabin. After the September 11, 2001 attacks, all major airlines fortified their cockpits against access by hijackers.
Etymology
The word cockpit seems to have been used as a nautical term in the 17th century, without reference to cock fighting. It referred to an area in the rear of a ship where the cockswain's station was located, the cockswain being the pilot of a smaller "boat" that could be dispatched from the ship to board another ship or to bring people ashore. The word "cockswain" in turn derives from the old English terms for "boat-servant" (coque is the French word for "shell"; and swain was old English for boy or servant). The midshipmen and master's mates were later berthed in the cockpit, and it served as the action station for the ship's surgeon and his mates during battle. Thus by the 18th century, "cockpit" had come to designate an area in the rear lower deck of a warship where the wounded were taken. The same term later came to designate the place from which a sailing vessel is steered, because it is also located in the rear, and is often in a well or "pit".
However, a convergent etymology does involve reference to cock fighting. According to the Barnhart Concise Dictionary of Etymology, the buildings in London where the king's cabinet worked (the Treasury and the Privy Council) were called the "Cockpit" because they were built on the site of a theater called The Cockpit (torn down in 1635), which itself was built in the place where a "cockpit" for cock-fighting had once stood prior to the 1580s. Thus the word Cockpit came to mean a control center.
The original meaning of "cockpit", first attested in the 1580s, is "a pit for fighting cocks", referring to the place where cockfights were held. This meaning no doubt influenced both lines of evolution of the term, since a cockpit in this sense was a tight enclosure where a great deal of stress or tension would occur. | Cockpit | Wikipedia | 492 | 160878 | https://en.wikipedia.org/wiki/Cockpit | Technology | Aircraft components | null |
From about 1935, cockpit came to be used informally to refer to the driver's cabin, especially in high performance cars, and this is official terminology used to describe the compartment that the driver occupies in a Formula One car.
In an airliner, the cockpit is usually referred to as the flight deck, the term deriving from its use by the RAF for the separate, upper platform in large flying boats where the pilot and co-pilot sat. In the USA and many other countries, however, the term cockpit is also used for airliners.
The seat of a powerboat racing craft is also referred to as the cockpit.
Ergonomics
The first airplane with an enclosed cabin appeared in 1912 on the Avro Type F; however, during the early 1920s there were many passenger aircraft in which the crew remained open to the air while the passengers sat in a cabin. Military biplanes and the first single-engined fighters and attack aircraft also had open cockpits, some as late as the Second World War when enclosed cockpits became the norm.
The largest impediment to having closed cabins was the material used to make the windows. Prior to Perspex becoming available in 1933, windows were either safety glass, which was heavy, or cellulose nitrate (i.e.: guncotton), which yellowed quickly and was extremely flammable. In the mid-1920s many aircraft manufacturers began using enclosed cockpits for the first time. Early airplanes with closed cockpits include the 1924 Fokker F.VII, the 1926 German Junkers W 34 transport, the 1926 Ford Trimotor, the 1927 Lockheed Vega, the Spirit of St. Louis and the passenger aircraft manufactured by the Douglas and Boeing companies during the mid-1930s. Open-cockpit airplanes were almost extinct by the mid-1950s, with the exception of training planes, crop-dusters and homebuilt aircraft designs.
Cockpit windows may be equipped with a sun shield. Most cockpits have windows that can be opened when the aircraft is on the ground. Nearly all glass windows in large aircraft have an anti-reflective coating, and an internal heating element to melt ice. Smaller aircraft may be equipped with a transparent aircraft canopy. | Cockpit | Wikipedia | 446 | 160878 | https://en.wikipedia.org/wiki/Cockpit | Technology | Aircraft components | null |
In most cockpits the pilot's control column or joystick is located centrally (centre stick), although in some military fast jets the side-stick is located on the right hand side. In some commercial airliners (i.e.: Airbus—which features the glass cockpit concept) both pilots use a side-stick located on the outboard side, so Captain's side-stick on the left and First-officer's seat on the right.
Except for some helicopters, the right seat in the cockpit of an aircraft is the seat used by the co-pilot. The captain or pilot in command sits in the , so that they can operate the throttles and other pedestal instruments with their right hand. The tradition has been maintained to this day, with the co-pilot on the right hand side.
The layout of the cockpit, especially in the military fast jet, has undergone standardisation, both within and between aircraft, manufacturers and even nations. An important development was the "Basic Six" pattern, later the "Basic T", developed from 1937 onwards by the Royal Air Force, designed to optimise pilot instrument scanning.
Ergonomics and Human Factors concerns are important in the design of modern cockpits. The layout and function of cockpit displays controls are designed to increase pilot situation awareness without causing information overload. In the past, many cockpits, especially in fighter aircraft, limited the size of the pilots that could fit into them. Now, cockpits are being designed to accommodate from the 1st percentile female physical size to the 99th percentile male size.
In the design of the cockpit in a military fast jet, the traditional "knobs and dials" associated with the cockpit are mainly absent. Instrument panels are now almost wholly replaced by electronic displays, which are themselves often re-configurable to save space. While some hard-wired dedicated switches must still be used for reasons of integrity and safety, many traditional controls are replaced by multi-function re-configurable controls or so-called "soft keys". Controls are incorporated onto the stick and throttle to enable the pilot to maintain a head-up and eyes-out position – the Hands On Throttle And Stick or HOTAS concept. These controls may be then further augmented by control media such as head pointing with a Helmet Mounted Sighting System or Direct voice input (DVI). Advances in auditory displays allow for Direct Voice Output of aircraft status information and for the spatial localisation of warning sounds for improved monitoring of aircraft systems. | Cockpit | Wikipedia | 510 | 160878 | https://en.wikipedia.org/wiki/Cockpit | Technology | Aircraft components | null |
The layout of control panels in modern airliners has become largely unified across the industry. The majority of the systems-related controls (such as electrical, fuel, hydraulics and pressurization) for example, are usually located in the ceiling on an overhead panel. Radios are generally placed on a panel between the pilot's seats known as the pedestal. Automatic flight controls such as the autopilot are usually placed just below the windscreen and above the main instrument panel on the glareshield. A central concept in the design of the cockpit is the Design Eye Position or "DEP", from which point all displays should be visible.
Most modern cockpits will also include some kind of integrated warning system.
A study undertaken in 2013, to assess methods for cockpit-user menu navigation, found that touchscreen produced the "best scores".
After the September 11, 2001 attacks, all major airlines fortified their cockpits against access by hijackers.
Flight instruments
In the modern electronic cockpit, the electronic flight instruments usually regarded as essential are MFD, PFD, ND, EICAS, FMS/CDU and back-up instruments.
MCP
A Mode control panel, usually a long narrow panel located centrally in front of the pilot, may be used to control heading, speed, altitude, vertical speed, vertical navigation and lateral navigation. It may also be used to engage or disengage both the autopilot and the autothrottle. The panel as an area is usually referred to as the "glareshield panel". MCP is a Boeing designation (that has been informally adopted as a generic name for the unit/panel) for a unit that allows for the selection and parameter setting of the different autoflight functions, the same unit on an Airbus aircraft is referred to as the FCU (Flight Control unit).
PFD
The primary flight display is usually located in a prominent position, either centrally or on either side of the cockpit. It will in most cases include a digitized presentation of the attitude indicator, air speed and altitude indicators (usually as a tape display) and the vertical speed indicator. It will in many cases include some form of heading indicator and ILS/VOR deviation indicators. In many cases an indicator of the engaged and armed autoflight system modes will be present along with some form of indication of the selected values for altitude, speed, vertical speed and heading. It may be pilot selectable to swap with the ND. | Cockpit | Wikipedia | 505 | 160878 | https://en.wikipedia.org/wiki/Cockpit | Technology | Aircraft components | null |
ND
A navigation display, which may be adjacent to the PFD, shows the route and information on the next waypoint, wind speed and wind direction. It may be pilot selectable to swap with the PFD.
EICAS/ECAM
The Engine Indication and Crew Alerting System (EICAS), used by Boeing and Embraer, or the Electronic Centralized Aircraft Monitor (ECAM), used by Airbus, allow the pilot to monitor the following information: values for N1, N2 and N3, fuel temperature, fuel flow, the electrical system, cockpit or cabin temperature and pressure, control surfaces and so on. The pilot may select display of information by means of button press.
FMS/MCDU
The flight management system/control and/or display unit may be used by the pilot to enter and check for the following information: flight plan, speed control, navigation control, etc.
Back-up instruments
In a less prominent part of the cockpit, in case of failure of the other instruments, there will be a battery-powered integrated standby instrument system along with a magnetic compass, showing essential flight information such as speed, altitude, attitude and heading.
Aerospace industry technologies
In the U.S. the Federal Aviation Administration (FAA) and the National Aeronautics and Space Administration (NASA) have researched the ergonomic aspects of cockpit design and have conducted investigations of airline industry accidents. Cockpit design disciplines include Cognitive science, Neuroscience, Human–computer interaction, Human Factors Engineering, Anthropometry and Ergonomics.
Aircraft designs have adopted the fully digital "glass cockpit". In such designs, instruments and gauges, including navigational map displays, use a user interface markup language known as ARINC 661. This standard defines the interface between an independent cockpit display system, generally produced by a single manufacturer, and the avionics equipment and user applications it is required to support, by means of displays and controls, often made by different manufacturers. The separation between the overall display system, and the applications driving it, allows for specialization and independence. | Cockpit | Wikipedia | 423 | 160878 | https://en.wikipedia.org/wiki/Cockpit | Technology | Aircraft components | null |
The fuselage (; from the French fuselé "spindle-shaped") is an aircraft's main body section. It holds crew, passengers, or cargo. In single-engine aircraft, it will usually contain an engine as well, although in some amphibious aircraft the single engine is mounted on a pylon attached to the fuselage, which in turn is used as a floating hull. The fuselage also serves to position the control and stabilization surfaces in specific relationships to lifting surfaces, which is required for aircraft stability and maneuverability.
Types of structures
Truss structure
This type of structure is still in use in many lightweight aircraft using welded steel tube trusses.
A box truss fuselage structure can also be built out of wood—often covered with plywood. Simple box structures may be rounded by the addition of supported lightweight stringers, allowing the fabric covering to form a more aerodynamic shape, or one more pleasing to the eye.
Geodesic construction
Geodesic structural elements were used by Barnes Wallis for British Vickers between the wars and into World War II to form the whole of the fuselage, including its aerodynamic shape. In this type of construction multiple flat strip stringers are wound about the formers in opposite spiral directions, forming a basket-like appearance. This proved to be light, strong, and rigid and had the advantage of being made almost entirely of wood. A similar construction using aluminum alloy was used in the Vickers Warwick with less material than would be required for other structural types. The geodesic structure is also redundant and so can survive localized damage without catastrophic failure. A fabric covering over the structure completed the aerodynamic shell (see the Vickers Wellington for an example of a large warplane which uses this process). The logical evolution of this is the creation of fuselages using molded plywood, in which several sheets are laid with the grain in differing directions to give the monocoque type below.
Monocoque shell | Fuselage | Wikipedia | 384 | 160891 | https://en.wikipedia.org/wiki/Fuselage | Technology | Aircraft components | null |
In this method, the exterior surface of the fuselage is also the primary structure. A typical early form of this (see the Lockheed Vega) was built using molded plywood, where the layers of plywood are formed over a "plug" or within a mold. A later form of this structure uses fiberglass cloth impregnated with polyester or epoxy resin as the skin, instead of plywood. A simple form of this used in some amateur-built aircraft uses rigid expanded foam plastic as the core, with a fiberglass covering, eliminating the necessity of fabricating molds, but requiring more effort in finishing (see the Rutan VariEze). An example of a larger molded plywood aircraft is the de Havilland Mosquito fighter/light bomber of World War II.
No plywood-skin fuselage is truly monocoque, since stiffening elements are incorporated into the structure to carry concentrated loads that would otherwise buckle the thin skin.
The use of molded fiberglass using negative ("female") molds (which give a nearly finished product) is prevalent in the series production of many modern sailplanes. The use of molded composites for fuselage structures is being extended to large passenger aircraft such as the Boeing 787 Dreamliner (using pressure-molding on female molds).
Semi-monocoque
This is the preferred method of constructing an all-aluminum fuselage. First, a series of formers in the shape of the fuselage cross sections are held in position on a rigid fixture. These formers are then joined with lightweight longitudinal elements called stringers. These are in turn covered with a skin of sheet aluminum, attached by riveting or by bonding with special adhesives. The fixture is then disassembled and removed from the completed fuselage shell, which is then fitted out with wiring, controls, and interior equipment such as seats and luggage bins. Most modern large aircraft are built using this technique, but use several large sections constructed in this fashion which are then joined with fasteners to form the complete fuselage. As the accuracy of the final product is determined largely by the costly fixture, this form is suitable for series production, where many identical aircraft are to be produced. Early examples of this type include the Douglas Aircraft DC-2 and DC-3 civil aircraft and the Boeing B-17 Flying Fortress. Most metal light aircraft are constructed using this process. | Fuselage | Wikipedia | 484 | 160891 | https://en.wikipedia.org/wiki/Fuselage | Technology | Aircraft components | null |
Both monocoque and semi-monocoque are referred to as "stressed skin" structures as all or a portion of the external load (i.e. from wings and empennage, and from discrete masses such as the engine) is taken by the surface covering. In addition, all the load from internal pressurization is carried (as skin tension) by the external skin.
The proportioning of loads between the components is a design choice dictated largely by the dimensions, strength, and elasticity of the components available for construction and whether or not a design is intended to be "self jigging", not requiring a complete fixture for alignment.
Materials
Early aircraft were constructed of wood frames covered in fabric. As monoplanes became popular, metal frames improved the strength, which eventually led to all-metal-structure aircraft, with metal covering for all its exterior surfaces - this was first pioneered in the second half of 1915. Some modern aircraft are constructed with composite materials for major control surfaces, wings, or the entire fuselage such as the Boeing 787. On the 787, it makes possible higher pressurization levels and larger windows for passenger comfort as well as lower weight to reduce operating costs. The Boeing 787 weighs less than if it were an all-aluminum assembly.
Windows
Cockpit windshields on the Airbus A320 must withstand bird strikes up to and are made of chemically strengthened glass. They are usually composed of three layers or plies, of glass or plastic : the inner two are 8 mm (0.3 in.) thick each and are structural, while the outer ply, about 3 mm thick, is a barrier against foreign object damage and abrasion, with often a hydrophobic coating. It must prevent fogging inside the cabin and de-ice from . This was previously done with thin wires similar to a rear car window but is now accomplished with a transparent, nanometers-thick coating of indium tin oxide sitting between plies, electrically conductive and thus transmitting heat. Curved glass improves aerodynamics but sight criteria also needs larger panes. A cockpit windshield is composed of 4–6 panels, 35 kg (77 lb) each on an Airbus A320. In its lifetime, an average aircraft goes through three or four windshields, and the market is shared evenly between OEM and higher margins aftermarket. | Fuselage | Wikipedia | 476 | 160891 | https://en.wikipedia.org/wiki/Fuselage | Technology | Aircraft components | null |
Cabin windows, made from much lighter than glass stretched acrylic glass, consists of multiple panes: an outer one built to support four times the maximum cabin pressure, an inner one for redundancy and a scratch pane near the passenger. Acrylic is susceptible to crazing : a network of fine cracks appears but can be polished to restore optical transparency, removal and polishing typically undergo every 2–3 years for uncoated windows.
Wing integration
"Flying wing" aircraft, such as the Northrop YB-49 Flying Wing and the Northrop B-2 Spirit bomber have no separate fuselage; instead what would be the fuselage is a thickened portion of the wing structure.
Conversely, there have been a small number of aircraft designs which have no separate wing, but use the fuselage to generate lift. Examples include National Aeronautics and Space Administration's experimental lifting body designs and the Vought XF5U-1 Flying Flapjack.
A blended wing body can be considered a mixture of the above. It carries the useful load in a fuselage producing lift. A modern example is Boeing X-48. One of the earliest aircraft using this design approach is Burnelli CBY-3, which fuselage was airfoil shaped to produce lift.
Gallery | Fuselage | Wikipedia | 254 | 160891 | https://en.wikipedia.org/wiki/Fuselage | Technology | Aircraft components | null |
In descriptive statistics, a box plot or boxplot is a method for demonstrating graphically the locality, spread and skewness groups of numerical data through their quartiles. In addition to the box on a box plot, there can be lines (which are called whiskers) extending from the box indicating variability outside the upper and lower quartiles, thus, the plot is also called the box-and-whisker plot and the box-and-whisker diagram. Outliers that differ significantly from the rest of the dataset may be plotted as individual points beyond the whiskers on the box-plot.
Box plots are non-parametric: they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution (though Tukey's boxplot assumes symmetry for the whiskers and normality for their length). The spacings in each subsection of the box-plot indicate the degree of dispersion (spread) and skewness of the data, which are usually described using the five-number summary. In addition, the box-plot allows one to visually estimate various L-estimators, notably the interquartile range, midhinge, range, mid-range, and trimean. Box plots can be drawn either horizontally or vertically.
History
The range-bar method was first introduced by Mary Eleanor Spear in her book "Charting Statistics" in 1952 and again in her book "Practical Charting Techniques" in 1969. The box-and-whisker plot was first introduced in 1970 by John Tukey, who later published on the subject in his book "Exploratory Data Analysis" in 1977.
Elements
A boxplot is a standardized way of displaying the dataset based on the five-number summary: the minimum, the maximum, the sample median, and the first and third quartiles. | Box plot | Wikipedia | 391 | 160960 | https://en.wikipedia.org/wiki/Box%20plot | Mathematics | Statistics | null |
Minimum (Q0 or 0th percentile): the lowest data point in the data set excluding any outliers
Maximum (Q4 or 100th percentile): the highest data point in the data set excluding any outliers
Median (Q2 or 50th percentile): the middle value in the data set
First quartile (Q1 or 25th percentile): also known as the lower quartile qn(0.25), it is the median of the lower half of the dataset.
Third quartile (Q3 or 75th percentile): also known as the upper quartile qn(0.75), it is the median of the upper half of the dataset.
In addition to the minimum and maximum values used to construct a box-plot, another important element that can also be employed to obtain a box-plot is the interquartile range (IQR), as denoted below:
Interquartile range (IQR) : the distance between the upper and lower quartiles
A box-plot usually includes two parts, a box and a set of whiskers as shown in Figure 2.
Box
The box is drawn from Q1 to Q3 with a horizontal line drawn inside it to denote the median. Some box plots include an additional character to represent the mean of the data.
Whiskers
The whiskers must end at an observed data point, but can be defined in various ways. In the most straightforward method, the boundary of the lower whisker is the minimum value of the data set, and the boundary of the upper whisker is the maximum value of the data set. Because of this variability, it is appropriate to describe the convention that is being used for the whiskers and outliers in the caption of the box-plot. | Box plot | Wikipedia | 376 | 160960 | https://en.wikipedia.org/wiki/Box%20plot | Mathematics | Statistics | null |
Another popular choice for the boundaries of the whiskers is based on the 1.5 IQR value. From above the upper quartile (Q3), a distance of 1.5 times the IQR is measured out and a whisker is drawn up to the largest observed data point from the dataset that falls within this distance. Similarly, a distance of 1.5 times the IQR is measured out below the lower quartile (Q1) and a whisker is drawn down to the lowest observed data point from the dataset that falls within this distance. Because the whiskers must end at an observed data point, the whisker lengths can look unequal, even though 1.5 IQR is the same for both sides. All other observed data points outside the boundary of the whiskers are plotted as outliers. The outliers can be plotted on the box-plot as a dot, a small circle, a star, etc. (see example below).
There are other representations in which the whiskers can stand for several other things, such as:
One standard deviation above and below the mean of the data set
The 9th percentile and the 91st percentile of the data set
The 2nd percentile and the 98th percentile of the data set
Rarely, box-plot can be plotted without the whiskers. This can be appropriate for sensitive information to avoid whiskers (and outliers) disclosing actual values observed.
The unusual percentiles 2%, 9%, 91%, 98% are sometimes used for whisker cross-hatches and whisker ends to depict the seven-number summary. If the data are normally distributed, the locations of the seven marks on the box plot will be equally spaced. On some box plots, a cross-hatch is placed before the end of each whisker.
Variations
Since the mathematician John W. Tukey first popularized this type of visual data display in 1969, several variations on the classical box plot have been developed, and the two most commonly found variations are the variable-width box plots and the notched box plots shown in Figure 4.
Variable-width box plots illustrate the size of each group whose data is being plotted by making the width of the box proportional to the size of the group. A popular convention is to make the box width proportional to the square root of the size of the group. | Box plot | Wikipedia | 497 | 160960 | https://en.wikipedia.org/wiki/Box%20plot | Mathematics | Statistics | null |
Notched box plots apply a "notch" or narrowing of the box around the median. Notches are useful in offering a rough guide of the significance of the difference of medians; if the notches of two boxes do not overlap, this will provide evidence of a statistically significant difference between the medians. The height of the notches is proportional to the interquartile range (IQR) of the sample and is inversely proportional to the square root of the size of the sample. However, there is an uncertainty about the most appropriate multiplier (as this may vary depending on the similarity of the variances of the samples). The width of the notch is arbitrarily chosen to be visually pleasing, and should be consistent amongst all box plots being displayed on the same page.
One convention for obtaining the boundaries of these notches is to use a distance of around the median.
Adjusted box plots are intended to describe skew distributions, and they rely on the medcouple statistic of skewness. For a medcouple value of MC, the lengths of the upper and lower whiskers on the box-plot are respectively defined to be:
For a symmetrical data distribution, the medcouple will be zero, and this reduces the adjusted box-plot to the Tukey's box-plot with equal whisker lengths of for both whiskers.
Other kinds of box plots, such as the violin plots and the bean plots can show the difference between single-modal and multimodal distributions, which cannot be observed from the original classical box-plot.
Examples
Example without outliers
A series of hourly temperatures were measured throughout the day in degrees Fahrenheit. The recorded values are listed in order as follows (°F): 57, 57, 57, 58, 63, 66, 66, 67, 67, 68, 69, 70, 70, 70, 70, 72, 73, 75, 75, 76, 76, 78, 79, 81.
A box plot of the data set can be generated by first calculating five relevant values of this data set: minimum, maximum, median (Q2), first quartile (Q1), and third quartile (Q3).
The minimum is the smallest number of the data set. In this case, the minimum recorded day temperature is 57°F.
The maximum is the largest number of the data set. In this case, the maximum recorded day temperature is 81°F. | Box plot | Wikipedia | 512 | 160960 | https://en.wikipedia.org/wiki/Box%20plot | Mathematics | Statistics | null |
The median is the "middle" number of the ordered data set. This means that exactly 50% of the elements are below the median and 50% of the elements are greater than the median. The median of this ordered data set is 70°F.
The first quartile value (Q1 or 25th percentile) is the number that marks one quarter of the ordered data set. In other words, there are exactly 25% of the elements that are less than the first quartile and exactly 75% of the elements that are greater than it. The first quartile value can be easily determined by finding the "middle" number between the minimum and the median. For the hourly temperatures, the "middle" number found between 57°F and 70°F is 66°F.
The third quartile value (Q3 or 75th percentile) is the number that marks three quarters of the ordered data set. In other words, there are exactly 75% of the elements that are less than the third quartile and 25% of the elements that are greater than it. The third quartile value can be easily obtained by finding the "middle" number between the median and the maximum. For the hourly temperatures, the "middle" number between 70°F and 81°F is 75°F.
The interquartile range, or IQR, can be calculated by subtracting the first quartile value (Q1) from the third quartile value (Q3):
Hence,
1.5 IQR above the third quartile is:
1.5 IQR below the first quartile is:
The upper whisker boundary of the box-plot is the largest data value that is within 1.5 IQR above the third quartile. Here, 1.5 IQR above the third quartile is 88.5°F and the maximum is 81°F. Therefore, the upper whisker is drawn at the value of the maximum, which is 81°F.
Similarly, the lower whisker boundary of the box plot is the smallest data value that is within 1.5 IQR below the first quartile. Here, 1.5 IQR below the first quartile is 52.5°F and the minimum is 57°F. Therefore, the lower whisker is drawn at the value of the minimum, which is 57°F.
Example with outliers | Box plot | Wikipedia | 501 | 160960 | https://en.wikipedia.org/wiki/Box%20plot | Mathematics | Statistics | null |
Above is an example without outliers. Here is a followup example for generating box-plot with outliers:
The ordered set for the recorded temperatures is (°F): 52, 57, 57, 58, 63, 66, 66, 67, 67, 68, 69, 70, 70, 70, 70, 72, 73, 75, 75, 76, 76, 78, 79, 89.
In this example, only the first and the last number are changed. The median, third quartile, and first quartile remain the same.
In this case, the maximum value in this data set is 89°F, and 1.5 IQR above the third quartile is 88.5°F. The maximum is greater than 1.5 IQR plus the third quartile, so the maximum is an outlier. Therefore, the upper whisker is drawn at the greatest value smaller than 1.5 IQR above the third quartile, which is 79°F.
Similarly, the minimum value in this data set is 52°F, and 1.5 IQR below the first quartile is 52.5°F. The minimum is smaller than 1.5 IQR minus the first quartile, so the minimum is also an outlier. Therefore, the lower whisker is drawn at the smallest value greater than 1.5 IQR below the first quartile, which is 57°F.
In the case of large datasets
An additional example for obtaining box-plot from a data set containing a large number of data points is:
General equation to compute empirical quantiles
Here stands for the general ordering of the data points (i.e. if , then )
Using the above example that has 24 data points (n = 24), one can calculate the median, first and third quartile either mathematically or visually.
Median
First quartile
Third quartile
Visualization | Box plot | Wikipedia | 401 | 160960 | https://en.wikipedia.org/wiki/Box%20plot | Mathematics | Statistics | null |
Although box plots may seem more primitive than histograms or kernel density estimates, they do have a number of advantages. First, the box plot enables statisticians to do a quick graphical examination on one or more data sets. Box-plots also take up less space and are therefore particularly useful for comparing distributions between several groups or sets of data in parallel (see Figure 1 for an example). Lastly, the overall structure of histograms and kernel density estimate can be strongly influenced by the choice of number and width of bins techniques and the choice of bandwidth, respectively.
Although looking at a statistical distribution is more common than looking at a box plot, it can be useful to compare the box plot against the probability density function (theoretical histogram) for a normal N(0,σ2) distribution and observe their characteristics directly (as shown in Figure 7). | Box plot | Wikipedia | 178 | 160960 | https://en.wikipedia.org/wiki/Box%20plot | Mathematics | Statistics | null |
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-eth" item in a sequence.
Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another.
Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers.
In the 3rd century BC Archimedes used what eventually came to be known as the method of indivisibles in his work The Method of Mechanical Theorems to find areas of regions and volumes of solids. In his formal published treatises, Archimedes solved the same problem using the method of exhaustion.
Infinitesimals regained popularity in the 20th century with Abraham Robinson's development of nonstandard analysis and the hyperreal numbers, which, after centuries of controversy, showed that a formal treatment of infinitesimal calculus was possible. Following this, mathematicians developed surreal numbers, a related formalization of infinite and infinitesimal numbers that include both hyperreal cardinal and ordinal numbers, which is the largest ordered field.
Vladimir Arnold wrote in 1990:
The crucial insight for making infinitesimals feasible mathematical entities was that they could still retain certain properties such as angle or slope, even if these entities were infinitely small. | Infinitesimal | Wikipedia | 378 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
Infinitesimals are a basic ingredient in calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective in mathematics, infinitesimal means infinitely small, smaller than any standard real number. Infinitesimals are often compared to other infinitesimals of similar size, as in examining the derivative of a function. An infinite number of infinitesimals are summed to calculate an integral.
The modern concept of infinitesimals was introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. The 15th century saw the work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular, the calculation of the area of a circle by representing the latter as an infinite-sided polygon. Simon Stevin's work on the decimal representation of all numbers in the 16th century prepared the ground for the real continuum. Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors. The method of indivisibles related to geometrical figures as being composed of entities of codimension 1. John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures into infinitely thin building blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus. He exploited an infinitesimal denoted 1/∞ in area calculations. | Infinitesimal | Wikipedia | 350 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
The use of infinitesimals by Leibniz relied upon heuristic principles, such as the law of continuity: what succeeds for the finite numbers succeeds also for the infinite numbers and vice versa; and the transcendental law of homogeneity that specifies procedures for replacing expressions involving unassignable quantities, by expressions involving only assignable ones. The 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange. Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function. As Cantor and Dedekind were developing more abstract versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions. Du Bois-Reymond's work inspired both Émile Borel and Thoralf Skolem. Borel explicitly linked du Bois-Reymond's work to Cauchy's work on rates of growth of infinitesimals. Skolem developed the first non-standard models of arithmetic in 1934. A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed nonstandard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955. The hyperreals implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity. The standard part function implements Fermat's adequality.
History of the infinitesimal
The notion of infinitely small quantities was discussed by the Eleatic School. The Greek mathematician Archimedes (c. 287 BC – c. 212 BC), in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals. His Archimedean property defines a number x as infinite if it satisfies the conditions ..., and infinitesimal if and a similar set of conditions holds for x and the reciprocals of the positive integers. A number system is said to be Archimedean if it contains no infinite or infinitesimal members. | Infinitesimal | Wikipedia | 465 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
The English mathematician John Wallis introduced the expression 1/∞ in his 1655 book Treatise on the Conic Sections. The symbol, which denotes the reciprocal, or inverse, of ∞, is the symbolic representation of the mathematical concept of an infinitesimal. In his Treatise on the Conic Sections, Wallis also discusses the concept of a relationship between the symbolic representation of infinitesimal 1/∞ that he introduced and the concept of infinity for which he introduced the symbol ∞. The concept suggests a thought experiment of adding an infinite number of parallelograms of infinitesimal width to form a finite area. This concept was the predecessor to the modern method of integration used in integral calculus. The conceptual origins of the concept of the infinitesimal 1/∞ can be traced as far back as the Greek philosopher Zeno of Elea, whose Zeno's dichotomy paradox was the first mathematical concept to consider the relationship between a finite interval and an interval approaching that of an infinitesimal-sized interval.
Infinitesimals were the subject of political and religious controversies in 17th century Europe, including a ban on infinitesimals issued by clerics in Rome in 1632. | Infinitesimal | Wikipedia | 238 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
Prior to the invention of calculus mathematicians were able to calculate tangent lines using Pierre de Fermat's method of adequality and René Descartes' method of normals. There is debate among scholars as to whether the method was infinitesimal or algebraic in nature. When Newton and Leibniz invented the calculus, they made use of infinitesimals, Newton's fluxions and Leibniz' differential. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The Analyst. Mathematicians, scientists, and engineers continued to use infinitesimals to produce correct results. In the second half of the nineteenth century, the calculus was reformulated by Augustin-Louis Cauchy, Bernard Bolzano, Karl Weierstrass, Cantor, Dedekind, and others using the (ε, δ)-definition of limit and set theory.
While the followers of Cantor, Dedekind, and Weierstrass sought to rid analysis of infinitesimals, and their philosophical allies like Bertrand Russell and Rudolf Carnap declared that infinitesimals are pseudoconcepts, Hermann Cohen and his Marburg school of neo-Kantianism sought to develop a working logic of infinitesimals. The mathematical study of systems containing infinitesimals continued through the work of Levi-Civita, Giuseppe Veronese, Paul du Bois-Reymond, and others, throughout the late nineteenth and the twentieth centuries, as documented by Philip Ehrlich (2006). In the 20th century, it was found that infinitesimals could serve as a basis for calculus and analysis (see hyperreal numbers). | Infinitesimal | Wikipedia | 341 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
First-order properties
In extending the real numbers to include infinite and infinitesimal quantities, one typically wishes to be as conservative as possible by not changing any of their elementary properties. This guarantees that as many familiar results as possible are still available. Typically, elementary means that there is no quantification over sets, but only over elements. This limitation allows statements of the form "for any number x..." For example, the axiom that states "for any number x, x + 0 = x" would still apply. The same is true for quantification over several numbers, e.g., "for any numbers x and y, xy = yx." However, statements of the form "for any set S of numbers ..." may not carry over. Logic with this limitation on quantification is referred to as first-order logic.
The resulting extended number system cannot agree with the reals on all properties that can be expressed by quantification over sets, because the goal is to construct a non-Archimedean system, and the Archimedean principle can be expressed by quantification over sets. One can conservatively extend any theory including reals, including set theory, to include infinitesimals, just by adding a countably infinite list of axioms that assert that a number is smaller than 1/2, 1/3, 1/4, and so on. Similarly, the completeness property cannot be expected to carry over, because the reals are the unique complete ordered field up to isomorphism.
We can distinguish three levels at which a non-Archimedean number system could have first-order properties compatible with those of the reals: | Infinitesimal | Wikipedia | 351 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
An ordered field obeys all the usual axioms of the real number system that can be stated in first-order logic. For example, the commutativity axiom x + y = y + x holds.
A real closed field has all the first-order properties of the real number system, regardless of whether they are usually taken as axiomatic, for statements involving the basic ordered-field relations +, ×, and ≤. This is a stronger condition than obeying the ordered-field axioms. More specifically, one includes additional first-order properties, such as the existence of a root for every odd-degree polynomial. For example, every number must have a cube root.
The system could have all the first-order properties of the real number system for statements involving any relations (regardless of whether those relations can be expressed using +, ×, and ≤). For example, there would have to be a sine function that is well defined for infinite inputs; the same is true for every real function.
Systems in category 1, at the weak end of the spectrum, are relatively easy to construct but do not allow a full treatment of classical analysis using infinitesimals in the spirit of Newton and Leibniz. For example, the transcendental functions are defined in terms of infinite limiting processes, and therefore there is typically no way to define them in first-order logic. Increasing the analytic strength of the system by passing to categories 2 and 3, we find that the flavor of the treatment tends to become less constructive, and it becomes more difficult to say anything concrete about the hierarchical structure of infinities and infinitesimals.
Number systems that include infinitesimals
Formal series | Infinitesimal | Wikipedia | 345 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
Laurent series
An example from category 1 above is the field of Laurent series with a finite number of negative-power terms. For example, the Laurent series consisting only of the constant term 1 is identified with the real number 1, and the series with only the linear term x is thought of as the simplest infinitesimal, from which the other infinitesimals are constructed. Dictionary ordering is used, which is equivalent to considering higher powers of x as negligible compared to lower powers. David O. Tall refers to this system as the super-reals, not to be confused with the superreal number system of Dales and Woodin. Since a Taylor series evaluated with a Laurent series as its argument is still a Laurent series, the system can be used to do calculus on transcendental functions if they are analytic. These infinitesimals have different first-order properties than the reals because, for example, the basic infinitesimal x does not have a square root.
The Levi-Civita field
The Levi-Civita field is similar to the Laurent series, but is algebraically closed. For example, the basic infinitesimal x has a square root. This field is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented in floating-point.
Transseries
The field of transseries is larger than the Levi-Civita field. An example of a transseries is:
where for purposes of ordering x is considered infinite.
Surreal numbers
Conway's surreal numbers fall into category 2, except that the surreal numbers form a proper class and not a set. They are a system designed to be as rich as possible in different sizes of numbers, but not necessarily for convenience in doing analysis, in the sense that every ordered field is a subfield of the surreal numbers. There is a natural extension of the exponential function to the surreal numbers.
Hyperreals | Infinitesimal | Wikipedia | 402 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
The most widespread technique for handling infinitesimals is the hyperreals, developed by Abraham Robinson in the 1960s. They fall into category 3 above, having been designed that way so all of classical analysis can be carried over from the reals. This property of being able to carry over all relations in a natural way is known as the transfer principle, proved by Jerzy Łoś in 1955. For example, the transcendental function sin has a natural counterpart *sin that takes a hyperreal input and gives a hyperreal output, and similarly the set of natural numbers has a natural counterpart , which contains both finite and infinite integers. A proposition such as carries over to the hyperreals as .
Superreals
The superreal number system of Dales and Woodin is a generalization of the hyperreals. It is different from the super-real system defined by David Tall.
Dual numbers
In linear algebra, the dual numbers extend the reals by adjoining one infinitesimal, the new element ε with the property ε2 = 0 (that is, ε is nilpotent). Every dual number has the form z = a + bε with a and b being uniquely determined real numbers.
One application of dual numbers is automatic differentiation. This application can be generalized to polynomials in n variables, using the Exterior algebra of an n-dimensional vector space.
Smooth infinitesimal analysis
Synthetic differential geometry or smooth infinitesimal analysis have roots in category theory. This approach departs from the classical logic used in conventional mathematics by denying the general applicability of the law of excluded middle – i.e., not (a ≠ b) does not have to mean a = b. A nilsquare or nilpotent infinitesimal can then be defined. This is a number x where x2 = 0 is true, but x = 0 need not be true at the same time. Since the background logic is intuitionistic logic, it is not immediately clear how to classify this system with regard to classes 1, 2, and 3. Intuitionistic analogues of these classes would have to be developed first. | Infinitesimal | Wikipedia | 429 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
Infinitesimal delta functions
Cauchy used an infinitesimal to write down a unit impulse, infinitely tall and narrow Dirac-type delta function satisfying in a number of articles in 1827, see Laugwitz (1989). Cauchy defined an infinitesimal in 1821 (Cours d'Analyse) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and Lazare Carnot's terminology.
Modern set-theoretic approaches allow one to define infinitesimals via the ultrapower construction, where a null sequence becomes an infinitesimal in the sense of an equivalence class modulo a relation defined in terms of a suitable ultrafilter. The article by Yamashita (2007) contains bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the hyperreals.
Logical properties
The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist. | Infinitesimal | Wikipedia | 234 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them. A consequence of this theorem is that if there is a number system in which it is true that for any positive integer n there is a positive number x such that 0 < x < 1/n, then there exists an extension of that number system in which it is true that there exists a positive number x such that for any positive integer n we have 0 < x < 1/n. The possibility to switch "for any" and "there exists" is crucial. The first statement is true in the real numbers as given in ZFC set theory : for any positive integer n it is possible to find a real number between 1/n and zero, but this real number depends on n. Here, one chooses n first, then one finds the corresponding x. In the second expression, the statement says that there is an x (at least one), chosen first, which is between 0 and 1/n for any n. In this case x is infinitesimal. This is not true in the real numbers (R) given by ZFC. Nonetheless, the theorem proves that there is a model (a number system) in which this is true. The question is: what is this model? What are its properties? Is there only one such model?
There are in fact many ways to construct such a one-dimensional linearly ordered set of numbers, but fundamentally, there are two different approaches:
Extend the number system so that it contains more numbers than the real numbers.
Extend the axioms (or extend the language) so that the distinction between the infinitesimals and non-infinitesimals can be made in the real numbers themselves.
In 1960, Abraham Robinson provided an answer following the first approach. The extended set is called the hyperreals and contains numbers less in absolute value than any positive real number. The method may be considered relatively complex but it does prove that infinitesimals exist in the universe of ZFC set theory. The real numbers are called standard numbers and the new non-real hyperreals are called nonstandard. | Infinitesimal | Wikipedia | 452 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal set theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system, we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number that is less, in absolute value, than any positive standard real number.
In 2006 Karel Hrbacek developed an extension of Nelson's approach in which the real numbers are stratified in (infinitely) many levels; i.e., in the coarsest level, there are no infinitesimals nor unlimited numbers. Infinitesimals are at a finer level and there are also infinitesimals with respect to this new level and so on.
Infinitesimals in teaching
Calculus textbooks based on infinitesimals include the classic Calculus Made Easy by Silvanus P. Thompson (bearing the motto "What one fool can do another can") and the German text Mathematik fur Mittlere Technische Fachschulen der Maschinenindustrie by R. Neuendorff. Pioneering works based on Abraham Robinson's infinitesimals include texts by Stroyan (dating from 1972) and Howard Jerome Keisler (Elementary Calculus: An Infinitesimal Approach). Students easily relate to the intuitive notion of an infinitesimal difference 1-"0.999...", where "0.999..." differs from its standard meaning as the real number 1, and is reinterpreted as an infinite terminating extended decimal that is strictly less than 1.
Another elementary calculus text that uses the theory of infinitesimals as developed by Robinson is Infinitesimal Calculus by Henle and Kleinberg, originally published in 1979. The authors introduce the language of first-order logic, and demonstrate the construction of a first order model of the hyperreal numbers. The text provides an introduction to the basics of integral and differential calculus in one dimension, including sequences and series of functions. In an Appendix, they also treat the extension of their model to the hyperhyperreals, and demonstrate some applications for the extended model. | Infinitesimal | Wikipedia | 475 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
An elementary calculus text based on smooth infinitesimal analysis is Bell, John L. (2008). A Primer of Infinitesimal Analysis, 2nd Edition. Cambridge University Press. ISBN 9780521887182.
A more recent calculus text utilizing infinitesimals is Dawson, C. Bryan (2022), Calculus Set Free: Infinitesimals to the Rescue, Oxford University Press. ISBN 9780192895608.
Functions tending to zero
In a related but somewhat different sense, which evolved from the original definition of "infinitesimal" as an infinitely small quantity, the term has also been used to refer to a function tending to zero. More precisely, Loomis and Sternberg's Advanced Calculus defines the function class of infinitesimals, , as a subset of functions between normed vector spaces by , as well as two related classes (see Big-O notation) by , and.The set inclusions generally hold. That the inclusions are proper is demonstrated by the real-valued functions of a real variable , , and : but and .As an application of these definitions, a mapping between normed vector spaces is defined to be differentiable at if there is a [i.e, a bounded linear map ] such that in a neighborhood of . If such a map exists, it is unique; this map is called the differential and is denoted by , coinciding with the traditional notation for the classical (though logically flawed) notion of a differential as an infinitely small "piece" of F. This definition represents a generalization of the usual definition of differentiability for vector-valued functions of (open subsets of) Euclidean spaces.
Array of random variables
Let be a probability space and let . An array of random variables is called infinitesimal if for every , we have:
The notion of infinitesimal array is essential in some central limit theorems and it is easily seen by monotonicity of the expectation operator that any array satisfying Lindeberg's condition is infinitesimal, thus playing an important role in Lindeberg's Central Limit Theorem (a generalization of the central limit theorem). | Infinitesimal | Wikipedia | 443 | 160990 | https://en.wikipedia.org/wiki/Infinitesimal | Mathematics | Basics_2 | null |
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.
Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence of term coefficients.
History
Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem.
George Pólya writes in Mathematics and plausible reasoning:
The name "generating function" is due to Laplace. Yet, without giving it a name, Euler used the device of generating functions long before Laplace [..]. He applied this mathematical tool to several problems in Combinatory Analysis and the Theory of Numbers.
Definition
Convergence
Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. Thus generating functions are not functions in the formal sense of a mapping from a domain to a codomain.
These expressions in terms of the indeterminate may involve arithmetic operations, differentiation with respect to and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of . Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of , and which has the formal series as its series expansion; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a convergent series when a nonzero numeric value is substituted for . | Generating function | Wikipedia | 499 | 160993 | https://en.wikipedia.org/wiki/Generating%20function | Mathematics | Sequences and series | null |
Limitations
Not all expressions that are meaningful as functions of are meaningful as expressions designating formal series; for example, negative and fractional powers of are examples of functions that do not have a corresponding formal power series.
Types
Ordinary generating function (OGF)
When the term generating function is used without qualification, it is usually taken to mean an ordinary generating function. The ordinary generating function of a sequence is:
If is the probability mass function of a discrete random variable, then its ordinary generating function is called a probability-generating function.
Exponential generating function (EGF)
The exponential generating function of a sequence is
Exponential generating functions are generally more convenient than ordinary generating functions for combinatorial enumeration problems that involve labelled objects.
Another benefit of exponential generating functions is that they are useful in transferring linear recurrence relations to the realm of differential equations. For example, take the Fibonacci sequence that satisfies the linear recurrence relation . The corresponding exponential generating function has the form
and its derivatives can readily be shown to satisfy the differential equation as a direct analogue with the recurrence relation above. In this view, the factorial term is merely a counter-term to normalise the derivative operator acting on .
Poisson generating function
The Poisson generating function of a sequence is
Lambert series
The Lambert series of a sequence is
Note that in a Lambert series the index starts at 1, not at 0, as the first term would otherwise be undefined.
The Lambert series coefficients in the power series expansions
for integers are related by the divisor sum
The main article provides several more classical, or at least well-known examples related to special arithmetic functions in number theory. As an example of a Lambert series identity not given in the main article, we can show that for we have that
where we have the special case identity for the generating function of the divisor function, , given by
Bell series
The Bell series of a sequence is an expression in terms of both an indeterminate and a prime and is given by:
Dirichlet series generating functions (DGFs)
Formal Dirichlet series are often classified as generating functions, although they are not strictly formal power series. The Dirichlet series generating function of a sequence is:
The Dirichlet series generating function is especially useful when is a multiplicative function, in which case it has an Euler product expression in terms of the function's Bell series: | Generating function | Wikipedia | 490 | 160993 | https://en.wikipedia.org/wiki/Generating%20function | Mathematics | Sequences and series | null |
If is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet -series. We also have a relation between the pair of coefficients in the Lambert series expansions above and their DGFs. Namely, we can prove that:
if and only if
where is the Riemann zeta function.
The sequence generated by a Dirichlet series generating function (DGF) corresponding to:has the ordinary generating function:
Polynomial sequence generating functions
The idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of binomial type are generated by:
where is a sequence of polynomials and is a function of a certain form. Sheffer sequences are generated in a similar way. See the main article generalized Appell polynomials for more information.
Examples of polynomial sequences generated by more complex generating functions include:
Appell polynomials
Chebyshev polynomials
Difference polynomials
Generalized Appell polynomials
-difference polynomials
Other generating functions
Other sequences generated by more complex generating functions include:
Double exponential generating functions. For example: Aitken's Array: Triangle of Numbers
Hadamard products of generating functions and diagonal generating functions, and their corresponding integral transformations
Convolution polynomials
Knuth's article titled "Convolution Polynomials" defines a generalized class of convolution polynomial sequences by their special generating functions of the form
for some analytic function with a power series expansion such that .
We say that a family of polynomials, , forms a convolution family if and if the following convolution condition holds for all , and for all :
We see that for non-identically zero convolution families, this definition is equivalent to requiring that the sequence have an ordinary generating function of the first form given above.
A sequence of convolution polynomials defined in the notation above has the following properties:
The sequence is of binomial type
Special values of the sequence include and , and
For arbitrary (fixed) , these polynomials satisfy convolution formulas of the form
For a fixed non-zero parameter , we have modified generating functions for these convolution polynomial sequences given by
where is implicitly defined by a functional equation of the form . Moreover, we can use matrix methods (as in the reference) to prove that given two convolution polynomial sequences, and , with respective corresponding generating functions, and , then for arbitrary we have the identity | Generating function | Wikipedia | 487 | 160993 | https://en.wikipedia.org/wiki/Generating%20function | Mathematics | Sequences and series | null |
Examples of convolution polynomial sequences include the binomial power series, , so-termed tree polynomials, the Bell numbers, , the Laguerre polynomials, and the Stirling convolution polynomials.
Ordinary generating functions
Examples for simple sequences
Polynomials are a special case of ordinary generating functions, corresponding to finite sequences, or equivalently sequences that vanish after a certain point. These are important in that many finite sequences can usefully be interpreted as generating functions, such as the Poincaré polynomial and others.
A fundamental generating function is that of the constant sequence , whose ordinary generating function is the geometric series
The left-hand side is the Maclaurin series expansion of the right-hand side. Alternatively, the equality can be justified by multiplying the power series on the left by , and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of are equal to 0). Moreover, there can be no other power series with this property. The left-hand side therefore designates the multiplicative inverse of in the ring of power series.
Expressions for the ordinary generating function of other sequences are easily derived from this one. For instance, the substitution gives the generating function for the geometric sequence for any constant :
(The equality also follows directly from the fact that the left-hand side is the Maclaurin series expansion of the right-hand side.) In particular,
One can also introduce regular gaps in the sequence by replacing by some power of , so for instance for the sequence (which skips over ) one gets the generating function
By squaring the initial generating function, or by finding the derivative of both sides with respect to and making a change of running variable , one sees that the coefficients form the sequence , so one has
and the third power has as coefficients the triangular numbers whose term is the binomial coefficient , so that
More generally, for any non-negative integer and non-zero real value , it is true that
Since
one can find the ordinary generating function for the sequence of square numbers by linear combination of binomial-coefficient generating sequences:
We may also expand alternately to generate this same sequence of squares as a sum of derivatives of the geometric series in the following form:
By induction, we can similarly show for positive integers that
where denote the Stirling numbers of the second kind and where the generating function
so that we can form the analogous generating functions over the integral th powers generalizing the result in the square case above. In particular, since we can write | Generating function | Wikipedia | 511 | 160993 | https://en.wikipedia.org/wiki/Generating%20function | Mathematics | Sequences and series | null |
we can apply a well-known finite sum identity involving the Stirling numbers to obtain that
Rational functions
The ordinary generating function of a sequence can be expressed as a rational function (the ratio of two finite-degree polynomials) if and only if the sequence is a linear recursive sequence with constant coefficients; this generalizes the examples above. Conversely, every sequence generated by a fraction of polynomials satisfies a linear recurrence with constant coefficients; these coefficients are identical to the coefficients of the fraction denominator polynomial (so they can be directly read off). This observation shows it is easy to solve for generating functions of sequences defined by a linear finite difference equation with constant coefficients, and then hence, for explicit closed-form formulas for the coefficients of these generating functions. The prototypical example here is to derive Binet's formula for the Fibonacci numbers via generating function techniques.
We also notice that the class of rational generating functions precisely corresponds to the generating functions that enumerate quasi-polynomial sequences of the form
where the reciprocal roots, , are fixed scalars and where is a polynomial in for all .
In general, Hadamard products of rational functions produce rational generating functions. Similarly, if
is a bivariate rational generating function, then its corresponding diagonal generating function,
is algebraic. For example, if we let
then this generating function's diagonal coefficient generating function is given by the well-known OGF formula
This result is computed in many ways, including Cauchy's integral formula or contour integration, taking complex residues, or by direct manipulations of formal power series in two variables.
Operations on generating functions
Multiplication yields convolution
Multiplication of ordinary generating functions yields a discrete convolution (the Cauchy product) of the sequences. For example, the sequence of cumulative sums (compare to the slightly more general Euler–Maclaurin formula)
of a sequence with ordinary generating function has the generating function
because is the ordinary generating function for the sequence . | Generating function | Wikipedia | 404 | 160993 | https://en.wikipedia.org/wiki/Generating%20function | Mathematics | Sequences and series | null |
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by , is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true; and the p-value of a result, , is the probability of obtaining a result at least as extreme, given that the null hypothesis is true. The result is statistically significant, by the standards of the study, when .<ref
name="Johnson"></ref> The significance level for a study is chosen before data collection, and is typically set to 5% or much lower—depending on the field of study.
In any experiment or observation that involves drawing a sample from a population, there is always the possibility that an observed effect would have occurred due to sampling error alone. But if the p-value of an observed effect is less than (or equal to) the significance level, an investigator may conclude that the effect reflects the characteristics of the whole population, thereby rejecting the null hypothesis.
This technique for testing the statistical significance of results was developed in the early 20th century. The term significance does not imply importance here, and the term statistical significance is not the same as research significance, theoretical significance, or practical significance. For example, the term clinical significance refers to the practical importance of a treatment effect.
History
Statistical significance dates to the 18th century, in the work of John Arbuthnot and Pierre-Simon Laplace, who computed the p-value for the human sex ratio at birth, assuming a null hypothesis of equal probability of male and female births; see for details.
In 1925, Ronald Fisher advanced the idea of statistical hypothesis testing, which he called "tests of significance", in his publication Statistical Methods for Research Workers. Fisher suggested a probability of one in twenty (0.05) as a convenient cutoff level to reject the null hypothesis. In a 1933 paper, Jerzy Neyman and Egon Pearson called this cutoff the significance level, which they named . They recommended that be set ahead of time, prior to any data collection.
Despite his initial suggestion of 0.05 as a significance level, Fisher did not intend this cutoff value to be fixed. In his 1956 publication Statistical Methods and Scientific Inference, he recommended that significance levels be set according to specific circumstances. | Statistical significance | Wikipedia | 491 | 160995 | https://en.wikipedia.org/wiki/Statistical%20significance | Mathematics | Statistics | null |
Related concepts
The significance level is the threshold for below which the null hypothesis is rejected even though by assumption it were true, and something else is going on. This means that is also the probability of mistakenly rejecting the null hypothesis, if the null hypothesis is true. This is also called false positive and type I error.
Sometimes researchers talk about the confidence level instead. This is the probability of not rejecting the null hypothesis given that it is true. Confidence levels and confidence intervals were introduced by Neyman in 1937.
Role in statistical hypothesis testing
Statistical significance plays a pivotal role in statistical hypothesis testing. It is used to determine whether the null hypothesis should be rejected or retained. The null hypothesis is the hypothesis that no effect exists in the phenomenon being studied. For the null hypothesis to be rejected, an observed result has to be statistically significant, i.e. the observed p-value is less than the pre-specified significance level .
To determine whether a result is statistically significant, a researcher calculates a p-value, which is the probability of observing an effect of the same magnitude or more extreme given that the null hypothesis is true. The null hypothesis is rejected if the p-value is less than (or equal to) a predetermined level, . is also called the significance level, and is the probability of rejecting the null hypothesis given that it is true (a type I error). It is usually set at or below 5%.
For example, when is set to 5%, the conditional probability of a type I error, given that the null hypothesis is true, is 5%, and a statistically significant result is one where the observed p-value is less than (or equal to) 5%. When drawing data from a sample, this means that the rejection region comprises 5% of the sampling distribution. These 5% can be allocated to one side of the sampling distribution, as in a one-tailed test, or partitioned to both sides of the distribution, as in a two-tailed test, with each tail (or rejection region) containing 2.5% of the distribution. | Statistical significance | Wikipedia | 425 | 160995 | https://en.wikipedia.org/wiki/Statistical%20significance | Mathematics | Statistics | null |
The use of a one-tailed test is dependent on whether the research question or alternative hypothesis specifies a direction such as whether a group of objects is heavier or the performance of students on an assessment is better. A two-tailed test may still be used but it will be less powerful than a one-tailed test, because the rejection region for a one-tailed test is concentrated on one end of the null distribution and is twice the size (5% vs. 2.5%) of each rejection region for a two-tailed test. As a result, the null hypothesis can be rejected with a less extreme result if a one-tailed test was used. The one-tailed test is only more powerful than a two-tailed test if the specified direction of the alternative hypothesis is correct. If it is wrong, however, then the one-tailed test has no power.
Significance thresholds in specific fields
In specific fields such as particle physics and manufacturing, statistical significance is often expressed in multiples of the standard deviation or sigma (σ) of a normal distribution, with significance thresholds set at a much stricter level (for example 5σ). For instance, the certainty of the Higgs boson particle's existence was based on the 5σ criterion, which corresponds to a p-value of about 1 in 3.5 million.
In other fields of scientific research such as genome-wide association studies, significance levels as low as are not uncommon—as the number of tests performed is extremely large.
Limitations
Researchers focusing solely on whether their results are statistically significant might report findings that are not substantive and not replicable. There is also a difference between statistical significance and practical significance. A study that is found to be statistically significant may not necessarily be practically significant.
Effect size
Effect size is a measure of a study's practical significance. A statistically significant result may have a weak effect. To gauge the research significance of their result, researchers are encouraged to always report an effect size along with p-values. An effect size measure quantifies the strength of an effect, such as the distance between two means in units of standard deviation (cf. Cohen's d), the correlation coefficient between two variables or its square, and other measures.
Reproducibility
A statistically significant result may not be easy to reproduce. In particular, some statistically significant results will in fact be false positives. Each failed attempt to reproduce a result increases the likelihood that the result was a false positive.
Challenges | Statistical significance | Wikipedia | 505 | 160995 | https://en.wikipedia.org/wiki/Statistical%20significance | Mathematics | Statistics | null |
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