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119-582: The purpose of the BICEP experiment is to measure the polarization of the cosmic microwave background. Specifically, it aims to measure the B -modes ( curl component) of the polarization of the CMB. BICEP operates from Antarctica , at the Amundsen–Scott South Pole Station . All three instruments have mapped the same part of the sky, around the south celestial pole . The institutions involved in

238-529: A i 1 , … , i k ∂ x j d x j ∧ d x i 1 ∧ ⋯ ∧ d x i k . {\displaystyle d\omega ^{(k)}=\sum _{\scriptstyle {j=1} \atop \scriptstyle {i_{1}<\cdots <i_{k}}}^{n}{\frac {\partial a_{i_{1},\ldots ,i_{k}}}{\partial x_{j}}}\,dx_{j}\wedge dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}}.} The exterior derivative of

357-596: A i 1 , … , i k d x i 1 ∧ ⋯ ∧ d x i k , {\displaystyle \omega ^{(k)}=\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}a_{i_{1},\ldots ,i_{k}}\,dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}},} then the exterior derivative d leads to d ω ( k ) = ∑ j = 1 i 1 < ⋯ < i k n ∂

476-509: A 12 d x ∧ d y + a 13 d x ∧ d z + a 23 d y ∧ d z ; {\displaystyle a_{12}\,dx\wedge dy+a_{13}\,dx\wedge dz+a_{23}\,dy\wedge dz;} and a differential 3-form is defined by a single term with one function as coefficient: a 123 d x ∧ d y ∧ d z . {\displaystyle a_{123}\,dx\wedge dy\wedge dz.} (Here

595-432: A confidence interval . They are mutually illuminating . A result is often significant when there is confidence in the sign of a relationship (the interval does not include 0). Whenever the sign of a relationship is important, statistical significance is a worthy goal. This also reveals weaknesses of significance testing: A result can be significant without a good estimate of the strength of a relationship; significance can be

714-410: A statistical model of what the data would look like if chance or random processes alone were responsible for the results. The hypothesis that chance alone is responsible for the results is called the null hypothesis . The model of the result of the random process is called the distribution under the null hypothesis . The obtained results are compared with the distribution under the null hypothesis, and

833-731: A 1-form is therefore a 2-form, and that of a 2-form is a 3-form. On the other hand, because of the interchangeability of mixed derivatives, ∂ 2 ∂ x i ∂ x j = ∂ 2 ∂ x j ∂ x i , {\displaystyle {\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}={\frac {\partial ^{2}}{\partial x_{j}\,\partial x_{i}}},} and antisymmetry, d x i ∧ d x j = − d x j ∧ d x i {\displaystyle dx_{i}\wedge dx_{j}=-dx_{j}\wedge dx_{i}}

952-486: A 68 cm aperture, providing roughly twice the optical throughput of the entire Keck array. One consequence of the large focal plane is a larger 28° field of view, which will necessarily mean scanning some foreground-contaminated portions of the sky. It was installed (with initial configuration) at the pole in January 2015. It was upgraded for the 2015-2016 Austral summer season to a full 2560 detector configuration. BICEP3

1071-434: A beneficial effect" is the more informative result of a one-tailed test. "The treatment has an effect, reducing the average length of hospitalization by 1.5 days" is the most informative report, combining a two-tailed significance test result with a numeric estimate of the relationship between treatment and effect. Explicitly reporting a numeric result eliminates a philosophical advantage of a one-tailed test. An underlying issue

1190-846: A closed line integral in a plane perpendicular to u ^ {\displaystyle \mathbf {\hat {u}} } divided by the area enclosed, as the path of integration is contracted indefinitely around the point. More specifically, the curl is defined at a point p as ( ∇ × F ) ( p ) ⋅ u ^   = d e f lim A → 0 1 | A | ∮ C F ⋅ d r {\displaystyle (\nabla \times \mathbf {F} )(p)\cdot \mathbf {\hat {u}} \ {\overset {\underset {\mathrm {def} }{}}{{}={}}}\lim _{A\to 0}{\frac {1}{|A|}}\oint _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {r} } where

1309-405: A currently useful regime to a different one. Nevertheless, if at this point the effect appears likely and/or large enough, there may be an incentive to further investigate, such as running a bigger sample. For instance, a certain drug may reduce the risk of having a heart attack. Possible null hypotheses are "this drug does not reduce the risk of having a heart attack" or "this drug has no effect on

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1428-407: A differential 0-form is a real-valued function f ( x , y , z ) ; a differential 1-form is the following expression, where the coefficients are functions: a 1 d x + a 2 d y + a 3 d z ; {\displaystyle a_{1}\,dx+a_{2}\,dy+a_{3}\,dz;} a differential 2-form is the formal sum, again with function coefficients:

1547-415: A large number of heads or a large number of tails, and our experiment with 5 heads would seem to belong to this class. However, the probability of 5 tosses of the same kind, irrespective of whether these are head or tails, is twice as much as that of the 5-head occurrence singly considered. Hence, under this two-tailed null hypothesis, the observation receives a probability value of 0.063. Hence again, with

1666-401: A modest goal. A weak relationship can also achieve significance with enough data. Reporting both significance and confidence intervals is commonly recommended. The varied uses of significance tests reduce the number of generalizations that can be made about all applications. The choice of the null hypothesis is associated with sparse and inconsistent advice. Fisher mentioned few constraints on

1785-455: A non-null. The used confidence level does absolutely certainly not correspond to the likelihood of null at failing to exclude; in fact in this case a high used confidence level expands the still plausible range. A non-null hypothesis can have the following meanings, depending on the author a) a value other than zero is used, b) some margin other than zero is used and c) the "alternative" hypothesis . Testing (excluding or failing to exclude)

1904-457: A null hypothesis at a certain confidence level. The confidence level should indicate the likelihood that much more and better data would still be able to exclude the null hypothesis on the same side. The concept of a null hypothesis is used differently in two approaches to statistical inference. In the significance testing approach of Ronald Fisher , a null hypothesis is rejected if the observed data are significantly unlikely to have occurred if

2023-428: A null hypothesis/alternative hypothesis pair. However, the results are not a full description of all the results of an experiment, merely a single result tailored to one particular purpose. For example, consider an H 0 that claims the population mean for a new treatment is an improvement on a well-established treatment with population mean = 10 (known from long experience), with the one-tailed alternative being that

2142-428: A particular quantity or difference is equal to a particular number. In classical science, it is most typically the statement that there is no effect of a particular treatment; in observations, it is typically that there is no difference between the value of a particular measured variable and that of a prediction. Most statisticians believe that it is valid to state direction as a part of null hypothesis, or as part of

2261-692: A prototype for future instruments, was first described in Keating et al. 2003 and started observing in January 2006 and ran until the end of 2008. The second-generation instrument was BICEP2. Featuring a greatly improved focal-plane transition edge sensor (TES) bolometer array of 512 sensors (256 pixels) operating at 150 GHz, this 26 cm aperture telescope replaced the BICEP1 instrument, and observed from 2010 to 2012. Reports stated in March 2014 that BICEP2 had detected B -modes from gravitational waves in

2380-503: A pseudo-Riemannian metric gives an isomorphism between vectors and covectors), and on an oriented vector space with a nondegenerate form (an isomorphism between vectors and covectors), there is an isomorphism between k -vectors and ( n − k ) -vectors; in particular on (the tangent space of) an oriented pseudo-Riemannian manifold. Thus on an oriented pseudo-Riemannian manifold, one can interchange k -forms, k -vector fields, ( n − k ) -forms, and ( n − k ) -vector fields; this

2499-420: A relationship does exist between two variables. The null hypothesis and the alternative hypothesis are types of conjectures used in statistical tests to make statistical inferences, which are formal methods of reaching conclusions and separating scientific claims from statistical noise. The statement being tested in a test of statistical significance is called the null hypothesis. The test of significance

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2618-417: A statistical alternative hypothesis and proceed: "Because H a expresses the effect that we wish to find evidence for, we often begin with H a and then set up H 0 as the statement that the hoped-for effect is not present." This advice is reversed for modeling applications where we hope not to find evidence against the null. A complex case example is as follows: The gold standard in clinical research

2737-433: A study of last year's weather reports indicates that rain in a region falls primarily on weekends, it is only valid to test that null hypothesis on weather reports from any other year. Testing hypotheses suggested by the data is circular reasoning that proves nothing; It is a special limitation on the choice of the null hypothesis. A routine procedure is as follows: Start from the scientific hypothesis. Translate this to

2856-933: A subscript 1, 2, 3 in cyclic permutation: 1 → 2, 2 → 3, and 3 → 1 (where the subscripts represent the relevant indices). If ( x 1 , x 2 , x 3 ) are the Cartesian coordinates and ( u 1 , u 2 , u 3 ) are the orthogonal coordinates, then h i = ( ∂ x 1 ∂ u i ) 2 + ( ∂ x 2 ∂ u i ) 2 + ( ∂ x 3 ∂ u i ) 2 {\displaystyle h_{i}={\sqrt {\left({\frac {\partial x_{1}}{\partial u_{i}}}\right)^{2}+\left({\frac {\partial x_{2}}{\partial u_{i}}}\right)^{2}+\left({\frac {\partial x_{3}}{\partial u_{i}}}\right)^{2}}}}

2975-421: A weak conclusion can be made: namely, that the observed data set provides insufficient evidence against the null hypothesis. In this case, because the null hypothesis could be true or false, in some contexts this is interpreted as meaning that the data give insufficient evidence to make any conclusion, while in other contexts, it is interpreted as meaning that there is not sufficient evidence to support changing from

3094-431: Is 5 heads. Let outcomes be considered unlikely with respect to an assumed distribution if their probability is lower than a significance threshold of 0.05. A potential null hypothesis implying a one-tailed test is "this coin is not biased toward heads". Beware that, in this context, the term "one-tailed" does not refer to the outcome of a single coin toss (i.e., whether or not the coin comes up "tails" instead of "heads");

3213-404: Is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space . The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field. A vector field whose curl

3332-406: Is a default hypothesis that a quantity to be measured is zero (null). Typically, the quantity to be measured is the difference between two situations. For instance, trying to determine if there is a positive proof that an effect has occurred or that samples derive from different batches. The null hypothesis is generally assumed to remain possibly true. Multiple analyses can be performed to show how

3451-461: Is also a prototype for the BICEP Array. The Keck array is being succeeded by the BICEP array, which consists of four BICEP3-like telescopes on a common mount, operating at 30/40, 95, 150 and 220/270 GHz. Installation began between the 2017 and 2018 observing seasons. It is scheduled to be fully installed by the 2020 observing season. According to the project website: "BICEP Array will measure

3570-426: Is designed to assess the strength of the evidence against the null hypothesis, or a statement of 'no effect' or 'no difference'. It is often symbolized as H 0 . The statement that is being tested against the null hypothesis is the alternative hypothesis. Symbols may include H 1 and H a . A statistical significance test starts with a random sample from a population. If the sample data are consistent with

3689-688: Is known as Hodge duality . Concretely, on R this is given by: Thus, identifying 0-forms and 3-forms with scalar fields, and 1-forms and 2-forms with vector fields: On the other hand, the fact that d = 0 corresponds to the identities ∇ × ( ∇ f ) = 0 {\displaystyle \nabla \times (\nabla f)=\mathbf {0} } for any scalar field f , and ∇ ⋅ ( ∇ × v ) = 0 {\displaystyle \nabla \cdot (\nabla \times \mathbf {v} )=0} for any vector field v . Grad and div generalize to all oriented pseudo-Riemannian manifolds, with

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3808-416: Is not automated (though the calculations of significance testing usually are). David Cox said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis". A statistical significance test is intended to test a hypothesis. If the hypothesis summarizes a set of data, there is no value in testing the hypothesis on that set of data. Example: If

3927-424: Is oriented via the right-hand rule . The above formula means that the component of the curl of a vector field along a certain axis is the infinitesimal area density of the circulation of the field in a plane perpendicular to that axis. This formula does not a priori define a legitimate vector field, for the individual circulation densities with respect to various axes a priori need not relate to each other in

4046-417: Is significant in every sense and should be reported and perhaps explained. Poor statistical reporting practices have contributed to disagreements over one-tailed tests. Statistical significance resulting from two-tailed tests is insensitive to the sign of the relationship; Reporting significance alone is inadequate. "The treatment has an effect" is the uninformative result of a two-tailed test. "The treatment has

4165-426: Is something for which infinite accuracy is needed as well as exactly zero effect, neither of which normally are realistic. Also measurements will never indicate a non-zero probability of exactly zero difference.) So failure of an exclusion of a null hypothesis amounts to a "don't know" at the specified confidence level; it does not immediately imply null somehow, as the data may already show a (less strong) indication for

4284-405: Is the randomized placebo-controlled double-blind clinical trial. But testing a new drug against a (medically ineffective) placebo may be unethical for a serious illness. Testing a new drug against an older medically effective drug raises fundamental philosophical issues regarding the goal of the test and the motivation of the experimenters. The standard "no difference" null hypothesis may reward

4403-720: Is the Feynman subscript notation, which considers only the variation due to the vector field F (i.e., in this case, v is treated as being constant in space). Another example is the curl of a curl of a vector field. It can be shown that in general coordinates ∇ × ( ∇ × F ) = ∇ ( ∇ ⋅ F ) − ∇ 2 F   , {\displaystyle \nabla \times \left(\mathbf {\nabla \times F} \right)=\mathbf {\nabla } (\mathbf {\nabla \cdot F} )-\nabla ^{2}\mathbf {F} \ ,} and this identity defines

4522-405: Is the appropriate form of an experimental science without numeric predictive theories: A model of numeric results is more informative than a model of effect signs (positive, negative or unknown) which is more informative than a model of simple significance (non-zero or unknown); in the absence of numeric theory signs may suffice. The history of the null and alternative hypotheses has much to do with

4641-486: Is the length of the coordinate vector corresponding to u i . The remaining two components of curl result from cyclic permutation of indices : 3,1,2 → 1,2,3 → 2,3,1. In practice, the two coordinate-free definitions described above are rarely used because in virtually all cases, the curl operator can be applied using some set of curvilinear coordinates , for which simpler representations have been derived. The notation ∇ × F has its origins in

4760-446: Is the space of sections of the exterior algebra Λ ( R ) vector bundle over R , whose dimension is the binomial coefficient ( k ) ; note that Ω ( R ) = 0 for k > 3 or k < 0 . Writing only dimensions, one obtains a row of Pascal's triangle : the 1-dimensional fibers correspond to scalar fields, and the 3-dimensional fibers to vector fields, as described below. Modulo suitable identifications,

4879-406: Is their potential subjectivity. A non-significant result can sometimes be converted to a significant result by the use of a one-tailed hypothesis (as the fair coin test, at the whim of the analyst). The flip side of the argument: One-sided tests are less likely to ignore a real effect. One-tailed tests can suppress the publication of data that differs in sign from predictions. Objectivity was a goal of

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4998-515: Is zero is called irrotational . The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem , which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. The notation curl F is more common in North America. In the rest of the world, particularly in 20th century scientific literature,

5117-801: The California Institute of Technology in 2002. In collaboration with the Jet Propulsion Laboratory , physicists Andrew Lange , Jamie Bock, Brian Keating , and William Holzapfel began the construction of the BICEP1 telescope which deployed to the Amundsen-Scott South Pole Station in 2005 for a three-season observing run. Immediately after deployment of BICEP1, the team, which now included Caltech postdoctoral fellows John Kovac and Chao-Lin Kuo, among others, began work on BICEP2. The telescope remained

5236-502: The Center for Astrophysics | Harvard & Smithsonian . The reported detection was of B-modes at the level of r = 0.20 +0.07 −0.05 , disfavouring the null hypothesis ( r = 0 ) at the level of 7 sigma (5.9 σ after foreground subtraction). However, on 19 June 2014, lowered confidence in confirming the cosmic inflation findings was reported; the accepted and reviewed version of the discovery paper contains an appendix discussing

5355-663: The European Space Agency announced that the signal can be entirely attributed to dust in the Milky Way. BICEP2 has combined their data with the Keck Array and Planck in a joint analysis. A March 2015 publication in Physical Review Letters set a limit on the tensor-to-scalar ratio of r < 0.12 . The BICEP2 affair forms the subject of a book by Brian Keating . Immediately next to

5474-912: The Levi-Civita tensor , ∇ the covariant derivative , g {\displaystyle g} is the determinant of the metric tensor and the Einstein summation convention implies that repeated indices are summed over. Due to the symmetry of the Christoffel symbols participating in the covariant derivative, this expression reduces to the partial derivative: ( ∇ × F ) = 1 g R k ε k ℓ m ∂ ℓ F m {\displaystyle (\nabla \times \mathbf {F} )={\frac {1}{\sqrt {g}}}\mathbf {R} _{k}\varepsilon ^{k\ell m}\partial _{\ell }F_{m}} where R k are

5593-512: The a -coefficients are real functions of three variables; the "wedge products", e.g. dx ∧ dy , can be interpreted as some kind of oriented area elements, dx ∧ dy = − dy ∧ dx , etc.) The exterior derivative of a k -form in R is defined as the ( k + 1) -form from above—and in R if, e.g., ω ( k ) = ∑ 1 ≤ i 1 < i 2 < ⋯ < i k ≤ n

5712-460: The early universe (called primordial gravitational waves ), a result reported by the four co-principal investigators of BICEP2: John M. Kovac of the Center for Astrophysics | Harvard & Smithsonian ; Chao-Lin Kuo of Stanford University ; Jamie Bock of the California Institute of Technology ; and Clem Pryke of the University of Minnesota . An announcement was made on 17 March 2014 from

5831-408: The gradient and divergence , curl as formulated in vector calculus does not generalize simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This deficiency is a direct consequence of the limitations of vector calculus; on the other hand, when expressed as an antisymmetric tensor field via

5950-474: The line integral is calculated along the boundary C of the area A in question, | A | being the magnitude of the area. This equation defines the component of the curl of F along the direction u ^ {\displaystyle \mathbf {\hat {u}} } . The infinitesimal surfaces bounded by C have u ^ {\displaystyle \mathbf {\hat {u}} } as their normal . C

6069-469: The musical isomorphisms , and ★ is the Hodge star operator . This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three-dimensional Riemannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed. Suppose

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6188-1193: The unit vectors for the x -, y -, and z -axes, respectively. This expands as follows: ∇ × F = ( ∂ F z ∂ y − ∂ F y ∂ z ) ı ^ + ( ∂ F x ∂ z − ∂ F z ∂ x ) ȷ ^ + ( ∂ F y ∂ x − ∂ F x ∂ y ) k ^ {\displaystyle \nabla \times \mathbf {F} =\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right){\boldsymbol {\hat {\imath }}}+\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right){\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right){\boldsymbol {\hat {k}}}} Although expressed in terms of coordinates,

6307-443: The vector Laplacian of F , symbolized as ∇ F . The curl of the gradient of any scalar field φ is always the zero vector field ∇ × ( ∇ φ ) = 0 {\displaystyle \nabla \times (\nabla \varphi )={\boldsymbol {0}}} which follows from the antisymmetry in the definition of the curl, and the symmetry of second derivatives . The divergence of

6426-402: The "Robinson gravitational wave background telescope") observed the sky at 100 and 150 GHz (3 mm and 2 mm wavelength) with an angular resolution of 1.0 and 0.7 degrees . It had an array of 98 detectors (50 at 100 GHz and 48 at 150 GHz), which were sensitive to the polarisation of the CMB. A pair of detectors constitutes one polarization-sensitive pixel. The instrument,

6545-409: The (unprovable) null hypothesis. (When it is proven that something is e.g. bigger than x , it does not necessarily imply it is plausible that it is smaller or equal than x ; it may instead be a poor quality measurement with low accuracy. Confirming the null hypothesis two-sided would amount to positively proving it is bigger or equal than 0 and to positively proving it is smaller or equal than 0; this

6664-763: The BICEP telescope at the Martin A. Pomerantz Observatory building at the South Pole was an unused telescope mount previously occupied by the Degree Angular Scale Interferometer . The Keck Array was built to take advantage of this larger telescope mount. This project was funded by $ 2.3 million from W. M. Keck Foundation , as well as funding from the National Science Foundation , the Gordon and Betty Moore Foundation ,

6783-608: The James and Nelly Kilroy Foundation and the Barzan Foundation. The Keck Array project was originally led by Andrew Lange . The Keck Array consists of five polarimeters , each very similar to the BICEP2 design, but using a pulse tube refrigerator rather than a large liquid helium cryogenic storage dewar . The first three started observations in the austral summer of 2010–11; another two started observing in 2012. All of

6902-626: The Keck Array (combined with BICEP2 data) were announced, using observations up to and including the 2015 season. These yielded an upper limit on cosmological B-modes of r < 0.07 {\displaystyle r<0.07} (95% confidence level), which reduces to r < 0.06 {\displaystyle r<0.06} in combination with Planck data. In October 2021, new results were announced giving r < 0.036 {\displaystyle r<0.036} (at 95% confidence level) based on BICEP/Keck 2018 observation season combined with Planck and WMAP data. Once

7021-465: The Keck array was completed in 2012, it was no longer cost-effective to continue to operate BICEP2. However, using the same technique as the Keck array to eliminate the large liquid helium dewar , a much larger telescope has been installed on the original BICEP telescope mount. BICEP3 consists of a single telescope with the same 2560 detectors (observing at 95 GHz) as the five-telescope Keck array, but

7140-439: The alternative notation rot F is traditionally used, which comes from the "rate of rotation" that it represents. To avoid confusion, modern authors tend to use the cross product notation with the del (nabla) operator, as in ∇ × F {\displaystyle \nabla \times \mathbf {F} } , which also reveals the relation between curl (rotor), divergence , and gradient operators. Unlike

7259-467: The basis of data, with certain error rates. It is used in formulating answers in research. Statistical inference can be done without a null hypothesis, by specifying a statistical model corresponding to each candidate hypothesis, and by using model selection techniques to choose the most appropriate model. (The most common selection techniques are based on either Akaike information criterion or Bayes factor ). Hypothesis testing requires constructing

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7378-407: The choice and stated that many null hypotheses should be considered and that many tests are possible for each. The variety of applications and the diversity of goals suggests that the choice can be complicated. In many applications the formulation of the test is traditional. A familiarity with the range of tests available may suggest a particular null hypothesis and test. Formulating the null hypothesis

7497-417: The claim that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data or variables being analyzed. If the null hypothesis is true, any experimentally observed effect is due to chance alone, hence the term "null". In contrast with the null hypothesis, an alternative hypothesis is developed, which claims that

7616-755: The context of differential forms, which involves a number of steps. In short, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifying bivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} , these all being 3-dimensional spaces. In 3 dimensions,

7735-429: The curl is not as obvious from the graph. However, taking the object in the previous example, and placing it anywhere on the line x = 3 , the force exerted on the right side would be slightly greater than the force exerted on the left, causing it to rotate clockwise. Using the right-hand rule, it can be predicted that the resulting curl would be straight in the negative z direction. Inversely, if placed on x = −3 ,

7854-419: The curl of a vector field at a point is implicitly through its components along various axes passing through the point: if u ^ {\displaystyle \mathbf {\hat {u}} } is any unit vector, the component of the curl of F along the direction u ^ {\displaystyle \mathbf {\hat {u}} } may be defined to be the limiting value of

7973-683: The curl of any vector field is equal to zero: ∇ ⋅ ( ∇ × F ) = 0. {\displaystyle \nabla \cdot (\nabla \times \mathbf {F} )=0.} If φ is a scalar valued function and F is a vector field, then ∇ × ( φ F ) = ∇ φ × F + φ ∇ × F {\displaystyle \nabla \times (\varphi \mathbf {F} )=\nabla \varphi \times \mathbf {F} +\varphi \nabla \times \mathbf {F} } The vector calculus operations of grad , curl, and div are most easily generalized in

8092-809: The curl vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume enclosed, as the shell is contracted indefinitely around p . More specifically, the curl may be defined by the vector formula ( ∇ × F ) ( p ) = d e f lim V → 0 1 | V | ∮ S n ^ × F   d S {\displaystyle (\nabla \times \mathbf {F} )(p){\overset {\underset {\mathrm {def} }{}}{{}={}}}\lim _{V\to 0}{\frac {1}{|V|}}\oint _{S}\mathbf {\hat {n}} \times \mathbf {F} \ \mathrm {d} S} where

8211-576: The curl would at all points be pointing in the negative z direction. The results of this equation align with what could have been predicted using the right-hand rule using a right-handed coordinate system . Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed. For the vector field F ( x , y , z ) = − x 2 ȷ ^ {\displaystyle \mathbf {F} (x,y,z)=-x^{2}{\boldsymbol {\hat {\jmath }}}}

8330-409: The data-set of a randomly selected representative sample is very unlikely relative to the null hypothesis (defined as being part of a class of sets of data that only rarely will be observed), the experimenter rejects the null hypothesis, concluding it (probably) is false. This class of data-sets is usually specified via a test statistic , which is designed to measure the extent of apparent departure from

8449-501: The developers of statistical tests. It is a common practice to use a one-tailed hypothesis by default. However, "If you do not have a specific direction firmly in mind in advance, use a two-sided alternative. Moreover, some users of statistics argue that we should always work with the two-sided alternative." One alternative to this advice is to use three-outcome tests. It eliminates the issues surrounding directionality of hypotheses by testing twice, once in each direction and combining

8568-529: The examples below. The vector field F ( x , y , z ) = y ı ^ − x ȷ ^ {\displaystyle \mathbf {F} (x,y,z)=y{\boldsymbol {\hat {\imath }}}-x{\boldsymbol {\hat {\jmath }}}} can be decomposed as F x = y , F y = − x , F z = 0. {\displaystyle F_{x}=y,F_{y}=-x,F_{z}=0.} Upon visual inspection,

8687-400: The field at the center of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point. The curl of the vector field at any point is given by the rotation of an infinitesimal area in the xy -plane (for z -axis component of the curl), zx -plane (for y -axis component of the curl) and yz -plane (for x -axis component of the curl vector). This can be seen in

8806-1003: The field can be described as "rotating". If the vectors of the field were to represent a linear force acting on objects present at that point, and an object were to be placed inside the field, the object would start to rotate clockwise around itself. This is true regardless of where the object is placed. Calculating the curl: ∇ × F = 0 ı ^ + 0 ȷ ^ + ( ∂ ∂ x ( − x ) − ∂ ∂ y y ) k ^ = − 2 k ^ {\displaystyle \nabla \times \mathbf {F} =0{\boldsymbol {\hat {\imath }}}+0{\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial }{\partial x}}(-x)-{\frac {\partial }{\partial y}}y\right){\boldsymbol {\hat {k}}}=-2{\boldsymbol {\hat {k}}}} The resulting vector field describing

8925-410: The hypothesis should either be rejected or excluded e.g. having a high confidence level, thus demonstrating a statistically significant difference. This is demonstrated by showing that zero is outside of the specified confidence interval of the measurement on either side, typically within the real numbers . Failure to exclude the null hypothesis (with any confidence) does not logically confirm or support

9044-423: The legal principle of presumption of innocence , in which a suspect or defendant is assumed to be innocent (null is not rejected) until proven guilty (null is rejected) beyond a reasonable doubt (to a statistically significant degree). In the hypothesis testing approach of Jerzy Neyman and Egon Pearson , a null hypothesis is contrasted with an alternative hypothesis , and the two hypotheses are distinguished on

9163-402: The likelihood of finding the obtained results is thereby determined. Hypothesis testing works by collecting data and measuring how likely the particular set of data is (assuming the null hypothesis is true), when the study is on a randomly selected representative sample. The null hypothesis assumes no relationship between variables in the population from which the sample is selected. If

9282-447: The local basis vectors. Equivalently, using the exterior derivative , the curl can be expressed as: ∇ × F = ( ⋆ ( d F ♭ ) ) ♯ {\displaystyle \nabla \times \mathbf {F} =\left(\star {\big (}{\mathrm {d} }\mathbf {F} ^{\flat }{\big )}\right)^{\sharp }} Here ♭ and ♯ are

9401-431: The mean male score is the same as the mean female score: where A stronger null hypothesis is that the two samples have equal variances and shapes of their respective distributions. The simple/composite distinction was made by Neyman and Pearson. Fisher required an exact null hypothesis for testing (see the quotations below). A one-tailed hypothesis (tested using a one-sided test) is an inexact hypothesis in which

9520-1028: The negative z direction when x is positive and vice versa. In this field, the intensity of rotation would be greater as the object moves away from the plane x = 0 . In general curvilinear coordinates (not only in Cartesian coordinates), the curl of a cross product of vector fields v and F can be shown to be ∇ × ( v × F ) = ( ( ∇ ⋅ F ) + F ⋅ ∇ ) v − ( ( ∇ ⋅ v ) + v ⋅ ∇ ) F   . {\displaystyle \nabla \times \left(\mathbf {v\times F} \right)={\Big (}\left(\mathbf {\nabla \cdot F} \right)+\mathbf {F\cdot \nabla } {\Big )}\mathbf {v} -{\Big (}\left(\mathbf {\nabla \cdot v} \right)+\mathbf {v\cdot \nabla } {\Big )}\mathbf {F} \ .} Interchanging

9639-419: The new treatment's mean > 10 . If the sample evidence obtained through x -bar equals −200 and the corresponding t-test statistic equals −50, the conclusion from the test would be that there is no evidence that the new treatment is better than the existing one: it would not report that it is markedly worse, but that is not what this particular test is looking for. To overcome any possible ambiguity in reporting

9758-404: The null hypothesis provides evidence that there are (or are not) statistically sufficient grounds to believe there is a relationship between two phenomena (e.g., that a potential treatment has a non-zero effect, either way). Testing the null hypothesis is a central task in statistical hypothesis testing in the modern practice of science. There are precise criteria for excluding or not excluding

9877-434: The null hypothesis is not necessarily the real goal of a significance tester. An adequate statistical model may be associated with a failure to reject the null; the model is adjusted until the null is not rejected. The numerous uses of significance testing were well known to Fisher who discussed many in his book written a decade before defining the null hypothesis. A statistical significance test shares much mathematics with

9996-480: The null hypothesis to hold, and the test refutes it. Since the coin is ostensibly neither fair nor biased toward tails, the conclusion of the experiment is that the coin is biased towards heads. Alternatively, a null hypothesis implying a two-tailed test is "this coin is fair". This one null hypothesis could be examined by looking out for either too many tails or too many heads in the experiments. The outcomes that would tend to refute this null hypothesis are those with

10115-402: The null hypothesis were true. In this case, the null hypothesis is rejected and an alternative hypothesis is accepted in its place. If the data are consistent with the null hypothesis statistically possibly true, then the null hypothesis is not rejected. In neither case is the null hypothesis or its alternative proven; with better or more data, the null may still be rejected. This is analogous to

10234-406: The null hypothesis, then you do not reject the null hypothesis; if the sample data are inconsistent with the null hypothesis, then you reject the null hypothesis and conclude that the alternative hypothesis is true. Consider the following example. Given the test scores of two random samples , one of men and one of women, does one group score better than the other? A possible null hypothesis is that

10353-409: The null hypothesis. The procedure works by assessing whether the observed departure, measured by the test statistic, is larger than a value defined, so that the probability of occurrence of a more extreme value is small under the null hypothesis (usually in less than either 5% or 1% of similar data-sets in which the null hypothesis does hold). If the data do not contradict the null hypothesis, then only

10472-713: The object would rotate counterclockwise and the right-hand rule would result in a positive z direction. Calculating the curl: ∇ × F = 0 ı ^ + 0 ȷ ^ + ∂ ∂ x ( − x 2 ) k ^ = − 2 x k ^ . {\displaystyle {\nabla }\times \mathbf {F} =0{\boldsymbol {\hat {\imath }}}+0{\boldsymbol {\hat {\jmath }}}+{\frac {\partial }{\partial x}}\left(-x^{2}\right){\boldsymbol {\hat {k}}}=-2x{\boldsymbol {\hat {k}}}.} The curl points in

10591-407: The pharmaceutical company for gathering inadequate data. "Difference" is a better null hypothesis in this case, but statistical significance is not an adequate criterion for reaching a nuanced conclusion which requires a good numeric estimate of the drug's effectiveness. A "minor" or "simple" proposed change in the null hypothesis ((new vs old) rather than (new vs placebo)) can have a dramatic effect on

10710-2431: The point. To this definition fits naturally another global formula (similar to the Kelvin-Stokes theorem) which equates the volume integral of the curl of a vector field to the above surface integral taken over the boundary of the volume. Whereas the above two definitions of the curl are coordinate free, there is another "easy to memorize" definition of the curl in curvilinear orthogonal coordinates , e.g. in Cartesian coordinates , spherical , cylindrical , or even elliptical or parabolic coordinates : ( curl ⁡ F ) 1 = 1 h 2 h 3 ( ∂ ( h 3 F 3 ) ∂ u 2 − ∂ ( h 2 F 2 ) ∂ u 3 ) , ( curl ⁡ F ) 2 = 1 h 3 h 1 ( ∂ ( h 1 F 1 ) ∂ u 3 − ∂ ( h 3 F 3 ) ∂ u 1 ) , ( curl ⁡ F ) 3 = 1 h 1 h 2 ( ∂ ( h 2 F 2 ) ∂ u 1 − ∂ ( h 1 F 1 ) ∂ u 2 ) . {\displaystyle {\begin{aligned}&(\operatorname {curl} \mathbf {F} )_{1}={\frac {1}{h_{2}h_{3}}}\left({\frac {\partial (h_{3}F_{3})}{\partial u_{2}}}-{\frac {\partial (h_{2}F_{2})}{\partial u_{3}}}\right),\\[5pt]&(\operatorname {curl} \mathbf {F} )_{2}={\frac {1}{h_{3}h_{1}}}\left({\frac {\partial (h_{1}F_{1})}{\partial u_{3}}}-{\frac {\partial (h_{3}F_{3})}{\partial u_{1}}}\right),\\[5pt]&(\operatorname {curl} \mathbf {F} )_{3}={\frac {1}{h_{1}h_{2}}}\left({\frac {\partial (h_{2}F_{2})}{\partial u_{1}}}-{\frac {\partial (h_{1}F_{1})}{\partial u_{2}}}\right).\end{aligned}}} The equation for each component (curl F ) k can be obtained by exchanging each occurrence of

10829-402: The polarized sky in five frequency bands to reach an ultimate sensitivity to the amplitude of IGW [inflationary gravitational waves] of σ(r) < 0.005" and "This measurement will be a definitive test of slow-roll models of inflation, which generally predict a gravitational-wave signal above approximately 0.01." Curl (mathematics) In vector calculus , the curl , also known as rotor ,

10948-630: The possible production of the signal by cosmic dust . In part because the large value of the tensor to scalar ratio, which contradicts limits from the Planck data, this is considered the most likely explanation for the detected signal by many scientists. For example, on June 5, 2014 at a conference of the American Astronomical Society , astronomer David Spergel argued that the B-mode polarization detected by BICEP2 could instead be

11067-413: The precise formulation of the null and alternative hypotheses. Fisher said, "the null hypothesis must be exact, that is free of vagueness and ambiguity, because it must supply the basis of the 'problem of distribution,' of which the test of significance is the solution", implying a more restrictive domain for H 0 . According to this view, the null hypothesis must be numerically exact—it must state that

11186-417: The receivers observed at 150 GHz until 2013, when two of them were converted to observe at 100 GHz. Each polarimeter consists of a refracting telescope (to minimise systematics) cooled by a pulse tube cooler to 4 K, and a focal-plane array of 512 transition edge sensors cooled to 250 mK, giving a total of 2560 detectors, or 1280 dual-polarization pixels. In October 2018, the first results from

11305-501: The result is invariant under proper rotations of the coordinate axes but the result inverts under reflection. In a general coordinate system, the curl is given by ( ∇ × F ) k = 1 g ε k ℓ m ∇ ℓ F m {\displaystyle (\nabla \times \mathbf {F} )^{k}={\frac {1}{\sqrt {g}}}\varepsilon ^{k\ell m}\nabla _{\ell }F_{m}} where ε denotes

11424-456: The result of light emitted from dust between the stars in our Milky Way galaxy. A preprint released by the Planck team in September 2014, eventually accepted in 2016, provided the most accurate measurement yet of dust, concluding that the signal from dust is the same strength as that reported from BICEP2. On January 30, 2015, a joint analysis of BICEP2 and Planck data was published and

11543-470: The result of the test of a null hypothesis, it is best to indicate whether the test was two-sided and, if one-sided, to include the direction of the effect being tested. The statistical theory required to deal with the simple cases of directionality dealt with here, and more complicated ones, makes use of the concept of an unbiased test . The directionality of hypotheses is not always obvious. The explicit null hypothesis of Fisher's Lady tasting tea example

11662-487: The results to produce three possible outcomes. Variations on this approach have a history, being suggested perhaps 10 times since 1950. Disagreements over one-tailed tests flow from the philosophy of science. While Fisher was willing to ignore the unlikely case of the Lady guessing all cups of tea incorrectly (which may have been appropriate for the circumstances), medicine believes that a proposed treatment that kills patients

11781-603: The risk of having a heart attack". The test of the hypothesis consists of administering the drug to half of the people in a study group as a controlled experiment . If the data show a statistically significant change in the people receiving the drug, the null hypothesis is rejected. There are many types of significance tests for one, two or more samples, for means, variances and proportions, paired or unpaired data, for different distributions, for large and small samples; all have null hypotheses. There are also at least four goals of null hypotheses for significance tests: Rejection of

11900-399: The same geometric interpretation, because the spaces of 0-forms and n -forms at each point are always 1-dimensional and can be identified with scalar fields, while the spaces of 1-forms and ( n − 1) -forms are always fiberwise n -dimensional and can be identified with vector fields. Null hypothesis In scientific research , the null hypothesis (often denoted H 0 ) is

12019-403: The same significance threshold used for the one-tailed test (0.05), the same outcome is not statistically significant. Therefore, the two-tailed null hypothesis will be preserved in this case, not supporting the conclusion reached with the single-tailed null hypothesis, that the coin is biased towards heads. This example illustrates that the conclusion reached from a statistical test may depend on

12138-474: The same way as the components of a vector do; that they do indeed relate to each other in this precise manner must be proven separately. To this definition fits naturally the Kelvin–Stokes theorem , as a global formula corresponding to the definition. It equates the surface integral of the curl of a vector field to the above line integral taken around the boundary of the surface. Another way one can define

12257-523: The same, but new detectors were inserted into BICEP2 using a completely different technology: a printed circuit board on the focal plane that could filter, process, image, and measure radiation from the cosmic microwave background. BICEP2 was deployed to the South Pole in 2009 to begin its three-season observing run which yielded the detection of B-mode polarization in the cosmic microwave background. The first BICEP instrument (known during development as

12376-491: The similarities to the 3-dimensional cross product , and it is useful as a mnemonic in Cartesian coordinates if ∇ is taken as a vector differential operator del . Such notation involving operators is common in physics and algebra . Expanded in 3-dimensional Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), ∇ × F is, for F composed of [ F x , F y , F z ] (where

12495-859: The subscripts indicate the components of the vector, not partial derivatives): ∇ × F = | ı ^ ȷ ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z | {\displaystyle \nabla \times \mathbf {F} ={\begin{vmatrix}{\boldsymbol {\hat {\imath }}}&{\boldsymbol {\hat {\jmath }}}&{\boldsymbol {\hat {k}}}\\[5mu]{\dfrac {\partial }{\partial x}}&{\dfrac {\partial }{\partial y}}&{\dfrac {\partial }{\partial z}}\\[5mu]F_{x}&F_{y}&F_{z}\end{vmatrix}}} where i , j , and k are

12614-452: The surface S , and points along the surface at right angles to the tangential projection of F . Integrating this cross product over the whole surface results in a vector whose magnitude measures the overall circulation of F around S , and whose direction is at right angles to this circulation. The above formula says that the curl of a vector field at a point is the infinitesimal volume density of this "circulation vector" around

12733-423: The surface integral is calculated along the boundary S of the volume V , | V | being the magnitude of the volume, and n ^ {\displaystyle \mathbf {\hat {n}} } pointing outward from the surface S perpendicularly at every point in S . In this formula, the cross product in the integrand measures the tangential component of F at each point on

12852-421: The term " one-tailed " refers to a specific way of testing the null hypothesis in which the critical region (also known as " region of rejection ") ends up in on only one side of the probability distribution. Indeed, with a fair coin the probability of this experiment outcome is 1/2 = 0.031, which would be even lower if the coin were biased in favour of tails. Therefore, the observations are not likely enough for

12971-419: The three nontrivial occurrences of the exterior derivative correspond to grad, curl, and div. Differential forms and the differential can be defined on any Euclidean space, or indeed any manifold, without any notion of a Riemannian metric. On a Riemannian manifold , or more generally pseudo-Riemannian manifold , k -forms can be identified with k -vector fields ( k -forms are k -covector fields, and

13090-1100: The twofold application of the exterior derivative yields 0 {\displaystyle 0} (the zero k + 2 {\displaystyle k+2} -form). Thus, denoting the space of k -forms by Ω ( R ) and the exterior derivative by d one gets a sequence: 0 ⟶ d Ω 0 ( R 3 ) ⟶ d Ω 1 ( R 3 ) ⟶ d Ω 2 ( R 3 ) ⟶ d Ω 3 ( R 3 ) ⟶ d 0. {\displaystyle 0\,{\overset {d}{\longrightarrow }}\;\Omega ^{0}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{1}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{2}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{3}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\,0.} Here Ω ( R )

13209-421: The use of one-tailed tests are complicated by the variety of tests. Some tests (for instance the χ goodness of fit test) are inherently one-tailed. Some probability distributions are asymmetric. The traditional tests of 3 or more groups are two-tailed. Advice concerning the use of one-tailed hypotheses has been inconsistent and accepted practice varies among fields. The greatest objection to one-tailed hypotheses

13328-399: The utility of a test for complex non-statistical reasons. The choice of null hypothesis ( H 0 ) and consideration of directionality (see " one-tailed test ") is critical. Consider the question of whether a tossed coin is fair (i.e. that on average it lands heads up 50% of the time) and an experiment where you toss the coin 5 times. A possible result of the experiment that we consider here

13447-422: The value of a parameter is specified as being either: A one-tailed hypothesis is said to have directionality . Fisher's original ( lady tasting tea ) example was a one-tailed test. The null hypothesis was asymmetric. The probability of guessing all cups correctly was the same as guessing all cups incorrectly, but Fisher noted that only guessing correctly was compatible with the lady's claim. The null hypothesis

13566-532: The various instruments are Caltech , Cardiff University , University of Chicago , Center for Astrophysics | Harvard & Smithsonian , Jet Propulsion Laboratory , CEA Grenoble (FR) , University of Minnesota and Stanford University (all experiments); UC San Diego (BICEP1 and 2); National Institute of Standards and Technology (NIST), University of British Columbia and University of Toronto (BICEP2, Keck Array and BICEP3); and Case Western Reserve University (Keck Array). The series of experiments began at

13685-583: The vector field v and ∇ operator, we arrive at the cross product of a vector field with curl of a vector field: v   × ( ∇ × F ) = ∇ F ( v ⋅ F ) − ( v ⋅ ∇ ) F   , {\displaystyle \mathbf {v\ \times } \left(\mathbf {\nabla \times F} \right)=\nabla _{\mathbf {F} }\left(\mathbf {v\cdot F} \right)-\left(\mathbf {v\cdot \nabla } \right)\mathbf {F} \ ,} where ∇ F

13804-399: The vector field describes the velocity field of a fluid flow (such as a large tank of liquid or gas ) and a small ball is located within the fluid or gas (the center of the ball being fixed at a certain point). If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of

13923-402: The wedge operator of geometric calculus , the curl generalizes to all dimensions. The circumstance is similar to that attending the 3-dimensional cross product , and indeed the connection is reflected in the notation ∇ × {\displaystyle \nabla \times } for the curl. The name "curl" was first suggested by James Clerk Maxwell in 1871 but the concept

14042-564: Was apparently first used in the construction of an optical field theory by James MacCullagh in 1839. The curl of a vector field F , denoted by curl F , or ∇ × F {\displaystyle \nabla \times \mathbf {F} } , or rot F , is an operator that maps C functions in R to C functions in R , and in particular, it maps continuously differentiable functions R → R to continuous functions R → R . It can be defined in several ways, to be mentioned below: One way to define

14161-426: Was that the Lady had no such ability, which led to a symmetric probability distribution. The one-tailed nature of the test resulted from the one-tailed alternate hypothesis (a term not used by Fisher). The null hypothesis became implicitly one-tailed. The logical negation of the Lady's one-tailed claim was also one-tailed. (Claim: Ability > 0; Stated null: Ability = 0; Implicit null: Ability ≤ 0). Pure arguments over

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