id
int32 0
100k
| text
stringlengths 21
3.54k
| source
stringlengths 1
124
| similarity
float32 0.78
0.88
|
|---|---|---|---|
0
|
{\displaystyle x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}.} The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ax/c = y). Muhammad ibn Musa al-Khwarizmi (9th century), possibly inspired by Brahmagupta, developed a set of formulas that worked for positive solutions.
|
Quadratic equation
| 0.881942
|
1
|
In the days before calculators, people would use mathematical tables—lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. Methods of numerical approximation existed, called prosthaphaeresis, that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots.
|
Quadratic equation
| 0.881942
|
2
|
Statistics show that waste collection is one of the most dangerous jobs, at times more dangerous than police work, but consistently less dangerous than commercial fishing and ranch and farm work. On-the-job hazards include broken glass, medical waste such as syringes, caustic chemicals, objects falling out of overloaded containers, diseases that may accompany solid waste, asbestos, dog attacks and pests, inhaling dust, smoke and chemical fumes, severe weather, traffic accidents, and unpleasant smells that can make someone physically sick.Risks also exist from working in close proximity to traffic hazards, and in using the heavy machinery (such as container lifters and compactors) found on collection vehicles.
|
Garbage collector
| 0.881522
|
3
|
Stories about Biology Stories about Biologists Games and Simulations Body Depot Biology Bits PLOSable Image Gallery Puzzles – Word Search & Crossword Coloring Pages Mysterious World of Dr. Biology comic book adventure activity Audio Podcasts Co-host Contest Ugly Bug Contest Bird Finder Tool Virtual Pocket Seed Experiment Biology Virtual Reality Tours
|
Ask a Biologist
| 0.875125
|
4
|
Students reconstructed a chain of events in the Dr. Biology laboratory and field site, writing their own narrative for the story. Early in 2007, Ask A Biologist became one of the early content channels on iTunes U with its audio podcast of the same name. Hosted by Dr. Biology, the program was soon listed as one of five great courses by Macworld.
|
Ask a Biologist
| 0.875124
|
5
|
The Virtual Bird Aviary, included the majority of bird species found in the Southwest United States including more than 400 vocal recordings and companion sonograms, bird images, text descriptions, and range maps. In 2005, the website was peer reviewed by the Multimedia Educational Resource for Learning and Online Teaching (MERLOT), earning a "five out of five star" rating.In 2006, the website introduced the Mysterious World of Dr. Biology a comic book adventure. The activity encouraged students to piece together a mystery.
|
Ask a Biologist
| 0.875124
|
6
|
Ask A Biologist is a pre-kindergarten through high school program dedicated to answering questions from students, their teachers, and parents. The primary focus of the program is to connect students and teachers with working scientists through a question and answer Web e-mail form. The companion website also includes a large collection of free content and activities that can be used inside, as well as outside, of the classroom. The award-winning program has been continuously running for more than 14 years, with the assistance of more than 150 volunteer scientists, faculty, and graduate students in biology and related fields. In 2010 the program released its new website interface and features that became the subject for articles in the journals Science and PLoS Biology.
|
Ask a Biologist
| 0.875124
|
7
|
In physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis ("as the extension, so the force" or "the extension is proportional to the force").
|
Spring Constant
| 0.873215
|
8
|
Since DNA is an informative macromolecule in terms of transmission from one generation to another, DNA sequencing is used in evolutionary biology to study how different organisms are related and how they evolved. In February 2021, scientists reported, for the first time, the sequencing of DNA from animal remains, a mammoth in this instance, over a million years old, the oldest DNA sequenced to date.
|
DNA Sequencing
| 0.87157
|
9
|
Systems of linear inequalities can be simplified by Fourier–Motzkin elimination.The cylindrical algebraic decomposition is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing them. The complexity of this algorithm is doubly exponential in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases.
|
System of inequalities
| 0.870745
|
10
|
E&M is equivalent to an introductory college course in electricity and magnetism for physics or engineering majors. The course modules are: Electrostatics Conductors, capacitors, and dielectrics Electric circuits Magnetic fields Electromagnetism.Methods of calculus are used wherever appropriate in formulating physical principles and in applying them to physical problems. Therefore, students should have completed or be concurrently enrolled in a calculus class.
|
AP Physics C: Electricity and Magnetism
| 0.868815
|
11
|
The AP examination for AP Physics C: Electricity and Magnetism is separate from the AP examination for AP Physics C: Mechanics. Before 2006, test-takers paid only once and were given the choice of taking either one or two parts of the Physics C test.
|
AP Physics C: Electricity and Magnetism
| 0.868815
|
12
|
The grade distributions for the Physics C: Electricity and Magnetism scores since 2010 were:
|
AP Physics C: Electricity and Magnetism
| 0.868815
|
13
|
The exam is typically administered on a Monday afternoon in May. The exam is configured in two categories: a 35-question multiple choice section and a 3-question free response section. Test takers are allowed to use an approved calculator during the entire exam. The test is weighted such that each section is worth half of the final score. This and AP Physics C: Mechanics are the shortest AP exams, with total testing time of 90 minutes (45 minutes for each section).The topics covered by the exam are as follows:
|
AP Physics C: Electricity and Magnetism
| 0.868815
|
14
|
Advanced Placement (AP) Physics C: Electricity and Magnetism (also known as AP Physics C: E&M or AP E&M) is an introductory physics course administered by the College Board as part of its Advanced Placement program. It is intended to proxy a second-semester calculus-based university course in electricity and magnetism. The content of Physics C: E&M overlaps with that of AP Physics 2, but Physics 2 is algebra-based and covers other topics outside of electromagnetism, while Physics C is calculus-based and only covers electromagnetism. Physics C: E&M may be combined with its mechanics counterpart to form a year-long course that prepares for both exams.
|
AP Physics C: Electricity and Magnetism
| 0.868815
|
15
|
Given three fields arranged in a tower, say K a subfield of L which is in turn a subfield of M, there is a simple relation between the degrees of the three extensions L/K, M/L and M/K: = ⋅ . {\displaystyle =\cdot .} In other words, the degree going from the "bottom" to the "top" field is just the product of the degrees going from the "bottom" to the "middle" and then from the "middle" to the "top". It is quite analogous to Lagrange's theorem in group theory, which relates the order of a group to the order and index of a subgroup — indeed Galois theory shows that this analogy is more than just a coincidence.
|
Degree of an extension
| 0.867439
|
16
|
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently.
|
Degree of an extension
| 0.867439
|
17
|
In the theory of quadratic forms, the parabola is the graph of the quadratic form x2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x2 + y2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x2 − y2. Generalizations to more variables yield further such objects. The curves y = xp for other values of p are traditionally referred to as the higher parabolas and were originally treated implicitly, in the form xp = kyq for p and q both positive integers, in which form they are seen to be algebraic curves. These correspond to the explicit formula y = xp/q for a positive fractional power of x. Negative fractional powers correspond to the implicit equation xp yq = k and are traditionally referred to as higher hyperbolas. Analytically, x can also be raised to an irrational power (for positive values of x); the analytic properties are analogous to when x is raised to rational powers, but the resulting curve is no longer algebraic and cannot be analyzed by algebraic geometry.
|
Parabola
| 0.865891
|
18
|
If one replaces the real numbers by an arbitrary field, many geometric properties of the parabola y = x 2 {\displaystyle y=x^{2}} are still valid: A line intersects in at most two points. At any point ( x 0 , x 0 2 ) {\displaystyle (x_{0},x_{0}^{2})} the line y = 2 x 0 x − x 0 2 {\displaystyle y=2x_{0}x-x_{0}^{2}} is the tangent.Essentially new phenomena arise, if the field has characteristic 2 (that is, 1 + 1 = 0 {\displaystyle 1+1=0} ): the tangents are all parallel. In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates (x, x2, x3, ..., xn); the standard parabola is the case n = 2, and the case n = 3 is known as the twisted cubic. A further generalization is given by the Veronese variety, when there is more than one input variable.
|
Parabola
| 0.865891
|
19
|
Let three tangents to a parabola form a triangle. Then Lambert's theorem states that the focus of the parabola lies on the circumcircle of the triangle. : Corollary 20 Tsukerman's converse to Lambert's theorem states that, given three lines that bound a triangle, if two of the lines are tangent to a parabola whose focus lies on the circumcircle of the triangle, then the third line is also tangent to the parabola.
|
Parabola
| 0.865891
|
20
|
Application: The 2-points–2-tangents property can be used for the construction of the tangent of a parabola at point P 2 {\displaystyle P_{2}} , if P 1 , P 2 {\displaystyle P_{1},P_{2}} and the tangent at P 1 {\displaystyle P_{1}} are given. Remark 1: The 2-points–2-tangents property of a parabola is an affine version of the 3-point degeneration of Pascal's theorem. Remark 2: The 2-points–2-tangents property should not be confused with the following property of a parabola, which also deals with 2 points and 2 tangents, but is not related to Pascal's theorem.
|
Parabola
| 0.865891
|
21
|
A short calculation shows: line Q 1 Q 2 {\displaystyle Q_{1}Q_{2}} has slope 2 x 0 {\displaystyle 2x_{0}} which is the slope of the tangent at point P 0 {\displaystyle P_{0}} . Application: The 3-points-1-tangent-property of a parabola can be used for the construction of the tangent at point P 0 {\displaystyle P_{0}} , while P 1 , P 2 , P 0 {\displaystyle P_{1},P_{2},P_{0}} are given. Remark: The 3-points-1-tangent-property of a parabola is an affine version of the 4-point-degeneration of Pascal's theorem.
|
Parabola
| 0.865891
|
22
|
At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola. Another hypothetical situation in which parabolas might arise, according to the theories of physics described in the 17th and 18th centuries by Sir Isaac Newton, is in two-body orbits, for example, the path of a small planetoid or other object under the influence of the gravitation of the Sun. Parabolic orbits do not occur in nature; simple orbits most commonly resemble hyperbolas or ellipses.
|
Parabola
| 0.865891
|
23
|
In nature, approximations of parabolas and paraboloids are found in many diverse situations. The best-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a ball flying through the air, neglecting air friction). The parabolic trajectory of projectiles was discovered experimentally in the early 17th century by Galileo, who performed experiments with balls rolling on inclined planes. He also later proved this mathematically in his book Dialogue Concerning Two New Sciences.
|
Parabola
| 0.865891
|
24
|
The same effects occur with sound and other waves. This reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. It is frequently used in physics, engineering, and many other areas.
|
Parabola
| 0.865891
|
25
|
Any parabola can be described in a suitable coordinate system by an equation y = a x 2 {\displaystyle y=ax^{2}} . Proof: straightforward calculation for the unit parabola y = x 2 {\displaystyle y=x^{2}} . Application: The 4-points property of a parabola can be used for the construction of point P 4 {\displaystyle P_{4}} , while P 1 , P 2 , P 3 {\displaystyle P_{1},P_{2},P_{3}} and Q 2 {\displaystyle Q_{2}} are given. Remark: the 4-points property of a parabola is an affine version of the 5-point degeneration of Pascal's theorem.
|
Parabola
| 0.865891
|
26
|
This can be done with calculus, or by using a line that is parallel to the axis of symmetry of the parabola and passes through the midpoint of the chord. The required point is where this line intersects the parabola. Then, using the formula given in Distance from a point to a line, calculate the perpendicular distance from this point to the chord. Multiply this by the length of the chord to get the area of the parallelogram, then by 2/3 to get the required enclosed area.
|
Parabola
| 0.865891
|
27
|
A theorem equivalent to this one, but different in details, was derived by Archimedes in the 3rd century BCE. He used the areas of triangles, rather than that of the parallelogram. See The Quadrature of the Parabola.
|
Parabola
| 0.865891
|
28
|
The above proofs of the reflective and tangent bisection properties use a line of calculus. Here a geometric proof is presented. In this diagram, F is the focus of the parabola, and T and U lie on its directrix. P is an arbitrary point on the parabola.
|
Parabola
| 0.865891
|
29
|
Tensor algebra, symmetric algebra and exterior algebra of a finite-dimensional vector space are graded quadratic (in fact, Koszul) algebras. Universal enveloping algebra of a finite-dimensional Lie algebra is a filtered quadratic algebra.
|
Quadratic algebra
| 0.864868
|
30
|
In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most important class of graded quadratic algebras is Koszul algebras.
|
Quadratic algebra
| 0.864868
|
31
|
A graded quadratic algebra A is determined by a vector space of generators V = A1 and a subspace of homogeneous quadratic relations S ⊂ V ⊗ V (Polishchuk & Positselski 2005, p. 6). Thus A = T ( V ) / ⟨ S ⟩ {\displaystyle A=T(V)/\langle S\rangle } and inherits its grading from the tensor algebra T(V). If the subspace of relations is instead allowed to also contain inhomogeneous degree 2 elements, i.e. S ⊂ k ⊕ V ⊕ (V ⊗ V), this construction results in a filtered quadratic algebra. A graded quadratic algebra A as above admits a quadratic dual: the quadratic algebra generated by V* and with quadratic relations forming the orthogonal complement of S in V* ⊗ V*.
|
Quadratic algebra
| 0.864868
|
32
|
Note that there are 3 translation "windows", or reading frames, depending on where you start reading the code. Finally, use the table at Genetic code to translate the above into a structural formula as used in chemistry. This will give you the primary structure of the protein.
|
Protein translation
| 0.864804
|
33
|
It developed over time to chromosomal analysis, then genetic marker analysis, and eventual genomic analysis. Identifying traits and their underlying genetics allowed for transferring useful genes and the traits they controlled from either wild or mutant plants to crop plants. Understanding and manipulating of plant genetics was in its heyday during the Green Revolution brought about by Norman Borlaug. During this time, the molecule of heredity, DNA, was also discovered, which allowed scientists to actually examine and manipulate genetic information directly.
|
Plant genetics
| 0.863828
|
34
|
Arabidopsis thaliana, also known as thale cress, has been the model organism for the study of plant genetics. As Drosophila, a species of fruit fly, was to the understanding of early genetics, so has been A. thaliana to the understanding of plant genetics. It was the first plant to ever have its genome sequenced in the year 2000. It has a small genome, making the initial sequencing more attainable.
|
Plant genetics
| 0.863828
|
35
|
Speciation can be easier in many plants due to unique genetic abilities, such as being well adapted to polyploidy. Plants are unique in that they are able to produce energy-dense carbohydrates via photosynthesis, a process which is achieved by use of chloroplasts. Chloroplasts, like the superficially similar mitochondria, possess their own DNA. Chloroplasts thus provide an additional reservoir for genes and genetic diversity, and an extra layer of genetic complexity not found in animals. The study of plant genetics has major economic impacts: many staple crops are genetically modified to increase yields, confer pest and disease resistance, provide resistance to herbicides, or to increase their nutritional value.
|
Plant genetics
| 0.863828
|
36
|
Much of Mendel's work with plants still forms the basis for modern plant genetics. Plants, like all known organisms, use DNA to pass on their traits. Animal genetics often focuses on parentage and lineage, but this can sometimes be difficult in plant genetics due to the fact that plants can, unlike most animals, be self-fertile.
|
Plant genetics
| 0.863828
|
37
|
Plant genetics is the study of genes, genetic variation, and heredity specifically in plants. It is generally considered a field of biology and botany, but intersects frequently with many other life sciences and is strongly linked with the study of information systems. Plant genetics is similar in many ways to animal genetics but differs in a few key areas. The discoverer of genetics was Gregor Mendel, a late 19th-century scientist and Augustinian friar.
|
Plant genetics
| 0.863828
|
38
|
The strict requirements for producing hybrid seed led to the development of careful population and inbred line maintenance, keeping plants isolated and unable to out-cross, which produced plants that better allowed researchers to tease out different genetic concepts. The structure of these populations allowed scientist such a T. Dobzhansky, S. Wright, and R.A. Fisher to develop evolutionary biology concepts as well as explore speciation over time and the statistics underlying plant genetics.
|
Plant genetics
| 0.863828
|
39
|
Castle discovered that while individual traits may segregate and change over time with selection, that when selection is stopped and environmental effects are taken into account, the genetic ratio stops changing and reach a sort of stasis, the foundation of Population Genetics. This was independently discovered by G. H. Hardy and W. Weinberg, which ultimately gave rise to the concept of Hardy–Weinberg equilibrium published in 1908.For a more thorough exploration of the history of population genetics, see History of Population Genetics by Bob Allard. Around this same time, genetic and plant breeding experiments in maize began.
|
Plant genetics
| 0.863828
|
40
|
His discoveries, deduction of segregation ratios, and subsequent laws have not only been used in research to gain a better understanding of plant genetics, but also play a large role in plant breeding. Mendel's works along with the works of Charles Darwin and Alfred Wallace on selection provided the basis for much of genetics as a discipline.
|
Plant genetics
| 0.863828
|
41
|
Mendel died in 1884. The significance of Mendel's work was not recognized until the turn of the 20th century. Its rediscovery prompted the foundation of modern genetics.
|
Plant genetics
| 0.863828
|
42
|
Mendel's work tracked many phenotypic traits of pea plants, such as their height, flower color, and seed characteristics. Mendel showed that the inheritance of these traits follows two particular laws, which were later named after him. His seminal work on genetics, “Versuche über Pflanzen-Hybriden” (Experiments on Plant Hybrids), was published in 1866, but went almost entirely unnoticed until 1900 when prominent botanists in the UK, like Sir Gavin de Beer, recognized its importance and re-published an English translation.
|
Plant genetics
| 0.863828
|
43
|
The earliest evidence of plant domestication found has been dated to 11,000 years before present in ancestral wheat. While initially selection may have happened unintentionally, it is very likely that by 5,000 years ago farmers had a basic understanding of heredity and inheritance, the foundation of genetics. This selection over time gave rise to new crop species and varieties that are the basis of the crops we grow, eat and research today. The field of plant genetics began with the work of Gregor Johann Mendel, who is often called the "father of genetics".
|
Plant genetics
| 0.863828
|
44
|
Genetic modification has been the cause for much research into modern plant genetics, and has also led to the sequencing of many plant genomes. Today there are two predominant procedures of transforming genes in organisms: the "Gene gun" method and the Agrobacterium method.
|
Plant genetics
| 0.863828
|
45
|
In plant breeding, people create hybrids between plant species for economic and aesthetic reasons. For example, the yield of Corn has increased nearly five-fold in the past century due in part to the discovery and proliferation of hybrid corn varieties. Plant genetics can be used to predict which combination of plants may produce a plant with Hybrid vigor, or conversely many discoveries in Plant genetics have come from studying the effects of hybridization.
|
Plant genetics
| 0.863828
|
46
|
The square root of 2, often known as root 2, radical 2, or Pythagoras' constant, and written as √2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational.
|
Mathematical constants
| 0.863742
|
47
|
In the computer science subfield of algorithmic information theory, Chaitin's constant is the real number representing the probability that a randomly chosen Turing machine will halt, formed from a construction due to Argentine-American mathematician and computer scientist Gregory Chaitin. Chaitin's constant, though not being computable, has been proven to be transcendental and normal. Chaitin's constant is not universal, depending heavily on the numerical encoding used for Turing machines; however, its interesting properties are independent of the encoding.
|
Mathematical constants
| 0.863742
|
48
|
Secosteroids (Latin seco, "to cut") are a subclass of steroidal compounds resulting, biosynthetically or conceptually, from scission (cleavage) of parent steroid rings (generally one of the four). Major secosteroid subclasses are defined by the steroid carbon atoms where this scission has taken place. For instance, the prototypical secosteroid cholecalciferol, vitamin D3 (shown), is in the 9,10-secosteroid subclass and derives from the cleavage of carbon atoms C-9 and C-10 of the steroid B-ring; 5,6-secosteroids and 13,14-steroids are similar.Norsteroids (nor-, L. norma; "normal" in chemistry, indicating carbon removal) and homosteroids (homo-, Greek homos; "same", indicating carbon addition) are structural subclasses of steroids formed from biosynthetic steps. The former involves enzymic ring expansion-contraction reactions, and the latter is accomplished (biomimetically) or (more frequently) through ring closures of acyclic precursors with more (or fewer) ring atoms than the parent steroid framework.Combinations of these ring alterations are known in nature.
|
Steroid
| 0.863628
|
49
|
The objective wave function evolves deterministically but, according to the Copenhagen interpretation, it deals with probabilities of observing, the outcome being explained by a wave function collapse when an observation is made. However, the loss of determinism for the sake of instrumentalism did not meet with universal approval. Albert Einstein famously remarked in a letter to Max Born: "I am convinced that God does not play dice". Like Einstein, Erwin Schrödinger, who discovered the wave function, believed quantum mechanics is a statistical approximation of an underlying deterministic reality. In some modern interpretations of the statistical mechanics of measurement, quantum decoherence is invoked to account for the appearance of subjectively probabilistic experimental outcomes.
|
Probability
| 0.863489
|
50
|
Physicists face the same situation in the kinetic theory of gases, where the system, while deterministic in principle, is so complex (with the number of molecules typically the order of magnitude of the Avogadro constant 6.02×1023) that only a statistical description of its properties is feasible. Probability theory is required to describe quantum phenomena. A revolutionary discovery of early 20th century physics was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of quantum mechanics.
|
Probability
| 0.863489
|
51
|
However, it is possible to define a conditional probability for some zero-probability events using a σ-algebra of such events (such as those arising from a continuous random variable).For example, in a bag of 2 red balls and 2 blue balls (4 balls in total), the probability of taking a red ball is 1 / 2 ; {\displaystyle 1/2;} however, when taking a second ball, the probability of it being either a red ball or a blue ball depends on the ball previously taken. For example, if a red ball was taken, then the probability of picking a red ball again would be 1 / 3 , {\displaystyle 1/3,} since only 1 red and 2 blue balls would have been remaining. And if a blue ball was taken previously, the probability of taking a red ball will be 2 / 3. {\displaystyle 2/3.}
|
Probability
| 0.863489
|
52
|
In Cox's theorem, probability is taken as a primitive (i.e., not further analyzed), and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, the laws of probability are the same, except for technical details. There are other methods for quantifying uncertainty, such as the Dempster–Shafer theory or possibility theory, but those are essentially different and not compatible with the usually-understood laws of probability.
|
Probability
| 0.863489
|
53
|
In probability theory and applications, Bayes' rule relates the odds of event A 1 {\displaystyle A_{1}} to event A 2 , {\displaystyle A_{2},} before (prior to) and after (posterior to) conditioning on another event B . {\displaystyle B.} The odds on A 1 {\displaystyle A_{1}} to event A 2 {\displaystyle A_{2}} is simply the ratio of the probabilities of the two events.
|
Probability
| 0.863489
|
54
|
The product of the prior and the likelihood, when normalized, results in a posterior probability distribution that incorporates all the information known to date. By Aumann's agreement theorem, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions, regardless of how much information the agents share.
|
Probability
| 0.863489
|
55
|
Augustus De Morgan and George Boole improved the exposition of the theory. In 1906, Andrey Markov introduced the notion of Markov chains, which played an important role in stochastic processes theory and its applications. The modern theory of probability based on measure theory was developed by Andrey Kolmogorov in 1931.On the geometric side, contributors to The Educational Times included Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin. See integral geometry for more information.
|
Probability
| 0.863489
|
56
|
The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict.In addition to financial assessment, probability can be used to analyze trends in biology (e.g., disease spread) as well as ecology (e.g., biological Punnett squares). As with finance, risk assessment can be used as a statistical tool to calculate the likelihood of undesirable events occurring, and can assist with implementing protocols to avoid encountering such circumstances. Probability is used to design games of chance so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play.Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacturer's decisions on a product's warranty.The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory.
|
Probability
| 0.863489
|
57
|
An example of the use of probability theory in equity trading is the effect of the perceived probability of any widespread Middle East conflict on oil prices, which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily rationally.
|
Probability
| 0.863489
|
58
|
Probability theory is applied in everyday life in risk assessment and modeling. The insurance industry and markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation, entitlement analysis, and financial regulation.
|
Probability
| 0.863489
|
59
|
Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.
|
Probability
| 0.863489
|
60
|
In quantum mechanics, the measurement problem is the problem of how, or whether, wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key set of questions that each interpretation must answer. The wave function in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states. However, actual measurements always find the physical system in a definite state.
|
Problem of measurement
| 0.86331
|
61
|
The heuristic method In optimization problems, heuristic algorithms can be used to find a solution close to the optimal solution in cases where finding the optimal solution is impractical. These algorithms work by getting closer and closer to the optimal solution as they progress. In principle, if run for an infinite amount of time, they will find the optimal solution.
|
Computer algorithms
| 0.863147
|
62
|
For some problems they can find the optimal solution while for others they stop at local optima, that is, at solutions that cannot be improved by the algorithm but are not optimum. The most popular use of greedy algorithms is for finding the minimal spanning tree where finding the optimal solution is possible with this method. Huffman Tree, Kruskal, Prim, Sollin are greedy algorithms that can solve this optimization problem.
|
Computer algorithms
| 0.863147
|
63
|
For optimization problems there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following: Linear programming When searching for optimal solutions to a linear function bound to linear equality and inequality constraints, the constraints of the problem can be used directly in producing the optimal solutions. There are algorithms that can solve any problem in this category, such as the popular simplex algorithm. Problems that can be solved with linear programming include the maximum flow problem for directed graphs. If a problem additionally requires that one or more of the unknowns must be an integer then it is classified in integer programming.
|
Computer algorithms
| 0.863147
|
64
|
In rapid succession the following appeared: Alonzo Church, Stephen Kleene and J.B. Rosser's λ-calculus a finely honed definition of "general recursion" from the work of Gödel acting on suggestions of Jacques Herbrand (cf. Gödel's Princeton lectures of 1934) and subsequent simplifications by Kleene. Church's proof that the Entscheidungsproblem was unsolvable, Emil Post's definition of effective calculability as a worker mindlessly following a list of instructions to move left or right through a sequence of rooms and while there either mark or erase a paper or observe the paper and make a yes-no decision about the next instruction. Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a- machine"—in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine". Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I", and a few years later Kleene's renaming his Thesis "Church's Thesis" and proposing "Turing's Thesis".
|
Computer algorithms
| 0.863147
|
65
|
Louis Georges Gouy in 1910 and David Leonard Chapman in 1913 both observed that capacitance was not a constant and that it depended on the applied potential and the ionic concentration. The "Gouy–Chapman model" made significant improvements by introducing a diffuse model of the DL. In this model, the charge distribution of ions as a function of distance from the metal surface allows Maxwell–Boltzmann statistics to be applied. Thus the electric potential decreases exponentially away from the surface of the fluid bulk.Gouy-Chapman layers may bear special relevance in bioelectrochemistry.
|
Electric surface potential
| 0.86177
|
66
|
Most research in the area of memory models revolves around: Designing a memory model that allows a maximal degree of freedom for compiler optimizations while still giving sufficient guarantees about race-free and (perhaps more importantly) race-containing programs. Proving program optimizations that are correct with respect to such a memory model.The Java Memory Model was the first attempt to provide a comprehensive threading memory model for a popular programming language. After it was established that threads could not be implemented safely as a library without placing certain restrictions on the implementation and, in particular, that the C and C++ standards (C99 and C++03) lacked necessary restrictions, the C++ threading subcommittee set to work on suitable memory model; in 2005, they submitted C working document n1131 to get the C Committee on board with their efforts. The final revision of the proposed memory model, C++ n2429, was accepted into the C++ draft standard at the October 2007 meeting in Kona. The memory model was then included in the next C++ and C standards, C++11 and C11.
|
Memory model (programming)
| 0.861157
|
67
|
The memory model stipulates that changes to the values of shared variables only need to be made visible to other threads when such a synchronization barrier is reached. Moreover, the entire notion of a race condition is defined over the order of operations with respect to these memory barriers.These semantics then give optimizing compilers a higher degree of freedom when applying optimizations: the compiler needs to make sure only that the values of (potentially shared) variables at synchronization barriers are guaranteed to be the same in both the optimized and unoptimized code. In particular, reordering statements in a block of code that contains no synchronization barrier is assumed to be safe by the compiler.
|
Memory model (programming)
| 0.861157
|
68
|
Or for some compilers assume no multi-threaded execution (so better optimized code can be produced), which can lead to optimizations that are incompatible with multi-threading - these can often lead to subtle bugs, that don't show up in early testing. Modern programming languages like Java therefore implement a memory model. The memory model specifies synchronization barriers that are established via special, well-defined synchronization operations such as acquiring a lock by entering a synchronized block or method.
|
Memory model (programming)
| 0.861157
|
69
|
A memory model allows a compiler to perform many important optimizations. Compiler optimizations like loop fusion move statements in the program, which can influence the order of read and write operations of potentially shared variables. Changes in the ordering of reads and writes can cause race conditions. Without a memory model, a compiler is not allowed to apply such optimizations to multi-threaded programs in general, or only in special cases.
|
Memory model (programming)
| 0.861157
|
70
|
Sometimes, a set is endowed with more than one feature simultaneously, which allows mathematicians to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topology on the set, and if a set has both a topology feature and a group feature, such that these two features are related in a certain way, then the structure becomes a topological group.Mappings between sets which preserve structures (i.e., structures in the domain are mapped to equivalent structures in the codomain) are of special interest in many fields of mathematics. Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures.
|
Mathematical structure
| 0.860057
|
71
|
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance. A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.
|
Mathematical structure
| 0.860057
|
72
|
By moving the two points closer together so that Δy and Δx decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using differential calculus, we can determine the limit, or the value that Δy/Δx approaches as Δy and Δx get closer to zero; it follows that this limit is the exact slope of the tangent. If y is dependent on x, then it is sufficient to take the limit where only Δx approaches zero.
|
Slope of a line
| 0.860053
|
73
|
For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve. For example, the slope of the secant intersecting y = x2 at (0,0) and (3,9) is 3. (The slope of the tangent at x = 3⁄2 is also 3 − a consequence of the mean value theorem.)
|
Slope of a line
| 0.860053
|
74
|
The concept of a slope is central to differential calculus. For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point. If we let Δx and Δy be the distances (along the x and y axes, respectively) between two points on a curve, then the slope given by the above definition, m = Δ y Δ x {\displaystyle m={\frac {\Delta y}{\Delta x}}} ,is the slope of a secant line to the curve.
|
Slope of a line
| 0.860053
|
75
|
In statistics, the gradient of the least-squares regression best-fitting line for a given sample of data may be written as: m = r s y s x {\displaystyle m={\frac {rs_{y}}{s_{x}}}} ,This quantity m is called as the regression slope for the line y = m x + c {\displaystyle y=mx+c} . The quantity r {\displaystyle r} is Pearson's correlation coefficient, s y {\displaystyle s_{y}} is the standard deviation of the y-values and s x {\displaystyle s_{x}} is the standard deviation of the x-values. This may also be written as a ratio of covariances: m = cov ( Y , X ) cov ( X , X ) {\displaystyle m={\frac {\operatorname {cov} (Y,X)}{\operatorname {cov} (X,X)}}}
|
Slope of a line
| 0.860053
|
76
|
When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic expression, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve. This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.
|
Slope of a line
| 0.860053
|
77
|
The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of inclination θ by the tangent function m = tan ( θ ) {\displaystyle m=\tan(\theta )} Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1. As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point.
|
Slope of a line
| 0.860053
|
78
|
Additionally, tables of equations, information, and constants are provided for all portions of the exam as of 2015. This and AP Physics C: Electricity and Magnetism are the shortest AP exams, with total testing time of 90 minutes.The topics covered by the exam are as follows: As a result of the 2019-20 coronavirus pandemic, the AP examination in 2020 was taken online. The topics of oscillations and gravitation were removed from the test.
|
AP Physics C: Mechanics
| 0.859946
|
79
|
Advanced Placement (AP) Physics C: Mechanics (also known as AP Mechanics) is an introductory physics course administered by the College Board as part of its Advanced Placement program. It is intended to proxy a one-semester calculus-based university course in mechanics. The content of Physics C: Mechanics overlaps with that of AP Physics 1, but Physics 1 is algebra-based, while Physics C is calculus-based. Physics C: Mechanics may be combined with its electricity and magnetism counterpart to form a year-long course that prepares for both exams.
|
AP Physics C: Mechanics
| 0.859946
|
80
|
The AP examination for AP Physics C: Mechanics is separate from the AP examination for AP Physics C: Electricity and Magnetism. Before 2006, test-takers paid only once and were given the choice of taking either one or two parts of the Physics C test.
|
AP Physics C: Mechanics
| 0.859946
|
81
|
Intended to be equivalent to an introductory college course in mechanics for physics or engineering majors, the course modules are: Kinematics Newton's laws of motion Work, energy and power Systems of particles and linear momentum Circular motion and rotation Oscillations and gravitation.Methods of calculus are used wherever appropriate in formulating physical principles and in applying them to physical problems. Therefore, students should have completed or be concurrently enrolled in a Calculus I class.This course is often compared to AP Physics 1: Algebra Based for its similar course material involving kinematics, work, motion, forces, rotation, and oscillations. However, AP Physics 1: Algebra Based lacks concepts found in Calculus I, like derivatives or integrals. This course may be combined with AP Physics C: Electricity and Magnetism to make a unified Physics C course that prepares for both exams.
|
AP Physics C: Mechanics
| 0.859946
|
82
|
In a similar fashion, constants appear in the solutions to differential equations where not enough initial values or boundary conditions are given. For example, the ordinary differential equation y' = y(x) has solution Cex where C is an arbitrary constant. When dealing with partial differential equations, the constants may be functions, constant with respect to some variables (but not necessarily all of them). For example, the PDE ∂ f ( x , y ) ∂ x = 0 {\displaystyle {\frac {\partial f(x,y)}{\partial x}}=0} has solutions f(x,y) = C(y), where C(y) is an arbitrary function in the variable y.
|
Mathematical constant
| 0.858972
|
83
|
Abbreviations used: R – Rational number, I – Irrational number (may be algebraic or transcendental), A – Algebraic number (irrational), T – Transcendental number Gen – General, NuT – Number theory, ChT – Chaos theory, Com – Combinatorics, Inf – Information theory, Ana – Mathematical analysis
|
Mathematical constant
| 0.858972
|
84
|
When unspecified, constants indicate classes of similar objects, commonly functions, all equal up to a constant—technically speaking, this may be viewed as 'similarity up to a constant'. Such constants appear frequently when dealing with integrals and differential equations. Though unspecified, they have a specific value, which often is not important.
|
Mathematical constant
| 0.858972
|
85
|
Kepler proved that it is the limit of the ratio of consecutive Fibonacci numbers. The golden ratio has the slowest convergence of any irrational number. It is, for that reason, one of the worst cases of Lagrange's approximation theorem and it is an extremal case of the Hurwitz inequality for Diophantine approximations.
|
Mathematical constant
| 0.858972
|
86
|
The number φ, also called the golden ratio, turns up frequently in geometry, particularly in figures with pentagonal symmetry. Indeed, the length of a regular pentagon's diagonal is φ times its side. The vertices of a regular icosahedron are those of three mutually orthogonal golden rectangles. Also, it appears in the Fibonacci sequence, related to growth by recursion.
|
Mathematical constant
| 0.858972
|
87
|
The term "imaginary" was coined because there is no (real) number having a negative square. There are in fact two complex square roots of −1, namely i and −i, just as there are two complex square roots of every other real number (except zero, which has one double square root). In contexts where the symbol i is ambiguous or problematic, j or the Greek iota (ι) is sometimes used. This is in particular the case in electrical engineering and control systems engineering, where the imaginary unit is often denoted by j, because i is commonly used to denote electric current.
|
Mathematical constant
| 0.858972
|
88
|
Euler's number e, also known as the exponential growth constant, appears in many areas of mathematics, and one possible definition of it is the value of the following expression: e = lim n → ∞ ( 1 + 1 n ) n {\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}} The constant e is intrinsically related to the exponential function x ↦ e x {\displaystyle x\mapsto e^{x}} . The Swiss mathematician Jacob Bernoulli discovered that e arises in compound interest: If an account starts at $1, and yields interest at annual rate R, then as the number of compounding periods per year tends to infinity (a situation known as continuous compounding), the amount of money at the end of the year will approach eR dollars. The constant e also has applications to probability theory, where it arises in a way not obviously related to exponential growth. As an example, suppose that a slot machine with a one in n probability of winning is played n times, then for large n (e.g., one million), the probability that nothing will be won will tend to 1/e as n tends to infinity.
|
Mathematical constant
| 0.858972
|
89
|
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, statistics, and calculus. Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle (π). Other constants are notable more for historical reasons than for their mathematical properties. The more popular constants have been studied throughout the ages and computed to many decimal places. All named mathematical constants are definable numbers, and usually are also computable numbers (Chaitin's constant being a significant exception).
|
Mathematical constant
| 0.858972
|
90
|
The constant π (pi) has a natural definition in Euclidean geometry as the ratio between the circumference and diameter of a circle. It may be found in many other places in mathematics: for example, the Gaussian integral, the complex roots of unity, and Cauchy distributions in probability. However, its ubiquity is not limited to pure mathematics. It appears in many formulas in physics, and several physical constants are most naturally defined with π or its reciprocal factored out.
|
Mathematical constant
| 0.858972
|
91
|
The Euler–Mascheroni constant is defined as the following limit: γ = lim n → ∞ ( ( ∑ k = 1 n 1 k ) − ln n ) {\displaystyle {\begin{aligned}\gamma &=\lim _{n\to \infty }\left(\left(\sum _{k=1}^{n}{\frac {1}{k}}\right)-\ln n\right)\\\end{aligned}}} The Euler–Mascheroni constant appears in Mertens' third theorem and has relations to the gamma function, the zeta function and many different integrals and series. It is yet unknown whether γ {\displaystyle \gamma } is rational or not. The numeric value of γ {\displaystyle \gamma } is approximately 0.57721.
|
Mathematical constant
| 0.858972
|
92
|
In mathematics, a function is a rule for taking an input (in the simplest case, a number or set of numbers) and providing an output (which may also be a number). A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. The most common symbol for the input is x, and the most common symbol for the output is y; the function itself is commonly written y = f(x).It is possible to have multiple independent variables or multiple dependent variables. For instance, in multivariable calculus, one often encounters functions of the form z = f(x,y), where z is a dependent variable and x and y are independent variables. Functions with multiple outputs are often referred to as vector-valued functions.
|
Response variable
| 0.858803
|
93
|
We distinguish between two kinds of features: Static features are in most cases some counts and statistics (e.g., clauses-to-variables ratio in SAT). These features ranges from very cheap features (e.g. number of variables) to very complex features (e.g., statistics about variable-clause graphs). Probing features (sometimes also called landmarking features) are computed by running some analysis of algorithm behavior on an instance (e.g., accuracy of a cheap decision tree algorithm on an ML data set, or running for a short time a stochastic local search solver on a Boolean formula). These feature often cost more than simple static features.
|
Algorithm selection
| 0.858412
|
94
|
In machine learning, algorithm selection is better known as meta-learning. The portfolio of algorithms consists of machine learning algorithms (e.g., Random Forest, SVM, DNN), the instances are data sets and the cost metric is for example the error rate. So, the goal is to predict which machine learning algorithm will have a small error on each data set.
|
Algorithm selection
| 0.858412
|
95
|
Algorithm selection is not limited to single domains but can be applied to any kind of algorithm if the above requirements are satisfied. Application domains include: hard combinatorial problems: SAT, Mixed Integer Programming, CSP, AI Planning, TSP, MAXSAT, QBF and Answer Set Programming combinatorial auctions in machine learning, the problem is known as meta-learning software design black-box optimization multi-agent systems numerical optimization linear algebra, differential equations evolutionary algorithms vehicle routing problem power systemsFor an extensive list of literature about algorithm selection, we refer to a literature overview.
|
Algorithm selection
| 0.858412
|
96
|
The algorithm selection problem is mainly solved with machine learning techniques. By representing the problem instances by numerical features f {\displaystyle f} , algorithm selection can be seen as a multi-class classification problem by learning a mapping f i ↦ A {\displaystyle f_{i}\mapsto {\mathcal {A}}} for a given instance i {\displaystyle i} . Instance features are numerical representations of instances. For example, we can count the number of variables, clauses, average clause length for Boolean formulas, or number of samples, features, class balance for ML data sets to get an impression about their characteristics.
|
Algorithm selection
| 0.858412
|
97
|
Algorithms were also used in Babylonian astronomy. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.Algorithms for arithmetic are also found in ancient Egyptian mathematics, dating back to the Rhind Mathematical Papyrus c. 1550 BC.
|
Algorithm
| 0.85781
|
98
|
Muhammad ibn Mūsā al-Khwārizmī, a Persian mathematician, wrote the Al-jabr in the 9th century. The terms "algorism" and "algorithm" are derived from the name al-Khwārizmī, while the term "algebra" is derived from the book Al-jabr. In Europe, the word "algorithm" was originally used to refer to the sets of rules and techniques used by Al-Khwarizmi to solve algebraic equations, before later being generalized to refer to any set of rules or techniques. This eventually culminated in Leibniz's notion of the calculus ratiocinator (c. 1680): A good century and a half ahead of his time, Leibniz proposed an algebra of logic, an algebra that would specify the rules for manipulating logical concepts in the manner that ordinary algebra specifies the rules for manipulating numbers.
|
Algorithm
| 0.85781
|
99
|
; Spillane, C. (1999) Biotechnology assisted participatory plant breeding: Complement or contradiction? CGIAR Program on Participatory Research and Gender Analysis, Working Document No.4, CIAT: Cali.
|
Plant breeding
| 0.857805
|
End of preview. Expand
in Data Studio
README.md exists but content is empty.
- Downloads last month
- 23