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Euler method Backward Euler method Trapezoidal rule (differential equations) Linear multistep methods Runge–Kutta methods Euler integration Multigrid methods (MG methods), a group of algorithms for solving differential equations using a hierarchy of discretizations Partial differential equation: Finite difference method Crank–Nicolson method for diffusion equations Lax–Wendroff for wave equations Verlet integration (French pronunciation: ): integrate Newton's equations of motion
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This method is an upgraded modification to combinatorial probe anchor ligation technology (cPAL) described by Complete Genomics which has since become part of Chinese genomics company BGI in 2013. The two companies have refined the technology to allow for longer read lengths, reaction time reductions and faster time to results. In addition, data are now generated as contiguous full-length reads in the standard FASTQ file format and can be used as-is in most short-read-based bioinformatics analysis pipelines.The two technologies that form the basis for this high-throughput sequencing technology are DNA nanoballs (DNB) and patterned arrays for nanoball attachment to a solid surface. DNA nanoballs are simply formed by denaturing double stranded, adapter ligated libraries and ligating the forward strand only to a splint oligonucleotide to form a ssDNA circle.
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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.
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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.
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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)}}}
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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.
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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.)
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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.
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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.
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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.
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Slope of a line
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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.
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AP Physics C: Mechanics
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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.
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AP Physics C: Mechanics
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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.
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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.
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telephone monopoly, presented its Bellboy radio paging system at the Seattle World's Fair. Bellboy was the first commercial system for personal paging. It also marked one of the first consumer applications of the transistor (invented by Bell Labs in 1947), for which three Bell Labs inventors received a Nobel Prize in Physics in 1956.
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Paging system
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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.
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Mathematical constant
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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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
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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.
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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.
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Response variable
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Although bubble sort is one of the simplest sorting algorithms to understand and implement, its O(n2) complexity means that its efficiency decreases dramatically on lists of more than a small number of elements. Even among simple O(n2) sorting algorithms, algorithms like insertion sort are usually considerably more efficient. Due to its simplicity, bubble sort is often used to introduce the concept of an algorithm, or a sorting algorithm, to introductory computer science students. However, some researchers such as Owen Astrachan have gone to great lengths to disparage bubble sort and its continued popularity in computer science education, recommending that it no longer even be taught.The Jargon File, which famously calls bogosort "the archetypical perversely awful algorithm", also calls bubble sort "the generic bad algorithm".
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Euclidean geometry has two fundamental types of measurements: angle and distance. The angle scale is absolute, and Euclid uses the right angle as his basic unit, so that, for example, a 45-degree angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction.
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2D geometry
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Angle bisector theorem Butterfly theorem Ceva's theorem Heron's formula Menelaus' theorem Nine-point circle Pythagorean theorem
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In recent years, multiple websites that maintain lists of conceptual questions have been created by instructors for various disciplines. Some books on physics provide many examples of conceptual questions as well.Multiple conceptual questions can be assembled into a concept inventory to test the working knowledge of students at the beginning of a course or to track the improvement in conceptual understanding throughout the course. == References ==
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Conceptual problems are often formulated as multiple-choice questions, making them easy to use during in-class discussions, particularly when utilizing active learning, peer instruction, and audience response. An example of a conceptual question in undergraduate thermodynamics is provided below: During adiabatic expansion of an ideal gas, its temperatureincreases decreases stays the same Impossible to tell/need more information The use of conceptual questions in physics was popularized by Eric Mazur, particularly in the form of multiple-choice tests that he called ConcepTests.
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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.
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Algorithm selection
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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.
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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.
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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.
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This method can be used to recombine structural elements or entire protein domains. This method is based on phosphorothioate chemistry which allows the specific cleavage of phosphorothiodiester bonds. The first step in the process begins with amplification of fragments that need to be recombined along with the vector backbone.
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Protein engineering
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Computing methods have been used to design a protein with a novel fold, named Top7, and sensors for unnatural molecules. The engineering of fusion proteins has yielded rilonacept, a pharmaceutical that has secured Food and Drug Administration (FDA) approval for treating cryopyrin-associated periodic syndrome. Another computing method, IPRO, successfully engineered the switching of cofactor specificity of Candida boidinii xylose reductase. Iterative Protein Redesign and Optimization (IPRO) redesigns proteins to increase or give specificity to native or novel substrates and cofactors.
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Protein engineering
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In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} . A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin. While a pair of real numbers R 2 {\displaystyle \mathbb {R} ^{2}} suffices to describe points on a plane, the relationship with out-of-plane points requires special consideration for their embedding in the ambient space R 3 {\displaystyle \mathbb {R} ^{3}} .
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Plane equation
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Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry. He selected a small core of undefined terms (called common notions) and postulates (or axioms) which he then used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the Elements, it may be thought of as part of the common notions.
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These restrictions were relaxed by Aiken et al.This extended lambda calculus was intended to serve as a provably memory-safe intermediate representation for compiling Standard ML programs into machine code, but building a translator that would produce good results on large programs faced a number of practical limitations which had to be resolved with new analyses, including dealing with recursive calls, tail calls, and eliminating regions which contained only a single value. This work was completed in 1995 and integrated into the ML Kit, a version of ML based on region allocation in place of garbage collection. This permitted a direct comparison between the two on medium-sized test programs, yielding widely varying results ("between 10 times faster and four times slower") depending on how "region-friendly" the program was; compile times, however, were on the order of minutes. The ML Kit was eventually scaled to large applications with two additions: a scheme for separate compilation of modules, and a hybrid technique combining region inference with tracing garbage collection.
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Region-based memory management
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In 1994, this work was generalized in a seminal work by Tofte and Talpin to support type polymorphism and higher-order functions in Standard ML, a functional programming language, using a different algorithm based on type inference and the theoretical concepts of polymorphic region types and the region calculus. Their work introduced an extension of the lambda calculus including regions, adding two constructs: e1 at ρ: Compute the result of the expression e1 and store it in region ρ; letregion ρ in e2 end: Create a region and bind it to ρ; evaluate e2; then deallocate the region.Due to this syntactic structure, regions are nested, meaning that if r2 is created after r1, it must also be deallocated before r1; the result is a stack of regions. Moreover, regions must be deallocated in the same function in which they are created.
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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.
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Algorithm
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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.
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Algorithm
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Plant breeding is the science of changing the traits of plants in order to produce desired characteristics. It has been used to improve the quality of nutrition in products for humans and animals. The goals of plant breeding are to produce crop varieties that boast unique and superior traits for a variety of applications. The most frequently addressed agricultural traits are those related to biotic and abiotic stress tolerance, grain or biomass yield, end-use quality characteristics such as taste or the concentrations of specific biological molecules (proteins, sugars, lipids, vitamins, fibers) and ease of processing (harvesting, milling, baking, malting, blending, etc.).Plant breeding can be performed through many different techniques ranging from simply selecting plants with desirable characteristics for propagation, to methods that make use of knowledge of genetics and chromosomes, to more complex molecular techniques.
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Improvements in nutritional value for forage crops from the use of analytical chemistry and rumen fermentation technology have been recorded since 1960; this science and technology gave breeders the ability to screen thousands of samples within a small amount of time, meaning breeders could identify a high performing hybrid quicker. The genetic improvement was mainly in vitro dry matter digestibility (IVDMD) resulting in 0.7-2.5% increase, at just 1% increase in IVDMD a single Bos Taurus also known as beef cattle reported 3.2% increase in daily gains. This improvement indicates plant breeding is an essential tool in gearing future agriculture to perform at a more advanced level.
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Plant breeding
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Cereal Genomics. Methods in Molecular Biology. Vol.
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; 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.
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(ISBN 9781439802427), CRC Press, Boca Raton, FL, USA, pp 584 Schlegel, Rolf (2007) Concise Encyclopedia of Crop Improvement: Institutions, Persons, Theories, Methods, and Histories (ISBN 9781560221463), CRC Press, Boca Raton, FL, USA, pp 423 Schlegel, Rolf (2014) Dictionary of Plant Breeding, 2nd ed., (ISBN 978-1439802427), CRC Press, Boca Raton, Taylor & Francis Group, Inc., New York, USA, pp 584 Schouten, Henk J.; Krens, Frans A.; Jacobsen, Evert (2006). "Do cisgenic plants warrant less stringent oversight?". Nature Biotechnology.
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In molecular biology, an actomyosin contractile ring is a prominent structure during cytokinesis. It forms perpendicular to the axis of the spindle apparatus towards the end of telophase, in which sister chromatids are identically separated at the opposite sides of the spindle forming nuclei (Figure 1). The actomyosin ring follows an orderly sequence of events: identification of the active division site, formation of the ring, constriction of the ring, and disassembly of the ring. It is composed of actin and myosin II bundles, thus the term actomyosin.
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Actomyosin ring
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In mathematics, a quadratic-linear algebra is an algebra over a field with a presentation such that all relations are sums of monomials of degrees 1 or 2 in the generators. They were introduced by Polishchuk and Positselski (2005, p.101). An example is the universal enveloping algebra of a Lie algebra, with generators a basis of the Lie algebra and relations of the form XY – YX – = 0.
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Quadratic-linear algebra
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In computer science, a search algorithm is an algorithm designed to solve a search problem. Search algorithms work to retrieve information stored within particular data structure, or calculated in the search space of a problem domain, with either discrete or continuous values. Although search engines use search algorithms, they belong to the study of information retrieval, not algorithmics.
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Search algorithms
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Specific applications of search algorithms include: Problems in combinatorial optimization, such as: The vehicle routing problem, a form of shortest path problem The knapsack problem: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. The nurse scheduling problem Problems in constraint satisfaction, such as: The map coloring problem Filling in a sudoku or crossword puzzle In game theory and especially combinatorial game theory, choosing the best move to make next (such as with the minmax algorithm) Finding a combination or password from the whole set of possibilities Factoring an integer (an important problem in cryptography) Optimizing an industrial process, such as a chemical reaction, by changing the parameters of the process (like temperature, pressure, and pH) Retrieving a record from a database Finding the maximum or minimum value in a list or array Checking to see if a given value is present in a set of values
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Protein structure prediction is the inference of the three-dimensional structure of a protein from its amino acid sequence—that is, the prediction of its folding and its secondary and tertiary structure from its primary structure. Structure prediction is fundamentally different from the inverse problem of protein design. Protein structure prediction is one of the most important goals pursued by bioinformatics and theoretical chemistry; it is highly important in medicine, in drug design, biotechnology and in the design of novel enzymes). Every two years, the performance of current methods is assessed in the CASP experiment (Critical Assessment of Techniques for Protein Structure Prediction). A continuous evaluation of protein structure prediction web servers is performed by the community project CAMEO3D.
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Protein chemistry
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For example, they could be used to identify and destroy cancer cells. Molecular nanotechnology is a speculative subfield of nanotechnology regarding the possibility of engineering molecular assemblers, biological machines which could re-order matter at a molecular or atomic scale. Nanomedicine would make use of these nanorobots, introduced into the body, to repair or detect damages and infections. Molecular nanotechnology is highly theoretical, seeking to anticipate what inventions nanotechnology might yield and to propose an agenda for future inquiry. The proposed elements of molecular nanotechnology, such as molecular assemblers and nanorobots are far beyond current capabilities.
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Molecular biophysics is a rapidly evolving interdisciplinary area of research that combines concepts in physics, chemistry, engineering, mathematics and biology. It seeks to understand biomolecular systems and explain biological function in terms of molecular structure, structural organization, and dynamic behaviour at various levels of complexity (from single molecules to supramolecular structures, viruses and small living systems). This discipline covers topics such as the measurement of molecular forces, molecular associations, allosteric interactions, Brownian motion, and cable theory. Additional areas of study can be found on Outline of Biophysics. The discipline has required development of specialized equipment and procedures capable of imaging and manipulating minute living structures, as well as novel experimental approaches.
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Computational biology involves the development and application of data-analytical and theoretical methods, mathematical modeling and computational simulation techniques to the study of biological, ecological, behavioral, and social systems. The field is broadly defined and includes foundations in biology, applied mathematics, statistics, biochemistry, chemistry, biophysics, molecular biology, genetics, genomics, computer science and evolution. Computational biology has become an important part of developing emerging technologies for the field of biology. Molecular modelling encompasses all methods, theoretical and computational, used to model or mimic the behaviour of molecules. The methods are used in the fields of computational chemistry, drug design, computational biology and materials science to study molecular systems ranging from small chemical systems to large biological molecules and material assemblies.
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Protein chemistry
| 0.85758
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156
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Molecular biophysics typically addresses biological questions similar to those in biochemistry and molecular biology, seeking to find the physical underpinnings of biomolecular phenomena. Scientists in this field conduct research concerned with understanding the interactions between the various systems of a cell, including the interactions between DNA, RNA and protein biosynthesis, as well as how these interactions are regulated. A great variety of techniques are used to answer these questions. Fluorescent imaging techniques, as well as electron microscopy, X-ray crystallography, NMR spectroscopy, atomic force microscopy (AFM) and small-angle scattering (SAS) both with X-rays and neutrons (SAXS/SANS) are often used to visualize structures of biological significance.
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Protein chemistry
| 0.85758
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157
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In graph theory, Graph equations are equations in which the unknowns are graphs. One of the central questions of graph theory concerns the notion of isomorphism. We ask: When are two graphs the same? (i.e., graph isomorphism) The graphs in question may be expressed differently in terms of graph equations.What are the graphs (solutions) G and H such that the line graph of G is same as the total graph of H?
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Graph equation
| 0.857529
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158
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The table below summarizes how algebraic expressions compare with several other types of mathematical expressions by the type of elements they may contain, according to common but not universal conventions. A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as x2 + 4x + 4. An irrational algebraic expression is one that is not rational, such as √x + 4.
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Algebraic expression
| 0.857402
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159
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Usually, π is constructed as a geometric relationship, and the definition of e requires an infinite number of algebraic operations. A rational expression is an expression that may be rewritten to a rational fraction by using the properties of the arithmetic operations (commutative properties and associative properties of addition and multiplication, distributive property and rules for the operations on the fractions). In other words, a rational expression is an expression which may be constructed from the variables and the constants by using only the four operations of arithmetic.
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Algebraic expression
| 0.857402
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160
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In mathematics, an algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, 3x2 − 2xy + c is an algebraic expression. Since taking the square root is the same as raising to the power 1/2, the following is also an algebraic expression: 1 − x 2 1 + x 2 {\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}} An algebraic equation is an equation involving only algebraic expressions. By contrast, transcendental numbers like π and e are not algebraic, since they are not derived from integer constants and algebraic operations.
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Algebraic expression
| 0.857402
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161
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In 1658, in the first edition of The New World of English Words, it says: Algorithme, (a word compounded of Arabick and Spanish,) the art of reckoning by Cyphers. In 1706, in the sixth edition of The New World of English Words, it says: Algorithm, the Art of computing or reckoning by numbers, which contains the five principle Rules of Arithmetick, viz. Numeration, Addition, Subtraction, Multiplication and Division; to which may be added Extraction of Roots: It is also call'd Logistica Numeralis. Algorism, the practical Operation in the several Parts of Specious Arithmetick or Algebra; sometimes it is taken for the Practice of Common Arithmetick by the ten Numeral Figures. In 1751, in the Young Algebraist's Companion, Daniel Fenning contrasts the terms algorism and algorithm as follows: Algorithm signifies the first Principles, and Algorism the practical Part, or knowing how to put the Algorithm in Practice. Since at least 1811, the term algorithm is attested to mean a "step-by-step procedure" in English.In 1842, in the Dictionary of Science, Literature and Art, it says: ALGORITHM, signifies the art of computing in reference to some particular subject, or in some particular way; as the algorithm of numbers; the algorithm of the differential calculus.
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Algorithmic problem
| 0.857278
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162
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Total weight that can be carried is no more than some fixed number X. So, the solution must consider weights of items as well as their value. Quantum algorithm They run on a realistic model of quantum computation. The term is usually used for those algorithms which seem inherently quantum, or use some essential feature of Quantum computing such as quantum superposition or quantum entanglement.
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Algorithmic problem
| 0.857278
|
163
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In mathematics and computer science, an algorithm ( ) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning), achieving automation eventually.
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Algorithmic problem
| 0.857278
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164
|
In digital electronics and computer science (fields of applied logic engineering and mathematics), truth tables can be used to reduce basic boolean operations to simple correlations of inputs to outputs, without the use of logic gates or code. For example, a binary addition can be represented with the truth table: where A is the first operand, B is the second operand, C is the carry digit, and R is the result. This truth table is read left to right: Value pair (A,B) equals value pair (C,R). Or for this example, A plus B equal result R, with the Carry C.Note that this table does not describe the logic operations necessary to implement this operation, rather it simply specifies the function of inputs to output values.
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Truth tables
| 0.857206
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165
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This may be due to a lack of mathematical knowledge; some problems were only solved after centuries of effort. But this also reflects that, in general, no such method can exist: some problems are known to be unsolvable by an algorithm, such as Hilbert's tenth problem, which was proved unsolvable in 1970. For several classes of equations, algorithms have been found for solving them, some of which have been implemented and incorporated in computer algebra systems, but often require no more sophisticated technology than pencil and paper. In some other cases, heuristic methods are known that are often successful but that are not guaranteed to lead to success.
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Equation solving
| 0.857109
|
166
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Equations involving linear or simple rational functions of a single real-valued unknown, say x, such as 8 x + 7 = 4 x + 35 or 4 x + 9 3 x + 4 = 2 , {\displaystyle 8x+7=4x+35\quad {\text{or}}\quad {\frac {4x+9}{3x+4}}=2\,,} can be solved using the methods of elementary algebra.
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Equation solving
| 0.857109
|
167
|
The most common type of equation is a polynomial equation (commonly called also an algebraic equation) in which the two sides are polynomials. The sides of a polynomial equation contain one or more terms.
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Mathematical equations
| 0.85657
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168
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An algebraic number is a number that is a solution of a non-zero polynomial equation in one variable with rational coefficients (or equivalently — by clearing denominators — with integer coefficients). Numbers such as π that are not algebraic are said to be transcendental. Almost all real and complex numbers are transcendental.
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Mathematical equations
| 0.85657
|
169
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A parametric equation for a curve expresses the coordinates of the points of the curve as functions of a variable, called a parameter. For example, x = cos t y = sin t {\displaystyle {\begin{aligned}x&=\cos t\\y&=\sin t\end{aligned}}} are parametric equations for the unit circle, where t is the parameter. Together, these equations are called a parametric representation of the curve. The notion of parametric equation has been generalized to surfaces, manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.).
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Mathematical equations
| 0.85657
|
170
|
In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
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Mathematical equations
| 0.85657
|
171
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A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in areas such as physics, chemistry, biology, and economics.
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Mathematical equations
| 0.85657
|
172
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The smallest and most basic number field is the field Q {\displaystyle \mathbb {Q} } of rational numbers. Many properties of general number fields are modeled after the properties of Q {\displaystyle \mathbb {Q} } . At the same time, many other properties of algebraic number fields are substantially different from the properties of rational numbers - one notable example is that the ring of algebraic integers of a number field is not a principal ideal domain, in general. The Gaussian rationals, denoted Q ( i ) {\displaystyle \mathbb {Q} (i)} (read as " Q {\displaystyle \mathbb {Q} } adjoined i {\displaystyle i} "), form the first (historically) non-trivial example of a number field.
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Degree of a number field
| 0.856411
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173
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Both types of functions encode the arithmetic behavior of Q {\displaystyle \mathbb {Q} } and K {\displaystyle K} , respectively. For example, Dirichlet's theorem asserts that in any arithmetic progression a , a + m , a + 2 m , … {\displaystyle a,a+m,a+2m,\ldots } with coprime a {\displaystyle a} and m {\displaystyle m} , there are infinitely many prime numbers. This theorem is implied by the fact that the Dirichlet L {\displaystyle L} -function is nonzero at s = 1 {\displaystyle s=1} . Using much more advanced techniques including algebraic K-theory and Tamagawa measures, modern number theory deals with a description, if largely conjectural (see Tamagawa number conjecture), of values of more general L-functions.
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Degree of a number field
| 0.856411
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174
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An integral basis for a number field K {\displaystyle K} of degree n {\displaystyle n} is a set B = {b1, …, bn}of n algebraic integers in K {\displaystyle K} such that every element of the ring of integers O K {\displaystyle {\mathcal {O}}_{K}} of K {\displaystyle K} can be written uniquely as a Z-linear combination of elements of B; that is, for any x in O K {\displaystyle {\mathcal {O}}_{K}} we have x = m1b1 + ⋯ + mnbn,where the mi are (ordinary) integers. It is then also the case that any element of K {\displaystyle K} can be written uniquely as m1b1 + ⋯ + mnbn,where now the mi are rational numbers. The algebraic integers of K {\displaystyle K} are then precisely those elements of K {\displaystyle K} where the mi are all integers. Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis, and it is now standard for computer algebra systems to have built-in programs to do this.
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Degree of a number field
| 0.856411
|
175
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In the United Kingdom, the original boxes (prior to the introduction of the Happy Meal-sized nugget boxes) were of 6, 9, and 20 nuggets. According to Schur's theorem, since 6, 9, and 20 are (setwise) relatively prime, any sufficiently large integer can be expressed as a (non-negative, integer) linear combination of these three. Therefore, there exists a largest non-McNugget number, and all integers larger than it are McNugget numbers.
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Coin problem
| 0.856384
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176
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One special case of the coin problem is sometimes also referred to as the McNugget numbers. The McNuggets version of the coin problem was introduced by Henri Picciotto, who placed it as a puzzle in Games Magazine in 1987, and included it in his algebra textbook co-authored with Anita Wah. Picciotto thought of the application in the 1980s while dining with his son at McDonald's, working out the problem on a napkin. A McNugget number is the total number of McDonald's Chicken McNuggets in any number of boxes.
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Coin problem
| 0.856384
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177
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Reasons for this may be that silicon is less versatile than carbon in forming compounds, that the compounds formed by silicon are unstable, and that it blocks the flow of heat.Even so, biogenic silica is used by some Earth life, such as the silicate skeletal structure of diatoms. According to the clay hypothesis of A. G. Cairns-Smith, silicate minerals in water played a crucial role in abiogenesis: they replicated their crystal structures, interacted with carbon compounds, and were the precursors of carbon-based life.Although not observed in nature, carbon–silicon bonds have been added to biochemistry by using directed evolution (artificial selection). A heme containing cytochrome c protein from Rhodothermus marinus has been engineered using directed evolution to catalyze the formation of new carbon–silicon bonds between hydrosilanes and diazo compounds.Silicon compounds may possibly be biologically useful under temperatures or pressures different from the surface of a terrestrial planet, either in conjunction with or in a role less directly analogous to carbon. Polysilanols, the silicon compounds corresponding to sugars, are soluble in liquid nitrogen, suggesting that they could play a role in very-low-temperature biochemistry.
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Hypothetical types of biochemistry
| 0.856269
|
178
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This may suggest a greater variety of complex carbon compounds throughout the cosmos, providing less of a foundation on which to build silicon-based biologies, at least under the conditions prevalent on the surface of planets. Also, even though Earth and other terrestrial planets are exceptionally silicon-rich and carbon-poor (the relative abundance of silicon to carbon in Earth's crust is roughly 925:1), terrestrial life is carbon-based. The fact that carbon is used instead of silicon may be evidence that silicon is poorly suited for biochemistry on Earth-like planets.
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Hypothetical types of biochemistry
| 0.856269
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179
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Silicon, on the other hand, interacts with very few other types of atoms. Moreover, where it does interact with other atoms, silicon creates molecules that have been described as "monotonous compared with the combinatorial universe of organic macromolecules". This is because silicon atoms are much bigger, having a larger mass and atomic radius, and so have difficulty forming double bonds (the double-bonded carbon is part of the carbonyl group, a fundamental motif of carbon-based bio-organic chemistry).
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Hypothetical types of biochemistry
| 0.856269
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180
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Silicon dioxide, also known as silica and quartz, is very abundant in the universe and has a large temperature range where it is liquid. However, its melting point is 1,600 to 1,725 °C (2,912 to 3,137 °F), so it would be impossible to make organic compounds in that temperature, because all of them would decompose. Silicates are similar to silicon dioxide and some have lower melting points than silica. Feinberg and Shapiro have suggested that molten silicate rock could serve as a liquid medium for organisms with a chemistry based on silicon, oxygen, and other elements such as aluminium.
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Hypothetical types of biochemistry
| 0.856269
|
181
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An algorithm is said to run in sub-linear time (often spelled sublinear time) if T ( n ) = o ( n ) {\displaystyle T(n)=o(n)} . In particular this includes algorithms with the time complexities defined above. The specific term sublinear time algorithm commonly refers to randomized algorithms that sample a small fraction of their inputs and process them efficiently to approximately infer properties of the entire instance. This type of sublinear time algorithm is closely related to property testing and Statistics.
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Polynomial time
| 0.85612
|
182
|
Whatever name is applied, it deals with the ways in which plants respond to their environment and so overlaps with the field of ecology. Environmental physiologists examine plant response to physical factors such as radiation (including light and ultraviolet radiation), temperature, fire, and wind.
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Plant Physiology
| 0.856078
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183
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The ripening of fruit and loss of leaves in the winter are controlled in part by the production of the gas ethylene by the plant. Finally, plant physiology includes the study of plant response to environmental conditions and their variation, a field known as environmental physiology. Stress from water loss, changes in air chemistry, or crowding by other plants can lead to changes in the way a plant functions. These changes may be affected by genetic, chemical, and physical factors.
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Plant Physiology
| 0.856078
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184
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Major subdisciplines of plant physiology include phytochemistry (the study of the biochemistry of plants) and phytopathology (the study of disease in plants). The scope of plant physiology as a discipline may be divided into several major areas of research. First, the study of phytochemistry (plant chemistry) is included within the domain of plant physiology.
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Plant Physiology
| 0.856078
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185
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Plant physiology is a subdiscipline of botany concerned with the functioning, or physiology, of plants. Closely related fields include plant morphology (structure of plants), plant ecology (interactions with the environment), phytochemistry (biochemistry of plants), cell biology, genetics, biophysics and molecular biology. Fundamental processes such as photosynthesis, respiration, plant nutrition, plant hormone functions, tropisms, nastic movements, photoperiodism, photomorphogenesis, circadian rhythms, environmental stress physiology, seed germination, dormancy and stomata function and transpiration, both parts of plant water relations, are studied by plant physiologists.
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Plant Physiology
| 0.856078
|
186
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Paradoxically, the subdiscipline of environmental physiology is on the one hand a recent field of study in plant ecology and on the other hand one of the oldest. Environmental physiology is the preferred name of the subdiscipline among plant physiologists, but it goes by a number of other names in the applied sciences. It is roughly synonymous with ecophysiology, crop ecology, horticulture and agronomy. The particular name applied to the subdiscipline is specific to the viewpoint and goals of research.
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Plant Physiology
| 0.856078
|
187
|
Baudrit, C., and D. Dubois (2006). Practical representations of incomplete probabilistic knowledge. Computational Statistics & Data Analysis 51: 86–108. Baudrit, C., D. Dubois, D. Guyonnet (2006).
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Probability box
| 0.855952
|
188
|
There may be uncertainty about the shape of a probability distribution because the sample size of the empirical data characterizing it is small. Several methods in traditional statistics have been proposed to account for this sampling uncertainty about the distribution shape, including Kolmogorov–Smirnov and similar confidence bands, which are distribution-free in the sense that they make no assumption about the shape of the underlying distribution. There are related confidence-band methods that do make assumptions about the shape or family of the underlying distribution, which can often result in tighter confidence bands. Constructing confidence bands requires one to select the probability defining the confidence level, which usually must be less than 100% for the result to be non-vacuous.
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Probability box
| 0.855952
|
189
|
P-boxes and probability bounds analysis have been used in many applications spanning many disciplines in engineering and environmental science, including: Engineering design Expert elicitation Analysis of species sensitivity distributions Sensitivity analysis in aerospace engineering of the buckling load of the frontskirt of the Ariane 5 launcher ODE models of chemical reactor dynamics Pharmacokinetic variability of inhaled VOCs Groundwater modeling Bounding failure probability for series systems Heavy metal contamination in soil at an ironworks brownfield Uncertainty propagation for salinity risk models Power supply system safety assessment Contaminated land risk assessment Engineered systems for drinking water treatment Computing soil screening levels Human health and ecological risk analysis by the U.S. EPA of PCB contamination at the Housatonic River Superfund site Environmental assessment for the Calcasieu Estuary Superfund site Aerospace engineering for supersonic nozzle thrust Verification and validation in scientific computation for engineering problems Toxicity to small mammals of environmental mercury contamination Modeling travel time of pollution in groundwater Reliability analysis Endangered species assessment for reintroduction of Leadbeater's possum Exposure of insectivorous birds to an agricultural pesticide Climate change projections Waiting time in queuing systems Extinction risk analysis for spotted owl on the Olympic Peninsula Biosecurity against introduction of invasive species or agricultural pests Finite-element structural analysis Cost estimates Nuclear stockpile certification Fracking risks to water pollution
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Probability box
| 0.855952
|
190
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Many algorithms where quantum speedups occur in quantum computing are instances of the hidden subgroup problem. The following list outlines important instances of the HSP, and whether or not they are solvable.
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Hidden subgroup problem
| 0.855947
|
191
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The hidden subgroup problem is especially important in the theory of quantum computing for the following reasons. Shor's quantum algorithm for factoring and discrete logarithm (as well as several of its extensions) relies on the ability of quantum computers to solve the HSP for finite Abelian groups. The existence of efficient quantum algorithms for HSPs for certain non-Abelian groups would imply efficient quantum algorithms for two major problems: the graph isomorphism problem and certain shortest vector problems (SVPs) in lattices. More precisely, an efficient quantum algorithm for the HSP for the symmetric group would give a quantum algorithm for the graph isomorphism. An efficient quantum algorithm for the HSP for the dihedral group would give a quantum algorithm for the poly ( n ) {\displaystyle \operatorname {poly} (n)} unique SVP.
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Hidden subgroup problem
| 0.855947
|
192
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The hidden subgroup problem (HSP) is a topic of research in mathematics and theoretical computer science. The framework captures problems such as factoring, discrete logarithm, graph isomorphism, and the shortest vector problem. This makes it especially important in the theory of quantum computing because Shor's quantum algorithm for factoring is an instance of the hidden subgroup problem for finite Abelian groups, while the other problems correspond to finite groups that are not Abelian.
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Hidden subgroup problem
| 0.855947
|
193
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Thus, |X| ≤ |F| and |F| ≤ |X|, so, by the Schröder–Bernstein theorem, |F| = |X|. This means precisely that there is a bijection j between X and F. Finally, for x, y ∈ X define x • y = j−1(j(x) Δ j(y)). This turns (X, •) into a group. Hence every set admits a group structure.
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Group structure and the axiom of choice
| 0.855938
|
194
|
Any nonempty finite set has a group structure as a cyclic group generated by any element. Under the assumption of the axiom of choice, every infinite set X is equipotent with a unique cardinal number |X| which equals an aleph. Using the axiom of choice, one can show that for any family S of sets |⋃S| ≤ |S| × sup { |s|: s ∈ S} (A). Moreover, by Tarski's theorem on choice, another equivalent of the axiom of choice, |X|n = |X| for all finite n (B).
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Group structure and the axiom of choice
| 0.855938
|
195
|
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N {\displaystyle N} of the group G {\displaystyle G} is normal in G {\displaystyle G} if and only if g n g − 1 ∈ N {\displaystyle gng^{-1}\in N} for all g ∈ G {\displaystyle g\in G} and n ∈ N . {\displaystyle n\in N.} The usual notation for this relation is N ◃ G .
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Normal subgroup
| 0.855827
|
196
|
{\displaystyle \sigma ^{2}>0.\,} Then the sequence of random variables Z n = ∑ i = 1 n ( X i − μ ) σ n {\displaystyle Z_{n}={\frac {\sum _{i=1}^{n}(X_{i}-\mu )}{\sigma {\sqrt {n}}}}\,} converges in distribution to a standard normal random variable. For some classes of random variables, the classic central limit theorem works rather fast, as illustrated in the Berry–Esseen theorem. For example, the distributions with finite first, second, and third moment from the exponential family; on the other hand, for some random variables of the heavy tail and fat tail variety, it works very slowly or may not work at all: in such cases one may use the Generalized Central Limit Theorem (GCLT).
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Mathematical probability
| 0.855788
|
197
|
The central limit theorem (CLT) explains the ubiquitous occurrence of the normal distribution in nature, and this theorem, according to David Williams, "is one of the great results of mathematics. "The theorem states that the average of many independent and identically distributed random variables with finite variance tends towards a normal distribution irrespective of the distribution followed by the original random variables. Formally, let X 1 , X 2 , … {\displaystyle X_{1},X_{2},\dots \,} be independent random variables with mean μ {\displaystyle \mu } and variance σ 2 > 0.
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Mathematical probability
| 0.855788
|
198
|
Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, and more.
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Mathematical probability
| 0.855788
|
199
|
In probability theory, there are several notions of convergence for random variables. They are listed below in the order of strength, i.e., any subsequent notion of convergence in the list implies convergence according to all of the preceding notions. Weak convergence A sequence of random variables X 1 , X 2 , … , {\displaystyle X_{1},X_{2},\dots ,\,} converges weakly to the random variable X {\displaystyle X\,} if their respective CDF F 1 , F 2 , … {\displaystyle F_{1},F_{2},\dots \,} converge to the CDF F {\displaystyle F\,} of X {\displaystyle X\,} , wherever F {\displaystyle F\,} is continuous. Weak convergence is also called convergence in distribution.Most common shorthand notation: X n → D X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {\mathcal {D}}}\,X} Convergence in probability The sequence of random variables X 1 , X 2 , … {\displaystyle X_{1},X_{2},\dots \,} is said to converge towards the random variable X {\displaystyle X\,} in probability if lim n → ∞ P ( | X n − X | ≥ ε ) = 0 {\displaystyle \lim _{n\rightarrow \infty }P\left(\left|X_{n}-X\right|\geq \varepsilon \right)=0} for every ε > 0.Most common shorthand notation: X n → P X {\displaystyle \displaystyle X_{n}\,{\xrightarrow {P}}\,X} Strong convergence The sequence of random variables X 1 , X 2 , … {\displaystyle X_{1},X_{2},\dots \,} is said to converge towards the random variable X {\displaystyle X\,} strongly if P ( lim n → ∞ X n = X ) = 1 {\displaystyle P(\lim _{n\rightarrow \infty }X_{n}=X)=1} .
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Mathematical probability
| 0.855788
|
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