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With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of group theory may be used when studying rings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. In general there is a balance between the amount of generality and the richness of the theory: more general structures have usually fewer nontrivial theorems and fewer applications. Examples of algebraic structures with a single binary operation are: Magma Quasigroup Monoid Semigroup GroupExamples involving several operations include:
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Abstract Algebra
| 0.864809
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101
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Sets with one or more operations that obey specific laws are called algebraic structures. When a new problem involves the same laws as such an algebraic structure, all the results that have been proved using only the laws of the structure can be directly applied to the new problem. In full generality, algebraic structures may involve an arbitrary collection of operations, including operations that combine more than two elements (higher arity operations) and operations that take only one argument (unary operations) or even zero arguments (nullary operations). The examples listed below are by no means a complete list, but include the most common structures taught in undergraduate courses.
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Algebraic system
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102
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In computer science, best, worst, and average cases of a given algorithm express what the resource usage is at least, at most and on average, respectively. Usually the resource being considered is running time, i.e. time complexity, but could also be memory or some other resource. Best case is the function which performs the minimum number of steps on input data of n elements. Worst case is the function which performs the maximum number of steps on input data of size n. Average case is the function which performs an average number of steps on input data of n elements.
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Average case analysis
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103
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In general, in the presence of both a charge density ρ(r, t) and current density J(r, t), there will be both an electric and a magnetic field, and both will vary in time. They are determined by Maxwell's equations, a set of differential equations which directly relate E and B to the electric charge density (charge per unit volume) ρ and current density (electric current per unit area) J.Alternatively, one can describe the system in terms of its scalar and vector potentials V and A. A set of integral equations known as retarded potentials allow one to calculate V and A from ρ and J, and from there the electric and magnetic fields are determined via the relations
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Field equations
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104
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Approximately 6,000 students take this first exam, which consists of 25 multiple choice questions to be solved in 75 minutes, focusing on algebra-based mechanics. In the past, a quarter point was deducted for each incorrect answer. From 2015 onwards, no points were deducted for incorrect answers. Prior to 2018, the exam was offered over multiple weeks at the discretion of the exam centers. From 2018 to 2023, the exam was changed to two single-day events with two different exams, F=ma A and F=ma B, to increase exam security. As of 2023, only one F=ma exam is given.
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United States Physics Olympiad
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105
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Of all SAT subject tests, the Biology E/M test was the only SAT II that allowed the test taker a choice between the ecological or molecular tests. A set of 60 questions was taken by all test takers for Biology and a choice of 20 questions was allowed between either the E or M tests. This test was graded on a scale between 200 and 800.
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SAT Subject Test in Biology E/M
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106
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The SAT Subject Test in Mathematics Level 1 (formerly known as Math I or MathIC (the "C" representing the use of a calculator)) was the name of a one-hour multiple choice test given on algebra, geometry, basic trigonometry, algebraic functions, elementary statistics and basic foundations of calculus by The College Board. A student chose whether to take the test depending upon college entrance requirements for the schools in which the student is planning to apply. Until 1994, the SAT Subject Tests were known as Achievement Tests; and from 1995 until January 2005, they were known as SAT IIs. Mathematics Level 1 was taken 109,048 times in 2006.
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SAT Subject Test in Mathematics Level 1
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107
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Also, the Parallel Computing Center (Khmelnytskyi, Ukraine) obtained an ensemble of only 5 convolutional neural networks which performs on MNIST at 0.21 percent error rate. Some images in the testing dataset are barely readable and may prevent reaching test error rates of 0%. In 2018, researchers from Department of System and Information Engineering, University of Virginia announced 0.18% error with simultaneous stacked three kind of neural networks (fully connected, recurrent and convolution neural networks).
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MNIST database
| 0.864005
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108
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Clearly, linear time is required for the described transformation of numbers into points and then extracting their sorted order. Therefore, in the general case the convex hull of n points cannot be computed more quickly than sorting. The standard Ω(n log n) lower bound for sorting is proven in the decision tree model of computing, in which only numerical comparisons but not arithmetic operations can be performed; however, in this model, convex hulls cannot be computed at all. Sorting also requires Ω(n log n) time in the algebraic decision tree model of computation, a model that is more suitable for convex hulls, and in this model convex hulls also require Ω(n log n) time. However, in models of computer arithmetic that allow numbers to be sorted more quickly than O(n log n) time, for instance by using integer sorting algorithms, planar convex hulls can also be computed more quickly: the Graham scan algorithm for convex hulls consists of a single sorting step followed by a linear amount of additional work.
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Convex hull algorithms
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109
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Data structure Data type Abstract data type Algebraic data type Composite type Array Associative array Deque List Linked list Queue Priority queue Skip list Stack Tree data structure Automatic garbage collection
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Outline of combinatorics
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110
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: 237–238 An object that can be electrically charged exhibits self capacitance, for which the electric potential is measured between the object and ground. Mutual capacitance is measured between two components, and is particularly important in the operation of the capacitor, an elementary linear electronic component designed to add capacitance to an electric circuit. The capacitance between two conductors is a function only of the geometry; the opposing surface area of the conductors and the distance between them, and the permittivity of any dielectric material between them.
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Stray capacitance
| 0.863911
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111
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Solutions Manual to Accompany Electricity and Magnetism: Berkeley Physics Course, Volume 2, Second Edition. McGraw-Hill. Purcell, Edward M.; Morin, David J.
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Electricity and Magnetism (book)
| 0.863694
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112
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(A) represents transport processes include the transport into the cell from the surrounding seawater of primary calcification substrates Ca2+ and HCO3− (black arrows) and the removal of the end product H+ from the cell (gray arrow). The transport of Ca2+ through the cytoplasm to the coccolith vesicle (CV) is the dominant cost associated with calcification. (B) represents metabolic processes include the synthesis of coccolith-associated polysaccharides (CAPs – gray rectangles) by the Golgi complex (white rectangles) that regulate the nucleation and geometry of CaCO3 crystals. The completed coccolith (gray plate) is a complex structure of intricately arranged CAPs and CaCO3 crystals. (C) Mechanical and structural processes account for the secretion of the completed coccoliths that are transported from their original position adjacent to the nucleus to the cell periphery, where they are transferred to the surface of the cell.
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Marine protozoans
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113
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DNA sequencing may be used to determine the sequence of individual genes, larger genetic regions (i.e. clusters of genes or operons), full chromosomes, or entire genomes of any organism. DNA sequencing is also the most efficient way to indirectly sequence RNA or proteins (via their open reading frames). In fact, DNA sequencing has become a key technology in many areas of biology and other sciences such as medicine, forensics, and anthropology.
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Next Generation Sequencing
| 0.863605
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114
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In computer science, hybrid algorithms are very common in optimized real-world implementations of recursive algorithms, particularly implementations of divide-and-conquer or decrease-and-conquer algorithms, where the size of the data decreases as one moves deeper in the recursion. In this case, one algorithm is used for the overall approach (on large data), but deep in the recursion, it switches to a different algorithm, which is more efficient on small data. A common example is in sorting algorithms, where the insertion sort, which is inefficient on large data, but very efficient on small data (say, five to ten elements), is used as the final step, after primarily applying another algorithm, such as merge sort or quicksort. Merge sort and quicksort are asymptotically optimal on large data, but the overhead becomes significant if applying them to small data, hence the use of a different algorithm at the end of the recursion.
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Hybrid algorithm
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115
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In materials science and engineering, one is often interested in understanding the forces or stresses involved in the deformation of a material. For instance, if the material were a simple spring, the answer would be given by Hooke's law, which says that the force experienced by a spring is proportional to the distance displaced from equilibrium. Stresses which can be attributed to the deformation of a material from some rest state are called elastic stresses. In other materials, stresses are present which can be attributed to the deformation rate over time.
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Viscous forces
| 0.86349
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116
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In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a: System of linear equations, System of nonlinear equations, System of bilinear equations, System of polynomial equations, System of differential equations, or a System of difference equations
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Simultaneous equation
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117
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quadratic function In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant: f ( x , y , z ) = a x 2 + b y 2 + c z 2 + d x y + e x z + f y z + g x + h y + i z + j , {\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,} with at least one of the coefficients a, b, c, d, e, or f of the second-degree terms being non-zero. A univariate (single-variable) quadratic function has the form f ( x ) = a x 2 + b x + c , a ≠ 0 {\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0} in the single variable x. The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the y-axis, as shown at right. If the quadratic function is set equal to zero, then the result is a quadratic equation.
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Glossary of calculus
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118
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x∗~x=e=~x∗x.To summarize, the usual definition has: a single binary operation (signature (2)) 1 equational law (associativity) 2 quantified laws (identity and inverse)while the universal algebra definition has: 3 operations: one binary, one unary, and one nullary (signature (2,1,0)) 3 equational laws (associativity, identity, and inverse) no quantified laws (except outermost universal quantifiers, which are allowed in varieties)A key point is that the extra operations do not add information, but follow uniquely from the usual definition of a group. Although the usual definition did not uniquely specify the identity element e, an easy exercise shows it is unique, as is each inverse element. The universal algebra point of view is well adapted to category theory.
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Equational theory
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119
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Even though comparison-sorting n items requires Ω(n log n) operations, selection algorithms can compute the kth-smallest of n items with only Θ(n) operations. This includes the median, which is the n/2th order statistic (or for an even number of samples, the arithmetic mean of the two middle order statistics).Selection algorithms still have the downside of requiring Ω(n) memory, that is, they need to have the full sample (or a linear-sized portion of it) in memory. Because this, as well as the linear time requirement, can be prohibitive, several estimation procedures for the median have been developed. A simple one is the median of three rule, which estimates the median as the median of a three-element subsample; this is commonly used as a subroutine in the quicksort sorting algorithm, which uses an estimate of its input's median. A more robust estimator is Tukey's ninther, which is the median of three rule applied with limited recursion: if A is the sample laid out as an array, and med3(A) = median(A, A, A),then ninther(A) = med3(med3(A), med3(A), med3(A))The remedian is an estimator for the median that requires linear time but sub-linear memory, operating in a single pass over the sample.
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Multivariate median
| 0.863001
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120
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As of 2020, the quantitative reasoning section contains 20 questions covering various areas of mathematics, such as geometry, algebra, percentages, averages, ratio questions, drawing conclusions from a diagram, and so on. The allotted time is 20 minutes. For the most part, the difficulty level of questions in the section increases – as the more difficult questions appear last. The mathematical knowledge required for the quantitative reasoning section is similar to that required for the high school matriculation examination in mathematics at the three-unit level – the minimum level required for obtaining a high school matriculation certificate and for admission to academic studies.
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Psychometric Entrance Test
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121
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The multiple-choice section is scored immediately after the exam by computer. One point is awarded for each correct answer, no points are credited or deducted for unanswered questions, and points are no longer deducted for having an incorrect answer.Students' answers to the free-response section are reviewed in early June by readers that include high school and college statistics teachers gathered in a designated location. The readers use a pre-made rubric to assess the answers and normally grade only one question in a given exam. Each question is graded on a scale from 0 to 4, with a 4 representing the most complete response. Communication and clarity in the answers receive a lot of emphasis in the grading.Both sections are weighted equally when the composite score is calculated. The composite score is reported on a scale from 1 to 5, with a score of 5 being the highest possible.
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AP Statistics
| 0.862753
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122
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(B) Metabolic processes include the synthesis of CAPs (gray rectangles) by the Golgi complex (white rectangles) that regulate the nucleation and geometry of CaCO3 crystals. The completed coccolith (gray plate) is a complex structure of intricately arranged CAPs and CaCO3 crystals. (C) Mechanical and structural processes account for the secretion of the completed coccoliths that are transported from their original position adjacent to the nucleus to the cell periphery, where they are transferred to the surface of the cell.
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Coccolithophore
| 0.862648
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123
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While for substantial resources Illumina NextSeq, NovaSeq, PacBio Sequel, Oxford Nanopore and PromethION are preferred. Moreover, for pathogen sequencing the use of controls is of fundamental importance ensuring mNGS assay quality and stability over time; PhiX is used as sequencing control, then the other controls include the positive control, an additional internal control (e.g., spiked DNA or other known pathogen) and a negative control (usually water sample). Bioinformatic analysis: Whereas the sequencing itself has been made widely accessible and more user friendly, the data analysis and interpretation that follows still requires specialized bioinformatics expertise and appropriate computational resources.
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Clinical metagenomic next-generation sequencing
| 0.862627
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124
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The Individual Round comprises the Algebra, Geometry, and Combinatorics exams for February, and the General and Theme exams for November. Each of the exams is 50 minutes in length and contains 10 questions. The exams are short-answer, meaning that the answers can be any real number or even an algebraic expression. Before 2012, competitors had the option to choose between a General exam or two exams in Algebra, Geometry, Combinatorics, or Calculus for the February tournament.
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HMMT
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125
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The areas of study in Units 3 and 4 extend content from Mathematical Methods Units 3 and 4 to include rational and other quotient functions as well as other advanced mathematics topics such as logic and proof, complex numbers, vectors, differential equations, kinematics, and statistical inference. They also provide background for advanced studies in mathematics and other STEM fields. Study of Specialist Mathematics Units 3 and 4 assumes concurrent study or previous completion of Mathematical Methods.
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Mathematics education in Australia
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126
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The number of comparisons needed to solve the problem of size n {\displaystyle n} , in a comparison-based model of computation such as a decision tree or algebraic decision tree, is Θ ( n log n ) {\displaystyle \Theta (n\log n)} . Here, Θ {\displaystyle \Theta } invokes big theta notation, meaning that the problem can be solved in a number of comparisons proportional to n log n {\displaystyle n\log n} (a linearithmic function) and that all solutions require this many comparisons. In these models of computation, the input numbers may not be used to index the computer's memory (as in the hash table solution) but may only be accessed by computing and comparing simple algebraic functions of their values. For these models, an algorithm based on comparison sort solves the problem within a constant factor of the best possible number of comparisons. The same lower bound applies as well to the expected number of comparisons in the randomized algebraic decision tree model.
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Element uniqueness problem
| 0.862361
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127
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This section tests the ability to apply scientific knowledge typically covered in school Science and Mathematics by the age of 16 (for example, GCSE in the UK and IGCSE internationally). It is made up of 27 questions, with 30 minutes to complete. The scope of scientific knowledge include that of Mathematics, Physics, Chemistry and Biology.
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BioMedical Admissions Test
| 0.86236
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128
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In computer science, a sorting algorithm is an algorithm that puts elements of a list into an order. The most frequently used orders are numerical order and lexicographical order, and either ascending or descending. Efficient sorting is important for optimizing the efficiency of other algorithms (such as search and merge algorithms) that require input data to be in sorted lists. Sorting is also often useful for canonicalizing data and for producing human-readable output. Formally, the output of any sorting algorithm must satisfy two conditions: The output is in monotonic order (each element is no smaller/larger than the previous element, according to the required order). The output is a permutation (a reordering, yet retaining all of the original elements) of the input.For optimum efficiency, the input data should be stored in a data structure which allows random access rather than one that allows only sequential access.
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Sorting Algorithm
| 0.862297
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129
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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.
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Parabola
| 0.862249
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130
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The content within the book is written using a question and answer format. It contains some 250 questions, which The Science Teacher states each are answered with a "concise and well-formulated essay that is informative and readable." The Science Teacher review goes on to state that many of the answers given in the book are "little gems of science writing". The Science Teacher summarizes by stating that each question is likely to be thought of by a student, and that "the answers are informative, well constructed, and thorough".The book covers information about the planets, the Earth, the Universe, practical astronomy, history, and awkward questions such as astronomy in the Bible, UFOs, and aliens. Also covered are subjects such as the Big Bang, comprehension of large numbers, and the Moon illusion.
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A Question and Answer Guide to Astronomy
| 0.862228
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131
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(Some authors also use the "closure" axiom that x ∗ y belongs to A whenever x and y do, but here this is already implied by calling ∗ a binary operation.) This definition of a group does not immediately fit the point of view of universal algebra, because the axioms of the identity element and inversion are not stated purely in terms of equational laws which hold universally "for all ..." elements, but also involve the existential quantifier "there exists ...". The group axioms can be phrased as universally quantified equations by specifying, in addition to the binary operation ∗, a nullary operation e and a unary operation ~, with ~x usually written as x−1.
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Equational theory
| 0.862062
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132
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Parametrization of planar curves is introduced, mainly focusing on lines, circles and parabolas. The plotting of cubic equations and solution of specific cases through polynomial long division and the remainder theorem enable a deeper understanding of polynomials. Mathematics Extension 2 (Must be studied concurrently with Mathematics Advanced and Mathematics Extension 1): A highly advanced mathematics course containing an introduction to complex numbers, advanced calculus, motion, and further work with vectors.
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Mathematics education in Australia
| 0.862041
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133
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Approximately the top 400 students from the F=ma exam are invited to take a free-response, calculus-based exam covering all topics in introductory physics, including mechanics, electricity and magnetism, thermodynamics, fluids, relativity, waves, and nuclear and atomic physics. There are two parts in the exam, each allotted 90 minutes, and 6 problems in total. Prior to 2017, the exam could be taken at any time during a two-week window in March.
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United States Physics Olympiad
| 0.862012
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134
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These expression levels may signify that the protein is performing a different function than previously known.The structure of a protein can also help determine its functions. Protein structure in turn may be elucidated with various techniques including X-ray crystallography or NMR. Dual-polarization interferometry may be used to measure changes in protein structure which may also give hints to the protein's function. Finally, application of systems biology approaches such as interactomics give clues to a proteins function based on what it interacts with.
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Protein moonlighting
| 0.861973
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135
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Many number systems, such as the integers and the rationals, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups.
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Elementary group theory
| 0.861818
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136
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DNA sequencing with commercially available NGS platforms is generally conducted with the following steps. First, DNA sequencing libraries are generated by clonal amplification by PCR in vitro. Second, the DNA is sequenced by synthesis, such that the DNA sequence is determined by the addition of nucleotides to the complementary strand rather than through chain-termination chemistry.
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Short-read sequencing
| 0.861756
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137
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The DAT comprises four sections: survey of the natural sciences (90 minutes), perceptual ability (often called the PAT, 60 minutes), reading comprehension (60 minutes), and quantitative reasoning (45 minutes). The mathematics of the quantitative exam is similar to that of the SAT. The first section is divided into questions about biology (40 questions), general chemistry (30 questions), and organic chemistry (30 questions). The second section is divided into six different problem sets designed to test perceptual ability, specifically in the areas of three-dimensional manipulation and spatial reasoning.
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Dental Admission Test
| 0.861709
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138
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There are several possible ways to specify the position of a line in the plane. A simple way is by the pair (m, b) where the equation of the line is y = mx + b. Here m is the slope and b is the y-intercept. This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler algebraically to use coordinates (l, m) where the equation of the line is lx + my + 1 = 0.
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Line geometry
| 0.861687
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139
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When a group G {\displaystyle G} has a normal subgroup N {\displaystyle N} other than { 1 } {\displaystyle \{1\}} and G {\displaystyle G} itself, questions about G {\displaystyle G} can sometimes be reduced to questions about N {\displaystyle N} and G / N {\displaystyle G/N} . A nontrivial group is called simple if it has no such normal subgroup. Finite simple groups are to finite groups as prime numbers are to positive integers: they serve as building blocks, in a sense made precise by the Jordan–Hölder theorem.
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Elementary group theory
| 0.861678
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140
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If all elements to be sorted are distinct, the expected number of comparisons performed in the average case by randomized bogosort is asymptotically equivalent to (e − 1)n!, and the expected number of swaps in the average case equals (n − 1)n!. The expected number of swaps grows faster than the expected number of comparisons, because if the elements are not in order, this will usually be discovered after only a few comparisons, no matter how many elements there are; but the work of shuffling the collection is proportional to its size. In the worst case, the number of comparisons and swaps are both unbounded, for the same reason that a tossed coin might turn up heads any number of times in a row. The best case occurs if the list as given is already sorted; in this case the expected number of comparisons is n − 1, and no swaps at all are carried out.For any collection of fixed size, the expected running time of the algorithm is finite for much the same reason that the infinite monkey theorem holds: there is some probability of getting the right permutation, so given an unbounded number of tries it will almost surely eventually be chosen.
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Bogosort
| 0.861635
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141
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Transfer of heat between a chemical system and its surroundings during change of phase or chemical reaction taking place called thermochemistry Study of colligative properties of number of species present in solution. Number of phases, number of components and degree of freedom (or variance) can be correlated with one another with help of phase rule.
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Physical Chemistry
| 0.861627
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142
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The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count the number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations.
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Monte-Carlo method
| 0.861578
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143
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Merge the sorted results A + A {\displaystyle A+A} , B + B {\displaystyle B+B} , and A + B {\displaystyle A+B} together.The number of comparisons C ( n ) {\displaystyle C(n)} needed to perform this recursive algorithm on an input of n {\displaystyle n} items can be analyzed using the recurrence relation where the 2 C ( n / 2 ) {\displaystyle 2C(n/2)} term of the recurrence counts the number of comparisons in the recursive calls to the algorithm to sort A + A {\displaystyle A+A} and B + B {\displaystyle B+B} , and the O ( n 2 ) {\displaystyle O(n^{2})} term counts the number of comparisons used to merge the results. The master theorem for recurrence relations of this form shows that C ( n ) = O ( n 2 ) . {\displaystyle C(n)=O(n^{2}).} The total time complexity is slower, O ( n 2 log n ) {\displaystyle O(n^{2}\log n)} , because of the steps of the algorithm that use already-made comparisons to infer orderings of other sets. These steps can be performed in time O ( n 2 log n ) {\displaystyle O(n^{2}\log n)} by using a standard comparison-sorting algorithm with its comparison steps replaced by the stated inferences.If only comparisons between elements of X + Y {\displaystyle X+Y} are allowed, then there is also a matching lower bound of Ω ( n 2 ) {\displaystyle \Omega (n^{2})} on the number of comparisons, but with more general comparisons involving linear combinations of constant numbers of elements, only O ( n log 2 n ) {\displaystyle O(n\log ^{2}n)} comparisons are needed.
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X + Y sorting
| 0.861534
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144
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StructureGroup (mathematics) Lagrange's theorem (group theory) Subgroup Coset Normal subgroup Characteristic subgroup Centralizer and normalizer subgroups Derived group Frattini subgroup Fitting subgroup Classification of finite simple groups Sylow theorems Local analysisConstructionsFree group Presentation of a group Word problem for groups Quotient group Extension problem Direct sum, direct product Semidirect product Wreath productTypesSimple group Finite group Abelian group Torsion subgroup Free abelian group Finitely generated abelian group Rank of an abelian group Cyclic group Locally cyclic group Solvable group Composition series Nilpotent group Divisible group Dedekind group, Hamiltonian groupExamplesExamples of groups Trivial group Additive group Permutation group Symmetric group Alternating group p-group List of small groups Klein four-group Quaternion group Dihedral group Dicyclic group Automorphism group Point group Circle group Linear group Orthogonal groupApplicationsGroup action Conjugacy class Inner automorphism Conjugate closure Stabilizer subgroup Orbit (group theory) Orbit-stabilizer theorem Cayley's theorem Burnside's lemma Burnside's problem Loop group Fundamental group
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List of abstract algebra topics
| 0.861526
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145
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Cost-efficiency as well as absolute speed can be critical, especially in cluster environments where lower node costs allow purchasing more nodes. Increasing software speed Some Sort Benchmark entrants use a variation on radix sort for the first phase of sorting: they separate data into one of many "bins" based on the beginning of its value. Sort Benchmark data is random and especially well-suited to this optimization. Compacting the input, intermediate files, and output can reduce time spent on I/O, but is not allowed in the Sort Benchmark. Because the Sort Benchmark sorts long (100-byte) records using short (10-byte) keys, sorting software sometimes rearranges the keys separately from the values to reduce memory I/O volume.
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External Sorting
| 0.861496
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146
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Semi-groups, quasi-groups, and monoids are algebraic structures similar to groups, but with less constraints on the operation. They comprise a set and a closed binary operation but do not necessarily satisfy the other conditions. A semi-group has an associative binary operation but might not have an identity element.
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Diagrammatic algebra
| 0.861416
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147
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As of 2018, the Illumina monopoly on high-quality next-generation sequencing reagents has meant that the sequencing reagents alone cost more than FDA-approved syndromic testing panels. Also additional direct costs of metagenomics such as extraction, library preparation, and computational analysis have to be considered. In general, metagenomic sequencing is most useful and cost efficient for pathogen discovery when at least one of the following criteria are met: the identification of the organism is not sufficient (one desires to go beyond discovery to produce data for genomic characterization), a coinfection is suspected, other simpler assays are ineffective or will take an inordinate amount of time, screening of environmental samples for previously undescribed or divergent pathogens.
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Clinical metagenomic next-generation sequencing
| 0.861413
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148
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Racket takes this further: the language is made fully safe-for-space, via live variable analysis. This complements the precise garbage collector and in some cases, like in the implementation of Lazy Racket, the two features are crucial for proper execution. This is in addition to additional compiler optimizations such as lambda lifting and just-in-time compilation.
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Racket features
| 0.861368
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149
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When the red homozygous flower is paired with the white homozygous flower, the result yields a pink snapdragon flower. The pink snapdragon is the result of incomplete dominance. A similar type of incomplete dominance is found in the four o'clock plant wherein pink color is produced when true-bred parents of white and red flowers are crossed. In quantitative genetics, where phenotypes are measured and treated numerically, if a heterozygote's phenotype is exactly between (numerically) that of the two homozygotes, the phenotype is said to exhibit no dominance at all, i.e. dominance exists only when the heterozygote's phenotype measure lies closer to one homozygote than the other. When plants of the F1 generation are self-pollinated, the phenotypic and genotypic ratio of the F2 generation will be 1:2:1 (Red:Pink:White).
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Homozygous dominant
| 0.861299
|
150
|
In the U.S., the SAT Subject Test in Mathematics Level 2 (formerly known as Math II or Math IIChest, the "C" representing chest) was a one-hour multiple choice test. The questions covered a broad range of topics. Approximately 10-14% of questions focused on numbers and operations, 48-52% focused on algebra and functions, 28-32% focused on geometry (coordinate, three-dimensional, and trigonometric geometry were covered; plane geometry was not directly tested), and 8-12% focused on data analysis, statistics and probability.
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SAT Subject Test in Mathematics Level 2
| 0.861125
|
151
|
The lengths of BM and CM are: Using the intersecting chords theorem on the chords BC and DE, we get Substituting: Rearranging: For any given cone and parabola, r and θ are constants, but x and y are variables that depend on the arbitrary height at which the horizontal cross-section BECD is made. This last equation shows the relationship between these variables. They can be interpreted as Cartesian coordinates of the points D and E, in a system in the pink plane with P as its origin. Since x is squared in the equation, the fact that D and E are on opposite sides of the y axis is unimportant. If the horizontal cross-section moves up or down, toward or away from the apex of the cone, D and E move along the parabola, always maintaining the relationship between x and y shown in the equation. The parabolic curve is therefore the locus of points where the equation is satisfied, which makes it a Cartesian graph of the quadratic function in the equation.
|
Parabolic curve
| 0.860848
|
152
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Task 1: "Answer retrieval" matching old post answers to newly posed questions, and Task 2: "Formula retrieval" matching old post formulae to new questions. Starting with the domain of mathematics, which involves formula language, the goal is to later extend the task to other domains (e.g., STEM disciplines, such as chemistry, biology, etc.), which employ other types of special notation (e.g., chemical formulae).The inverse of mathematical question answering—mathematical question generation—has also been researched. The PhysWikiQuiz physics question generation and test engine retrieves mathematical formulae from Wikidata together with semantic information about their constituting identifiers (names and values of variables).
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Answer engine
| 0.860808
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153
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Tasks tend to require the application of number sense and spatial sense; recognizing and working with mathematical relationships, patterns, and proportions expressed in verbal or numerical form; and interpretation and basic analysis of data and statistics in texts, tables and graphs.Level 4 326 – 375 Tasks at this level require the respondent to understand a broad range of mathematical information that may be complex, abstract or embedded in unfamiliar contexts. These tasks involve undertaking multiple steps and choosing relevant problem-solving strategies and processes.
|
Programme for the International Assessment of Adult Competencies
| 0.860681
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154
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In three-space, the most general equation of the second degree in x, y and z has the form where the quantities A , B , C , … , J {\displaystyle A,B,C,\ldots ,J} are positive or negative numbers or zero. The points in space satisfying such an equation all lie on a surface. Any second-degree equation which does not reduce to a cylinder, plane, line, or point corresponds to a surface which is called quadric.As in the case of plane analytic geometry, the method of translation of axes may be used to simplify second-degree equations, thereby making evident the nature of certain quadric surfaces. The principal tool in this process is "completing the square."
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Translation of axes
| 0.86067
|
155
|
Physical chemistry: free energy, phase diagram, phase rule, transport phenomena Statistical mechanics: statistical ensemble, phase space, chemical potential, Gibbs entropy, Gibbs paradox Mathematics: Vector Analysis, convex analysis, Gibbs phenomenon Electromagnetism: Maxwell's equations, birefringence
|
Josiah Willard Gibbs
| 0.860575
|
156
|
Consider a simple example drawn from physics. A spring should obey Hooke's law which states that the extension of a spring y is proportional to the force, F, applied to it. y = f ( F , k ) = k F {\displaystyle y=f(F,k)=kF\!} constitutes the model, where F is the independent variable.
|
Least-squares fit
| 0.860503
|
157
|
1 comment line: tools must account for all code and comments regardless of comment placement.Even the "logical" and "physical" SLOC values can have a large number of varying definitions. Robert E. Park (while at the Software Engineering Institute) and others developed a framework for defining SLOC values, to enable people to carefully explain and define the SLOC measure used in a project. For example, most software systems reuse code, and determining which (if any) reused code to include is important when reporting a measure.
|
Unified Code Count (UCC)
| 0.860469
|
158
|
This is the topic of the scientific field of structural biology, which employs techniques such as X-ray crystallography, NMR spectroscopy, cryo-electron microscopy (cryo-EM) and dual polarisation interferometry, to determine the structure of proteins. Protein structures range in size from tens to several thousand amino acids.
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Protein Structure
| 0.860449
|
159
|
Q1 General biology Q2 Cytology Q3 Genetics Q4 Physiology Q5 Biochemistry Q6 Biophysics Q7 Molecular biology Q8 Bioengineering Q9 Zoology and botany
|
Chinese Library Classification
| 0.860366
|
160
|
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
| 0.860366
|
161
|
In this experiment the coin was tossed by balancing it on the forefinger, flipping it using the thumb so that it spun through the air for about a foot before landing on a flat cloth spread over a table. Edwin Thompson Jaynes claimed that when a coin is caught in the hand, instead of being allowed to bounce, the physical bias in the coin is insignificant compared to the method of the toss, where with sufficient practice a coin can be made to land heads 100% of the time. Exploring the problem of checking whether a coin is fair is a well-established pedagogical tool in teaching statistics.
|
Fair coin
| 0.860275
|
162
|
Second, average-case complexity analysis provides tools and techniques to generate hard instances of problems which can be utilized in areas such as cryptography and derandomization. Third, average-case complexity allows discriminating the most efficient algorithm in practice among algorithms of equivalent best case complexity (for instance Quicksort). Average-case analysis requires a notion of an "average" input to an algorithm, which leads to the problem of devising a probability distribution over inputs.
|
Average case complexity
| 0.860177
|
163
|
The Math Test – No Calculator section has 20 questions (15 multiple choice and 5 grid-in) and lasts 25 minutes. The Math Test – Calculator section has 38 questions (30 multiple choice and 8 grid-in) and lasts 55 minutes.Several scores are provided to the test taker for the math test. A subscore (on a scale of 1 to 15) is reported for each of three categories of math content: "Heart of Algebra" (linear equations, systems of linear equations, and linear functions) "Problem Solving and Data Analysis" (statistics, modeling, and problem-solving skills) "Passport to Advanced Math" (non-linear expressions, radicals, exponentials and other topics that form the basis of more advanced math).A test score for the math test is reported on a scale of 10 to 40, with an increment of 0.5, and a section score (equal to the test score multiplied by 20) is reported on a scale of 200 to 800.
|
Scholastic aptitude test
| 0.859848
|
164
|
Examples of the need for merging include external sorting and streaming results from distributed data such as a log structured merge tree. The inner loop is obtaining the min element, replacing with the next element for the corresponding input stream, then doing a sift-down heap operation. (Alternatively the replace function.) (Using extract-max and insert functions of a priority queue are much less efficient.) Order statistics: The Heap data structure can be used to efficiently find the kth smallest (or largest) element in an array.
|
Minimum-heap property
| 0.859688
|
165
|
Experiment: Rosalind Franklin used pure DNA to perform X-ray diffraction to produce photo 51. The results showed an X-shape. Analysis: When Watson saw the detailed diffraction pattern, he immediately recognized it as a helix. He and Crick then produced their model, using this information along with the previously known information about DNA's composition, especially Chargaff's rules of base pairing.The discovery became the starting point for many further studies involving the genetic material, such as the field of molecular genetics, and it was awarded the Nobel Prize in 1962. Each step of the example is examined in more detail later in the article.
|
Experimental confirmation
| 0.859543
|
166
|
The geometry consists of two parallel plates as perfect conductors (PEC), an idealized structure, filled by two stacked planar slabs of homogeneous and isotropic materials with their respective constitutive parameters ε1, ε2, u1, u2. Each slab has thickness = d, slab 1 = d1, and slab 2 = d2. Choosing which combination of parameters to employ involves pairing DPS and DNG or ENG and MNG materials. As mentioned previously, this is one pair of oppositely-signed constitutive parameters, combined.
|
Metamaterial antennas
| 0.859532
|
167
|
Students in computer science and economics might have the option of taking algorithmic game theory. Students in the physical sciences and engineering need to understand error analysis for their laboratory sessions and courses. Advanced undergraduates and beginning graduate students in physics may take a course on advanced mathematical methods for physics.
|
Mathematics education in the United States
| 0.859521
|
168
|
The class of all semigroups forms a variety of algebras of signature (2), meaning that a semigroup has a single binary operation. A sufficient defining equation is the associative law: x ( y z ) = ( x y ) z . {\displaystyle x(yz)=(xy)z.} The class of groups forms a variety of algebras of signature (2,0,1), the three operations being respectively multiplication (binary), identity (nullary, a constant) and inversion (unary).
|
Finitary algebraic category
| 0.859424
|
169
|
The tendency of people to provide the answer 1/2 is likely due to a tendency to ignore context that may seem unimpactful. For example, how the question is posed to the warden can affect the answer. This can be shown by considering a modified case, where P ( A ) = 1 4 , P ( B ) = 1 4 , P ( C ) = 1 2 {\displaystyle P(A)={\frac {1}{4}},P(B)={\frac {1}{4}},P(C)={\frac {1}{2}}} and everything else about the problem remains the same. Using Bayes' Theorem once again: P ( A | b ) = 1 2 × 1 4 1 2 × 1 4 + 0 × 1 4 + 1 × 1 2 = 1 5 .
|
Three Prisoners problem
| 0.859411
|
170
|
In classical physics, a spring can be seen as a device that stores potential energy, specifically elastic potential energy, by straining the bonds between the atoms of an elastic material. Hooke's law of elasticity states that the extension of an elastic rod (its distended length minus its relaxed length) is linearly proportional to its tension, the force used to stretch it. Similarly, the contraction (negative extension) is proportional to the compression (negative tension). This law actually holds only approximately, and only when the deformation (extension or contraction) is small compared to the rod's overall length.
|
Ideal spring
| 0.859392
|
171
|
Proposition. Consider polynomial algebra Pol = C and matrix M with elements in some algebra EndPol. The elements y i = ∑ k M i k ⊗ x k ∈ E n d P o l ⊗ P o l {\displaystyle y_{i}=\sum _{k}M_{ik}\otimes x_{k}\in EndPol\otimes Pol} commute among themselves if and only if M is a Manin matrix.
|
Manin matrices
| 0.859297
|
172
|
The rays are focused onto an area the size of a cooking pot and can reach 4,000 °C (7,230 °F), depending on the process installed, for example: about 1,000 °C (1,830 °F) for metallic receivers producing hot air for the next generation solar towers as it will be tested at the Themis plant with the Pegase project about 1,400 °C (2,550 °F) to produce hydrogen by cracking methane molecules up to 2,500 °C (4,530 °F) to test materials for extreme environment such as nuclear reactors or space vehicle atmospheric reentry up to 3,500 °C (6,330 °F) to produce nanomaterials by solar induced sublimation and controlled cooling, such as carbon nanotubes or zinc nanoparticlesIt has been suggested that solar furnaces could be used in space to provide energy for manufacturing purposes. Their reliance on sunny weather is a limiting factor as a source of renewable energy on Earth but could be tied to thermal energy storage systems for energy production through these periods and into the night.
|
Solar furnace
| 0.859219
|
173
|
He showed that if 1/2 were correct, it would lead to a contradiction, so 1/2 cannot be correct. This simple but counterintuitive puzzle is used as a standard example in teaching probability theory. The solution illustrates some basic principles, including the Kolmogorov axioms.
|
Bertrand's box paradox
| 0.859126
|
174
|
Typically cocktail sort is less than two times faster than bubble sort. Another optimization can be that the algorithm remembers where the last actual swap has been done. In the next iteration, there will be no swaps beyond this limit and the algorithm has shorter passes. As the cocktail shaker sort goes bidirectionally, the range of possible swaps, which is the range to be tested, will reduce per pass, thus reducing the overall running time slightly.
|
Cocktail shaker sort
| 0.859115
|
175
|
Probability Theory 9. Sequences and Series 10. Topology 11. Set Theory Recently a similar competition has been started in France.
|
Miklós Schweitzer Competition
| 0.859029
|
176
|
Sudoku: How many puzzles have exactly one solution? How many puzzles with exactly one solution are minimal? What is the maximum number of givens for a minimal puzzle? Tic-tac-toe variants: Given a width of tic-tac-toe board, what is the smallest dimension such that X is guaranteed a winning strategy? (See also: Hales-Jewett theorem) What is the Turing completeness status of all unique elementary cellular automata?
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Open problem in mathematics
| 0.858933
|
177
|
In addition, several languages have been designed to run natively on the JVM, including Clojure, Groovy, and Scala. Java syntax borrows heavily from C and C++, but object-oriented features are modeled after Smalltalk and Objective-C. Java eschews certain low-level constructs such as pointers and has a very simple memory model where objects are allocated on the heap (while some implementations e.g. all currently supported by Oracle, may use escape analysis optimization to allocate on the stack instead) and all variables of object types are references. Memory management is handled through integrated automatic garbage collection performed by the JVM.
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Oracle Java
| 0.85885
|
178
|
ECAT consists of 100 multiple choice questions. Each question carries 4 marks, with 1 mark deducted for each wrong answer. Total marks are 400. 30 questions are from Mathematics, 30 from Chemistry, 30 from Physics, 10 from English language. According to the updated rules of Pakistan Engineering Council, from entry 2023 onwards, 33% marks (132 out of 400) are required to pass ECAT.
|
ECAT Pakistan
| 0.858848
|
179
|
Math League runs three contest formats: Grades 4-5: 30 multiple-choice questions to solve in 30 minutes, covering arithmetic and basic principles Grades 6-8: 35 multiple-choice questions to solve in 30 minutes, covering advanced arithmetic and basic topics in geometry and algebra Grades 9-12: Series of 6 contests. Each contest contains 6 short-answer questions to solve in 30 minutes, covering geometry, algebra, trigonometry, and other advanced pre-calculus topics.Only plain paper, pencil or pen, and a calculator without QWERTY keyboard are allowed.Students who score above 12 points in grades 4 and 5, and above 15 points in grades 6-8 are awarded a 'Certificate of Merit." Which means they win
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Math League
| 0.858842
|
180
|
For some questions, there is no known way to find an answer quickly, but if one is provided with information showing what the answer is, it is possible to verify the answer quickly. The class of questions for which an answer can be verified in polynomial time is NP, which stands for "nondeterministic polynomial time".An answer to the P versus NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If it turns out that P ≠ NP, which is widely believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be verified in polynomial time. The problem has been called the most important open problem in computer science. Aside from being an important problem in computational theory, a proof either way would have profound implications for mathematics, cryptography, algorithm research, artificial intelligence, game theory, multimedia processing, philosophy, economics and many other fields.It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute, each of which carries a US$1,000,000 prize for the first correct solution.
|
P versus NP problem
| 0.858804
|
181
|
According to Snopes.com, more recent (1999 and 1988) versions identify the problem as a question in "a physics degree exam at the University of Copenhagen" and the student was Niels Bohr, and includes the following answers: Tying a piece of string to the barometer, lowering the barometer from the roof to the ground, and measuring the length of the string and barometer. Dropping the barometer off the roof, measuring the time it takes to hit the ground, and calculating the building's height assuming constant acceleration under gravity. When the sun is shining, standing the barometer up, measuring the height of the barometer and the lengths of the shadows of both barometer and building, and finding the building's height using similar triangles. Tying a piece of string to the barometer, and swinging it like a pendulum both on the ground and on the roof, and from the known pendulum length and swing period, calculate the gravitational field for the two cases.
|
Barometer question
| 0.858701
|
182
|
In a slight abuse of notation, the word "structure" can also refer to just the operations on a structure, instead of the underlying set itself. For example, the sentence, "We have defined a ring structure on the set A {\displaystyle A} ", means that we have defined ring operations on the set A {\displaystyle A} . For another example, the group ( Z , + ) {\displaystyle (\mathbb {Z} ,+)} can be seen as a set Z {\displaystyle \mathbb {Z} } that is equipped with an algebraic structure, namely the operation + {\displaystyle +} .
|
Carrier set
| 0.8587
|
183
|
Theorem: The identity element e for a binary operation (addition or multiplication) of a ring is unique.
|
Proofs of elementary ring properties
| 0.858696
|
184
|
They might also depend on the way in which the data is arranged; for example, some sorting algorithms perform poorly on data which is already sorted, or which is sorted in reverse order. In practice, there are other factors which can affect the efficiency of an algorithm, such as requirements for accuracy and/or reliability. As detailed below, the way in which an algorithm is implemented can also have a significant effect on actual efficiency, though many aspects of this relate to optimization issues.
|
Computationally efficient
| 0.858693
|
185
|
They attribute this to the relatively small amount of research put into patience sort, and develop several optimizations that bring its performance to within a factor of two of that of quicksort.If values of cards are in the range 1, . . . , n, there is an efficient implementation with O ( n log n ) {\displaystyle O(n\log n)} worst-case running time for putting the cards into piles, relying on a Van Emde Boas tree.
|
Patience sort
| 0.858567
|
186
|
Some researchers have achieved "near-human performance" on the MNIST database, using a committee of neural networks; in the same paper, the authors achieve performance double that of humans on other recognition tasks. The highest error rate listed on the original website of the database is 12 percent, which is achieved using a simple linear classifier with no preprocessing.In 2004, a best-case error rate of 0.42 percent was achieved on the database by researchers using a new classifier called the LIRA, which is a neural classifier with three neuron layers based on Rosenblatt's perceptron principles.Some researchers have tested artificial intelligence systems using the database put under random distortions. The systems in these cases are usually neural networks and the distortions used tend to be either affine distortions or elastic distortions. Sometimes, these systems can be very successful; one such system achieved an error rate on the database of 0.39 percent.In 2011, an error rate of 0.27 percent, improving on the previous best result, was reported by researchers using a similar system of neural networks.
|
MNIST database
| 0.858542
|
187
|
The College Board suggested a year-long course in biology at the college preparatory level, as well as a one-year course in algebra, and lab experience as preparation for the test. The test required understanding of biological data and concepts, science-related terms, and the ability to effectively synthesize and interpret data from charts, maps, and other visual media. However, most questions from this test were derived from, or are similar to, the pre-2012 AP Biology multiple choice questions. By taking an AP class or a class with similar rigor, one's chances at doing well on this test should have improved.
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SAT Subject Test in Biology E/M
| 0.858485
|
188
|
So in our previous example, we might say that the problem requires O ( n ) {\displaystyle O(n)} steps to solve. Perhaps the most important open problem in all of computer science is the question of whether a certain broad class of problems denoted NP can be solved efficiently. This is discussed further at Complexity classes P and NP, and P versus NP problem is one of the seven Millennium Prize Problems stated by the Clay Mathematics Institute in 2000. The Official Problem Description was given by Turing Award winner Stephen Cook.
|
Theory of algorithms
| 0.858481
|
189
|
Solutions Manual to Accompany Electricity and Magnetism: Berkeley Physics Course, Volume 2, First Edition. McGraw-Hill. Purcell, Edward M.
|
Electricity and Magnetism (book)
| 0.858403
|
190
|
(2) Despite their large size, some proteins crystallize readily into symmetric crystals, consistent with the idea of symmetric faces that match up upon association. (3) Proteins bind metal ions; since metal-binding sites must have specific bond geometries (e.g., octahedral), it was plausible to assume that the entire protein also had similarly crystalline geometry. (4) As described above, the cyclol model provided a simple chemical explanation of denaturation and the difficulty of cleaving folded proteins with proteases.
|
Cyclol
| 0.858337
|
191
|
The following module-like structures have the common feature of having two sets, A and B, so that there is a binary operation from A×A into A and another operation from A×B into A. Modules, counting the ring operations, have at least three binary operations. Group with operators: a group G with a set Ω and a binary operation Ω × G → G satisfying certain axioms. Module: an abelian group M and a ring R acting as operators on M. The members of R are sometimes called scalars, and the binary operation of scalar multiplication is a function R × M → M, which satisfies several axioms. Special types of modules, including free modules, projective modules, injective modules and flat modules are studied in abstract algebra.
|
Outline of algebraic structures
| 0.85828
|
192
|
But if some characteristic of the items is exploitable (for example, they are already arranged in some particular order), a different method can be used, or even a custom-made sort routine. After the programmer is reasonably sure that the best algorithm is selected, code optimization can start. Loops can be unrolled (for lower loop overhead, although this can often lead to lower speed if it overloads the CPU cache), data types as small as possible can be used, integer arithmetic can be used instead of floating-point, and so on.
|
Software optimization
| 0.858279
|
193
|
As another example from physics, consider a parallel-plate capacitor, in which the plates can move relative to one another. Such a capacitor would allow transfer of the electric energy which is stored in the capacitor into external mechanical work, done by the force acting on the plates. One may think of the electric charge as analogous to the "charge" of a gas in a cylinder, with the resulting mechanical force exerted on a piston. Compute the force on the plates as a function of x, the distance which separates them.
|
Legendre transformation
| 0.858271
|
194
|
There are two rounds in the Senior Section: a written round (Round 1) and an invitational round (Round 2). The paper in Round 1 comprises 5 multiple-choice questions, each with five options, and 20 short answer questions. The Senior section is geared towards Upper Secondary students, and topics tested include number theory, combinatorics, geometry, algebra, and probability. The second round, the Senior Invitational Round consists of a 5-question, 4-hour long paper requiring full-length solutions. Only the top 10% of students from Round 1 are eligible to take Round 2.
|
Singapore Mathematical Olympiad
| 0.858265
|
195
|
There was a hard, universally used pass-fail criterion for the Eddy Test, and a second chance was normally never allowed. Eddy described the test as having questions with multiple-choice answers, with each of the answers giving some indication of the test-taker's mathematics/physics knowledge, creativity, reasoning ability, and general aptitude. Most answers were weighted – not simply right or wrong – and speed certainly affected the results. No copies of the actual test have been found.
|
Eddy Test
| 0.858251
|
196
|
An example of dimensional analysis can be found for the case of the mechanics of a thin, solid and parallel-sided rotating disc. There are five variables involved which reduce to two non-dimensional groups. The relationship between these can be determined by numerical experiment using, for example, the finite element method.The theorem has also been used in fields other than physics, for instance in sports science.
|
Buckingham pi theorem
| 0.858235
|
197
|
For any algebraic structure it is possible to consider the minimal cardinality of generators of the structure. For algebraic extensions, algebraic degree and separable degree are often employed (note that the algebraic degree equals the dimension of the extension as a vector space over the smaller field). For non-algebraic field extensions, transcendence degree is likewise used.
|
Bounding number
| 0.85823
|
198
|
Given groups G (with operation *) and H (with operation ∆), the direct product G × H is defined as follows: The resulting algebraic object satisfies the axioms for a group. Specifically: Associativity The binary operation on G × H is associative. Identity The direct product has an identity element, namely (1G, 1H), where 1G is the identity element of G and 1H is the identity element of H. Inverses The inverse of an element (g, h) of G × H is the pair (g−1, h−1), where g−1 is the inverse of g in G, and h−1 is the inverse of h in H.
|
Direct product of groups
| 0.858118
|
199
|
Modern quantum mechanics explains this in terms of electron shells and subshells which can each hold a number of electrons determined by the Pauli exclusion principle. Thus the n = 1 state can hold one or two electrons, while the n = 2 state can hold up to eight electrons in 2s and 2p subshells. In helium, all n = 1 states are fully occupied; the same is true for n = 1 and n = 2 in neon. In argon, the 3s and 3p subshells are similarly fully occupied by eight electrons; quantum mechanics also allows a 3d subshell but this is at higher energy than the 3s and 3p in argon (contrary to the situation for hydrogen) and remains empty.
|
Sp orbital
| 0.858075
|
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