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The test had 50 multiple choice questions that were to be answered in one hour. All questions had five answer choices. Students received 1 point for every correct answer, lost ¼ of a point for each incorrect answer, and received 0 points for questions left blank. The questions covered a broad range of topics. Approxima...
SAT Subject Test in Mathematics Level 1
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1
The first part contains 60 multiple-choice questions. Each question has four answer choices. The questions are loosely grouped into 10 sets of 6 items; each set corresponds to a different chemistry topic.
United States National Chemistry Olympiad
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2
Simple structures: no binary operation: Set: a degenerate algebraic structure S having no operations.Group-like structures: one binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers. Group: a monoid with a unary op...
Pointed unary system
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3
Sorting algorithms are prevalent in introductory computer science classes, where the abundance of algorithms for the problem provides a gentle introduction to a variety of core algorithm concepts, such as big O notation, divide-and-conquer algorithms, data structures such as heaps and binary trees, randomized algorithm...
Sorted list
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4
Two metal electrodes held at different electric potential V {\displaystyle V} and separated by a finite distance will induce an electric field E {\displaystyle E} in the region between and surrounding them. The field distribution is determined by the geometry of the problem and the constitutive medium properties such a...
Electrical capacitance volume tomography
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5
In analytic geometry, the graph of any quadratic function is a parabola in the xy-plane. Given a quadratic polynomial of the form the numbers h and k may be interpreted as the Cartesian coordinates of the vertex (or stationary point) of the parabola. That is, h is the x-coordinate of the axis of symmetry (i.e. the axis...
Completing the square
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There exist sets of 3 points that can indeed be shattered using this model (any 3 points that are not collinear can be shattered). However, no set of 4 points can be shattered: by Radon's theorem, any four points can be partitioned into two subsets with intersecting convex hulls, so it is not possible to separate one o...
Vapnik–Chervonenkis dimension
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7
In some versions, they compete to be the first to guess correctly; in others, they can work out a strategy beforehand to cooperate and maximize the probability of correct guesses.One variation received some new publicity as a result of Todd Ebert's 1998 Ph.D. thesis at the University of California, Santa Barbara. It is...
Induction puzzles
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8
The problem of managing the deallocation of garbage is well-known in computer science. Several approaches are taken: Many operating systems reclaim the memory and resources used by a process or program when it terminates. Simple or short-lived programs which are designed to run in such environments can exit and allow t...
Garbage (computer science)
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9
Hestenes (1998) found that while "nearly 80% of the could state Newton's Third Law at the beginning of the course, FCI data showed that less than 15% of them fully understood it at the end". These results have been replicated in a number of studies involving students at a range of institutions (see sources section bel...
Force Concept Inventory
0.883803
10
Many programming languages require garbage collection, either as part of the language specification (e.g., RPL, Java, C#, D, Go, and most scripting languages) or effectively for practical implementation (e.g., formal languages like lambda calculus). These are said to be garbage-collected languages. Other languages, suc...
Object pinning
0.883114
11
Typically, the topics are, in order, descriptive chemistry/laboratory techniques, stoichiometry, gases/liquids/solids, thermodynamics, kinetics, equilibrium, electrochemistry, electronic structure/periodic trends, bonding theories, and organic chemistry. There is no penalty for guessing; a student's score is equal to t...
United States National Chemistry Olympiad
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b. What did the scientist criticize a proof of? c. What did the scientist criticize the proof of? d. Why did the scientist criticize Max's proof of the theorem?
Grammaticality
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In computer science, garbage collection (GC) is a form of automatic memory management. The garbage collector attempts to reclaim memory which was allocated by the program, but is no longer referenced; such memory is called garbage. Garbage collection was invented by American computer scientist John McCarthy around 1959...
Root set
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In terms of coordinate geometry, a parabola is a curve whose (x, y)-coordinates are described by a second-degree polynomial, i.e. any equation of the form: where p represents the polynomial of degree 2 and a0, a1, and a2 ≠ 0 are constant coefficients whose subscripts correspond to their respective term's degree. The ge...
Quadratic formula
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15
There are fundamental limits on the performance of comparison sorts. A comparison sort must have an average-case lower bound of Ω(n log n) comparison operations, which is known as linearithmic time. This is a consequence of the limited information available through comparisons alone — or, to put it differently, of the ...
Comparison sort
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The competition consists of 15 questions of increasing difficulty, where each answer is an integer between 0 and 999 inclusive. Thus the competition effectively removes the element of chance afforded by a multiple-choice test while preserving the ease of automated grading; answers are entered onto an OMR sheet, similar...
American Invitational Mathematics Examination
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17
A prototypical elastic component is a coiled spring. The linear elastic performance of a spring is parametrized by a constant of proportionality, called the spring constant. This constant is usually denoted as k (see also Hooke's Law) and depends on the geometry, cross-sectional area, undeformed length and nature of th...
Elastic energy
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The paper consists of single correct type questions. There are questions from high school level physics, mathematics and some questions from general astronomy.
National Standard Examination in Astronomy
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19
The classical integer sorting algorithms of pigeonhole sort, counting sort, and radix sort are widely used and practical. Much of the subsequent research on integer sorting algorithms has focused less on practicality and more on theoretical improvements in their worst case analysis, and the algorithms that come from th...
Integer sorting
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The SAT Subject Test in Physics had 75 questions and consisted of two parts: Part A and Part B. Part A: First 12 or 13 questions 4 groups of two to four questions each The questions within any one group all relate to a single situation. Five possible answer choices are given before the question. An answer choice can be...
SAT Subject Test in Physics
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21
Most mathematics questions, or calculation questions from subjects such as chemistry, physics, or economics employ a style which does not fall into any of the above categories, although some papers, notably the Maths Challenge papers in the United Kingdom employ multiple choice. Instead, most mathematics questions stat...
Aptitude test
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The answer sheet had room for 115 answers; however, no test had more than 95 questions. 1–100 were standard multiple-choice bubbles and 101–115 were for 'relationship analysis questions', which were only used for the chemistry exam. The biology test was the only test to use answers 96–100; questions 1–60 were common to...
SAT Subject Tests
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In computer science, manual memory management refers to the usage of manual instructions by the programmer to identify and deallocate unused objects, or garbage. Up until the mid-1990s, the majority of programming languages used in industry supported manual memory management, though garbage collection has existed since...
Manual memory management
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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 eff...
Bubble Sort
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More general structures may be defined by relaxing some of the axioms defining a group. The table gives a list of several structures generalizing groups. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N {\display...
Elementary group theory
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The NSEC contains only multiple choice questions. The questions include physical chemistry, organic chemistry, and inorganic chemistry. The stress on biochemistry is more in the NSEC than in the typical school syllabi.
National Standard Examination in Chemistry
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The second part contains 8 free response questions. Complete written explanations and calculations are required for full credit on a question, and partial credit is awarded. More thorough knowledge of basic theories is required, and often there are questions on less-emphasized portions of normal high school chemistry c...
United States National Chemistry Olympiad
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In computer science, garbage includes data, objects, or other regions of the memory of a computer system (or other system resources), which will not be used in any future computation by the system, or by a program running on it. Because every computer system has a finite amount of memory, and most software produces gar...
Garbage (computer science)
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A common optimization is to put the unsorted elements of the buckets back in the original array first, then run insertion sort over the complete array; because insertion sort's runtime is based on how far each element is from its final position, the number of comparisons remains relatively small, and the memory hierarc...
Bucket sorting
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30
Any set of things that obeys all the rules for one (or two) operation(s) is, by definition, a group (or ring), and obeys all theorems about groups (or rings). Integer numbers, and the operations of addition and multiplication, are just one example. For example, the elements might be computer data words, where the first...
Emmy Noether
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31
": 46 She also contended that the two studies that had failed to replicate the Uhlmann paper's results were flawed for two reasons: because they looked at cells of children with autism rather than in their GI tract, and because they did not test children with autism with gastrointestinal dysfunction. : 629A Immunologis...
Autism omnibus trial
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32
At the beginning of the contest, teams have a choice between three problems. Problem A involves a system that requires the use of continuous mathematics, and thus often involves concepts from geometry, physics, or engineering. Problem B involves a system that requires the use of discrete mathematics. In 2016, a "data i...
Mathematical Contest in Modeling
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Next-generation sequencing (NGS) allows for the rapid sequencing of large amounts of DNA, significantly advancing the study of genetics, and replacing older methods such as Sanger sequencing. This technology is starting to become more common in healthcare and research not only because it is a reliable method of determi...
Exome
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Weighted reference counts are a good solution for garbage collecting a distributed system. Tracing garbage collection cycles are triggered too often if the set of live objects fills most of the available memory; it requires extra space to be efficient. Reference counting performance does not deteriorate as the total am...
Reference counting
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35
They were interested in trait inheritance in the sweet pea and were studying two genes—the gene for flower colour (P, purple, and p, red) and the gene affecting the shape of pollen grains (L, long, and l, round). They crossed the pure lines PPLL and ppll and then self-crossed the resulting PpLl lines. According to Mend...
Recombination frequency
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36
This test consisted of 85 questions. The first 23 questions numbered 1-23 were 'classification questions'. The next 15 questions, numbered 101-115, were called 'relationship analysis questions'. The SAT Subject Test in Chemistry was currently the only SAT that incorporates the relationship analysis questions.
SAT Subject Test in Chemistry
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37
Historically, and in current teaching, the study of algebra starts with the solving of equations, such as the quadratic equation above. Then more general questions, such as "does an equation have a solution? ", "how many solutions does an equation have?
Diagrammatic algebra
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38
Next-Generation Sequencing (NGS) is the most recent body identification method in the field of genetics. The process of NGS includes three fundamental steps; “library preparation, sequencing, and data interpretation”. Its success is due to its ability to “target a larger number of PCR amplicons in a single assay”.
Body identification
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39
Two of the most basic objects in abstract algebra are groups and rings. A group consists of a set of elements and a single operation which combines a first and a second element and returns a third. The operation must satisfy certain constraints for it to determine a group: It must be closed (when applied to any pair of...
Emmy Noether
0.873859
40
The problem of computing the kth smallest (or largest) element of a list is called the selection problem and is solved by a selection algorithm. Although this problem is difficult for very large lists, sophisticated selection algorithms have been created that can solve this problem in time proportional to the number of...
Order statistic
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41
Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax2 + bx + c), the set of all two dimensional vectors of a plane, and the various finite groups such as the cyclic groups, which are the groups of integers modulo n. Set theory is a branch of logic and not techn...
Algebra
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42
Theorem: Let G be a group, and let H be a subgroup. Let G / H {\displaystyle G/H} be the set of left cosets of H in G. Let N be the normal core of H in G, defined to be the intersection of the conjugates of H in G. Then the quotient group G / N {\displaystyle G/N} is isomorphic to a subgroup of Sym ⁡ ( G / H ) {\displa...
Cayley's theorem
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43
The compiled code is additionally optimized (and re-optimized) dynamically at runtime, based on heuristics of the code's execution profile. Optimization techniques used include inlining, elision of expensive runtime properties, and inline caching. The garbage collector is a generational incremental collector.
Chrome V8
0.873474
44
In algebra, a quadratic equation (from Latin quadratus 'square') is any equation that can be rearranged in standard form as where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coeffici...
Quadratic equations
0.872698
45
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 expans...
Conceptual question
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46
In sorting n objects, merge sort has an average and worst-case performance of O(n log n). If the running time of merge sort for a list of length n is T(n), then the recurrence relation T(n) = 2T(n/2) + n follows from the definition of the algorithm (apply the algorithm to two lists of half the size of the original list...
Merge sort
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47
For example, if binary tree sort is implemented with a self-balancing BST, we have a very simple-to-describe yet asymptotically optimal O ( n log ⁡ n ) {\displaystyle O(n\log n)} sorting algorithm. Similarly, many algorithms in computational geometry exploit variations on self-balancing BSTs to solve problems such as t...
Balanced binary search tree
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48
Quadratic equations are studied and students learn techniques to reduce special quintic and exponential equations to quadratics. Mathematics Extension 1 (Must be studied concurrently with Mathematics Advanced): A more advanced course building on concepts in calculus, trigonometry, polynomials, basic combinatorics, vect...
Mathematics education in Australia
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49
For this reason selection sort may be preferable in cases where writing to memory is significantly more expensive than reading, such as with EEPROM or flash memory. While some divide-and-conquer algorithms such as quicksort and mergesort outperform insertion sort for larger arrays, non-recursive sorting algorithms such...
Insertion Sort
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50
The student's score was based entirely on his or her performance in answering the multiple-choice questions. The questions covered a broad range of topics in general biology. There were more specific questions related respectively on ecological concepts (such as population studies and general Ecology) on the E test and...
SAT Subject Test in Biology E/M
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51
However, for any degree there are some polynomial equations that have algebraic solutions; for example, the equation x 10 = 2 {\displaystyle x^{10}=2} can be solved as x = ± 2 10 . {\displaystyle x=\pm {\sqrt{2}}.} The eight other solutions are nonreal complex numbers, which are also algebraic and have the form x = ± r...
Solution in radicals
0.871192
52
Steroids can be classified based on their chemical composition. One example of how MeSH performs this classification is available at the Wikipedia MeSH catalog. Examples of this classification include: In biology, it is common to name the above steroid classes by the number of carbon atoms present when referring to hor...
Steroid biosynthesis
0.871136
53
These are conceptual ranking-task questions that help the student before embarking on numerical calculations. The textbook covers most of the basic topics in physics: Mechanics Waves Thermodynamics Electromagnetism Optics Special RelativityThe extended edition also contains introductions to topics such as quantum mecha...
Fundamentals of Physics
0.871011
54
Consider a simple physics problem: a car is moving such that it covers a distance of 1 mile in every 2 minutes, what is its velocity in metres per second? With some conversion and calculation, one can come up with the answer "13.41m/s"; on the other hand, one can instead answer "0, relative to itself". The first answer...
Quantum reference frame
0.870845
55
By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied are sets, to which the theorems of set theory apply. Those sets that have a certain binary operation defined on them form magmas, t...
Abstract Algebra
0.870814
56
In abstract algebra, a semigroup with three elements is an object consisting of three elements and an associative operation defined on them. The basic example would be the three integers 0, 1, and −1, together with the operation of multiplication. Multiplication of integers is associative, and the product of any two of...
Semigroup with three elements
0.870731
57
Sylow's test: Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1 is the only divisor of n that is congruent to 1 modulo p, then there does not exist a simple group of order n. Proof: If n is a prime-power, then a group of order n has a nontrivial center and, therefore, is not simple....
Simple groups
0.870414
58
If this distance term were to decrease to zero, the value of the axis of symmetry would be the x value of the only zero, that is, there is only one possible solution to the quadratic equation. Algebraically, this means that √b2 − 4ac = 0, or simply b2 − 4ac = 0 (where the left-hand side is referred to as the discrimina...
Quadratic formula
0.870334
59
Finally, selection sort is greatly outperformed on larger arrays by Θ ( n log ⁡ n ) {\displaystyle \Theta (n\log n)} divide-and-conquer algorithms such as mergesort. However, insertion sort or selection sort are both typically faster for small arrays (i.e. fewer than 10–20 elements). A useful optimization in practice f...
Selection sort
0.870064
60
Multiple methods are available for the isolation and study of human viruses: Deep sequencing is a rapid DNA sequencing technique that is useful for characterizing virome richness, stability, gene function and the association with disease phenotypes. This technology creates large amounts of sequence information and is c...
Human virome
0.869919
61
Lines in a Cartesian plane, or more generally, in affine coordinates, can be described algebraically by linear equations. In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form: where: m is the slope or gradient of the line. b is the y-intercept of the line. x is the independe...
Coordinate geometry
0.869796
62
The following question was posed to Jeff Hawkins in September 2011 with regard to cortical learning algorithms: "How do you know if the changes you are making to the model are good or not?" To which Jeff's response was "There are two categories for the answer: one is to look at neuroscience, and the other is methods fo...
Hierarchical Temporal Memory
0.869724
63
Based on the amortized analysis of splay trees, the worst case running time of splaysort, on an input with n data items, is O(n log n), matching the time bounds for efficient non-adaptive algorithms such as quicksort, heap sort, and merge sort. For an input sequence in which most items are placed close to their predece...
Splaysort
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64
Neither of two available options (pass or fail) was morally acceptable.By mutual agreement with the student and the examiner, Calandra gave the student another opportunity to answer, warning the student the answer would require demonstrating some knowledge of physics. The student came up with several possible answers, ...
Barometer question
0.869467
65
This method of performing selection in a heap has been applied to problems of listing multiple solutions to combinatorial optimization problems, such as finding the k shortest paths in a weighted graph, by defining a state space of solutions in the form of an implicitly defined heap-ordered tree, and then applying this...
Selection algorithm
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66
In full generality, an algebraic structure may use any number of sets and any number of axioms in its definition. The most commonly studied structures, however, usually involve only one or two sets and one or two binary operations. The structures below are organized by how many sets are involved, and how many binary op...
Outline of algebraic structures
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67
Once the annotation is completed, it is submitted to the National Center for Biotechnology Information's (NCBI) DNA sequence database GenBank. If there is still time in the semester or the sent DNA was not able to be sequenced, the class could request genome file from the University of Pittsburgh that had yet to be seq...
SEA-PHAGES
0.869018
68
This process is known as post-translational processing or post-translational modification, because it takes place on the protein after the DNA is translated. The role of post-translational processing in gene regulation is the subject of the growing field of study, epigenetics. One modification mechanism is methylation.
Citrullination
0.869003
69
Any parabola can be described in a suitable coordinate system by an equation y = a x 2 {\displaystyle y=ax^{2}} . Proof: straightforward calculation for the unit parabola y = x 2 {\displaystyle y=x^{2}} . Application: The 4-points property of a parabola can be used for the construction of point P 4 {\displaystyle P_{4}...
Parabola
0.868909
70
Randomized algorithms that solve the problem in linear time are known, in Euclidean spaces whose dimension is treated as a constant for the purposes of asymptotic analysis. This is significantly faster than the O ( n 2 ) {\displaystyle O(n^{2})} time (expressed here in big O notation) that would be obtained by a naive ...
Closest pair of points problem
0.868774
71
It is important to understand the terms "heterozygous", "homozygous", "double heterozygote" (or homozygote), "dominant allele" and "recessive allele" when using the Punnett square method. For multiple traits, using the "forked-line method" is typically much easier than the Punnett square. Phenotypes may be predicted wi...
Allele chart
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72
Massive parallel sequencing or massively parallel sequencing is any of several high-throughput approaches to DNA sequencing using the concept of massively parallel processing; it is also called next-generation sequencing (NGS) or second-generation sequencing. Some of these technologies emerged between 1993 and 1998 and...
Massive parallel sequencing
0.86856
73
In chemistry, values n = 1, 2, 3, 4, 5, 6, 7 are used in relation to the electron shell theory, with expected inclusion of n = 8 (and possibly 9) for yet-undiscovered period 8 elements. In atomic physics, higher n sometimes occur for description of excited states. Observations of the interstellar medium reveal atomic h...
Principal quantum level
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74
backtracking bag Baillie–PSW primality test balanced binary search tree balanced binary tree balanced k-way merge sort balanced merge sort balanced multiway merge balanced multiway tree balanced quicksort balanced tree balanced two-way merge sort BANG file Batcher sort Baum Welch algorithm BB α tree BDD BD-tree Bellman...
List of terms relating to algorithms and data structures
0.868221
75
Proceedings of the National Academy of Sciences USA 81: 2060–2064 Botts J, Thomason JF, Morales MF. (1989) On the origin and transmission of force in actomyosin subfragment 1. Proceedings of the National Academy of Sciences USA 86: 2204–2208 Onishi H, Mochizuki N, Morales MF. (2004) On the Myosin Catalysis of ATP Hydro...
Manuel Morales
0.868126
76
Thus, for example, groups have a signature containing two operators: the multiplication operator m, taking two arguments, and the inverse operator i, taking one argument, and the identity element e, a constant, which may be considered an operator that takes zero arguments. Given a (countable) set of variables x, y, z, ...
Carrier set
0.868096
77
In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation, or locus. For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal. These points form a line, and y = x is sa...
Analytical geometry
0.867903
78
An elementary physics professor was teaching about how close you could get to the sun. He laid the foundation of heat and distance, and said that is as close as you can get FAPP. A boy asked, "what does that mean?"
For all practical purposes
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79
An underdetermined linear system has either no solution or infinitely many solutions. For example, x + y + z = 1 x + y + z = 0 {\displaystyle {\begin{aligned}x+y+z&=1\\x+y+z&=0\end{aligned}}} is an underdetermined system without any solution; any system of equations having no solution is said to be inconsistent. On the...
Underdetermined system
0.867771
80
A compiler can use the results of escape analysis as a basis for optimizations: Converting heap allocations to stack allocations. If an object is allocated in a subroutine, and a pointer to the object never escapes, the object may be a candidate for stack allocation instead of heap allocation. In garbage-collected lang...
Escape analysis
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81
The sets being studied may also be subsets of algebraic structures other than the integers, for example, groups, rings and fields.
Additive Combinatorics
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82
These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group. Some structures do not form varieties, because either: It is necessary that 0 ≠ 1, 0 being the additive identity element and 1 being a multiplicative identity element, but this is a nonident...
Carrier set
0.867427
83
The 2014 AP Chemistry exam was the first administration of a redesigned test as a result of a redesigning of the AP Chemistry course. The exam format is now different from the previous years, with 60 multiple choice questions (now with only four answer choices per question), 3 long free response questions, and 4 short ...
AP Chemistry
0.867393
84
The questions are divided into four categories: arithmetic, algebra, geometry and problem solving, and the number of questions that the student answered correctly for each category is listed along with the regional mean. Every school receives a more comprehensive analysis, with a complete record of answers given by all...
Australian Mathematics Competition
0.867339
85
Multiple Choice and Free Response Sections of the AP® Physics 1 exam are also assessed on scientific practices. Below are tables representing the practices assessed and their weighting for both parts of the exam
AP Physics 1
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86
Carefully written justifications are required for each problem.Prior to academic year 2010–2011 the competition consisted of four rounds of five problems each, covering all non-calculus topics. Students were given approximately one month to solve the questions. Each question is scored out of five points; thus, a perfec...
United States of America Mathematical Talent Search
0.866768
87
cache thrashing). Furthermore, cache-optimization, usually via cache-aware or cache-oblivious data structures and algorithms, can often lead to orders of magnitude improvements in performance as well as avoiding time-complexity degeneracy that is characteristic of many cache-pessimizing algorithms, and is therefore one...
Comparison of Java and C++
0.866496
88
This method also produces a sorted version of the collection, which may be useful for other later computations, and in particular for selection with other choices of k {\displaystyle k} . For a sorting algorithm that generates one item at a time, such as selection sort, the scan can be done in tandem with the sort, and...
Selection algorithm
0.866056
89
A parabola with equation y = a x 2 + b x + c , a ≠ 0 {\displaystyle y=ax^{2}+bx+c,\ a\neq 0} is uniquely determined by three points ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) {\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})} with different x coordinates. The usual procedure to determine the coefficients a , ...
Parabolic curve
0.866051
90
Unusually for a probability theory book, this book does not use the phrase "random variable", instead referring to random processes as games. The first five chapters of the book concern counting problems, and include material on the exponential function, binomial coefficients, factorials, games of cards, dice, and coin...
Fat Chance: Probability from 0 to 1
0.865639
91
Suppose that a mathematician is studying geometry and shapes, and she wishes to prove certain theorems about them. She conjectures that "All rectangles are squares", and she is interested in knowing whether this statement is true or false. In this case, she can either attempt to prove the truth of the statement using d...
Counterexample
0.865632
92
Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: (a·a)(b·b) = (a×b)·(a×b). This defines the...
Four squares formula
0.86555
93
NBDE I consists of 400 multiple choice questions emphasizing basic sciences: 1. Human Anatomy, Embryology, and Histology 2. Biochemistry and Physiology 3.
National Board Dental Examination
0.865427
94
Ask A Biologist is a pre-kindergarten through high school program dedicated to answering questions from students, their teachers, and parents. The primary focus of the program is to connect students and teachers with working scientists through a question and answer Web e-mail form. The companion website also includes a...
Ask a Biologist
0.865411
95
Tables can also define binary operations on groups, fields, rings, and other algebraic systems. In such contexts they are called Cayley tables. Here are the addition and multiplication tables for the finite field Z5: for every natural number n, there are also addition and multiplication tables for the ring Zn. For othe...
Multiplication Table
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96
Here are some of the most common existential axioms. Identity element A binary operation ∗ {\displaystyle *} has an identity element if there is an element e such that for all x in the structure. Here, the auxiliary operation is the operation of arity zero that has e as its result. Inverse element Given a binary operat...
Carrier set
0.865273
97
Algorithms with running time O(n log n) are sometimes called linearithmic. Some examples of algorithms with running time O(log n) or O(n log n) are: Average time quicksort and other comparison sort algorithms Searching in balanced binary search trees Exponentiation by squaring Longest increasing subsequenceBinary logar...
Logarithmus binaris
0.865122
98
Consider the quadratic equation x 2 − 4 x + 1 = 0. {\displaystyle x^{2}-4x+1=0.} By using the quadratic formula, we find that the two roots are A = 2 + 3 , B = 2 − 3 . {\displaystyle {\begin{aligned}A&=2+{\sqrt {3}},\\B&=2-{\sqrt {3}}.\end{aligned}}} Examples of algebraic equations satisfied by A and B include A + B = ...
Solvability by radicals
0.864878
99
The fungal, algal, or cyanobacterial component of a lichen can be grown by itself in culture. When growing by themselves, the fungus, algae, or cyanobacteria have very different properties than those of the lichen. Lichen properties such as growth form, physiology, and biochemistry, are very different from the combinat...
Lichenized fungus
0.864825