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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. Approximately 10-14% of questions focused on Numbers and Operations, 38-42% focused on Algebra and functions, 38-42% focused on Geometry (including Euclidean, coordinate, three-dimensional, and trigonometry), and 6-10% focused on Data analysis, Statistics, and probability.
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SAT Subject Test in Mathematics Level 1
| 0.905157
|
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.
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United States National Chemistry Olympiad
| 0.902024
|
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 operation (inverse), giving rise to inverse elements.
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Pointed unary system
| 0.895174
|
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 algorithms, best, worst and average case analysis, time–space tradeoffs, and upper and lower bounds. Sorting small arrays optimally (in fewest comparisons and swaps) or fast (i.e. taking into account machine specific details) is still an open research problem, with solutions only known for very small arrays (<20 elements). Similarly optimal (by various definitions) sorting on a parallel machine is an open research topic.
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Sorted list
| 0.891738
|
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 as permittivity ε {\displaystyle \varepsilon } and conductivity σ {\displaystyle \sigma } . Assuming a static or quasi-static regime and the presence of a lossless dielectric medium, such as a perfect insulator, in the region between the plates, the field obeys the following equation: ∇ .
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Electrical capacitance volume tomography
| 0.888878
|
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 of symmetry has equation x = h), and k is the minimum value (or maximum value, if a < 0) of the quadratic function. One way to see this is to note that the graph of the function f(x) = x2 is a parabola whose vertex is at the origin (0, 0).
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Completing the square
| 0.888771
|
6
|
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 of these two subsets from the other. Thus, the VC dimension of this particular classifier is 3.
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Vapnik–Chervonenkis dimension
| 0.886073
|
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 a strategy question about a cooperative game, which has connections to algebraic coding theory.Three players are told that each of them will receive either a red hat or a blue hat.
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Induction puzzles
| 0.886001
|
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 the operating system to perform any necessary reclamation. In systems or programming languages with manual memory management, the programmer must explicitly arrange for memory to be deallocated when it is no longer used.
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Garbage (computer science)
| 0.885159
|
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 below), and have led to greater recognition in the physics education research community of the importance of students' "active engagement" with the materials to be mastered. The 1995 version has 30 five-way multiple choice questions.Example question (question 4): A large truck collides head-on with a small compact car.
|
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, such as C and C++, were designed for use with manual memory management, but have garbage-collected implementations available. Some languages, like Ada, Modula-3, and C++/CLI, allow both garbage collection and manual memory management to co-exist in the same application by using separate heaps for collected and manually managed objects.
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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 the number of questions answered correctly. One and a half hours (90 minutes) are allotted for this first part.
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United States National Chemistry Olympiad
| 0.882279
|
12
|
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
| 0.882243
|
13
|
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 to simplify manual memory management in Lisp.Garbage collection relieves the programmer from doing manual memory management, where the programmer specifies what objects to de-allocate and return to the memory system and when to do so. Other, similar techniques include stack allocation, region inference, and memory ownership, and combinations thereof.
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Root set
| 0.882153
|
14
|
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 geometrical interpretation of the quadratic formula is that it defines the points on the x-axis where the parabola will cross the axis. Additionally, if the quadratic formula was looked at as two terms, the axis of symmetry appears as the line x = −b/2a. The other term, √b2 − 4ac/2a, gives the distance the zeros are away from the axis of symmetry, where the plus sign represents the distance to the right, and the minus sign represents the distance to the left.
|
Quadratic formula
| 0.880539
|
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 vague algebraic structure of totally ordered sets. In this sense, mergesort, heapsort, and introsort are asymptotically optimal in terms of the number of comparisons they must perform, although this metric neglects other operations.
|
Comparison sort
| 0.88048
|
16
|
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 to the way grid-in math questions are answered on the SAT. Leading zeros must be gridded in; for example, answers of 7 and 43 must be written and gridded in as 007 and 043, respectively. Concepts typically covered in the competition include topics in elementary algebra, geometry, trigonometry, as well as number theory, probability, and combinatorics.
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American Invitational Mathematics Examination
| 0.880129
|
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 the material from which the coil is fashioned.
|
Elastic energy
| 0.880039
|
18
|
The paper consists of single correct type questions. There are questions from high school level physics, mathematics and some questions from general astronomy.
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National Standard Examination in Astronomy
| 0.879551
|
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 this line of research are not believed to be practical for current 64-bit computer architectures, although experiments have shown that some of these methods may be an improvement on radix sorting for data with 128 or more bits per key. Additionally, for large data sets, the near-random memory access patterns of many integer sorting algorithms can handicap them compared to comparison sorting algorithms that have been designed with the memory hierarchy in mind.Integer sorting provides one of the six benchmarks in the DARPA High Productivity Computing Systems Discrete Mathematics benchmark suite, and one of eleven benchmarks in the NAS Parallel Benchmarks suite.
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Integer sorting
| 0.879418
|
20
|
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 used once, more than once, or not at all in each group.Part B: Last 62 or 63 questions Each question has five possible answer choice with one correct answer. Some questions may be in groups of two or three.
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SAT Subject Test in Physics
| 0.879415
|
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 state a mathematical problem or exercise that requires a student to write a freehand response. Marks are given more for the steps taken than for the correct answer. If the question has multiple parts, later parts may use answers from previous sections, and marks may be granted if an earlier incorrect answer was used but the correct method was followed, and an answer which is correct (given the incorrect input) is returned. Higher-level mathematical papers may include variations on true/false, where the candidate is given a statement and asked to verify its validity by direct proof or stating a counterexample.
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Aptitude test
| 0.878944
|
22
|
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 both the E and M tests, in addition, the E used 61–80, and the M used 81–100.
|
SAT Subject Tests
| 0.87855
|
23
|
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 1959, when it was introduced with Lisp. Today, however, languages with garbage collection such as Java are increasingly popular and the languages Objective-C and Swift provide similar functionality through Automatic Reference Counting. The main manually managed languages still in widespread use today are C and C++ – see C dynamic memory allocation.
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Manual memory management
| 0.878479
|
24
|
Although bubble sort is one of the simplest sorting algorithms to understand and implement, its O(n2) complexity means that its efficiency decreases dramatically on lists of more than a small number of elements. Even among simple O(n2) sorting algorithms, algorithms like insertion sort are usually considerably more efficient. Due to its simplicity, bubble sort is often used to introduce the concept of an algorithm, or a sorting algorithm, to introductory computer science students. However, some researchers such as Owen Astrachan have gone to great lengths to disparage bubble sort and its continued popularity in computer science education, recommending that it no longer even be taught.The Jargon File, which famously calls bogosort "the archetypical perversely awful algorithm", also calls bubble sort "the generic bad algorithm".
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Bubble Sort
| 0.878453
|
25
|
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 {\displaystyle \mathbb {N} } (including zero) under addition form a monoid, as do the nonzero integers under multiplication ( Z ∖ { 0 } , ⋅ ) {\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )} .
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Elementary group theory
| 0.878384
|
26
|
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
| 0.878078
|
27
|
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 curricula, such as organic chemistry and coordination chemistry. One hour and 45 minutes (105 minutes) are allowed for this section.
|
United States National Chemistry Olympiad
| 0.878039
|
28
|
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 garbage, it is frequently necessary to deallocate memory that is occupied by garbage and return it to the heap, or memory pool, for reuse.
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Garbage (computer science)
| 0.876088
|
29
|
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 hierarchy is better exploited by storing the list contiguously in memory.If the input distribution is known or can be estimated, buckets can often be chosen which contain constant density (rather than merely having constant size). This allows O ( n ) {\displaystyle O(n)} average time complexity even without uniformly distributed input.
|
Bucket sorting
| 0.875803
|
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 combining operation is exclusive or and the second is logical conjunction.
|
Emmy Noether
| 0.875711
|
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 Immunologist Vera Byers testified that Michelle Cedillo had a dysregulated immune system, which allowed the measles virus to persist in her system, and that her malfunctioning immune system was in part a result of the virus itself. : 32 She also stated that this dysregulation was caused by "a combination of genetics and the measles virus vaccination and the thimerosal-containing vaccines that she had received.
|
Autism omnibus trial
| 0.87547
|
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 insights" problem was added, where teams are given access to database files and tasked with using them to answer a question.
|
Mathematical Contest in Modeling
| 0.875259
|
33
|
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 determining genetic variations, but also because it is cost effective and allows researchers to sequence entire genomes in anywhere between days to weeks. This compares to former methods which may have taken months. Next-gen sequencing includes both whole-exome sequencing and whole-genome sequencing.
|
Exome
| 0.875245
|
34
|
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 amount of free space decreases.Reference counts are also useful information to use as input to other runtime optimizations.
|
Reference counting
| 0.875074
|
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 Mendelian genetics, the expected phenotypes would occur in a 9:3:3:1 ratio of PL:Pl:pL:pl.
|
Recombination frequency
| 0.874908
|
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
| 0.874573
|
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
| 0.874277
|
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”.
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Body identification
| 0.873926
|
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 elements of the associated set, the generated element must also be a member of that set), it must be associative, there must be an identity element (an element which, when combined with another element using the operation, results in the original element, such as adding zero to a number or multiplying it by one), and for every element there must be an inverse element. A ring likewise, has a set of elements, but now has two operations.
|
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 elements in the list, even if the list is totally unordered. If the data is stored in certain specialized data structures, this time can be brought down to O(log n). In many applications all order statistics are required, in which case a sorting algorithm can be used and the time taken is O(n log n).
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Order statistic
| 0.873768
|
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 technically a branch of algebra. Binary operations: The notion of addition (+) is generalized to the notion of binary operation (denoted here by ∗). The notion of binary operation is meaningless without the set on which the operation is defined.
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Algebra
| 0.873757
|
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 ) {\displaystyle \operatorname {Sym} (G/H)} . The special case H = 1 {\displaystyle H=1} is Cayley's original theorem.
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Cayley's theorem
| 0.873508
|
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.
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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 coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the expression on its left-hand side. A quadratic equation has at most two solutions.
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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 expansion of an ideal gas, its temperatureincreases decreases stays the same Impossible to tell/need more information The use of conceptual questions in physics was popularized by Eric Mazur, particularly in the form of multiple-choice tests that he called ConcepTests.
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Conceptual question
| 0.872689
|
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, and add the n steps taken to merge the resulting two lists). The closed form follows from the master theorem for divide-and-conquer recurrences. The number of comparisons made by merge sort in the worst case is given by the sorting numbers.
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Merge sort
| 0.872629
|
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 the line segment intersection problem and the point location problem efficiently. (For average-case performance, however, self-balancing BSTs may be less efficient than other solutions.
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Balanced binary search tree
| 0.872443
|
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, vectors, and further statistics. Students learn the binomial theorem to extend their knowledge of probability, along with using circle geometry to prove a greater family of statements.
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Mathematics education in Australia
| 0.871808
|
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 as insertion sort or selection sort are generally faster for very small arrays (the exact size varies by environment and implementation, but is typically between 7 and 50 elements). Therefore, a useful optimization in the implementation of those algorithms is a hybrid approach, using the simpler algorithm when the array has been divided to a small size.
|
Insertion Sort
| 0.871429
|
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 molecular concepts such as DNA structure, translation, and biochemistry on the M test.
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SAT Subject Test in Biology E/M
| 0.871202
|
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 2 10 , {\displaystyle x=\pm r{\sqrt{2}},} where r is a fifth root of unity, which can be expressed with two nested square roots.
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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 hormones: C18-steroids for the estranes (mostly estrogens), C19-steroids for the androstanes (mostly androgens), and C21-steroids for the pregnanes (mostly corticosteroids). The classification "17-ketosteroid" is also important in medicine. The gonane (steroid nucleus) is the parent 17-carbon tetracyclic hydrocarbon molecule with no alkyl sidechains.
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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 mechanics, atomic theory, solid-state physics, nuclear physics and cosmology. A solutions manual and a study guide are also available.
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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 is correct because it recognises a reference frame is implied in the problem. The second one, albeit pedantic, is also correct because it exploits the fact that there is not a particular reference frame specified by the problem.
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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, to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to form semigroups); identity, and inverses (to form groups); and other more complex structures.
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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 these three integers is again one of these three integers. There are 18 inequivalent ways to define an associative operation on three elements: while there are, altogether, a total of 39 = 19683 different binary operations that can be defined, only 113 of these are associative, and many of these are isomorphic or antiisomorphic so that there are essentially only 18 possibilities.One of these is C3, the cyclic group with three elements.
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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. If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal. Since it is a proper, non-identity subgroup, the group is not simple. Burnside: A non-Abelian finite simple group has order divisible by at least three distinct primes. This follows from Burnside's theorem.
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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 discriminant). This is one of three cases, where the discriminant indicates how many zeros the parabola will have.
|
Quadratic formula
| 0.870334
|
59
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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 for the recursive algorithms is to switch to insertion sort or selection sort for "small enough" sublists.
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Selection sort
| 0.870064
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60
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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 capable of detecting rare components of a microbial community. Current methods combining the removal of human and bacterial DNA from samples, large scale sequencing, and bioinformatics are very efficient in the identification of unknown viruses. Unlike other discovery methods, viruses do not need to be grown in cell cultures.
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Human virome
| 0.869919
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61
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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 independent variable of the function y = f(x).In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".
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Coordinate geometry
| 0.869796
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62
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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 for machine intelligence. In the neuroscience realm, there are many predictions that we can make, and those can be tested. If our theories explain a vast array of neuroscience observations then it tells us that we’re on the right track.
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Hierarchical Temporal Memory
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63
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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 predecessor in the sorted order, or are out of order with only a small number of other items, splaysort can be faster than O(n log n), showing that it is an adaptive sort. To quantify this, let dx be the number of positions in the input that separate x from its predecessor, and let ix be the number of items that appear on one side of x in the input and on the other side of x in the output (the number of inversions that involve x). Then it follows from the dynamic finger theorem for splay trees that the total time for splaysort is bounded by ∑ x log d x {\displaystyle \sum _{x}\log d_{x}} and by ∑ x log i x {\displaystyle \sum _{x}\log i_{x}} .Splaysort can also be shown to be adaptive to the entropy of the input sequence.
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Splaysort
| 0.869547
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64
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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, but settled on dropping the barometer from the top of the building, timing its fall, and using the equation of motion d = 1 2 a t 2 {\displaystyle d={\tfrac {1}{2}}{a}t^{2}} to derive the height. The examiner agreed that this satisfied the requirement and gave the student “almost full credit”.When Calandra asked about the other answers, the student gave the examples: using the proportion between the lengths of the building's shadow and that of the barometer to calculate the building's height from the height of the barometer using the barometer as a measuring rod to mark off its height on the wall while climbing the stairs, then counting the number of marks suspending the barometer from a string to create a pendulum, then using the pendulum to measure the strength of Earth's gravity at the top and bottom of the building, and calculating the height of the building from the difference in the two measurements (see Newton's law of universal gravitation)There were, the student said, many other possible solutions.
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Barometer question
| 0.869467
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65
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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 to this tree. In the other direction, linear time selection algorithms have been used as a subroutine in a priority queue data structure related to the heap, improving the time for extracting its k {\displaystyle k} th item from O ( log n ) {\displaystyle O(\log n)} to O ( log ∗ n + log k ) {\displaystyle O(\log ^{*}n+\log k)} ; here log ∗ n {\displaystyle \log ^{*}n} is the iterated logarithm. For a collection of data values undergoing dynamic insertions and deletions, the order statistic tree augments a self-balancing binary search tree structure with a constant amount of additional information per tree node, allowing insertions, deletions, and selection queries that ask for the k {\displaystyle k} th element in the current set to all be performed in O ( log n ) {\displaystyle O(\log n)} time per operation. Going beyond the comparison model of computation, faster times per operation are possible for values that are small integers, on which binary arithmetic operations are allowed. It is not possible for a streaming algorithm with memory sublinear in both n {\displaystyle n} and k {\displaystyle k} to solve selection queries exactly for dynamic data, but the count–min sketch can be used to solve selection queries approximately, by finding a value whose position in the ordering of the elements (if it were added to them) would be within ε n {\displaystyle \varepsilon n} steps of k {\displaystyle k} , for a sketch whose size is within logarithmic factors of 1 / ε {\displaystyle 1/\varepsilon } .
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Selection algorithm
| 0.869404
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66
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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 operations are used. Increased indentation is meant to indicate a more exotic structure, and the least indented levels are the most basic.
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Outline of algebraic structures
| 0.869213
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67
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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 sequenced. In addition to the laboratory and bioinformatic skills acquired, students have the opportunity to publish their work in academic journals and attend the national SEA-PHAGES conference in Washington, D.C. or a regional symposium.
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SEA-PHAGES
| 0.869018
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68
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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.
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Citrullination
| 0.869003
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69
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Any parabola can be described in a suitable coordinate system by an equation y = a x 2 {\displaystyle y=ax^{2}} . Proof: straightforward calculation for the unit parabola y = x 2 {\displaystyle y=x^{2}} . Application: The 4-points property of a parabola can be used for the construction of point P 4 {\displaystyle P_{4}} , while P 1 , P 2 , P 3 {\displaystyle P_{1},P_{2},P_{3}} and Q 2 {\displaystyle Q_{2}} are given. Remark: the 4-points property of a parabola is an affine version of the 5-point degeneration of Pascal's theorem.
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Parabola
| 0.868909
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70
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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 algorithm of finding distances between all pairs of points and selecting the smallest. It is also possible to solve the problem without randomization, in random-access machine models of computation with unlimited memory that allow the use of the floor function, in near-linear O ( n log log n ) {\displaystyle O(n\log \log n)} time. In even more restricted models of computation, such as the algebraic decision tree, the problem can be solved in the somewhat slower O ( n log n ) {\displaystyle O(n\log n)} time bound, and this is optimal for this model, by a reduction from the element uniqueness problem. Both sweep line algorithms and divide-and-conquer algorithms with this slower time bound are commonly taught as examples of these algorithm design techniques.
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Closest pair of points problem
| 0.868774
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71
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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 with at least better-than-chance accuracy using a Punnett square, but the phenotype that may appear in the presence of a given genotype can in some instances be influenced by many other factors, as when polygenic inheritance and/or epigenetics are at work.
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Allele chart
| 0.868771
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72
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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 have been commercially available since 2005. These technologies use miniaturized and parallelized platforms for sequencing of 1 million to 43 billion short reads (50 to 400 bases each) per instrument run. Many NGS platforms differ in engineering configurations and sequencing chemistry.
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Massive parallel sequencing
| 0.86856
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73
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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 hydrogen spectral lines involving n on order of hundreds; values up to 766 were detected.
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Principal quantum level
| 0.868327
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74
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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–Ford algorithm Benford's law best case best-case cost best-first search biconnected component biconnected graph bidirectional bubble sort big-O notation binary function binary fuse filter binary GCD algorithm binary heap binary insertion sort binary knapsack problem binary priority queue binary relation binary search binary search tree binary tree binary tree representation of trees bingo sort binomial heap binomial tree bin packing problem bin sort bintree bipartite graph bipartite matching bisector bitonic sort bit vector Bk tree bdk tree (not to be confused with k-d-B-tree) block block addressing index blocking flow block search Bloom filter blossom (graph theory) bogosort boogol boolean boolean expression boolean function bottleneck traveling salesman bottom-up tree automaton boundary-based representation bounded error probability in polynomial time bounded queue bounded stack Bounding volume hierarchy, also referred to as bounding volume tree (BV-tree, BVT) Boyer–Moore string-search algorithm Boyer–Moore–Horspool algorithm bozo sort B+ tree BPP (complexity) Bradford's law branch (as in control flow) branch (as in revision control) branch and bound breadth-first search Bresenham's line algorithm brick sort bridge British Museum algorithm brute-force attack brute-force search brute-force string search brute-force string search with mismatches BSP-tree B*-tree B-tree bubble sort bucket bucket array bucketing method bucket sort bucket trie buddy system buddy tree build-heap Burrows–Wheeler transform (BWT) busy beaver Byzantine generals
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List of terms relating to algorithms and data structures
| 0.868221
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75
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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 Hydrolysis. Biochemistry 43: 3757–3763 (doi:10.1021/bi040002m) == References ==
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Manuel Morales
| 0.868126
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76
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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, etc. the term algebra is the collection of all possible terms involving m, i, e and the variables; so for example, m(i(x), m(x, m(y,e))) would be an element of the term algebra. One of the axioms defining a group is the identity m(x, i(x)) = e; another is m(x,e) = x. The axioms can be represented as trees.
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Carrier set
| 0.868096
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77
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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 said to be the equation for this line. In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.Usually, a single equation corresponds to a curve on the plane.
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Analytical geometry
| 0.867903
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78
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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?"
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For all practical purposes
| 0.867805
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79
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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 other hand, the system x + y + z = 1 x + y + 2 z = 3 {\displaystyle {\begin{aligned}x+y+z&=1\\x+y+2z&=3\end{aligned}}} is consistent and has an infinitude of solutions, such as (x, y, z) = (1, −2, 2), (2, −3, 2), and (3, −4, 2). All of these solutions can be characterized by first subtracting the first equation from the second, to show that all solutions obey z = 2; using this in either equation shows that any value of y is possible, with x = −1 − y. More specifically, according to the Rouché–Capelli theorem, any system of linear equations (underdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix.
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Underdetermined system
| 0.867771
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80
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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 languages this can reduce how often the collector needs to run.
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Escape analysis
| 0.867706
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81
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The sets being studied may also be subsets of algebraic structures other than the integers, for example, groups, rings and fields.
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Additive Combinatorics
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82
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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 nonidentity; Structures such as fields have some axioms that hold only for nonzero members of S. For an algebraic structure to be a variety, its operations must be defined for all members of S; there can be no partial operations.Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and division rings. Structures with nonidentities present challenges varieties do not. For example, the direct product of two fields is not a field, because ( 1 , 0 ) ⋅ ( 0 , 1 ) = ( 0 , 0 ) {\displaystyle (1,0)\cdot (0,1)=(0,0)} , but fields do not have zero divisors.
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Carrier set
| 0.867427
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83
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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 free response questions. The new exam has a focus on longer, more in depth, lab-based questions. The penalty for incorrect answers on the multiple choice section was also removed. More detailed information can be found at the related link.
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AP Chemistry
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84
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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 students, as well as the percentage of students choosing any given answer for a given question, and a comparison to the percentage of students choosing any given answer for a given question in the whole region. Schools also receive an analysis of their students by mathematical topic, compared to the entire region.
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Australian Mathematics Competition
| 0.867339
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85
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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
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AP Physics 1
| 0.867053
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86
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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 perfect score is 4 × 5 × 5 = 100 {\displaystyle 4\times 5\times 5=100} .
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United States of America Mathematical Talent Search
| 0.866768
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87
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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 of the most important forms of optimization; reference-semantics, as mandated in Java, makes such optimizations impossible to realize in practice (by neither the programmer nor the JIT compiler). Garbage collection, as this form of automatic memory management introduces memory overhead.However, there are a number of benefits to Java's design, some realized, some only theorized: Java garbage collection may have better cache coherence than the usual use of malloc/new for memory allocation.
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Comparison of Java and C++
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88
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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 the sort can be terminated once the k {\displaystyle k} th element has been found. One possible design of a consolation bracket in a single-elimination tournament, in which the teams who lost to the eventual winner play another mini-tournament to determine second place, can be seen as an instance of this method. Applying this optimization to heapsort produces the heapselect algorithm, which can select the k {\displaystyle k} th smallest value in time O ( n + k log n ) {\displaystyle O(n+k\log n)} . This is fast when k {\displaystyle k} is small relative to n {\displaystyle n} , but degenerates to O ( n log n ) {\displaystyle O(n\log n)} for larger values of k {\displaystyle k} , such as the choice k = n / 2 {\displaystyle k=n/2} used for median finding.
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Selection algorithm
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89
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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 , b , c {\displaystyle a,b,c} is to insert the point coordinates into the equation. The result is a linear system of three equations, which can be solved by Gaussian elimination or Cramer's rule, for example. An alternative way uses the inscribed angle theorem for parabolas.
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Parabolic curve
| 0.866051
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90
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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 coins, and the birthday paradox. After an interlude involving the binomial theorem, Pascal's triangle, and the Catalan numbers, the second part of the book concerns probability more directly. Its chapters concern the expected value, conditional probability and Bayes' theorem, events with unequal probabilities (biased coins and loaded dice), geometric probability, the law of large numbers, and normal distributions. The third part moves from probability to statistics, with topics including the central limit theorem and the meaning of false positives and false negatives in medical testing.
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Fat Chance: Probability from 0 to 1
| 0.865639
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91
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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 deductive reasoning, or she can attempt to find a counterexample of the statement if she suspects it to be false. In the latter case, a counterexample would be a rectangle that is not a square, such as a rectangle with two sides of length 5 and two sides of length 7.
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Counterexample
| 0.865632
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92
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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 quaternion multiplication rule a×b, which simply reflects Euler's identity, and some mathematics of quaternions. Quaternions are, so to say, the "square root" of the four-square identity.
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Four squares formula
| 0.86555
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93
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NBDE I consists of 400 multiple choice questions emphasizing basic sciences: 1. Human Anatomy, Embryology, and Histology 2. Biochemistry and Physiology 3.
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National Board Dental Examination
| 0.865427
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94
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Ask A Biologist is a pre-kindergarten through high school program dedicated to answering questions from students, their teachers, and parents. The primary focus of the program is to connect students and teachers with working scientists through a question and answer Web e-mail form. The companion website also includes a large collection of free content and activities that can be used inside, as well as outside, of the classroom. The award-winning program has been continuously running for more than 14 years, with the assistance of more than 150 volunteer scientists, faculty, and graduate students in biology and related fields. In 2010 the program released its new website interface and features that became the subject for articles in the journals Science and PLoS Biology.
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Ask a Biologist
| 0.865411
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95
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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 other examples, see group, and octonion.
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Multiplication Table
| 0.865347
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96
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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 operation ∗ {\displaystyle *} that has an identity element e, an element x is invertible if it has an inverse element, that is, if there exists an element inv ( x ) {\displaystyle \operatorname {inv} (x)} such that For example, a group is an algebraic structure with a binary operation that is associative, has an identity element, and for which all elements are invertible.
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Carrier set
| 0.865273
|
97
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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 logarithms also occur in the exponents of the time bounds for some divide and conquer algorithms, such as the Karatsuba algorithm for multiplying n-bit numbers in time O(nlog2 3), and the Strassen algorithm for multiplying n × n matrices in time O(nlog2 7). The occurrence of binary logarithms in these running times can be explained by reference to the master theorem for divide-and-conquer recurrences.
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Logarithmus binaris
| 0.865122
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98
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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 = 4 , {\displaystyle A+B=4,} and A B = 1.
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Solvability by radicals
| 0.864878
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99
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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 combination of the properties of the fungus and the algae or cyanobacteria.
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Lichenized fungus
| 0.864825
|
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