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100 | With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of group theory may be used when studying rings (algebraic objects that have two binary ope... | Abstract Algebra | 0.864809 |
101 | Sets with one or more operations that obey specific laws are called algebraic structures. When a new problem involves the same laws as such an algebraic structure, all the results that have been proved using only the laws of the structure can be directly applied to the new problem. In full generality, algebraic structu... | Algebraic system | 0.864723 |
102 | In computer science, best, worst, and average cases of a given algorithm express what the resource usage is at least, at most and on average, respectively. Usually the resource being considered is running time, i.e. time complexity, but could also be memory or some other resource. Best case is the function which perfor... | Average case analysis | 0.864654 |
103 | In general, in the presence of both a charge density ρ(r, t) and current density J(r, t), there will be both an electric and a magnetic field, and both will vary in time. They are determined by Maxwell's equations, a set of differential equations which directly relate E and B to the electric charge density (charge per ... | Field equations | 0.864562 |
104 | Approximately 6,000 students take this first exam, which consists of 25 multiple choice questions to be solved in 75 minutes, focusing on algebra-based mechanics. In the past, a quarter point was deducted for each incorrect answer. From 2015 onwards, no points were deducted for incorrect answers. Prior to 2018, the exa... | United States Physics Olympiad | 0.864497 |
105 | Of all SAT subject tests, the Biology E/M test was the only SAT II that allowed the test taker a choice between the ecological or molecular tests. A set of 60 questions was taken by all test takers for Biology and a choice of 20 questions was allowed between either the E or M tests. This test was graded on a scale betw... | SAT Subject Test in Biology E/M | 0.864397 |
106 | The SAT Subject Test in Mathematics Level 1 (formerly known as Math I or MathIC (the "C" representing the use of a calculator)) was the name of a one-hour multiple choice test given on algebra, geometry, basic trigonometry, algebraic functions, elementary statistics and basic foundations of calculus by The College Boar... | SAT Subject Test in Mathematics Level 1 | 0.864158 |
107 | Also, the Parallel Computing Center (Khmelnytskyi, Ukraine) obtained an ensemble of only 5 convolutional neural networks which performs on MNIST at 0.21 percent error rate. Some images in the testing dataset are barely readable and may prevent reaching test error rates of 0%. In 2018, researchers from Department of Sys... | MNIST database | 0.864005 |
108 | Clearly, linear time is required for the described transformation of numbers into points and then extracting their sorted order. Therefore, in the general case the convex hull of n points cannot be computed more quickly than sorting. The standard Ω(n log n) lower bound for sorting is proven in the decision tree model o... | Convex hull algorithms | 0.86399 |
109 | Data structure Data type Abstract data type Algebraic data type Composite type Array Associative array Deque List Linked list Queue Priority queue Skip list Stack Tree data structure Automatic garbage collection | Outline of combinatorics | 0.863934 |
110 | : 237–238 An object that can be electrically charged exhibits self capacitance, for which the electric potential is measured between the object and ground. Mutual capacitance is measured between two components, and is particularly important in the operation of the capacitor, an elementary linear electronic component de... | Stray capacitance | 0.863911 |
111 | Solutions Manual to Accompany Electricity and Magnetism: Berkeley Physics Course, Volume 2, Second Edition. McGraw-Hill. Purcell, Edward M.; Morin, David J. | Electricity and Magnetism (book) | 0.863694 |
112 | (A) represents transport processes include the transport into the cell from the surrounding seawater of primary calcification substrates Ca2+ and HCO3− (black arrows) and the removal of the end product H+ from the cell (gray arrow). The transport of Ca2+ through the cytoplasm to the coccolith vesicle (CV) is the domina... | Marine protozoans | 0.863608 |
113 | DNA sequencing may be used to determine the sequence of individual genes, larger genetic regions (i.e. clusters of genes or operons), full chromosomes, or entire genomes of any organism. DNA sequencing is also the most efficient way to indirectly sequence RNA or proteins (via their open reading frames). In fact, DNA se... | Next Generation Sequencing | 0.863605 |
114 | In computer science, hybrid algorithms are very common in optimized real-world implementations of recursive algorithms, particularly implementations of divide-and-conquer or decrease-and-conquer algorithms, where the size of the data decreases as one moves deeper in the recursion. In this case, one algorithm is used fo... | Hybrid algorithm | 0.863497 |
115 | In materials science and engineering, one is often interested in understanding the forces or stresses involved in the deformation of a material. For instance, if the material were a simple spring, the answer would be given by Hooke's law, which says that the force experienced by a spring is proportional to the distance... | Viscous forces | 0.86349 |
116 | In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single equations, namely as a: System of linear equations, System of nonlinear eq... | Simultaneous equation | 0.863476 |
117 | quadratic function In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusivel... | Glossary of calculus | 0.863438 |
118 | x∗~x=e=~x∗x.To summarize, the usual definition has: a single binary operation (signature (2)) 1 equational law (associativity) 2 quantified laws (identity and inverse)while the universal algebra definition has: 3 operations: one binary, one unary, and one nullary (signature (2,1,0)) 3 equational laws (associativity, id... | Equational theory | 0.863275 |
119 | Even though comparison-sorting n items requires Ω(n log n) operations, selection algorithms can compute the kth-smallest of n items with only Θ(n) operations. This includes the median, which is the n/2th order statistic (or for an even number of samples, the arithmetic mean of the two middle order statistics).Selection... | Multivariate median | 0.863001 |
120 | As of 2020, the quantitative reasoning section contains 20 questions covering various areas of mathematics, such as geometry, algebra, percentages, averages, ratio questions, drawing conclusions from a diagram, and so on. The allotted time is 20 minutes. For the most part, the difficulty level of questions in the secti... | Psychometric Entrance Test | 0.862759 |
121 | The multiple-choice section is scored immediately after the exam by computer. One point is awarded for each correct answer, no points are credited or deducted for unanswered questions, and points are no longer deducted for having an incorrect answer.Students' answers to the free-response section are reviewed in early J... | AP Statistics | 0.862753 |
122 | (B) Metabolic processes include the synthesis of CAPs (gray rectangles) by the Golgi complex (white rectangles) that regulate the nucleation and geometry of CaCO3 crystals. The completed coccolith (gray plate) is a complex structure of intricately arranged CAPs and CaCO3 crystals. (C) Mechanical and structural processe... | Coccolithophore | 0.862648 |
123 | While for substantial resources Illumina NextSeq, NovaSeq, PacBio Sequel, Oxford Nanopore and PromethION are preferred. Moreover, for pathogen sequencing the use of controls is of fundamental importance ensuring mNGS assay quality and stability over time; PhiX is used as sequencing control, then the other controls incl... | Clinical metagenomic next-generation sequencing | 0.862627 |
124 | The Individual Round comprises the Algebra, Geometry, and Combinatorics exams for February, and the General and Theme exams for November. Each of the exams is 50 minutes in length and contains 10 questions. The exams are short-answer, meaning that the answers can be any real number or even an algebraic expression. Befo... | HMMT | 0.862575 |
125 | The areas of study in Units 3 and 4 extend content from Mathematical Methods Units 3 and 4 to include rational and other quotient functions as well as other advanced mathematics topics such as logic and proof, complex numbers, vectors, differential equations, kinematics, and statistical inference. They also provide bac... | Mathematics education in Australia | 0.862571 |
126 | The number of comparisons needed to solve the problem of size n {\displaystyle n} , in a comparison-based model of computation such as a decision tree or algebraic decision tree, is Θ ( n log n ) {\displaystyle \Theta (n\log n)} . Here, Θ {\displaystyle \Theta } invokes big theta notation, meaning that the problem ca... | Element uniqueness problem | 0.862361 |
127 | This section tests the ability to apply scientific knowledge typically covered in school Science and Mathematics by the age of 16 (for example, GCSE in the UK and IGCSE internationally). It is made up of 27 questions, with 30 minutes to complete. The scope of scientific knowledge include that of Mathematics, Physics, C... | BioMedical Admissions Test | 0.86236 |
128 | In computer science, a sorting algorithm is an algorithm that puts elements of a list into an order. The most frequently used orders are numerical order and lexicographical order, and either ascending or descending. Efficient sorting is important for optimizing the efficiency of other algorithms (such as search and mer... | Sorting Algorithm | 0.862297 |
129 | In the theory of quadratic forms, the parabola is the graph of the quadratic form x2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x2 + y2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x2 − y2. Generalizations to more... | Parabola | 0.862249 |
130 | The content within the book is written using a question and answer format. It contains some 250 questions, which The Science Teacher states each are answered with a "concise and well-formulated essay that is informative and readable." The Science Teacher review goes on to state that many of the answers given in the boo... | A Question and Answer Guide to Astronomy | 0.862228 |
131 | (Some authors also use the "closure" axiom that x ∗ y belongs to A whenever x and y do, but here this is already implied by calling ∗ a binary operation.) This definition of a group does not immediately fit the point of view of universal algebra, because the axioms of the identity element and inversion are not stated p... | Equational theory | 0.862062 |
132 | Parametrization of planar curves is introduced, mainly focusing on lines, circles and parabolas. The plotting of cubic equations and solution of specific cases through polynomial long division and the remainder theorem enable a deeper understanding of polynomials. Mathematics Extension 2 (Must be studied concurrently w... | Mathematics education in Australia | 0.862041 |
133 | Approximately the top 400 students from the F=ma exam are invited to take a free-response, calculus-based exam covering all topics in introductory physics, including mechanics, electricity and magnetism, thermodynamics, fluids, relativity, waves, and nuclear and atomic physics. There are two parts in the exam, each all... | United States Physics Olympiad | 0.862012 |
134 | These expression levels may signify that the protein is performing a different function than previously known.The structure of a protein can also help determine its functions. Protein structure in turn may be elucidated with various techniques including X-ray crystallography or NMR. Dual-polarization interferometry may... | Protein moonlighting | 0.861973 |
135 | Many number systems, such as the integers and the rationals, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields.... | Elementary group theory | 0.861818 |
136 | DNA sequencing with commercially available NGS platforms is generally conducted with the following steps. First, DNA sequencing libraries are generated by clonal amplification by PCR in vitro. Second, the DNA is sequenced by synthesis, such that the DNA sequence is determined by the addition of nucleotides to the compl... | Short-read sequencing | 0.861756 |
137 | The DAT comprises four sections: survey of the natural sciences (90 minutes), perceptual ability (often called the PAT, 60 minutes), reading comprehension (60 minutes), and quantitative reasoning (45 minutes). The mathematics of the quantitative exam is similar to that of the SAT. The first section is divided into ques... | Dental Admission Test | 0.861709 |
138 | There are several possible ways to specify the position of a line in the plane. A simple way is by the pair (m, b) where the equation of the line is y = mx + b. Here m is the slope and b is the y-intercept. This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler alg... | Line geometry | 0.861687 |
139 | When a group G {\displaystyle G} has a normal subgroup N {\displaystyle N} other than { 1 } {\displaystyle \{1\}} and G {\displaystyle G} itself, questions about G {\displaystyle G} can sometimes be reduced to questions about N {\displaystyle N} and G / N {\displaystyle G/N} . A nontrivial group is called simple if it ... | Elementary group theory | 0.861678 |
140 | If all elements to be sorted are distinct, the expected number of comparisons performed in the average case by randomized bogosort is asymptotically equivalent to (e − 1)n!, and the expected number of swaps in the average case equals (n − 1)n!. The expected number of swaps grows faster than the expected number of compa... | Bogosort | 0.861635 |
141 | Transfer of heat between a chemical system and its surroundings during change of phase or chemical reaction taking place called thermochemistry Study of colligative properties of number of species present in solution. Number of phases, number of components and degree of freedom (or variance) can be correlated with one ... | Physical Chemistry | 0.861627 |
142 | The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times... | Monte-Carlo method | 0.861578 |
143 | Merge the sorted results A + A {\displaystyle A+A} , B + B {\displaystyle B+B} , and A + B {\displaystyle A+B} together.The number of comparisons C ( n ) {\displaystyle C(n)} needed to perform this recursive algorithm on an input of n {\displaystyle n} items can be analyzed using the recurrence relation where the 2 C (... | X + Y sorting | 0.861534 |
144 | StructureGroup (mathematics) Lagrange's theorem (group theory) Subgroup Coset Normal subgroup Characteristic subgroup Centralizer and normalizer subgroups Derived group Frattini subgroup Fitting subgroup Classification of finite simple groups Sylow theorems Local analysisConstructionsFree group Presentation of a group ... | List of abstract algebra topics | 0.861526 |
145 | Cost-efficiency as well as absolute speed can be critical, especially in cluster environments where lower node costs allow purchasing more nodes. Increasing software speed Some Sort Benchmark entrants use a variation on radix sort for the first phase of sorting: they separate data into one of many "bins" based on the b... | External Sorting | 0.861496 |
146 | Semi-groups, quasi-groups, and monoids are algebraic structures similar to groups, but with less constraints on the operation. They comprise a set and a closed binary operation but do not necessarily satisfy the other conditions. A semi-group has an associative binary operation but might not have an identity element. | Diagrammatic algebra | 0.861416 |
147 | As of 2018, the Illumina monopoly on high-quality next-generation sequencing reagents has meant that the sequencing reagents alone cost more than FDA-approved syndromic testing panels. Also additional direct costs of metagenomics such as extraction, library preparation, and computational analysis have to be considered.... | Clinical metagenomic next-generation sequencing | 0.861413 |
148 | Racket takes this further: the language is made fully safe-for-space, via live variable analysis. This complements the precise garbage collector and in some cases, like in the implementation of Lazy Racket, the two features are crucial for proper execution. This is in addition to additional compiler optimizations such ... | Racket features | 0.861368 |
149 | When the red homozygous flower is paired with the white homozygous flower, the result yields a pink snapdragon flower. The pink snapdragon is the result of incomplete dominance. A similar type of incomplete dominance is found in the four o'clock plant wherein pink color is produced when true-bred parents of white and r... | Homozygous dominant | 0.861299 |
150 | In the U.S., the SAT Subject Test in Mathematics Level 2 (formerly known as Math II or Math IIChest, the "C" representing chest) was a one-hour multiple choice test. The questions covered a broad range of topics. Approximately 10-14% of questions focused on numbers and operations, 48-52% focused on algebra and function... | SAT Subject Test in Mathematics Level 2 | 0.861125 |
151 | The lengths of BM and CM are: Using the intersecting chords theorem on the chords BC and DE, we get Substituting: Rearranging: For any given cone and parabola, r and θ are constants, but x and y are variables that depend on the arbitrary height at which the horizontal cross-section BECD is made. This last equation show... | Parabolic curve | 0.860848 |
152 | Task 1: "Answer retrieval" matching old post answers to newly posed questions, and Task 2: "Formula retrieval" matching old post formulae to new questions. Starting with the domain of mathematics, which involves formula language, the goal is to later extend the task to other domains (e.g., STEM disciplines, such as che... | Answer engine | 0.860808 |
153 | Tasks tend to require the application of number sense and spatial sense; recognizing and working with mathematical relationships, patterns, and proportions expressed in verbal or numerical form; and interpretation and basic analysis of data and statistics in texts, tables and graphs.Level 4 326 – 375 Tasks at this leve... | Programme for the International Assessment of Adult Competencies | 0.860681 |
154 | In three-space, the most general equation of the second degree in x, y and z has the form where the quantities A , B , C , … , J {\displaystyle A,B,C,\ldots ,J} are positive or negative numbers or zero. The points in space satisfying such an equation all lie on a surface. Any second-degree equation which does not reduc... | Translation of axes | 0.86067 |
155 | Physical chemistry: free energy, phase diagram, phase rule, transport phenomena Statistical mechanics: statistical ensemble, phase space, chemical potential, Gibbs entropy, Gibbs paradox Mathematics: Vector Analysis, convex analysis, Gibbs phenomenon Electromagnetism: Maxwell's equations, birefringence | Josiah Willard Gibbs | 0.860575 |
156 | Consider a simple example drawn from physics. A spring should obey Hooke's law which states that the extension of a spring y is proportional to the force, F, applied to it. y = f ( F , k ) = k F {\displaystyle y=f(F,k)=kF\!} constitutes the model, where F is the independent variable. | Least-squares fit | 0.860503 |
157 | 1 comment line: tools must account for all code and comments regardless of comment placement.Even the "logical" and "physical" SLOC values can have a large number of varying definitions. Robert E. Park (while at the Software Engineering Institute) and others developed a framework for defining SLOC values, to enable peo... | Unified Code Count (UCC) | 0.860469 |
158 | This is the topic of the scientific field of structural biology, which employs techniques such as X-ray crystallography, NMR spectroscopy, cryo-electron microscopy (cryo-EM) and dual polarisation interferometry, to determine the structure of proteins. Protein structures range in size from tens to several thousand amino... | Protein Structure | 0.860449 |
159 | Q1 General biology Q2 Cytology Q3 Genetics Q4 Physiology Q5 Biochemistry Q6 Biophysics Q7 Molecular biology Q8 Bioengineering Q9 Zoology and botany | Chinese Library Classification | 0.860366 |
160 | In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently. | Degree of an extension | 0.860366 |
161 | In this experiment the coin was tossed by balancing it on the forefinger, flipping it using the thumb so that it spun through the air for about a foot before landing on a flat cloth spread over a table. Edwin Thompson Jaynes claimed that when a coin is caught in the hand, instead of being allowed to bounce, the physica... | Fair coin | 0.860275 |
162 | Second, average-case complexity analysis provides tools and techniques to generate hard instances of problems which can be utilized in areas such as cryptography and derandomization. Third, average-case complexity allows discriminating the most efficient algorithm in practice among algorithms of equivalent best case co... | Average case complexity | 0.860177 |
163 | The Math Test – No Calculator section has 20 questions (15 multiple choice and 5 grid-in) and lasts 25 minutes. The Math Test – Calculator section has 38 questions (30 multiple choice and 8 grid-in) and lasts 55 minutes.Several scores are provided to the test taker for the math test. A subscore (on a scale of 1 to 15) ... | Scholastic aptitude test | 0.859848 |
164 | Examples of the need for merging include external sorting and streaming results from distributed data such as a log structured merge tree. The inner loop is obtaining the min element, replacing with the next element for the corresponding input stream, then doing a sift-down heap operation. (Alternatively the replace fu... | Minimum-heap property | 0.859688 |
165 | Experiment: Rosalind Franklin used pure DNA to perform X-ray diffraction to produce photo 51. The results showed an X-shape. Analysis: When Watson saw the detailed diffraction pattern, he immediately recognized it as a helix. He and Crick then produced their model, using this information along with the previously known... | Experimental confirmation | 0.859543 |
166 | The geometry consists of two parallel plates as perfect conductors (PEC), an idealized structure, filled by two stacked planar slabs of homogeneous and isotropic materials with their respective constitutive parameters ε1, ε2, u1, u2. Each slab has thickness = d, slab 1 = d1, and slab 2 = d2. Choosing which combination ... | Metamaterial antennas | 0.859532 |
167 | Students in computer science and economics might have the option of taking algorithmic game theory. Students in the physical sciences and engineering need to understand error analysis for their laboratory sessions and courses. Advanced undergraduates and beginning graduate students in physics may take a course on advan... | Mathematics education in the United States | 0.859521 |
168 | The class of all semigroups forms a variety of algebras of signature (2), meaning that a semigroup has a single binary operation. A sufficient defining equation is the associative law: x ( y z ) = ( x y ) z . {\displaystyle x(yz)=(xy)z.} The class of groups forms a variety of algebras of signature (2,0,1), the three op... | Finitary algebraic category | 0.859424 |
169 | The tendency of people to provide the answer 1/2 is likely due to a tendency to ignore context that may seem unimpactful. For example, how the question is posed to the warden can affect the answer. This can be shown by considering a modified case, where P ( A ) = 1 4 , P ( B ) = 1 4 , P ( C ) = 1 2 {\displaystyle P(A)=... | Three Prisoners problem | 0.859411 |
170 | In classical physics, a spring can be seen as a device that stores potential energy, specifically elastic potential energy, by straining the bonds between the atoms of an elastic material. Hooke's law of elasticity states that the extension of an elastic rod (its distended length minus its relaxed length) is linearly p... | Ideal spring | 0.859392 |
171 | Proposition. Consider polynomial algebra Pol = C and matrix M with elements in some algebra EndPol. The elements y i = ∑ k M i k ⊗ x k ∈ E n d P o l ⊗ P o l {\displaystyle y_{i}=\sum _{k}M_{ik}\otimes x_{k}\in EndPol\otimes Pol} commute among themselves if and only if M is a Manin matrix. | Manin matrices | 0.859297 |
172 | The rays are focused onto an area the size of a cooking pot and can reach 4,000 °C (7,230 °F), depending on the process installed, for example: about 1,000 °C (1,830 °F) for metallic receivers producing hot air for the next generation solar towers as it will be tested at the Themis plant with the Pegase project about 1... | Solar furnace | 0.859219 |
173 | He showed that if 1/2 were correct, it would lead to a contradiction, so 1/2 cannot be correct. This simple but counterintuitive puzzle is used as a standard example in teaching probability theory. The solution illustrates some basic principles, including the Kolmogorov axioms. | Bertrand's box paradox | 0.859126 |
174 | Typically cocktail sort is less than two times faster than bubble sort. Another optimization can be that the algorithm remembers where the last actual swap has been done. In the next iteration, there will be no swaps beyond this limit and the algorithm has shorter passes. As the cocktail shaker sort goes bidirectionall... | Cocktail shaker sort | 0.859115 |
175 | Probability Theory 9. Sequences and Series 10. Topology 11. Set Theory Recently a similar competition has been started in France. | Miklós Schweitzer Competition | 0.859029 |
176 | Sudoku: How many puzzles have exactly one solution? How many puzzles with exactly one solution are minimal? What is the maximum number of givens for a minimal puzzle? Tic-tac-toe variants: Given a width of tic-tac-toe board, what is the smallest dimension such that X is guaranteed a winning strategy? (See also: Hales-J... | Open problem in mathematics | 0.858933 |
177 | In addition, several languages have been designed to run natively on the JVM, including Clojure, Groovy, and Scala. Java syntax borrows heavily from C and C++, but object-oriented features are modeled after Smalltalk and Objective-C. Java eschews certain low-level constructs such as pointers and has a very simple memor... | Oracle Java | 0.85885 |
178 | ECAT consists of 100 multiple choice questions. Each question carries 4 marks, with 1 mark deducted for each wrong answer. Total marks are 400. 30 questions are from Mathematics, 30 from Chemistry, 30 from Physics, 10 from English language. According to the updated rules of Pakistan Engineering Council, from entry 2023... | ECAT Pakistan | 0.858848 |
179 | Math League runs three contest formats: Grades 4-5: 30 multiple-choice questions to solve in 30 minutes, covering arithmetic and basic principles Grades 6-8: 35 multiple-choice questions to solve in 30 minutes, covering advanced arithmetic and basic topics in geometry and algebra Grades 9-12: Series of 6 contests. Each... | Math League | 0.858842 |
180 | For some questions, there is no known way to find an answer quickly, but if one is provided with information showing what the answer is, it is possible to verify the answer quickly. The class of questions for which an answer can be verified in polynomial time is NP, which stands for "nondeterministic polynomial time".A... | P versus NP problem | 0.858804 |
181 | According to Snopes.com, more recent (1999 and 1988) versions identify the problem as a question in "a physics degree exam at the University of Copenhagen" and the student was Niels Bohr, and includes the following answers: Tying a piece of string to the barometer, lowering the barometer from the roof to the ground, an... | Barometer question | 0.858701 |
182 | In a slight abuse of notation, the word "structure" can also refer to just the operations on a structure, instead of the underlying set itself. For example, the sentence, "We have defined a ring structure on the set A {\displaystyle A} ", means that we have defined ring operations on the set A {\displaystyle A} . For a... | Carrier set | 0.8587 |
183 | Theorem: The identity element e for a binary operation (addition or multiplication) of a ring is unique. | Proofs of elementary ring properties | 0.858696 |
184 | They might also depend on the way in which the data is arranged; for example, some sorting algorithms perform poorly on data which is already sorted, or which is sorted in reverse order. In practice, there are other factors which can affect the efficiency of an algorithm, such as requirements for accuracy and/or reliab... | Computationally efficient | 0.858693 |
185 | They attribute this to the relatively small amount of research put into patience sort, and develop several optimizations that bring its performance to within a factor of two of that of quicksort.If values of cards are in the range 1, . . . , n, there is an efficient implementation with O ( n log n ) {\displaystyle O(... | Patience sort | 0.858567 |
186 | Some researchers have achieved "near-human performance" on the MNIST database, using a committee of neural networks; in the same paper, the authors achieve performance double that of humans on other recognition tasks. The highest error rate listed on the original website of the database is 12 percent, which is achieved... | MNIST database | 0.858542 |
187 | The College Board suggested a year-long course in biology at the college preparatory level, as well as a one-year course in algebra, and lab experience as preparation for the test. The test required understanding of biological data and concepts, science-related terms, and the ability to effectively synthesize and inter... | SAT Subject Test in Biology E/M | 0.858485 |
188 | So in our previous example, we might say that the problem requires O ( n ) {\displaystyle O(n)} steps to solve. Perhaps the most important open problem in all of computer science is the question of whether a certain broad class of problems denoted NP can be solved efficiently. This is discussed further at Complexity cl... | Theory of algorithms | 0.858481 |
189 | Solutions Manual to Accompany Electricity and Magnetism: Berkeley Physics Course, Volume 2, First Edition. McGraw-Hill. Purcell, Edward M. | Electricity and Magnetism (book) | 0.858403 |
190 | (2) Despite their large size, some proteins crystallize readily into symmetric crystals, consistent with the idea of symmetric faces that match up upon association. (3) Proteins bind metal ions; since metal-binding sites must have specific bond geometries (e.g., octahedral), it was plausible to assume that the entire p... | Cyclol | 0.858337 |
191 | The following module-like structures have the common feature of having two sets, A and B, so that there is a binary operation from A×A into A and another operation from A×B into A. Modules, counting the ring operations, have at least three binary operations. Group with operators: a group G with a set Ω and a binary ope... | Outline of algebraic structures | 0.85828 |
192 | But if some characteristic of the items is exploitable (for example, they are already arranged in some particular order), a different method can be used, or even a custom-made sort routine. After the programmer is reasonably sure that the best algorithm is selected, code optimization can start. Loops can be unrolled (f... | Software optimization | 0.858279 |
193 | As another example from physics, consider a parallel-plate capacitor, in which the plates can move relative to one another. Such a capacitor would allow transfer of the electric energy which is stored in the capacitor into external mechanical work, done by the force acting on the plates. One may think of the electric c... | Legendre transformation | 0.858271 |
194 | There are two rounds in the Senior Section: a written round (Round 1) and an invitational round (Round 2). The paper in Round 1 comprises 5 multiple-choice questions, each with five options, and 20 short answer questions. The Senior section is geared towards Upper Secondary students, and topics tested include number th... | Singapore Mathematical Olympiad | 0.858265 |
195 | There was a hard, universally used pass-fail criterion for the Eddy Test, and a second chance was normally never allowed. Eddy described the test as having questions with multiple-choice answers, with each of the answers giving some indication of the test-taker's mathematics/physics knowledge, creativity, reasoning abi... | Eddy Test | 0.858251 |
196 | An example of dimensional analysis can be found for the case of the mechanics of a thin, solid and parallel-sided rotating disc. There are five variables involved which reduce to two non-dimensional groups. The relationship between these can be determined by numerical experiment using, for example, the finite element m... | Buckingham pi theorem | 0.858235 |
197 | For any algebraic structure it is possible to consider the minimal cardinality of generators of the structure. For algebraic extensions, algebraic degree and separable degree are often employed (note that the algebraic degree equals the dimension of the extension as a vector space over the smaller field). For non-algeb... | Bounding number | 0.85823 |
198 | Given groups G (with operation *) and H (with operation ∆), the direct product G × H is defined as follows: The resulting algebraic object satisfies the axioms for a group. Specifically: Associativity The binary operation on G × H is associative. Identity The direct product has an identity element, namely (1G, 1H), whe... | Direct product of groups | 0.858118 |
199 | Modern quantum mechanics explains this in terms of electron shells and subshells which can each hold a number of electrons determined by the Pauli exclusion principle. Thus the n = 1 state can hold one or two electrons, while the n = 2 state can hold up to eight electrons in 2s and 2p subshells. In helium, all n = 1 st... | Sp orbital | 0.858075 |
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