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"The examiner and Calandra, who was called to advise on the case, faced a moral dilemma. According to the format of the exam, a correct answer deserved a full credit. But issuing a full credit would have violated academic standards by rewarding a student who had not demonstrated competence in the academic field that had been tested (physics).
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The teacher can steer the students either through careful design of the questions (this rules out barometer questions), or through guiding the students to the desired choices. In case of the original barometer question, the examiner may explicitly say that the problem has more than one solution, insist on applying the laws of physics, or give them the "ending point" of the solution: "How did I discover that the building was 410 feet in height with only a barometer? "Herson used the Calandra account as an illustration of the difference between academic tests and assessment in education. Tests, even the ones designed for reliability and validity, are useful, but they are not sufficient in real-world education.Sanders interpreted Calandra's story as a conflict between perfection and optimal solutions: "We struggle to determine a 'best' answer, when a simple call to a building superintendent (the resource man) would quickly provide adequate information."
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Professor of physics Mark Silverman used what he called "The Barometer-Story formula" precisely for explaining the subject of pressure and recommended it to physics teachers. Silverman called Calandra's story "a delightful essay that I habitually read to my class whenever we study fluids ... the essay is short, hilarious and satisfying (at least to me and my class). "Financial advisor Robert G. Allen presented Calandra's essay to illustrate the process and role of creativity in finance. "Creativity is born when you have a problem to solve.
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van Emde Boas priority queue vehicle routing problem Veitch diagram Venn diagram vertex vertex coloring vertex connectivity vertex cover vertical visibility map virtual hashing visibility map visible (geometry) Viterbi algorithm VP-tree VRP (vehicle routing problem)
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packing (see set packing) padding argument pagoda pairing heap PAM (point access method) parallel computation thesis parallel prefix computation parallel random-access machine (PRAM) parametric searching parent partial function partially decidable problem partially dynamic graph problem partially ordered set partially persistent data structure partial order partial recursive function partition (set theory) passive data structure patience sorting path (graph theory) path cover path system problem Patricia tree pattern pattern element P-complete PCP Peano curve Pearson's hashing perfect binary tree perfect hashing perfect k-ary tree perfect matching perfect shuffle performance guarantee performance ratio permutation persistent data structure phonetic coding pile (data structure) pipelined divide and conquer planar graph planarization planar straight-line graph PLOP-hashing point access method pointer jumping pointer machine poissonization polychotomy polyhedron polylogarithmic polynomial polynomial-time approximation scheme (PTAS) polynomial hierarchy polynomial time polynomial-time Church–Turing thesis polynomial-time reduction polyphase merge polyphase merge sort polytope poset postfix traversal Post machine (see Post–Turing machine) postman's sort postorder traversal Post correspondence problem potential function (see potential method) predicate prefix prefix code prefix computation prefix sum prefix traversal preorder traversal primary clustering primitive recursive Prim's algorithm principle of optimality priority queue prisoner's dilemma PRNG probabilistic algorithm probabilistically checkable proof probabilistic Turing machine probe sequence Procedure (computer science) process algebra proper (see proper subset) proper binary tree proper coloring proper subset property list prune and search pseudorandom number generator pth order Fibonacci numbers P-tree purely functional language pushdown automaton (PDA) pushdown transducer p-way merge sort
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objective function occurrence octree odd–even sort offline algorithm offset (computer science) omega omicron one-based indexing one-dimensional online algorithm open addressing optimal optimal cost optimal hashing optimal merge optimal mismatch optimal polygon triangulation problem optimal polyphase merge optimal polyphase merge sort optimal solution optimal triangulation problem optimal value optimization problem or oracle set oracle tape oracle Turing machine orders of approximation ordered array ordered binary decision diagram (OBDD) ordered linked list ordered tree order preserving hash order preserving minimal perfect hashing oriented acyclic graph oriented graph oriented tree orthogonal drawing orthogonal lists orthogonally convex rectilinear polygon oscillating merge sort out-branching out-degree overlapping subproblems
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Identity function ideal merge implication implies implicit data structure in-branching inclusion–exclusion principle inclusive or incompressible string incremental algorithm in-degree independent set (graph theory) index file information theoretic bound in-place algorithm in-order traversal in-place sort insertion sort instantaneous description integer linear program integer multi-commodity flow integer polyhedron interactive proof system interface interior-based representation internal node internal sort interpolation search interpolation-sequential search interpolation sort intersection (set theory) interval tree intractable introsort introspective sort inverse Ackermann function inverted file index inverted index irreflexive isomorphic iteration
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tail tail recursion tango tree target temporal logic terminal (see Steiner tree) terminal node ternary search ternary search tree (TST) text searching theta threaded binary tree threaded tree three-dimensional three-way merge sort three-way radix quicksort time-constructible function time/space complexity top-down radix sort top-down tree automaton top-node topological order topological sort topology tree total function totally decidable language totally decidable problem totally undecidable problem total order tour tournament towers of Hanoi tractable problem transducer transition (see finite-state machine) transition function (of a finite-state machine or Turing machine) transitive relation transitive closure transitive reduction transpose sequential search travelling salesman problem (TSP) treap tree tree automaton tree contraction tree editing problem tree sort tree transducer tree traversal triangle inequality triconnected graph trie trinary function tripartition Turbo-BM Turbo Reverse Factor Turing machine Turing reduction Turing transducer twin grid file two-dimensional two-level grid file 2–3 tree 2–3–4 tree Two Way algorithm two-way linked list two-way merge sort
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saguaro stack saturated edge SBB tree scan scapegoat tree search algorithm search tree search tree property secant search secondary clustering memory segment select algorithm select and partition selection problem selection sort select kth element select mode self-loop self-organizing heuristic self-organizing list self-organizing sequential search semidefinite programming separate chaining hashing separator theorem sequential search set set cover set packing shadow heap shadow merge shadow merge insert shaker sort Shannon–Fano coding shared memory Shell sort Shift-Or Shor's algorithm shortcutting shortest common supersequence shortest common superstring shortest path shortest spanning tree shuffle shuffle sort sibling Sierpiński curve Sierpinski triangle sieve of Eratosthenes sift up signature Simon's algorithm simple merge simple path simple uniform hashing simplex communication simulated annealing simulation theorem single-destination shortest-path problem single-pair shortest-path problem single program multiple data single-source shortest-path problem singly linked list singularity analysis sink sinking sort skd-tree skew-symmetry skip list skip search slope selection Smith algorithm Smith–Waterman algorithm smoothsort solvable problem sort algorithm sorted array sorted list sort in-place sort merge soundex space-constructible function spanning tree sparse graph sparse matrix sparsification sparsity spatial access method spectral test splay tree SPMD square matrix square root SST (shortest spanning tree) stable stack (data structure) stack tree star-shaped polygon start state state state machine state transition static data structure static Huffman encoding s-t cut st-digraph Steiner minimum tree Steiner point Steiner ratio Steiner tree Steiner vertex Steinhaus–Johnson–Trotter algorithm Stirling's approximation Stirling's formula stooge sort straight-line drawing strand sort strictly decreasing strictly increasing strictly lower triangular matrix strictly upper triangular matrix string string editing problem string matching string matching on ordered alphabets string matching with errors string matching with mismatches string searching strip packing strongly connected component strongly connected graph strongly NP-hard subadditive ergodic theorem subgraph isomorphism sublinear time algorithm subsequence subset substring subtree succinct data structure suffix suffix array suffix automaton suffix tree superimposed code superset supersink supersource symmetric relation symmetrically linked list symmetric binary B-tree symmetric set difference symmetry breaking symmetric min max heap
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qm sort qsort quadratic probing quadtree quadtree complexity theorem quad trie quantum computation queue quicksort
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labeled graph language last-in, first-out (LIFO) Las Vegas algorithm lattice (group) layered graph LCS leaf least common multiple (LCM) leftist tree left rotation left-child right-sibling binary tree also termed first-child next-sibling binary tree, doubly chained tree, or filial-heir chain Lempel–Ziv–Welch (LZW) level-order traversal Levenshtein distance lexicographical order linear linear congruential generator linear hash linear insertion sort linear order linear probing linear probing sort linear product linear program linear quadtree linear search link linked list list list contraction little-o notation Lm distance load factor (computer science) local alignment local optimum logarithm, logarithmic scale longest common subsequence longest common substring Lotka's law lower bound lower triangular matrix lowest common ancestor l-reduction
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cactus stack Calculus of Communicating Systems (CCS) calendar queue candidate consistency testing candidate verification canonical complexity class capacitated facility location capacity capacity constraint Cartesian tree cascade merge sort caverphone Cayley–Purser algorithm C curve cell probe model cell tree cellular automaton centroid certificate chain (order theory) chaining (algorithm) child Chinese postman problem Chinese remainder theorem Christofides algorithm Christofides heuristic chromatic index chromatic number Church–Turing thesis circuit circuit complexity circuit value problem circular list circular queue clique clique problem clustering (see hash table) clustering free coalesced hashing coarsening cocktail shaker sort codeword coding tree collective recursion collision collision resolution scheme Colussi combination comb sort Communicating Sequential Processes commutative compact DAWG compact trie comparison sort competitive analysis competitive ratio complement complete binary tree complete graph completely connected graph complete tree complexity complexity class computable concave function concurrent flow concurrent read, concurrent write concurrent read, exclusive write configuration confluently persistent data structure conjunction connected components connected graph co-NP constant function continuous knapsack problem Cook reduction Cook's theorem counting sort covering CRCW Crew (algorithm) critical path problem CSP (communicating sequential processes) CSP (constraint satisfaction problem) CTL cuckoo hashing cuckoo filter cut (graph theory) cut (logic programming) cutting plane cutting stock problem cutting theorem cut vertex cycle sort cyclic redundancy check (CRC)
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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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Galil–Giancarlo Galil–Seiferas gamma function GBD-tree geometric optimization problem global optimum gnome sort goobi graph graph coloring graph concentration graph drawing graph isomorphism graph partition Gray code greatest common divisor (GCD) greedy algorithm greedy heuristic grid drawing grid file Grover's algorithm
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Rabin–Karp string-search algorithm radix quicksort radix sort ragged matrix Raita algorithm random-access machine random number generation randomization randomized algorithm randomized binary search tree randomized complexity randomized polynomial time randomized rounding randomized search tree Randomized-Select random number generator random sampling range (function) range sort Rank (graph theory) Ratcliff/Obershelp pattern recognition reachable rebalance recognizer rectangular matrix rectilinear rectilinear Steiner tree recurrence equations recurrence relation recursion recursion termination recursion tree recursive (computer science) recursive data structure recursive doubling recursive language recursively enumerable language recursively solvable red–black tree reduced basis reduced digraph reduced ordered binary decision diagram (ROBDD) reduction reflexive relation regular decomposition rehashing relation (mathematics) relational structure relative performance guarantee relaxation relaxed balance rescalable restricted universe sort result cache Reverse Colussi Reverse Factor R-file Rice's method right rotation right-threaded tree root root balance rooted tree rotate left rotate right rotation rough graph RP R+-tree R*-tree R-tree run time
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Malhotra–Kumar–Maheshwari blocking flow (ru.) Manhattan distance many-one reduction Markov chain marriage problem (see assignment problem) Master theorem (analysis of algorithms) matched edge matched vertex matching (graph theory) matrix matrix-chain multiplication problem max-heap property maximal independent set maximally connected component Maximal Shift maximum bipartite matching maximum-flow problem MAX-SNP Mealy machine mean median meld (data structures) memoization merge algorithm merge sort Merkle tree meromorphic function metaheuristic metaphone midrange Miller–Rabin primality test min-heap property minimal perfect hashing minimum bounding box (MBB) minimum cut minimum path cover minimum spanning tree minimum vertex cut mixed integer linear program mode model checking model of computation moderately exponential MODIFIND monotone priority queue monotonically decreasing monotonically increasing Monte Carlo algorithm Moore machine Morris–Pratt move (finite-state machine transition) move-to-front heuristic move-to-root heuristic multi-commodity flow multigraph multilayer grid file multiplication method multiprefix multiprocessor model multiset multi suffix tree multiway decision multiway merge multiway search tree multiway tree Munkres' assignment algorithm
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In physics, this phenomenon is known as Friedel oscillations, and applies both to surface and bulk screening. In each case the net electric field does not fall off exponentially in space, but rather as an inverse power law multiplied by an oscillatory term. Theoretical calculations can be obtained from quantum hydrodynamics and density functional theory (DFT).
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In neurobiology, the length constant (λ) is a mathematical constant used to quantify the distance that a graded electric potential will travel along a neurite via passive electrical conduction. The greater the value of the length constant, the farther the potential will travel. A large length constant can contribute to spatial summation—the electrical addition of one potential with potentials from adjacent areas of the cell. The length constant can be defined as: λ = r m r i + r o {\displaystyle \lambda ={\sqrt {\frac {r_{m}}{r_{i}+r_{o}}}}} where rm is the membrane resistance (the force that impedes the flow of electric current from the outside of the membrane to the inside, and vice versa), ri is the axial resistance (the force that impedes current flow through the axoplasm, parallel to the membrane), and ro is the extracellular resistance (the force that impedes current flow through the extracellular fluid, parallel to the membrane).
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This harmonic condition is frequently used by physicists when working with gravitational waves. This condition is also frequently used to derive the post-Newtonian approximation. Although the harmonic coordinate condition is not generally covariant, it is Lorentz covariant. This coordinate condition resolves the ambiguity of the metric tensor g μ ν {\displaystyle g_{\mu \nu }\!} by providing four additional differential equations that the metric tensor must satisfy.
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They are frequently used in cosmology.The synchronous coordinate condition is neither generally covariant nor Lorentz covariant. This coordinate condition resolves the ambiguity of the metric tensor g μ ν {\displaystyle g_{\mu \nu }\!} by providing four algebraic equations that the metric tensor must satisfy.
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In interactions of proteins with nucleic acids, arginine residues are important hydrogen bond donors for the phosphate backbone — many arginine-methylated proteins have been found to interact with DNA or RNA.Enzymes that facilitate histone acetylation as well as histones themselves can be arginine methylated. Arginine methylation affects the interactions between proteins and has been implicated in a variety of cellular processes, including protein trafficking, signal transduction and transcriptional regulation. In epigenetics, arginine methylation of histones H3 and H4 is associated with a more accessible chromatin structure and thus higher levels of transcription. The existence of arginine demethylases that could reverse arginine methylation is controversial.
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Protein methylation is a type of post-translational modification featuring the addition of methyl groups to proteins. It can occur on the nitrogen-containing side-chains of arginine and lysine, but also at the amino- and carboxy-termini of a number of different proteins. In biology, methyltransferases catalyze the methylation process, activated primarily by S-adenosylmethionine. Protein methylation has been most studied in histones, where the transfer of methyl groups from S-adenosyl methionine is catalyzed by histone methyltransferases. Histones that are methylated on certain residues can act epigenetically to repress or activate gene expression.
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Further branches crucially applying groups include algebraic geometry and number theory.In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as physics, chemistry and computer science benefit from the concept.
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Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group.
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Examples and applications of groups abound. A starting point is the group Z {\displaystyle \mathbb {Z} } of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra.
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Artin, Michael (2018), Algebra, Prentice Hall, ISBN 978-0-13-468960-9, Chapter 2 contains an undergraduate-level exposition of the notions covered in this article. Cook, Mariana R. (2009), Mathematicians: An Outer View of the Inner World, Princeton, N.J.
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In this group, the order of r 1 {\displaystyle r_{1}} is 4, as is the order of the subgroup R {\displaystyle R} that this element generates. The order of the reflection elements f v {\displaystyle f_{\mathrm {v} }} etc. is 2. Both orders divide 8, as predicted by Lagrange's theorem. The groups F p × {\displaystyle \mathbb {F} _{p}^{\times }} of multiplication modulo a prime p {\displaystyle p} have order p − 1 {\displaystyle p-1} .
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The Sylow theorems give a partial converse. The dihedral group D 4 {\displaystyle \mathrm {D} _{4}} of symmetries of a square is a finite group of order 8.
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In infinite groups, such an n {\displaystyle n} may not exist, in which case the order of a {\displaystyle a} is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element. More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group G {\displaystyle G} the order of any finite subgroup H {\displaystyle H} divides the order of G {\displaystyle G} .
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The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (factorial of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group S N {\displaystyle \mathrm {S} _{N}} for a suitable integer N {\displaystyle N} , according to Cayley's theorem.
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Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved. Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.
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These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.Group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons.
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For example, an element of the (2,3,7) triangle group acts on a triangular tiling of the hyperbolic plane by permuting the triangles. By a group action, the group pattern is connected to the structure of the object being acted on. In chemistry, point groups describe molecular symmetries, while space groups describe crystal symmetries in crystallography.
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Symmetry groups are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below). Conceptually, group theory can be thought of as the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element can be associated to some operation on X and the composition of these operations follows the group law.
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211 (Revised third ed. ), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 Lang, Serge (2005), Undergraduate Algebra (3rd ed. ), Berlin, New York: Springer-Verlag, ISBN 978-0-387-22025-3.
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), Lexington, Mass. : Xerox College Publishing, MR 0356988. Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol.
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), Upper Saddle River, NJ: Prentice Hall Inc., ISBN 978-0-13-374562-7, MR 1375019. Herstein, Israel Nathan (1975), Topics in Algebra (2nd ed.
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: Princeton University Press, ISBN 978-0-691-13951-7 Hall, G. G. (1967), Applied Group Theory, American Elsevier Publishing Co., Inc., New York, MR 0219593, an elementary introduction. Herstein, Israel Nathan (1996), Abstract Algebra (3rd ed.
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Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the semidirect product construction; D 4 {\displaystyle \mathrm {D} _{4}} is an example. The first isomorphism theorem implies that any surjective homomorphism ϕ: G → H {\displaystyle \phi \colon G\to H} factors canonically as a quotient homomorphism followed by an isomorphism: G → G / ker ϕ → ∼ H {\displaystyle G\to G/\ker \phi \;{\stackrel {\sim }{\to }}\;H} . Surjective homomorphisms are the epimorphisms in the category of groups.
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The relationship between the two types of constant is given in association and dissociation constants. In biochemistry, an oxygen molecule can bind to an iron(II) atom in a heme prosthetic group in hemoglobin. The equilibrium is usually written, denoting hemoglobin by Hb, as Hb + O2 ⇌ HbO2but this representation is incomplete as the Bohr effect shows that the equilibrium concentrations are pH-dependent.
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Equilibrium chemistry is concerned with systems in chemical equilibrium. The unifying principle is that the free energy of a system at equilibrium is the minimum possible, so that the slope of the free energy with respect to the reaction coordinate is zero. This principle, applied to mixtures at equilibrium provides a definition of an equilibrium constant. Applications include acid–base, host–guest, metal–complex, solubility, partition, chromatography and redox equilibria.
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In aqueous solution H+ denotes a solvated hydronium ion.The Brønsted–Lowry definition applies to other solvents, such as dimethyl sulfoxide: the solvent S acts as a base, accepting a proton and forming the conjugate acid SH+. A broader definition of acid dissociation includes hydrolysis, in which protons are produced by the splitting of water molecules. For example, boric acid, B(OH)3, acts as a weak acid, even though it is not a proton donor, because of the hydrolysis equilibrium B(OH)3 + H2O ⇌ B(OH)−4 + H+.Similarly, metal ion hydrolysis causes ions such as 3+ to behave as weak acids: 3+ ⇌ 2+ + H+.Acid–base equilibria are important in a very wide range of applications, such as acid–base homeostasis, ocean acidification, pharmacology and analytical chemistry.
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Brønsted and Lowry characterized an acid–base equilibrium as involving a proton exchange reaction: acid + base ⇌ conjugate base + conjugate acid.An acid is a proton donor; the proton is transferred to the base, a proton acceptor, creating a conjugate acid. For aqueous solutions of an acid HA, the base is water; the conjugate base is A− and the conjugate acid is the solvated hydrogen ion. In solution chemistry, it is usual to use H+ as an abbreviation for the solvated hydrogen ion, regardless of the solvent.
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The general equilibrium can be written as p A + q B ⇌ ApBqThe study of these complexes is important for supramolecular chemistry and molecular recognition. The objective of these studies is often to find systems with a high binding selectivity of a host (receptor) for a particular target molecule or ion, the guest or ligand. An application is the development of chemical sensors. Finding a drug which either blocks a receptor, an antagonist which forms a strong complex the receptor, or activate it, an agonist, is an important pathway to drug discovery.
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A host–guest complex, also known as a donor–acceptor complex, may be formed from a Lewis base, B, and a Lewis acid, A. The host may be either a donor or an acceptor. In biochemistry host–guest complexes are known as receptor-ligand complexes; they are formed primarily by non-covalent bonding. Many host–guest complexes has 1:1 stoichiometry, but many others have more complex structures.
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Writing for Nature, Virginia Dignum gave the book a positive review, favorably comparing it to Kate Crawford's Atlas of AI: Power, Politics, and the Planetary Costs of Artificial Intelligence.In 2021, journalist Ezra Klein had Christian on his podcast, The Ezra Klein Show, writing in The New York Times, "The Alignment Problem is the best book on the key technical and moral questions of A.I. that I’ve read." Later that year, the book was listed in a Fast Company feature, "5 books that inspired Microsoft CEO Satya Nadella this year".
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The book is divided into three sections: Prophecy, Agency, and Normativity. Each section covers researchers and engineers working on different challenges in the alignment of artificial intelligence with human values.
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The Alignment Problem: Machine Learning and Human Values is a 2020 non-fiction book by the American writer Brian Christian. It is based on numerous interviews with experts trying to build artificial intelligence systems, particular machine learning systems, that are aligned with human values.
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In the second section, Christian similarly interweaves the history of the psychological study of reward, such as behaviorism and dopamine, with the computer science of reinforcement learning, in which AI systems need to develop policy ("what to do") in the face of a value function ("what rewards or punishment to expect"). He calls the DeepMind AlphaGo and AlphaZero systems "perhaps the single most impressive achievement in automated curriculum design." He also highlights the importance of curiosity, in which reinforcement learners are intrinsically motivated to explore their environment, rather than exclusively seeking the external reward.
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In the first section, Christian interweaves discussions of the history of artificial intelligence research, particularly the machine learning approach of artificial neural networks such as the Perceptron and AlexNet, with examples of how AI systems can have unintended behavior. He tells the story of Julia Angwin, a journalist whose ProPublica investigation of the COMPAS algorithm, a tool for predicting recidivism among criminal defendants, led to widespread criticism of its accuracy and bias towards certain demographics. One of AI's main alignment challenges is its black box nature (inputs and outputs are identifiable but the transformation process in between is undetermined). The lack of transparency makes it difficult to know where the system is going right and where it is going wrong.
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For example, 8 x − 5 {\displaystyle 8x-5} is an expression, while 8 x − 5 ≥ 5 x − 8 {\displaystyle 8x-5\geq 5x-8} is a formula. However, in modern mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to true or false, depending on the values that are given to the variables occurring in the expressions. For example 8 x − 5 ≥ 5 x − 8 {\displaystyle 8x-5\geq 5x-8} takes the value false if x is given a value less than –1, and the value true otherwise.
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Semantics is the study of meaning. Formal semantics is about attaching meaning to expressions. In algebra, an expression may be used to designate a value, which might depend on values assigned to variables occurring in the expression. The determination of this value depends on the semantics attached to the symbols of the expression.
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Recently, chemists and those involved in nanotechnology have begun to explore the possibility of creating molecular motors de novo. These synthetic molecular motors currently suffer many limitations that confine their use to the research laboratory. However, many of these limitations may be overcome as our understanding of chemistry and physics at the nanoscale increases. One step toward understanding nanoscale dynamics was made with the study of catalyst diffusion in the Grubb's catalyst system.
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Molecular motor
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353
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In experimental biophysics, the activity of molecular motors is observed with many different experimental approaches, among them: Fluorescent methods: fluorescence resonance energy transfer (FRET), fluorescence correlation spectroscopy (FCS), total internal reflection fluorescence (TIRF). Magnetic tweezers can also be useful for analysis of motors that operate on long pieces of DNA. Neutron spin echo spectroscopy can be used to observe motion on nanosecond timescales. Optical tweezers (not to be confused with molecular tweezers in context) are well-suited for studying molecular motors because of their low spring constants. Scattering techniques: single particle tracking based on dark field microscopy or interferometric scattering microscopy (iSCAT) Single-molecule electrophysiology can be used to measure the dynamics of individual ion channels.Many more techniques are also used. As new technologies and methods are developed, it is expected that knowledge of naturally occurring molecular motors will be helpful in constructing synthetic nanoscale motors.
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Molecular motor
| 0.851305
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354
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Stability and other properties can be predicted using energy calculations and computational chemistry. " the Born–Haber cycle to estimate ... the heat of formation ... can be used to determine whether a hypothetical compound is stable." However, "a negative formation enthalpy does not automatically imply the existence of a hypothetical compound." The method predicts that NaCl is stable but NeCl is not. It predicted XePtF6 based on the stability of O2PtF6.
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Hypothetical chemical compound
| 0.851164
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355
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For D > 0, ω is a positive irrational real number, and the corresponding quadratic integer ring is a set of algebraic real numbers. The solutions of the Pell's equation X 2 − DY 2 = 1, a Diophantine equation that has been widely studied, are the units of these rings, for D ≡ 2, 3 (mod 4). For D = 5, ω = 1+√5/2 is the golden ratio. This ring was studied by Peter Gustav Lejeune Dirichlet.
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Quadratic integers
| 0.851042
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356
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For D < 0, ω is a complex (imaginary or otherwise non-real) number. Therefore, it is natural to treat a quadratic integer ring as a set of algebraic complex numbers. A classic example is Z {\displaystyle \mathbf {Z} } , the Gaussian integers, which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law. The elements in O Q ( − 3 ) = Z {\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-3}}\,)}=\mathbf {Z} \left} are called Eisenstein integers.Both rings mentioned above are rings of integers of cyclotomic fields Q(ζ 4) and Q(ζ 3) correspondingly.
|
Quadratic integers
| 0.851042
|
357
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Another common example is the non-real cubic root of unity −1 + √−3/2, which generates the Eisenstein integers. Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations, and other questions related to integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory.
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Quadratic integers
| 0.851042
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358
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In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form x2 + bx + c = 0with b and c (usual) integers. When algebraic integers are considered, the usual integers are often called rational integers. Common examples of quadratic integers are the square roots of rational integers, such as √2, and the complex number i = √−1, which generates the Gaussian integers.
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Quadratic integers
| 0.851042
|
359
|
For the execution of a single thread, the rules are simple. The Java Language Specification requires a Java virtual machine to observe within-thread as-if-serial semantics. The runtime (which, in this case, usually refers to the dynamic compiler, the processor and the memory subsystem) is free to introduce any useful execution optimizations as long as the result of the thread in isolation is guaranteed to be exactly the same as it would have been had all the statements been executed in the order the statements occurred in the program (also called program order).The major caveat of this is that as-if-serial semantics do not prevent different threads from having different views of the data. The memory model provides clear guidance about what values are allowed to be returned when the data is read.
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Java Memory Model
| 0.851016
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360
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For example, consider two threads with the following instructions, executing concurrently, where the variables x and y are both initialized to 0: If no reorderings are performed, and the read of y in Thread 2 returns the value 2, then the subsequent read of x should return the value 1, because the write to x was performed before the write to y. However, if the two writes are reordered, then the read of y can return the value 2, and the read of x can return the value 0. The Java Memory Model (JMM) defines the allowable behavior of multithreaded programs, and therefore describes when such reorderings are possible. It places execution-time constraints on the relationship between threads and main memory in order to achieve consistent and reliable Java applications. By doing this, it makes it possible to reason about code execution in a multithreaded environment, even in the face of optimizations performed by the dynamic compiler, the processor(s), and the caches.
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Java Memory Model
| 0.851016
|
361
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The Java memory model describes how threads in the Java programming language interact through memory. Together with the description of single-threaded execution of code, the memory model provides the semantics of the Java programming language. The original Java memory model developed in 1995, was widely perceived as broken, preventing many runtime optimizations and not providing strong enough guarantees for code safety. It was updated through the Java Community Process, as Java Specification Request 133 (JSR-133), which took effect back in 2004, for Tiger (Java 5.0).
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Java Memory Model
| 0.851016
|
362
|
The binary tetrahedral group was used in the context of Yang–Mills theory in 1956 by Chen Ning Yang and others. It was first used in flavor physics model building by Paul Frampton and Thomas Kephart in 1994. In 2012 it was shown that a relation between two neutrino mixing angles, derived by using this binary tetrahedral flavor symmetry, agrees with experiment.
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Binary tetrahedral group
| 0.850957
|
363
|
The cladistic method takes a systematic approach to characters, distinguishing between those that carry no information about shared evolutionary history – such as those evolved separately in different groups (homoplasies) or those left over from ancestors (plesiomorphies) – and derived characters, which have been passed down from innovations in a shared ancestor (apomorphies). Only derived characters, such as the spine-producing areoles of cacti, provide evidence for descent from a common ancestor. The results of cladistic analyses are expressed as cladograms: tree-like diagrams showing the pattern of evolutionary branching and descent.From the 1990s onwards, the predominant approach to constructing phylogenies for living plants has been molecular phylogenetics, which uses molecular characters, particularly DNA sequences, rather than morphological characters like the presence or absence of spines and areoles.
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Plant science
| 0.850953
|
364
|
While scientists do not always agree on how to classify organisms, molecular phylogenetics, which uses DNA sequences as data, has driven many recent revisions along evolutionary lines and is likely to continue to do so. The dominant classification system is called Linnaean taxonomy. It includes ranks and binomial nomenclature.
|
Plant science
| 0.850953
|
365
|
Systematic botany is part of systematic biology, which is concerned with the range and diversity of organisms and their relationships, particularly as determined by their evolutionary history. It involves, or is related to, biological classification, scientific taxonomy and phylogenetics. Biological classification is the method by which botanists group organisms into categories such as genera or species. Biological classification is a form of scientific taxonomy.
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Plant science
| 0.850953
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366
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Botanists also study weeds, which are a considerable problem in agriculture, and the biology and control of plant pathogens in agriculture and natural ecosystems. Ethnobotany is the study of the relationships between plants and people. When applied to the investigation of historical plant–people relationships ethnobotany may be referred to as archaeobotany or palaeoethnobotany. Some of the earliest plant-people relationships arose between the indigenous people of Canada in identifying edible plants from inedible plants. This relationship the indigenous people had with plants was recorded by ethnobotanists.
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Plant science
| 0.850953
|
367
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So in our previous example, we might say that the problem requires O ( n ) {\displaystyle O(n)} steps to solve. Perhaps the most important open problem in all of computer science is the question of whether a certain broad class of problems denoted NP can be solved efficiently. This is discussed further at Complexity classes P and NP, and P versus NP problem is one of the seven Millennium Prize Problems stated by the Clay Mathematics Institute in 2000. The Official Problem Description was given by Turing Award winner Stephen Cook.
|
Theory of algorithms
| 0.850913
|
368
|
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree (e.g., approximate solutions versus precise ones). The field is divided into three major branches: automata theory and formal languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers? ".In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation.
|
Theory of algorithms
| 0.850913
|
369
|
Plant genetics played a key role in the modern-day theories of heredity, beginning with Gregor Mendel's study of pea plants in the 19th century. The occupation has since grown to encompass advancements in biotechnology that have led to greater understanding of plant breeding and hybridization. Commercially, plant geneticists are sometimes employed to develop methods of making produce more nutritious, or altering plant pigments to make the food more enticing to consumers.
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Plant geneticist
| 0.85085
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370
|
A plant geneticist is a scientist involved with the study of genetics in botany. Typical work is done with genes in order to isolate and then develop certain plant traits. Once a certain trait, such as plant height, fruit sweetness, or tolerance to cold, is found, a plant geneticist works to improve breeding methods to ensure that future plant generations possess the desired traits.
|
Plant geneticist
| 0.85085
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371
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Other examples of emerging RNA-Seq applications due to the advancement of bioinformatics algorithms are copy number alteration, microbial contamination, transposable elements, cell type (deconvolution) and the presence of neoantigens.Prior to RNA-Seq, gene expression studies were done with hybridization-based microarrays. Issues with microarrays include cross-hybridization artifacts, poor quantification of lowly and highly expressed genes, and needing to know the sequence a priori. Because of these technical issues, transcriptomics transitioned to sequencing-based methods. These progressed from Sanger sequencing of Expressed sequence tag libraries, to chemical tag-based methods (e.g., serial analysis of gene expression), and finally to the current technology, next-gen sequencing of complementary DNA (cDNA), notably RNA-Seq.
|
RNA sequencing
| 0.850788
|
372
|
Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by Eduard Study.There exist two such two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, 1 (the identity element) and a. According to the definition of an identity element, 1 ⋅ 1 = 1 , 1 ⋅ a = a , a ⋅ 1 = a . {\displaystyle \textstyle 1\cdot 1=1\,,\quad 1\cdot a=a\,,\quad a\cdot 1=a\,.} It remains to specify a a = 1 {\displaystyle \textstyle aa=1} for the first algebra, a a = 0 {\displaystyle \textstyle aa=0} for the second algebra.There exist five such three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the identity element), a and b. Taking into account the definition of an identity element, it is sufficient to specify a a = a , b b = b , a b = b a = 0 {\displaystyle \textstyle aa=a\,,\quad bb=b\,,\quad ab=ba=0} for the first algebra, a a = a , b b = 0 , a b = b a = 0 {\displaystyle \textstyle aa=a\,,\quad bb=0\,,\quad ab=ba=0} for the second algebra, a a = b , b b = 0 , a b = b a = 0 {\displaystyle \textstyle aa=b\,,\quad bb=0\,,\quad ab=ba=0} for the third algebra, a a = 1 , b b = 0 , a b = − b a = b {\displaystyle \textstyle aa=1\,,\quad bb=0\,,\quad ab=-ba=b} for the fourth algebra, a a = 0 , b b = 0 , a b = b a = 0 {\displaystyle \textstyle aa=0\,,\quad bb=0\,,\quad ab=ba=0} for the fifth algebra.The fourth of these algebras is non-commutative, and the others are commutative.
|
An algebra
| 0.850706
|
373
|
Algebras over fields come in many different types. These types are specified by insisting on some further axioms, such as commutativity or associativity of the multiplication operation, which are not required in the broad definition of an algebra. The theories corresponding to the different types of algebras are often very different.
|
An algebra
| 0.850706
|
374
|
A non-associative algebra (or distributive algebra) over a field K is a K-vector space A equipped with a K-bilinear map A × A → A {\displaystyle A\times A\rightarrow A} . The usage of "non-associative" here is meant to convey that associativity is not assumed, but it does not mean it is prohibited – that is, it means "not necessarily associative". Examples detailed in the main article include: Euclidean space R3 with multiplication given by the vector cross product Octonions Lie algebras Jordan algebras Alternative algebras Flexible algebras Power-associative algebras
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An algebra
| 0.850706
|
375
|
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
| 0.850698
|
376
|
This is equivalent to: Śrīdharācāryya (870–930 AD), an Indian mathematician also came up with a similar algorithm for solving quadratic equations, though there is no indication that he considered both the roots. The 9th-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī solved quadratic equations algebraically. : 42 The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. In 1637 René Descartes published La Géométrie containing special cases of the quadratic formula in the form we know today.
|
Quadratic formula
| 0.850698
|
377
|
In his work Arithmetica, the Greek mathematician Diophantus (circa 250 AD) solved quadratic equations with a method more recognizably algebraic than the geometric algebra of Euclid. : 39 His solution gives only one root, even when both roots are positive.The Indian mathematician Brahmagupta (597–668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD, but written in words instead of symbols. His solution of the quadratic equation ax2 + bx = c was as follows: "To the absolute number multiplied by four times the square, add the square of the middle term; the square root of the same, less the middle term, being divided by twice the square is the value."
|
Quadratic formula
| 0.850698
|
378
|
In elementary algebra, the quadratic formula is a formula that provides the two solutions to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as completing the square. Given a general quadratic equation of the form whose discriminant b 2 − 4 a c {\displaystyle b^{2}-4ac} is positive, with x representing an unknown, with a, b and c representing constants, and with a ≠ 0, the quadratic formula is: where the plus–minus symbol "±" indicates that the quadratic equation has two solutions. Written separately, they become: Each of these two solutions is also called a root (or zero) of the quadratic equation.
|
Quadratic formula
| 0.850698
|
379
|
An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents, which is an early part of Galois theory. This method can be generalized to give the roots of cubic polynomials and quartic polynomials, and leads to Galois theory, which allows one to understand the solution of algebraic equations of any degree in terms of the symmetry group of their roots, the Galois group. This approach focuses on the roots more than on rearranging the original equation. Given a monic quadratic polynomial assume that it factors as Expanding yields where p = −(α + β) and q = αβ.
|
Quadratic formula
| 0.850698
|
380
|
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
| 0.850698
|
381
|
In the latter 19th and early 20th centuries, many scientists believed that all motor control came from the spinal cord, as experiments with stimulation in frogs displayed patterned movement ("motor primitives"), and spinalized cats were shown to be able to walk. This tradition was closely tied with the strict nervous system localizationism advocated during that period; since stimulation of the frog spinal cord in different places produced different movements, it was thought that all motor impulses were localized in the spinal cord. However, fixed structure and localizationism were slowly broken down as the central dogma of neuroscience. It is now known that the primary motor cortex and premotor cortex at the highest level are responsible for most voluntary movements. Animal models, though, remain relevant in motor control and spinal cord reflexes and central pattern generators are still a topic of study.
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Degrees of freedom problem
| 0.850479
|
382
|
In neuroscience and motor control , the degrees of freedom problem or motor equivalence problem states that there are multiple ways for humans or animals to perform a movement in order to achieve the same goal. In other words, under normal circumstances, no simple one-to-one correspondence exists between a motor problem (or task) and a motor solution to the problem. The motor equivalence problem was first formulated by the Russian neurophysiologist Nikolai Bernstein: "It is clear that the basic difficulties for co-ordination consist precisely in the extreme abundance of degrees of freedom, with which the centre is not at first in a position to deal.
|
Degrees of freedom problem
| 0.850479
|
383
|
Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, and completeness. For example, in most systems of logic (but not in intuitionistic logic) Peirce's law (((P→Q)→P)→P) is a theorem. For classical logic, it can be easily verified with a truth table. The study of mathematical proof is particularly important in logic, and has accumulated to automated theorem proving and formal verification of software.
|
Discrete structure
| 0.850266
|
384
|
Targeted analysis sequencing (sometimes called target amplicon sequencing) (TAS) is a next-generation DNA sequencing technique focusing on amplicons and specific genes. It is useful in population genetics since it can target a large diversity of organisms. The TAS approach incorporates bioinformatics techniques to produce a large amount of data at a fraction of the cost involved in Sanger sequencing. TAS is also useful in DNA studies because it allows for amplification of the needed gene area via PCR, which is followed by next-gen sequencing platforms.
|
Targeted analysis sequencing
| 0.850186
|
385
|
Ionic potential is the ratio of the electrical charge (z) to the radius (r) of an ion. As such, this ratio is a measure of the charge density at the surface of the ion; usually the denser the charge, the stronger the bond formed by the ion with ions of opposite charge.The ionic potential gives an indication of how strongly, or weakly, the ion will be electrostatically attracted by ions of opposite charge; and to what extent the ion will be repelled by ions of the same charge. Victor Moritz Goldschmidt, the father of modern geochemistry found that the behavior of an element in its environment could be predicted from its ionic potential and illustrated this with a diagram (plot of the bare ionic radius as a function of the ionic charge). For instance, the solubility of dissolved iron is highly dependent on its redox state.
|
Ionic potential
| 0.850153
|
386
|
The problem of finding or estimating the number of graphs with a given degree sequence is a problem from the field of graph enumeration. More generally, the degree sequence of a hypergraph is the non-increasing sequence of its vertex degrees. A sequence is k {\displaystyle k} -graphic if it is the degree sequence of some k {\displaystyle k} -uniform hypergraph. In particular, a 2 {\displaystyle 2} -graphic sequence is graphic. Deciding if a given sequence is k {\displaystyle k} -graphic is doable in polynomial time for k = 2 {\displaystyle k=2} via the Erdős–Gallai theorem but is NP-complete for all k ≥ 3 {\displaystyle k\geq 3} .
|
Degree sequence
| 0.850125
|
387
|
The construction of such a graph is straightforward: connect vertices with odd degrees in pairs (forming a matching), and fill out the remaining even degree counts by self-loops. The question of whether a given degree sequence can be realized by a simple graph is more challenging. This problem is also called graph realization problem and can be solved by either the Erdős–Gallai theorem or the Havel–Hakimi algorithm.
|
Degree sequence
| 0.850125
|
388
|
In algebraic geometry, solution sets are called algebraic sets if there are no inequalities. Over the reals, and with inequalities, there are called semialgebraic sets.
|
Solution set
| 0.85012
|
389
|
In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities. For example, for a set { f i } {\displaystyle \{f_{i}\}} of polynomials over a ring R {\displaystyle R} , the solution set is the subset of R {\displaystyle R} on which the polynomials all vanish (evaluate to 0), formally { x ∈ R: ∀ i ∈ I , f i ( x ) = 0 } {\displaystyle \{x\in R:\forall i\in I,f_{i}(x)=0\}} The feasible region of a constrained optimization problem is the solution set of the constraints.
|
Solution set
| 0.85012
|
390
|
The first natural problem proven to be NP-complete was the Boolean satisfiability problem, also known as SAT. As noted above, this is the Cook–Levin theorem; its proof that satisfiability is NP-complete contains technical details about Turing machines as they relate to the definition of NP. However, after this problem was proved to be NP-complete, proof by reduction provided a simpler way to show that many other problems are also NP-complete, including the game Sudoku discussed earlier.
|
P versus NP problem
| 0.850069
|
391
|
NP-hard problems are those at least as hard as NP problems; i.e., all NP problems can be reduced (in polynomial time) to them. NP-hard problems need not be in NP; i.e., they need not have solutions verifiable in polynomial time. For instance, the Boolean satisfiability problem is NP-complete by the Cook–Levin theorem, so any instance of any problem in NP can be transformed mechanically into an instance of the Boolean satisfiability problem in polynomial time.
|
P versus NP problem
| 0.850069
|
392
|
For some questions, there is no known way to find an answer quickly, but if one is provided with information showing what the answer is, it is possible to verify the answer quickly. The class of questions for which an answer can be verified in polynomial time is NP, which stands for "nondeterministic polynomial time".An answer to the P versus NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If it turns out that P ≠ NP, which is widely believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be verified in polynomial time. The problem has been called the most important open problem in computer science. Aside from being an important problem in computational theory, a proof either way would have profound implications for mathematics, cryptography, algorithm research, artificial intelligence, game theory, multimedia processing, philosophy, economics and many other fields.It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute, each of which carries a US$1,000,000 prize for the first correct solution.
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P versus NP problem
| 0.850069
|
393
|
The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved. The informal term quickly, used above, means the existence of an algorithm solving the task that runs in polynomial time, such that the time to complete the task varies as a polynomial function on the size of the input to the algorithm (as opposed to, say, exponential time). The general class of questions for which some algorithm can provide an answer in polynomial time is "P" or "class P".
|
P versus NP problem
| 0.850069
|
394
|
Process Biochemistry is a monthly peer-reviewed scientific journal that covers the study of biochemical processes and their applications in industries, such as food, pharmaceuticals, and biotechnology. The journal was established in 1966 and is published by Elsevier. The editor-in-chief is Joseph Boudrant (University of Lorraine). The journal covers a wide range of topics related to biochemical processes, including enzyme and microbial technology, protein engineering, metabolic engineering, biotransformations, and bioseparations. The journal publishes research articles, review articles, and case studies.
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Process Biochemistry
| 0.849991
|
395
|
The volume of each infinitesimal disc is therefore πf(y)2 dy. The limit of the Riemann sum of the volumes of the discs between a and b becomes integral (1). Assuming the applicability of Fubini's theorem and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D): V = ∭ D d V = ∫ a b ∫ g ( z ) f ( z ) ∫ 0 2 π r d θ d r d z = 2 π ∫ a b ∫ g ( z ) f ( z ) r d r d z = 2 π ∫ a b 1 2 r 2 ‖ g ( z ) f ( z ) d z = π ∫ a b f ( z ) 2 − g ( z ) 2 d z {\displaystyle V=\iiint _{D}dV=\int _{a}^{b}\int _{g(z)}^{f(z)}\int _{0}^{2\pi }r\,d\theta \,dr\,dz=2\pi \int _{a}^{b}\int _{g(z)}^{f(z)}r\,dr\,dz=2\pi \int _{a}^{b}{\frac {1}{2}}r^{2}\Vert _{g(z)}^{f(z)}\,dz=\pi \int _{a}^{b}f(z)^{2}-g(z)^{2}\,dz}
|
Solids of revolution
| 0.84992
|
396
|
particle and nuclear physics nuclear properties radioactive decay fission and fusion reactions fundamental properties of elementary particles condensed matter crystal structure x-ray diffraction thermal properties electron theory of metals semiconductors superconductors mathematical methods single and multivariate calculus coordinate systems (rectangular, cylindrical, spherical) vector algebra and vector differential operators Fourier series partial differential equations boundary value problems matrices and determinants functions of complex variables miscellaneous astrophysics computer applications
|
GRE Physics Test
| 0.849733
|
397
|
The GRE physics test is an examination administered by the Educational Testing Service (ETS). The test attempts to determine the extent of the examinees' understanding of fundamental principles of physics and their ability to apply them to problem solving. Many graduate schools require applicants to take the exam and base admission decisions in part on the results. The scope of the test is largely that of the first three years of a standard United States undergraduate physics curriculum, since many students who plan to continue to graduate school apply during the first half of the fourth year. It consists of 100 five-option multiple-choice questions covering subject areas including classical mechanics, electromagnetism, wave phenomena and optics, thermal physics, relativity, atomic and nuclear physics, quantum mechanics, laboratory techniques, and mathematical methods. The table below indicates the relative weights, as asserted by ETS, and detailed contents of the major topics.
|
GRE Physics Test
| 0.849733
|
398
|
data and error analysis electronics instrumentation radiation detection counting statistics interaction of charged particles with matter laser and optical interferometers dimensional analysis fundamental applications of probability and statistics
|
GRE Physics Test
| 0.849733
|
399
|
This is an extraordinarily strong concentration of mathematical education – up to 16 hours a week – in which elementary analytic geometry and mechanics, and recently infinitesimal calculus also, are thoroughly studied and are made into a securely mastered tool by means of many exercises.Sylvestre Lacroix was a gifted teacher and expositor. His book on descriptive geometry uses sections labelled "Probleme" to exercise the reader’s understanding. In 1816 he wrote Essays on Teaching in General, and on Mathematics Teaching in Particular which emphasized the need to exercise and test: The examiner, obliged, in the short-term, to multiply his questions enough to cover the subjects that he asks, to the greater part of the material taught, cannot be less thorough, since if, to abbreviate, he puts applications aside, he will not gain anything for the pupils’ faculties this way.Andrew Warwick has drawn attention to the historical question of exercises: The inclusion of illustrative exercises and problems at the end of chapters in textbooks of mathematical physics is now so commonplace as to seem unexceptional, but it is important to appreciate that this pedagogical device is of relatively recent origin and was introduced in a specific historical context.
|
Mathematical exercise
| 0.84963
|
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