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17,400 | This is "special relativity" as usually understood. A description of physics without gravity using arbitrary curvilinear coordinates. This is non-gravitational physics plus general covariance. | Formulations of special relativity | 0.81297 |
17,401 | The three things are: A description of physics without gravity using just "inertial frames", i.e. non-accelerating Cartesian coordinate systems. These coordinate systems are all related to each other by the linear Lorentz transformations. The physical laws may be described more simply in these frames than in the others... | Formulations of special relativity | 0.81297 |
17,402 | Strings admit the following interpretation as nodes on a graph, where k is the number of symbols in Σ: Fixed-length strings of length n can be viewed as the integer locations in an n-dimensional hypercube with sides of length k-1. Variable-length strings (of finite length) can be viewed as nodes on a perfect k-ary tree... | Character string (computer science) | 0.812968 |
17,403 | Algorithms for calculating variance play a major role in computational statistics. A key difficulty in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values. | Algorithms for calculating variance | 0.812967 |
17,404 | One can also find there similar formulas for covariance. Choi and Sweetman offer two alternative methods to compute the skewness and kurtosis, each of which can save substantial computer memory requirements and CPU time in certain applications. The first approach is to compute the statistical moments by separating the ... | Algorithms for calculating variance | 0.812967 |
17,405 | It is also common to denote M k = x ¯ k {\displaystyle M_{k}={\bar {x}}_{k}} and S k = M 2 , k {\displaystyle S_{k}=M_{2,k}} .An example Python implementation for Welford's algorithm is given below. This algorithm is much less prone to loss of precision due to catastrophic cancellation, but might not be as efficient be... | Algorithms for calculating variance | 0.812967 |
17,406 | It is often useful to be able to compute the variance in a single pass, inspecting each value x i {\displaystyle x_{i}} only once; for example, when the data is being collected without enough storage to keep all the values, or when costs of memory access dominate those of computation. For such an online algorithm, a re... | Algorithms for calculating variance | 0.812967 |
17,407 | In software engineering, basis path testing, or structured testing, is a white box method for designing test cases. The method analyzes the control-flow graph of a program to find a set of linearly independent paths of execution. The method normally uses McCabe cyclomatic complexity to determine the number of linearly ... | Basis path testing | 0.812967 |
17,408 | Figure eight knots (41), three-twist knots (52), and Stevedore knots (61) have also been discovered. Recently, use of machine learning techniques for predicting protein structure, resulted in highly accurate prediction of 6₃ knot. Furthermore, using same techniques, composite knots (namely 31#31) were found. | Knotted protein | 0.812967 |
17,409 | Post-quantum cryptography studies quantum attacks on classical cryptographic schemes. As a random oracle is an abstraction of a hash function, it makes sense to assume that a quantum attacker can access the random oracle in quantum superposition. Many of the classical security proofs break down in that quantum random o... | Random oracle | 0.812963 |
17,410 | Random oracles as a mathematical abstraction were first used in rigorous cryptographic proofs in the 1993 publication by Mihir Bellare and Phillip Rogaway (1993). They are typically used when the proof cannot be carried out using weaker assumptions on the cryptographic hash function. A system that is proven secure when... | Random oracle | 0.812963 |
17,411 | In cryptography, a random oracle is an oracle (a theoretical black box) that responds to every unique query with a (truly) random response chosen uniformly from its output domain. If a query is repeated, it responds the same way every time that query is submitted. Stated differently, a random oracle is a mathematical f... | Random oracle | 0.812963 |
17,412 | As chemistry developed into a science, it became clear that metals formed the majority of the periodic table of the elements, and great progress was made in the description of the salts that can be formed in reactions with acids. With the advent of electrochemistry, it became clear that metals generally go into solutio... | Metallic bonds | 0.812962 |
17,413 | One could call these materials metal compounds. But, because materials with metallic bonding are typically not molecular, Dalton's law of integral proportions is not valid; and often a range of stoichiometric ratios can be achieved. It is better to abandon such concepts as 'pure substance' or 'solute' in such cases and... | Metallic bonds | 0.812962 |
17,414 | Anatomical planes may be lateral ("sagittal"), frontal ("coronal") or transversal. In geology, beds (layers of sediments) often are planar. Planes are involved in different forms of imaging, such as the focal plane, picture plane, and image plane. | Planar surface | 0.812962 |
17,415 | A plane serves as a mathematical model for many physical phenomena, such as specular reflection in a plane mirror or wavefronts in a traveling plane wave. The free surface of undisturbed liquids tends to be nearly flat (see flatness). The flattest surface ever manufactured is a quantum-stabilized atom mirror. In astron... | Planar surface | 0.812962 |
17,416 | The paper did not appear at the time in a primary publication in English. An English translation of the seminal paper was published in the American Journal of Physics in 1968.Fermi found the initial rejection of the paper so troubling that he decided to take some time off from theoretical physics, and do only experimen... | Fermi's constant | 0.81296 |
17,417 | In particle physics, Fermi's interaction (also the Fermi theory of beta decay or the Fermi four-fermion interaction) is an explanation of the beta decay, proposed by Enrico Fermi in 1933. The theory posits four fermions directly interacting with one another (at one vertex of the associated Feynman diagram). This intera... | Fermi's constant | 0.812959 |
17,418 | These transformations can cause the displacement of the triangle in the plane, while leaving the vertex angle and the distances between vertices unchanged. Kinematics is often described as applied geometry, where the movement of a mechanical system is described using the rigid transformations of Euclidean geometry. | Exact constraint | 0.812958 |
17,419 | The movement of components of a mechanical system are analyzed by attaching a reference frame to each part and determining how the various reference frames move relative to each other. If the structural stiffness of the parts are sufficient, then their deformation can be neglected and rigid transformations can be used ... | Exact constraint | 0.812958 |
17,420 | Kinematic analysis is the process of measuring the kinematic quantities used to describe motion. In engineering, for instance, kinematic analysis may be used to find the range of movement for a given mechanism and, working in reverse, using kinematic synthesis to design a mechanism for a desired range of motion. In add... | Exact constraint | 0.812958 |
17,421 | In mechanical engineering, robotics, and biomechanics kinematics is used to describe the motion of systems composed of joined parts (multi-link systems) such as an engine, a robotic arm or the human skeleton. Geometric transformations, also called rigid transformations, are used to describe the movement of components i... | Exact constraint | 0.812958 |
17,422 | The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics. Kinematics is used in astrophysics to describe the motion of celestial bodies and collections of such bodies. | Exact constraint | 0.812958 |
17,423 | Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasio... | Exact constraint | 0.812958 |
17,424 | Then, the angle θ around this axis in the X–Y plane can be used to define the trajectory as, where the constant distance from the center is denoted as R, and θ = θ(t) is a function of time. The cylindrical coordinates for r(t) can be simplified by introducing the radial and tangential unit vectors, and their time deriv... | Exact constraint | 0.812958 |
17,425 | The topology defined in this way is called the Alexandrov topology determined by ≤. The equivalence between preorders and finite topologies can be interpreted as a version of Birkhoff's representation theorem, an equivalence between finite distributive lattices (the lattice of open sets of the topology) and partial ord... | Finite space | 0.812958 |
17,426 | Going in the other direction, suppose (X, ≤) is a preordered set. Define a topology τ on X by taking the open sets to be the upper sets with respect to ≤. Then the relation ≤ will be the specialization preorder of (X, τ). | Finite space | 0.812958 |
17,427 | Topologies on a finite set X are in one-to-one correspondence with preorders on X. Recall that a preorder on X is a binary relation on X which is reflexive and transitive. Given a (not necessarily finite) topological space X we can define a preorder on X by x ≤ y if and only if x ∈ cl{y}where cl{y} denotes the closure ... | Finite space | 0.812958 |
17,428 | A finite space X is hyperconnected if and only if there is a greatest element with respect to the specialization preorder. This is an element whose closure is the whole space X. ultraconnected if and only if there is a least element with respect to the specialization preorder. This is an element whose only neighborhood... | Finite space | 0.812958 |
17,429 | Let X = {a,b} be a set with 2 elements. There are four distinct topologies on X: {∅, {a,b}} (the trivial topology) {∅, {a}, {a,b}} {∅, {b}, {a,b}} {∅, {a}, {b}, {a,b}} (the discrete topology)The second and third topologies above are easily seen to be homeomorphic. The function from X to itself which swaps a and b is a ... | Finite space | 0.812958 |
17,430 | Perhaps surprisingly, there are finite topological spaces with nontrivial fundamental groups. A simple example is the pseudocircle, which is space X with four points, two of which are open and two of which are closed. There is a continuous map from the unit circle S1 to X which is a weak homotopy equivalence (i.e. it i... | Finite space | 0.812958 |
17,431 | In 2004, Journal of Algebra announced (vol. 276, no. 1 and 2) the creation of a new section on computational algebra, with a separate editorial board. The first issue completely devoted to computational algebra was vol. 292, no. 1 (October 2005). The Editor-in-Chief of the Journal of Algebra is Michel Broué, Université... | Journal of Algebra | 0.812953 |
17,432 | Journal of Algebra (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. Journal of Algebra was founded by Graham Higman, who was its editor from 1964 to 1984. From 1985 until 2000, Walter Feit served as its editor-in-chief. | Journal of Algebra | 0.812953 |
17,433 | For example, if a researcher wanted to perform a combustion reaction in a bomb calorimeter, the volume is kept constant throughout the course of a reaction. Therefore, the heat of the reaction is a direct measure of the free energy change, q = Δ U {\displaystyle q=\Delta U} . In solution chemistry, on the other hand, m... | Thermodynamic free energy | 0.812952 |
17,434 | Just like the general concept of energy, free energy has a few definitions suitable for different conditions. In physics, chemistry, and biology, these conditions are thermodynamic parameters (temperature T {\displaystyle T} , volume V {\displaystyle V} , pressure p {\displaystyle p} , etc.). Scientists have come up wi... | Thermodynamic free energy | 0.812952 |
17,435 | The Helmholtz free energy has a special theoretical importance since it is proportional to the logarithm of the partition function for the canonical ensemble in statistical mechanics. (Hence its utility to physicists; and to gas-phase chemists and engineers, who do not want to ignore p dV work.) Historically, the term ... | Thermodynamic free energy | 0.812952 |
17,436 | According to the second law of thermodynamics, for any process that occurs in a closed system, the inequality of Clausius, ΔS > q/Tsurr, applies. For a process at constant temperature and pressure without non-PV work, this inequality transforms into Δ G < 0 {\displaystyle \Delta G<0} . Similarly, for a process at const... | Thermodynamic free energy | 0.812952 |
17,437 | "Molecules to the MAX!" is a 41-minute fully animated 3D IMAX film for the Giant Screen. The film re-imagines the characters and story developed for "Molecularium- Riding Snowflakes" for an older audience and a different medium. The film's simulations and rendering were partially computed at the Computational Center fo... | Molecularium Project | 0.812947 |
17,438 | In the spring of 2012, the Molecularium Project launched NanoSpace, an online molecular theme park. Visitors to NanoSpace learn scientific concepts with games, activities and movies. Areas within Nanospace include the Hall of Atoms and Molecules, H2O park (the water cycle), Sizes in the Universe (scale and scientific n... | Molecularium Project | 0.812947 |
17,439 | When Dantzig arranged a meeting with John von Neumann to discuss his simplex method, Neumann immediately conjectured the theory of duality by realizing that the problem he had been working in game theory was equivalent. Dantzig provided formal proof in an unpublished report "A Theorem on Linear Inequalities" on January... | 0-1 integer programming | 0.812946 |
17,440 | Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optim... | 0-1 integer programming | 0.812946 |
17,441 | They translated stack code sequences into equivalent sequences of RISC code. Minor 'local' optimizations removed much of the overhead of a stack architecture. Spare registers were used to factor out repeated address calculations. | Zero address arithmetic | 0.812944 |
17,442 | There are some cases which allow exact solutions. For very small microscopic systems, the ensembles can be directly computed by simply enumerating over all possible states of the system (using exact diagonalization in quantum mechanics, or integral over all phase space in classical mechanics). Some large systems consis... | Non-equilibrium statistical mechanics | 0.812942 |
17,443 | However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is giv... | Non-equilibrium statistical mechanics | 0.812942 |
17,444 | In physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts: The complete state of the mechanical system at a given time, mathematically encoded as a phase point (classical mechanics) or a... | Non-equilibrium statistical mechanics | 0.812942 |
17,445 | At the same time, 1,755 scientists had volunteered from all 50 US states and 17 other countries across 12 time zones. By February 2019, a total of 14,312 classrooms had been matched with over 7,000 scientists. Participants were from 43 countries and sessions occurred in 14 languages, including American Sign Language.In... | Skype a Scientist | 0.812942 |
17,446 | Sarah McAnulty came up with the idea for Skype a Scientist in 2016 while she was a graduate student in molecular and cell biology at the University of Connecticut in Storrs. Inspiration for the program came from the growing distrust of scientists and the rise of anti-intellectualism in US society. McAnulty was concerne... | Skype a Scientist | 0.812942 |
17,447 | In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, after which an element ϕ ∈ T {\displaystyle \phi \in T} of a deductively closed theory T {\displaystyle T} is then called a theorem of the theo... | Theory (mathematical logic) | 0.812939 |
17,448 | There are many formal derivation ("proof") systems for first-order logic. These include Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method and resolution. | Theory (mathematical logic) | 0.812939 |
17,449 | A theory is said to be a deductive theory if T {\displaystyle {\mathcal {T}}} is an inductive class, which is to say that its content is based on some formal deductive system and that some of its elementary statements are taken as axioms. In a deductive theory, any sentence that is a logical consequence of one or more ... | Theory (mathematical logic) | 0.812939 |
17,450 | A formula A is a syntactic consequence of a first-order theory Q S {\displaystyle {\mathcal {QS}}} if there is a derivation of A using only formulas in Q S {\displaystyle {\mathcal {QS}}} as non-logical axioms. Such a formula A is also called a theorem of Q S {\displaystyle {\mathcal {QS}}} . The notation " Q S ⊢ A {\d... | Theory (mathematical logic) | 0.812939 |
17,451 | If k = 0 {\displaystyle k=0} then it is immediate. Otherwise p {\displaystyle p} lies on a line segment in S {\displaystyle S} which can be maximally extended (because S {\displaystyle S} is closed and bounded). If the endpoints of the segment are q {\displaystyle q} and r , {\displaystyle r,} then their extreme rank m... | Extreme points | 0.812936 |
17,452 | If S {\displaystyle S} is closed, bounded, and n {\displaystyle n} -dimensional, and if p {\displaystyle p} is a point in S , {\displaystyle S,} then p {\displaystyle p} is k {\displaystyle k} -extreme for some k ≤ n . {\displaystyle k\leq n.} The theorem asserts that p {\displaystyle p} is a convex combination of extr... | Extreme points | 0.812936 |
17,453 | {\displaystyle S.} More generally, for any convex set S , {\displaystyle S,} the k {\displaystyle k} -extreme points are partitioned into k {\displaystyle k} -dimensional open faces. The finite-dimensional Krein-Milman theorem, which is due to Minkowski, can be quickly proved using the concept of k {\displaystyle k} -e... | Extreme points | 0.812936 |
17,454 | The extreme points of a compact convex set form a Baire space (with the subspace topology) but this set may fail to be closed in X . {\displaystyle X.} | Extreme points | 0.812936 |
17,455 | The Krein–Milman theorem is arguably one of the most well-known theorems about extreme points. | Extreme points | 0.812936 |
17,456 | As the CR occurs, electromagnetic radiation is emitted in a cone shape, fanning out in the forward direction. CR and the 1937 theory has led to a large array of applications in high energy physics. A notable application are the Cherenkov counters. | Double negative metamaterial | 0.812933 |
17,457 | A theoretical work published in 1967 by Soviet physicist Victor Veselago showed that a refractive index with negative values is possible and that this does not violate the laws of physics. As discussed previously (above), the first metamaterial had a range of frequencies over which the refractive index was predicted to... | Double negative metamaterial | 0.812933 |
17,458 | Specifications for the response of each unit cell are predetermined prior to construction and are based on the intended response of the entire, newly constructed, material. In other words, each cell is individually tuned to respond in a certain way, based on the desired output of the NIM. The aggregate response is main... | Double negative metamaterial | 0.812933 |
17,459 | The nested sampling algorithm is a computational approach to the Bayesian statistics problems of comparing models and generating samples from posterior distributions. It was developed in 2004 by physicist John Skilling. | Nested sampling algorithm | 0.812931 |
17,460 | PolyChord is another nested sampling software package available on GitHub. PolyChord's computational efficiency scales better with an increase in the number of parameters than MultiNest, meaning PolyChord can be more efficient for high dimensional problems. NestedSamplers.jl a Julia package for implementing single- and... | Nested sampling algorithm | 0.812931 |
17,461 | An example in C++, named Diamonds, is on GitHub. A highly modular Python parallel example for statistical physics and condensed matter physics uses is on GitHub. | Nested sampling algorithm | 0.812931 |
17,462 | Since nested sampling was proposed in 2004, it has been used in many aspects of the field of astronomy. One paper suggested using nested sampling for cosmological model selection and object detection, as it "uniquely combines accuracy, general applicability and computational feasibility." A refinement of the algorithm ... | Nested sampling algorithm | 0.812931 |
17,463 | The relationship between biology and sexual orientation is a subject of research. While scientists do not know the exact cause of sexual orientation, they theorize that it is caused by a complex interplay of genetic, hormonal, and environmental influences. Hypotheses for the impact of the post-natal social environment ... | Biology and sexual orientation | 0.812931 |
17,464 | A number of twin studies have attempted to compare the relative importance of genetics and environment in the determination of sexual orientation. In a 1991 study, Bailey and Pillard conducted a study of male twins recruited from "homophile publications", and found that 52% of monozygotic (MZ) brothers (of whom 59 were... | Biology and sexual orientation | 0.812931 |
17,465 | As both blood type and Rh factor are genetically inherited traits controlled by alleles located on chromosome 9 and chromosome 1 respectively, the study indicates a potential link between genes on autosomes and homosexuality.The biology of sexual orientation has been studied in detail in several animal model systems. I... | Biology and sexual orientation | 0.81293 |
17,466 | Meta-analysis of all available linkage data indicates a significant link to Xq28, but also indicates that additional genes must be present to account for the full heritability of sexual orientation.Mustanski et al. (2005) performed a full-genome scan (instead of just an X chromosome scan) on individuals and families pr... | Biology and sexual orientation | 0.81293 |
17,467 | They cited additional research demonstrating that root-emitted molecules have the potential to induce physiological responses in neighboring plants either directly or indirectly by modifying the soil chemistry. Moreover, Kegge et al. demonstrated that plants perceive the presence of neighbors through changes in water/n... | Plant communication | 0.81293 |
17,468 | In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua. | Continuum theory | 0.812929 |
17,469 | Cantor, using Qt, can interface with Maxima (along with SageMath, R, and KAlgebra) The GNU TeXmacs and LyX mathematical editor programs can be used to provide an interactive GUI for Maxima, as can SageMath. Other options include the Imaxima front end, as well as an Emacs and XEmacs interaction mode which is activated b... | Maxima computer algebra system | 0.812927 |
17,470 | Like most computer algebra systems, Maxima supports a variety of ways of reorganizing symbolic algebraic expressions, such as polynomial factorization, polynomial greatest common divisor calculation, expansion, separation into real and imaginary parts, and transformation of trigonometric functions to exponential and vi... | Maxima computer algebra system | 0.812927 |
17,471 | Maxima () is a computer algebra system (CAS) based on a 1982 version of Macsyma. It is written in Common Lisp and runs on all POSIX platforms such as macOS, Unix, BSD, and Linux, as well as under Microsoft Windows and Android. It is free software released under the terms of the GNU General Public License (GPL). | Maxima computer algebra system | 0.812927 |
17,472 | The satisfiability problem for a sentence of monadic second-order logic is the problem of determining whether there exists at least one graph (possibly within a restricted family of graphs) for which the sentence is true. For arbitrary graph families, and arbitrary sentences, this problem is undecidable. However, satis... | Logic of graphs | 0.812923 |
17,473 | The fixed point of a logical formula can be constructed in polynomial time, by an algorithm that repeatedly adds tuples to the set of values for which the predicate is true until reaching a fixed point, so deciding whether a graph models a sentence in this logic can always be decided in polynomial time. Not every polyn... | Logic of graphs | 0.812923 |
17,474 | In another variant with equivalent logical power, inflationary fixed point logic, the formula need not be monotone but the resulting fixed point is defined as the one obtained by repeatedly applying implications derived from the defining formula starting from the all-false predicate. Other variants, allowing negative i... | Logic of graphs | 0.812923 |
17,475 | As a partial converse, Seese (1991) proved that, whenever a family of graphs has a decidable MSO2 satisfiability problem, the family must have bounded treewidth. The proof is based on a theorem of Robertson and Seymour that the families of graphs with unbounded treewidth have arbitrarily large grid minors. Seese also c... | Logic of graphs | 0.812923 |
17,476 | Glebskiĭ et al. (1969) and, independently, Fagin (1976) proved a zero–one law for first-order graph logic; Fagin's proof used the compactness theorem. According to this result, every first-order sentence is either almost always true or almost always false for random graphs in the Erdős–Rényi model. That is, let S {\dis... | Logic of graphs | 0.812923 |
17,477 | According to Courcelle's theorem, every fixed MSO2 property can be tested in linear time on graphs of bounded treewidth, and every fixed MSO1 property can be tested in linear time on graphs of bounded clique-width. The version of this result for graphs of bounded treewidth can also be implemented in logarithmic space. ... | Logic of graphs | 0.812923 |
17,478 | The algorithmic problem of satisfiability concerns testing whether there exists a graph that models a given sentence. Although both model checking and satisfiability are hard in general, several major algorithmic meta-theorems show that properties expressed in this way can be tested efficiently for important classes of... | Logic of graphs | 0.812923 |
17,479 | In the mathematical fields of graph theory and finite model theory, the logic of graphs deals with formal specifications of graph properties using sentences of mathematical logic. There are several variations in the types of logical operation that can be used in these sentences. The first-order logic of graphs concerns... | Logic of graphs | 0.812923 |
17,480 | The SVD is typically computed using large matrix methods (for example, Lanczos methods) but may also be computed incrementally and with greatly reduced resources via a neural network-like approach, which does not require the large, full-rank matrix to be held in memory. A fast, incremental, low-memory, large-matrix SVD... | Latent semantic indexing | 0.81292 |
17,481 | LSI uses common linear algebra techniques to learn the conceptual correlations in a collection of text. In general, the process involves constructing a weighted term-document matrix, performing a Singular Value Decomposition on the matrix, and using the matrix to identify the concepts contained in the text. | Latent semantic indexing | 0.81292 |
17,482 | It is generally acknowledged that the ability to work with text on a semantic basis is essential to modern information retrieval systems. As a result, the use of LSI has significantly expanded in recent years as earlier challenges in scalability and performance have been overcome. LSI is being used in a variety of info... | Latent semantic indexing | 0.81292 |
17,483 | A drawback of NMR crystallography is that the method is typically more time-consuming and more expensive (due to spectrometer costs and isotope labelling) than X-ray crystallography, it often elucidates only part of the structure, and isotope labelling and experiments may have to be tailored to obtain key structural in... | NMR crystallography | 0.812913 |
17,484 | Nuclear magnetic resonance crystallography (NMR crystallography) is a method which utilizes primarily NMR spectroscopy to determine the structure of solid materials on the atomic scale. Thus, solid-state NMR spectroscopy would be used primarily, possibly supplemented by quantum chemistry calculations (e.g. density func... | NMR crystallography | 0.812913 |
17,485 | Since R is a subring of R', I ∩ R = 0 {\displaystyle I\cap R=0} and so ht I = dim R I = dim Q ( R ) I = 1 − t r . d e g Q ( R ) κ ( I ) = 1 {\displaystyle \operatorname {ht} I=\dim R_{I}=\dim Q(R)_{I}=1-\operatorname {tr.deg} _{Q(R)}\kappa (I)=1} since κ ( I ) = Q ( R ′ ) {\displaystyle \kappa (I)=Q(R')} is a... | Dimension theory (algebra) | 0.812912 |
17,486 | Suppose not. Then R ′ {\displaystyle R'} is algebraic over R and so t r . d e g R R ′ = 0 {\displaystyle \operatorname {tr.deg} _{R}R'=0} . | Dimension theory (algebra) | 0.812912 |
17,487 | The theorem suggests that we consider a sort of a dual of a global dimension: It was originally called the weak global dimension of R but today it is more commonly called the Tor dimension of R. Remark: for any ring R, w . g l . d i m R ≤ g l . d i m R {\displaystyle \operatorname {w.gl.dim} R\leq \operatorname {gl... | Dimension theory (algebra) | 0.812912 |
17,488 | Indeed, by the above theorem, Tor s R ( k , k ) ≠ 0 {\displaystyle \operatorname {Tor} _{s}^{R}(k,k)\neq 0} and thus g l . d i m R ≥ s {\displaystyle \operatorname {gl.dim} R\geq s} . On the other hand, as g l . | Dimension theory (algebra) | 0.812912 |
17,489 | A Koszul complex is a powerful computational tool. For instance, it follows from the theorem and the corollary (Here, one uses the self-duality of a Koszul complex; see Proposition 17.15. of Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry.) Another instance would be Remark: The theorem can be used t... | Dimension theory (algebra) | 0.812912 |
17,490 | As a matter of notation, for any R-module M, we let One sees without difficulty that Γ m {\displaystyle \Gamma _{\mathfrak {m}}} is a left-exact functor and then let H m j = R j Γ m {\displaystyle H_{\mathfrak {m}}^{j}=R^{j}\Gamma _{\mathfrak {m}}} be its j-th right derived functor, called the local cohomology of R. Si... | Dimension theory (algebra) | 0.812912 |
17,491 | {\displaystyle {\mathfrak {m}}\in \operatorname {Supp} (N).} It follows by linear algebra that there is a nonzero homomorphism from N to M modulo m {\displaystyle {\mathfrak {m}}} ; hence, one from N to M by Nakayama's lemma. | Dimension theory (algebra) | 0.812912 |
17,492 | 4. Corollary 2 to Proposition 16, a divisorial ideal is principal if it admits a finite free resolution, which is indeed the case by the theorem. Q.E.D. | Dimension theory (algebra) | 0.812912 |
17,493 | It is a standard algebra exercise to show this implies that R is an integrally closed domain. Now, we need to show every divisorial ideal is principal; i.e., the divisor class group of R vanishes. But, according to Bourbaki, Algèbre commutative, chapitre 7, §. | Dimension theory (algebra) | 0.812912 |
17,494 | Let R be a noetherian ring. The projective dimension of a finite R-module M is the shortest length of any projective resolution of M (possibly infinite) and is denoted by pd R M {\displaystyle \operatorname {pd} _{R}M} . We set g l . d i m R = sup { pd R M ∣ M is a finite module } {\displaystyle \operatorname {gl... | Dimension theory (algebra) | 0.812912 |
17,495 | Let R be a noetherian ring or valuation ring. Then If R is noetherian, this follows from the fundamental theorem below (in particular, Krull's principal ideal theorem), but it is also a consequence of a more precise result. For any prime ideal p {\displaystyle {\mathfrak {p}}} in R, for any prime ideal q ⊋ p R {\displ... | Dimension theory (algebra) | 0.812912 |
17,496 | The converse was shown in the course of the proof of the fundamental theorem.) Proof: Let x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} generate a m A {\displaystyle {\mathfrak {m}}_{A}} -primary ideal and y 1 , … , y m {\displaystyle y_{1},\dots ,y_{m}} be such that their images generate a m B / m A B {\displaystyl... | Dimension theory (algebra) | 0.812912 |
17,497 | (Krull's principal ideal theorem) The height of the ideal generated by elements x 1 , … , x s {\displaystyle x_{1},\dots ,x_{s}} in a noetherian ring is at most s. Conversely, a prime ideal of height s is minimal over an ideal generated by s elements. (Proof: Let p {\displaystyle {\mathfrak {p}}} be a prime ideal minim... | Dimension theory (algebra) | 0.812912 |
17,498 | We set d ( R ) = d {\displaystyle d(R)=d} . We also set δ ( R ) {\displaystyle \delta (R)} to be the minimum number of elements of R that can generate an m {\displaystyle {\mathfrak {m}}} -primary ideal of R. Our ambition is to prove the fundamental theorem: Since we can take s to be δ ( R ) {\displaystyle \delta (R)} ... | Dimension theory (algebra) | 0.812912 |
17,499 | By the Hilbert–Serre theorem, F is a rational function with exactly one pole at t = 1 {\displaystyle t=1} of order d ≤ s {\displaystyle d\leq s} . Since we find that the coefficient of t n {\displaystyle t^{n}} in F ( t ) = ( 1 − t ) d F ( t ) ( 1 − t ) − d {\displaystyle F(t)=(1-t)^{d}F(t)(1-t)^{-d}} is of the form Th... | Dimension theory (algebra) | 0.812912 |
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