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19,000 | Neuropsychology, in exploring the neural correlates of consciousness, interfaces with neuroscience, although the complexity of the central nervous system is a challenge to its study (that is, its operation resists easy reduction). Context-dependent memory and state-dependent memory show hysteretic aspects of neurocogni... | Tipping point (physics) | 0.811804 |
19,001 | Therefore, multidimensional (2, 3 or 4D) experiments have been devised to deal with this problem. To facilitate these experiments, it is desirable to isotopically label the protein with 13C and 15N because the predominant naturally occurring isotope 12C is not NMR-active and the nuclear quadrupole moment of the predomi... | NMR spec | 0.811803 |
19,002 | Much of the innovation within NMR spectroscopy has been within the field of protein NMR spectroscopy, an important technique in structural biology. A common goal of these investigations is to obtain high resolution 3-dimensional structures of the protein, similar to what can be achieved by X-ray crystallography. In con... | NMR spec | 0.811803 |
19,003 | {\displaystyle V.} The square of the distance to P of a point of V {\displaystyle V} is a differential map such that each connected component of V {\displaystyle V} contains at least a critical point, where the distance is minimal. It follows that the number of connected components of V {\displaystyle V} is bounded abo... | Critical point (mathematics) | 0.811802 |
19,004 | Critical points are fundamental for studying the topology of manifolds and real algebraic varieties. In particular, they are the basic tool for Morse theory and catastrophe theory. The link between critical points and topology already appears at a lower level of abstraction. For example, let V {\displaystyle V} be a su... | Critical point (mathematics) | 0.811802 |
19,005 | {\displaystyle {\tfrac {\partial g}{\partial y}}(x,y)=0.} This means that the tangent of the curve is parallel to the y-axis, and that, at this point, g does not define an implicit function from x to y (see implicit function theorem). If (x0, y0) is such a critical point, then x0 is the corresponding critical value. | Critical point (mathematics) | 0.811802 |
19,006 | Let T be a spanning tree for Y and define the fundamental group Γ to be the group generated by the vertex groups Gx and elements y for each edge of Y with the following relations: y = y−1 if y is the edge y with the reverse orientation. y φy,0(x) y−1 = φy,1(x) for all x in Gy. y = 1 if y is an edge in T.This definition... | Graph of groups | 0.811797 |
19,007 | In computer science, robustness is the ability of a computer system to cope with errors during execution and cope with erroneous input. Robustness can encompass many areas of computer science, such as robust programming, robust machine learning, and Robust Security Network. Formal techniques, such as fuzz testing, are ... | Robustness (computer science) | 0.811796 |
19,008 | Robust machine learning typically refers to the robustness of machine learning algorithms. For a machine learning algorithm to be considered robust, either the testing error has to be consistent with the training error, or the performance is stable after adding some noise to the dataset. Recently, consistently with the... | Robustness (computer science) | 0.811796 |
19,009 | In these, pupils can learn about the use of Physics in Medicine, and do hands-on experiments with light and sound using good physics lab equipment. These are guided by local volunteers and an IoP scientist. | Mobile laboratory | 0.811784 |
19,010 | Mobile laboratories have recently been gaining favor from the National Institutes of Health, in the form of the particular mobile laboratories such as the Biobus located in New York and Georgia State University's Bio-Bus Program located in Atlanta, GA and the Boston University MobileLab, as a way to develop knowledge a... | Mobile laboratory | 0.811784 |
19,011 | In molecules whose vibrational mode involves a rotational or pseudorotational mechanism (such as the Berry mechanism or the Bartell mechanism), Van der Waals strain can cause significant differences in potential energy, even between molecules with identical geometry. PF5, for example, has significantly lower potential ... | Steric strain | 0.811779 |
19,012 | Passing–Bablok regression is a method from robust statistics for nonparametric regression analysis suitable for method comparison studies introduced by Wolfgang Bablok and Heinrich Passing in 1983. The procedure is adapted to fit linear errors-in-variables models. It is symmetrical and is robust in the presence of one ... | Passing–Bablok regression | 0.811778 |
19,013 | The term medical technology may also refer to the duties performed by clinical laboratory professionals or medical technologists in various settings within the public and private sectors. The work of these professionals encompasses clinical applications of chemistry, genetics, hematology, immunohematology (blood bankin... | Medical technology | 0.811778 |
19,014 | Artificial Intelligence (AI) is a program that enables computers to sense, reason, act and adapt. AI is not new, but it is growing rapidly and tremendously. AI can now deal with large data sets, solve problems, and provide more efficient operation. AI will be more potential in healthcare because it provides easier acce... | Medical technology | 0.811778 |
19,015 | 3D printing is another major development in healthcare. It can be used to produce specialized splints, prostheses, parts for medical devices and inert implants. The end goal of 3D printing is being able to print out customized replaceable body parts. In the following section, it will explain more about 3D printing in h... | Medical technology | 0.811778 |
19,016 | The IMMC is overseen by an international Organizing Committee of the following people: Solomon Garfunkel, COMAP, USA – Chair Keng Cheng Ang, National Institute of Education, Singapore Fengshan Bai, Deputy Dean of Faculty of Science, Tsinghua University, China Alfred Cheung, NeoUnion ESC Organization, Hong Kong SAR Fred... | International Mathematical Modeling Challenge | 0.811774 |
19,017 | The 2015 IMMC winning teams from the USA, Singapore, and China were brought to Hong Kong for the IMMC Award Presentation Ceremony on 5 July 2015 at Hong Kong University, sponsored by Guangdong Qtone Education Co., Knowledge Magazine, Hong Kong Federation of Education Workers, Competition on System Modeling & Optimizati... | International Mathematical Modeling Challenge | 0.811774 |
19,018 | The Hamiltonian is H = P θ θ ˙ + P ϕ ϕ ˙ − L {\displaystyle H=P_{\theta }{\dot {\theta }}+P_{\phi }{\dot {\phi }}-L} where conjugate momenta are P θ = ∂ L ∂ θ ˙ = m l 2 ⋅ θ ˙ {\displaystyle P_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=ml^{2}\cdot {\dot {\theta }}} and P ϕ = ∂ L ∂ ϕ ˙ = m l 2 sin 2 θ ⋅ ϕ ... | Spherical pendulum | 0.811773 |
19,019 | In physics, a spherical pendulum is a higher dimensional analogue of the pendulum. It consists of a mass m moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity. Owing to the spherical geometry of the problem, spherical coordinates are used t... | Spherical pendulum | 0.811773 |
19,020 | Theoretical problems that need computational investigation are often the concern of computational physics. Theoretical advances may consist in setting aside old, incorrect paradigms (e.g., aether theory of light propagation, caloric theory of heat, burning consisting of evolving phlogiston, or astronomical bodies revol... | Theoretical Physics | 0.811773 |
19,021 | In this regard, theoretical particle physics forms a good example. For instance: "phenomenologists" might employ (semi-) empirical formulas and heuristics to agree with experimental results, often without deep physical understanding. "Modelers" (also called "model-builders") often appear much like phenomenologists, but... | Theoretical Physics | 0.811773 |
19,022 | Other examples include entropy as a measure of the uncertainty regarding the positions and motions of unseen particles and the quantum mechanical idea that (action and) energy are not continuously variable. Theoretical physics consists of several different approaches. | Theoretical Physics | 0.811773 |
19,023 | A physical theory is a model of physical events. It is judged by the extent to which its predictions agree with empirical observations. The quality of a physical theory is also judged on its ability to make new predictions which can be verified by new observations. A physical theory differs from a mathematical theorem ... | Theoretical Physics | 0.811773 |
19,024 | Stack Attack was the game for the 2003 FIRST Robotics Competition. Two teams of two robots compete by moving large Sterilite bins into their zones and arranging them into stacks. | Stack Attack | 0.811771 |
19,025 | The Department has changed names three times in its history from the Department of Chemical Hygiene, when it was founded, to the Department of Biochemistry by the time Roger H. Herriott was its Chair in 1948, to the Department of Biochemistry and Molecular Biology by the time Roger McMacken was its chair in 1990. Yet, ... | JHSPH Department of Biochemistry and Molecular Biology | 0.811765 |
19,026 | The Johns Hopkins School of Public Health Department of Biochemistry and Molecular Biology (BMB) was established in 1916, as the Department of Chemical Hygiene. That same year, the Johns Hopkins School of Hygiene and Public Health was founded, as it was named then. Today, the school is named the Bloomberg School of Pub... | JHSPH Department of Biochemistry and Molecular Biology | 0.811765 |
19,027 | The Division of Reproductive Biology was established in the School of Public Health in 1972. With its emphasis on biochemistry, molecular biology, cell biology, physiology and genetics of reproductive processes. Prior to the Division becoming part of BMB in 1998, it was part of the Department of Population Dynamics, al... | JHSPH Department of Biochemistry and Molecular Biology | 0.811765 |
19,028 | For a rotation matrix R, either proper or improper, the following mathematical equation is always true: ( R v 1 ) × ( R v 2 ) = ( det R ) ( R ( v 1 × v 2 ) ) {\displaystyle (R\mathbf {v_{1}} )\times (R\mathbf {v_{2}} )=(\det R)(R(\mathbf {v_{1}} \times \mathbf {v_{2}} ))} ,where v1 and v2 are any three-dimensional vect... | Polar vector | 0.811765 |
19,029 | A basic and rather concrete example is that of row and column vectors under the usual matrix multiplication operator: in one order they yield the dot product, which is just a scalar and as such a rank zero tensor, while in the other they yield the dyadic product, which is a matrix representing a rank two mixed tensor, ... | Polar vector | 0.811765 |
19,030 | This important requirement is what distinguishes a vector (which might be composed of, for example, the x-, y-, and z-components of velocity) from any other triplet of physical quantities (For example, the length, width, and height of a rectangular box cannot be considered the three components of a vector, since rotati... | Polar vector | 0.811765 |
19,031 | The definition of a "vector" in physics (including both polar vectors and pseudovectors) is more specific than the mathematical definition of "vector" (namely, any element of an abstract vector space). Under the physics definition, a "vector" is required to have components that "transform" in a certain way under a prop... | Polar vector | 0.811765 |
19,032 | The pattern of the spikes matches the pattern of the blue light stimuli. By inserting channelrhodopsin gene sequences into mouse DNA, researchers can control spikes and therefore certain behaviors of the mouse (e.g., making the mouse turn left). Researchers, through optogenetics, have the tools to effect different temp... | Sparse code | 0.811765 |
19,033 | The specificity of temporal coding requires highly refined technology to measure informative, reliable, experimental data. Advances made in optogenetics allow neurologists to control spikes in individual neurons, offering electrical and spatial single-cell resolution. For example, blue light causes the light-gated ion ... | Sparse code | 0.811765 |
19,034 | Population coding is a method to represent stimuli by using the joint activities of a number of neurons. In population coding, each neuron has a distribution of responses over some set of inputs, and the responses of many neurons may be combined to determine some value about the inputs. From the theoretical point of vi... | Sparse code | 0.811765 |
19,035 | Above, pseudovectors have been discussed using active transformations. An alternate approach, more along the lines of passive transformations, is to keep the universe fixed, but switch "right-hand rule" with "left-hand rule" everywhere in math and physics, including in the definition of the cross product and the curl. ... | Polar vector | 0.811765 |
19,036 | The data of the real vector space and the J matrix is exactly the same as the data of the complex vector space, as the J matrix allows one to define complex multiplication. At the level of Lie algebras and Lie groups, this corresponds to the inclusion of gl(n,C) in gl(2n,R) (Lie algebras – matrices, not necessarily inv... | Linear complex structure | 0.811762 |
19,037 | Since C is algebraically closed, J is guaranteed to have eigenvalues which satisfy λ2 = −1, namely λ = ±i. Thus we may write V C = V + ⊕ V − {\displaystyle V^{\mathbb {C} }=V^{+}\oplus V^{-}} where V+ and V− are the eigenspaces of +i and −i, respectively. Complex conjugation interchanges V+ and V−. | Linear complex structure | 0.811762 |
19,038 | This algebra is realized concretely as which corresponds to i2 = −1. Then a representation of C is a real vector space V, together with an action of C on V (a map C → End(V)). | Linear complex structure | 0.811762 |
19,039 | A complex structure on a real vector space V is a real linear transformation such that Here J2 means J composed with itself and IdV is the identity map on V. That is, the effect of applying J twice is the same as multiplication by −1. This is reminiscent of multiplication by the imaginary unit, i. A complex structure a... | Linear complex structure | 0.811762 |
19,040 | The exterior algebra is perhaps the most important application of this decomposition. In general, if a vector space U admits a decomposition U = S ⊕ T then the exterior powers of U can be decomposed as follows: Λ r U = ⨁ p + q = r ( Λ p S ) ⊗ ( Λ q T ) . {\displaystyle \Lambda ^{r}U=\bigoplus _{p+q=r}(\Lambda ^{p}S)\ot... | Linear complex structure | 0.811762 |
19,041 | Let V be a real vector space with a complex structure J. The dual space V* has a natural complex structure J* given by the dual (or transpose) of J. The complexification of the dual space (V*)C therefore has a natural decomposition ( V ∗ ) C = ( V ∗ ) + ⊕ ( V ∗ ) − {\displaystyle (V^{*})^{\mathbb {C} }=(V^{*})^{+}\oplu... | Linear complex structure | 0.811762 |
19,042 | Concretely, this is just an action of i, as this generates the algebra, and the operator representing i (the image of i in End(V)) is exactly J. If VJ has complex dimension n then V must have real dimension 2n. That is, a finite-dimensional space V admits a complex structure only if it is even-dimensional. It is not ha... | Linear complex structure | 0.811762 |
19,043 | In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space. Every complex vector space can be equipped with a ... | Linear complex structure | 0.811762 |
19,044 | Different contact forms whose kernels give the same contact structure will yield different Reeb vector fields, whose dynamics are in general very different. The various flavors of contact homology depend a priori on the choice of a contact form, and construct algebraic structures the closed trajectories of their Reeb v... | Contact geometry | 0.811761 |
19,045 | As a prime example, consider R3, endowed with coordinates (x,y,z) and the one-form dz − y dx. The contact plane ξ at a point (x,y,z) is spanned by the vectors X1 = ∂y and X2 = ∂x + y ∂z. By replacing the single variables x and y with the multivariables x1, ..., xn, y1, ..., yn, one can generalize this example to any R2... | Contact geometry | 0.811761 |
19,046 | In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and ... | Contact geometry | 0.811761 |
19,047 | (In Boolean algebra, exchanging AND ↔ OR and 1 ↔ 0 throughout an equation yields an equally valid equation.) The identities remain invariant regardless of which interpretation is chosen, so the transformations or modes of calculation remain the same; only the interpretation of each form would be different. Example: J1a... | Boundary algebra | 0.81176 |
19,048 | This description of 2 is simpler than the conventional one, namely an ⟨ B , + , × , ¬ , 1 , 0 ⟩ {\displaystyle \langle B,+,\times ,\lnot ,1,0\rangle } algebra of type ⟨ 2 , 2 , 1 , 0 , 0 ⟩ {\displaystyle \langle 2,2,1,0,0\rangle } . The two possible interpretations are dual to each other in the Boolean sense. | Boundary algebra | 0.81176 |
19,049 | The expressive adequacy of the Sheffer stroke points to the primary algebra also being a ⟨ B , − − | ¯ , | ¯ ⟩ {\displaystyle \langle B,{\overline {-\ -\ |}},{\overline {\ \ |}}\rangle } algebra of type ⟨ 2 , 0 ⟩ {\displaystyle \langle 2,0\rangle } . In both cases, the identities are J1a, J0, C2, and ACD=CDA. Since the... | Boundary algebra | 0.81176 |
19,050 | The primary arithmetic suggests the following arithmetic axiomatization of 2: 1+1=1+0=0+1=1=~0, and 0+0=0=~1. The set B = { {\displaystyle \ B=\{} , {\displaystyle ,} } {\displaystyle \ \}} is the Boolean domain or carrier. In the language of universal algebra, the primary algebra is the algebraic structure ⟨ B , − − ,... | Boundary algebra | 0.81176 |
19,051 | The primary algebra is an elegant minimalist notation for the two-element Boolean algebra 2. Let: One of Boolean join (+) or meet (×) interpret concatenation; The complement of A interpret 0 (1) interpret the empty Mark if join (meet) interprets concatenation (because a binary operation applied to zero operands may be ... | Boundary algebra | 0.81176 |
19,052 | This is the defining arithmetical identity of group theory, in boundary notation. The primary algebra differs from an abelian group in two ways: From A2, it follows that ≠ . If the primary algebra were a group, = would hold, and one of a = or a = a would have to be a primary algebra consequence. Note that and are mutua... | Boundary algebra | 0.81176 |
19,053 | Let denote the inverse of a. Let denote the group identity element. Then groups and the primary algebra have the same signatures, namely they are both ⟨ − − , − | ¯ , | ¯ ⟩ {\displaystyle \langle -\ -,{\overline {-\ |}},{\overline {\ \ |}}\rangle } algebras of type 〈2,1,0〉. Hence the primary algebra is a boundary algeb... | Boundary algebra | 0.81176 |
19,054 | For a few authors, mereology is simply a model of Boolean algebra and hence of the primary algebra as well. Mereotopology, which is inherently richer than Boolean algebra; The system of Whitehead (1934), whose fundamental primitive is "indication".The primary arithmetic and algebra are a minimalist formalism for senten... | Boundary algebra | 0.81176 |
19,055 | The biologists and cognitive scientists Humberto Maturana and his student Francisco Varela both discuss LoF in their writings, which identify "distinction" as the fundamental cognitive act. The Berkeley psychologist and cognitive scientist Eleanor Rosch has written extensively on the closely related notion of categoriz... | Boundary algebra | 0.81176 |
19,056 | Second-generation cognitive science emerged in the 1970s, after LoF was written. On cognitive science and its relevance to Boolean algebra, logic, and set theory, see Lakoff (1987) (see index entries under "Image schema examples: container") and Lakoff and Núñez (2001). Neither book cites LoF. | Boundary algebra | 0.81176 |
19,057 | Craig (1979) argued that the world, and how humans perceive and interact with that world, has a rich Boolean structure. Craig was an orthodox logician and an authority on algebraic logic. | Boundary algebra | 0.81176 |
19,058 | The above formalisms are, like the primary algebra, all instances of boundary mathematics, i.e., mathematics whose syntax is limited to letters and brackets (enclosing devices). A minimalist syntax of this nature is a "boundary notation". Boundary notation is free of infix operators, prefix, or postfix operator symbols... | Boundary algebra | 0.81176 |
19,059 | The syntax of his alpha existential graphs is merely concatenation, read as conjunction, and enclosure by ovals, read as negation. If primary algebra concatenation is read as conjunction, then these graphs are isomorphic to the primary algebra (Kauffman 2001).Ironically, LoF cites vol. 4 of Peirce's Collected Papers, t... | Boundary algebra | 0.81176 |
19,060 | The semantics of the streamer are identical to those of the Cross, except that Peirce never wrote a streamer with nothing under it. An excerpt from one of these papers was published in 1976, but they were not published in full until 1993. In a 1902 encyclopedia article, Peirce notated Boolean algebra and sentential log... | Boundary algebra | 0.81176 |
19,061 | Gottfried Leibniz, in memoranda not published before the late 19th and early 20th centuries, invented Boolean logic. His notation was isomorphic to that of LoF: concatenation read as conjunction, and "non-(X)" read as the complement of X. Recognition of Leibniz's pioneering role in algebraic logic was foreshadowed by L... | Boundary algebra | 0.81176 |
19,062 | Thus the flow of time may be introduced into the primary algebra. Turney (1986) shows how these recursive formulae can be interpreted via Alonzo Church's Restricted Recursive Arithmetic (RRA). Church introduced RRA in 1955 as an axiomatic formalization of finite automata. | Boundary algebra | 0.81176 |
19,063 | Appendix 2 of LoF shows how to translate traditional syllogisms and sorites into the primary algebra. A valid syllogism is simply one whose primary algebra translation simplifies to an empty Cross. Let A* denote a literal, i.e., either A or A | ¯ {\displaystyle {\overline {A|}}} , indifferently. Then every syllogism th... | Boundary algebra | 0.81176 |
19,064 | Another consequence of self-duality is the irrelevance of De Morgan's laws; those laws are built into the syntax of the primary algebra from the outset. The true nature of the distinction between the primary algebra on the one hand, and 2 and sentential logic on the other, now emerges. In the latter formalisms, complem... | Boundary algebra | 0.81176 |
19,065 | All that is required is that the interpretation of concatenation change from OR to AND. IF A THEN B now translates as instead of . More generally, the primary algebra is "self-dual", meaning that any primary algebra formula has two sentential or Boolean readings, each the dual of the other. | Boundary algebra | 0.81176 |
19,066 | Schwartz (1981) proved that the primary algebra is equivalent — syntactically, semantically, and proof theoretically — with the classical propositional calculus. Likewise, it can be shown that the primary algebra is syntactically equivalent with expressions built up in the usual way from the classical truth values true... | Boundary algebra | 0.81176 |
19,067 | This is an example of a decision procedure, one more or less in the spirit of conventional truth tables. Given some primary algebra formula containing N variables, this decision procedure requires simplifying 2N primary arithmetic formulae. For a less tedious decision procedure more in the spirit of Quine's "truth valu... | Boundary algebra | 0.81176 |
19,068 | Equivalently, the primary algebra interprets sentential logic. Given an assignment of every variable to the Marked or Unmarked states, this primary algebra translation reduces to a primary arithmetic expression, which can be simplified. Repeating this exercise for all possible assignments of the two primitive values to... | Boundary algebra | 0.81176 |
19,069 | Let the blank page denote False, and let a Cross be read as Not. Then the primary arithmetic has the following sentential reading: = False = True = not False = Not True = FalseThe primary algebra interprets sentential logic as follows. A letter represents any given sentential expression. Thus: interprets Not A interpre... | Boundary algebra | 0.81176 |
19,070 | The primary algebra embodies a point noted by Huntington in 1933: Boolean algebra requires, in addition to one unary operation, one, and not two, binary operations. Hence the seldom-noted fact that Boolean algebras are magmas. (Magmas were called groupoids until the latter term was appropriated by category theory.) To ... | Boundary algebra | 0.81176 |
19,071 | If the Marked and Unmarked states are read as the Boolean values 1 and 0 (or True and False), the primary algebra interprets 2 (or sentential logic). LoF shows how the primary algebra can interpret the syllogism. Each of these interpretations is discussed in a subsection below. Extending the primary algebra so that it ... | Boundary algebra | 0.81176 |
19,072 | But an empty Cross is a well-formed primary algebra expression, denoting the Marked state, a primitive value. Hence a nonempty Cross is an operator, while an empty Cross is an operand because it denotes a primitive value. Thus the primary algebra reveals that the heretofore distinct mathematical concepts of operator an... | Boundary algebra | 0.81176 |
19,073 | Over the past decade many scientists argued that the hierarchy problem is a specific application of Bayesian statistics. Studying renormalization in hierarchy problems is difficult, because such quantum corrections are usually power-law divergent, which means that the shortest-distance physics are most important. Becau... | Gauge hierarchy problem | 0.811759 |
19,074 | In theoretical physics, the hierarchy problem is the problem concerning the large discrepancy between aspects of the weak force and gravity. There is no scientific consensus on why, for example, the weak force is 1024 times stronger than gravity. | Gauge hierarchy problem | 0.811759 |
19,075 | The method has been applied to several combinatorial optimization problems including the Job Shop Scheduling problems, Flow-Shop Problems, Vehicle Routing Problems as well as many others. == References == | Iterated local search | 0.811758 |
19,076 | Iterated Local Search (ILS) is a term in applied mathematics and computer science defining a modification of local search or hill climbing methods for solving discrete optimization problems. Local search methods can get stuck in a local minimum, where no improving neighbors are available. A simple modification consists... | Iterated local search | 0.811758 |
19,077 | A bibliography at the Website for systolic geometry and topology currently contains over 170 articles. Systolic geometry is a rapidly developing field, featuring a number of recent publications in leading journals. Recently, an intriguing link has emerged with the Lusternik–Schnirelmann category. The existence of such ... | Introduction to systolic geometry | 0.811757 |
19,078 | The explanation of the strengthened version of Loewner's inequality is somewhat more technical than the rest of this article. It seems worth including it here for the sake of completeness. The strengthened version is the inequality a r e a ( g ) − 3 2 s y s ( g ) 2 ≥ V a r ( f ) , {\displaystyle \mathrm {area} (g)-{\tf... | Introduction to systolic geometry | 0.811757 |
19,079 | In the room example, absent other features, the systole would be the circumference of the pillar. Systolic geometry gives lower bounds for various attributes of the space in terms of its systole. It is known that the Fubini–Study metric is the natural metric for the geometrisation of quantum mechanics. In an intriguing... | Introduction to systolic geometry | 0.811757 |
19,080 | For example, a room with a pillar in the middle, connecting floor to ceiling, is not simply connected. In geometry, a systole is a distance which is characteristic of a compact metric space which is not simply connected. It is the length of a shortest loop in the space that cannot be contracted to a point in the space. | Introduction to systolic geometry | 0.811757 |
19,081 | Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve C, and the length or perimeter of C. Since the area A may be small while the length l is large, when C looks elongated, the relationship can only take the... | Introduction to systolic geometry | 0.811757 |
19,082 | In projective geometry, the real projective plane R P 2 {\displaystyle \mathbb {RP} ^{2}} is defined as the collection of lines through the origin in R 3 {\displaystyle \mathbb {R} ^{3}} . The distance function on R P 2 {\displaystyle \mathbb {RP} ^{2}} is most readily understood from this point of view. Namely, the di... | Introduction to systolic geometry | 0.811757 |
19,083 | The International Journal of Pattern Recognition and Artificial Intelligence was founded in 1987 and is published by World Scientific. The journal covers developments in artificial intelligence, and its sub-field, pattern recognition. This includes articles on image and language processing, robotics and neural networks... | International Journal of Pattern Recognition and Artificial Intelligence | 0.811755 |
19,084 | Spectroscopy is the study of objects based on the spectrum of color they emit, absorb or reflect. Visible-light spectroscopy is an important tool in astronomy (as is spectroscopy at other wavelengths), where scientists use it to analyze the properties of distant objects. Chemical elements and small molecules can be det... | Visible frequency | 0.811749 |
19,085 | In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field. | Regular semi-algebraic system | 0.811747 |
19,086 | Define P > := { p > 0 ∣ p ∈ P } {\displaystyle P_{>}:=\{p>0\mid p\in P\}} . Let Q {\displaystyle {\mathcal {Q}}} be a quantifier-free formula of k {\displaystyle \mathbf {k} } involving only the variables of u {\displaystyle \mathbf {u} } . We say that R := {\displaystyle R:=} is a regular semi-algebraic system if th... | Regular semi-algebraic system | 0.811747 |
19,087 | Let T {\displaystyle T} be a regular chain of k {\displaystyle \mathbf {k} } for some ordering of the variables x = x 1 , … , x n {\displaystyle \mathbf {x} =x_{1},\ldots ,x_{n}} and a real closed field k {\displaystyle \mathbf {k} } . Let u = u 1 , … , u d {\displaystyle \mathbf {u} =u_{1},\ldots ,u_{d}} and y = y 1 ... | Regular semi-algebraic system | 0.811747 |
19,088 | Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. The notion of a regular semi-algebraic system is an adaptation of the concept of a regular chain focusing on solutions of the real analogue: semi-algebraic systems. Any semi-... | Regular semi-algebraic system | 0.811747 |
19,089 | A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in kinematics and Partial Differential Equations – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the ... | Mathematical singularity | 0.81174 |
19,090 | Percy Williams Bridgman received a Nobel Prize in 1946 for advancing this area of physics by two magnitudes of pressure (400 MPa to 40 GPa). The list of founding fathers of this field includes also the names of Harry George Drickamer, Tracy Hall, Francis P. Bundy, Leonid F. Vereschagin, and Sergey M. Stishov. It was by... | High pressure | 0.811738 |
19,091 | To define a formal reduction potential for a biochemical reaction, the pH value, the concentrations values and the hypotheses made on the activity coefficients must always be explicitly indicated. When using, or comparing, several formal reduction potentials they must also be internally consistent. Problems may occur w... | Formal reduction potential | 0.811738 |
19,092 | Formal standard reduction potentials ( E red ⊖ ′ {\displaystyle E_{\text{red}}^{\ominus '}} ) are also commonly used in biochemistry and cell biology for referring to standard reduction potentials measured at pH 7, a value closer to the pH of most physiological and intracellular fluids than the standard state pH of 0. ... | Formal reduction potential | 0.811738 |
19,093 | Quantities here are given per molecule, not per mole, and so Boltzmann constant k and the electron charge e are used instead of the gas constant R and Faraday's constant F. To convert to the molar quantities given in most chemistry textbooks, it is simply necessary to multiply by the Avogadro constant: R = kNA and F = ... | Formal reduction potential | 0.811738 |
19,094 | In mathematics, specifically abstract algebra, if ( G , + ) {\displaystyle (G,+)} is an (abelian) group with identity element e {\displaystyle e} then ν: G → R {\displaystyle \nu \colon G\to \mathbb {R} } is said to be a norm on ( G , + ) {\displaystyle (G,+)} if: Positive definiteness: ν ( g ) > 0 for all g ≠ e and ν ... | Norm (abelian group) | 0.811735 |
19,095 | A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form. Over an algebraically closed field, any matrix A has a Jordan normal form and therefore ad... | Characteristic root | 0.811729 |
19,096 | One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an algebra representation – an associative algebra acting on a module. The study of such actions is the field of representation theory. The representation-theoretical concept of weight i... | Characteristic root | 0.811729 |
19,097 | Let λi be an eigenvalue of an n by n matrix A. The algebraic multiplicity μA(λi) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that (λ − λi)k divides evenly that polynomial.Suppose a matrix A has dimension n and d ≤ n distinct eigenvalues. Whereas ... | Characteristic root | 0.811729 |
19,098 | Computational mechanics (CM) is interdisciplinary. Its three pillars are mechanics, mathematics, and computer science and physics. | Tangent stiffness matrix | 0.811727 |
19,099 | The areas of mathematics most related to computational mechanics are partial differential equations, linear algebra and numerical analysis. The most popular numerical methods used are the finite element, finite difference, and boundary element methods in order of dominance. In solid mechanics finite element methods are... | Tangent stiffness matrix | 0.811727 |
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