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2,700 | Some of these algorithms, such as BLAST, are still used in modern bioinformatics. Scientists in the HGP also had to address complexities often associated with biomedical data including noisy data, such as DNA read errors, and privacy rights of the research subjects. The HGP, completed in 2004, has had immense impact bo... | Biomedical data science | 0.833368 |
2,701 | The Human Genome Project (HGP), which uncovered the DNA sequences that compose human genes, would not have been possible without biomedical data science. Significant computational resources were required to process the data in the HGP, as the human genome contains over 6 billion DNA base pairs. Scientists constructed t... | Biomedical data science | 0.833368 |
2,702 | In 1972, the department of engineering mechanics at the Virginia Polytechnic Institute and State University changed its name and undergraduate program to engineering science and mechanics. In 1974, the department of engineering mechanics at the Pennsylvania State University merged with engineering science program and t... | Engineering science and mechanics | 0.83336 |
2,703 | A Bachelor of Science, Master of Science, Master of Engineering, or Ph.D. degree in engineering science, engineering mechanics, or engineering science and mechanics is awarded upon completion of the respective program. Areas of specialization include aerodynamics, biomechanics, bionanotechnology, biosensors and bioelec... | Engineering science and mechanics | 0.83336 |
2,704 | The most widely used learning algorithms are: Support-vector machines Linear regression Logistic regression Naive Bayes Linear discriminant analysis Decision trees K-nearest neighbor algorithm Neural networks (Multilayer perceptron) Similarity learning | Supervised classification | 0.833325 |
2,705 | Computational learning theory Inductive bias Overfitting (machine learning) (Uncalibrated) class membership probabilities Unsupervised learning Version spaces | Supervised classification | 0.833325 |
2,706 | Analytical learning Artificial neural network Backpropagation Boosting (meta-algorithm) Bayesian statistics Case-based reasoning Decision tree learning Inductive logic programming Gaussian process regression Genetic programming Group method of data handling Kernel estimators Learning automata Learning classifier system... | Supervised classification | 0.833325 |
2,707 | A wide range of supervised learning algorithms are available, each with its strengths and weaknesses. There is no single learning algorithm that works best on all supervised learning problems (see the No free lunch theorem). There are four major issues to consider in supervised learning: | Supervised classification | 0.833325 |
2,708 | Supervised learning (SL) is a paradigm in machine learning where input objects (for example, a vector of predictor variables) and a desired output value (also known as human-labeled supervisory signal) train a model. The training data is processed, building a function that maps new data on expected output values. An op... | Supervised classification | 0.833325 |
2,709 | This "yes-no" format is often equivalently stated as "accept-reject"; that is, an algorithm "accepts" an input string if the answer to the decision problem is "yes" and "rejects" if the answer is "no". While some problems cannot easily be expressed as decision problems, they nonetheless encompass a broad range of compu... | Complexity class | 0.833321 |
2,710 | The most commonly analyzed problems in theoretical computer science are decision problems—the kinds of problems that can be posed as yes–no questions. The primality example above, for instance, is an example of a decision problem as it can be represented by the yes–no question "is the natural number n {\displaystyle n}... | Complexity class | 0.833321 |
2,711 | The complexity classes PSPACE and NPSPACE are the space analogues to P and NP. That is, PSPACE is the class of problems solvable in polynomial space by a deterministic Turing machine and NPSPACE is the class of problems solvable in polynomial space by a nondeterministic Turing machine. More formally, P S P A C E = ⋃ k ... | Complexity class | 0.833321 |
2,712 | (Imagine a soap film on a wire loop, which can be deformed by blowing on the film). Which of those surfaces is to be chosen? If the loop does not lie in a single plane, for example, there is no one obvious choice. The answer is that it does not matter: in the magnetostatic case, the current density is solenoidal (see n... | Ampère-Maxwell equation | 0.833293 |
2,713 | Proof that the formulations of the circuital law in terms of free current are equivalent to the formulations involving total current In this proof, we will show that the equation ∇ × H = J f + ∂ D ∂ t {\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}} is e... | Ampère-Maxwell equation | 0.833293 |
2,714 | Lazy funnelsort is a modification of the funnelsort, introduced by Gerth Stølting Brodal and Rolf Fagerberg in 2002. The modification is that when a merger is invoked, it does not have to fill each of its buffers. Instead, it lazily fills a buffer only when it is empty. This modification has the same asymptotic runtime... | Funnelsort | 0.833286 |
2,715 | In graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can be drawn in the plane without edge intersections). This is a well-studied problem in computer science for which many practical algorithms have emerged, many taking advant... | Planarity testing | 0.833271 |
2,716 | Planarity testing algorithms typically take advantage of theorems in graph theory that characterize the set of planar graphs in terms that are independent of graph drawings. These include Kuratowski's theorem that a graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (the complete ... | Planarity testing | 0.833271 |
2,717 | PP is also equal to another quantum complexity class known as PQP, which is the unbounded error analog of BQP. It denotes the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of less than 1/2 for all instances. Even if all amplitudes used for PQP-computation are dr... | Probabilistic polynomial-time | 0.83324 |
2,718 | The set of all possible values of a sum type is the set-theoretic sum, i.e., the disjoint union, of the sets of all possible values of its variants. Enumerated types are a special case of sum types in which the constructors take no arguments, as exactly one value is defined for each constructor. Values of algebraic typ... | Algebraic data types | 0.833238 |
2,719 | In computer programming, especially functional programming and type theory, an algebraic data type (ADT) is a kind of composite type, i.e., a type formed by combining other types. Two common classes of algebraic types are product types (i.e., tuples and records) and sum types (i.e., tagged or disjoint unions, coproduct... | Algebraic data types | 0.833238 |
2,720 | Do not confuse these patterns with regular expression patterns used in string pattern matching. The purpose is similar: to check whether a piece of data matches certain constraints, and if so, extract relevant parts of it for processing. However, the mechanism is very different. This kind of pattern matching on algebra... | Algebraic data types | 0.833238 |
2,721 | The recursion in patterns in this example are trivial, but a possible more complex recursive pattern would be something like Node (Node (Leaf 4) x) (Node y (Node Empty z)). Recursive patterns several layers deep are used for example in balancing red–black trees, which involve cases that require looking at colors severa... | Algebraic data types | 0.833238 |
2,722 | To do something with a value of this Tree algebraic data type, it is deconstructed using a process called pattern matching. It involves matching the data with a series of patterns. The example function "depth" above pattern-matches its argument with three patterns. | Algebraic data types | 0.833238 |
2,723 | Algebraic data types are used to represent values that can be one of several types of things. Each type of thing is associated with an identifier called a constructor, which can be viewed as a kind of tag for that kind of data. Each constructor can carry with it a different type of data. A constructor could carry no da... | Algebraic data types | 0.833238 |
2,724 | A general algebraic data type is a possibly recursive sum type of product types. Each constructor tags a product type to separate it from others, or if there is only one constructor, the data type is a product type. Further, the parameter types of a constructor are the factors of the product type. A parameterless const... | Algebraic data types | 0.833238 |
2,725 | In many situations where the congruence subgroup problem is expected to have a positive solution it has been proven that this is indeed the case. Here is a list of algebraic groups such that the congruence subgroup property is known to hold for the associated arithmetic lattices, in case the rank of the associated Lie ... | Congruence subgroup | 0.833236 |
2,726 | In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible 2 × 2 integer matrices of determinant 1 in which the off-diagonal entries are even. More generally, the notion of congruence sub... | Congruence subgroup | 0.833236 |
2,727 | RNA interference (RNAi) screen is essentially a forward genetics screen using a reverse genetics technique. Similar to classical genetic screens in the past, large-scale RNAi surveys success depends on a careful development of phenotypic assays and their interpretation. In Drosophila, RNAi has been applied in cultured ... | Genetic screens | 0.833177 |
2,728 | In molecular biology, a protein domain is a region of a protein's polypeptide chain that is self-stabilizing and that folds independently from the rest. Each domain forms a compact folded three-dimensional structure. Many proteins consist of several domains, and a domain may appear in a variety of different proteins. | Protein domain | 0.833162 |
2,729 | Methods of calculus are often used in problems in which the variables are continuous, for example in continuous optimization problems.In statistical theory, the probability distributions of continuous variables can be expressed in terms of probability density functions. In continuous-time dynamics, the variable time is... | Continuous variables | 0.833147 |
2,730 | Software engineer and blogger Jeff Atwood described Code as a "love letter to the computer".Publishers Weekly, shortly after Code's publication, said "Initial response, at least among traditional tech book readers, has been positive" and quotes the book's editor, Ben Ryan, as saying "We're trying to cross the boundary ... | Code: The Hidden Language of Computer Hardware and Software | 0.833147 |
2,731 | Although the process had been going on throughout the nineteenth century, it was this work that finally displaced action at a distance physics and substituted the physics of the field. "Mark P. Silverman (1998) "I studied the principles on my own – in this case with Maxwell's Treatise as both my inspiration and textboo... | A Treatise on Electricity and Magnetism | 0.833142 |
2,732 | Pearce Williams (1991): "In 1873, James Clerk Maxwell published a rambling and difficult two-volume Treatise on Electricity and Magnetism that was destined to change the orthodox picture of physical reality. This treatise did for electromagnetism what Newton's Principia had done for classical mechanics. It not only pro... | A Treatise on Electricity and Magnetism | 0.833142 |
2,733 | A third edition was prepared by J. J. Thomson for publication in 1892. The treatise is said to be notoriously hard to read, containing plenty of ideas but lacking both the clear focus and orderliness that may have allowed it catch on more easily. It was noted by one historian of science that Maxwell's attempt at a comp... | A Treatise on Electricity and Magnetism | 0.833142 |
2,734 | Elementary Mathematical Theory of Electricity. On Electrical Work and Energy in a System of Conductors. General Theorems. | A Treatise on Electricity and Magnetism | 0.833142 |
2,735 | This change in the conception of reality is the most profound and fruitful one that has come to physics since Newton; but it has at the same time to be admitted that the program has by no means been completely carried out yet. "Richard P. Feynman (1964): "From a long view of the history of mankind—seen from, say, ten t... | A Treatise on Electricity and Magnetism | 0.833142 |
2,736 | Maxwell is the scientific ancestor of Hertz, Hertz of Marconi and all other workers at wireless telegraphy.Oliver Lodge (1907) "Then comes Maxwell, with his keen penetration and great grasp of thought, combined with mathematical subtlety and power of expression; he assimilates the facts, sympathizes with the philosophi... | A Treatise on Electricity and Magnetism | 0.833142 |
2,737 | Chemical physics is a subdiscipline of chemistry and physics that investigates physicochemical phenomena using techniques from atomic and molecular physics and condensed matter physics; it is the branch of physics that studies chemical processes from the point of view of physics. While at the interface of physics and c... | Chemical physicist | 0.833142 |
2,738 | The goals of chemical physics research include understanding chemical structures and reactions at the quantum mechanical level, elucidating the structure and reactivity of gas phase ions and radicals, and discovering accurate approximations to make the physics of chemical phenomena computationally accessible. Chemical ... | Chemical physicist | 0.833142 |
2,739 | The Journal of Chemical Physics Journal of Physical Chemistry Letters Journal of Physical Chemistry A Journal of Physical Chemistry B Journal of Physical Chemistry C Physical Chemistry Chemical Physics Chemical Physics Letters Chemical Physics ChemPhysChem Molecular Physics (journal) | Chemical physicist | 0.833142 |
2,740 | In 2007, SGI worked with BP to commercialize microbial-based processes for increasing the conversion and recovery of subsurface hydrocarbons. In 2009, SGI received funding from ExxonMobil to produce biofuels on an industrial-scale using recombinant algae and other microorganisms. The company purchased an 81-acre site (... | Synthetic Genomics | 0.833142 |
2,741 | Synthetic Genomics was founded in the spring of 2005 by J. Craig Venter, Nobel Laureate Hamilton O. Smith, Juan Enriquez, and David Kiernan. Venter (and Smith)'s previous company, Celera Genomics, was a driving force in the race to sequence the human genome. The firm takes its name from the phrase synthetic genomics wh... | Synthetic Genomics | 0.833142 |
2,742 | Rational mechanics may refer to: mécanique rationelle, a historical (19th century) term for classical mechanics a school of thought within physics advocated by Clifford Truesdell in the 1960s | Rational mechanics | 0.833137 |
2,743 | In particular, unlike Contains it does not distinguish between points in the boundary and in the interior of geometries. For most situations, Covers should be used in preference to Contains.Similarly, the mask T*F**F*** occurs in the definition of both Within and CoveredBy. For most situations, CoveredBy should be used... | Spatial predicates | 0.833129 |
2,744 | Predicates "meets" is a synonym for touches; "inside" is a synonym for within Oracle's "ANYINTERACT" is a synonym for intersects and "OVERLAPBDYINTERSECT" is a synonym for overlaps. Its "OVERLAPBDYDISJOINT" does not have a corresponding named predicate. In Region connection calculus operators offer some synonyms for pr... | Spatial predicates | 0.833129 |
2,745 | Sets with one or more operations that obey specific laws are called algebraic structures. When a new problem involves the same laws as such an algebraic structure, all the results that have been proved using only the laws of the structure can be directly applied to the new problem. In full generality, algebraic structu... | Algebraic system | 0.833127 |
2,746 | Addition and multiplication are prototypical examples of operations that combine two elements of a set to produce a third element of the same set. These operations obey several algebraic laws. For example, a + (b + c) = (a + b) + c and a(bc) = (ab)c are associative laws, and a + b = b + a and ab = ba are commutative la... | Algebraic system | 0.833127 |
2,747 | (it follows that fields do not form a variety in the sense of universal algebra.) It can be stated: "Every nonzero element of a field is invertible;" or, equivalently: the structure has a unary operation inv such that ∀ x , x = 0 or x ⋅ inv ( x ) = 1. {\displaystyle \forall x,\quad x=0\quad {\text{or}}\quad x\cdot \o... | Algebraic system | 0.833127 |
2,748 | The axioms of an algebraic structure can be any first-order formula, that is a formula involving logical connectives (such as "and", "or" and "not"), and logical quantifiers ( ∀ , ∃ {\displaystyle \forall ,\exists } ) that apply to elements (not to subsets) of the structure. Such a typical axiom is inversion in fields.... | Algebraic system | 0.833127 |
2,749 | Cambridge Univ. Press, Cambridge 2008, ISBN 978-0-521-86149-6. Schwenn, Rainer, Space Weather, Living Reviews in Solar Physics 3, (2006), 2, online article. | Solar weather | 0.83312 |
2,750 | These models take less resources than their more sophisticated descendants. Later models use physics to account for as many phenomena as possible. No model can yet reliably predict the environment from the surface of the Sun to the bottom of the Earth's ionosphere. | Solar weather | 0.83312 |
2,751 | Space weather models are simulations of the space weather environment. Models use sets of mathematical equations to describe physical processes. These models take a limited data set and attempt to describe all or part of the space weather environment in or to predict how weather evolves over time. Early models were heu... | Solar weather | 0.83312 |
2,752 | Space weather is a branch of space physics and aeronomy, or heliophysics, concerned with the time varying conditions within the Solar System, including the solar wind, emphasizing the space surrounding the Earth, including conditions in the magnetosphere, ionosphere, thermosphere, and exosphere. Space weather is distin... | Solar weather | 0.83312 |
2,753 | Since Sylow's theorem ensures the existence of p-subgroups of a finite group, it's worthwhile to study groups of prime power order more closely. Most of the examples use Sylow's theorem to prove that a group of a particular order is not simple. For groups of small order, the congruence condition of Sylow's theorem is o... | Sylow 2-subgroup | 0.833119 |
2,754 | In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R, that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to ex... | Boolean rings | 0.833089 |
2,755 | There are at least four different and incompatible systems of notation for Boolean rings and algebras: In commutative algebra the standard notation is to use x + y = (x ∧ ¬ y) ∨ (¬ x ∧ y) for the ring sum of x and y, and use xy = x ∧ y for their product. In logic, a common notation is to use x ∧ y for the meet (same as... | Boolean rings | 0.833089 |
2,756 | Not every unital associative algebra over F2 is a Boolean ring: consider for instance the polynomial ring F2. The quotient ring R/I of any Boolean ring R modulo any ideal I is again a Boolean ring. Likewise, any subring of a Boolean ring is a Boolean ring. | Boolean rings | 0.833089 |
2,757 | Every Boolean ring R satisfies x ⊕ x = 0 for all x in R, because we know x ⊕ x = (x ⊕ x)2 = x2 ⊕ x2 ⊕ x2 ⊕ x2 = x ⊕ x ⊕ x ⊕ xand since (R,⊕) is an abelian group, we can subtract x ⊕ x from both sides of this equation, which gives x ⊕ x = 0. A similar proof shows that every Boolean ring is commutative: x ⊕ y = (x ⊕ y)2 ... | Boolean rings | 0.833089 |
2,758 | The analogous result holds beginning with a Boolean algebra. A map between two Boolean rings is a ring homomorphism if and only if it is a homomorphism of the corresponding Boolean algebras. Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order idea... | Boolean rings | 0.833089 |
2,759 | Since the join operation ∨ in a Boolean algebra is often written additively, it makes sense in this context to denote ring addition by ⊕, a symbol that is often used to denote exclusive or. Given a Boolean ring R, for x and y in R we can define x ∧ y = xy,x ∨ y = x ⊕ y ⊕ xy,¬x = 1 ⊕ x.These operations then satisfy all ... | Boolean rings | 0.833089 |
2,760 | One example of a Boolean ring is the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection. As another example, we can also consider the set of all finite or cofinite subsets of X, again with symmetric difference and intersection as operations. More gener... | Boolean rings | 0.833089 |
2,761 | Precision is the fraction of the documents retrieved that are relevant to the user's information need. precision = | { relevant documents } ∩ { retrieved documents } | | { retrieved documents } | {\displaystyle {\mbox{precision}}={\frac {|\{{\mbox{relevant documents}}\}\cap \{{\mbox{retrieved documents}}\}|}{|\{{\mbox{... | Precision at k | 0.833082 |
2,762 | Some authors choose to interpolate the p ( r ) {\displaystyle p(r)} function to reduce the impact of "wiggles" in the curve. For example, the PASCAL Visual Object Classes challenge (a benchmark for computer vision object detection) until 2010 computed the average precision by averaging the precision over a set of evenl... | Precision at k | 0.833082 |
2,763 | Young, Hugh D.; Freedman, Roger A. (2010). Sears and Zemansky's University Physics: With Modern Physics (13th ed.). Addison-Wesley (Pearson). ISBN 978-0-321-69686-1. | Coulomb force constant | 0.833079 |
2,764 | 23–27. Tipler, Paul A.; Mosca, Gene (2008). Physics for Scientists and Engineers (6th ed.). | Coulomb force constant | 0.833079 |
2,765 | Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that calculates the amount of force between two electrically charged particles at rest. This electric force is conventionally called electrostatic force or Coulomb force. Although the law was known earlier, it was first published i... | Coulomb force constant | 0.833079 |
2,766 | The constant ε 0 {\displaystyle \varepsilon _{0}} is the vacuum electric permittivity (also known as the electric constant). It should not be confused with ε r {\displaystyle \varepsilon _{r}} , which is the dimensionless relative permittivity of the material in which the charges are immersed, or with their product ε a... | Coulomb force constant | 0.833079 |
2,767 | A derivation on A is a map D with the property D ( x ⋅ y ) = D ( x ) ⋅ y + x ⋅ D ( y ) . {\displaystyle D(x\cdot y)=D(x)\cdot y+x\cdot D(y)\ .} The derivations on A form a subspace DerK(A) in EndK(A). The commutator of two derivations is again a derivation, so that the Lie bracket gives DerK(A) a structure of Lie algeb... | Nonassociative algebra | 0.833069 |
2,768 | Ring-like structures with two binary operations and no other restrictions are a broad class, one which is too general to study. For this reason, the best-known kinds of non-associative algebras satisfy identities, or properties, which simplify multiplication somewhat. These include the following ones. | Nonassociative algebra | 0.833069 |
2,769 | A chemical species is a chemical substance or ensemble composed of chemically identical molecular entities that can explore the same set of molecular energy levels on a characteristic or delineated time scale. These energy levels determine the way the chemical species will interact with others (engaging in chemical bon... | Chemical specie | 0.833047 |
2,770 | Complements need not exist, and if they do they need not be unique. That is, H could have two distinct complements K1 and K2 in G. If there are several complements of a normal subgroup, then they are necessarily isomorphic to each other and to the quotient group. If K is a complement of H in G then K forms both a left ... | Complement (group theory) | 0.833035 |
2,771 | As previously mentioned, complements need not exist. A p-complement is a complement to a Sylow p-subgroup. Theorems of Frobenius and Thompson describe when a group has a normal p-complement. Philip Hall characterized finite soluble groups amongst finite groups as those with p-complements for every prime p; these p-comp... | Complement (group theory) | 0.833035 |
2,772 | In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup H in a group G is a subgroup K of G such that G = H K = { h k: h ∈ H , k ∈ K } and H ∩ K = { e } . {\displaystyle G=HK=\{hk:h\in H,k\in K\}{\text{ and }}H\cap K=\{e\}.} Equivalently, every element of G has a unique expre... | Complement (group theory) | 0.833035 |
2,773 | The classification of energy into different types often follows the boundaries of the fields of study in the natural sciences. Chemical energy is the kind of potential energy "stored" in chemical bonds and is studied in chemistry. Nuclear energy is energy stored in interactions between the particles in the atomic nucle... | Conservation of mechanical energy | 0.833022 |
2,774 | In some versions, they compete to be the first to guess correctly; in others, they can work out a strategy beforehand to cooperate and maximize the probability of correct guesses.One variation received some new publicity as a result of Todd Ebert's 1998 Ph.D. thesis at the University of California, Santa Barbara. It is... | Induction puzzles | 0.833021 |
2,775 | When they hear no gunshot, they will realize that the case above did not apply, thus there must be more than 2 unfaithful husbands and as before their own husband is the only candidate to be the extra one. In general, if there are n unfaithful husbands, each of their wives will believe there to be n-1 and will expect t... | Induction puzzles | 0.833021 |
2,776 | The Smith report has influenced debate on U.K. educational policy. A particular concern of the report was where and how statistics should be taught: the report recommended that statistics should be embedded in application subjects and taught by teachers of those subjects where it is applied. The government decision was... | Making Mathematics Count | 0.833017 |
2,777 | Making Mathematics Count is the title of a report on mathematics education in the United Kingdom (U.K.). The report was written by Adrian Smith as leader of an "Inquiry into Post–14 Mathematics Education", which was commissioned by the UK Government in 2002. The report recommended an increase in mathematics schooling; ... | Making Mathematics Count | 0.833017 |
2,778 | In that convention, Lie algebra elements are multiplied by i, the exponential map (below) is defined with an extra factor of i in the exponent and the structure constants remain the same, but the definition of them acquires a factor of i. Likewise, commutation relations acquire a factor of i. The commutation relations ... | 3D rotation | 0.832989 |
2,779 | These are related to the Pauli matrices by t i ⟷ 1 2 i σ i . {\displaystyle {\boldsymbol {t}}_{i}\longleftrightarrow {\frac {1}{2i}}\sigma _{i}.} The Pauli matrices abide by the physicists' convention for Lie algebras. | 3D rotation | 0.832989 |
2,780 | The Lie algebras s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} and s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} are isomorphic. One basis for s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} is given by t 1 = 1 2 , t 2 = 1 2 , t 3 = 1 2 . {\displaystyle {\boldsymbol {t}}_{1}={\frac {1}{2}}{\begin{bmatrix}0&-i\\-i&0\en... | 3D rotation | 0.832989 |
2,781 | For every unitary irreducible representations Dj there is an equivalent one, D−j−1. All infinite-dimensional irreducible representations must be non-unitary, since the group is compact. In quantum mechanics, the Casimir invariant is the "angular-momentum-squared" operator; integer values of spin j characterize bosonic ... | 3D rotation | 0.832989 |
2,782 | In Lie algebra representations, the group SO(3) is compact and simple of rank 1, and so it has a single independent Casimir element, a quadratic invariant function of the three generators which commutes with all of them. The Killing form for the rotation group is just the Kronecker delta, and so this Casimir invariant ... | 3D rotation | 0.832989 |
2,783 | The definitions for graded modules and algebras can also be extended this way replacing the indexing set N {\displaystyle \mathbb {N} } with any monoid G. Remarks: If we do not require that the ring have an identity element, semigroups may replace monoids.Examples: A group naturally grades the corresponding group ring;... | Graded algebra | 0.832986 |
2,784 | (cf. Serre's twisting sheaf in algebraic geometry.) Let M and N be graded modules. If f: M → N {\displaystyle f\colon M\to N} is a morphism of modules, then f is said to have degree d if f ( M n ) ⊆ N n + d {\displaystyle f(M_{n})\subseteq N_{n+d}} . An exterior derivative of differential forms in differential geometry... | Graded algebra | 0.832986 |
2,785 | The kernel and the image of a morphism of graded modules are graded submodules. Remark: To give a graded morphism from a graded ring to another graded ring with the image lying in the center is the same as to give the structure of a graded algebra to the latter ring. Given a graded module M {\displaystyle M} , the ℓ {\... | Graded algebra | 0.832986 |
2,786 | By the Cameron–Martin theorem, γ {\displaystyle \gamma } would then be quasi-invariant under translation by any element of H , {\displaystyle H,} which implies that either H {\displaystyle H} is finite-dimensional or that γ {\displaystyle \gamma } is the zero measure. In either case, we have a contradiction. Sazonov's ... | Cylindrical measure | 0.83297 |
2,787 | An alternative proof that the Gaussian cylinder set measure is not a measure uses the Cameron–Martin theorem and a result on the quasi-invariance of measures. If γ H = γ {\displaystyle \gamma ^{H}=\gamma } really were a measure, then the identity function on H {\displaystyle H} would radonify that measure, thus making ... | Cylindrical measure | 0.83297 |
2,788 | The cylinder sets are the pre-images in E {\displaystyle E} of measurable sets in F T {\displaystyle F_{T}}: if B T {\displaystyle {\mathcal {B}}_{T}} denotes the σ {\displaystyle \sigma } -algebra on F T {\displaystyle F_{T}} on which μ T {\displaystyle \mu _{T}} is defined, then In practice, one often takes B T {\dis... | Cylindrical measure | 0.83297 |
2,789 | Light sheet fluorescence microscopy for the observation of development and LILBID mass spectrometry for the analysis of membrane complexes was improved. PELDOR-EPR was developed to a resolution that allows in-cell measurements. The Cluster promoted scientific exchange through a range of programmes as well as through wo... | Cluster of Excellence Frankfurt Macromolecular Complexes | 0.832958 |
2,790 | They delineated the role of sumoylation in ribosome quality control and characterized the process of genetic quality control in oocytes. The efforts in these three research areas were accompanied by approaches to design or reprogram macromolecular complexes and new methods developed to expand the already strong experti... | Cluster of Excellence Frankfurt Macromolecular Complexes | 0.832958 |
2,791 | CEF scientists in collaboration with the Max Planck Institute for Biophysical Chemistry visualized the RNA Polymerase I (Pol I) in the process of actively transcribing ribosome genes in a cellular environment and solved its structure with and without nucleic acids at 3.8 Å resolution by cryo-EM. Their structures explai... | Cluster of Excellence Frankfurt Macromolecular Complexes | 0.832958 |
2,792 | The Cluster of Excellence Frankfurt "Macromolecular Complexes" (CEF) was established in 2006 by Goethe University Frankfurt together with the Max Planck Institute of Biophysics and the Max Planck Institute for Brain Research in the context of the German Universities Excellence Initiative. Funding by the Deutsche Forsch... | Cluster of Excellence Frankfurt Macromolecular Complexes | 0.832958 |
2,793 | The Cluster also strongly support new developments in advanced light microscopy. A particularly important technique CEF added to the research technique portfolio in Frankfurt is light sheet fluorescence microscopy (LSFM)). In LSFM, optical sectioning in the excitation process minimizes fluorophore bleaching and phototo... | Cluster of Excellence Frankfurt Macromolecular Complexes | 0.832958 |
2,794 | They also develops algorithms to solve problems in molecular biology, ranging from atomic protein structure analysis to computational systems biology. Their tools leverage on graph theory, Petri nets and Boolean networks with broad applications within CEF. Their collaborations cover diverse topics from plant metabolomi... | Cluster of Excellence Frankfurt Macromolecular Complexes | 0.832958 |
2,795 | Their computational and theoretical studies aided in the interpretation of increasingly complex measurements, and guided the design of future experiments. The interdisciplinary field of bioinformatics opened new perspectives on molecular processes and cellular function. CEF scientists used custom-tailored code and pipe... | Cluster of Excellence Frankfurt Macromolecular Complexes | 0.832958 |
2,796 | Method development in theoretical biophysics plays an increasingly important role in the study of macromolecular complexes and has made essential contributions to many studies in the other research areas of CEF. Bridging between fundamental physics, chemistry and biology, CEF scientists studied biomolecular processes o... | Cluster of Excellence Frankfurt Macromolecular Complexes | 0.832958 |
2,797 | The origin of optogenetics lies in the work of the Bamberg group at the MPI of Biophysics in Frankfurt, who showed that channelrhodopsin-2 (ChR2) is a light-gated cation channel that can depolarize the cells in which it is expressed. During CEF, the Bamberg lab continued to work in this field and contributed several se... | Cluster of Excellence Frankfurt Macromolecular Complexes | 0.832958 |
2,798 | Femtosecond time-resolved spectroscopy was used by CEF scientists to study molecular dynamics and function. This method enables the observation of extremely fast chemical and biological reactions in real time involving a wide variety of molecules from small organic compounds to complex enzymes. Studies included molecul... | Cluster of Excellence Frankfurt Macromolecular Complexes | 0.832958 |
2,799 | Cryo-electron microscopy, Nature Method of the Year 2015 and the method for which a Nobel prize was awarded in 2017, was extensively employed by several CEF groups, at the MPI of Biophysics as well as at Goethe University's Buchmann Institute for Molecular Biology. Direct electron detectors, in the development of which... | Cluster of Excellence Frankfurt Macromolecular Complexes | 0.832958 |
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