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98,100 | Geology has been used in many military intelligence investigations. During World War II, the American Military Geology Unit discovered the origin of balloon bombs which had been dispatched towards North America from Japan. They accomplished this by determining from which beach the sand in the balloon's ballast originat... | Military geology | 0.777451 |
98,101 | Geologists have been involved in the construction of forts, tunnels, and bunkers both during military conflicts and in peacetime. This included digging tunnels in northeastern Italy and Austria during the so-called mine war in World War I. The rocks of the Dolomites are different from those in other theaters and specia... | Military geology | 0.777451 |
98,102 | Geologists have been employed since the Napoleonic Wars to provide an analysis of terrain which was expected to become a war theater, both in case of an upcoming battle and to assess the difficulty of logistical supply. Academically, it has been found that battles are likely to occur on rocks of Permian, Triassic, or U... | Military geology | 0.777451 |
98,103 | In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or (6,3,7 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,7}. | Order-8 hexagonal tiling honeycomb | 0.77745 |
98,104 | By the 18th century maritime exploration had become safer and more efficient with technical innovations that vastly improved navigation and cartography: improvements were made to the theodolite, octant, precision clocks, as well as the compass, telescope, and general shipbuilding techniques. From the mid-18th century t... | European and American voyages of scientific exploration | 0.777449 |
98,105 | (five volumes, 1838–1843),R. FitzRoy (editor), Narrative of the surveying voyages of His Majesty's Ships Adventure and Beagle between the years 1826 and 1836, describing their examination of the southern shores of South America, and the Beagle's circumnavigation of the globe. (volume 2 and appendix by FitzRoy, Proceedi... | European and American voyages of scientific exploration | 0.777449 |
98,106 | A French expedition exploring Western Australia and islands of Timor, Molucca, Samoa and Hawaii. L'Uranie visited Rio de Janeiro to take a series of pendulum measurements as well as other observations, not only in geography and ethnology, but in astronomy, terrestrial magnetism, and meteorology, and for the collection ... | European and American voyages of scientific exploration | 0.777449 |
98,107 | All the iron(II) sulfates dissolve in water to give the same aquo complex 2+, which has octahedral molecular geometry and is paramagnetic. The name copperas dates from times when the copper(II) sulfate was known as blue copperas, and perhaps in analogy, iron(II) and zinc sulfate were known respectively as green and whi... | Ferrous sulfate | 0.777447 |
98,108 | A C*-algebra M acting on H is said to act non-degenerately if for h in H, Mh = {0} implies h = 0. In this case, it can be shown using an approximate identity in M that the identity operator I lies in the strong closure of M. Therefore, the conclusion of the bicommutant theorem holds for M. | Von Neumann double commutant theorem | 0.777446 |
98,109 | From this it follows that PTP = TP for all T in M.By using the closure of M under the adjoint we further have, for every T in M and all x, y ∈ H: ⟨ x , T P y ⟩ = ⟨ x , P T P y ⟩ = ⟨ P x , T P y ⟩ = ⟨ T ∗ P x , P y ⟩ = ⟨ P T ∗ P x , y ⟩ = ⟨ T ∗ P x , y ⟩ = ⟨ P x , T y ⟩ = ⟨ x , P T y ⟩ {\displaystyle \langle x,TPy\rangl... | Von Neumann double commutant theorem | 0.777446 |
98,110 | Fix X ∈ M′′. We will show X ∈ clS(M). Fix an open neighborhood U of X in the strong operator topology. By definition of the strong operator topology, U contains a finite intersection U(h1,ε1) ∩...∩U(hn,εn) of subbasic open sets of the form U(h,ε) = {O ∈ L(H): ||Oh - Xh|| < ε}, where h is in H and ε > 0. | Von Neumann double commutant theorem | 0.777446 |
98,111 | If the structure assumed over variables is in the form of a 1D, 2D or 3D grid, then submodular functions based on overlapping groups can be considered as norms, leading to stable sets equal to rectangular or convex shapes. Such methods have applications in computer vision | Structured sparsity regularization | 0.777446 |
98,112 | Hierarchical models using Bayesian non-parametric methods have been used to learn topic models, which are statistical models for discovering the abstract "topics" that occur in a collection of documents. Hierarchies have also been considered in the context of kernel methods. Hierarchical norms have been applied to bioi... | Structured sparsity regularization | 0.777446 |
98,113 | Structured Sparsity regularization can be applied in the context of multiple kernel learning. Multiple kernel learning refers to a set of machine learning methods that use a predefined set of kernels and learn an optimal linear or non-linear combination of kernels as part of the algorithm. In the algorithms mentioned a... | Structured sparsity regularization | 0.777446 |
98,114 | Subsequently, NATO has issued EW policy and doctrine and is addressing the other NATO defense lines of development. Primary EW activities have been developed over time to exploit the opportunities and vulnerabilities that are inherent in the physics of EM energy. Activities used in EW include electro-optical, infrared ... | Electronic Warfare | 0.777443 |
98,115 | In electrical engineering, Millman's theorem (or the parallel generator theorem) is a method to simplify the solution of a circuit. Specifically, Millman's theorem is used to compute the voltage at the ends of a circuit made up of only branches in parallel. It is named after Jacob Millman, who proved the theorem. | Millman's Theorem | 0.777442 |
98,116 | If one of the branches is an ideal voltage source, Millman's theorem cannot be used, but in this case the solution is trivial, the voltage at the output is forced to the voltage of the ideal voltage source. The theorem does not work with ideal voltage sources because such sources have zero resistance (infinite conducta... | Millman's Theorem | 0.777442 |
98,117 | One method of deriving Millman's theorem starts by converting all the branches to current sources (which can be done using Norton's theorem). A branch that is already a current source is simply not converted. In the expression above, this is equivalent to replacing the e k / R k {\displaystyle e_{k}/R_{k}} term in the ... | Millman's Theorem | 0.777442 |
98,118 | The American Diabetes Association, European Association for the Study of Diabetes, and International Diabetes Federation have agreed that, in the future, HbA1c is to be reported in the International Federation of Clinical Chemistry and Laboratory Medicine (IFCC) units. IFCC reporting was introduced in Europe except for... | Glycated haemoglobin | 0.777441 |
98,119 | Textbooks in physical chemistry generally give two derivations of the title equation. One is the conventional derivation that goes back to Van der Waals, a mechanical equation of state that cannot be used to specify all thermodynamic functions; the other is a statistical mechanics derivation that makes explicit the int... | Real gas law | 0.77744 |
98,120 | After initial pursuit of teaching credentials, Van der Waals' undergraduate coursework in mathematics and physics at the University of Leiden in the Netherlands led (with significant hurdles) to his acceptance for doctoral studies at Leiden under Pieter Rijke. While his dissertation helps to explain the experimental ob... | Real gas law | 0.77744 |
98,121 | The Van der Waals equation is a thermodynamic equation of state based on the theory that fluids are composed of particles with non-zero volumes, and subject to a (not necessarily pairwise) inter-particle attractive force. It was based on work in theoretical physical chemistry performed in the late 19th century by Johan... | Real gas law | 0.77744 |
98,122 | The superconducting tunnel junction (STJ) — also known as a superconductor–insulator–superconductor tunnel junction (SIS) — is an electronic device consisting of two superconductors separated by a very thin layer of insulating material. Current passes through the junction via the process of quantum tunneling. The STJ i... | Superconducting tunnel junction | 0.777439 |
98,123 | Superconducting quantum computing utilizes STJ-based circuits, including charge qubits, flux qubits and phase qubits. | Superconducting tunnel junction | 0.777439 |
98,124 | Consequently, landlines used by broadcasters for stereo transmissions are equalised to very tight differential phase specifications. Another property of the lattice filter is that it is an intrinsically balanced topology. This is useful when used with landlines which invariably use a balanced format. Many other types o... | Lattice phase equaliser | 0.777437 |
98,125 | A lattice phase equaliser or lattice filter is an example of an all-pass filter. That is, the attenuation of the filter is constant at all frequencies but the relative phase between input and output varies with frequency. The lattice filter topology has the particular property of being a constant-resistance network and... | Lattice phase equaliser | 0.777437 |
98,126 | If g l ( k , R ) {\displaystyle {\mathfrak {gl}}(k,\mathbb {R} )} is the set of k × k matrices with entries in R {\displaystyle \mathbb {R} } , then the subalgebra consisting of strictly upper triangular matrices is a nilpotent Lie algebra. | Nilpotent Lie algebra | 0.777436 |
98,127 | If a Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent, then all subalgebras and homomorphic images are nilpotent. | Nilpotent Lie algebra | 0.777436 |
98,128 | If a Lie algebra g {\displaystyle {\mathfrak {g}}} has an automorphism of prime period with no fixed points except at 0, then g {\displaystyle {\mathfrak {g}}} is nilpotent. | Nilpotent Lie algebra | 0.777436 |
98,129 | Every nilpotent Lie algebra is solvable. This is useful in proving the solvability of a Lie algebra since, in practice, it is usually easier to prove nilpotency (when it holds!) rather than solvability. However, in general, the converse of this property is false. For example, the subalgebra of g l ( k , R ) {\displayst... | Nilpotent Lie algebra | 0.777436 |
98,130 | Engel's theorem: A finite dimensional Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if and only if all elements of g {\displaystyle {\mathfrak {g}}} are ad-nilpotent. | Nilpotent Lie algebra | 0.777436 |
98,131 | Conversely, one may write , ⋯ , X 2 ] , X 1 ] = a d , ⋯ , X 2 ] ( X 1 ) , {\displaystyle ,\cdots ,X_{2}],X_{1}]=\mathrm {ad} ,\cdots ,X_{2}](X_{1}),} and since ad is a Lie algebra homomorphism, a d , ⋯ , X 2 ] = , ⋯ X 3 ] , a d X 2 ] = … = , ⋯ a d X 2 ] . {\displaystyle {\begin{aligned}\mathrm {ad} ,\cdots ,X_{2}]... | Nilpotent Lie algebra | 0.777436 |
98,132 | Remarkably, if g {\displaystyle {\mathfrak {g}}} is finite dimensional, the apparently much weaker condition (2) is actually equivalent to (1), as stated by Engel's theorem: A finite dimensional Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if and only if all elements of g {\displaystyle {\mathfrak {g}}} a... | Nilpotent Lie algebra | 0.777436 |
98,133 | A very special consequence of (1) is that ⋯ ] = a d X n Y ∈ g n = 0 ∀ X , Y ∈ g . ( 2 ) {\displaystyle \cdots ]={\mathrm {ad} _{X}}^{n}Y\in {\mathfrak {g}}_{n}=0\quad \forall X,Y\in {\mathfrak {g}}.\qquad (2)} Thus (adX)n = 0 for all X ∈ g {\displaystyle X\in {\mathfrak {g}}} . That is, adX is a nilpotent endomorphism... | Nilpotent Lie algebra | 0.777436 |
98,134 | A nilpotent Lie algebra has an outer automorphism, that is, an automorphism that is not in the image of Ad. | Nilpotent Lie algebra | 0.777436 |
98,135 | A Heisenberg algebra is nilpotent. For example, in dimension 3, the commutator of two matrices , ] = {\displaystyle \left={\begin{bmatrix}0&0&a''\\0&0&0\\0&0&0\end{bmatrix}}} where a ″ = a c ′ − a ′ c {\displaystyle a''=ac'-a'c} . | Nilpotent Lie algebra | 0.777436 |
98,136 | A Cartan subalgebra c {\displaystyle {\mathfrak {c}}} of a Lie algebra l {\displaystyle {\mathfrak {l}}} is nilpotent and self-normalizing page 80. The self-normalizing condition is equivalent to being the normalizer of a Lie algebra. This means c = N l ( c ) = { x ∈ l: ⊂ c for c ∈ c } {\displaystyle {\mathfrak {c}}=N... | Nilpotent Lie algebra | 0.777436 |
98,137 | The derived subalgebra of a finite dimensional solvable Lie algebra over a field of characteristic 0 is nilpotent. | Nilpotent Lie algebra | 0.777436 |
98,138 | Let g {\displaystyle {\mathfrak {g}}} be a Lie algebra. One says that g {\displaystyle {\mathfrak {g}}} is nilpotent if the lower central series terminates, i.e. if g n = 0 {\displaystyle {\mathfrak {g}}_{n}=0} for some n ∈ N . {\displaystyle n\in \mathbb {N} .} Explicitly, this means that ⋯ ] ] = a d X 1 a d X 2 ⋯ a ... | Nilpotent Lie algebra | 0.777436 |
98,139 | The lower central series for Lie algebras is analogous to the lower central series in group theory, and nilpotent Lie algebras are analogs of nilpotent groups. The nilpotent Lie algebras are precisely those that can be obtained from abelian Lie algebras, by successive central extensions. Note that the definition means ... | Nilpotent Lie algebra | 0.777436 |
98,140 | . {\displaystyle {\mathfrak {g}}\geq \geq ]\geq ]]\geq ...} We write g 0 = g {\displaystyle {\mathfrak {g}}_{0}={\mathfrak {g}}} , and g n = {\displaystyle {\mathfrak {g}}_{n}=} for all n > 0 {\displaystyle n>0} . If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nil... | Nilpotent Lie algebra | 0.777436 |
98,141 | In mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras g ≥ ≥ ] ≥ ] ] ≥ . . | Nilpotent Lie algebra | 0.777436 |
98,142 | The Killing form of a nilpotent Lie algebra is 0. | Nilpotent Lie algebra | 0.777436 |
98,143 | If the quotient algebra g / Z ( g ) {\displaystyle {\mathfrak {g}}/Z({\mathfrak {g}})} , where Z ( g ) {\displaystyle Z({\mathfrak {g}})} is the center of g {\displaystyle {\mathfrak {g}}} , is nilpotent, then so is g {\displaystyle {\mathfrak {g}}} . This is to say that a central extension of a nilpotent Lie algebra b... | Nilpotent Lie algebra | 0.777436 |
98,144 | A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map. The inverse function theorem implies that a smooth map f: M → N {\displaystyle f:M\to N} is a local diffeomorphism if and only if the derivative D f p: T p M → T f ( p ) N {\displaystyle Df_{p}:T_{p}M\to T_... | Local diffeomorphism | 0.777435 |
98,145 | In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. | Local diffeomorphism | 0.777435 |
98,146 | In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f: X → Y {\displaystyle f:X\to Y} is a local homeomorphism, X {\displaystyle X} is said to be an étale space over Y . {\displaystyle... | Local homeomorphism | 0.777435 |
98,147 | Consider a complex vector space V {\displaystyle V} with either even complex dimension 2 n {\displaystyle 2n} or odd complex dimension 2 n + 1 {\displaystyle 2n+1} and a nondegenerate complex scalar product Q {\displaystyle Q} , with values Q ( u , v ) {\displaystyle Q(u,v)} on pairs of vectors ( u , v ) {\displaystyle... | Pure spinor | 0.777433 |
98,148 | Pure spinors were introduced in string quantization by Nathan Berkovits.Nigel Hitchin introduced generalized Calabi–Yau manifolds, where the generalized complex structure is defined by a pure spinor. These spaces describe the geometry of flux compactifications in string theory. | Pure spinor | 0.777433 |
98,149 | In the domain of mathematics known as representation theory, pure spinors (or simple spinors) are spinors that are annihilated under the Clifford action by a maximal isotropic subspace of the space V {\displaystyle V} of vectors with respect to the scalar product determining the Clifford algebra. They were introduced b... | Pure spinor | 0.777433 |
98,150 | Spinors are modules of the Clifford algebra, and so in particular there is an action of the elements of V {\displaystyle V} on the space of spinors. The complex subspace V ψ 0 ⊂ V {\displaystyle V_{\psi }^{0}\subset V} that annihilates a given nonzero spinor ψ {\displaystyle \psi } has dimension m ≤ n {\displaystyle m\... | Pure spinor | 0.777433 |
98,151 | Mir was the first modular space station and was assembled in orbit from 1986 to 1996. It had a greater mass than any previous spacecraft. Until 21 March 2001 it was the largest artificial satellite in orbit, succeeded by the International Space Station after Mir's orbit decayed. The station served as a microgravity res... | Human spaceflight programs | 0.777432 |
98,152 | This section compares the butterfly network with linear array, ring, 2-D mesh and hypercube networks. Linear array can be considered as a 1-D mesh topology. Relevant parameters are compiled in the table (‘p’ represents the number of processor nodes). | Butterfly network | 0.77743 |
98,153 | They can be uni-directional or bi-directional.These multistage networks have lower cost than a cross bar, but obtain lower contention than a bus. The ratio of switching nodes to processor nodes is greater than one in a butterfly network. Such topology, where the ratio of switching nodes to processor nodes is greater th... | Butterfly network | 0.77743 |
98,154 | Butterfly networks are more complex and costlier than other topologies due to the higher number of links required to sustain the network.The difference between hypercube and butterfly lies within their implementation. Butterfly network has a symmetric structure where all processor nodes between two ranks are equidistan... | Butterfly network | 0.77743 |
98,155 | A butterfly network is a technique to link multiple computers into a high-speed network. This form of multistage interconnection network topology can be used to connect different nodes in a multiprocessor system. The interconnect network for a shared memory multiprocessor system must have low latency and high bandwidth... | Butterfly network | 0.77743 |
98,156 | Several parameters help evaluate a network topology. The prominent ones relevant in designing large-scale multi-processor systems are summarized below and an explanation of how they are calculated for a butterfly network with 8 processor nodes as shown in figure 1 is provided. Bisection Bandwidth: The maximum bandwidth... | Butterfly network | 0.77743 |
98,157 | In category theory, the notion of Frobenius object is an abstract definition of a Frobenius algebra in a category. A Frobenius object ( A , μ , η , δ , ε ) {\displaystyle (A,\mu ,\eta ,\delta ,\varepsilon )} in a monoidal category ( C , ⊗ , I ) {\displaystyle (C,\otimes ,I)} consists of an object A of C together with f... | Symmetric Frobenius algebra | 0.777429 |
98,158 | For a Frobenius algebra A with σ as above, the automorphism ν of A such that σ(a, b) = σ(ν(b), a) is Nakayama automorphism associated to A and σ. | Symmetric Frobenius algebra | 0.777429 |
98,159 | A finite-dimensional, unital, associative algebra A defined over a field k is said to be a Frobenius algebra if A is equipped with a nondegenerate bilinear form σ: A × A → k that satisfies the following equation: σ(a·b, c) = σ(a, b·c). This bilinear form is called the Frobenius form of the algebra. Equivalently, one ma... | Symmetric Frobenius algebra | 0.777429 |
98,160 | Frobenius algebras originally were studied as part of an investigation into the representation theory of finite groups, and have contributed to the study of number theory, algebraic geometry, and combinatorics. They have been used to study Hopf algebras, coding theory, and cohomology rings of compact oriented manifolds... | Symmetric Frobenius algebra | 0.777429 |
98,161 | Consistent with other aspects of Mn(II) chemistry, manganocene (Mn(C5H5)2) is high-spin. In contrast, its neighboring metal iron forms an air-stable, low-spin derivative in the form of ferrocene (Fe(C5H5)2). When conducted under an atmosphere of carbon monoxide, reduction of Mn(II) salts gives dimanganese decacarbonyl ... | Alloy of aluminium and manganese | 0.777428 |
98,162 | Manganese forms a large variety of organometallic derivatives, i.e., compounds with Mn-C bonds. The organometallic derivatives include numerous examples of Mn in its lower oxidation states, i.e. Mn(-III) up through Mn(I). This area of organometallic chemistry is attractive because Mn is inexpensive and of relatively lo... | Alloy of aluminium and manganese | 0.777428 |
98,163 | (The word "googly" is a term used in the game of cricket for a ball bowled with right-handed helicity using the apparent action that would normally give rise to left-handed helicity.) The most recent proposal in this direction by Penrose in 2015 was based on noncommutative geometry on twistor space and referred to as p... | Twistor Theory | 0.777427 |
98,164 | In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a widely studied branch of theoretical and mathematical physics. Penrose's idea was that twistor space should be the basic arena for physics from which space-time itself should emerge.... | Twistor Theory | 0.777427 |
98,165 | Projective twistor space P T {\displaystyle \mathbb {PT} } is projective 3-space C P 3 {\displaystyle \mathbb {CP} ^{3}} , the simplest 3-dimensional compact algebraic variety. It has a physical interpretation as the space of massless particles with spin. It is the projectivisation of a 4-dimensional complex vector spa... | Twistor Theory | 0.777427 |
98,166 | Beg solved the early LDDMM algorithms by solving the variational matching taking variations with respect to the vector fields. Another solution by Vialard, reparameterizes the optimization problem in terms of the state q t ≐ I ∘ ϕ t − 1 , q 0 = I {\displaystyle q_{t}\doteq I\circ \phi _{t}^{-1},q_{0}=I} , for image I (... | Large Deformation Diffeomorphic Metric Mapping | 0.777425 |
98,167 | The Calculus of variations was used in Beg to derive the iterative algorithm as a solution which when it converges satisfies the necessary maximizer conditions given by the necessary conditions for a first order variation requiring the variation of the endpoint with respect to a first order variation of the vector fiel... | Large Deformation Diffeomorphic Metric Mapping | 0.777425 |
98,168 | The disadvantages of these methods is that there was no associated global least-action property which could score the flows of minimum energy. This contrasts the geodesic motions which are central to the study of Rigid body kinematics and the many problems solved in Physics via Hamilton's principle of least action. In ... | Large Deformation Diffeomorphic Metric Mapping | 0.777425 |
98,169 | Diffeomorphic mapping 3-dimensional information across coordinate systems is central to high-resolution Medical imaging and the area of Neuroinformatics within the newly emerging field of bioinformatics. Diffeomorphic mapping 3-dimensional coordinate systems as measured via high resolution dense imagery has a long hist... | Large Deformation Diffeomorphic Metric Mapping | 0.777425 |
98,170 | The vector fields are guaranteed to be 1-time continuously differentiable v t ∈ C 1 {\displaystyle v_{t}\in C^{1}} by modelling them to be in a smooth Hilbert space v ∈ V {\displaystyle v\in V} supporting 1-continuous derivative. The inverse ϕ t − 1 , t ∈ {\displaystyle \phi _{t}^{-1},t\in } is defined by the Eulerian... | Large Deformation Diffeomorphic Metric Mapping | 0.777425 |
98,171 | If C ~ {\displaystyle {\tilde {C}}} is a contraction that extends CA and its projection onto its domain is self-adjoint, then it is clear that its inverse Cayley transform A ~ = ( 1 + C ~ ) ( 1 − C ~ ) − 1 {\displaystyle {\tilde {A}}=(1+{\tilde {C}})(1-{\tilde {C}})^{-1}} defined on R a n ( 1 − C ~ ) {\displaystyle Ran... | Extensions of symmetric operators | 0.777424 |
98,172 | In functional analysis, one is interested in extensions of symmetric operators acting on a Hilbert space. Of particular importance is the existence, and sometimes explicit constructions, of self-adjoint extensions. This problem arises, for example, when one needs to specify domains of self-adjointness for formal expres... | Extensions of symmetric operators | 0.777424 |
98,173 | If f ∈ Dom(Aα), then f is absolutely continuous and | f ( 0 ) f ( 1 ) | = | e α − 1 α − e | = 1. {\displaystyle \left|{\frac {f(0)}{f(1)}}\right|=\left|{\frac {e\alpha -1}{\alpha -e}}\right|=1.} Conversely, if f is absolutely continuous and f(0) = γf(1) for some complex γ with |γ| = 1, then f lies in the above domain. ... | Extensions of symmetric operators | 0.777424 |
98,174 | The unitarity criterion of the Cayley transform is replaced by self-adjointness for positive operators. Theorem A symmetric positive operator A is self-adjoint if and only if its Cayley transform is a self-adjoint contraction defined on all of H, i.e. when Ran(A + 1) = H. Therefore, finding self-adjoint extension for a... | Extensions of symmetric operators | 0.777424 |
98,175 | The following is true. Theorem A closed symmetric operator A is self-adjoint if and only if Ran (A ± i) = H, i.e. when its Cayley transform UA is a unitary operator on H. Corollary The self-adjoint extensions of a closed symmetric operator A is in one-to-one correspondence with the unitary extensions of its Cayley tran... | Extensions of symmetric operators | 0.777424 |
98,176 | The map A ↦ U A {\displaystyle A\mapsto U_{A}} is also called the Cayley transform of the symmetric operator A. Given UA, A can be recovered by A = − i ( U + 1 ) ( U − 1 ) − 1 , {\displaystyle A=-i(U+1)(U-1)^{-1},\,} defined on Dom(A) = Ran(U - 1). Now if U ~ {\displaystyle {\tilde {U}}} is an isometric extension of UA... | Extensions of symmetric operators | 0.777424 |
98,177 | In the present context, it is a convenient fact that every symmetric operator A is closable. That is, A has the smallest closed extension, called the closure of A. This can be shown by invoking the symmetric assumption and Riesz representation theorem. Since A and its closure have the same closed extensions, it can alw... | Extensions of symmetric operators | 0.777424 |
98,178 | Let H be a Hilbert space. A linear operator A acting on H with dense domain Dom(A) is symmetric if ⟨ A x , y ⟩ = ⟨ x , A y ⟩ {\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle } for all x, y in Dom(A).If Dom(A) = H, the Hellinger-Toeplitz theorem says that A is a bounded operator, in which case A is self-adjoint a... | Extensions of symmetric operators | 0.777424 |
98,179 | Multitenancy allows for cost savings over and above the basic economies of scale achievable from consolidating IT resources into a single operation. An application instance usually incurs a certain amount of memory and processing overhead which can be substantial when multiplied by many customers, especially if the cus... | Multitenancy | 0.777424 |
98,180 | In molecular biology, the ACT domain is a protein domain that is found in a variety of proteins involved in metabolism. ACT domains are linked to a wide range of metabolic enzymes that are regulated by amino acid concentration. The ACT domain is named after three of the proteins that contain it: aspartate kinase, chori... | ACT domain | 0.777424 |
98,181 | In the mathematical discipline of graph theory the Tutte–Berge formula is a characterization of the size of a maximum matching in a graph. It is a generalization of Tutte theorem on perfect matchings, and is named after W. T. Tutte (who proved Tutte's theorem) and Claude Berge (who proved its generalization). | Tutte–Berge formula | 0.777421 |
98,182 | Tutte's theorem characterizes the graphs with perfect matchings as being the ones for which deleting any subset U {\displaystyle U} of vertices creates at most | U | {\displaystyle |U|} odd components. (A subset U {\displaystyle U} that creates at least | U | {\displaystyle |U|} odd components can always be found in th... | Tutte–Berge formula | 0.777421 |
98,183 | The theorem states that the size of a maximum matching of a graph G = ( V , E ) {\displaystyle G=(V,E)} equals where odd ( H ) {\displaystyle \operatorname {odd} (H)} counts how many of the connected components of the graph H {\displaystyle H} have an odd number of vertices. Equivalently, the number of unmatched vert... | Tutte–Berge formula | 0.777421 |
98,184 | In mathematics, any vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on V , {\displaystyle V,} together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is def... | Topological dual space | 0.777421 |
98,185 | in Computer Science from Carnegie Mellon University in 2002. John was the program co-chair for the 2012 International Conference on Machine Learning (ICML), general chair for the 2016 ICML, and is the President of ICML from 2019–2021. == References == | John Langford (computer scientist) | 0.77742 |
98,186 | John Langford (born January 2, 1975) is a computer scientist working in machine learning and learning theory, a field that he says, "is shifting from an academic discipline to an industrial tool".He is well known for work on the Isomap embedding algorithm, CAPTCHA challenges, Cover Trees for nearest neighbor search, Co... | John Langford (computer scientist) | 0.77742 |
98,187 | Thus, some have appealed to theories of sociocultural evolution to assert that optimizing the ecology and the social harmony of closely knit groups is more desirable or necessary than the progression to "civilization." A 2002 poll of experts on Neoarctic and Neotropic indigenous peoples (reported in Harper's magazine) ... | Sociocultural evolution | 0.777418 |
98,188 | Current political theories of the new tribalists consciously mimic ecology and the life-ways of indigenous peoples, augmenting them with modern sciences. Ecoregional Democracy attempts to confine the "shifting groups", or tribes, within "more or less clear boundaries" that a society inherits from the surrounding ecolog... | Sociocultural evolution | 0.777418 |
98,189 | Moreover, they rejected directional change (i.e. orthogenetic, teleological or progressive change). Most archaeologists work within the framework of multilineal evolution. Other contemporary approaches to social change include neoevolutionism, sociobiology, dual inheritance theory, modernisation theory and postindustri... | Sociocultural evolution | 0.777418 |
98,190 | The most comprehensive attempt to develop a general theory of social evolution centering on the development of sociocultural systems, the work of Talcott Parsons (1902–1979), operated on a scale which included a theory of world history. Another attempt, on a less systematic scale, originated from the 1970s with the wor... | Sociocultural evolution | 0.777418 |
98,191 | Sociocultural evolution, sociocultural evolutionism or social evolution are theories of sociobiology and cultural evolution that describe how societies and culture change over time. Whereas sociocultural development traces processes that tend to increase the complexity of a society or culture, sociocultural evolution a... | Sociocultural evolution | 0.777418 |
98,192 | Alternatively, this fusion could create a higher-level superorganism employing a low-conflict division of labor in performing informational tasks...humans already embrace fusions of biology and technology. We spend most of our waking time communicating through digitally mediated channels, ...most transactions on the st... | Sociocultural evolution | 0.777418 |
98,193 | It increases exponentially, exhibits high-fidelity replication, evolves through differential fitness, is expressed through artificial intelligence (AI), and has facility for virtually limitless recombination. Like previous evolutionary transitions, the potential symbiosis between biological and digital information will... | Sociocultural evolution | 0.777418 |
98,194 | Other anthropologists, building on or responding to work by White and Steward, developed theories of cultural ecology and ecological anthropology. The most prominent examples are Peter Vayda and Roy Rappaport. By the late 1950s, students of Steward such as Eric Wolf and Sidney Mintz turned away from cultural ecology to... | Sociocultural evolution | 0.777418 |
98,195 | Another method of producing power to the grid is through virtual net metering (also called peer-to-peer (P2P) energy trading, wheeling and sometimes local energy trading). Peer-to-peer energy trading is a novel paradigm of power system operation, where sellers can generate their own energy in dwellings, offices and fac... | Solar Guerrilla | 0.777418 |
98,196 | Net purchase and sale is a different method of providing power to the electricity grid that does not offer the price symmetry of net metering, making this system a lot less profitable for home users of small renewable electricity systems. Under this arrangement, two uni-directional meters are installed—one records elec... | Solar Guerrilla | 0.777418 |
98,197 | The IMAGINARY photo gallery with algebraic surfaces and mathematical works of art from Germany, Austria, the United States, Canada, Belgium and France. The programme Morenaments which enables interactive painting of symmetrical patterns. The Penrose Puzzle Game, a logic puzzle game and the video game jReality with a 3D... | MiMa Mineralogy and Mathematics Museum | 0.777416 |
98,198 | To understand what happens in a ring laser cavity, it is helpful to discuss the physics of the laser process in a laser setup with continuous generation of light. As the laser excitation is started, the molecules inside the cavity emit photons, but since the molecules have a thermal velocity, the light inside the laser... | Sagnac effect | 0.777415 |
98,199 | It follows that the time difference for completing a cycle for the two beams isRemarkably, the time difference is independent of the refraction index n {\displaystyle n} and the velocity of light in the fiber. Imagine a screen for viewing fringes placed at the light source (alternatively, use a beamsplitter to send lig... | Sagnac effect | 0.777415 |
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