But I think Slave Play allows Kaneisha to get a little closer to subspace, if "subspace" here can be thought of as a moment of metaphorical purging, an out-of-colonized body experience.
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Within this space, in Furey's model, particles are mathematical "ideals": elements of a subspace that, when multiplied by other elements, stay in that subspace, allowing particles to stay particles even as they move, rotate, interact and transform.
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" Rachel, his partner, said, "People are much more open [at Subspace events].
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She was drifting in subspace, and probably able to take more pain now.
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But communication is key to proper BDSM and reaching subspace (for me, at least).
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He then took this subspace and combined it with the graph of a Diophantine equation.
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The ball on Saturday was one of the last large-format events Subspace will be holding.
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Imagine that multiplied by about 20, and that's what subspace feels like; floaty, calm, happy, and trusting.
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Brawl's Subspace Emissary mode was — although there are cutscenes and an overarching storyline — but it fills a similar role.
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You can stream the whole project below and probably use it to open doors to Subspace if you wanted to.
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The Easter Fetish Ball is an annual event held by Subspace, a fetish community in Toronto that was founded in 2006.
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But what about the psychological subspace felt by those experiencing non-physical play, such as humiliation, pet play, and other fetishes?
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Kaneisha is clearly nowhere near subspace at the end of Slave Play — there's nothing transcendent about this ending, nor should there be.
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Data: It is possible that the convergence of three tachyon pulses could have ruptured the subspace barrier and created an anti-time reaction.
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Twarock applied this concept by importing symmetry from a higher-dimensional space—in this case, from a lattice in six dimensions—into a three-dimensional subspace.
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I've experienced that subspace in the past through roleplay and pain, so I hoped the Muse could capture this unique mindset that can result from BDSM.
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And unlike, say, Subspace Emissary, which awkwardly grafted Smash gameplay onto a platforming and boss fight mode, Spirits is still, at its core, the same Smash Bros.
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And though there is an undeniable sexually charged atmosphere at Subspace events, it is not a place where you could find men aggressively prowling on uninterested women.
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There's even a word for the state of a submissive's mind and body during and after consensual kinky play: subspace, often described as a "floaty" or "flying" feeling.
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Once someone is in bondage, Yin says, they might enter something called "subspace" and might no longer feel comfortable negotiating what they do and don't want to try.
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The two-part debut ("The Vulcan Hello" and "Battle at the Binary Stars") gave fans the first new TV Trek since Star Trek: Enterprise ceased subspace transmission in 2005.
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Madison: The scene I just shot last week was a BDSM impact scene where I was the submissive, so I had an authentic experience where I went into subspace.
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Similarly, those who practice role-playing BDSM often describe entering an altered state of mind called "domspace" (for dominants) or "subspace" (for submissives) that seems similar, as Dunkley described it.
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After being in subspace, especially with hard impact and pain play, you go through a high followed by a low called "sub drop" over the next 24 hours or so.
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Being the one to show someone what subspace is all about—most people don't even know about the positive impact it can have on your mental health, which is quite sad.
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The mathematician Claude Chabauty discovered that inside a larger geometric space he constructed (using an expanded universe of numbers called the p-adic numbers), the rational numbers form their own symmetric subspace.
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The arrival of Hekaran scientists who warn that the warp drive was gradually destroying the fabric of subspace—the metaphor that was the whole point of the episode—came late in the hour.
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Subspace, which is named for the altered mental state a sub can go into during a scene with a dom, has had many themed events over the years: everything from military, to medical, to alien abduction.
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Galbraith's sentiments were echoed by literally every single person I spoke to at the event, including a couple I met named Stuart and Rachel, who have been going to Subspace events for as long as they've been together—seven years.
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He then started to build a setting with Subspace events (listed on Fetlife) where his guests could feel like they were in a safe space to express their fantasies, including the implementation of dungeon monitors to ensure attendees were abiding by rules.
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Her research lab at the university—called the "Science of BDSM" and founded by Northern Illinois University psychology professor Brad Sagarin—is also studying the relationship between BDSM and altered states of consciousness like "subspace," or the meditative flow that submissives can achieve when they fully relinquish control.
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But subspace does exist: Dr. Brad Sagarin, founder of the Science of BDSM research team and a professor of social and evolutionary psychology at Northern Illinois University, has compared it to runner's high, the sense of euphoria and increased tolerance for pain that some joggers feel after a long run.
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The main room at the Subspace Easter Fetish Ball, which was held at the Great Hall in Toronto, didn't seem that different at first from some of the raves I've attended in my life, except for the increased amount of skin attendees were showing and an increased amount of leather being donned.
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When a person who's acting the submissive role in a domination session is able to push themselves past the limits of what they think they can take emotionally and physically, they will sometimes enter a free-floating blissful state known as "subspace" where their body responds to the intensity of what's happening by flooding them with chemicals, and they basically leave their body and lose their ability to feel pain of any kind.
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This carries evaporation energy from the warmer first subspace to the colder second subspace. At the second subspace the energy flows by conduction through the tube walls to the colder next space.
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In Star Wars, a subspace transceiver, also known as a subspace comm, subspace radio, and hypertransceiver, was a standard device used for instantaneous, faster-than- light communications between nearby systems. Similar to its shorter-ranged cousin, the comlink, the subspace transceiver relied on energy to broadcast signals. Starships carried these units to broadcast distress signals and other important messages. They used subspace as the communications medium.
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This subspace is known as the Krylov subspace. It can be computed by Arnoldi iteration or Lanczos iteration.
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"Iterative methods by space decomposition and subspace correction." SIAM review 34, no. 4 (1992): 581-613. BPX preconditioner is a parallel subspace correction method where as the classic V-cycle is a successive subspace correction method.
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In control theory, a controlled invariant subspace of the state space representation of some system is a subspace such that, if the state of the system is initially in the subspace, it is possible to control the system so that the state is in the subspace at all times. This concept was introduced by Giuseppe Basile and Giovanni Marro .
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In this case, a linear subspace contains the zero vector, while an affine subspace does not necessarily contain it. Subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set S of vectors is called its span, and it is the smallest subspace of V containing the set S. Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of S. A linear subspace of dimension 1 is a vector line. A linear subspace of dimension 2 is a vector plane.
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Subspace weapons are a class of directed energy weapons that directly affect subspace. The weapons can produce actual tears in subspace, and are extremely unpredictable. These weapons were banned under the second Khitomer Accord. The Son'a equipped their vessels with these types of weapons.
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The subspace of null sequences c0 consists of all sequences whose limit is zero. This is a closed subspace of c, and so again a Banach space. The subspace of eventually zero sequences c00 consists of all sequences which have only finitely many nonzero elements. This is not a closed subspace and therefore is not a Banach space with respect to the infinity norm.
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In the Adventure mode titled "The Subspace Emissary", Mario and Kirby face each other on a stadium located in the Smash Bros. world. In this world, when a fighter is defeated, they become a trophy which can be revived by touching the base. Suddenly, the Battleship Halberd appears, releasing a stream of black, purple-clouded Shadow Bugs that form the soldiers of the Subspace Army. The Ancient Minister, the cloaked, mysterious Subspace General, arrives and detonates a Subspace Bomb, which can only be detonated by the sacrifice of two R.O.B units, and transports the stadium into Subspace, an alternate dimension where the Subspace Army resides.
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However, if is a TVS then every vector subspace of has the extension property if and only if every vector subspace of has the separation property.
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A subspace V ⊂ Rn is a controlled invariant subspace if and only if AV ⊂ V + Im B. If V is a controlled invariant subspace, then there exists a matrix K such that the input u(t) = Kx(t) keeps the state within V; this is a simple feedback control .
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What does it all mean intuitively? It means the Hamiltonian and constraint flows all commute with each other on the constrained subspace; or alternatively, that if we start on a point on the constrained subspace, then the Hamiltonian and constraint flows all bring the point to another point on the constrained subspace. Since we wish to restrict ourselves to the constrained subspace only, this suggests that the Hamiltonian, or any other physical observable, should only be defined on that subspace. Equivalently, we can look at the equivalence class of smooth functions over the symplectic manifold, which agree on the constrained subspace (the quotient algebra by the ideal generated by the 's, in other words).
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Original Image Blurred Image: obtained after the convolution of original image with blur kernel. Original image lies in fixed subspace of wavelet transform and blur lies in random subspace.
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Given an affine subspace A in a linear space L, a straight line in A may be defined as the intersection of A with a linear subspace of L that intersects A: in other words, with a plane through the origin that is not parallel to A. More generally, a affine subspace of A is the intersection of A with a linear subspace of L that intersects A. Every point of the affine subspace A is the intersection of A with a linear subspace of L. However, some subspaces of L are parallel to A; in some sense, they intersect A at infinity. The set of all linear subspaces of a linear space is, by definition, a projective space. And the affine subspace A is embedded into the projective space as a proper subset. However, the projective space itself is homogeneous.
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Cover of the first edition, published by Berkley Books. Subspace Encounter is a 1983 science fiction novel by American writer E. E. Smith, a posthumously published sequel to his Subspace Explorers.
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The combined technique is implemented in open-source libraries OctaveOctave function subspace and SciPySciPy linear-algebra function subspace_angles and contributed MATLAB FileExchange function subspace and MATLAB FileExchange function subspacea to MATLAB.
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In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace.Borsuk (1931). The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a subspace.
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Suppose that is a vector space and that is its the algebraic dual. Then every -bounded subset of is contained in a finite dimensional vector subspace and every vector subspace of is -closed.
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This therefore defines the volume form in the linear subspace.
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In mathematics, an invariant subspace of a linear mapping T : V -> V from some vector space V to itself, is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.
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Any subspace of a countably generated space is again countably generated.
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In the projective space, each projective subspace of dimension k intersects the ideal hyperplane in a projective subspace "at infinity" whose dimension is . A pair of non-parallel affine hyperplanes intersect at an affine subspace of dimension , but a parallel pair of affine hyperplanes intersect at a projective subspace of the ideal hyperplane (the intersection lies on the ideal hyperplane). Thus, parallel hyperplanes, which did not meet in the affine space, intersect in the projective completion due to the addition of the hyperplane at infinity.
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Super Smash Bros. Brawl features a single-player mode known as The Subspace Emissary. This mode features unique character storylines along with numerous side-scrolling levels and multiple bosses to fight, as well as CG cut scenes explaining the storyline. The Subspace Emissary features a new group of antagonists called the Subspace Army, who are led by the Ancient Minister.
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The catch is, the Hamiltonian flows on the constrained subspace depend on the gradient of the Hamiltonian there, not its value. But there's an easy way out of this. Look at the orbits of the constrained subspace under the action of the symplectic flows generated by the 's. This gives a local foliation of the subspace because it satisfies integrability conditions (Frobenius theorem).
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In functional analysis, one of the most prominent problems was the invariant subspace problem, which required the evaluation of the truth of the following proposition: :Given a complex Banach space H of dimension > 1 and a bounded linear operator T : H → H, then H has a non-trivial closed T-invariant subspace, i.e. there exists a closed linear subspace W of H which is different from {0} and H such that T(W) ⊆ W. For Banach spaces, the first example of an operator without an invariant subspace was constructed by Enflo. (For Hilbert spaces, the invariant subspace problem remains open.) Enflo proposed a solution to the invariant subspace problem in 1975, publishing an outline in 1976. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987Beauzamy 1988; Yadav.
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It turns out if we start with two different points on a same orbit on the constrained subspace and evolve both of them under two different Hamiltonians, respectively, which agree on the constrained subspace, then the time evolution of both points under their respective Hamiltonian flows will always lie in the same orbit at equal times. It also turns out if we have two smooth functions A1 and B1, which are constant over orbits at least on the constrained subspace (i.e. physical observables) (i.e. {A1,f}={B1,f}=0 over the constrained subspace)and another two A2 and B2, which are also constant over orbits such that A1 and B1 agrees with A2 and B2 respectively over the restrained subspace, then their Poisson brackets {A1, B1} and {A2, B2} are also constant over orbits and agree over the constrained subspace.
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The group of units R× of R is a topological group when endowed with the topology coming from the embedding of R× into the product R × R as (x,x−1). However, if the unit group is endowed with the subspace topology as a subspace of R, it may not be a topological group, because inversion on R× need not be continuous with respect to the subspace topology. An example of this situation is the adele ring of a global field; its unit group, called the idele group, is not a topological group in the subspace topology. If inversion on R× is continuous in the subspace topology of R then these two topologies on R× are the same.
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Many special cases of this invariant subspace problem have already been proven.
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Images of a convex, Lambertian surface under varying illuminations span a low- dimensional subspace. This is one of the reasons for effectiveness of low- dimensional models for imagery data. In particular, it is easy to approximate images of a human's face by a low-dimensional subspace. To be able to correctly retrieve this subspace is crucial in many applications such as face recognition and alignment.
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The space is a minimal Banach space.see , p. 54. This means that every infinite-dimensional Banach subspace of contains a further subspace isomorphic to . Prior to the construction of , the only known examples of minimal spaces were ℓ p and 0.
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When the dimension of H is infinite, then S1(H) (the trace class) is not reflexive, because it contains a subspace isomorphic to ℓ1, and S∞(H) = L(H) (the bounded linear operators on H) is not reflexive, because it contains a subspace isomorphic to ℓ∞. In both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of H.
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Continuum Interface Continuum, first released in 2001, was developed as a clone of the SubSpace client, but now contains new original features exclusive to the client over the original. Continuum is the official client of the SubSpace Central Billing Server. It was developed primarily because of the original SubSpace client's failure to prevent hacking. As such, it has been adopted by several zones as a requirement in order to play.
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For non-positive definite matrices, this method may suffer from stagnation in convergence as the restarted subspace is often close to the earlier subspace. The shortcomings of GMRES and restarted GMRES are addressed by the recycling of Krylov subspace in the GCRO type methods such as GCROT and GCRODR. Recycling of Krylov subspaces in GMRES can also speed up convergence when sequences of linear systems need to be solved.
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Every intersection of projective subspaces is a projective subspace. It follows that for every subset of a projective space, there is a smallest projective subspace containing , the intersection of all projective subspaces containing . This projective subspace is called the projective span of , and is a spanning set for it. A set of points is projectively independent if its span is not the span of any proper subset of .
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The subspace of supersingular K3 surfaces with Artin invariant e has dimension e − 1.
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A completely normal space' or a ' is a topological space X such that every subspace of X with subspace topology is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods. Also, X is completely normal if and only if every open subset of X is normal with the subspace topology. A completely T4 space, or T5 space is a completely normal T1 space topological space X, which implies that X is Hausdorff; equivalently, every subspace of X must be a T4 space.
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It follows that oriented curves can be studied in a Lie invariant way via their contact lifts, which may be characterized, generically as Legendrian curves in Z3. More precisely, the tangent space to Z3 at the point corresponding to a null 2-dimensional subspace π of R3,2 is the subspace of those linear maps (A mod π):π → R3,2/π with : A(x) · y + x · A(y) = 0 and the contact distribution is the subspace Hom(π,π⊥/π) of this tangent space in the space Hom(π,R3,2/π) of linear maps. It follows that an immersed Legendrian curve λ in Z3 has a preferred Lie cycle associated to each point on the curve: the derivative of the immersion at t is a 1-dimensional subspace of Hom(π,π⊥/π) where π=λ(t); the kernel of any nonzero element of this subspace is a well defined 1-dimensional subspace of π, i.e., a point on the Lie quadric.
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In other words, all complex vector spaces with complex conjugation are the complexification of a real vector space. For example, when with the standard complex conjugation :\chi(z_1,\ldots,z_n) = (\bar z_1,\ldots,\bar z_n) the invariant subspace is just the real subspace .
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See, for example, . Enflo's works inspired a similar construction of an operator without an invariant subspace for example by Beauzamy, who acknowledged Enflo's ideas. In the 1990s, Enflo developed a "constructive" approach to the invariant subspace problem on Hilbert spaces.Page 401 in .
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Spectral based algorithmic solutions can be further classified into beamforming techniques and subspace- based techniques.
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He has made various contributions to SubSpace under the username PriitK, particularly the Continuum client.
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Any open subset of an n-manifold is an n-manifold with the subspace topology.
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It has been asserted that the relaxed solution of -means clustering, specified by the cluster indicators, is given by the principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace. However, that PCA is a useful relaxation of -means clustering was not a new result, and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.
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A video or an image sequence represented as a third-order tensor of column x row x time for multilinear subspace learning. Multilinear subspace learning is an approach to dimensionality reduction.M. A. O. Vasilescu, D. Terzopoulos (2003) "Multilinear Subspace Analysis of Image Ensembles", "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’03), Madison, WI, June, 2003"M. A. O. Vasilescu, D. Terzopoulos (2002) "Multilinear Analysis of Image Ensembles: TensorFaces", Proc.
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A retraction of a metric space X is a function ƒ mapping X to a subspace of itself, such that # for all x, ƒ(ƒ(x)) = ƒ(x); that is, ƒ is the identity function on its image (i. e. it is idempotent), and # for all x and y, d(ƒ(x), ƒ(y)) ≤ d(x, y); that is, ƒ is nonexpansive. A retract of a space X is a subspace of X that is an image of a retraction. A metric space X is said to be injective if, whenever X is isometric to a subspace Z of a space Y, that subspace Z is a retract of Y.
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A subspace Y of a geodesic metric space X is said to be quasiconvex if there is a constant C such that any geodesic in x between two points of Y stays within distance C of Y. :A quasi-convex subspace of an hyperbolic space is hyperbolic.
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The operations intersection and sum make the set of all subspaces a bounded modular lattice, where the {0} subspace, the least element, is an identity element of the sum operation, and the identical subspace V, the greatest element, is an identity element of the intersection operation.
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Brawl, the game's story mode, The Subspace Emissary, suggests Mr. Game & Watch is made of a primordial substance that can take on any number of forms. Mr. Game & Watch was harvested for this reason to create the Subspace Army.Shadow Bugs - Trophy Description. Super Smash Bros. Brawl. Nintendo.
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S. Ouyang and Y. Hua, "Bi-iterative least square method for subspace tracking," IEEE Transactions on Signal Processing, pp. 2948-2996, Vol. 53, No. 8, August 2005. Y. Hua and T. Chen, "On convergence of the NIC algorithm for subspace computation," IEEE Transactions on Signal Processing, pp. 1112-1115, Vol. 52, No. 4, April 2004. Y. Hua, “Asymptotical orthonormalization of subspace matrices without square root,” IEEE Signal Processing Magazine, Vol. 21, No. 4, pp. 56-61, July 2004.
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Screenshot of a generic zone portraying the basic game play system and graphical interface, which bear resemblance to those of the game SubSpace. Cosmic Rift is provided as part of the Station Pass package offered by Sony Online Entertainment, which also includes Infantry and Tanarus. Its gameplay is comparable to Virgin Interactive Entertainment's Subspace, another two-dimensional space shooter written by Jeff Petersen prior to joining Sony's team. Today, Subspace remains available in a freeware flavor known as Continuum.
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SUBCLU uses a monotonicity criteria: if a cluster is found in a subspace S, then each subspace T \subseteq S also contains a cluster. However, a cluster C \subseteq DB in subspace S is not necessarily a cluster in T \subseteq S, since clusters are required to be maximal, and more objects might be contained in the cluster in T that contains C. However, a density-connected set in a subspace S is also a density-connected set in T \subseteq S. This downward-closure property is utilized by SUBCLU in a way similar to the Apriori algorithm: first, all 1-dimensional subspaces are clustered. All clusters in a higher-dimensional subspace will be subsets of the clusters detected in this first clustering. SUBCLU hence recursively produces k+1-dimensional candidate subspaces by combining k-dimensional subspaces with clusters sharing k-1 attributes.
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The reader can find a good review of this topic in the article about the typical subspace.
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His results in this field include Milman's reverse Brunn–Minkowski inequality and the quotient of subspace theorem.
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A concrete example of an operator without an invariant subspace was produced in 1985 by Charles Read.
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The set- theoretical inclusion binary relation specifies a partial order on the set of all subspaces (of any dimension). A subspace cannot lie in any subspace of lesser dimension. If dim U = k, a finite number, and U ⊂ W, then dim W = k if and only if U = W.
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A straight line in the projective space corresponds to a linear subspace of the (n+1)-dimensional linear space. More generally, a projective subspace of the projective space corresponds to a linear subspace of the (n+1)-dimensional linear space, and is isomorphic to the projective space. Defined this way, affine and projective spaces are of algebraic nature; they can be real, complex, and more generally, over any field. Every real or complex affine or projective space is also a topological space.
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1.8 Another characterization: q has Arf invariant 0 if and only if the underlying 2k-dimensional vector space over the field F2 has a k-dimensional subspace on which q is identically 0 – that is, a totally isotropic subspace of half the dimension. In other words, a nonsingular quadratic form of dimension 2k has Arf invariant 0 if and only if its isotropy index is k (this is the maximum dimension of a totally isotropic subspace of a nonsingular form).
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This is the only central configuration for these masses that does not lie in a lower- dimensional subspace.
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Both L1-PCA and standard PCA seek a collection of orthogonal directions (principal components) that define a subspace wherein data representation is maximized according to the selected criterion. Standard PCA quantifies data representation as the aggregate of the L2-norm of the data point projections into the subspace, or equivalently the aggregate Euclidean distance of the original points from their subspace-projected representations. L1-PCA uses instead the aggregate of the L1-norm of the data point projections into the subspace. In PCA and L1-PCA, the number of principal components (PCs) is lower than the rank of the analyzed matrix, which coincides with the dimensionality of the space defined by the original data points.
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The ovals are projection operators which "tie" together two spin 1/2s into a single spin 1, projecting out the spin 0 or singlet subspace and keeping only the spin 1 or triplet subspace. The symbols "+", "0" and "−" label the standard spin 1 basis states (eigenstates of the S^z operator).
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Every topological space is an open dense subspace of a compact space having at most one point more than , by the Alexandroff one- point compactification. By the same construction, every locally compact Hausdorff space is an open dense subspace of a compact Hausdorff space having at most one point more than .
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Son'a vessels carried and used isolytic burst weapons, a type of subspace weapon. They were seen using this weapon against the Enterprise-E in Star Trek: Insurrection. The Enterprise was only able to escape the weapon's effect by ejecting its warp core and detonating it to seal a subspace rift.
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Subspace Explorers is a science fiction novel by American writer E. E. "Doc" Smith. It was first published in 1965 by Canaveral Press in an edition of 1,460 copies. The novel is an expansion of Smith's story "Subspace Survivors" which first appeared in the July 1960 issue of the magazine Astounding.
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Any element of the group O(3,2) of orthogonal transformations of R3,2 maps any one-dimensional subspace of null vectors in R3,2 to another such subspace. Hence the group O(3,2) acts on the Lie quadric. These transformations of cycles are called "Lie transformations". They preserve the incidence relation between cycles.
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For example, two half-moon shaped clusters intertwined in space do not separate well when projected onto PCA subspace. k-means should not be expected to do well on this data. It is straightforward to produce counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.
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I, June 2005, pp. 526–532. Multilinear Subspace Learning employ different types of data tensor analysis tools for dimensionality reduction. Multilinear Subspace learning can be applied to observations whose measurements were vectorized and organized into a data tensor, or whose measurements are treated as a matrix and concatenated into a tensor.
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After pruning irrelevant candidates, DBSCAN is applied to the candidate subspace to find out if it still contains clusters. If it does, the candidate subspace is used for the next combination of subspaces. In order to improve the runtime of DBSCAN, only the points known to belong to clusters in one k-dimensional subspace (which is chosen to contain as little clusters as possible) are considered. Due to the downward-closure property, other point cannot be part of a k+1-dimensional cluster anyway.
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Vidal has been a prominent scientist in the fields of machine learning, computer vision, medical image computing, robotics and control theory since the 2000s. In machine learning, Vidal has made many contributions to subspace clustering, including his work on Generalized Principal Component Analysis (GPCA), Sparse Subspace Clustering (SSC) and Low Rank Subspace Clustering (LRSC). Much of his work in machine learning is summarized in his book Generalized Principal Component Analysis. Currently, he is working on understanding the mathematical foundations of deep learning, specifically conditions for global optimality.
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A line passing through the origin (blue, thick) in is a linear subspace. It is the intersection of two planes (green and yellow). A nonempty subset W of a vector space V that is closed under addition and scalar multiplication (and therefore contains the 0-vector of V) is called a linear subspace of V, or simply a subspace of V, when the ambient space is unambiguously a vector space.This is typically the case when a vector space is also considered as an affine space.
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A linear subspace that contains all elements but one of a basis of the ambient space is a vector hyperplane. In a vector space of finite dimension , a vector hyperplane is thus a subspace of dimension . The counterpart to subspaces are quotient vector spaces. Given any subspace , the quotient space V/W ("V modulo W") is defined as follows: as a set, it consists of where v is an arbitrary vector in V. The sum of two such elements and is and scalar multiplication is given by .
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In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.
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Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets. A space such that every subspace of it is a paracompact space is called hereditarily paracompact.
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The glitch is removed when you play the Song of Time. Many other glitches may also include background music being played at the time it was not intended to play, such as the Western Super Mario Bros. 2 (based on the Japan-only game Doki Doki Panic) has a glitch where the Subspace music (the Overworld background music from the original Super Mario Bros.) can be played outside of Subspace after the player becomes invincible and enters Subspace and leaves before invincibility drains away (this was fixed for Super Mario All-Stars and Super Mario Advance so Subspace music can only be played in Subspace). One of the best known examples for glitching in an online game is Grand Theft Auto IV, where people can glitch into rooms they're not supposed to, or get under the map (for example by glitching a helicopter to spawn under the map and fly into beta rooms).
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In the latter case, SPIKE is used as a preconditioner for iterative schemes like Krylov subspace methods and iterative refinement.
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More generally, on each subdomain, the coarse space needs to only contain the nullspace of the problem as a subspace.
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In most Star Trek series, subspace communications are a means to establish nearly instantaneous contact with people and places that are light years away. The physics of Star Trek describe infinite speed (expressed as warp factor 10) as an impossibility; as such, even subspace communications which travel at speeds over Warp 9.9 may take hours or weeks to reach certain destinations. In the Star Trek universe subspace signals do not degrade with the square of the distance as do other methods of communication utilizing conventional bands of the electromagnetic spectrum (i.e. radio waves), so signals sent from a great distance can be expected to reach their destination at a predictable time and with little relative degradation (barring any random subspace interference or spatial anomalies).
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PROCLUS uses a similar approach with a k-medoid clustering. Initial medoids are guessed, and for each medoid the subspace spanned by attributes with low variance is determined. Points are assigned to the medoid closest, considering only the subspace of that medoid in determining the distance. The algorithm then proceeds as the regular PAM algorithm.
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The space of convergent sequences c is a sequence space. This consists of all x ∈ KN such that limn→∞ xn exists. Since every convergent sequence is bounded, c is a linear subspace of ℓ∞. It is, moreover, a closed subspace with respect to the infinity norm, and so a Banach space in its own right.
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The Birkhoff polytope lies within an dimensional affine subspace of the n2-dimensional space of all matrices: this subspace is determined by the linear equality constraints that the sum of each row and of each column be one. Within this subspace, it is defined by n2 linear inequalities, one for each coordinate of the matrix, specifying that the coordinate be non-negative. Therefore, for n\ge 3, it has exactly n2 facets. For n = 2, there are two facets, given by a11 = a22 = 0, and a12 = a21 = 0.
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They therefore either coincide or intersect in a 2-dimensional subspace. In the latter case, the 2-dimensional subspace can either have signature (2,0), (1,0), (1,1), in which case the corresponding two circles in S intersect in zero, one or two points respectively. Hence they have first order contact if and only if the 2-dimensional subspace is degenerate (signature (1,0)), which holds if and only if the span of v and w is degenerate. By Lagrange's identity, this holds if and only if (v · w)2 = (v · v)(w · w) = 1, i.e.
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A visualization of a warp field--the ship rests in a bubble of normal space. In the Star Trek fictional universe, subspace is a feature of space-time that facilitates faster-than-light transit, in the form of interstellar travel or the transmission of information. Faster-than-light warp drive travel via subspace works similarly to the Alcubierre Drive, but obeys different laws of physics. Subspace has also been adopted and used in other fictional settings, such as the Stargate franchise, The Hitchhiker's Guide to the Galaxy series, and Descent: Freespace.
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Two subspaces and of the same dimension in a Euclidean space are parallel if they have the same direction. Equivalently, they are parallel, if there is a translation vector that maps one to the other: :T= S+v. Given a point and a subspace , there exists exactly one subspace that contains and is parallel to , which is P + \overrightarrow S. In the case where is a line (subspace of dimension one), this property is Playfair's axiom. It follows that in a Euclidean plane, two lines either meet in one point or are parallel.
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By giving a collection of subspaces of some vector space a topological structure, it is possible to talk about a continuous choice of subspace or open and closed collections of subspaces; by giving them the structure of a differential manifold one can talk about smooth choices of subspace. A natural example comes from tangent bundles of smooth manifolds embedded in Euclidean space. Suppose we have a manifold of dimension embedded in . At each point in , the tangent space to can be considered as a subspace of the tangent space of , which is just .
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SubSpace utilizes a client–server architecture. Initially, both the client and server were provided by VIE. The client executable was titled SubSpace while the server was called SubGame. A new client, titled Continuum, was created by reverse engineering without access to the original source code by the players PriitK (one of the creators of Kazaa) and Mr Ekted.
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In mathematics, in abstract algebra, a multivariate polynomial over a field such that the Laplacian of is zero is termed a harmonic polynomial. The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In fact, they form a graded subspace. For the real field, the harmonic polynomials are important in mathematical physics.
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The number v (resp. p) is the maximal dimension of a vector subspace on which the scalar product g is positive-definite (resp. negative-definite), and r is the dimension of the radical of the scalar product g or the null subspace of symmetric matrix of the scalar product. Thus a nondegenerate scalar product has signature , with .
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SIAM Int. Conf. on Data Mining (SDM'04), pp. 246–257, 2004. Ideas from density-based clustering methods (in particular the DBSCAN/OPTICS family of algorithms) have been adapted to subspace clustering (HiSC, hierarchical subspace clustering and DiSH) and correlation clustering (HiCO, hierarchical correlation clustering, 4C using "correlation connectivity" and ERiC exploring hierarchical density-based correlation clusters).
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A linear functional is non-trivial if and only if it is surjective (i.e. its range is all of ).This follows since just as the image of a vector subspace under a linear transformation is a vector subspace, so is the image of under . However, the only vector subspaces (that is, -subspaces) of are } and itself.
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The original server software, heavily modified, is still the most common, although an open source alternative, A Small Subspace Server, is now available.
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His main research interests are Krylov subspace methods, non-normal operators and spectral perturbation theory, Toeplitz matrices, random matrices, and damped wave operators.
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Scott and Ramona affirm their desire to face the challenges of a relationship together, and walk hand-in-hand into a subspace door.
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The vertical space is therefore a vector subspace of TeE. A horizontal space HeE is then a choice of a subspace of TeE such that TeE is the direct sum of VeE and HeE. The disjoint union of the vertical spaces VeE for each e in E is the subbundle VE of TE: this is the vertical bundle of E. Likewise, a horizontal bundle is the disjoint union of the horizontal subspaces HeE. The use of the words "the" and "a" in this definition is crucial: the vertical subspace is unique, it is determined solely by the fibration.
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Commonalities among the different chemical groups are studied as they are often reflective of a particular chemical subspace. Additional chemistry work may be needed to further optimize the chemical library in the active portion of the subspace. When it is needed, more synthesis is completed to extend out the chemical library in that particular subspace by generating more compounds that are very similar to the original hits. This new selection of compounds within this narrow range are further screened and then taken on to more sophisticated models for further validation in the Drug Discovery Hit to Lead process.
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From the definition of vector spaces, it follows that subspaces are nonempty, and are closed under sums and under scalar multiples. Equivalently, subspaces can be characterized by the property of being closed under linear combinations. That is, a nonempty set W is a subspace if and only if every linear combination of finitely many elements of W also belongs to W. The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time. In a topological vector space X, a subspace W need not be topologically closed, but a finite-dimensional subspace is always closed.
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If is given the discrete topology, and if is given the product topology, and is viewed as a subspace of and is given the subspace topology, then acts densely on if and only if is dense set in with this topology.It turns out this topology is the same as the compact-open topology in this case. Herstein, p. 41 uses this description.
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Every pure spinor is annihilated by a half-dimensional subspace of C2n. Conversely given a half-dimensional subspace it is possible to determine the pure spinor that it annihilates up to multiplication by a complex number. Pure spinors defined up to complex multiplication are called projective pure spinors. The space of projective pure spinors is the homogeneous space :SO(2n)/U(n).
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Let be a TVS. Say that a vector subspace of has the extension property if any continuous linear functional on can be extended to a continuous linear functional on . Say that has the Hahn-Banach extension property (HBEP) if every vector subspace of has the extension property. The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP.
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In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W⊥ of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement. It is a subspace of V.
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A topological space X is said to be locally regular if and only if each point, x, of X has a neighbourhood that is regular under the subspace topology. Equivalently, a space X is locally regular if and only if the collection of all open sets that are regular under the subspace topology forms a base for the topology on X.
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In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the metric envelope, or tight span, which are basic (injective) objects of the category of metric spaces. Following , a notion of a metric space Y aimed at its subspace X is defined.
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The pressure in a space cannot be in equilibrium with the temperatures of the walls of both subspaces. It has an intermediate pressure. Then the pressure is too low or the temperature too high in the first subspace, and the water evaporates. In the second subspace, the pressure is too high or the temperature too low, and the vapor condenses.
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Fig. 5: Relations between mathematical spaces: affine, projective etc It is convenient to introduce affine and projective spaces by means of linear spaces, as follows. A linear subspace of a linear space, being itself a linear space, is not homogeneous; it contains a special point, the origin. Shifting it by a vector external to it, one obtains a affine subspace. It is homogeneous.
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In fact this generalizes to Rn whereby deleting a -dimensional subspace from Rn leaves a non-simply connected space). 4\. If A is a strong deformation retract of a topological space X, then the inclusion map from A to X induces an isomorphism between fundamental groups (so the fundamental group of X can be described using only loops in the subspace A).
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The extreme points of a compact convex form a Baire space (with the subspace topology) but this set may fail to be closed in .
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The Poisson random measure generalizes to the Poisson-type random measures, where members of the PT family are invariant under restriction to a subspace.
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If distinguishes points of and if denotes the range of the injection then is a vector subspace of the algebraic dual space of and the pairing becomes canonically identified with the canonical pairing (where is the natural evaluation map). In particular, in this situation we can assume without loss of generality that is a vector subspace of 's algebraic dual and is the evaluation map. :Convention: Often, whenever is injective (especially when forms a dual pair) then we will use the common practice of assuming without loss of generality that is a vector subspace of the algebraic dual space of , that is the natural evaluation map, and we may also denote by . In a completely analogous manner, if distinguishes points of then it is possible for to be identified as a vector subspace of 's algebraic dual space.
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The space C of continuous functions on E is a subspace of D. The Skorokhod topology relativized to C coincides with the uniform topology there.
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The last step will try to prune the solution subspace gained in the second step. We will introduce each step in details in the following.
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The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions. By virtue of their order, the rationals carry an order topology. The rational numbers, as a subspace of the real numbers, also carry a subspace topology.
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Several bases can span the same space. Of course also dependent vectors span a space, but the linear combinations of the latter can give only rise to the set of vectors lying on a straight line. As we are searching for a \,kdimensional subspace, we are interested in finding \,k linearly independent vectors that span the \,kdimensional subspace we want to project our data on.
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Newton–Krylov methods are numerical methods for solving non-linear problems using Krylov subspace linear solvers. The Newton method, when generalised to systems of multiple variables, includes the inverse of a Jacobian matrix in the iteration formula. Calculation of the inverse of the Jacobian matrix is bypassed by employing a Krylov subspace method, e.g. the Generalized minimal residual method (GMRES), to solve the iteration formula.
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Suppose that is a vector space and X# is the algebraic dual space of (i.e. the vector space of all linear functionals on ). If is endowed with the weak topology induced by X# then the continuous dual space of is , every bounded subset of is contained in a finite-dimensional vector subspace of , every vector subspace of is closed and has a topological complement.
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A topological space X is said to be locally normal if and only if each point, x, of X has a neighbourhood that is normal under the subspace topology. Note that not every neighbourhood of x has to be normal, but at least one neighbourhood of x has to be normal (under the subspace topology). Note however, that if a space were called locally normal if and only if each point of the space belonged to a subset of the space that was normal under the subspace topology, then every topological space would be locally normal. This is because, the singleton {x} is vacuously normal and contains x.
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They further introduced linear subspace learning and imputation schemes for streaming tensors; online categorical subspace learning; and kernel-based nonlinear subspace trackers on a budget. Graphs underpin the structure and operation of networks everywhere: from the Internet to the power grid, financial markets, social media, gene regulation, and brain functionality. Whether graph edges capture physical interconnections or interdependencies among nodes or variables, learning a graph and carrying out inference of processes on a graph, are two tasks of paramount importance in data science, network science, and applications. Giannakis and collaborators established conditions to first identify topologies of directed graphs using sparse linear or nonlinear, and static or dynamic structural equation models.
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Examples for such clustering algorithms are CLIQUE and SUBCLU.Karin Kailing, Hans-Peter Kriegel and Peer Kröger. Density-Connected Subspace Clustering for High-Dimensional Data. In: Proc.
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In functional analysis and related areas of mathematics, the beta-dual or -dual is a certain linear subspace of the algebraic dual of a sequence space.
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In addition, multilinear principal component analysis in multilinear subspace learning involves the same mathematical operations as Tucker decomposition, being used in a different context of dimensionality reduction.
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SIAM Int. Conf. on Data Mining (SDM'04), pp. 246-257, 2004. It is a subspace clustering algorithm that builds on the density-based clustering algorithm DBSCAN.
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Every real, separable Banach space is isometrically isomorphic to a closed subspace of , the space of all continuous functions from the unit interval into the real line.
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In mathematics, the subspace theorem says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by .
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The formal definition does not use isometries, but almost isometries. A Banach space Y is finitely representableJames, Robert C. (1972), "Super-reflexive Banach spaces", Can. J. Math. 24:896-904. in a Banach space X if for every finite-dimensional subspace Y0 of Y and every , there is a subspace X0 of X such that the multiplicative Banach-Mazur distance between X0 and Y0 satisfies :d(X_0, Y_0) < 1 + \varepsilon.
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The result is a universal space of types in which, subject to specified consistency conditions, each type corresponds to the infinite hierarchy of his probabilistic beliefs about others' probabilistic beliefs. They also showed that any subspace can be approximated arbitrarily closely by a finite subspace. Another popular example of the usage of the construction is the Prisoners and hats puzzle. And so is Robert Aumann's construction of common knowledge.
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Other approaches use global dimension minimization to reveal the clusters corresponding to the underlying subspace. These approaches use only two frames for motion segmentation even if multiple frames are available as they cannot use multi frame information. Multiview-based approaches utilize the trajectory of feature points unlike two-view based approaches. A number of approaches have been provided which include Principle Angles Configuration (PAC) and Sparse Subspace Clustering (SSC) methods.
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Flying through a nebula involves impaired vision, and occasional disruptions to flight electronics. Nebulae have become known as an eerie and suspenseful arena of play. Journeys between star systems are achieved by "jumping" through jump nodes and traveling through subspace, while shorter intra-system distances are done by "hopping" into subspace at any time. All ships in a mission either "jump" or "hop" to make their entries and exits.
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A combination of HOSVD and SVD also has been applied for real-time event detection from complex data streams (multivariate data with space and time dimensions) in disease surveillance. It is also used in tensor product model transformation-based controller design. In multilinear subspace learning,Haiping Lu, K.N. Plataniotis and A.N. Venetsanopoulos, "A Survey of Multilinear Subspace Learning for Tensor Data", Pattern Recognition, Vol. 44, No. 7, pp.
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Kain's remains were destroyed by Kiyone with the Dimensional Cannon which destroyed the subspace that Kain was in after Achika barely broke away from Kain during their escape.
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If M is a solid vector subspace of a vector lattice X, then the order topology of X/M is the quotient of the order topology on X.
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In the case of a system of linear equations, the two main classes of iterative methods are the stationary iterative methods, and the more general Krylov subspace methods.
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These subspaces must be generated by a single nonzero vector v and all its images by repeated application of the linear operator associated to the matrix; such subspaces are called cyclic subspaces (by analogy with cyclic subgroups) and they are clearly stable under the linear operator. A basis of such a subspace is obtained by taking v and its successive images as long as they are linearly independent. The matrix of the linear operator with respect to such a basis is the companion matrix of a monic polynomial; this polynomial (the minimal polynomial of the operator restricted to the subspace, which notion is analogous to that of the order of a cyclic subgroup) determines the action of the operator on the cyclic subspace up to isomorphism, and is independent of the choice of the vector v generating the subspace. A direct sum decomposition into cyclic subspaces always exists, and finding one does not require factoring polynomials.
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In functional analysis, a branch of mathematics, a strictly singular operator is a bounded linear operator between normed spaces which is not bounded below on any infinite-dimensional subspace.
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If a Banach space Y is isomorphic to a reflexive Banach space X, then Y is reflexive.Proposition 1.11.8 in . Every closed linear subspace of a reflexive space is reflexive.
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Since every subspace of a compact Hausdorff space is Tychonoff one has: :A topological space is Tychonoff if and only if it can be embedded in a Tychonoff cube.
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Choose a subspace V_n \subset V of dimension n and solve the projected problem: : Find u_n\in V_n such that for all v_n\in V_n, a(u_n,v_n) = f(v_n). We call this the Galerkin equation. Notice that the equation has remained unchanged and only the spaces have changed. Reducing the problem to a finite-dimensional vector subspace allows us to numerically compute u_n as a finite linear combination of the basis vectors in V_n .
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A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace R (the apex of the cone) and an arbitrary subset A (the basis) of some other subspace S, disjoint from R. In the special case that R is a single point, S is a plane, and A is a conic section on S, the projective cone is a conical surface; hence the name.
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Towards the end of The Subspace Emissary, it is revealed that Tabuu is the entity that controlled Master Hand and the true antagonist behind the events of the story. The Subspace Emissary also features other boss characters, like Petey Piranha, Ridley, Meta Ridley, Porky, and Rayquaza. Ridley eventually became playable in Super Smash Bros. Ultimate, with Meta Ridley as an alternate costume for him, while Petey Piranha became DLC character Piranha Plant's Final Smash.
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Sc. 199, 337 (1934). A typical example of a Chebyshev space is the subspace of Chebyshev polynomials of order n in the space of real continuous functions on an interval, C[a, b]. The polynomial of best approximation within a given subspace is defined to be the one that minimizes the maximum absolute difference between the polynomial and the function. In this case, the form of the solution is precised by the equioscillation theorem.
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Consider linear operators on a finite-dimensional vector space over a field. An operator T is semisimple if every T-invariant subspace has a complementary T-invariant subspace (if the underlying field is algebraically closed, this is the same as the requirement that the operator be diagonalizable). An operator x is nilpotent if some power xm of it is the zero operator. An operator x is unipotent if x − 1 is nilpotent.
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For Banach spaces, the first example of an operator without an invariant subspace was constructed by Per Enflo. He proposed a counterexample to the invariant subspace problem in 1975, publishing an outline in 1976. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987; . Enflo's long "manuscript had a world-wide circulation among mathematicians" and some of its ideas were described in publications besides Enflo (1976).
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This method was proposed by J. Yang and H. Tang and it is based in hallucinating of High-Resolution face image by taking Low-Resolution input value. The method exploits the facial features by using a Non-negative Matrix factorization (NMF) approach to learn localized part-based subspace. That subspace is effective for super-resolving the incoming face. For further enhance the detailed facial structure by using a local patch method based on sparse representation.
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In the first example (picture of shapes), recovered image was very fine, exactly similar to original image because L > K + N. In the second example (picture of a girl), L < K + N, so essential condition is violated, hence recovered image is far different from original image. Blurred Image, obtained by convolution of original image with blur kernel. Input image lies in fixed subspace of wavelet transform and blur kernel lies in random subspace.
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In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° (π/2 radians), or one of the vectors is zero. Hence orthogonality of vectors is an extension of the concept of perpendicular vectors to spaces of any dimension. The orthogonal complement of a subspace is the space of all vectors that are orthogonal to every vector in the subspace.
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If Y is a subset of X, then Y inherits a total order from X. The set Y therefore has an order topology, the induced order topology. As a subset of X, Y also has a subspace topology. The subspace topology is always at least as fine as the induced order topology, but they are not in general the same. For example, consider the subset Y = {–1} ∪ {1/n}n∈N in the rationals.
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The Arnoldi iteration reduces to the Lanczos iteration for symmetric matrices. The corresponding Krylov subspace method is the minimal residual method (MinRes) of Paige and Saunders. Unlike the unsymmetric case, the MinRes method is given by a three-term recurrence relation. It can be shown that there is no Krylov subspace method for general matrices, which is given by a short recurrence relation and yet minimizes the norms of the residuals, as GMRES does.
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Krylov subspace methods work by forming a basis of the sequence of successive matrix powers times the initial residual (the Krylov sequence). The approximations to the solution are then formed by minimizing the residual over the subspace formed. The prototypical method in this class is the conjugate gradient method (CG) which assumes that the system matrix A is symmetric positive-definite. For symmetric (and possibly indefinite) A one works with the minimal residual method (MINRES).
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In the Star Trek franchise, subspace communications have a limit of just over 20 light years before they must be boosted, although this limitation has been ignored in several storylines.
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Kowalsky's theorem, named after Hans-Joachim Kowalsky, states that any metrizable space of weight K can be represented as a topological subspace of the product of countably many K-hedgehog spaces.
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The book was published almost twenty years after Smith's death, and edited by Lloyd Arthur Eshbach.Smith, E.E. Subspace Encounter. Edited and with an introduction by Lloyd Arthur Eshbach. Berkley Books, 1983. .
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This follows directly from their definition since . They act on the subspace the span in the passive sense, according to In , the second equality follows from property of the Clifford algebra.
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In higher dimensional spaces, a pencil of hyperplanes consists of all the hyperplanes that contain a subspace of codimension 2. Such a pencil is determined by any two of its members.
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Goldberg, Jeff (1988). Anatomy of a Scientific Discovery. Bantam Books, 1988. ; (British edition); The corresponding trance-like mental state is also called subspace, for the submissive, and topspace, for the dominant.
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Since the relation of two functions f, g\colon X\to Y being homotopic relative to a subspace is an equivalence relation, we can look at the equivalence classes of maps between a fixed X and Y. If we fix X = [0,1]^n, the unit interval [0, 1] crossed with itself n times, and we take its boundary \partial([0,1]^n) as a subspace, then the equivalence classes form a group, denoted \pi_n(Y,y_0), where y_0 is in the image of the subspace \partial([0,1]^n). We can define the action of one equivalence class on another, and so we get a group. These groups are called the homotopy groups. In the case n = 1, it is also called the fundamental group.
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Nigel Hitchin introduced the notion of a generalized almost complex structure on the manifold M, which was elaborated in the doctoral dissertations of his students Marco Gualtieri and Gil Cavalcanti. An ordinary almost complex structure is a choice of a half-dimensional subspace of each fiber of the complexified tangent bundle TM. A generalized almost complex structure is a choice of a half-dimensional isotropic subspace of each fiber of the direct sum of the complexified tangent and cotangent bundles. In both cases one demands that the direct sum of the subbundle and its complex conjugate yield the original bundle. An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the Lie bracket.
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The vector x is an eigenvector of the matrix A. Every operator on a non- trivial complex finite dimensional vector space has an eigenvector, solving the invariant subspace problem for these spaces. In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many variants of the problem have been solved, by restricting the class of bounded operators considered or by specifying a particular class of Banach spaces. The problem is still open for separable Hilbert spaces (in other words, all the examples found of operators with no non-trivial invariant subspaces act on Banach spaces which are not separable Hilbert spaces).
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In the 1990s, Enflo developed a "constructive" approach to the invariant subspace problem on Hilbert spaces.Page 401 in . Enflo's method of ("forward") "minimal vectors" is also noted in the review of this research article by Gilles Cassier in Mathematical Reviews: Enflo's method of minimal vector is described in greater detail in a survey article on the invariant subspace problem by Enflo and Victor Lomonosov, which appears in the Handbook of the Geometry of Banach Spaces (2001).
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Plot of the first two Principal Components (left) and a two-dimension hidden layer of a Linear Autoencoder (Right) applied to the Fashion MNIST dataset. The two models being both linear learn to span the same subspace. The projection of the data points is indeed identical, apart from rotation of the subspace - to which PCA is invariant.Dimensionality Reduction was one of the first applications of deep learning, and one of the early motivations to study autoencoders.
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Trophies unavailable in Coin Launcher mode are obtained by using an item called the Trophy Stand on weakened enemy characters and bosses within The Subspace Emissary. Trophies obtained in this manner may contain information on the backstory of the Subspace Emissary. In addition to trophies, players can now collect stickers of video game artwork. Players can place stickers and trophies onto virtual backgrounds and take snapshots, which can be sent to other players via Nintendo Wi-Fi Connection.
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System sizes were typically small. For more general circuits, the method was considered impractical for all but these very small circuits until the mid-1990s, when Krylov subspace methods were applied to the problem. The application of preconditioned Krylov subspace methods allowed much larger systems to be solved, both in size of circuit and in numbers of harmonics. This made practical the present-day use of harmonic balance methods to analyze radio-frequency integrated circuits (RFICs).
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Blowup of the affine plane. In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion.
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Then the map given by is a fibration. Furthermore, is homotopy equivalent to as follows: Embed as a subspace of by where is the constant path at . Then deformation retracts to this subspace by contracting the paths. The fiber of this fibration (which is only well-defined up to homotopy equivalence) is the homotopy fiber , which can be defined as the set of all with and a path such that and , where is some fixed basepoint of .
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T. Chen, Y. Hua and W. Y. Yan, "Global convergence of Oja's subspace algorithm for principal component extraction," IEEE Transactions on Neural Networks, Vol. 9, No. 1, pp. 58-67, Jan 1998.
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In mathematics, an ovoid O of a (finite) polar space of rank r is a set of points, such that every subspace of rank r-1 intersects O in exactly one point..
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De-normalization, as in, repeating the same piece of data in multiple subspace is common practice. It allows to create secondary representation (also called indices) that will allow to speed up queries.
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For any other (k − 1)-dimensional subspace G, some f in the linear span of the first k eigenvectors must be orthogonal to G. Hence λ(G) ≤ (Df,f)/(f,f) ≤ λk.
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The reverse is true for subsets of Euclidean space (with the subspace topology), but not in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded.
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Under the subspace topology, the singleton set {–1} is open in Y, but under the induced order topology, any open set containing –1 must contain all but finitely many members of the space.
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Thereafter, he received his Ph.D. in Civil Engineering from the University of California, Berkeley in 1971. His thesis was on numerical solution of large eigenvalue problems, where he developed the subspace iteration method.
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This study investigated a high- resolution subspace source localization approach, FINE (first principle vectors) to image the locations and estimate the extents of current sources from the scalp EEG. A thresholding technique was applied to the resulting tomography of subspace correlation values in order to identify epileptogenic sources. This method was tested in three pediatric patients with intractable epilepsy, with encouraging clinical results. Each patient was evaluated using structural MRI, long-term video EEG monitoring with scalp electrodes, and subsequently with subdural electrodes.
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The scalars and vectors have their usual interpretation, and make up distinct subspaces of a GA. Bivectors provide a more natural representation of the pseudovector quantities in vector algebra such as oriented area, oriented angle of rotation, torque, angular momentum, electromagnetic field and the Poynting vector. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of V and orthogonal projections onto that subspace. Rotations and reflections are represented as elements.
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Meanwhile, Meta Knight leaves his group to reclaim his stolen ship, the Halberd. He allies with Lucario and Snake, reaching the cockpit and destroying the Mr. Game & Watch clones piloting the battleship. The Ancient Minister's true identity is found to be that of the Master R.O.B. unit, who rebels against Ganondorf to join the allied characters. Bowser and Ganondorf detonate several bombs on their Isle of the Ancients base to create a subspace portal large enough to summon their Subspace Gunship weapon.
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Chow's theorem, proved by Wei-Liang Chow, is an example of the most immediately useful kind of comparison available. It states that an analytic subspace of complex projective space that is closed (in the ordinary topological sense) is an algebraic subvariety. This can be rephrased as "any analytic subspace of complex projective space which is closed in the strong topology is closed in the Zariski topology." This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry.
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The MUSIC method is considered to be a poor performer in SAR applications. This method uses a constant instead of the clutter subspace. In this method, the denominator is equated to zero when a sinusoidal signal corresponding to a point in the SAR image is in alignment to one of the signal subspace eigenvectors which is the peak in image estimate. Thus this method does not accurately represent the scattering intensity at each point, but show the particular points of the image.
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X. He, D. Cai, P. Niyogi, Tensor subspace analysis, in: Advances in Neural Information Processing Systemsc 18 (NIPS), 2005. Here are some examples of data tensors whose observations are vectorized or whose observations are matrices concatenated into data tensor images (2D/3D), video sequences (3D/4D), and hyperspectral cubes (3D/4D). The mapping from a high- dimensional vector space to a set of lower dimensional vector spaces is a multilinear projection.. When observations are retained in the same organizational structure as the sensor provides them; as matrices or higher order tensors, their representations are computed by performing N multiple linear projections. Multilinear subspace learning algorithms are higher-order generalizations of linear subspace learning methods such as principal component analysis (PCA), independent component analysis (ICA), linear discriminant analysis (LDA) and canonical correlation analysis (CCA).
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Matrix inversion plays a significant role in computer graphics, particularly in 3D graphics rendering and 3D simulations. Examples include screen-to-world ray casting, world-to-subspace- to-world object transformations, and physical simulations.
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This method is also applicable to one-class classifiers. Recently, the random subspace method has been used in a portfolio selection problem showing its superiority to the conventional resampled portfolio essentially based on Bagging.
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A Jordan operator algebra is a norm-closed subspace of the space of operators on a complex Hilbert space, closed under the Jordan product a ∘ b = ½(ab + ba) and closed in the operator norm.
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The axiom of determinacy implies that for every subspace X of the real numbers, the Banach–Mazur game BM(X) is determined (and therefore that every set of reals has the property of Baire).
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Every sequential space (in particular, every metrizable space) is countably generated. An example of a space which is countably generated but not sequential can be obtained, for instance, as a subspace of Arens–Fort space.
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As the name suggests, it only covers prediction models, a particular data mining task of high importance to business applications. However, extensions to cover (for example) subspace clustering have been proposed independently of the DMG.
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141-142, June 1998. Y. Miao and Y. Hua, "Fast subspace tracking and neural network learning by a novel information criterion," IEEE Transactions on Signal Processing, Vol. 46, No. 7, pp. 1967-1979, July 1998.
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SUBCLU is an algorithm for clustering high-dimensional data by Karin Kailing, Hans-Peter Kriegel and Peer Kröger.Karin Kailing, Hans-Peter Kriegel and Peer Kröger. Density-Connected Subspace Clustering for High-Dimensional Data. In: Proc.
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By convention, the term shift is understood to refer to the full n-shift. A subshift is then any subspace of the full shift that is shift-invariant (that is, a subspace that is invariant under the action of the shift operator), non-empty, and closed for the product topology defined below. Some subshifts can be characterized by a transition matrix, as above; such subshifts are then called subshifts of finite type. Often, subshifts of finite type are called simply shifts of finite type.
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The number is called the regulator of the algebraic number field (it does not depend on the choice of generators ). It measures the "density" of the units: if the regulator is small, this means that there are "lots" of units. The regulator has the following geometric interpretation. The map taking a unit to the vector with entries has an image in the -dimensional subspace of consisting of all vectors whose entries have sum 0, and by Dirichlet's unit theorem the image is a lattice in this subspace.
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MUSIC detects frequencies in a signal by performing an eigen decomposition on the covariance matrix of a data vector of the samples obtained from the samples of the received signal. When all of the eigenvectors are included in the clutter subspace (model order = 0) the EV method becomes identical to the Capon method. Thus the determination of model order is critical to operation of the EV method. The eigenvalue of the R matrix decides whether its corresponding eigenvector corresponds to the clutter or to the signal subspace.
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A decoherence-free subspace (DFS) is a subspace of a quantum system's Hilbert space that is invariant to non-unitary dynamics. Alternatively stated, they are a small section of the system Hilbert space where the system is decoupled from the environment and thus its evolution is completely unitary. DFSs can also be characterized as a special class of quantum error correcting codes. In this representation they are passive error-preventing codes since these subspaces are encoded with information that (possibly) won't require any active stabilization methods.
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If C is a non-empty convex cone in X, then the linear span of C is equal to C - C and the largest vector subspace of X contained in C is equal to C ∩ (-C).
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The continuous dual of a reflexive space is reflexive. Every quotient of a reflexive space by a closed subspace is reflexive.. Let X be a Banach space. The following are equivalent. # The space X is reflexive.
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Even in the trivial case T=I and B=I the resulting approximation with i>3 will be different from that obtained by the Lanczos algorithm, although both approximations will belong to the same Krylov subspace.
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A JC algebra is a norm-closed self-adjoint subspace of the space of operators on a complex Hilbert space, closed under the operator Jordan product a ∘ b = ½(ab + ba) and closed in the operator norm.
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Every locally compact Hausdorff space is a Baire space. That is, the conclusion of the Baire category theorem holds: the interior of every union of countably many nowhere dense subsets is empty. A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets of Y. As a corollary, a dense subspace X of a locally compact Hausdorff space Y is locally compact if and only if X is an open subset of Y. Furthermore, if a subspace X of any Hausdorff space Y is locally compact, then X still must be the difference of two closed subsets of Y, although the converse needn't hold in this case. Quotient spaces of locally compact Hausdorff spaces are compactly generated.
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The difference in dimension between a subspace and its ambient space is known as the codimension of with respect to . Therefore, a necessary condition for to be a hyperplane in is for to have codimension one in .
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If V and W are topological vector spaces such that W is finite-dimensional, then a linear operator L: V → W is continuous if and only if the kernel of L is a closed subspace of V.
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A JC algebra is a real subspace of the space of self-adjoint operators on a real or complex Hilbert space, closed under the operator Jordan product a ∘ b = ½(ab + ba) and closed in the operator norm.
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Keys are hashes: there is no notion of semantic closeness when speaking of key closeness. Therefore, there will be no correlation between key closeness and similar popularity of data as there might be if keys did exhibit some semantic meaning, thus avoiding bottlenecks caused by popular subjects. There are two main varieties of keys in use on Freenet, the Content Hash Key (CHK) and the Signed Subspace Key (SSK). A subtype of SSKs is the Updatable Subspace Key (USK) which adds versioning to allow secure updating of content.
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They span a wide range of topics, including shipbuilding, magnetism, artillery, mathematics, astronomy, and geodesy. His floodability tables have been used worldwide. Of note are his works in hydrodynamics including theory of ships moving in shallow water (he was the first to explain and calculate the significant increase of hydrodynamic resistance in shallow water) and the theory of solitons. In 1904 he built the first machine in Russia for integrating Ordinary differential equations. Krylov in the 1930s In 1931 he published a paper on what is now called the Krylov subspace and Krylov subspace methods.
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The tricobalt warhead is a subspace weapon whose high- yield detonations can tear holes in subspace. Tricobalt devices are not a standard armament of Federation vessels and yields are calculated in Tera- Cochranes, indicating that its mechanism is somewhat similar to the general reaction in a warp field. In TOS: "A Taste of Armageddon", the Eminian Union classified the USS Enterprise as 'destroyed' when it was hit by virtual tricobalt satellites. In DS9: "Trials and Tribble-ations", Arne Darvin plants a tricobalt explosive in a dead Tribble in an attempt to kill Kirk.
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In the absence of (true) eigenvectors, one can look for subspaces consisting of almost eigenvectors. In the above example, for example, where [A \varphi](t) = t \varphi(t), \; we might consider the subspace of functions supported on a small interval [a,a+\varepsilon] inside [0,1]. This space is invariant under A and for any \varphi in this subspace, A\varphi is very close to a\varphi. In this approach to the spectral theorem, if A is a bounded self-adjoint operator, one looks for large families of such "spectral subspaces".
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The three major groups converge and use the Halberd to battle the Gunship. Although the Halberd is destroyed, all of the characters escape unscathed and Ganondorf and Bowser retreat after Kirby destroys the Gunship with his Dragoon. Ganondorf betrays Bowser and turns him into a trophy, only to learn that Master Hand was being manipulated by the actual Subspace Army leader, Tabuu, who turns Ganondorf into a trophy and defeats Master Hand. The allied characters enter Tabuu's chamber, but Tabuu annihilates them all with powerful "Off Waves" and scatters them all throughout Subspace.
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We now show how every non-Hausdorff TVS can be TVS-embedded onto a dense vector subspace of a complete TVS. The proof that every Hausdorff TVS has a Hausdorff completion is widely available so we use its conclusion to prove that every non-Hausdorff TVS also has a completion. These details are sometimes useful for extending results from Hausdorff TVSs to non-Hausdorff TVSs. Let us write } for the closure of the origin in , where is endowed with its subspace topology induced by (so that has the indiscrete topology).
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This shows that s generates V as a k-algebra and thus the S-stable k-linear subspaces of V are ideals of V, i.e. they are 0, J and V. We see that J is an S-invariant subspace of V which has no complement S-invariant subspace, contrary to the assumption that S is semisimple. Thus, there is no decomposition of T as a sum of commuting k-linear operators that are respectively semisimple and nilpotent. Note that minimal polynomial of T is inseparable over k and is a square in k[X].
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Importantly, the subspace topology C^k(K;U) inherits from C^k(U) (when it is viewed as a subset of C^k(U)) is identical to the subspace topology that it inherits from C^k(V) (when C^k(K;U) is viewed instead as a subset of C^k(V) via the identification). Thus the topology on C^k(K;U) is independent of the open subset of \R^n that contains . This justifies our practice of using C^k(K) instead of C^k(K;U).
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Charles Read Charles John Read (16 February 1958 - 14 August 2015) was a British mathematician known for his work in functional analysis. In operator theory, he is best known for his work in the 1980s on the invariant subspace problem, where he constructed operators with only trivial invariant subspaces on particular Banach spaces, especially on \ell_1. He won the 1985 Junior Berwick Prize for his work on the invariant subspace problem. Read has also published on Banach algebras and hypercyclicity; in particular, he constructed the first example of an amenable, commutative, radical Banach algebra.
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However, proved a complex-number analogue of the Sylvester–Gallai theorem: whenever the points of a Sylvester–Gallai configuration are embedded into a complex projective space, the points must all lie in a two-dimensional subspace. Equivalently, a set of points in three- dimensional complex space whose affine hull is the whole space must have an ordinary line, and in fact must have a linear number of ordinary lines. Similarly, showed that whenever a Sylvester–Gallai configuration is embedded into a space defined over the quaternions, its points must lie in a three- dimensional subspace.
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MPCA features: Supervised MPCA feature selection is used in object recognitionM. A. O. Vasilescu, D. Terzopoulos (2003) "Multilinear Subspace Analysis of Image Ensembles", "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’03), Madison, WI, June, 2003" while unsupervised MPCA feature selection is employed in visualization task.H. Lu, H.-L. Eng, M. Thida, and K.N. Plataniotis, "Visualization and Clustering of Crowd Video Content in MPCA Subspace," in Proceedings of the 19th ACM Conference on Information and Knowledge Management (CIKM 2010), Toronto, ON, Canada, October, 2010.
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With Jonathan Pila he developed a method (now known as the Pila-Zannier method) of applying O-minimality to number- theoretical and algebro-geometric problems. Thus they gave a new proof of the Manin–Mumford conjecture (which was first proved by Michel Raynaud and Ehud Hrushovski). Zannier and Pietro Corvaja in 2002 gave a new proof of Siegel's theorem on integral points by using a new method based upon the subspace theorem.P. Corvaja and Zannier, U. "A subspace theorem approach to integral points on curves", Compte Rendu Acad. Sci.
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Let be a topological vector space. A vector subspace of has the extension property if any continuous linear functional on can be extended to a continuous linear functional on , and we say that has the Hahn–Banach extension property (HBEP) if every vector subspace of has the extension property. The Hahn–Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable topological vector spaces there is a converse, due to Kalton: every complete metrizable TVS with the Hahn–Banach extension property is locally convex.
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In 1992, with Timothy Gowers, Maurey resolved the "unconditional basic sequence problem" in the theory of Banach spaces, by showing that not every infinite-dimensional Banach space has an infinite-dimensional subspace that admits an unconditional Schauder basis.
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The best known Krylov subspace methods are the Arnoldi, Lanczos, Conjugate gradient, IDR(s) (Induced dimension reduction), GMRES (generalized minimum residual), BiCGSTAB (biconjugate gradient stabilized), QMR (quasi minimal residual), TFQMR (transpose-free QMR), and MINRES (minimal residual) methods.
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Popular recognition algorithms include principal component analysis using eigenfaces, linear discriminant analysis, elastic bunch graph matching using the Fisherface algorithm, the hidden Markov model, the multilinear subspace learning using tensor representation, and the neuronal motivated dynamic link matching.
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In mathematics and control theory, H2, or H-square is a Hardy space with square norm. It is a subspace of L2 space, and is thus a Hilbert space. In particular, it is a reproducing kernel Hilbert space.
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Roketz is a 360 degree scrolling shoot 'em up released for the Amiga 1200 in 1995 and MS-DOS in 1996. It features gravity environment and thrust-and-turn gameplay similar to the likes of Thrust, Virus, and SubSpace.
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The space Aim(X) is injective (hyperconvex in the sense of Aronszajn-Panitchpakdi) – given a metric space M, which contains Aim(X) as a metric subspace, there is a canonical (and explicit) metric retraction of M onto Aim(X) .
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C. Ding, X. He, H. Zha, H.D. Simon, Adaptive Dimension Reduction for Clustering High Dimensional Data, Proceedings of International Conference on Data Mining, 2002 For multidimensional data, tensor representation can be used in dimensionality reduction through multilinear subspace learning.
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Therefore, given a subspace V, there may be many projections whose range (or kernel) is V. If a projection is nontrivial it has minimal polynomial x^2-x = x(x-1), which factors into distinct roots, and thus P is diagonalizable.
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An orthogonal array is linear if X is a finite field of order q, Fq (q a prime power) and the rows of the array form a subspace of the vector space (Fq)k. Every linear orthogonal array is simple.
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Let C ⊆ (Fq)n, be a linear code of dimension m with minimum distance d. Then C⊥ (the orthogonal complement of the vector subspace C) is a (linear) (d − 1)-(q, n, λ) orthogonal array where λ = qn − m − d + 1.
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Because the vectors usually soon become almost linearly dependent due to the properties of power iteration, methods relying on Krylov subspace frequently involve some orthogonalization scheme, such as Lanczos iteration for Hermitian matrices or Arnoldi iteration for more general matrices.
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Chao et al. (2015) In these applications, exponential integrators show the advantage of large time stepping capability and high accuracy. To accelerate the evaluation of matrix functions in such complex scenarios, exponential integrators are often combined with Krylov subspace projection methods.
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"Relations between the statistics of natural images and the response properties of cortical cells." J. Opt. Soc. am. A 4:2479-2394, 1987. but correctly identifying this subspace using traditional techniques is complicated by the correlations that exist within natural images.
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Thus for all and , we have . :: by which is a TVS-embedding of onto a dense vector subspace of the complete TVS . Moreover, observe that the closure of the origin in is equal to }, and that } and are topological complements in .
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The relation between the two is simply: algebraic dimension = geometric dimension + 1. Also the - (vector) dimensional subspaces of represent the ()- (geometric) dimensional hyperplanes of projective -space over , i.e., . A nonzero vector in also determines an - geometric dimensional subspace (hyperplane) , by :.
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The Birkhoff–von Neumann theorem states that this polytope can be described by two types of linear inequality or equality. First, for each matrix cell, there is a constraint that this cell has a non-negative value. And second, for each row or column of the matrix, there is a constraint that the sum of the cells in that row or column equal one. The row and column constraints define a linear subspace of dimension n2 − 2n + 1 in which the Birkhoff polytope lies, and the non- negativity constraints define facets of the Birkhoff polytope within that subspace.
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For instance, the completion of a metric space M involves an isometry from M into M', a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space. An isometric surjective linear operator on a Hilbert space is called a unitary operator.
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In mathematics, more specifically algebraic topology, a pair (X,A) is shorthand for an inclusion of topological spaces i\colon A\hookrightarrow X. Sometimes i is assumed to be a cofibration. A morphism from (X,A) to (X',A') is given by two maps f\colon X\rightarrow X' and g\colon A\rightarrow A' such that i' \circ g =f \circ i . A pair of spaces is an ordered pair where is a topological space and a subspace (with the subspace topology). The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of by .
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Stickers can be applied to characters to power up their abilities in the Subspace Emissary. Other stickers or trophies which cannot be collected through the Coin Launcher minigame, Subspace Emissary, or Vs. matches can be unlocked from the Challenges menu, an interactive display which catalogs unlocked features and items in gridded windows. Once a window has been broken and its contents are unlocked, horizontally adjacent windows display the conditions necessary to unlock them. Brawl contains demo versions of several Nintendo games, named "Masterpieces", which were originally released for older consoles and feature characters playable in Brawl.
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SpectreRF pioneered a variety of periodic small-signal analyses, including periodic AC (pac), periodic noise (pnoise), periodic transfer function (pxf), periodic s-parameter (psp) and periodic stability (pstb). After its introduction, SpectreRF quickly became the dominant simulator for RF integrated circuits, and was instrumental in establishing Spectre as the most popular circuit simulator for integrated circuits. Eventually the dominance of SpectreRF faded as the use of Krylov subspace methods propagated to other simulators, particularly those based on harmonic balance. SpectreRF now provides harmonic balance in addition to shooting methods, both of which are accelerated using Krylov subspace methods.
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DIIS (direct inversion in the iterative subspace or direct inversion of the iterative subspace), also known as Pulay mixing, is a technique for extrapolating the solution to a set of linear equations by directly minimizing an error residual (e.g. a Newton-Raphson step size) with respect to a linear combination of known sample vectors. DIIS was developed by Peter Pulay in the field of computational quantum chemistry with the intent to accelerate and stabilize the convergence of the Hartree–Fock self-consistent field method. At a given iteration, the approach constructs a linear combination of approximate error vectors from previous iterations.
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So the family of separable states is the closed convex hull of pure product states. We will make use of the following variant of Hahn–Banach theorem: Theorem Let S_1 and S_2 be disjoint convex closed sets in a real Banach space and one of them is compact, then there exists a bounded functional f separating the two sets. This is a generalization of the fact that, in real Euclidean space, given a convex set and a point outside, there always exists an affine subspace separating the two. The affine subspace manifests itself as the functional f.
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The dimension of the subspace in which the image data resides is a direct consequence of two factors: # The type of camera that projects the scene (for example, affine or perspective) # The nature of inspected object (for instance, rigid or non-rigid). The low-dimensionality of the subspace is mirrored (captured) trivially as reduced rank of the measurement matrix. This reduced rank of measurement matrix can be motivated from the fact that, the position of the projection of an object point on the image plane is constrained as the motion of each point is globally described by a precise geometric model.
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In these systems, when spin-orbit interaction is ignored, the degeneracy of conical intersection is lifted through first order by displacements in a two dimensional subspace of the nuclear coordinate space. The two-dimensional degeneracy lifting subspace is referred to as the branching space or branching plane. This space is spanned by two vectors, the difference of energy gradient vectors of the two intersecting electronic states (the g vector), and the non-adiabatic coupling vector between these two states (the h vector). Because the electronic states are degenerate, the wave functions of the two electronic states are subject to an arbitrary rotation.
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In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms --given a topological space X and subspaces A and U such that U is also a subspace of A, the theorem says that under certain circumstances, we can cut out (excise) U from both spaces such that the relative homologies of the pairs (X \setminus U,A \setminus U ) into (X, A) are isomorphic. This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute.
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The Sparse Subspace Learning (SSL) method leverages the use of hierarchical collocation to approximate the numerical solution of parametric models. With respect to traditional projection-based reduced order modeling, the use of a collocation enables non- intrusive approach based on sparse adaptive sampling of the parametric space. This allows to recover the lowdimensional structure of the parametric solution subspace while also learning the functional dependency from the parameters in explicit form. A sparse low-rank approximate tensor representation of the parametric solution can be built through an incremental strategy that only needs to have access to the output of a deterministic solver.
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This again is reasonable, as projection operators give the appropriate mathematical description of quantum measurements. After a measurement by Alice, the state of the total system is said to have collapsed to a state P(σ). The goal of the theorem is to prove that Bob cannot in any way distinguish the pre-measurement state σ from the post-measurement state P(σ). This is accomplished mathematically by comparing the trace of σ and the trace of P(σ), with the trace being taken over the subspace HA. Since the trace is only over a subspace, it is technically called a partial trace.
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In R3, the intersection of two distinct two-dimensional subspaces is one-dimensional Given subspaces U and W of a vector space V, then their intersection U ∩ W := {v ∈ V : v is an element of both U and W} is also a subspace of V. Proof: # Let v and w be elements of U ∩ W. Then v and w belong to both U and W. Because U is a subspace, then v + w belongs to U. Similarly, since W is a subspace, then v + w belongs to W. Thus, v + w belongs to U ∩ W. # Let v belong to U ∩ W, and let c be a scalar. Then v belongs to both U and W. Since U and W are subspaces, cv belongs to both U and W. # Since U and W are vector spaces, then 0 belongs to both sets. Thus, 0 belongs to U ∩ W. For every vector space V, the set {0} and V itself are subspaces of V.
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Victor Lomonosov (7 February 1946 - 29 March 2018) was a Russian-American mathematician known for his work in functional analysis. In operator theory, he is best known for his work in 1973 on the invariant subspace problem, which was described by Walter Rudin in his classical book on Functional Analysis as "Lomonosov's spectacular invariant subspace theorem". The Theorem Lomonosov gives a very short proof, using the Schauder fixed point theorem, that if the bounded linear operator T on a Banach space commutes with a non-zero compact operator then T has a non-trivial invariant subspace.. Lomonosov has also published on the Bishop–Phelps theorem and Burnside's Theorem Lomonosov received his master's degree from Moscow State University in 1969 and his Ph.D. from National University of Kharkiv in 1974 (adviser Vladimir Matsaev). He was appointed at the rank of Associate Professor at Kent State University in fall 1991, becoming Professor at the same university in 1999.
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The span of is also the intersection of all linear subspaces containing . In other words, it is the (smallest for the inclusion relation) linear subspace containing . A set of vectors is linearly independent if none is in the span of the others.
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Schlickerei's theorem implies the Thue-Siegel-Roth theorem, whose p-adic analogue was already proved in 1958 by David Ridout. In 1998 Schlickewei was an invited speaker with talk The Subspace Theorem and Applications at the International Congress of Mathematicians in Berlin.
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A generalized almost complex structure integrates to a generalized complex structure if the subspace is closed under the Courant bracket. If furthermore this half-dimensional space is the annihilator of a nowhere vanishing pure spinor then M is a generalized Calabi–Yau manifold.
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In quantum information theory, the idea of a typical subspace plays an important role in the proofs of many coding theorems (the most prominent example being Schumacher compression). Its role is analogous to that of the typical set in classical information theory.
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ACM SIGGRAPH 2004 Conference Los Angeles, CA, August, 2004, in Computer Graphics Proceedings, Annual Conference Series, 2004, 336–342. computer vision,Vasilescu, M.A.O.; Terzopoulos, D. (2003) "Multilinear Subspace Analysis for Image Ensembles," Proc. Computer Vision and Pattern Recognition Conf. (CVPR '03), Vol.
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In a vector space, a vector hyperplane is a subspace of codimension 1, only possibly shifted from the origin by a vector, in which case it is referred to as a flat. Such a hyperplane is the solution of a single linear equation.
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Suppose that (X, ℬ) is a bounded structure and S be a subset of X. The subspace bornology 𝒜 on S is the finest bornology on S making the natural inclusion map of S into X, Id : (S, 𝒜) -> (X, ℬ), locally bounded.
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Amplitude amplification is a technique that allows the amplification of a chosen subspace of a quantum state. Applications of amplitude amplification usually lead to quadratic speedups over the corresponding classical algorithms. It can be considered to be a generalization of Grover's algorithm.
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The Tychonoff plank is a compact Hausdorff space and is therefore a normal space. However, the deleted Tychonoff plank is non-normal. Therefore the Tychonoff plank is not completely normal. This shows that a subspace of a normal space need not be normal.
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Convexity can be extended for a totally ordered set endowed with the order topology.Munkres, James; Topology, Prentice Hall; 2nd edition (December 28, 1999). . Let . The subspace is a convex set if for each pair of points in such that , the interval is contained in .
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Alternatively, each number, when written in binary, can be identified with a non-zero vector of length three over the binary field. Three vectors that generate a subspace form a line; in this case, that is equivalent to their vector sum being the zero vector.
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Let be a subspace of that generates as an algebra and which is minimal with respect to this property. Let be an orthonormal basis of with respect to . Then orthonormality implies that: :e_i^2 =-1, \quad e_i e_j = - e_j e_i. If , then is isomorphic to .
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In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N).
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Keep the same field and vector space as before, but now consider the set Diff(R) of all differentiable functions. The same sort of argument as before shows that this is a subspace too. Examples that extend these themes are common in functional analysis.
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Lounesto (2001) p. 193 More generally every real geometric algebra is isomorphic to a matrix algebra. These contain bivectors as a subspace, though often in a way which is not especially useful. These matrices are mainly of interest as a way of classifying Clifford algebras.
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By definition, f is a quotient map. The most common example of this is to consider an equivalence relation on X, with Y the set of equivalence classes and f the natural projection map. This construction is dual to the construction of the subspace topology.
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In machine learning the random subspace method, also called attribute bagging or feature bagging, is an ensemble learning method that attempts to reduce the correlation between estimators in an ensemble by training them on random samples of features instead of the entire feature set.
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If is a vector subspace of the algebraic dual space of then we will assume that they are associated with the canonical pairing . In this case, the weak topology on (resp. the weak topology on ), denoted by (resp. by ) is the weak topology on (resp.
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Long short-term memory (LSTM) recurrent units are typically incorporated after the CNN to account for inter-frame or inter-clip dependencies. Unsupervised learning schemes for training spatio-temporal features have been introduced, based on Convolutional Gated Restricted Boltzmann Machines and Independent Subspace Analysis.
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A 1979 paper by Ralph O. Schmidt of Electromagnetic Systems Laboratory (ESL, a supplier of strategic reconnaissance systems) described the multiple signal classification (MUSIC) algorithm for estimating signals’ angle of arrival. Schmidt used a signal subspace method based on geometric modeling to derive a solution assuming the absence of noise and then extended the method to provide a good approximation in the presence of noise. Schmidt's paper became the most cited and his signal subspace method became the focus of ongoing research. Jack Winters showed in 1984 that received signals from multiple antennas can be combined (using the optimum combining technique) to reduce co-channel interference in digital mobile networks.
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Not all algorithms try to either find a unique cluster assignment for each point or all clusters in all subspaces; many settle for a result in between, where a number of possibly overlapping, but not necessarily exhaustive set of clusters are found. An example is FIRES, which is from its basic approach a subspace clustering algorithm, but uses a heuristic too aggressive to credibly produce all subspace clusters. Another hybrid approach is to include a human-into-the-algorithmic-loop: Human domain expertise can help to reduce an exponential search space through heuristic selection of samples. This can be beneficial in the health domain where, e.g.
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Trivially, the zero module {0} is injective. Given a field k, every k-vector space Q is an injective k-module. Reason: if Q is a subspace of V, we can find a basis of Q and extend it to a basis of V. The new extending basis vectors span a subspace K of V and V is the internal direct sum of Q and K. Note that the direct complement K of Q is not uniquely determined by Q, and likewise the extending map h in the above definition is typically not unique. The rationals Q (with addition) form an injective abelian group (i.e.
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Some of these enemy characters appeared in previous Nintendo video games, such as Petey Piranha from the Super Mario series and a squadron of R.O.B.s based on classic Nintendo hardware. The Subspace Emissary also boasts a number of original enemies, such as the Roader, a robotic unicycle; the Bytan, a one-eyed ball- like creature which can replicate itself if left alone; and the Primid, enemies that come in many variations. Though primarily a single-player mode, The Subspace Emissary allows for cooperative multiplayer. There are five difficulty levels for each stage, and there is a method of increasing characters' powers during the game.
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If V is a vector space over a field K and if W is a subset of V, then W is a subspace of V if under the operations of V, W is a vector space over K. Equivalently, a nonempty subset W is a subspace of V if, whenever w_1, w_2 are elements of W and \alpha, \beta are elements of K, it follows that \alpha w_1 + \beta w_2 is in W. As a corollary, all vector spaces are equipped with at least two subspaces: the singleton set with the zero vector and the vector space itself. These are called the trivial subspaces of the vector space.
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When there is no degeneracy, this subspace is one-dimensional and so all such linear transformations commute (because they are just multiplications by a phase factor). When there is degeneracy and this subspace has higher dimension, then these linear transformations need not commute (just as matrix multiplication does not). Gregory Moore, Nicholas Read, and Xiao-Gang Wen pointed out that non-Abelian statistics can be realized in the fractional quantum Hall effect (FQHE). While at first non-abelian anyons were generally considered a mathematical curiosity, physicists began pushing toward their discovery when Alexei Kitaev showed that non-abelian anyons could be used to construct a topological quantum computer.
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In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1, ... , vk} in an inner product space (most commonly the Euclidean space Rn), orthogonalization results in a set of orthogonal vectors {u1, ... , uk} that generate the same subspace as the vectors v1, ... , vk. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span. In addition, if we want the resulting vectors to all be unit vectors, then the procedure is called orthonormalization.
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The power of isospin symmetry and related methods comes from the observation that families of particles with similar masses tend to correspond to the invariant subspaces associated with the irreducible representations of the Lie algebra SU(2). In this context, an invariant subspace is spanned by basis vectors which correspond to particles in a family. Under the action of the Lie algebra SU(2), which generates rotations in isospin space, elements corresponding to definite particle states or superpositions of states can be rotated into each other, but can never leave the space (since the subspace is in fact invariant). This is reflective of the symmetry present.
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In the context of schemes, the importance of ideal sheaves lies mainly in the correspondence between closed subschemes and quasi-coherent ideal sheaves. Consider a scheme X and a quasi-coherent ideal sheaf J in OX. Then, the support Z of OX/J is a closed subspace of X, and (Z, OX/J) is a scheme (both assertions can be checked locally). It is called the closed subscheme of X defined by J. Conversely, let i: Z → X be a closed immersion, i.e., a morphism which is a homeomorphism onto a closed subspace such that the associated map : i#: OX → i⋆OZ is surjective on the stalks.
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The submanifold topology on an immersed submanifold need not be the relative topology inherited from M. In general, it will be finer than the subspace topology (i.e. have more open sets). Immersed submanifolds occur in the theory of Lie groups where Lie subgroups are naturally immersed submanifolds.
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If X is some topological space, such as the unit interval [0,1], we can consider the space of all continuous functions from X to R. This is a vector subspace of RX since the sum of any two continuous functions is continuous and scalar multiplication is continuous.
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However, there exist examples of sequentially compact, first-countable spaces which are not compact (these are necessarily non-metric spaces). One such space is the ordinal space [0,ω1). Every first-countable space is compactly generated. Every subspace of a first-countable space is first-countable.
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In 2011 he became professor emeritus. Bombieri is also known for his pro bono service on behalf of the mathematics profession, e.g. for serving on external review boards and for peer-reviewing extraordinarily complicated manuscripts (like the paper of Per Enflo on the invariant subspace problem).
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Pick some mathematical object that has an underlying set, for instance a group, ring, vector space, etc. For any subset of , let be the smallest subobject of that contains , i.e. the subgroup, subring or subspace generated by . For any subobject of , let be the underlying set of .
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In mathematics, specifically in topology and functional analysis, a subspace of a uniform space is said to be sequentially complete or semi-complete if every Cauchy sequence in converges to an element in . We call sequentially complete if it is a sequentially complete subset of itself.
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Let Hn be the subspace of all states with ghost number n. Then, Q restricted to Hn maps Hn to Hn+1. Since Q2 = 0, we have a cochain complex describing a cohomology. The physical states are identified as elements of cohomology of the operator Q, i.e.
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There are many existing methods for dimension reduction, both graphical and numeric. For example, sliced inverse regression (SIR) and sliced average variance estimation (SAVE) were introduced in the 1990s and continue to be widely used.Li, K-C. (1991) Sliced Inverse Regression for Dimension Reduction In: Journal of the American Statistical Association, 86(414): 316–327 Although SIR was originally designed to estimate an effective dimension reducing subspace, it is now understood that it estimates only the central subspace, which is generally different. More recent methods for dimension reduction include likelihood-based sufficient dimension reduction,Cook, R.D. and Forzani, L. (2009) Likelihood-Based Sufficient Dimension Reduction In: Journal of the American Statistical Association, 104(485): 197–208 estimating the central subspace based on the inverse third moment (or kth moment),Yin, X. and Cook, R.D. (2003) Estimating Central Subspaces via Inverse Third Moments In: Biometrika, 90(1): 113–125 estimating the central solution space,Li, B. and Dong, Y.D. (2009) Dimension Reduction for Nonelliptically Distributed Predictors In: Annals of Statistics, 37(3): 1272–1298 graphical regression,envelope model, and the principal support vector machine.
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In particular, a vector space is an affine space over itself, by the map :. If W is a vector space, then an affine subspace is a subset of W obtained by translating a linear subspace V by a fixed vector ; this space is denoted by (it is a coset of V in W) and consists of all vectors of the form for An important example is the space of solutions of a system of inhomogeneous linear equations :Ax = b generalizing the homogeneous case above. The space of solutions is the affine subspace where x is a particular solution of the equation, and V is the space of solutions of the homogeneous equation (the nullspace of A). The set of one-dimensional subspaces of a fixed finite-dimensional vector space V is known as projective space; it may be used to formalize the idea of parallel lines intersecting at infinity. Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension k and flags of subspaces, respectively.
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The set of evil numbers (numbers n with t_n=0) forms a subspace of the nonnegative integers under nim-addition (bitwise exclusive or). For the game of Kayles, evil nim-values occur for few (finitely many) positions in the game, with all remaining positions having odious nim- values.
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Grothendieck & Raynaud, SGA 1, Exposé XII. (The first group here is defined using the Zariski topology, and the second using the classical (Euclidean) topology.) For example, the equivalence between algebraic and analytic coherent sheaves on projective space implies Chow's theorem that every closed analytic subspace of CPn is algebraic.
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Block locally optimal multi-step steepest descent for eigenvalue problems was described in. Local minimization of the Rayleigh quotient on the subspace spanned by the current approximation, the current residual and the previous approximation, as well as its block version, appeared in. The preconditioned version was analyzed in and.
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The most common map elements in SubSpace are prizes, or "greens" (for their green color). Prizes allow players to upgrade their ships and gain special weapons or abilities. While prizes are generally plentifully scattered throughout the map, the upgrades or abilities they award are randomly selected by the zone.
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In science fiction, ultrawaves (or hyperwaves or subwaves) are transmissions or signals that may propagate faster than light through either normal space, or alternate space, such as hyperspace or subspace. Ultrawaves are also sometimes a form of energy transmission or weapon such as a beam weapon or death-ray.
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In an infinite-dimensional space V, as used in functional analysis, the flag idea generalises to a subspace nest, namely a collection of subspaces of V that is a total order for inclusion and which further is closed under arbitrary intersections and closed linear spans. See nest algebra.
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In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like a normal space. More precisely, a locally normal space satisfies the property that each point of the space belongs to a neighbourhood of the space that is normal under the subspace topology.
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Ultimate, each of which refers to R.O.B. as male within their respective universes. In Brawls adventure mode, The Subspace Emissary, R.O.B. plays a major role in the story's plot. As part of Super Smash Bros. for Nintendo 3DS and Wii U, R.O.B. is among the Super Smash Bros.
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Energy reserves allow the player to upgrade units, maintain facilities, and attempt to win by the Global Energy Market scenario. Bases are military strongpoints and objectives that are vital for all winning strategies. They produce military units, house the population, collect energy, and build secret projects and Subspace Generators.
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In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations.
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For the higher order elements of the Clifford algebra in general and their transformation rules, see the article Dirac algebra. But it is noted here that the Clifford algebra has no subspace being the representation space of a spin representation of the Lorentz group in the context used here.
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A map is called a quasi-isometric embedding if it satisfies the first condition but not necessarily the second (i.e. it is coarsely Lipschitz but may fail to be coarsely surjective). In other words, if through the map, (M_1,d_1) is quasi-isometric to a subspace of (M_2,d_2).
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Mertens and ZamirAn exposition for the general reader is by Shmuel Zamir, 2008: "Bayesian games: Games with incomplete information," Discussion Paper 486, Center for Rationality, Hebrew University. implemented John Harsanyi's proposal to model games with incomplete information by supposing that each player is characterized by a privately known type that describes his feasible strategies and payoffs as well as a probability distribution over other players' types. They constructed a universal space of types in which, subject to specified consistency conditions, each type corresponds to the infinite hierarchy of his probabilistic beliefs about others' probabilistic beliefs. They also showed that any subspace can be approximated arbitrarily closely by a finite subspace, which is the usual tactic in applications.
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A group action of a topological group G on a topological space X is said to be strongly continuous if for all x in X, the map is continuous with respect to the respective topologies. Such an action induces an action on the space of continuous functions on X by defining for every g in G, f a continuous function on X, and x in X. Note that, while every continuous group action is strongly continuous, the converse is not in general true. The subspace of smooth points for the action is the subspace of X of points x such that is smooth, that is, it is continuous and all derivatives are continuous.
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A subspace W of V that is invariant under the group action is called a subrepresentation. If V has exactly two subrepresentations, namely the zero-dimensional subspace and V itself, then the representation is said to be irreducible; if it has a proper subrepresentation of nonzero dimension, the representation is said to be reducible. The representation of dimension zero is considered to be neither reducible nor irreducible, just like the number 1 is considered to be neither composite nor prime. Under the assumption that the characteristic of the field K does not divide the size of the group, representations of finite groups can be decomposed into a direct sum of irreducible subrepresentations (see Maschke's theorem).
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In order to define the fundamental group, one needs the notion of homotopy relative to a subspace. These are homotopies which keep the elements of the subspace fixed. Formally: if f and g are continuous maps from X to Y and K is a subset of X, then we say that f and g are homotopic relative to K if there exists a homotopy between f and g such that for all and Also, if g is a retraction from X to K and f is the identity map, this is known as a strong deformation retract of X to K. When K is a point, the term pointed homotopy is used.
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Reconstruction of 28x28pixel images by an Autoencoder with a code size of two (two-units hidden layer) and the reconstruction from the first two Principal Components of PCA. Images come from the Fashion MNIST dataset. If linear activations are used, or only a single sigmoid hidden layer, then the optimal solution to an autoencoder is strongly related to principal component analysis (PCA). The weights of an autoencoder with a single hidden layer of size p (where p is less than the size of the input) span the same vector subspace as the one spanned by the first p principal components, and the output of the autoencoder is an orthogonal projection onto this subspace.
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The Ancient Minister's advance prompts the heroes to ally and attempt to repel the enemy, while villains harvest the power of the allied characters by using dark cannons to convert them into trophies and using Shadow Bugs on some of them to create powerful doppelgängers. King Dedede begins independently gathering some fallen fighters' trophies, placing golden brooches on them. The Ancient Minister is revealed as a subordinate to Ganondorf, Bowser, and Wario, who are under orders from Master Hand to draw the world into Subspace. Wario, who had stolen Ness with his dark cannon, is turned into a trophy by a Pokémon Trainer and Lucas, and is then helplessly sucked in by a Subspace Bomb.
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If each of the bonding maps f_{i}^{j} is an embedding of TVSs onto proper vector subspaces and if the system is directed by with its natural ordering, then the resulting limit is called a strict (countable) direct limit. In such a situation we may assume without loss of generality that each is a vector subspace of and that the subspace topology induced on by is identical to the original topology on . In the category of locally convex topological vector spaces, the topology on a strict inductive limit of Fréchet spaces can be described by specifying that an absolutely convex subset is a neighborhood of if and only if is an absolutely convex neighborhood of in for every .
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Frederick Gehring showed in 1977 that U is the interior of the closed subset of Schwarzian derivatives of univalent functions. For a compact Riemann surface S of genus greater than 1, its universal covering space is the unit disc D on which its fundamental group Γ acts by Möbius transformations. The Teichmüller space of S can be identified with the subspace of the universal Teichmüller space invariant under Γ. The holomorphic functions g have the property that :g(z) \, dz^2 is invariant under Γ, so determine quadratic differentials on S. In this way, the Teichmüller space of S is realized as an open subspace of the finite-dimensional complex vector space of quadratic differentials on S.
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In topology, a branch of mathematics, a topological space X is said to be simply connected at infinity if for any compact subset C of X, there is a compact set D in X containing C so that the induced map : \pi_1(X-D) \to \pi_1(X-C) is the zero map. Intuitively, this is the property that loops far away from a small subspace of X can be collapsed, no matter how bad the small subspace is. The Whitehead manifold is an example of a 3-manifold that is contractible but not simply connected at infinity. Since this property is invariant under homeomorphism, this proves that the Whitehead manifold is not homeomorphic to R3.
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The approach of Cartan and Weyl, using connection 1-forms on the frame bundle of , gives a third way to understand the Riemannian connection. They noticed that parallel transport dictates that a path in the surface be lifted to a path in the frame bundle so that its tangent vectors lie in a special subspace of codimension one in the three- dimensional tangent space of the frame bundle. The projection onto this subspace is defined by a differential 1-form on the orthonormal frame bundle, the connection form. This enabled the curvature properties of the surface to be encoded in differential forms on the frame bundle and formulas involving their exterior derivatives.
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The explanation of the forest method's resistance to overtraining can be found in Kleinberg's theory of stochastic discrimination. The early development of Breiman's notion of random forests was influenced by the work of Amit and Geman who introduced the idea of searching over a random subset of the available decisions when splitting a node, in the context of growing a single tree. The idea of random subspace selection from Ho was also influential in the design of random forests. In this method a forest of trees is grown, and variation among the trees is introduced by projecting the training data into a randomly chosen subspace before fitting each tree or each node.
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We call -complete or (if no ambiguity can arise) weakly- complete if is a complete vector space. Note that there exist Banach spaces that are not weakly-complete (despite being complete). In particular, if is a vector subspace of such that separates points of , then is complete if and only if .
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Note that if a linear operator T : X → Y is almost open then because T(X) is a vector subspace of Y that contains a neighborhood of 0 in Y, T : X → Y is necessarily surjective. For this reason many authors require surjectivity as part of the definition of "almost open".
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Y. Hua, Y. Xiang, T. Chen, K. Abed-Meraim and Y. Miao, "A new look at the power method for fast subspace tracking," Digital Signal Processing, Vol. 9. pp. 297-314, 1999. Y. Hua and W. Liu, "Generalized Karhunen-Loeve Transform", IEEE Signal Processing Letters, Vol. 5, No. 6, pp.
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The concept of "space fold" used in the Japanese Macross franchise is actually hyperspace travel, which is carried out by first swapping the location of the spacecraft with Super Dimension space or subspace ("fold in"), and then swapping the Super Dimension space with the space at the destination ("fold out").
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In the case where there are more than two classes, the analysis used in the derivation of the Fisher discriminant can be extended to find a subspace which appears to contain all of the class variability.Garson, G. D. (2008). Discriminant function analysis. . This generalization is due to C. R. Rao.
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In mathematics, particularly topology, a topological space X is locally regular if intuitively it looks locally like a regular space. More precisely, a locally regular space satisfies the property that each point of the space belongs to an open subset of the space that is regular under the subspace topology.
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Conic optimization is a subfield of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone. The class of conic optimization problems includes some of the most well known classes of convex optimization problems, namely linear and semidefinite programming.
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In mathematics, or more specifically in spectral theory, the Riesz projector is the projector onto the eigenspace corresponding to a particular eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an isolated part of the spectrum). It was introduced by Frigyes Riesz in 1912.
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In the theory of complex-analytic spaces, the Oka-Cartan theorem states that a closed subset A of a complex space is analytic if and only if the ideal sheaf of functions vanishing on A is coherent. This ideal sheaf also gives A the structure of a reduced closed complex subspace.
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The being would be able to discern all points in a 3-dimensional subspace simultaneously, including the inner structure of solid 3-dimensional objects, things obscured from human viewpoints in three dimensions on two-dimensional projections. Brains receive images in two dimensions and use reasoning to help picture three-dimensional objects.
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To illustrate an actual application of the Hahn–Banach theorem, we will now prove a result that follows almost entirely from the Hahn–Banach theorem. One may use the above result to show that every closed vector subspace of is complemented and either finite dimensional or else TVS-isomorphic to .
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Enflo's long "manuscript had a world-wide circulation among mathematicians"Yadav, page 292. and some of its ideas were described in publications besides Enflo (1976).For example, Radjavi and Rosenthal (1982). Enflo's works inspired a similar construction of an operator without an invariant subspace for example by Beauzamy, who acknowledged Enflo's ideas.
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In topology, a branch of mathematics, an excisive triad is a triple (X; A, B) of topological spaces such that A, B are subspaces of X and X is the union of the interior of A and the interior of B. Note B is not required to be a subspace of A.
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Paibok somehow escaped from subspace and struck a deal with Centaurian scientists to enhance his powers. Their treatment succeeded — but also affected his appearance, leaving him in a cadaverous, zombie-like form. Still viewed as a traitor, he fled to Earth and assembled a band of Skrull renegades.Thing: Freakshow #3 (October, 2002).
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However, once you know the first n − 1 components, the constraint tells you the value of the nth component. Therefore, this vector has n − 1 degrees of freedom. Mathematically, the first vector is the orthogonal, or least-squares, projection of the data vector onto the subspace spanned by the vector of 1's.
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In 2014, Vidal was elected IEEE Fellow for contributions to subspace clustering and motion segmentation in computer vision. In 2016, Vidal was elected IAPR fellow for contributions to computer vision and pattern recognition. In 2020, Vidal was inducted into AIMBE College of Fellows for outstanding contributions to medical image analysis and medical robotics.
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In mathematics, particularly topology, a comb space is a particular subspace of \R^2 that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb space. The deleted comb space is a variation on the comb space.
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An algebraic proof, based on the variational interpretation of eigenvalues, has been published in Magnus' Matrix Differential Calculus with Applications in Statistics and Econometrics. From the geometric point of view, B'AB can be considered as the orthogonal projection of A onto the linear subspace spanned by B, so the above results follow immediately.
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V is not a geometry, as the closure of any nontrivial vector is a subspace of size at least 2. The associated geometry of a \kappa-dimensional vector space over F is the (\kappa-1)-dimensional projective space over F. It is easy to see that this pregeometry is a projective geometry.
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It follows that an open connected subspace of a locally path connected space is necessarily path connected.Willard, Theorem 27.5, p. 199 Moreover, if a space is locally path connected, then it is also locally connected, so for all x in X, C_x is connected and open, hence path connected, i.e., C_x = PC_x.
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In terms of the dual space, it is quite evident why dimensions add. The subspaces can be defined by the vanishing of a certain number of linear functionals, which if we take to be linearly independent, their number is the codimension. Therefore, we see that U is defined by taking the union of the sets of linear functionals defining the Wi. That union may introduce some degree of linear dependence: the possible values of j express that dependence, with the RHS sum being the case where there is no dependence. This definition of codimension in terms of the number of functions needed to cut out a subspace extends to situations in which both the ambient space and subspace are infinite dimensional.
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Let \rho be a representation i.e. a homomorphism \rho: G\to GL(V) of a group G where V is a vector space over a field F. If we pick a basis B for V, \rho can be thought of as a function (a homomorphism) from a group into a set of invertible matrices and in this context is called a matrix representation. However, it simplifies things greatly if we think of the space V without a basis. A linear subspace W\subset V is called G-invariant if \rho(g)w\in W for all g\in G and all w\in W. The restriction of \rho to a G-invariant subspace W\subset V is known as a subrepresentation.
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Two years of playing for free became problematic as many players refused to pay for a game that they had beta tested for two years, and instead opted to pirate the software. SubSpace server software being distributed with the commercial release of the game allowed users to host their own servers on their own computers, enabling them to preserve the game. Once VIE went under in 1998, many of its remaining US assets were purchased by Electronic Arts, but the SubSpace license was not. This caused all of the commercially hosted servers, including the official VIE servers, to eventually go offline permanently, and independent user-run servers became the only choice for hosting zones, including original zones previously hosted by VIE.
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In mathematics, the Lévy–Steinitz theorem identifies the set of values to which rearrangements of an infinite series of vectors in Rn can converge. It was proved by Paul Lévy in his first published paper when he was 19 years old.. In 1913 Ernst Steinitz filled in a gap in Lévy's proof and also proved the result by a different method.. In an expository article, Peter Rosenthal stated the theorem in the following way.. : The set of all sums of rearrangements of a given series of vectors in a finite-dimensional real Euclidean space is either the empty set or a translate of a subspace (i.e., a set of the form v + M, where v is a given vector and M is a linear subspace).
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In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is associated to a contact manifold and one of its Legendrian submanifolds. It is a part of a more general invariant known as symplectic field theory, and is defined using pseudoholomorphic curves.
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Even though the formal analysis by Gander and Vandewalle covers only linear problems with constant coefficients, the problem also arises when Parareal is applied to the nonlinear Navier–Stokes equations when the viscosity coefficient becomes too small and the Reynolds number too large. Different approaches exist to stabilise Parareal, one being Krylov-subspace enhanced Parareal.
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Other structures considered on include the one of a pseudo-Euclidean space, symplectic structure (even ), and contact structure (odd ). All these structures, although can be defined in a coordinate-free manner, admit standard (and reasonably simple) forms in coordinates. is also a real vector subspace of which is invariant to complex conjugation; see also complexification.
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The subset of the space of all functions from R to R consisting of (sufficiently differentiable) functions that satisfy a certain differential equation is a subspace of RR if the equation is linear. This is because differentiation is a linear operation, i.e., (a f + b g)′ = a f′ + b g′, where ′ is the differentiation operator.
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Disillusion was founded in 1994 by Andy Schmidt, Tobias Spier, Alex Motz, Markus Espenhain and Jan Stölzel. The band played Thrash Metal. Despite the departure of bassist Markus Espenhain in early 1996, the first demo Subspace Insanity was recorded in March the same year. 1997 followed Red, where the band already developed their own style.
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The Theory of Functional Connections (TFC) is a mathematical framework generalizing interpolation. TFC derives analytical functionals representing all possible functions subject to a set of constraints. These functionals restrict the whole space of functions to just the subspace that fully satisfies the constraints. Using these functionals, constrained optimization problems are transformed into unconstrained problems.
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The dihedral angle between two non-parallel hyperplanes of a Euclidean space is the angle between the corresponding normal vectors. The product of the transformations in the two hyperplanes is a rotation whose axis is the subspace of codimension 2 obtained by intersecting the hyperplanes, and whose angle is twice the angle between the hyperplanes.
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It is simple linear algebra to show that GL4 acts transitively on those. We can parameterize them by line co-ordinates: these are the 2×2 minors of the 4×2 matrix with columns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the line geometry of Julius Plücker.
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In functional analysis, a reflexive operator algebra A is an operator algebra that has enough invariant subspaces to characterize it. Formally, A is reflexive if it is equal to the algebra of bounded operators which leave invariant each subspace left invariant by every operator in A. This should not be confused with a reflexive space.
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In mathematics, specifically in spectral theory, an eigenvalue of a closed linear operator is called normal if the space admits a decomposition into a direct sum of a finite-dimensional generalized eigenspace and an invariant subspace where A-\lambda I has a bounded inverse. The set of normal eigenvalues coincides with the discrete spectrum.
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Here is an example of the technique. Consider the problem of determining the Euler characteristic of the Grassmannian of -dimensional subspaces of . Fix a -dimensional subspace and consider the partition of into those -dimensional subspaces of that contain and those that do not. The former is and the latter is a -dimensional vector bundle over .
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The editors were Michael Wakin, Christopher Rozell, Mark Davenport and Jason Laska. After almost two years since the inaugural issue, the second issue was published in June 2011 and contains topics such as subspace classification and distributions of pseudoprimes. , the original website is no longer online, but an archival copy is hosted on GitHub.
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An affine space need not be included into a linear space, but is isomorphic to an affine subspace of a linear space. All affine spaces are mutually isomorphic. In the words of John Baez, "an affine space is a vector space that's forgotten its origin". In particular, every linear space is also an affine space.
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This domain decomposition condenses the error of the FSI problem into a subspace related to the interface. The FSI problem can hence be written as either a root finding problem or a fixed point problem, with the interface’s position as unknowns. Interface Newton–Raphson methods solve this root-finding problem with Newton–Raphson iterations, e.g.
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If a correlation φ is an involution (that is, two applications of the correlation equals the identity: for all points P) then it is called a polarity. Polarities of projective spaces lead to polar spaces, which are defined by taking the collection of all subspace which are contained in their image under the polarity.
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In mathematical representation theory, a (Zuckerman) translation functor is a functor taking representations of a Lie algebra to representations with a possibly different central character. Translation functors were introduced independently by and . Roughly speaking, the functor is given by taking a tensor product with a finite-dimensional representation, and then taking a subspace with some central character.
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A space with quadratic form is split (or metabolic) if there is a subspace which is equal to its own orthogonal complement; equivalently, the index of isotropy is equal to half the dimension. The hyperbolic plane is an example, and over a field of characteristic not equal to 2, every split space is a direct sum of hyperbolic planes.
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The integers k,s,m and the real numbers \sigma_i are uniquely determined. Note that k+s+m=d. The factor I_m \oplus 0_s corresponds to the maximal invariant subspace on which P acts as an orthogonal projection (so that P itself is orthogonal if and only if k=0) and the \sigma_i-blocks correspond to the oblique components.
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A topological space X is called countably generated if V is closed in X whenever for each countable subspace U of X the set V \cap U is closed in U. Equivalently, X is countably generated if and only if the closure of any A \subset X equals the union of closures of all countable subsets of A.
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Any locally convex space having the extension property is injective. If is an injective Banach space, then for every Banach space , every continuous linear operator from a vector subspace of into has a continuous linear extension to all of . In 1953, Alexander Grothendieck that any Banach space with the extension property is either finite-dimensional or else not separable.
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Tabuu is the villain of Super Smash Bros. Brawl story mode, The Subspace Emissary. He is a humanoid apparition composed of pure energy, with a single eye-shaped object located where a person's stomach would be. He can conjure several weapons for use in battle, including a rapier and large chakram; change his size at will, and teleport.
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In mathematics, a scattered space is a topological space X that contains no nonempty dense-in-itself subset.Steen & Seebach, p. 33Engelking, p. 59 Equivalently, every nonempty subset A of X contains a point isolated in A. A subset of a topological space is called a scattered set if it is a scattered space with the subspace topology.
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A derivation on A is a map D with the property :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 algebra.
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Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive. If is a dense proper vector subspace of a reflexive Banach space then is a normed space that not semi- reflexive but its strong dual space is a reflexive Banach space. There exists a semi-reflexive countably barrelled space that is not barrelled.
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The basic idea has been extended to hierarchical clustering by the OPTICS algorithm. DBSCAN is also used as part of subspace clustering algorithms like PreDeCon and SUBCLU. HDBSCAN is a hierarchical version of DBSCAN which is also faster than OPTICS, from which a flat partition consisting of the most prominent clusters can be extracted from the hierarchy.
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Quoted in D. N. McCloskey, The Bourgeois Virtues (2006) p. 190 Among other contexts, ego reduction has been seen as a goal in Alcoholics Anonymous; as a part of BDSM play,B. A. Firestein, Becoming Visible (2007) p. 365 providing a means of entering "subspace"; and as a way of attaining religious humility and freedom from desire in Buddhism.
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Analogies for the plane would be lifting a string up off the surface, or removing a dot from inside a circle. In fact, in four dimensions, any non-intersecting closed loop of one- dimensional string is equivalent to an unknot. First "push" the loop into a three-dimensional subspace, which is always possible, though technical to explain.
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In mathematics, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander, and subsequently further developed, particularly by Pavel Alexandrov and Lev Pontryagin. It applies to the homology theory properties of the complement of a subspace X in Euclidean space, a sphere, or other manifold. It is generalized by Spanier–Whitehead duality.
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41 ;Discrete topology: See discrete space. ;Disjoint union topology: See Coproduct topology. ;Dispersion point: If X is a connected space with more than one point, then a point x of X is a dispersion point if the subspace X − {x} is hereditarily disconnected (its only connected components are the one- point sets). ;Distance: See metric space.
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If not properly disposed of, it may destroy subspace and render warp travel impossible. In Star Trek: Voyager, during the episode The Omega Directive, Voyager encounters Omega particles and Captain Janeway must comply with the Omega Directive and destroy the particles. Later in the episode, they spontaneously stabilize for a brief moment before they are destroyed.
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The approximating operator that appears in stationary iterative methods can also be incorporated in Krylov subspace methods such as GMRES (alternatively, preconditioned Krylov methods can be considered as accelerations of stationary iterative methods), where they become transformations of the original operator to a presumably better conditioned one. The construction of preconditioners is a large research area.
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A two-dimensional Poincaré section of the forced Duffing equation In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensional subspace, called the Poincaré section, transversal to the flow of the system. More precisely, one considers a periodic orbit with initial conditions within a section of the space, which leaves that section afterwards, and observes the point at which this orbit first returns to the section. One then creates a map to send the first point to the second, hence the name first recurrence map. The transversality of the Poincaré section means that periodic orbits starting on the subspace flow through it and not parallel to it.
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One way to visualize this identification with the real numbers as usually viewed is that the equivalence class consisting of those Cauchy sequences of rational numbers that "ought" to have a given real limit is identified with that real number. The truncations of the decimal expansion give just one choice of Cauchy sequence in the relevant equivalence class. For a prime p, the p-adic numbers arise by completing the rational numbers with respect to a different metric. If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, the result is a Hilbert space containing the original space as a dense subspace.
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Now define an action of on the , and the linear subspace they span in , given by The last equality in , which follows from and the property of the gamma matrices, shows that the constitute a representation of since the commutation relations in are exactly those of . The action of can either be thought of as six-dimensional matrices multiplying the basis vectors , since the space in spanned by the is six-dimensional, or be thought of as the action by commutation on the . In the following, The and the are both (disjoint) subsets of the basis elements of Cℓ4(C), generated by the four-dimensional Dirac matrices in four spacetime dimensions. The Lie algebra of is thus embedded in Cℓ4(C) by as the real subspace of Cℓ4(C) spanned by the .
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Given a Gale diagram of a polytope, that is, a set of n unit vectors in an (n-d-1)-dimensional space, one can choose a (n-d-2)-dimensional subspace S through the origin that avoids all of the vectors, and a parallel subspace S' that does not pass through the origin. Then, a central projection from the origin to S' will produce a set of (n-d-2)-dimensional points. This projection loses the information about which vectors lie above S and which lie below it, but this information can be represented by assigning a sign (positive, negative, or zero) or equivalently a color (black, white, or gray) to each point. The resulting set of signed or colored points is the affine Gale diagram of the given polytope.
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This is the condition that fails to hold for non-Abelian gauge groups. If one ignores the problem and attempts to use the Feynman rules obtained from "naive" functional quantization, one finds that one's calculations contain unremovable anomalies. The problem of perturbative calculations in QCD was solved by introducing additional fields known as Faddeev–Popov ghosts, whose contribution to the gauge-fixed Lagrangian offsets the anomaly introduced by the coupling of "physical" and "unphysical" perturbations of the non-Abelian gauge field. From the functional quantization perspective, the "unphysical" perturbations of the field configuration (the gauge transformations) form a subspace of the space of all (infinitesimal) perturbations; in the non-Abelian case, the embedding of this subspace in the larger space depends on the configuration around which the perturbation takes place.
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A subspace of a Hilbert space is a Hilbert space if it is closed. In summary, the set of all possible normalizable wave functions for a system with a particular choice of basis, together with the null vector, constitute a Hilbert space. Not all functions of interest are elements of some Hilbert space, say . The most glaring example is the set of functions .
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A subalgebra of an algebra over a commutative ring or field is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. to associative algebras or to Lie algebras.
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Its extent is the union over Φ. A paracompactifying family of supports that satisfies further that any Y in Φ is, with the subspace topology, a paracompact space; and has some Z in Φ which is a neighbourhood. If X is a locally compact space, assumed Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying.
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Original characters were created for the story mode. Pictured is the Ancient Minister, who starts as an antagonist but later becomes the playable fighter R.O.B. Super Smash Bros. Brawl features a new Adventure mode titled "The Subspace Emissary", abbreviated to "SSE". This mode features a unique storyline and numerous side-scrolling levels and bosses, as well as cutscenes explaining the plot.
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In 2 dimensions, a point reflection is a 180 degree rotation. Reflection symmetry can be generalized to other isometries of -dimensional space which are involutions, such as : in a certain system of Cartesian coordinates. This reflects the space along an -dimensional affine subspace. If = , then such a transformation is known as a point reflection, or an inversion through a point.
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In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even- dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the manifold is a multiple of four. It is an instance of a Dirac-type operator.
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It may have a rather poor accuracy, but the method is iterative and improves with the steps below. Decomposition of the system into left and right blocks, according to DMRG. The candidate ground state that has been found is projected into the Hilbert subspace for each block using a density matrix, hence the name. Thus, the relevant states for each block are updated.
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Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense. The lemma may also be called the Riesz lemma or Riesz inequality. It can be seen as a substitute for orthogonality when one is not in an inner product space.
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Ali Naci Akansu (born May 6, 1958) is a Turkish-American electrical engineer and scientist. He is best known for his seminal contributions to the theory and applications of linear subspace methods including sub-band and wavelet transforms, particularly the binomial QMFA.N. Akansu, An Efficient QMF-Wavelet Structure (Binomial-QMF Daubechies Wavelets), Proc. 1st NJIT Symposium on Wavelets, April 1990.
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In terms of the affine subspace , an isotropic line through the origin is :x_2 = \pm i x_1 . In projective geometry, the isotropic lines are the ones passing through the circular points at infinity. In the real orthogonal geometry of Emil Artin, isotropic lines occur in pairs: :A non-singular plane which contains an isotropic vector shall be called a hyperbolic plane.
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The formalism does not allow to determine what the relevant variables are, these can typiclly be obtained from the properties of the system. The observables describing the system form a Hilbert space. The projection operator then projects the dynamics onto the subspace spanned by the relevant variables..Robert Zwanzig Nonequilibrium Statistical Mechanics 3rd ed., Oxford University Press, New York, 2001, S.144 ff.
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A matrix or, equivalently, a linear operator T on a finite- dimensional vector space V is called semi-simple if every T-invariant subspace has a complementary T-invariant subspace.Lam (2001), [ p. 39] This is equivalent to the minimal polynomial of T being square-free. For vector spaces over an algebraically closed field F, semi-simplicity of a matrix is equivalent to diagonalizability.
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Super-resolution imaging (SR) is a class of techniques that enhance (increase) the resolution of an imaging system. In optical SR the diffraction limit of systems is transcended, while in geometrical SR the resolution of digital imaging sensors is enhanced. In some radar and sonar imaging applications (e.g. magnetic resonance imaging (MRI), high-resolution computed tomography), subspace decomposition-based methods (e.g.
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This is equivalent to requiring that every open subspace be paracompact. Tychonoff's theorem (which states that the product of any collection of compact topological spaces is compact) does not generalize to paracompact spaces in that the product of paracompact spaces need not be paracompact. However, the product of a paracompact space and a compact space is always paracompact. Every metric space is paracompact.
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In mathematics, specifically in control theory, subspace identification (SID) aims at identifying linear time invariant (LTI) state space models from input- output data. SID does not require that the user parametrizes the system matrices before solving a parametric optimization problem and, as a consequence, SID methods do not suffer from problems related to local minima that often lead to unsatisfactory identification results.
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Given a Hilbert space H and a set S of mutually orthogonal vectors in H, we can take the smallest closed linear subspace V of H containing S. Then S will be an orthogonal basis of V; which may of course be smaller than H itself, being an incomplete orthogonal set, or be H, when it is a complete orthogonal set.
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The Federation starship Enterprise arrives at a subspace communications relay station near the Klingon border on a resupply mission. However, when an away team boards the relay there is no sign of the two officers assigned there. Lieutenant Aquiel Uhnari, Lieutenant Rocha, and the station's shuttlecraft are missing. While searching the station, the away team finds the dog that belongs to Lieutenant Uhnari.
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This wraps an object in a kind of subspace bubble, and teleports it to another location using spatial folding. The range was 40,000 light-years. However, the technology was not compatible with the warp core and almost destroyed Voyager when it was used. Three years later, in the episode Vis à Vis, Voyager discovered a stranded spaceship with a coaxial warp drive.
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This would mean that is an embedded non- compact Lie subgroup of the compact group . This is impossible with the subspace topology on since all embedded Lie subgroups of a Lie group are closed If were closed, it would be compact, Lemma A.17 (c). Closed subsets of compact sets are compact. and then would be compact, Lemma A.17 (a).
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The kernel of a matrix A over a field K is a linear subspace of Kn. That is, the kernel of A, the set Null(A), has the following three properties: # Null(A) always contains the zero vector, since . # If and , then . This follows from the distributivity of matrix multiplication over addition. # If and c is a scalar , then , since .
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In mathematics, a linear map is a mapping between two modules (including vector spaces) that preserves the operations of addition and scalar multiplication. By studying the linear maps between two modules one can gain insight into their structures. If the modules have additional structure, like topologies or bornologies, then one can study the subspace of linear maps that preserve this structure.
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In systems with limited control authority, it is often no longer possible to move any initial state to any final state inside the controllable subspace. This phenomenon is caused by constraints on the input that could be inherent to the system (e.g. due to saturating actuator) or imposed on the system for other reasons (e.g. due to safety-related concerns).
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These numbers occur also in other, related areas. In matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real n×n matrices, for which each non-zero matrix is a similarity transformation, i.e. a product of an orthogonal matrix and a scalar matrix. In quadratic forms, the Hurwitz problem asks for multiplicative identities between quadratic forms.
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The element of this subspace that has the smallest length (that is, is closest to the origin) is the answer we are looking for. It can be found by taking an arbitrary member of and projecting it orthogonally onto the orthogonal complement of the kernel of . This description is closely related to the Minimum norm solution to a linear system.
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Every -dimensional subspace of determines an -dimensional quotient space of . This gives the natural short exact sequence: :. Taking the dual to each of these three spaces and linear transformations yields an inclusion of in with quotient : :. Using the natural isomorphism of a finite-dimensional vector space with its double dual shows that taking the dual again recovers the original short exact sequence.
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The one-sided limit to a point p corresponds to the general definition of limit, with the domain of the function restricted to one side, by either allowing that the function domain is a subset of the topological space, or by considering a one-sided subspace, including p. Alternatively, one may consider the domain with a half-open interval topology.
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In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, is a bounded linear operator on the space L2[0,1] of complex-valued square-integrable functions on the interval [0,1]. On the subspace C[0,1] of continuous functions it represents indefinite integration. It is the operator corresponding to the Volterra integral equations.
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Let V be a \kappa-dimensional affine space over a field F. Given a set define its closure to be its affine hull (i.e. the smallest affine subspace containing it). This forms a homogeneous (\kappa+1)-dimensional geometry. An affine space is not modular (for example, if X and Y be parallel lines then the formula in the definition of modularity fails).
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In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers.. A common point of confusion is that while a complex line has dimension one over C (hence the term "line"), it has dimension two over the real numbers R, and is topologically equivalent to a real plane, not a real line..
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In terms of Ramona's height, O'Malley said that it was "supposed to be tallish" meaning around or . Ramona's subspace bag, like her hair, changes colour periodically. Throughout her dating history, Ramona had one "non-evil" ex, Doug, who O'Malley named after one of his real life friends. Ramona Flowers was ranked 70th in Comics Buyer's Guides "100 Sexiest Women in Comics" list.
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In 2003 Universal Poplab made their live debut at the Scandinavian Alternative Music Awards (SAMA), followed by a number of summer gigs. The SubSpace Encounter in Malmö and the support show for German synth-pop stars Melotron in Gothenburg. In 2004 their debut album Universal Poplab is released by independent record label SubSpace Communications. On the album, one can find a wide spectre of music, from gentle love songs as ”Dice Roller” and ”Any More Than This” weaved between electrifying tracks as “I Can’t Help Myself” (with Paul as the lead vocalist) and “Days Astray” to floor fillers as ”Extasy” and the first single, the melancholic yet bouncy “Casanova Fall”. The record also includes a cover version of Morrissey’s ”We Hate It When Our Friends Become Successful” featuring the backing vocals of one of Sweden's most famous pop stars, Håkan Hellström.
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In mathematics, the usual convention for any Riemannian manifold is to use a positive-definite metric tensor (meaning that after diagonalization, elements on the diagonal are all positive). In theoretical physics, spacetime is modeled by a pseudo-Riemannian manifold. The signature counts how many time-like or space-like characters are in the spacetime, in the sense defined by special relativity: as used in particle physics, the metric has an eigenvalue on the time-like subspace, and its mirroring eigenvalue on the space-like subspace. In the specific case of the Minkowski metric, : ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 , the metric signature is (1, 3, 0)^+ or (+, −, −, −) if its eigenvalue is defined in the time direction, or (1, 3, 0)^- or (−, +, +, +) if the eigenvalue is defined in the three spatial directions x, y and z.
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In mathematics, more precisely in functional analysis, an energetic space is, intuitively, a subspace of a given real Hilbert space equipped with a new "energetic" inner product. The motivation for the name comes from physics, as in many physical problems the energy of a system can be expressed in terms of the energetic inner product. An example of this will be given later in the article.
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A subgraph of a graph X is a subspace Y \subseteq X which is also a graph and whose nodes are all contained in the 0-skeleton of X. Y is a subgraph if and only if it consists of vertices and edges from X and is closed. A subgraph T \subseteq X is called a tree iff it is contractible as a topological space.
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To get the Moonshine Module, one takes the fixed point subspace of h in the direct sum of VL and its twisted module. Frenkel, Lepowsky, and Meurman then showed that the automorphism group of the moonshine module, as a vertex operator algebra, is M. Furthermore, they determined that the graded traces of elements in the subgroup 21+24.Co1 match the functions predicted by Conway and Norton ().
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The matching requirement for the function spaces is a natural one. The Hilbert space property of the abstract state space was originally extracted from the observation that the function spaces forming normalizable solutions to the Schrödinger equation are Hilbert spaces. The space is a Hilbert space, with inner product presented later. The function space of the example of the figure is a subspace of .
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The algorithm is written using matrix notation (1 based arrays instead of 0 based). 5\. When implementing the algorithm, the part specified using matrix notation must be performed simultaneously. 6\. This implementation does not correctly account for the case in which one dimension is an independent subspace. For example, if given a diagonal matrix, the above implementation will never terminate, as none of the eigenvalues will change.
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The Hawaiian earring is not semi-locally simply connected. A simple example of a space that is not semi-locally simply connected is the Hawaiian earring: the union of the circles in the Euclidean plane with centers (1/n, 0) and radii 1/n, for n a natural number. Give this space the subspace topology. Then all neighborhoods of the origin contain circles that are not nullhomotopic.
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Given a subspace , the quotient space with the usual quotient topology is a Hausdorff topological vector space if and only if M is closed.In particular, is Hausdorff if and only if the set {0} is closed (i.e., is a T1 space). This permits the following construction: given a topological vector space (that is probably not Hausdorff), form the quotient space where M is the closure of {0}.
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Also known as Megha or Meg from Alcorn, contributing to Stability and Peace over the Planet Om, and the Princess/Prayer Maiden of the Planet Anova. Korton, captain of the flying saucer, the fleet communications officer in charge of maintaining subspace communications for the fleet. Esola, captain of the flying saucer Starship #77. Merku, of the planet Alcorn, subcommander of one of the wings of the fleet.
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The surrounding space matters: a set may be nowhere dense when considered as a subset of a topological space , but not when considered as a subset of another topological space . Notably, a set is always dense in its own subspace topology. A countable union of nowhere dense sets is called a meagre set. Meager sets play an important role in the formulation of the Baire category theorem.
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In that case, B may be taken to be the Banach space of continuous functions on [0,T] with the supremum norm. In this case, the measure on B is the Wiener measure describing Brownian motion starting at the origin. The original subspace H\subset B is called the Cameron–Martin space, which forms a set of measure zero with respect to the Wiener measure.
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In linear algebra, if a linear transformation T has an eigenvector v, then the line through 0 and v is an invariant set under T, in which case, the eigenvectors span an invariant subspace which is stable under T. When T is a screw displacement, the screw axis is an invariant line, though if the pitch is non-zero, T has no fixed points.
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However, this drive does not work on the basis of transwarp conduits, as the transwarp drive of the Borg, but is a further development of the conventional warp drive. The mention of a second transwarp technology took place in the episode Descent of the series Star Trek: The Next Generation. A group of renegade Borg used transwarp conduits. These are wormhole-like tunnels through subspace.
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FanFiction.net also hosts one of the longest works of fiction ever written. The Subspace Emissary's Worlds Conquest, a Super Smash Bros. fanfiction written by FanFiction.net user AuraChannelerChris, gained media attention for its length of over four million words at the time of notice, more than three times as long as In Search of Lost Time written by Marcel Proust, and is still being written.
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In terms of 3-dimensional geometric vectors, these affine subspaces are all the "lines" or "planes" parallel to the subspace, which is a line or plane going through the origin. For example, consider the plane ℝ2. If is a line through the origin , then is a subgroup of the abelian group ℝ2. If is in ℝ2, then the coset is a line parallel to and passing through .
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It also influenced how other restarted methods are analyzed. Theoretical results have shown that convergence improves with an increase in the Krylov subspace dimension n. However, an a-priori value of n which would lead to optimal convergence is not known. Recently a dynamic switching strategy has been proposed which fluctuates the dimension n before each restart and thus leads to acceleration in the rate of convergence.
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If A were a subspace of (R, T) containing K, K would have no limit point in A so that A can not be limit point compact. Therefore, A cannot be compact 8\. The quotient space of (R, T) obtained by collapsing K to a point is not Hausdorff. K is distinct from 0, but can't be separated from 0 by disjoint open sets.
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The subalgebra generated by the bivectors is the even subalgebra of the geometric algebra, written . This algebra results from considering all products of scalars and bivectors generated by the geometric product. It has dimension , and contains Λ2ℝn as a linear subspace with dimension (a triangular number). In two and three dimensions the even subalgebra contains only scalars and bivectors, and each is of particular interest.
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Here, she and her faction of Replicators had found a way to "digitally ascend" by downloading their consciousnesses into subspace. They planned to ascend from there, only to fail miserably. Once they determined that they needed organic bodies to ascend, they sought out technology that would allow them to regain solid form. But when they couldn't find the technology they needed, Elizabeth infiltrated the Atlantis system.
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If X is a compact subspace of Euclidean space, the cone on X is homeomorphic to the union of segments from X to any fixed point v ot\in X such that these segments intersect only by v itself. That is, the topological cone agrees with the geometric cone for compact spaces when the latter is defined. However, the topological cone construction is more general.
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29 ;Path-connected: A space X is path-connected if, for every two points x, y in X, there is a path f from x to y, i.e., a path with initial point f(0) = x and terminal point f(1) = y. Every path-connected space is connected. ;Path-connected component: A path-connected component of a space is a maximal nonempty path-connected subspace.
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Gameplay screenshot Shred Nebula takes its inspiration from Asteroids and SubSpace. Like the aforementioned games, the game is a top down shooter in which the player can only shoot in the direction the ship is facing. The game features over 20 ships, each with its own unique speed, durability, and special attacks. In multiplayer, players can choose from a selection of 8 different ships.
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150px Orthographic projection of 10-simplex with 11 vertices, 55 edges. The abstract 11-cell contains the same number of vertices and edges as the 10-dimensional 10-simplex, and contains 1/3 of its 165 faces. Thus it can be drawn as a regular figure in 11-space, although then its hemi-icosahedral cells are skew; that is, each cell is not contained within a flat 3-dimensional subspace.
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In mathematics, S-space is a regular topological space that is hereditarily separable but is not a Lindelöf space. L-space is a regular topological space that is hereditarily Lindelöf but not separable. A space is separable if it has a countable dense set and hereditarily separable if every subspace is separable. It had been believed for a long time that S-space problem and L-space problem are dual, i.e.
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In mathematics, an ω-bounded space is a topological space in which the closure of every countable subset is compact. More generally, if P is some property of subspaces, then a P-bounded space is one in which every subspace with property P has compact closure. Every compact space is ω-bounded, and every ω-bounded space is countably compact. The long line is ω-bounded but not compact.
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The Zwanzig projection operator is a mathematical device used in statistical mechanics. It operates in the linear space of phase space functions and projects onto the linear subspace of "slow" phase space functions. It was introduced by Robert Zwanzig to derive a generic master equation. It is mostly used in this or similar context in a formal way to derive equations of motion for some "slow" collective variables.
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If any basis of (and therefore every basis) has a finite number of elements, is a finite-dimensional vector space. If is a subspace of , then . In the case where is finite-dimensional, the equality of the dimensions implies . If U1 and U2 are subspaces of V, then :\dim(U_1 + U_2) = \dim U_1 + \dim U_2 - \dim(U_1 \cap U_2), where U_1+U_2denotes the span of U_1\cup U_2.
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Hans Peter Schlickewei, Oberwolfach 2007 Hans Peter Schlickewei (born 1947) is a German mathematician, specializing in number theory and, in particular, the theory of transcendental numbers. Schlickewei received his doctorate in 1975 at the University of Freiburg under the supervision of Theodor Schneider. Schlickewei is a professor at the University of Marburg. He proved in 1976 the p-adic generalization of the subspace theorem of Wolfgang M. Schmidt.
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A topological space X is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.
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It is then straightforward to show that contains V and satisfies the above universal property, so that Cl is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra . It also follows from this construction that i is injective. One usually drops the i and considers V as a linear subspace of . The universal characterization of the Clifford algebra shows that the construction of is functorial in nature.
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For a subspace A\subset X, the relative homology Hn(X, A) is understood to be the homology of the quotient of the chain complexes, that is, :H_n(X,A)=H_n(C_\bullet(X)/C_\bullet(A)) where the quotient of chain complexes is given by the short exact sequence :0\to C_\bullet(A) \to C_\bullet(X) \to C_\bullet(X)/C_\bullet(A) \to 0.
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A densely defined linear operator T from one topological vector space, X, to another one, Y, is a linear operator that is defined on a dense linear subspace dom(T) of X and takes values in Y, written T : dom(T) ⊆ X → Y. Sometimes this is abbreviated as T : X → Y when the context makes it clear that X might not be the set- theoretic domain of T.
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Elements of a spin representation are called spinors. They play an important role in the physical description of fermions such as the electron. The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a complexification of the vector representation.
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SubSpace players control one of eight ships which are equipped with weapons and a variety of special abilities. Players interact with each other in zones, which are typically split into multiple arenas. Players in each arena are then divided into teams; friendly ships appear as yellow, while enemy ships appear blue. The keyboard is used exclusively for control of the ship and use of in-game chat functions.
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Many SubSpace players are organized into squads. These squadrons serve the same purpose as clans or teams do in other online games and allow players to cooperate and improve their skills, as well as to become more familiar with fellow players. In addition, many squads compete in competitive leagues hosted by various zones. Dueling is another favorite pastime and many zones have separate arenas for this purpose alone.
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A zone is a server to which players can connect using a client. Perhaps the most attractive feature of SubSpace is the extremely high degree of customization that zone sysops can implement. Almost every element of the game can be replaced, from the ship graphics to colors and sounds. Apart from a few basic settings, many game settings, such as ship speeds, energy levels, and such, can be changed.
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In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every subset of a relatively compact set is relatively compact. Every compact subset of a Hausdorff space is relatively compact.
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Suppose that N is a subfactor of M , and that both are finite von Neumann algebras. The GNS construction produces a Hilbert space L^2(M) acted on by M with a cyclic vector \Omega. Let e_N be the projection onto the subspace N \Omega. Then M and e_N generate a new von Neumann algebra \langle M, e_N \rangle acting on L^2(M) , containing M as a subfactor.
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In numerical linear algebra, the biconjugate gradient stabilized method, often abbreviated as BiCGSTAB, is an iterative method developed by H. A. van der Vorst for the numerical solution of nonsymmetric linear systems. It is a variant of the biconjugate gradient method (BiCG) and has faster and smoother convergence than the original BiCG as well as other variants such as the conjugate gradient squared method (CGS). It is a Krylov subspace method.
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A vector subspace of a Riesz space is called an ideal if it is solid, meaning if for and , we have: implies that . The intersection of an arbitrary collection of ideals is again an ideal, which allows for the definition of a smallest ideal containing some non-empty subset of , and is called the ideal generated by . An Ideal generated by a singleton is called a principal ideal.
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The coordinates (x0, x1, x2) of a point in PG(2,K) are called homogeneous coordinates. Each triple (x0, x1, x2) represents a well-defined point in PG(2,K), except for the triple (0, 0, 0), which represents no point. Each point in PG(2,K), however, is represented by many triples. If K is a topological space, then KP2, inherits a topology via the product, subspace, and quotient topologies.
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Target space is a subspace of free space which denotes where we want the robot to move to. In global motion planning, target space is observable by the robot's sensors. However, in local motion planning, the robot cannot observe the target space in some states. To solve this problem, the robot goes through several virtual target spaces, each of which is located within the observable area (around the robot).
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Another form of transwarp used in Star Trek is called Quantum Slipstream. Similar to the Borg transwarp conduits, the slipstream is a narrowly focused, directed field that is initiated by manipulating the fabric of the space-time continuum using the starship's navigational deflector array. This creates a subspace tunnel, which is projected ahead of the vessel. Once a ship has entered this tunnel, the forces inside propel it at incredible speed.
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This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, hyperplanes may have different properties. For instance, a hyperplane of an n-dimensional affine space is a flat subset with dimension n − 1 and it separates the space into two half spaces. While a hyperplane of an n-dimensional projective space does not have this property.
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Projective hyperplanes, are used in projective geometry. A projective subspace is a set of points with the property that for any two points of the set, all the points on the line determined by the two points are contained in the set. Projective geometry can be viewed as affine geometry with vanishing points (points at infinity) added. An affine hyperplane together with the associated points at infinity forms a projective hyperplane.
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The integration functional defined above defines a linear functional on the subspace of polynomials of degree . If are distinct points in , then there are coefficients for which :I(f) = a_0 f(x_0) + a_1 f(x_1) + \dots + a_n f(x_n) for all . This forms the foundation of the theory of numerical quadrature. This follows from the fact that the linear functionals defined above form a basis of the dual space of .
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Celledoni earned a master's degree at the University of Trieste in 1993. She completed a Ph.D. at the University of Padua in 1997. Her dissertation, Krylov Subspace Methods For Linear Systems Of ODEs, was jointly supervised by Igor Moret and Alfredo Bellen. Before becoming a faculty member at NTNU in 2004, she was a posdoctoral researcher at the University of Cambridge, at the Mathematical Sciences Research Institute, and at NTNU.
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Tychonoff spaces are precisely those spaces that can be embedded in compact Hausdorff spaces. More precisely, for every Tychonoff space X, there exists a compact Hausdorff space K such that X is homeomorphic to a subspace of K. In fact, one can always choose K to be a Tychonoff cube (i.e. a possibly infinite product of unit intervals). Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem.
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Zanardi and Rasetti called these DF states "error avoiding codes". Subsequently, Daniel A. Lidar proposed the title "decoherence-free subspace" for the space in which these DF states exist. Lidar studied the strength of DF states against perturbations and discovered that the coherence prevalent in DF states can be upset by evolution of the system Hamiltonian. This observation discerned another prerequisite for the possible use of DF states for quantum computation.
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In fact, this is another way to state the Lie–Kolchin theorem. Lie's theorem states that any nonzero representation of a solvable Lie algebra on a finite dimensional vector space over an algebraically closed field of characteristic 0 has a one-dimensional invariant subspace. The result for Lie algebras was proved by and for algebraic groups was proved by . The Borel fixed point theorem generalizes the Lie–Kolchin theorem.
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The first algorithm for random decision forests was created by Tin Kam Ho using the random subspace method, which, in Ho's formulation, is a way to implement the "stochastic discrimination" approach to classification proposed by Eugene Kleinberg. An extension of the algorithm was developed by Leo Breiman and Adele Cutler, who registeredU.S. trademark registration number 3185828, registered 2006/12/19. "Random Forests" as a trademark (, owned by Minitab, Inc.).
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Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a contraction semigroup if and only ifEngel and Nagel Theorem II.3.15, Arent et al. Theorem 3.4.5, Staffans Theorem 3.4.8 # D(A) is dense in X, # A is closed, # A is dissipative, and # A − λ0I is surjective for some λ0> 0, where I denotes the identity operator.
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Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a quasi contraction semigroup if and only if # D(A) is dense in X, # A is closed, # A is quasidissipative, i.e. there exists a ω ≥ 0 such that A − ωI is dissipative, and # A − λ0I is surjective for some λ0 > ω, where I denotes the identity operator.
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In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the classification of Lie groups and especially semi-simple Lie groups. Maximal compact subgroups of Lie groups are not in general unique, but are unique up to conjugation – they are essentially unique.
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Similarly, starting from an affine space A, every class of parallel lines can be associated with a point at infinity. The union over all classes of parallels constitute the points of the hyperplane at infinity. Adjoining the points of this hyperplane (called ideal points) to A converts it into an n-dimensional projective space, such as the real projective space . By adding these ideal points, the entire affine space A is completed to a projective space P, which may be called the projective completion of A. Each affine subspace S of A is completed to a projective subspace of P by adding to S all the ideal points corresponding to the directions of the lines contained in S. The resulting projective subspaces are often called affine subspaces of the projective space P, as opposed to the infinite or ideal subspaces, which are the subspaces of the hyperplane at infinity (however, they are projective spaces, not affine spaces).
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Plochingen is about nine kilometres east-southeast of the district town Esslingen am Neckar and in the same direction about 19 kilometres from the state capital Stuttgart. The town is situated on the right banks of the Fils and the outflowing Neckar. In the area of the town, three Natural areas collide, the Foreland of the central Swabian Alb in the southeast, the subspace Schurwald of the natural area Schurwald and Welzheimer Wald in the northeast, and the subspace Nürtinger-Esslinger Neckartal, which is part of the Filder, along the larger of the two rivers in the west. The lowest point in the city area is in the very west at the outflow of the Neckar at 247 metres above sea level, the highest point is in the very north at the White Stone on the Schurwaldkamm at about 448 metres above sea level, from which forest areas cover a large part of the city area.
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Let the field be R again, but now let the vector space V be the Cartesian plane R2. Take W to be the set of points (x, y) of R2 such that x = y. Then W is a subspace of R2. Example II Illustrated Proof: #Let p = (p1, p2) and q = (q1, q2) be elements of W, that is, points in the plane such that p1 = p2 and q1 = q2. Then p + q = (p1+q1, p2+q2); since p1 = p2 and q1 = q2, then p1 + q1 = p2 + q2, so p + q is an element of W. #Let p = (p1, p2) be an element of W, that is, a point in the plane such that p1 = p2, and let c be a scalar in R. Then cp = (cp1, cp2); since p1 = p2, then cp1 = cp2, so cp is an element of W. In general, any subset of the real coordinate space Rn that is defined by a system of homogeneous linear equations will yield a subspace.
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Smith tested the car by driving it on a back road at illegally high speeds with their heads pressed tightly against the roof columns to listen for chassis squeaks by bone conduction—a process apparently improvised on the spot. In his nonseries novels written after his professional retirement, Galaxy Primes, Subspace Explorers, and Subspace Encounter, E. E. Smith explores themes of telepathy and other mental abilities collectively called "psionics", and of the conflict between libertarian and socialistic/communistic influences in the colonization of other planets. Galaxy Primes was written after critics such as Groff Conklin and P. Schuyler Miller in the early '50s accused his fiction of being passé, and he made an attempt to do something more in line with the concepts about which Astounding editor John W. Campbell encouraged his writers to make stories. Despite this, it was rejected by Campbell, and it was eventually published by Amazing Stories in 1959.
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With the recent need to process larger and larger data sets (also known as big data), the willingness to trade semantic meaning of the generated clusters for performance has been increasing. This led to the development of pre-clustering methods such as canopy clustering, which can process huge data sets efficiently, but the resulting "clusters" are merely a rough pre- partitioning of the data set to then analyze the partitions with existing slower methods such as k-means clustering. For high-dimensional data, many of the existing methods fail due to the curse of dimensionality, which renders particular distance functions problematic in high-dimensional spaces. This led to new clustering algorithms for high-dimensional data that focus on subspace clustering (where only some attributes are used, and cluster models include the relevant attributes for the cluster) and correlation clustering that also looks for arbitrary rotated ("correlated") subspace clusters that can be modeled by giving a correlation of their attributes.
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In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V. The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can be given in coordinates as the solution of a single (due to the "codimension 1" constraint) algebraic equation of degree 1\. If V is a vector space, one distinguishes "vector hyperplanes" (which are linear subspaces, and therefore must pass through the origin) and "affine hyperplanes" (which need not pass through the origin; they can be obtained by translation of a vector hyperplane). A hyperplane in a Euclidean space separates that space into two half spaces, and defines a reflection that fixes the hyperplane and interchanges those two half spaces.
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In general topology, an embedding is a homeomorphism onto its image.. . More explicitly, an injective continuous map f : X \to Y between topological spaces X and Y is a topological embedding if f yields a homeomorphism between X and f(X) (where f(X) carries the subspace topology inherited from Y). Intuitively then, the embedding f : X \to Y lets us treat X as a subspace of Y. Every embedding is injective and continuous. Every map that is injective, continuous and either open or closed is an embedding; however there are also embeddings which are neither open nor closed. The latter happens if the image f(X) is neither an open set nor a closed set in Y. For a given space Y, the existence of an embedding X \to Y is a topological invariant of X. This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.
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The generalized finite element method (GFEM) uses local spaces consisting of functions, not necessarily polynomials, that reflect the available information on the unknown solution and thus ensure good local approximation. Then a partition of unity is used to “bond” these spaces together to form the approximating subspace. The effectiveness of GFEM has been shown when applied to problems with domains having complicated boundaries, problems with micro- scales, and problems with boundary layers.
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These two transformations map any two points at unit distance from each other to two different points in the dense subspace, and from there map them to two different points of the simplex, which are necessarily at unit distance apart. Therefore, their composition preserves unit distances. However, it is not an isometry, because it maps every pair of points, no matter their original distance, either to the same point or to a unit distance.
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In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line. A hyperplane is a subspace of one dimension less than the dimension of the full space.
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Her Final Smash in Brawl is unique in that it does not kill opponents outright. Instead, it puts them to sleep and spawns peaches across the battlefield that restore Peach's health. Additionally, in the Subspace Emissary story mode, Princess Peach is a very prominent character, being present throughout most of the storyline. Peach returned in the recent installments released for Nintendo 3DS and Wii U, as well as Super Smash Bros. Ultimate.
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They work by delivering destructive subspace compression pulse explosion. Upon detonation the pulse is delivered in asymmetric superposition of multiple phase states. Since the shields can block only one subcomponent of the pulse the remaining majority is delivered to the target. On the top of it every torpedo has different transphasic configuration generated randomly by a dissonant feedback effect to prevent Borg or any other enemy to predict the configuration of the phase states.
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Along with Melee Adventure Mode came the inclusion of minor, generic enemies, such as Goombas from the Super Mario series and Octoroks from The Legend of Zelda series. This trend continues into Super Smash Bros. Brawl, which also includes an assortment of original characters to serve as non-playable generic enemies led by the Subspace Army. Many generic enemies from various games appear as part of the "Smash Run" mode in Super Smash Bros.
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There is thus a group homomorphism whose kernel has two elements denoted , where is the identity element. Thus, the group elements and of are equivalent after the homomorphism to ; that is, for any in . The groups and are all Lie groups, and for fixed they have the same Lie algebra, . If is real, then is a real vector subspace of its complexification , and the quadratic form extends naturally to a quadratic form on .
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Game Informer highlighted Brawls "finely tuned balance, core fighting mechanics, and local multiplayer modes". Edge concluded that, while the Smash Bros. games have often been "derided as button-mashing", Brawl features "one of the most enduringly innovative and deep systems of any fighter". IGN editor Matt Casamassina, however, noted that, although Brawl is "completely engrossing and wholly entertaining", it suffers from "long loading times" and "uninspired enemies and locales" in the Subspace Emissary adventure mode.
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The no-hiding theorem proves that if information is lost from a system via decoherence, then it moves to the subspace of the environment and it cannot remain in the correlation between the system and the environment. This is a fundamental consequence of the linearity and unitarity of quantum mechanics. Thus, information is never lost. This has implications in black hole information paradox and in fact any process that tends to lose information completely.
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A Brief Subspace History - In the beginning there were negs... on subspace.legendzones.com (Retrieved January 30, 2008, archived) The game was originally developed by Burst, led by Jeff Petersen, Rod Humble and Juan Sanchez, for the US branch of the now-defunct Virgin Interactive.prehistory by epinephrine on oocities.org When the game was officially released, it was not a commercial success due to a lack of marketing and the relative newness of internet gaming.
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Standard VIE Settings, SVS for short, (also referred to as Standard SubSpace SettingsSubspace Continuum – Freeware Game on beginner.getcontinuum.com) is a server configuration conforming to the physics and rules used in non-special game types hosted by Virgin Interactive Entertainment (VIE) before the company's dissolution.SubSpaceDownloads.com – Free Online Multiplayer Action Game The term is sometimes used informally to describe servers which seek, through other means, to preserve the spirit of the game as it was originally played.
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Maximally informative dimensions is a dimensionality reduction technique used in the statistical analyses of neural responses. Specifically, it is a way of projecting a stimulus onto a low-dimensional subspace so that as much information as possible about the stimulus is preserved in the neural response. It is motivated by the fact that natural stimuli are typically confined by their statistics to a lower-dimensional space than that spanned by white noiseD.J. Field.
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Devos promptly summoned his troops, took personal command of his flagship (the Death Cruiser) and attempted to destroy Throneworld. The Empress blamed Paibok for this and repaid him by ordering his death. Seeking to redeem himself, Paibok made his way on board the Death Cruiser and confronted Devos. The ship's stardrive was damaged during their confrontation and the cruiser fell into subspace — Devos and Paibok were both believed lost along with the ship.
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For every balanced neighborhood of in , let ::}. If is Hausdorff then the collection of all sets , as ranges over all balanced neighborhoods of in , forms a vector topology on making into a complete Hausdorff TVS. Moreover, the map is a TVS- embedding onto a dense vector subspace of . If is a metrizable TVS then a Hausdorff completion of can be constructed using equivalence classes of Cauchy sequences instead of minimal Cauchy filters.
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Using the appropriate "angle", and a radial vector, any one of these planes can be given a polar decomposition. Any one of these decompositions, or Lie algebra renderings, may be necessary for rendering the Lie subalgebra of a 2 × 2 real matrix. There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere. Its Lie algebra is the subspace of quaternion vectors.
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Vector quantization, also called "block quantization" or "pattern matching quantization" is often used in lossy data compression. It works by encoding values from a multidimensional vector space into a finite set of values from a discrete subspace of lower dimension. A lower-space vector requires less storage space, so the data is compressed. Due to the density matching property of vector quantization, the compressed data has errors that are inversely proportional to density.
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In Metal Gear Solid V: The Phantom Pain, Zero tasked EVA to arrange Big Boss's confinement at a hospital in Cyprus after XOF's attack on Mother Base, revealing this to Ocelot. EVA and Olga Gurlukovich's outfits can be used as costumes in Rumble Roses XX.METAL GEAR SOLID 3 SUBSISTENCE EVA also makes a cameo appearance in Super Smash Bros. Brawl as an obtainable sticker, usable exclusively by Solid Snake in The Subspace Emissary.
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A moving wave surface in special relativity may be regarded as a hypersurface (a 3D subspace) in spacetime, formed by all the events passed by the wave surface. A wavetrain (denoted by some variable X) can be regarded as a one-parameter family of such hypersurfaces in spacetime. This variable X is a scalar function of position in spacetime. The derivative of this scalar is a vector that characterizes the wave, the four-wavevector.
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The projection of each data point to a linear subspace spanned by those vectors groups points originating from the same distribution very close together, while points from different distributions stay far apart. One distinctive feature of the spectral method is that it allows us to prove that if distributions satisfy certain separation condition (e.g., not too close), then the estimated mixture will be very close to the true one with high probability.
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Nightcrawler appears in the X-Men/Star Trek crossover novel Planet X. In it, Geordi La Forge determines that his teleportation ability works by sending Nightcrawler through the same subspace dimension as warp drive. When the Enterprise is attacked by an unknown enemy ship, Nightcrawler proves key to their victory when his ability allows him to teleport himself and Data onto the enemy ship, whose shields are up, something that transporters cannot do.
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In addition to the possibility to let a spaceship glide through space in a warp field, there is also space folding in Star Trek. Spatial folding means that two points of space-time are directly connected and an instantaneous change takes place. The space between is simply folded into a higher-dimensional hyperspace or subspace. In the episode That Which Survives of The Original Series, the Enterprise encountered the remains of people called Kalandans.
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The 15-deck (257 rooms), 700,000-metric-ton Voyager was built at the Utopia Planitia Fleet Yards and launched from Earth Station McKinley. Voyager was equipped with 47 bio-neural gel packs and two holodecks. It was the first ship with a class-9 warp drive, allowing for a maximum sustainable speed of Warp 9.975. Variable geometry pylons allowed Voyager and other Intrepid-class ships to exceed warp 5 without damaging subspace.
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There is another infinite-dimensional unitary group, of major significance in homotopy theory, that to which the Bott periodicity theorem applies. It is certainly not contractible. The difference from Kuiper's group can be explained: Bott's group is the subgroup in which a given operator acts non-trivially only on a subspace spanned by the first N of a fixed orthonormal basis {ei}, for some N, being the identity on the remaining basis vectors.
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Melee, his trophy profile states that he is an "extremely skilled technician". Kirby plays a prominent role in the Subspace Emissary plot as a protagonist, the story mode of Super Smash Bros. Brawl. Although he appears cute and innocent, many commercials and ads have contrasted this with his extreme fighting skills when he takes on the abilities of an enemy. Kirby is a character of few words and rarely speaks in- game.
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In mathematics, specifically in order theory and functional analysis, a band in a vector lattice X is a subspace M of X that is solid and such that for all S ⊆ M such that x = sup S exists in X, we have x ∈ M. The smallest band containing a subset S of X is called the band generated by S in X. A band generated by a singleton set is called a principal band.
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Linear subspace learning algorithms are traditional dimensionality reduction techniques that represent input data as vectors and solve for an optimal linear mapping to a lower-dimensional space. Unfortunately, they often become inadequate when dealing with massive multidimensional data. They result in very-high-dimensional vectors, lead to the estimation of a large number of parameters.H. Lu, K. N. Plataniotis, and A. N. Venetsanopoulos, "MPCA: Multilinear principal component analysis of tensor objects," IEEE Trans.
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Consequently there is a one-to-one correspondence between -dimensional subspaces of and -dimensional subspaces of . In terms of the Grassmannian, this is a canonical isomorphism :. Choosing an isomorphism of with therefore determines a (non-canonical) isomorphism of and . An isomorphism of with is equivalent to a choice of an inner product, and with respect to the chosen inner product, this isomorphism of Grassmannians sends an -dimensional subspace into its -dimensional orthogonal complement.
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In the branch of mathematical logic called model theory, infinite finitary matroids, there called "pregeometries" (and "geometries" if they are simple matroids), are used in the discussion of independence phenomena. It turns out that many fundamental concepts of linear algebra – closure, independence, subspace, basis, dimension – are preserved in the framework of abstract geometries. The study of how pregeometries, geometries, and abstract closure operators influence the structure of first-order models is called geometric stability theory.
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An analysis of how bagging and random subspace projection contribute to accuracy gains under different conditions is given by Ho. Typically, for a classification problem with features, (rounded down) features are used in each split. For regression problems the inventors recommend (rounded down) with a minimum node size of 5 as the default. In practice the best values for these parameters will depend on the problem, and they should be treated as tuning parameters.
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As a result, any surface is nonorientable if and only if it contains a Möbius band as a subspace. The Möbius strip is also a standard example used to illustrate the mathematical concept of a fiber bundle. Specifically, it is a nontrivial bundle over the circle S1 with its fiber equal to the unit interval, . Looking only at the edge of the Möbius strip gives a nontrivial two point (or Z2) bundle over S1.
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If is a vector subspace of then the restriction of to is called the canonical pairing where if this pairing is a duality then we will instead call it the canonical duality. Clearly, always distinguishes points of so the canonical pairing is a dual system if and only if separates points of . The following notation is now nearly ubiquitous in duality theory. :Notation: The evaluation map will be denoted by (rather than by ) and we will write rather than .
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The Poisson-type random measures (PT) are a family of three random counting measures which are closed under restriction to a subspace, i.e. closed under Point_process_operation#Thinning. These random measures are examples of the mixed binomial process and share the distributional self-similarity property of the Poisson random measure. They are the only members of the canonical non-negative power series family of distributions to possess this property and include the Poisson distribution, negative binomial distribution, and binomial distribution.
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More generally, given a map between normed vector spaces T\colon V \to W, one can analogously ask for this map to be an isometry on the orthogonal complement of the kernel: that (\ker T)^\perp \to W be an isometry (compare Partial isometry); in particular it must be onto. The case of an orthogonal projection is when W is a subspace of V. In Riemannian geometry, this is used in the definition of a Riemannian submersion.
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In other words, a variety is defined as the intersection of k hypersurfaces, and the normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point. The normal (affine) space at a point P of the variety is the affine subspace passing through P and generated by the normal vector space at P. These definitions may be extended verbatim to the points where the variety is not a manifold.
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That is, the equation of the projective completion is P(x_0, x_1, \ldots, x_n) = 0, with :P(x_0, x_1, \ldots, x_n) = x_0^dp(x_1/x_0, \ldots, x_n/x_0), where is the degree of . These two processes projective completion and restriction to an affine subspace are inverse one to the other. Therefore, an affine hypersurface and its projective completion have essentially the same properties, and are often considered as two points-of- view for the same hypersurface.
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The linear least squares problem is to find the that minimizes , which is equivalent to projecting to the subspace spanned by the columns of . Assuming the columns of (and hence ) are independent, the projection solution is found from . Now is square () and invertible, and also equal to . But the lower rows of zeros in are superfluous in the product, which is thus already in lower-triangular upper-triangular factored form, as in Gaussian elimination (Cholesky decomposition).
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In order for a TVS to have the extension property, it must be complete (since it must be possible to extend the identity map from to the completion of ; i.e. to the map ). If is a continuous linear map from a vector subspace of into a complete space , then there always exists a unique continuous linear extension of from to the closure of in . Consequently, it suffices to only consider maps from closed vector subspaces into complete spaces.
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Actually, it does not depend much even on the linear structure: there are many non-linear diffeomorphisms (and other homeomorphisms) of onto itself, or its parts such as a Euclidean open ball or the interior of a hypercube). has the topological dimension . An important result on the topology of , that is far from superficial, is Brouwer's invariance of domain. Any subset of (with its subspace topology) that is homeomorphic to another open subset of is itself open.
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The ring is also programmed to react to other rings. Fellow Lanterns Kyle Rayner and Guy Gardner rescue Natu and convince her to stay in the Corps. Natu, Rayner, and Gardner rendezvous with Corps trainer Kilowog and new recruits Vath Sarn and Isamot Kol. It is learned that the recent rash of stars collapsing into black holes is caused by the subspace web created by the inhabitants of the Vega star system known as the Spider Guild.
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The weights of the neurons are initialized either to small random values or sampled evenly from the subspace spanned by the two largest principal component eigenvectors. With the latter alternative, learning is much faster because the initial weights already give a good approximation of SOM weights. The network must be fed a large number of example vectors that represent, as close as possible, the kinds of vectors expected during mapping. The examples are usually administered several times as iterations.
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When hearing about the Code Crown and the dreams of being Digimon King, she goes with it, thinking that if Shoutmon becomes King, they will be able to go home. After Shoutmon defeats Tactimon, Mikey returns to the Digital World, while Angie stays in the human world. After forming the Fusion Fighters United Army and retreating to a separate subspace, they reunite with Cutemon. After MegaDarknessBagramon's defeat, Angie moved to another town, but is still in touch with Mikey.
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Atari would also stop production of other upcoming titles for the Jaguar before merging with JT Storage on April 1996, ultimately resulting with the game not being released. Although a prototype cartridge is rumored to exist, no ROM image of the title has managed to surface online. The only known gameplay footage of the game that exists as of date was shown by Atari Explorer Online on the "AEO at E3 1995" VHS release by Subspace Publishing.
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In geometry, a flat or Euclidean subspace is a subset of a Euclidean space that is itself a Euclidean space (of lower dimension). The flats in two- dimensional space are points and lines, and the flats in three-dimensional space are points, lines, and planes. In a -dimensional space, there are flats of every dimension from 0 to .In addition, a whole -dimensional space, being a subset of itself, may also be considered as an -dimensional flat.
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The subspace of K3 surfaces with Picard number a has dimension 20−a.Griffiths & Harris (1994), p. 594. (Thus, for most projective K3 surfaces, the intersection of with H1,1(X) is isomorphic to Z, but for "special" K3 surfaces the intersection can be bigger.) This example suggests several different roles played by Hodge theory in complex algebraic geometry. First, Hodge theory gives restrictions on which topological spaces can have the structure of a smooth complex projective variety.
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Stated another way, two continuous functions f,g : M \to N are homotopic if they represent points in the same path-components of the mapping space C(M,N), given the compact-open topology. The space of immersions is the subspace of C(M,N) consisting of immersions, denote it by Imm(M,N). Two immersions f,g:M \to N are regularly homotopic if they represent points in the same path-component of Imm(M,N).
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Again take the field to be R, but now let the vector space V be the set RR of all functions from R to R. Let C(R) be the subset consisting of continuous functions. Then C(R) is a subspace of RR. Proof: #We know from calculus that . #We know from calculus that the sum of continuous functions is continuous. #Again, we know from calculus that the product of a continuous function and a number is continuous.
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Chattering can be reduced through the use of deadbands or boundary layers around the sliding surface, or other compensatory methods. Although the system is nonlinear in general, the idealized (i.e., non-chattering) behavior of the system in Figure 1 when confined to the s=0 surface is an LTI system with an exponentially stable origin. Intuitively, sliding mode control uses practically infinite gain to force the trajectories of a dynamic system to slide along the restricted sliding mode subspace.
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A popular extension of Matching Pursuit (MP) is its orthogonal version: Orthogonal Matching Pursuit (OMP). The main difference from MP is that after every step, all the coefficients extracted so far are updated, by computing the orthogonal projection of the signal onto the subspace spanned by the set of atoms selected so far. This can lead to results better than standard MP, but requires more computation. Extensions such as Multichannel MP"Piecewise linear source separation", R. Gribonval, Proc.
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The representation is, however, unitary when restricted to the rotation subgroup , but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an representation have -invariant subspaces of highest weight (spin) , where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) is -dimensional. So for example, the (, ) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.
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The standard ways of circumventing relativity in 1950s and 1960s science fiction were hyperspace, subspace and spacewarp. Harrison's contribution was the "Bloater Drive". This enlarges the gaps between the atoms of the ship until it spans the distance to the destination, whereupon the atoms are moved back together again, reconstituting the ship at its previous size but in the new location. An occasional side-effect is that the occupants see a planet drifting, in miniature, through the hull.
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Let be any non-empty open subset of (e.g. could be a non-empty bounded open interval in ) and let denote the subspace topology on that inherits from (so ). Then the topology generated by on is equal to the union (see this footnote for an explanation),Since is a topology on and is an open subset of , it is easy to verify that is a topology on . Since isn't a topology on , is clearly the smallest topology on containing ).
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This hypercube graph is the of the tesseract. :This article is not about the topological skeleton concept of computer graphics In mathematics, particularly in algebraic topology, the of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that is the union of the simplices of X (resp. cells of X) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the .
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Scott awakens without memories of the previous night, though Lisa informs him that he confessed that he loved Ramona. The samurai is revealed to be Knives's father Mr. Chau, who disliked that his daughter was dating a white person. When he attacks Scott again, Scott escapes via a subspace portal and ends up in Ramona's mind, where he sees her as a slave to a shadowy figure. Upon exiting, Scott encounters his dark self – "NegaScott" – and rejects it.
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Scott returns to Toronto to fight Gideon at his newly-opened club, where Envy is making her solo debut. Gideon invites Scott to join League Of Evil Exes; when Scott refuses, Gideon kills him with the Power of Love sword. Scott awakens in a desert, where he encounters Ramona. They reconcile and Scott uses his extra life to return to the club, where he confronts Gideon inside Ramona's subspace and encourages Ramona to overcome Gideon's influence.
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Poisson-type random measures are a family of three random counting measures which are closed under restriction to a subspace, i.e. closed under thinning, that are examples of mixed binomial processes. They are the only distributions in the canonical non-negative power series family of distributions to possess this property and include the Poisson distribution, negative binomial distribution, and binomial distribution. Poisson-type (PT) random measures include the Poisson random measure, negative binomial random measure, and binomial random measure.
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Theorem Let X be a weakly locally connected space. Then X is locally connected. Proof It is sufficient to show that the components of open sets are open. Let U be open in X and let C be a component of U. Let x be an element of C. Then x is an element of U so that there is a connected subspace A of X contained in U and containing a neighbourhood V of x.
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It was only after a combined effort with the Galaxy Police and the supreme sacrifice of the Jurai emperor, that Kain was finally captured and locked away in the subspace room of G.P.H.Q. However, one hundred years later, Kain broke out of the subspace room, destroyed G.P.H.Q. (but not after getting off a warning), and retreated twenty-six years in the past, where he hopes to kill Achika Masaki, which would eliminate Tenchi and affect the futures of Ryoko, Ayeka, Sasami, Ryo-Ohki, Mihoshi, and Kiyone. Despite being said to be immensely powerful, Kain was sealed up rather easily by one of Washu's creations in a sub-space reality. Though he dragged Achika and Nobuyuki in with him, he was forced to show his draconian face when Tenchi, Ryoko and Ayeka entered his prison-world to save Achika. Though Kain easily overpowered them, he was in turn destroyed when Achika awakened her Jurai powers and used the Tenchi-ken – which became a naginata – to bisect Kain and end his reign of terror.
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Informally, imagine terrain Y, and its part X, such that wherever in Y you place a sharpshooter, and an apple at another place in Y, and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of X, or at least it will fly arbitrarily close to points of X – then we say that Y is aimed at X. A priori, it may seem plausible that for a given X the superspaces Y that aim at X can be arbitrarily large or at least huge. We will see that this is not the case. Among the spaces which aim at a subspace isometric to X, there is a unique (up to isometry) universal one, Aim(X), which in a sense of canonical isometric embeddings contains any other space aimed at (an isometric image of) X. And in the special case of an arbitrary compact metric space X every bounded subspace of an arbitrary metric space Y aimed at X is totally bounded (i.e. its metric completion is compact).
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As implied by the definition, Voronoi cells can be defined for metrics other than Euclidean, such as the Mahalanobis distance or Manhattan distance. However, in these cases the boundaries of the Voronoi cells may be more complicated than in the Euclidean case, since the equidistant locus for two points may fail to be subspace of codimension 1, even in the two-dimensional case. Approximate Voronoi diagram of a set of points. Notice the blended colors in the fuzzy boundary of the Voronoi cells.
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PCA is often used in this manner for dimensionality reduction. PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest "variance" (as defined above). This advantage, however, comes at the price of greater computational requirements if compared, for example, and when applicable, to the discrete cosine transform, and in particular to the DCT-II which is simply known as the "DCT". Nonlinear dimensionality reduction techniques tend to be more computationally demanding than PCA.
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In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality. There are five homeomorphism classes of countable Toronto spaces, namely: the discrete topology, the indiscrete topology, the cofinite topology and the upper and lower topologies on the natural numbers. The only countable Hausdorff Toronto space is the discrete space.. The Toronto space problem asks for an uncountable Toronto Hausdorff space that is not discrete..
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The subspace transceiver of an Imperial Star Destroyer had a range of 100 light- years. Devices for shorter-range communications, such as the comlink, can be either hand-held (as seen in A New Hope) or strapped to the wrist (as seen in The Empire Strikes Back, during the early scenes on the planet Hoth). These devices can also be tuned with encryption algorithms for private communication. Most humanoid droids, such as C-3PO, communicate long distances using these comlinks.
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Two vectors of Rn are in the same congruence class modulo the subspace if and only if they are identical in the last n−m coordinates. The quotient space Rn/ Rm is isomorphic to Rn−m in an obvious manner. More generally, if V is an (internal) direct sum of subspaces U and W, :V=U\oplus W then the quotient space V/U is naturally isomorphic to W . An important example of a functional quotient space is a Lp space.
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Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm. Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1] / M is isomorphic to R. If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.
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Let the field K be the set R of real numbers, and let the vector space V be the real coordinate space R3. Take W to be the set of all vectors in V whose last component is 0. Then W is a subspace of V. Proof: #Given u and v in W, then they can be expressed as u = (u1, u2, 0) and v = (v1, v2, 0). Then u + v = (u1+v1, u2+v2, 0+0) = (u1+v1, u2+v2, 0).
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The numerical methods such as FEM or FDM derive a matrix equation as shown in the previous section. To solve this equation faster, a method called Model order reduction can be employed to find an approximation of lower order. This method is based on the fact that a high-dimensional state vector belongs to a low-dimensional subspace . Figure below shows the concept of the MOR approximation: finding matrix V, the dimension of the system can be reduced to solve a simplified system.
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On the other hand, in positive characteristic it is possible for a group scheme to be non-reduced at every point so that the dimension is less than the dimension of any tangent space, which is what happens in Igusa's example. Mumford shows that the tangent space to the Picard variety is the subspace of H0,1 annihilated by all Bockstein operations from H0,1 to H0,2, so the irregularity q is equal to h0,1 if and only if all these Bockstein operations vanish.
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What have been described are irreducible first-class constraints. Another complication is that Δf might not be right invertible on subspaces of the restricted submanifold of codimension 1 or greater (which violates the stronger assumption stated earlier in this article). This happens, for example in the cotetrad formulation of general relativity, at the subspace of configurations where the cotetrad field and the connection form happen to be zero over some open subset of space. Here, the constraints are the diffeomorphism constraints.
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In functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is a continuous linear map between topological vector spaces (TVSs) such that the induced map is an open mapping when , which is the range or image of , is given the subspace topology induced by Y. This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homorphism.
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In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.
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Hypnosis is an increasingly popular practice within a dominance and submission relationship to reinforce power exchange and as a form of play. Some people report that being hypnotized produces a strong feeling of giving up control, for example the feeling that they must obey the hypnotist's commands. Several practitioners have compared the sensation of trance to subspace. Major BDSM events in North America now frequently contain classes and workshops dedicated to erotic hypnosis, such as events run by Black Rose and TES.
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On an interstellar journey in the future, a medical doctor and a priest debate the existence of God in the wonders of the universe. Dr. Chandler (Donald Moffat) believes the universe to be random, but Father Matthew Costigan (Fritz Weaver), who's also an astrophysicist, believes in God's grand but ineffable design. During their friendly debate, their spaceship picks up a subspace signal from a long-dead world. Father Matthew claims it is impossible that a civilization could have survived its star going supernova.
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In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space. That is, the points that are not incident to the hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any subspace connecting a point in one set to a point in the other must intersect the hyperplane.
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The Bluetooth ComBadge also has a Cos-Play mode which when pressed activates the same Chirp sound effect as seen on the show. No real-world equivalent to subspace communication has been developed, proposed, or theorized. However, many other aspects of Starfleet communications technology are commonplace. For example, locator/transponder functionality is implemented via GPS, LoJack, RFID, and radio direction finder devices, and cloud-based digital assistants perform in a way similar to the artificial intelligence of a Starfleet ship's computer.
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The large amount of electrical subspace distortions and anti-tachyon particle emissions from the planet would make it difficult for other ships to accurately scan the area. It was the perfect hiding place. At least it would have been if the weapons computer hadn't been accidentally crosswired to one of the recreational systems running a copy of Doom XXVII. Things were tense when a small patrol was spotted, but there was no possible way they could have detected the ship at that range.
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He begins a relationship with Envy, promoting her new solo album and setting her debut up as his club's opening act. During this time, he meets up with Scott, who flees instead of fighting him, though their altercation is unavoidably inevitable. He has apparent control over subspace, a side effect of his ability to induce "emotional warfare" via The Glow, which essentially traps people in their minds with their psychological issues. Using this allows him to literally be inside Ramona's head.
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Ramona Victoria "Rammy" Flowers is an American expatriate from New York, a "ninja delivery girl" for Amazon and Scott's main love interest. Her age is unknown until the end of the 4th volume, where she reveals that she is 24 years old. She reveals very little and is very guarded about her past in New York City before she moved to Toronto. She is capable of traveling through subspace and has seven evil exes who challenge Scott for her affection.
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The high rank matrix completion in general is NP-Hard. However, with certain assumptions, some incomplete high rank matrix or even full rank matrix can be completed. Eriksson, Balzano and Nowak have considered the problem of completing a matrix with the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces. Since the columns belong to a union of subspaces, the problem may be viewed as a missing-data version of the subspace clustering problem.
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For the homogeneous case in one dimension, all points are uniformly and independently placed in the window or interval \textstyle W. For higher dimensions in a Cartesian coordinate system, each coordinate is uniformly and independently placed in the window \textstyle W. If the window is not a subspace of Cartesian space (for example, inside a unit sphere or on the surface of a unit sphere), then the points will not be uniformly placed in \textstyle W, and suitable change of coordinates (from Cartesian) are needed.
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So, this w does not lie in the span of the dual set. The dual of an infinite-dimensional space has greater dimensionality (this being a greater infinite cardinality) than the original space has, and thus these cannot have a basis with the same indexing set. However, a dual set of vectors exists, which defines a subspace of the dual isomorphic to the original space. Further, for topological vector spaces, a continuous dual space can be defined, in which case a dual basis may exist.
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In Star Trek, tractor beams are imagined to work by placing a target in the focus of a subspace/graviton interference pattern created by two beams from an emitter. When the beams are manipulated correctly the target is drawn along with the interference pattern. The target may be moved toward or away from the emitter by changing the polarity of the beams. Range of the beam affects the maximum mass that can be moved by the emitter, and the emitter subjects its anchoring structure to significant force.
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If the restriction of Q to a subspace U of V is identically zero, U is totally singular. The orthogonal group of a non-singular quadratic form Q is the group of the linear automorphisms of V that preserve Q, that is, the group of isometries of into itself. If a quadratic space has a product so that A is an algebra over a field, and satisfies :\forall x, y \isin A \quad Q(x y) = Q(x) Q(y) , then it is a composition algebra.
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In a field 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 of vectors. They were introduced by Élie Cartan in the 1930s to classify complex structures. Pure spinors were introduced into the realm of theoretical physics, and elevated in their importance in the study of spin geometry more generally, by Roger Penrose in the 1960s, where they became among the basic objects of study in twistor theory.
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SubSpace evolved from a game originally called Sniper (1995), a project to test the effects and severity of lag in a massively multiplayer environment over dialup connections.Subspace for PC Review on cnet.com (Retrieved January 30, 2008) After its creators realized its viability as an actual game, public beta testing began in February, 1996, and it became fully public later that year. The game was released commercially in December 1997 with a list price of US$27.99 for unlimited play, requiring no monthly or hourly fees.
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Any upgrades, weapons, or special abilities are lost. The energy mechanic forces players to be cautious of their energy usage, as reckless weapons fire could result in a quick death. It is usually not possible for players to commit suicide; if a player's own weapon causes more damage than that player has energy, his energy will simply be reduced to one and begin recharging. However, suicide was possible in early beta versions of the game and the offline practice mode included with the original SubSpace client.
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SpectreRF is an option to the Spectre Circuit Simulator from Cadence Design Systems. It adds a series of analyses that are particularly useful for RF circuits to the basic capabilities of Spectre. SpectreRF was first released in 1996 and was notable for three reasons. First, it was arguably the first RF simulator in that it was the first to be designed for large bipolar and CMOS RF circuits; it used shooting methods as its base algorithm; and it pioneered the use of Krylov subspace methods.
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The light-like vectors of Minkowski space are null vectors. The four linearly independent biquaternions , , , and are null vectors and } can serve as a basis for the subspace used to represent spacetime. Null vectors are also used in the Newman–Penrose formalism approach to spacetime manifolds.Patrick Dolan (1968) A Singularity-free solution of the Maxwell-Einstein Equations, Communications in Mathematical Physics 9(2):161–8, especially 166, link from Project Euclid A composition algebra splits when it has a null vector; otherwise it is a division algebra.
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A Baire space is a topological space on infinite sequences of natural numbers. The infinite continued fraction provides a homeomorphism from the Baire space to the space of irrational real numbers (with the subspace topology inherited from the usual topology on the reals). The infinite continued fraction also provides a map between the quadratic irrationals and the dyadic rationals, and from other irrationals to the set of infinite strings of binary numbers (i.e. the Cantor set); this map is called the Minkowski question mark function.
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The quotient of the 24-dimensional linear representation of the permutation representation by its 1-dimensional fixed subspace gives a 23-dimensional representation, which is irreducible over any field of characteristic not 2 or 3, and gives the smallest faithful representation over such fields. Reducing the 24-dimensional representation mod 2 gives an action on F. This has invariant subspaces of dimension 1, 12 (the Golay code), and 23. The subquotients give two irreducible representations of dimension 11 over the field with 2 elements.
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In 2004, Vidal was recognized with the National Science Foundation CAREER Awards. In 2009, Vidal was recognized by the Office of Naval Research with an award from the Young Investigator Program. In 2009, Vidal was recognized with a Sloan Research Fellowship in computer science by the Alfred P. Sloan Foundation. In 2012, Vidal was recognized by the International Association for Pattern Recognition by winning the J.K. Aggarwal Prize for outstanding contributions to generalized principal component analysis (GPCA) and subspace clustering in computer vision and pattern recognition.
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Since a nontrivial quadratic equation has at most two solutions, this line actually lies in the Lie quadric, and any point [q] on this line defines a cycle incident with [x], [y] and [z]. Thus there are infinitely many solutions in this case. If instead x, y and z are linearly independent then the subspace V orthogonal to all three is 2-dimensional. It can have signature (2,0), (1,0), or (1,1), in which case there are zero, one or two solutions for [q] respectively.
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Note that the cycles are all incident with each other as well. In terms of the Lie quadric, this means that a pencil of cycles is a (projective) line lying entirely on the Lie quadric, i.e., it is the projectivization of a totally null two dimensional subspace of R3,2: the representative vectors for the cycles in the pencil are all orthogonal to each other. The set of all lines on the Lie quadric is a 3-dimensional manifold called the space of contact elements Z3.
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Let K be a convex body in n-dimensional Euclidean space Rn containing the origin in its interior. Let S be an (n − 2)-dimensional linear subspace of Rn. For each unit vector θ in S⊥, the orthogonal complement of S, let Sθ denote the (n − 1)-dimensional hyperplane containing θ and S. Define r(θ) to be the (n − 1)-dimensional volume of K ∩ Sθ. Let C be the curve {θr(θ)} in S⊥. Then C forms the boundary of a convex body in S⊥.
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Every normed space can be isometrically embedded onto a dense vector subspace of some Banach space, where this Banach space is called a completion of the normed space. This completion is unique up to isometric isomorphism. More precisely, for every normed space , there exist a Banach space and a mapping such that T is an isometric mapping and is dense in . If is another Banach space such that there is an isometric isomorphism from onto a dense subset of , then is isometrically isomorphic to .
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A eutactic star consisting of 5 pairs of vectors in three-dimensional space (n = 3, s = 5) In Euclidean geometry, a eutactic star is a geometrical figure in a Euclidean space. A star is a figure consisting of any number of opposing pairs of vectors (or arms) issuing from a central origin. A star is eutactic if it is the orthogonal projection of plus and minus the set of standard basis vectors (i.e., the vertices of a cross-polytope) from a higher-dimensional space onto a subspace.
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Therefore, and are second class constraints while is a first class constraint. Note that these constraints satisfy the regularity condition. Here, we have a symplectic space where the Poisson bracket does not have "nice properties" on the constrained subspace. However, Dirac noticed that we can turn the underlying differential manifold of the symplectic space into a Poisson manifold using his eponymous modified bracket, called the Dirac bracket, such that this Dirac bracket of any (smooth) function with any of the second class constraints always vanishes.
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Returning to Japan, Jyan is ambushed by Nuchaku Banki before he, Ran, and Retsu are trapped in the subspace before the Go-ongers come to their aid. After his new friends lost their engines, and being tricked into giving away the sealed Long, Jyan helps Sōsuke train and develop his own Firece Ki before Long is able to fully manifest. With the aid of a revived Rio, Long is defeated and Jyan reaffirms his vow with Rio to become stronger. Jyan is portrayed by .
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In mathematics, the Hilbert projection theorem is a famous result of convex analysis that says that for every vector x in a Hilbert space H and every nonempty closed convex C \subset H, there exists a unique vector y \in C for which \lVert x - z \rVert is minimized over the vectors z \in C. This is, in particular, true for any closed subspace M of H. In that case, a necessary and sufficient condition for y is that the vector x-y be orthogonal to M.
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A second arc covers James Kirk and his crew, just after the successful conference on admitting Coridan into the Federation. Kirk is hauled onto the carpet by a Starfleet admiral demanding that he explain a subspace message showing "dead" Commissioner Nancy Hedford. Kirk discovers that Cochrane was kidnapped from his and Nancy's home at Gamma Canaris. A third arc covers Jean- Luc Picard and his crew, just after dropping off Sarek of Vulcan to another ship for his voyage home from the Legaran home world.
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Given an inner product space , we can form the orthogonal complement of any subspace of . This yields an antitone Galois connection between the set of subspaces of and itself, ordered by inclusion; both polarities are equal to . Given a vector space and a subset of we can define its annihilator , consisting of all elements of the dual space of that vanish on . Similarly, given a subset of , we define its annihilator This gives an antitone Galois connection between the subsets of and the subsets of .
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He then became interested in different problems of statistical signal processing. In particular, it contributes to the development of subspaces methods for the identification of multivariate linear systemsE Moulines, P Duhamel, JF Cardoso, S Mayrargue, « Subspace methods for the blind identification of multichannel FIR filters », IEEE Transactions on signal processing,, 1995, pp. 516–525 and source separationBelouchrani, Adel and Abed-Meraim, Karim and Cardoso, J-F and Moulines, Eric, « A blind source separation technique using second-order statistics », IEEE Transactions on signal processing, 1997, pp.
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To endow the Grassmannian with the structure of a differentiable manifold, choose a basis for . This is equivalent to identifying it with with the standard basis, denoted (e_1, \dots, e_n) , viewed as column vectors. Then for any -dimensional subspace , viewed as an element of , we may choose a basis consisting of linearly independent column vectors (W_1, \dots, W_k) . The homogeneous coordinates of the element consist of the components of the rectangular matrix of maximal rank whose th column vector is W_i, \quad i=1, \dots, k .
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The kth exterior power of V, denoted Λk(V), is the vector subspace of Λ(V) spanned by elements of the form :x_1\wedge x_2\wedge\cdots\wedge x_k,\quad x_i\in V, i=1,2,\ldots, k. If , then α is said to be a k-vector. If, furthermore, α can be expressed as an exterior product of k elements of V, then α is said to be decomposable. Although decomposable k-vectors span Λk(V), not every element of Λk(V) is decomposable.
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Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sumConvex Analysis and Minimization Algorithms by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, , pp. 101, 102Mathematical Programming, by Melvyn W. Jeter (1986) , p. 68 of these vectors is a vector of the form : \alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n where \alpha_i are non-negative real numbers. The name derives from the fact that a conical sum of vectors defines a cone (possibly in a lower-dimensional subspace).
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Many curves, for example hyperelliptic curves, may be presented abstractly, as ramified covers of the projective line. According to the Riemann–Hurwitz formula, the genus then depends only on the type of ramification. A rational curve is a curve that is birationally equivalent to a projective line (see rational variety); its genus is 0. A rational normal curve in projective space Pn is a rational curve that lies in no proper linear subspace; it is known that there is only one example (up to projective equivalence),.
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But in the Euclidean plane, every finite set of points is either collinear, or includes a pair of points whose line does not contain any other points of the set; this is the Sylvester–Gallai theorem. Because the Hesse configuration disobeys the Sylvester–Gallai theorem, it has no Euclidean realization. This example also shows that the Sylvester–Gallai theorem cannot be generalized to the complex projective plane. However, in complex spaces, the Hesse configuration and all Sylvester–Gallai configurations must lie within a two-dimensional flat subspace..
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There is a natural correlation induced between a projective space P(V) and its dual P(V∗) by the natural pairing between the underlying vector spaces V and its dual V∗, where every subspace W of V∗ is mapped to its orthogonal complement W⊥ in V, defined as Composing this natural correlation with an isomorphism of projective spaces induced by a semilinear map produces a correlation of P(V) to itself. In this way, every nondegenerate semilinear map induces a correlation of a projective space to itself.
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A first immediate consequence of the definition is that whenever or . This may be seen by writing the zero vector 0V as (and similarly for 0W) and moving the scalar 0 "outside", in front of B, by linearity. The set of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from into X. associates of this are taken to the other three possibilities using duality and the musical isomorphism If V, W, X are finite-dimensional, then so is .
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Peripheral loops live in U ∪ γ Let Y be a subspace of the path- connected topological space X, whose complement X − Y is path-connected. Fix a basepoint x ∈ X − Y. For each path component Vi of X − Y∩Y, choose a path γi from x to a point in Vi. An element [α] ∈ π1(X − Y, x) is called peripheral with respect to this choice if it is represented by a loop in U ∪ ∪ iγi for every neighborhood U of Y. The set of all peripheral elements with respect to a given choice forms a subgroup of π1(X − Y, x), called a peripheral subgroup. In the diagram, a peripheral loop would start at the basepoint x and travel down the path γ until it's inside the neighborhood U of the subspace Y. Then it would move around through U however it likes (avoiding Y). Finally it would return to the basepoint x via γ. Since U can be a very tight envelope around Y, the loop has to stay close to Y. Any two peripheral subgroups of π1(X − Y, x), resulting from different choices of paths γi, are conjugate in π1(X − Y, x).
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In mathematics, a Carnot group is a simply connected nilpotent Lie group, together with a derivation of its Lie algebra such that the subspace with eigenvalue 1 generates the Lie algebra. The subbundle of the tangent bundle associated to this eigenspace is called horizontal. On a Carnot group, any norm on the horizontal subbundle gives rise to a Carnot–Carathéodory metric. Carnot–Carathéodory metrics have metric dilations; they are asymptotic cones (see Ultralimit) of finitely-generated nilpotent groups, and of nilpotent Lie groups, as well as tangent cones of sub-Riemannian manifolds.
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Theses concepts can be reformulated in purely topological terms, using the Zariski topology, for which the closed sets are the algebraic subsets: A topological space is irreducible if it is not the union of two proper closed subsets, and an irreducible component is a maximal subspace (necessarily closed) that is irreducible for the induced topology. Although these concepts may be considered for every topological space, this is rarely done outside algebraic geometry, since most common topological spaces are Hausdorff spaces, and, in a Hausdorff space, the irreducible components are the singletons.
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The starting point for the standard derivation of a Langevin equation is the identity 1 = P + Q, where Q projects onto the fast subspace. Consider discrete small time steps \tau with evolution operator U\cong 1 +i\tau L, where L is the Liouville operator. The goal is to express U^n in terms of U^kP and Q(UQ)^m. The motivation is that U^kP is a functional of slow variables and that Q(UQ)^m generates expressions which are fast variables at every time step.
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If is a pairing over and is a vector topology on then we say that is a topology of the pairing and that it is compatible (or consistent) with the pairing if it is locally convex and if the continuous dual space of .Of course, there is an analogous definition for topologies on to be "compatible it a pairing" but this article will only deal with topologies on . If distinguishes points of then by identifying as a vector subspace of 's algebraic dual, the defining condition becomes: . Note that some authors (e.g.
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There are two naturally isomorphic functors that are typically used to quantize bosonic strings. In both cases, one starts with positive-energy representations of the Virasoro algebra of central charge 26, equipped with Virasoro-invariant bilinear forms, and ends up with vector spaces equipped with bilinear forms. Here, "Virasoro-invariant" means Ln is adjoint to L−n for all integers n. The first functor historically is "old canonical quantization", and it is given by taking the quotient of the weight 1 primary subspace by the radical of the bilinear form.
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Here, "primary subspace" is the set of vectors annihilated by Ln for all strictly positive n, and "weight 1" means L0 acts by identity. A second, naturally isomorphic functor, is given by degree 1 BRST cohomology. Older treatments of BRST cohomology often have a shift in the degree due to a change in choice of BRST charge, so one may see degree −1/2 cohomology in papers and texts from before 1995. A proof that the functors are naturally isomorphic can be found in Section 4.4 of Polchinski's String Theory text.
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This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally-- that is, in a neighborhood of each point--into corresponding statements that hold throughout the space, and many theorems are of this character. The term compact set is sometimes used as a synonym for compact space, but often refers to a compact subspace of a topological space as well.
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In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is locally connected, which neither implies nor follows from connectedness.
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In mathematical representation theory, coherence is a property of sets of characters that allows one to extend an isometry from the degree-zero subspace of a space of characters to the whole space. The general notion of coherence was developed by , as a generalization of the proof by Frobenius of the existence of a Frobenius kernel of a Frobenius group and of the work of Brauer and Suzuki on exceptional characters. developed coherence further in the proof of the Feit–Thompson theorem that all groups of odd order are solvable.
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In the category of unital rings, there are no kernels in the category-theoretic sense; indeed, this category does not even have zero morphisms. Nevertheless, there is still a notion of kernel studied in ring theory that corresponds to kernels in the category of non-unital rings. In the category of pointed topological spaces, if f : X → Y is a continuous pointed map, then the preimage of the distinguished point, K, is a subspace of X. The inclusion map of K into X is the categorical kernel of f.
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Meanwhile, with Scott in command, the Enterprise is investigating what happened to the Hood, science officer Follet suggests it may be connected to the subspace anomaly they were investigating. Back at Corinth IV, Gray informs Kirk and Spock of a complication. Commander Diana Garrett, 1st officer of Earth's Space Dock, had applied for the Captaincy of the Hood but was turned down, she has now launched a formal appeal, stating she has been overlooked because she's a woman. Kirk and Spock are shocked at such an assertion in the 23rd century.
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The Jacquet module J(V) of a representation (π,V) of a group N is the space of co-invariants of N; or in other words the largest quotient of V on which N acts trivially, or the zeroth homology group H0(N,V). In other words, it is the quotient V/VN where VN is the subspace of V generated by elements of the form π(n)v - v for all n in N and all v in V. The Jacquet functor J is the functor taking V to its Jacquet module J(V).
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Dedede's brooches are revealed as a fail-safe against Tabuu's Off Waves and revive Luigi, Dedede, Ness, and Kirby, who rescue the characters scattered across Subspace and navigate a great maze where Tabuu is located. As Tabuu is about to use his Off Waves to turn the characters back into trophies, he is ambushed by Sonic, who weakens him; the fighters defeat Tabuu and save the Smash Bros. universe. In the final scene, the fighters look at a great luminous cross on the horizon where the Isle of the Ancients once resided.
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Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace (the analog of "dropping the altitude" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane.
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This paragraph was adapted from The weak- coupling limit derivation assumes a quantum system with a finite number of degrees of freedom coupled to a bath containing an infinite number of degrees of freedom. The system and bath each possess a Hamiltonian written in terms of operators acting only on the respective subspace of the total Hilbert space. These Hamiltonians govern the internal dynamics of the uncoupled system and bath. There is a third Hamiltonian that contains products of system and bath operators, thus coupling the system and bath.
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Consider a smooth surface in 3-dimensional Euclidean space. Near to any point, can be approximated by its tangent plane at that point, which is an affine subspace of Euclidean space. Differential geometers in the 19th century were interested in the notion of development in which one surface was rolled along another, without slipping or twisting. In particular, the tangent plane to a point of can be rolled on : this should be easy to imagine when is a surface like the 2-sphere, which is the smooth boundary of a convex region.
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The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector (see the third axiom in the Vector space article). Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is different from the null space of a linear operator L, which is the kernel of L.
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Since has the trivial topology, it is easily shown that every vector subspace of that is an algebraic complement of in is necessarily a topological complement of in . Let denote any topological complement of in , which is necessarily a Hausdorff TVS (since it is TVS-isomorphic to the quotient TVS This particular quotient map is in fact also a closed map.). Since is the topological direct sum of and (which means that in the category of TVSs), the canonical map ::, given by is a TVS-isomorphism. Let denote the inverse of this canonical map.
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Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in particular its algorithmic aspects are still of current interest. The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For details see Schubert variety.
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One is to use a pseudo inverse instead of the usual matrix inverse in the above formulae. However, better numeric stability may be achieved by first projecting the problem onto the subspace spanned by \Sigma_b . Another strategy to deal with small sample size is to use a shrinkage estimator of the covariance matrix, which can be expressed mathematically as : \Sigma = (1-\lambda) \Sigma+\lambda I\, where I is the identity matrix, and \lambda is the shrinkage intensity or regularisation parameter. This leads to the framework of regularized discriminant analysis or shrinkage discriminant analysis.
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Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed. Closed sets also give a useful characterization of compactness: a topological space X is compact if and only if every collection of nonempty closed subsets of X with empty intersection admits a finite subcollection with empty intersection. A topological space X is disconnected if there exist disjoint, nonempty, open subsets A and B of X whose union is X. Furthermore, X is totally disconnected if it has an open basis consisting of closed sets.
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Each "view" (i.e., frame) of the animation is an orthogonal projection of the data set onto a 2-dimensional subspace of the Euclidean space Rp where the data resides. The subspaces are selected by taking small steps along a continuous curve, parametrized by time, in the space of all 2-dimensional subspaces of Rp, known as the Grassmannian G(2,p). To display these views on a computer screen, it is necessary to pick one particular rotated position of each view (in the plane of the computer screen) for display.
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A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space. This highly entangled, encoded state corrects for local noisy errors. A quantum error-correcting code makes quantum computation and quantum communication practical by providing a way for a sender and receiver to simulate a noiseless qubit channel given a noisy qubit channel whose noise conforms to a particular error model. The stabilizer theory of quantum error correction allows one to import some classical binary or quaternary codes for use as a quantum code.
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Once they return they are scolded by their coach, but because of their importance to the missions success as well as building feelings between the coach and them, they are let go with a simple warning. Jung later apologizes for her challenge, and thus begins a friendship with the two. As they move farther into space, the young pilots are placed in their quarters for subspace traveling. On a dare, Takaya is sent into the hangars and meets a male space pilot named Toren Smith (voice: Kazuki Yao).
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The gap-filling versions of SSA can be used to analyze data sets that are unevenly sampled or contain missing data (Schoellhamer, 2001; Golyandina and Osipov, 2007). Schoellhamer (2001) shows that the straightforward idea to formally calculate approximate inner products omitting unknown terms is workable for long stationary time series. Golyandina and Osipov (2007) uses the idea of filling in missing entries in vectors taken from the given subspace. The recurrent and vector SSA forecasting can be considered as particular cases of filling in algorithms described in the paper.
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There were many works to attempt to design a robust and fast multigrid method for such nearly singular problems. A general guide has been provided as a design principle to achieve parameters (e.g., mesh size and physical parameters such as Poisson's ratio that appear in the nearly singular operator) independent convergence rate of the multigrid method applied to such nearly singular systems,Young-Ju Lee, Jinbiao Wu, Jinchao Xu and Ludmil Zikatanov, Robust Subspace Correction Methods for Nearly Singular Systems, Mathematical Models and Methods in Applied Sciences, Vol. 17, No 11, pp.
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The purpose of the coarse problem is to propagate information throughout the whole problem globally. In multigrid methods for partial differential equations, the coarse problem is typically obtained as a discretization of the same equation on a coarser grid (usually, in finite difference methods) or by a Galerkin approximation on a subspace, called a coarse space. In finite element methods, the Galerkin approximation is typically used, with the coarse space generated by larger elements on the same domain. Typically, the coarse problem corresponds to a grid that is twice or three times coarser.
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The Tomasi–Kanade factorization is the seminal work by Carlo Tomasi and Takeo Kanade in the early 1990s. It charted out an elegant and simple solution based on a SVD-based factorization scheme for analysing image measurements of a rigid object captured from different views using a weak perspective camera model. The crucial observation made by authors was that if all the measurements (i.e., image co-ordinates of all the points in all the views) are collected in a single matrix, the point trajectories will reside in a certain subspace.
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Theorem: Let M and N be compact, locally symmetric Riemannian manifolds with everywhere non-positive curvature having no closed one or two dimensional geodesic subspace which are direct factor locally. If f : M → N is a homotopy equivalence then f is homotopic to an isometry. Theorem (Mostow's theorem for hyperbolic n-manifolds, n ≥ 3): If M and N are complete hyperbolic n-manifolds, n ≥ 3 with finite volume and f : M → N is a homotopy equivalence then f is homotopic to an isometry. These results are named after George Mostow.
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In mathematics, specifically the field of algebraic number theory, a Minkowski space is a Euclidean space associated with an algebraic number field. If K is a number field of degree d then there are d distinct embeddings of K into C. We let KC be the image of K in the product Cd, considered as equipped with the usual Hermitian inner product. If c denotes complex conjugation, let KR denote the subspace of KC fixed by c, equipped with a scalar product. This is the Minkowski space of K.
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On the planet Centaurus, the planetary capital of New Athens has been annihilated by a terrorist antimatter bomb. Millions are dead; because of a computer malfunction, the planetary defense system is preventing any rescue ships from approaching the planet. No subspace communication is possible, and traditional speed-of-light radio is blanketed with heavy static. Despite an emergency do-not-approach warning (known as Code 7-10, which went unheard), the first three relief ships, carrying hundreds of medical personnel, are destroyed by ground-to-air missiles as they assumed standard orbit.
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Together with the corresponding initial conditions these equations fix the evolution of the running couplings g_\alpha(k), and thus determine \Gamma_k completely. As one can see, the FRGE gives rise to a system of infinitely many coupled differential equations since there are infinitely many couplings, and the \beta-functions can depend on all of them. This makes it very hard to solve the system in general. A possible way out is to restrict the analysis on a finite-dimensional subspace as an approximation of the full theory space.
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In 1988, Jürg Fröhlich showed that it was valid under the spin–statistics theorem for the particle exchange to be monoidal (non-abelian statistics). In particular, this can be achieved when the system exhibits some degeneracy, so that multiple distinct states of the system have the same configuration of particles. Then an exchange of particles can contribute not just a phase change, but can send the system into a different state with the same particle configuration. Particle exchange then corresponds to a linear transformation on this subspace of degenerate states.
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Figure-Oh, No. 158 He battles Gokai Red and Gosei Red within his own subspace, using the Bibi Soldiers to hold them off while having a Bibi Bug tail them so he can unleash dead-on hits. But once his strategy is exposed, the two red Sentai warriors defeat him. In a final gambit by the Black Cross Colossus, Brajira is revived again along with his Buredoran guises before they are all ultimately destroyed by the combined finishers of Gokaioh, Gosei Great, and many of the previous Sentai teams' giant robots.
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The armor stored a "solar charge" that could be used as a weapon and could drain power sources by mere contact. It gave him the ability to summon three pieces of equipment stored in "subspace": ;Neutralizer :Rom's primary weapon, which is designed to banish Dire Wraiths to Limbo by opening a dimensional portal. Unfortunately, the process leaves considerable waste material (ash, etc.) that makes it appear to an uninformed observer that the weapon kills its target. This handheld weapon could fire energy beams that could be deadly at a high setting.
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Therefore, these models still require a UV completion, of the kind that string theory is intended to provide. In particular, superstring theory requires six compact dimensions (6D hyperspace) forming a Calabi–Yau manifold. Thus Kaluza-Klein theory may be considered either as an incomplete description on its own, or as a subset of string theory model building. In addition to small and curled up extra dimensions, there may be extra dimensions that instead aren't apparent because the matter associated with our visible universe is localized on a subspace.
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Sisko begins by fortifying the garrison's defenses. The Dominion have left a set of booby-traps: "Houdini" anti-personnel mines that can pass in and out of subspace at random. Kellin and Dax work out a way to force them into normal space, so that they can be moved out of the camp and used to halt attacking Jem'Hadar. He also sends out a scouting party to locate the Jem'Hadar base; Lt. Larkin leads, with Reese as survival expert and Nog as talent: his Ferengi ears will work where jammed tricorders will not.
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If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of βX. For general topological spaces X, the map from X to βX need not be injective. A form of the axiom of choice is required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces X, an accessible concrete description of βX often remains elusive.
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For instance, the subspace theorem proved by demonstrates that points of small height (i.e. small complexity) in projective space lie in a finite number of hyperplanes and generalizes Siegel's theorem on integral points and solution of the S-unit equation. Height functions were crucial to the proofs of the Mordell–Weil theorem and Faltings's theorem by and respectively. Several outstanding unsolved problems about the heights of rational points on algebraic varieties, such as the Manin conjecture and Vojta's conjecture, have far-reaching implications for problems in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic.
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Just as the fundamental theorem of algebra ensures that every linear transformation acting on a finite dimensional complex vector space has a nontrivial invariant subspace, the fundamental theorem of noncommutative algebra asserts that Lat(Σ) contains nontrivial elements for certain Σ. Theorem (Burnside) Assume V is a complex vector space of finite dimension. For every proper subalgebra Σ of L(V), Lat(Σ) contains a nontrivial element. Burnside's theorem is of fundamental importance in linear algebra. One consequence is that every commuting family in L(V) can be simultaneously upper-triangularized.
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All cones are path-connected since every point can be connected to the vertex point. Furthermore, every cone is contractible to the vertex point by the homotopy :h_t(x,s) = (x, (1-t)s). The cone is used in algebraic topology precisely because it embeds a space as a subspace of a contractible space. When X is compact and Hausdorff (essentially, when X can be embedded in Euclidean space), then the cone CX can be visualized as the collection of lines joining every point of X to a single point.
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We give the more general definition of when a -valued function or multifunction defined on a subset of has a closed graph since this generality is needed in the study of closed linear operators that are defined on a dense subspace of a topological vector space (and not necessarily defined on all of ). This particular case is one of the main reasons why functions with closed graphs are studied in functional analysis. :Assumptions: Throughout, and are topological spaces, , and is a -valued function or multifunction on (i.e. or ).
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On board Pod 1, Commander Tucker and Lieutenant Reed are attempting to locate Enterprise in an asteroid field that Captain Archer had intended to map. Just then, Reed spots an impact crater surrounded by a debris field; with only one piece large enough to be identifiable as part of Enterprise, they conclude that the ship has somehow been destroyed. They are now alone, with ten days' worth of air left. Tucker orders Reed to head to Echo Three, a subspace amplifier, using the stars for reference as navigation is down.
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In the special case of two topological vector spaces V and W, the notion of uniform continuity of a map f:V\to W becomes: for any neighborhood B of zero in W, there exists a neighborhood A of zero in V such that v_1-v_2\in A implies f(v_1)-f(v_2)\in B. For linear transformations f:V\to W, uniform continuity is equivalent to continuity. This fact is frequently used implicitly in functional analysis to extend a linear map off a dense subspace of a Banach space.
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In ensemble learning one tries to combine the models produced by several learners into an ensemble that performs better than the original learners. One way of combining learners is bootstrap aggregating or bagging, which shows each learner a randomly sampled subset of the training points so that the learners will produce different models that can be sensibly averaged. In bagging, one samples training points with replacement from the full training set. The random subspace method is similar to bagging except that the features ("attributes", "predictors", "independent variables") are randomly sampled, with replacement, for each learner.
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An occupied Borg "alcove" prop on display at the Hollywood Entertainment Museum Borg civilization is based on a hive or group mind known as the Collective. Each Borg drone is linked to the collective by a sophisticated subspace network that ensures each member is given constant supervision and guidance. The mental energy of the group consciousness can help an injured or damaged drone heal or regenerate damaged body parts or technology. The collective consciousness gives them the ability not only to "share the same thoughts", but also to adapt quickly to new tactics.
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In the film adaptation, Todd is never shown to cheat on Envy, and Scott gets the Vegan Police to Todd by tricking him into drinking half and half. 4\. Roxanne "Roxie" Richter is Ramona's fourth evil ex and her former college roommate. She is a "half-ninja" (kunoichi) and an accomplished fine artist, who taught Ramona much of what she knows of her ninja abilities and subspace. Scott is usually against fighting girls (or anyone with a sword), so Ramona fights Roxie, for part of the time using Scott's body as a weapon.
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Let A be a linear operator defined on a linear subspace D(A) of the reflexive Banach space X. Then A generates a contraction semigroup if and only ifEngel and Nagel Corollary II.3.20 # A is dissipative, and # A − λ0I is surjective for some λ0> 0, where I denotes the identity operator. Note that the conditions that D(A) is dense and that A is closed are dropped in comparison to the non-reflexive case. This is because in the reflexive case they follow from the other two conditions.
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It is the automorphism group of the Fano plane and of the group Z, and is also known as . More generally, one can count points of Grassmannian over F: in other words the number of subspaces of a given dimension k. This requires only finding the order of the stabilizer subgroup of one such subspace and dividing into the formula just given, by the orbit-stabilizer theorem. These formulas are connected to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians.
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The Hahn–Banach theorem is a central tool in functional analysis (a field of mathematics). It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.
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Rosenthal graduated from Queens College, City University of New York with a B.S. in Mathematics in 1962. In 1963 he obtained an MA in Mathematics and in 1967 a Ph.D. in Mathematics from the University of Michigan; his Ph.D. thesis advisor was Paul Halmos. His thesis, "On lattices of invariant subspaces" concerns operators on Hilbert space, and most of his subsequent research has been in operator theory and related fields. Much of his work has been related to the invariant subspace problem, the still-unsolved problem of the existence of invariant subspaces for bounded linear operators on Hilbert space.
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In November 2006, Client signed with Out of Line in German-speaking Europe, SubSpace Communications in Scandinavia, Metropolis Records in North America, Noiselab in Latin America and their own label Loser Friendly Records in the UK and Ireland. It was confirmed in December 2010 that Blackwood had left the band and a new member was being recruited to front the band. In July 2011, Holmes announced that Xan Tyler—with whom Holmes formed the synthpop duo Technique in the mid-1990s—would take over live lead vocals for the band, replacing Blackwood. However, this never came to fruition.
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In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds. Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space.
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A knot K is an embedding of the one-sphere S1 in three-dimensional space R3. (Alternatively, the ambient space can also be taken to be the three-sphere S3, which does not make a difference for the purposes of the Wirtinger presentation.) The open subspace which is the complement of the knot, S^3 \setminus K is the knot complement. Its fundamental group \pi_1(S^3 \setminus K) is an invariant of the knot in the sense that equivalent knots have isomorphic knot groups. It is therefore interesting to understand this group in an accessible way.
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Openness of V in the subspace topology is automatic.) Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space. A map which is not a homeomorphism onto its image: with g(t) = (t2 − 1, t3 − t) It is of crucial importance that both domain and range of f are contained in Euclidean space of the same dimension. Consider for instance the map f : (0,1) → ℝ2 defined by . This map is injective and continuous, the domain is an open subset of , but the image is not open in .
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An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. It is often useful to embed topological spaces in compact spaces, because of the special properties compact spaces have. Embeddings into compact Hausdorff spaces may be of particular interest. Since every compact Hausdorff space is a Tychonoff space, and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space. In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.
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The free algebra generated by V may be written as the tensor algebra , that is, the sum of the tensor product of n copies of V over all n, and so a Clifford algebra would be the quotient of this tensor algebra by the two-sided ideal generated by elements of the form for all elements . The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g. uv). Its associativity follows from the associativity of the tensor product. The Clifford algebra has a distinguished subspace V, being the image of the embedding map.
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The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term linear for describing this type of equations. More generally, the solutions of a linear equation in variables form a hyperplane (a subspace of dimension ) in the Euclidean space of dimension . Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations.
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Consider a complex vector space C2n with even complex dimension 2n and a quadratic form Q, which maps a vector v to complex number Q(v). The Clifford algebra Cℓ2n(C) is the ring generated by products of vectors in C2n subject to the relation :v^2=Q(v). \, Spinors are modules of the Clifford algebra, and so in particular there is an action of C2n on the space of spinors. The subset of C2n that annihilates a given spinor ψ is a complex subspace Cm. If ψ is nonzero then m is less than or equal to n.
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According to basic results of linear algebra, any two complete flags in an n-dimensional vector space V over a field F are no different from each other from a geometric point of view. That is to say, the general linear group acts transitively on the set of all complete flags. Fix an ordered basis for V, identifying it with Fn, whose general linear group is the group GL(n,F) of n × n invertible matrices. The standard flag associated with this basis is the one where the i th subspace is spanned by the first i vectors of the basis.
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The SSE introduces a group of antagonists called the Subspace Army, led by the Ancient Minister. Some of these enemies appeared in previous Nintendo video games, such as Petey Piranha from the Mario series and a squadron of R.O.B.s based on classic Nintendo hardware. The SSE boasts a number of original enemies, such as the Roader, a robotic unicycle; the Bytan, a one-eyed spherical creature which can replicate itself if left alone; and the Primid, enemies that fight with a variety of weapons. Though the game is primarily played as a single-player mode, cooperative multiplayer is available.
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In 2006 he was awarded a knighthood of the Order of the Netherlands Lion. Henk van der Vorst is a Fellow of Society for Industrial and Applied Mathematics (SIAM). His major contributions include preconditioned iterative methods, in particular the ICCG (incomplete Cholesky conjugate gradient) method (developed together with Koos Meijerink), a version of preconditioned conjugate gradient method, the BiCGSTAB and (together with Kees Vuik) GMRESR Krylov subspace methods and (together with Gerard Sleijpen) the Jacobi-Davidson method for solving ordinary, generalized, and nonlinear eigenproblems. He has analyzed convergence behavior of the conjugate gradient and Lanczos methods.
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Thus we can see that there is a commutative diagram including A ∩ B into A and B and then another inclusion from A and B into S^2 and that there is a corresponding diagram of homomorphisms between the fundamental groups of each subspace. Applying Van Kampen's theorem gives the result :\pi_1(S^2)=\pi_1(A)\cdot\pi_1(B)/\ker(\Phi). However A and B are both homeomorphic to R2 which is simply connected, so both A and B have trivial fundamental groups. It is clear from this that the fundamental group of S^2 is trivial.
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She also makes a cameo appearance alongside Maxima in the ending of K' from NeoGeo Battle Coliseum. She is playable in the mobile phone game Kimi wa Hero (as an adult) and Brave Frontier, while also making a cameo in the dating sim Days of Memories should the player interact with K'. She has since made guest appearances in more games, including the role-playing game Valkyrie Connect. She also appears as a downloadble character in Koei Tecmo's fighting game Dead or Alive 6. In her story chapter, NiCO pulls her into the Dead or Alive dimension while experimenting with subspace portals.
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Then (X,tp) and (X,tq) are homeomorphic incomparable topologies on the same set. ; No nonempty subset dense-in-itself : Let S be a nonempty subset of X. If S contains p, then p is isolated in S (since it is an isolated point of X). If S does not contain p, any x in S is isolated in S. ; Not first category : Any set containing p is dense in X. Hence X is not a union of nowhere dense subsets. ; Subspaces : Every subspace of a set given the particular point topology that doesn't contain the particular point, inherits the discrete topology.
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An important example arising in the context of linear algebra itself is the vector space of linear maps. Let L(V,W) denote the set of all linear maps from V to W (both of which are vector spaces over F). Then L(V,W) is a subspace of WV since it is closed under addition and scalar multiplication. Note that L(Fn,Fm) can be identified with the space of matrices Fm×n in a natural way. In fact, by choosing appropriate bases for finite-dimensional spaces V and W, L(V,W) can also be identified with Fm×n.
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In terms of the fundamental group in algebraic topology, the HNN extension is the construction required to understand the fundamental group of a topological space X that has been 'glued back' on itself by a mapping f (see e.g. Surface bundle over the circle). That is, HNN extensions stand in relation of that aspect of the fundamental group, as free products with amalgamation do with respect to the Seifert-van Kampen theorem for gluing spaces X and Y along a connected common subspace. Between the two constructions essentially any geometric gluing can be described, from the point of view of the fundamental group.
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More formally, there is a vector space of such polynomials, for each given value of d = 0, 1, 2, ..., and we write nd for its vector space dimension, or in other words the number of linearly independent homogeneous invariants of a given degree. In more algebraic terms, take the d-th symmetric power of V, and the representation of G on it arising from ρ. The invariants form the subspace consisting of all vectors fixed by all elements of G, and nd is its dimension. The Molien series is then by definition the formal power series :M(t) = \sum_d n_d t^d.
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In the game's story mode, the "Subspace Emissary", he teams up with Ike from Fire Emblem: Path of Radiance and Fire Emblem: Radiant Dawn and Meta Knight from the Kirby series. His final smash, Critical Hit, strikes opponents with a blow that instantly knocks them out of the screen unless they hit a barrier. During this attack, a quickly- depleting 60 HP health bar is shown, as an homage to the Fire Emblem series' battle system. Marth is also playable in Super Smash Bros for Nintendo 3DS and Wii U, using his character design from New Mystery of the Emblem.
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For V embedded in a projective space Pn and defined over some algebraically closed field K, the degree d of V is the number of points of intersection of V, defined over K, with a linear subspace L in general position, when :\dim(V) + \dim(L) = n. Here dim(V) is the dimension of V, and the codimension of L will be equal to that dimension. The degree d is an extrinsic quantity, and not intrinsic as a property of V. For example, the projective line has an (essentially unique) embedding of degree n in Pn.
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Since the real line R is complete, the Cauchy-continuous functions on R are the same as the continuous ones. On the subspace Q of rational numbers, however, matters are different. For example, define a two-valued function so that f(x) is 0 when x2 is less than 2 but 1 when x2 is greater than 2. (Note that x2 is never equal to 2 for any rational number x.) This function is continuous on Q but not Cauchy-continuous, since it cannot be extended continuously to R. On the other hand, any uniformly continuous function on Q must be Cauchy-continuous.
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A reducible pair can be decomposed into a direct product of irreducible ones, and for many purposes, it is enough to restrict one's attention to the irreducible case. Several classes of reductive dual pairs had appeared earlier in the work of André Weil. Roger Howe proved a classification theorem, which states that in the irreducible case, those pairs exhaust all possibilities. An irreducible reductive dual pair (G, G′) in Sp(W) is said to be of type II if there is a lagrangian subspace X in W that is invariant under both G and G′, and of type I otherwise.
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Discretization in the space produces a system of ordinary differential equations for unsteady problems and algebraic equations for steady problems. Implicit or semi-implicit methods are generally used to integrate the ordinary differential equations, producing a system of (usually) nonlinear algebraic equations. Applying a Newton or Picard iteration produces a system of linear equations which is nonsymmetric in the presence of advection and indefinite in the presence of incompressibility. Such systems, particularly in 3D, are frequently too large for direct solvers, so iterative methods are used, either stationary methods such as successive overrelaxation or Krylov subspace methods.
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Because spherical elliptic geometry can be modeled as, for example, a spherical subspace of a Euclidean space, it follows that if Euclidean geometry is self-consistent, so is spherical elliptic geometry. Therefore it is not possible to prove the parallel postulate based on the other four postulates of Euclidean geometry. Tarski proved that elementary Euclidean geometry is complete: there is an algorithm which, for every proposition, can show it to be either true or false.Tarski (1951) (This does not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.
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Let V be a finite-dimensional vector space over a field k not of characteristic 2 equipped with a non-degenerate sesquilinear form that is \varepsilon-symmetric (i.e. \varepsilon = 1 if the form is symmetric and \varepsilon = -1 if the form is skew-symmetric. Let W be a non-degenerate subspace of V such that V = W \oplus W^\perp of dimension (\varepsilon + 1)/2. Then let G = G(V) \times G(W), where G(V) is the unitary group preserving the form on V, and let H = \Delta G(W) be the diagonal subgroup of G(W).
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The integers with their usual topology are a discrete subgroup of the real numbers. In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of G in which every open subset contains exactly one element of H ; in other words, the subspace topology of H in G is the discrete topology. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not. A discrete group is a topological group G equipped with the discrete topology.
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First, when V is not locally convex, the continuous dual may be equal to {0} and the map Ψ trivial. However, if V is Hausdorff and locally convex, the map Ψ is injective from V to the algebraic dual of the continuous dual, again as a consequence of the Hahn–Banach theorem.If V is locally convex but not Hausdorff, the kernel of Ψ is the smallest closed subspace containing {0}. Second, even in the locally convex setting, several natural vector space topologies can be defined on the continuous dual , so that the continuous double dual is not uniquely defined as a set.
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For example, in the line geometry case, we can identify H as a 12-dimensional subgroup of the 16-dimensional general linear group, GL(4), defined by conditions on the matrix entries :h13 = h14 = h23 = h24 = 0, by looking for the stabilizer of the subspace spanned by the first two standard basis vectors. That shows that X has dimension 4. Since the homogeneous coordinates given by the minors are 6 in number, this means that the latter are not independent of each other. In fact, a single quadratic relation holds between the six minors, as was known to nineteenth- century geometers.
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Multilinear subspace learning algorithms aim to learn low-dimensional representations directly from tensor representations for multidimensional data, without reshaping them into higher- dimensional vectors. Deep learning algorithms discover multiple levels of representation, or a hierarchy of features, with higher-level, more abstract features defined in terms of (or generating) lower-level features. It has been argued that an intelligent machine is one that learns a representation that disentangles the underlying factors of variation that explain the observed data. Feature learning is motivated by the fact that machine learning tasks such as classification often require input that is mathematically and computationally convenient to process.
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To reach the rendezvous with Degra in three days, Enterprise approaches a nebula that contains a subspace corridor defended by Kovaalan vessels. The corridor lets them traverse the distance in minutes. Suddenly, an older yet enhanced copy of Enterprise appears, captained by a half-Vulcan man named Lorian, who explains that after the Enterprise enters the corridor, it will destabilize, causing Captain Archer's version to travel 117 years into the past. Confronted with this situation, and not wanting to contaminate Earth's time stream, it then turns itself into a generational ship to await the Xindi crisis.
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The river Erms drains an area of about 179 km², mainly in the middle Swabian Alb and its foreland. Its upper part belongs to the Mittleren Kuppenalb, the lower part almost entirely to the foothills of the central Swabian Alb, where it passes successively through the Neuffen-Vorberge and the Erms-Steinach-Albvorland. Finally, a gusset close to the mouth of the river counts towards the subspace Nürtinger-Esslinger Neckartal of the Fildern. The highest point in the area reaches about 870 m above sea level in the area of the Römerstein on the Alb plateau.
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An ensemble of models employing the random subspace method can be constructed using the following algorithm: # Let the number of training points be N and the number of features in the training data be D. # Choose L to be the number of individual models in the ensemble. # For each individual model l, choose n (n < N) to be the number of input points for l. It is common to have only one value of n for all the individual models. # For each individual model l, create a training set by choosing d features from D with replacement and train the model.
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We now consider the special case where is a vector subspace of the algebraic dual space of (i.e. a vector space of linear functionals on ). There is a pairing, denoted by (X,Y,\langle\cdot, \cdot\rangle) or (X,Y), called the canonical pairing whose bilinear map \langle\cdot, \cdot\rangle is the canonical evaluation map, defined by \langle x,x'\rangle =x'(x) for all x\in X and x'\in Y. Note in particular that \langle \cdot,x'\rangle is just another way of denoting x' i.e. \langle \cdot,x'\rangle=x'(\cdot). :Assumption.
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Scott Pilgrim, a 23-year-old slacker, lives in Toronto with his roommate Wallace Wells. He is the bass player for Sex Bob- omb, an unsuccessful band consisting of himself, guitarist Stephen Stills, and drummer Kim Pine. To the discomfort of Scott's friends, he has recently begun dating Knives Chau, a 17-year-old high school student. After having a dream about a woman on rollerblades, Scott encounters her in real life and discovers that she is Ramona Flowers, who can travel through subspace and who has recently moved to Toronto after breaking up with a man named Gideon Graves.
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He revives six monsters, Vict Lugiel, Hyper Zetton, King Joe, Gudon, Twin Tail and Birdon as distractions for the two before leaving. After dealing with two of them, Zero sent Orb to chase Reibatos before his father Seven, Zoffy and Ultraman Jack arrive to his aid. Orb stopped Reibatos in the midway and managed to deal fatal blow but the villain simply shrugs it off due to his regenerative ability. Through his words, Zoffy deduced that he is trying to pursue the Giga Battle Nizer in the Monster Graveyard and leaves with Jack while Orb received a special training in Zero's subspace dimension, .
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The new World of Light mode was inspired by Brawl (2008) Subspace Emissary, and Sakurai chose to start it with a cataclysmic event because he thought it would leave a greater impact on players. The team conceived the Spirits mechanic because they wanted to create an enjoyable single-player mode but did not have enough resources to create character models. While it did not let them tell stories for individual fighters or create new locations and rules, the Spirits let them use a variety of characters and assets. One part of the team chose Spirits to include in the game and had to thoroughly research them.
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According to the above definition, if C is a convex cone, then C ∪ {0} is a convex cone, too. A convex cone is said to be ' if 0 is in C, and ' if 0 is not in C. Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β. A cone is called flat if it contains some nonzero vector x and its opposite -x, meaning C contains a linear subspace of dimension at least one, and salient otherwise. A blunt convex cone is necessarily salient, but the converse is not necessarily true.
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Every topological space is a dense subset of itself. For a set X equipped with the discrete topology, the whole space is the only dense subset. Every non- empty subset of a set X equipped with the trivial topology is dense, and every topology for which every non-empty subset is dense must be trivial. Denseness is transitive: Given three subsets A, B and C of a topological space X with such that A is dense in B and B is dense in C (in the respective subspace topology) then A is also dense in C. The image of a dense subset under a surjective continuous function is again dense.
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The density of a topological space (the least of the cardinalities of its dense subsets) is a topological invariant. A topological space with a connected dense subset is necessarily connected itself. Continuous functions into Hausdorff spaces are determined by their values on dense subsets: if two continuous functions into a Hausdorff space Y agree on a dense subset of X then they agree on all of X. For metric spaces there are universal spaces, into which all spaces of given density can be embedded: a metric space of density is isometric to a subspace of , the space of real continuous functions on the product of copies of the unit interval.
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The maximization/minimization of the Rayleigh quotient in a 3-dimensional subspace can be performed numerically by the Rayleigh–Ritz method. As the iterations converge, the vectors x^i and x^{i-1} become nearly linearly dependent, making the Rayleigh–Ritz method numerically unstable in the presence of round-off errors. It is possible to substitute the vector x^{i-1} with an explicitly computed difference p^i=x^{i-1}-x^i making the Rayleigh–Ritz method more stable; see. This is a single-vector version of the LOBPCG method—one of possible generalization of the preconditioned conjugate gradient linear solvers to the case of symmetric eigenvalue problems.
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The objects are the topological vector spaces over and the morphisms are the continuous -linear maps from one object to another. :Definition: A TVS homomorphism or topological homomorphism is a continuous linear map between topological vector spaces (TVSs) such that the induced map is an open mapping when , which is the range or image of , is given the subspace topology induced by Y. :Definition: A TVS embedding or a topological monomorphism is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding. :Definition: A TVS isomorphism or an isomorphism in the category of TVSs is a bijective linear homeomorphism.
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The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For details see Schubert variety. The intersection theory of these cells, which can be seen as the product structure in the cohomology ring of the Grassmannian of associated cohomology classes, in principle allows the prediction of the cases where intersections of cells results in a finite set of points, which are potentially concrete answers to enumerative questions. A supporting theoretical result is that the Schubert cells (or rather, their classes) span the whole cohomology ring.
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The USS Defiant is sent on a mission to investigate a subspace anomaly, which could yield significant technological advances for the Federation. Jadzia Dax (Terry Farrell), Miles O'Brien (Colm Meaney) and Julian Bashir (Alexander Siddig) take the runabout USS Rubicon into the anomaly in order to take readings, tethered to the Defiant by a tractor beam. As a side- effect of entering the anomaly, the entire runabout is shrunk to only several centimeters high, but the crew expects to return to normal once they exit. However, during the procedure, the Defiant is fired upon by a Jem'Hadar attack ship, disabled, and boarded by Jem'Hadar soldiers.
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In the Euclidean plane, the real projective plane, higher-dimensional Euclidean spaces or real projective spaces, or spaces with coordinates in an ordered field, the Sylvester–Gallai theorem shows that the only possible Sylvester–Gallai configurations are one-dimensional: they consist of three or more collinear points. was inspired by this fact and by the example of the Hesse configuration to ask whether, in spaces with complex-number coordinates, every Sylvester–Gallai configuration is at most two-dimensional. repeated the question. answered Serre's question affirmatively; simplified Kelly's proof, and proved analogously that in spaces with quaternion coordinates, all Sylvester–Gallai configurations must lie within a three-dimensional subspace.
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In contrast, the analytic approach is to define projective space based on linear algebra and utilizing homogeneous co-ordinates. The propositions of incidence are derived from the following basic result on vector spaces: given subspaces and of a (finite dimensional) vector space , the dimension of their intersection is . Bearing in mind that the geometric dimension of the projective space associated to is and that the geometric dimension of any subspace is positive, the basic proposition of incidence in this setting can take the form: linear subspaces and of projective space meet provided .Joel G. Broida & S. Gill Williamson (1998) A Comprehensive Introduction to Linear Algebra, Theorem 2.11, p 86, Addison-Wesley .
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Let be a finite-dimensional vector space over a field k of characteristic different from 2 together with a non-degenerate symmetric or skew-symmetric bilinear form. If is an isometry between two subspaces of V then f extends to an isometry of V. Witt's theorem implies that the dimension of a maximal totally isotropic subspace (null space) of V is an invariant, called the index' or ' of b, and moreover, that the isometry group of acts transitively on the set of maximal isotropic subspaces. This fact plays an important role in the structure theory and representation theory of the isometry group and in the theory of reductive dual pairs.
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Krylov subspaces are used in algorithms for finding approximate solutions to high-dimensional linear algebra problems. Modern iterative methods for finding one (or a few) eigenvalues of large sparse matrices or solving large systems of linear equations avoid matrix-matrix operations, but rather multiply vectors by the matrix and work with the resulting vectors. Starting with a vector, b, one computes A b, then one multiplies that vector by A to find A^2 b and so on. All algorithms that work this way are referred to as Krylov subspace methods; they are among the most successful methods currently available in numerical linear algebra.
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The Digital World, a computer-generated subspace that exists between all forms of digital devices, and home of the creatures known as "Digimon" is under attack by a malevolent and powerful force known as Millenniummon, who seeks to corrupt all the data present in the world and modify it to his own designs. In response, several Digimon and their human companions have set out to stop Millenniummon and his minions before any irreparable harm can be done. This is accomplished in a very round-about and a-typical way to the fighting video game genre, by finding and defeating as many opponents as possible on the way.
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The class of n\times n doubly stochastic matrices is a convex polytope known as the Birkhoff polytope B_n. Using the matrix entries as Cartesian coordinates, it lies in an (n-1)^2-dimensional affine subspace of n^2-dimensional Euclidean space defined by 2n-1 independent linear constraints specifying that the row and column sums all equal one. (There are 2n-1 constraints rather than 2n because one of these constraints is dependent, as the sum of the row sums must equal the sum of the column sums.) Moreover, the entries are all constrained to be non-negative and less than or equal to one.
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In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system (X_n, i_{nm}) of Fréchet spaces. This means that X is a direct limit of a direct system (X_n, i_{nm}) in the category of locally convex topological vector spaces and each X_n is a Fréchet space. If each of the bonding maps i_{nm} is an embedding of TVSs then the LF-space is called a strict LF-space. This means that the subspace topology induced on by is identical to the original topology on .
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In the case of the group SU(3), all the irreducible representations can be generated from the standard 3-dimensional representation and its dual, as follows. To generate the representation with label (m_1,m_2), one takes the tensor product of m_1 copies of the standard representation and m_2 copies of the dual of the standard representation, and then takes the invariant subspace generated by the tensor product of the highest weight vectors. Proof of Proposition 6.17 In contrast to the situation for SU(2), in the Clebsch–Gordan decomposition for SU(3), a given irreducible representation W may occur more than once in the decomposition of U\otimes V.
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In topology, a compactly generated space (or k-space) is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it satisfies the following condition: :A subspace A is closed in X if and only if A ∩ K is closed in K for all compact subspaces K ⊆ X. Equivalently, one can replace closed with open in this definition. If X is coherent with any cover of compact subspaces in the above sense then it is, in fact, coherent with all compact subspaces. A compactly generated Hausdorff space is a compactly generated space that is also Hausdorff.
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Rod Humble (born 1 June 1964) is the former Chief Executive Officer of Second Life creator Linden Lab, Chief Creative Officer at ToyTalk and former Executive Vice President for the EA Play label of the video game company Electronic Arts. He is the General Manager for the San Francisco studio of SGN (now Jam City, Inc). He has been contributing to the development of games since 1990, and is recently best known for his work on the Electronic Arts titles, The Sims 2 and The Sims 3. Previously he worked at Sony Online where he worked on EverQuest and before that Virgin Interactive's SubSpace.
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Paul Halmos writes in "Invariant subspaces", American Mathematical Monthly 85 (1978) 182-183 as follows: :"the extension to polynomially compact operators was obtained by Bernstein and Robinson (1966). They presented their result in the metamathematical language called non-standard analysis, but, as it was realized very soon, that was a matter of personal preference, not necessity." Halmos writes in (Halmos 1985) as follows (p. 204): :The Bernstein–Robinson proof [of the invariant subspace conjecture of Halmos] uses non-standard models of higher order predicate languages, and when [Robinson] sent me his reprint I really had to sweat to pinpoint and translate its mathematical insight.
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Each installment gradually made more differences between the two. Eiji Aonuma said that his design team submitted designs for Ganon based on Twilight Princess to the developers of Super Smash Bros. Brawl. Ganon appears in the single-player The Subspace Emissary mode, allied with Bowser and Wario (in which, near the end they leave Tabuu's side to join the heroes when his deception is revealed, as they believed they were working under Master Hand) as well as being a servant of Master Hand. His Final Smash is his "Dark Beast: Ganon" form from Twilight Princess, in which he transforms, charges across the screen, and then warps himself back onto the stage.
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In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag (See Chapter 11 for construction.) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces . It is dual to the mapping cone. In particular, given such a map, define the mapping path space to be the set of pairs where and is a path such that . We give a topology by giving it the subspace topology as a subset of (where is the space of paths in which as a function space has the compact-open topology).
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We say G is a (\tau, \kappa) -disjoint collection if G is the union of at most \tau subcollections G_\alpha, where for each \alpha, G_\alpha is a disjoint collection of cardinality at most \kappa It was proven by Petr Simon that X is a Boolean space with the generating set G of CO(X) being (\tau, \kappa) -disjoint if and only if X is homeomorphic to a closed subspace of \alpha \kappa ^ \tau. The Ramsey-like property for polyadic spaces as stated by Murray Bell for Boolean spaces is then as follows: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint.
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That is, an ordered n-tuple is in the same orbit as any other that is a re-ordered version of it. A path in the n-fold symmetric product is the abstract way of discussing n points of X, considered as an unordered n-tuple, independently tracing out n strings. Since we must require that the strings never pass through each other, it is necessary that we pass to the subspace Y of the symmetric product, of orbits of n-tuples of distinct points. That is, we remove all the subspaces of X^n defined by conditions x_i = x_j for all 1\le i.
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Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case. Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other. In this construction, each "point" of the real projective plane is the one-dimensional subspace (a geometric line) through the origin in a 3-dimensional vector space, and a "line" in the projective plane arises from a (geometric) plane through the origin in the 3-space. This idea can be generalized and made more precise as follows.
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This operation preserves the necessary conditions for Minkowski's theorem on the existence of a polytope described by the resulting set of vectors, and this polytope is the Blaschke sum. The two polytopes need not have the same dimension as each other, as long as they are both defined in a common space of high enough dimension to contain both: lower-dimensional polytopes in a higher-dimensional space are defined in the same way by sets of vectors that span a lower-dimensional subspace of the higher-dimensional space, and these sets of vectors can be combined without regard to the dimensions of the spaces they span.
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Several episodes of The Original Series placed the Enterprise in peril by having it travel at high warp factors. However, the velocity (in present dimensional units) of any given warp factor is rarely the subject of explicit expression, and travel times for specific interstellar distances are not consistent through the various series. In the Star Trek: The Next Generation Technical Manual it was written that the real warp speed depends on external factors such as particle density or electromagnetic fields and only roughly corresponds with the calculated speed of current warp factor. The reference work Star Trek Maps established the theory of subspace (or warp) highways.
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In a similar way to Differential evolution, MPS uses difference vectors between the members of the population in order to generate new solutions. It attempts to provide an efficient use of function evaluations by maintaining a small population size. If the population size is smaller than the dimensionality of the search space, then the solutions generated through difference vectors will be constrained to the n - 1 dimensional hyperplane. A smaller population size will lead to a more restricted subspace. With a population size equal to the dimensionality of the problem (n = d), the “line/hyperplane points” in MPS will be generated within a d - 1 dimensional hyperplane.
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Following that acquisition, the ship and crew is approached by a version of Enterprise that had been sent 117 years into the past and has become a generational ship attempting not to disrupt the timeline. The crew of the two versions of Enterprise work together and enable the present-day Enterprise to enter a subspace corridor to allow Captain Jonathan Archer (Scott Bakula) to meet with the Xindi Council. The whereabouts of the future Enterprise are unknown, but there is speculation as to whether it was destroyed or ceased to exist. After locating the Xindi weapon in "The Council", the weapon was launched in "Countdown" with Enterprise in pursuit.
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The variable is called the conjugate variable to . In classical mechanics, the physical state of a particle (existing in one dimension, for simplicity of exposition) would be given by assigning definite values to both and simultaneously. Thus, the set of all possible physical states is the two-dimensional real vector space with a -axis and a -axis called the phase space. In contrast, quantum mechanics chooses a polarisation of this space in the sense that it picks a subspace of one-half the dimension, for example, the -axis alone, but instead of considering only points, takes the set of all complex-valued "wave functions" on this axis.
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Then multiplying a global section of O(D) by a nonzero scalar in k does not change its zero locus. As a result, the projective space of lines in the k-vector space of global sections H0(X, O(D)) can be identified with the set of effective divisors linearly equivalent to D, called the complete linear system of D. A projective linear subspace of this projective space is called a linear system of divisors. One reason to study the space of global sections of a line bundle is to understand the possible maps from a given variety to projective space. This is essential for the classification of algebraic varieties.
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Observe that a set is absorbing in if and only if } is absorbing in every 1-dimensional vector subspace , where . Thus, it is necessary and sufficient to show that contains an open -ball around the origin in . The condition implies that every "open ray" in starting at the origin (i.e. a set of the form for some ) contains an element of so that in particular, and (where so that ) so that now the convexity of makes it clear that for every , the convex set is a line segment (possibly open, closed, or half-closed, and possibly bounded or unbounded) containing an open sub-interval that contains the origin.
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A generalization of the Kakeya conjecture is to consider sets that contain, instead of segments of lines in every direction, but, say, portions of k-dimensional subspaces. Define an (n, k)-Besicovitch set K to be a compact set in Rn containing a translate of every k-dimensional unit disk which has Lebesgue measure zero. That is, if B denotes the unit ball centered at zero, for every k-dimensional subspace P, there exists x ∈ Rn such that (P ∩ B) + x ⊆ K. Hence, a (n, 1)-Besicovitch set is the standard Besicovitch set described earlier. :The (n, k)-Besicovitch conjecture: There are no (n, k)-Besicovitch sets for k > 1.
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You are advised to leave the system immediately! In the ending cutscene the player's commanding officer, Admiral Petrarch, delivers a speech about everything the Alliance has lost, speculating on the nature of the Shivans and why they destroyed the Capella star, and if the player decides to stay, a small tribute is paid to the player's heroic actions as Petrarch informs his wingmen of his sacrifice. The Admiral concludes by saying that the Alliance now has the means to recreate the Ancient subspace gate, implying that there's a chance the node to Earth can be restored and that this conflict didn't bring only sorrow, before signing off.
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Given a CW-complex A, consider the set of all pairs of CW-complexes (X, A) such that the inclusion of A into X is a homotopy equivalence. Two pairs (X1, A) and (X2, A) are said to be equivalent, if there is a simple homotopy equivalence between X1 and X2 relative to A. The set of such equivalence classes form a group where the addition is given by taking union of X1 and X2 with common subspace A. This group is natural isomorphic to the Whitehead group Wh(A) of the CW- complex A. The proof of this fact is similar to the proof of s-cobordism theorem.
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Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can be given the product topology, which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space.
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An element of the interior of a set S is an interior point of S. ; Interior point: See Interior. ; Isolated point: A point x is an isolated point if the singleton {x} is open. More generally, if S is a subset of a space X, and if x is a point of S, then x is an isolated point of S if {x} is open in the subspace topology on S. ; Isometric isomorphism: If M1 and M2 are metric spaces, an isometric isomorphism from M1 to M2 is a bijective isometry f : M1 → M2. The metric spaces are then said to be isometrically isomorphic.
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"Todd" (Christopher Heyerdahl) informs Atlantis that a group of Wraith have multiple ZPMs and they plan to use one of them to power a new super Hive Ship. "Todd" urges the expedition to destroy it, but when the team reaches the hive, they realize that the ZPM has already made the ship far more powerful than expected, and causes significant damage to Daedalus before jumping into hyperspace. The team then finds out that the Hive has picked up a subspace transmission from the alternate reality depicted in "Vegas", giving the Wraith the location of Earth. Earth sends Apollo and Sun Tzu to stop them, but the two ships are quickly disabled.
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In mathematics, the Farrell–Markushevich theorem, proved independently by O. J. Farrell (1899–1981) and A. I. Markushevich (1908–1979) in 1934, is a result concerning the approximation in mean square of holomorphic functions on a bounded open set in the complex plane by complex polynomials. It states that complex polynomials form a dense subspace of the Bergman space of a domain bounded by a simple closed Jordan curve. The Gram–Schmidt process can be used to construct an orthonormal basis in the Bergman space and hence an explicit form of the Bergman kernel, which in turn yields an explicit Riemann mapping function for the domain.
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Informally, this causes individual learners to not over-focus on features that appear highly predictive/descriptive in the training set, but fail to be as predictive for points outside that set. For this reason, random subspaces are an attractive choice for problems where the number of features is much larger than the number of training points, such as learning from fMRI data or gene expression data. The random subspace method has been used for decision trees; when combined with "ordinary" bagging of decision trees, the resulting models are called random forests. It has also been applied to linear classifiers, support vector machines, nearest neighbours and other types of classifiers.
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Frequent measurement prohibits the transition. It can be a transition of a particle from one half-space to another (which could be used for an atomic mirror in an atomic nanoscope ) as in the time-of-arrival problem, a transition of a photon in a waveguide from one mode to another, and it can be a transition of an atom from one quantum state to another. It can be a transition from the subspace without decoherent loss of a qubit to a state with a qubit lost in a quantum computer. In this sense, for the qubit correction, it is sufficient to determine whether the decoherence has already occurred or not.
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This holds for a geometry over any field, and more generally over any division ring. In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point).
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In general n-dimensional projective space, a correlation takes a point to a hyperplane. This context was described by Paul Yale: :A correlation of the projective space P(V) is an inclusion-reversing permutation of the proper subspaces of P(V).Paul B. Yale (1968, 1988. 2004) Geometry and Symmetry, chapter 6.9 Correlations and semi-bilinear forms, Dover Publications He proves a theorem stating that a correlation φ interchanges joins and intersections, and for any projective subspace W of P(V), the dimension of the image of W under φ is , where n is the dimension of the vector space V used to produce the projective space P(V).
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It allows crew members to contact starships in orbit without relying on an artificial satellite to relay the signal. Communicators use subspace transmissions that do not conform to normal rules of physics in that signals can bypass EM interference, and the devices allow nearly instantaneous communication at distances that would otherwise require more time to traverse. In Star Trek: The Original Series (TOS), communicators functioned as a plot device, stranding characters in challenging situations when they malfunctioned, were lost or stolen, or went out of range. Otherwise, the transporter could have allowed characters to return to the ship at the first sign of trouble, ending the storyline prematurely.
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In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on X that makes those functions continuous. The subspace topology and product topology constructions are both special cases of initial topologies. Indeed, the initial topology construction can be viewed as a generalization of these. The dual notion is the final topology, which for a given family of functions mapping to a set X is the finest topology on X that makes those functions continuous.
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The Hahn-Banach theorem is the first sign of an important philosophy in functional analysis: to understand a space, one should understand its continuous functionals. For example, linear subpaces are characterized by functionals: if is a normed vector space with linear subspace (not necessarily closed) and if is an element of not in the closure of , then there exists a continuous linear map with for all in , , and . (To see this, note that is a sublinear function.) Moreover, if is an element of , then there exists a continuous linear map such that and . This implies that the natural injection from a normed space into its double dual is isometric.
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Then one may define an abstract integration map assigning to each function an element of or the symbol , : f\mapsto\int_E f \,d\mu, \, that is compatible with linear combinations. In this situation, the linearity holds for the subspace of functions whose integral is an element of (i.e. "finite"). The most important special cases arise when is , , or a finite extension of the field of p-adic numbers, and is a finite-dimensional vector space over , and when and is a complex Hilbert space. Linearity, together with some natural continuity properties and normalisation for a certain class of "simple" functions, may be used to give an alternative definition of the integral.
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Beckman and Quarles observe that the theorem is not true for the real line (one- dimensional Euclidean space). For, the function that returns if is an integer and returns otherwise obeys the preconditions of the theorem (it preserves unit distances) but is not an isometry. Beckman and Quarles also provide a counterexample for Hilbert space, the space of square-summable sequences of real numbers. This example involves the composition of two discontinuous functions: one that maps every point of the Hilbert space onto a nearby point in a countable dense subspace, and a second that maps this dense set into a countable unit simplex (an infinite set of points all at unit distance from each other).
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The Goddard–Thorn theorem amounts to the assertion that this quantization functor more or less cancels the addition of two free bosons, as conjectured by Lovelace in 1971. Lovelace's precise claim was that at critical dimension 26, Virasoro-type Ward identities cancel two full sets of oscillators. Mathematically, this is the following claim: Let V be a unitarizable Virasoro representation of central charge 24 with Virasoro-invariant bilinear form, and let π1,1λ be the irreducible module of the R1,1 Heisenberg Lie algebra attached to a nonzero vector λ in R1,1. Then the image of V ⊗ π1,1λ under quantization is canonically isomorphic to the subspace of V on which L0 acts by 1-(λ,λ).
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The bijective correspondence between points on two lines in a plane determined by a point of that plane not on either line has higher-dimensional analogues which will also be called perspectivities. Let Sm and Tm be two distinct m-dimensional projective spaces contained in an n-dimensional projective space Rn. Let Pn−m−1 be an (n − m − 1)-dimensional subspace of Rn with no points in common with either Sm or Tm. For each point X of Sm, the space L spanned by X and Pn-m-1 meets Tm in a point . This correspondence fP is also called a perspectivity. The central perspectivity described above is the case with and .
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The notion of ideal generalises to any Mal'cev algebra (as linear subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ideals in the case of rings, and submodule in the case of modules). It turns out that ker f is not a subalgebra of A, but it is an ideal. Then it makes sense to speak of the quotient algebra G/(ker f). The first isomorphism theorem for Mal'cev algebras states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B). The connection between this and the congruence relation for more general types of algebras is as follows.
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The TOS episode "Obsession" however, appears to indicate that the transporters' maximum range, during that time period in Star Trek history, is actually around 30,000 kilometers. Transporter operations have been disrupted or prevented by dense metals (TNG: "Contagion"), solar flares (TNG: "Symbiosis"), and other forms of radiation, including electromagnetic (TNG: "The Enemy"; TNG: "Power Play") and nucleonic (TNG: "Schisms"), and affected by ion storms (TOS: "Mirror, Mirror"). Transporting, in progress, has also been stopped by telekinetic powers (TNG: "Skin of Evil") and by brute strength (TNG: "The Hunted"). The TNG episode "Bloodlines" features a dangerous and experimental "subspace transporter" capable of interstellar distances and the Dominion had the ability to transport over great distances (DS9: "Covenant").
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However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possess an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable. For instance, a bosonic field can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural state space of a boson might seem to be a non-separable space. However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined).
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Canonical analysis belongs to a group of methods which involve solving the characteristic equation for its latent roots and vectors. It describes formal structures in hyperspace invariant with respect to the rotation of their coordinates. In this type of solution, rotation leaves many optimizing properties preserved, provided it takes place in certain ways and in a subspace of its corresponding hyperspace. This rotation from the maximum intervariate correlation structure into a different, simpler and more meaningful structure increases the interpretability of the canonical weights C and D. In this the canonical analysis differs from Harold Hotelling’s (1936) canonical variate analysis (also called the canonical correlation analysis), designed to obtain maximum (canonical) correlations between the predictor and criterion canonical variates.
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Rather than dealing with ammunition counts and hit points separately, SubSpace combines both of these elements into a single unit of measure: energy. Each ship is equipped with a certain amount of energy, from which it must draw its health as well as its weapons power. When a ship's energy is reduced from its capacity (whether from firing weapons or enduring enemy fire), the ship will automatically recharge back to its maximum capacity over a period of time; however, sustained weapons fire or enemy fire will inevitably cause the energy to drop lower. Once the ship's energy drops below zero, the ship is destroyed and the player is respawned elsewhere in the area.
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A densely defined operator T is symmetric, if the subspace (defined in a previous section) is orthogonal to its image under J (where J(x,y):=(y,-x)).Follows from and the definition via adjoint operators. Equivalently, an operator T is self-adjoint if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators , are surjective, that is, map the domain of T onto the whole space H. In other words: for every x in H there exist y and z in the domain of T such that and . An operator T is self-adjoint, if the two subspaces , are orthogonal and their sum is the whole space H \oplus H .
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The origins of SSA and, more generally, of subspace-based methods for signal processing, go back to the eighteenth century (Prony's method). A key development was the formulation of the fspectral decomposition of the covariance operator of stochastic processes by Kari Karhunen and Michel Loève in the late 1940s (Loève, 1945; Karhunen, 1947). Broomhead and King (1986a, b) and Fraedrich (1986) proposed to use SSA and multichannel SSA (M-SSA) in the context of nonlinear dynamics for the purpose of reconstructing the attractor of a system from measured time series. These authors provided an extension and a more robust application of the idea of reconstructing dynamics from a single time series based on the embedding theorem.
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Dimension is being used here in two different senses. When referring to a projective space, the term is used in the common geometric way where lines are 1-dimensional and planes are 2-dimensional objects. However, when applied to a vector space, dimension means the number of vectors in a basis, and a basis for a vector subspace, thought of as a line, has two vectors in it, while a basis for a vector space, thought of as a plane, has three vectors in it. If the meaning is not clear from the context, the terms projective or geometric are applied to the projective space concept while algebraic or vector are applied to the vector space one.
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The reviewers praised the variety and depth of the single-player content, the unpredictability of Final Smashes, and the dynamic fighting styles of the characters. Thunderbolt Games gave the game 10 out of 10, calling it "a vastly improved entry into the venerable series". Chris Slate of Nintendo Power also awarded Brawl a perfect score in its March 2008 issue, calling it "one of the very best games that Nintendo has ever produced". IGN critic Matt Casamassina, in his February 11 Wii-k in Review podcast, noted that although Brawl is a "solid fighter," it does have "some issues that need to be acknowledged," including "long loading times" and repetition in The Subspace Emissary.
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Suppose that K is a field (for example, the real numbers) and V is a vector space over K. As usual, we call elements of V vectors and call elements of K scalars. If v1,...,vn are vectors and a1,...,an are scalars, then the linear combination of those vectors with those scalars as coefficients is :a_1 \vec v_1 + a_2 \vec v_2 + a_3 \vec v_3 + \cdots + a_n \vec v_n. \, There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, as in the assertion "the set of all linear combinations of v1,...,vn always forms a subspace".
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This means that the coordinate "ring" associated to the geometry must be a division ring (skewfield) K, and the projective geometry is isomorphic to the one constructed from the vector space Kd+1, i.e. PG(d,K). As in the construction given earlier, the points of the d-dimensional projective space PG(d,K) are the lines through the origin in Kd + 1 and a line in PG(d,K) corresponds to a plane through the origin in Kd + 1. In fact, each i-dimensional object in PG(d,K), with i < d, is an (i + 1)-dimensional (algebraic) vector subspace of Kd + 1 ("goes through the origin"). The projective spaces in turn generalize to the Grassmannian spaces.
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Theorem 1: The subspace spanned by a non-empty subset S of a vector space V is the set of all linear combinations of vectors in S. This theorem is so well known that at times, it is referred to as the definition of span of a set. Theorem 2: Every spanning set S of a vector space V must contain at least as many elements as any linearly independent set of vectors from V. Theorem 3: Let V be a finite-dimensional vector space. Any set of vectors that spans V can be reduced to a basis for V, by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set).
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According to the above paragraph, there are subspaces with spin both and in the last two cases, so these representations cannot likely represent a single physical particle which must be well-behaved under . It cannot be ruled out in general, however, that representations with multiple subrepresentations with different spin can represent physical particles with well-defined spin. It may be that there is a suitable relativistic wave equation that projects out unphysical components, leaving only a single spin. Construction of pure spin representations for any (under ) from the irreducible representations involves taking tensor products of the Dirac-representation with a non-spin representation, extraction of a suitable subspace, and finally imposing differential constraints.
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In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite- dimensional vector spaces.
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Many game features from Civilization II are present, but renamed or slightly tweaked: players establish bases (Civilization II's cities), build facilities (buildings) and secret projects (Wonders of the World), explore territory, research technology, and conquer other factions (civilizations). In addition to conquering all non-allied factions, players may also win by obtaining votes from three quarters of the total population (similar to Civilization IVs Diplomatic victory), "cornering the Global Energy Market", completing the Ascent to Transcendence secret project, or for alien factions, constructing six Subspace Generators.Shah (2000), p.5. The main map (the upper two thirds of the screen) is divided into squares, on which players can establish bases, move units and engage in combat.
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In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method. Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non- Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices. The Arnoldi method belongs to a class of linear algebra algorithms that give a partial result after a small number of iterations, in contrast to so-called direct methods which must complete to give any useful results (see for example, Householder transformation). The partial result in this case being the first few vectors of the basis the algorithm is building.
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In the paper , he precised by using a triangular mesh as an increasing net approximating the open domain, defining positive and negative variations whose sum is the total variation, i.e. the area functional. His inspiring point of view, as he explicitly admitted, was those of Giuseppe Peano, as expressed by the Peano-Jordan Measure: to associate to every portion of a surface an oriented plane area in a similar way as an approximating chord is associated to a curve. Also, another theme found in this theory was the extension of a functional from a subspace to the whole ambient space: the use of theorems generalizing the Hahn–Banach theorem is frequently encountered in Caccioppoli research.
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Linear normality may also be expressed geometrically: V as projective variety cannot be obtained by an isomorphic linear projection from a projective space of higher dimension, except in the trivial way of lying in a proper linear subspace. Projective normality may similarly be translated, by using enough Veronese mappings to reduce it to conditions of linear normality. Looking at the issue from the point of view of a given very ample line bundle giving rise to the projective embedding of V, such a line bundle (invertible sheaf) is said to be normally generated if V as embedded is projectively normal. Projective normality is the first condition N0 of a sequence of conditions defined by Green and Lazarsfeld.
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1-4: Trevor Jamieson is stranded in a deadly jungle on the planet Eristan II with an ezwal, a 3-ton, six-legged saurian-like creature that dislikes humans and wants them to leave his native world, Carson's Planet. Having bailed out of a crashing spaceship, Jamieson and the telepathic ezwal must make their way to the wreckage in hopes that the subspace radio survived and they can call for help. Their journey is interrupted by a cruiser belonging to the Rull, creatures that appear to have evolved from chameleon-like worms and who are implacably hostile to all intelligent life. The Rull capture the ezwal, but must lie hidden when a Terran battleship engages their cruiser.
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A function in H_0^1, with zero values at the endpoints (blue), and a piecewise linear approximation (red) P1 and P2 are ready to be discretized which leads to a common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem: :Find u \in H_0^1 such that :\forall v \in H_0^1, \; -\phi(u,v)=\int fv with a finite- dimensional version: :(3) Find u \in V such that :\forall v \in V, \; -\phi(u,v)=\int fv where V is a finite-dimensional subspace of H_0^1. There are many possible choices for V (one possibility leads to the spectral method). However, for the finite element method we take V to be a space of piecewise polynomial functions.
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When a plan to convert spiritual energy from Tokyo's five-color Fudo shrines to generate enough Jurai power to recapture Kain fails, Washu sends the dimensional cannon, a weapon with enough destructive power to raze entire galaxies. Kiyone is charged with operating the machine, and fires just as Achika exits subspace with Kain in close pursuit, killing him and ending the villain's threat to the Masaki family. Tenchi Forever! Though Kiyone is not present over the course of most of the final film, the GP detective's role is still an important one; she and Mihoshi break into the Galaxy Academy to hack into their dimension stabilizing system so that Washu will be able to create a gateway to track down the missing Tenchi.
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After consoling her in the subspace, she realizes what Chifuyu sees in him and falls in love with him. Upon learning Ichika's strength is his freedom to choose and helping those around him and Chifuyu's advice, Laura decides to become Ichika's bodyguard to protect him against the other girls by kissing him in front of his class and professes that he is her "bride", a misunderstanding of her knowledge of Japanese culture, and she will not accept any objections. ; : :Tatenashi is the Student Council President of the IS Academy, as well as the 17th leader of the Sarashiki Family and the IS Representative of Russia. The long history of her traditional family, it's actually a secret organization that's been protesting Phantom Task for over 50 years.
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De Branges' proof of the Bieberbach conjecture was not initially accepted by the mathematical community. Rumors of his proof began to circulate in March 1984, but many mathematicians were skeptical because de Branges had earlier announced some false results, including a claimed proof of the invariant subspace conjecture in 1964 (incidentally, in December 2008 he published a new claimed proof for this conjecture on his website). It took verification by a team of mathematicians at Steklov Institute of Mathematics in Leningrad to validate de Branges' proof, a process that took several months and led later to significant simplification of the main argument. The original proof uses hypergeometric functions and innovative tools from the theory of Hilbert spaces of entire functions, largely developed by de Branges.
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At this point Daniel Stahl became the executive producer for STO. In September 2011, it was announced on the Star Trek Online website that he would be leaving Cryptic Studios and in an interview with Trek Radio, he revealed is taking a position at Zynga In December 2011, Stahl posts on the Star Trek Online forums that he has returned to Cryptic and will now be the Sr. Producer managing the Foundry (UGC) tools at Cryptic Studios. This announcement was discussed in detail as it was revealed on Subspace Radio. In January 2014, Cryptic Studios announced that Stahl would once again be stepping down as Sr. Producer for Star Trek Online, to pursue a yet unannounced project that Cryptic is working on.
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The most commonly used example of this is the kernel Fisher discriminant. LDA can be generalized to multiple discriminant analysis, where c becomes a categorical variable with N possible states, instead of only two. Analogously, if the class-conditional densities p(\vec x\mid c=i) are normal with shared covariances, the sufficient statistic for P(c\mid\vec x) are the values of N projections, which are the subspace spanned by the N means, affine projected by the inverse covariance matrix. These projections can be found by solving a generalized eigenvalue problem, where the numerator is the covariance matrix formed by treating the means as the samples, and the denominator is the shared covariance matrix. See “Multiclass LDA” above for details.
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In domain decomposition methods, the construction of a coarse problem follows the same principles as in multigrid methods, but the coarser problem has much fewer unknowns, generally only one or just a few unknowns per subdomain or substructure, and the coarse space can be of a quite different type that the original finite element space, e.g. piecewise constants with averaging in balancing domain decomposition or built from energy minimal functions in BDDC. The construction of the coarse problem in FETI is unusual in that it is not obtained as a Galerkin approximation of the original problem, however. In Algebraic Multigrid Methods and in iterative aggregation methods in mathematical economics and Markov chains, the coarse problem is generally obtained by the Galerkin approximation on a subspace.
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Let us write for . In 1995, Luis Rodríguez-Piazza proved that the isometry can be chosen so that every non-zero function in the image is nowhere differentiable. Put another way, if consists of functions that are differentiable at at least one point of , then can be chosen so that This conclusion applies to the space itself, hence there exists a linear map that is an isometry onto its image, such that image under of (the subspace consisting of functions that are everywhere differentiable with continuous derivative) intersects only at : thus the space of smooth functions (with respect to the uniform distance) is isometrically isomorphic to a space of nowhere-differentiable functions. Note that the (metrically incomplete) space of smooth functions is dense in .
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Locating the offending application code for complex transient timing errors could be a very-difficult operating-system analyst problem. These shortcomings persisted for multiple new releases of CICS over a period of more than 20 years, in spite of their severity and the fact that top-quality CICS skills were in high demand and short supply. They were addressed in TS V3.3, V4.1 and V5.2 with the Storage Protection, Transaction Isolation and Subspace features respectively, which utilize operating system hardware features to protect the application code and the data within the same address space even though the applications were not written to be separated. CICS application transactions remain mission-critical for many public utility companies, large banks and other multibillion-dollar financial institutions.
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He has exhibitioned his art throughout Detroit at venues such as Susanne Hilberry Gallery, and at the KW Institute for Contemporary Art and Subspace Gallery and the Kunsthaus Dresden in Germany. His collaborative art project "Detroit Tree of Heaven Woodshop" is permanently installed at the World Workers Museum in Steyr, Austria. In 2005, Cope became the first American artist to officially travel on a U.S. Embassy Cultural Envoy to Ashgabat, Turkmenistan, where he traveled throughout the country lecturing about his work, and visiting and working with Turkmen artists on a joint American- Turkmen exhibition. Most recently, Cope was involved in the foundation and planning of the new Museum of Contemporary Art Detroit where he also filled the role of Assistant Curator until March 2007.
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STAP has been extended for MIMO radar to improve spatial resolution for clutter, using modified SIMO radar STAP techniques.Li, J. and Stoica, P., MIMO Radar Signal Processing, John Wiley & Sons, 2009. . New algorithms and formulations are required that depart from the standard technique due to the large rank of the jammer-clutter subspace created by MIMO radar virtual arrays, which typically involves exploiting the block diagonal structure of the MIMO interference covariance matrix to break the large matrix inversion problem into smaller ones. In comparison with SIMO radar systems, which will have M transmit degrees of freedom, and NL receive degrees of freedom, for a total of M+NL, MIMO radar systems have MNL degrees of freedom, allowing for much greater adaptive spatial resolution for clutter mitigation.
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The Tsirelson space is reflexive () and finitely universal, which means that for some constant , the space contains -isomorphic copies of every finite-dimensional normed space, namely, for every finite-dimensional normed space , there exists a subspace of the Tsirelson space with multiplicative Banach-Mazur distance to less than . Actually, every finitely universal Banach space contains almost-isometric copies of every finite- dimensional normed space,this is because for every , and ε, there exists such that every -isomorph of ℓ∞ contains a -isomorph of ℓ∞n, by James' blocking technique (see Lemma 2.2 in Robert C. James "Uniformly Non-Square Banach Spaces", Annals of Mathematics, Vol. 80, 1964, pp. 542-550), and because every finite-dimensional normed space -embeds in ℓ∞ when is large enough.
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Scott's close work with the designers, and co-ordination of the voice recording process helped to tightly integrate the story into the missions, giving a more sophisticated feel to the story. Due to time constraints, a lot of the initial ideas were dropped from the final version of the game, such as atmospheric battles, and new weapons types like a "subspace missile artillery strike". The team made major improvements to the same FreeSpace engine from the first game. By revamping the core of the graphical engine, and adding 32-bit support, they sped up the interface screens and graphic processing. Hardware acceleration for the graphics was also decided to be a requirement to target the high-end machines of 1999.
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Mapping cylinders are quite common homotopical tools. One use of mapping cylinders is to apply theorems concerning inclusions of spaces to general maps, which might not be injective. Consequently, theorems or techniques (such as homology, cohomology or homotopy theory) which are only dependent on the homotopy class of spaces and maps involved may be applied to f\colon X\rightarrow Y with the assumption that X \subset Y and that f is actually the inclusion of a subspace. Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as "sending" points of X to points of Y, and hence of embedding X within Y, despite the fact that the function need not be one-to-one.
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Water: - It is one of the largest water areas in the state of the subspace of the 1M so it has become clear that every home has a deep water well and a family has access to water, but it is a hygienic problem. Electricity: - There is no generic power but it is used by the powerful force of sunshine in the eastern and north-east of the city to provide better service to the people. Livestock: - It is a good pasture livestock and they are tired of a severe drought and have been shown to all part of the city who have been relocating to the drought- stricken people of the state. Agriculture: - During the drought season, local residents produced nutrients and dried fruit and vegetables.
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The map assigning to its tangent space defines a map from to . (In order to do this, we have to translate the tangent space at each ∈ so that it passes through the origin rather than , and hence defines a -dimensional vector subspace. This idea is very similar to the Gauss map for surfaces in a 3-dimensional space.) This idea can with some effort be extended to all vector bundles over a manifold , so that every vector bundle generates a continuous map from to a suitably generalised Grassmannian--although various embedding theorems must be proved to show this. We then find that the properties of our vector bundles are related to the properties of the corresponding maps viewed as continuous maps.
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Thus, face recognition is transformed to a multivariate, statistical pattern recognition problem. In a similar fashion to appearance-based face recognition, an appearance-based gait recognition approach considers gait as a holistic pattern and uses a full-body representation of a human subject as silhouettes or contours. Gait video sequences are naturally three-dimensional objects, formally named tensor objects, and they are very difficult to deal with using traditional vector-based learning algorithms. In order to deal with these tensor objects effectively, Venetsanopoulos and his research team developed a framework of multilinear subspace learning, so that computation and memory demands are reduced, natural structure and correlation in the original data are preserved, and more compact and useful features can be obtained.
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Let X be an n \times N matrix whose (complete) columns lie in a union of at most k subspaces, each of rank \leq r < n, and assume N \gg kn. Eriksson, Balzano and Nowak showed that under mild assumptions each column of X can be perfectly recovered with high probability from an incomplete version so long as at least CrN\log^2(n) entries of X are observed uniformly at random, with C>1 a constant depending on the usual incoherence conditions, the geometrical arrangement of subspaces, and the distribution of columns over the subspaces. The algorithm involves several steps : (1) local neighborhoods; (2) local subspaces; (3) subspace refinement; (4) full matrix completion. This method can be applied to Internet distance matrix completion and topology identification.
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On the other hand, a vector space of uncountable dimension, endowed with the finest vector topology, then this is a topological vector spaces with the Hahn-Banach extension property that is neither locally convex nor metrizable. A vector subspace of a TVS has the separation property if for every element of such that , there exists a continuous linear functional on such that and for all . Clearly, the continuous dual space of a TVS separates points on if and only if } has the separation property. In 1992, Kakol proved that any infinite dimensional vector space , there exist TVS-topologies on that do not have the HBEP despite having enough continuous linear functionals for the continuous dual space to separate points on .
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To understand why particle statistics work the way that they do, note first that particles are point-localized excitations and that particles that are spacelike separated do not interact. In a flat d-dimensional space M, at any given time, the configuration of two identical particles can be specified as an element of M × M. If there is no overlap between the particles, so that they do not interact directly, then their locations must belong to the space the subspace with coincident points removed. The element describes the configuration with particle I at x and particle II at y, while describes the interchanged configuration. With identical particles, the state described by ought to be indistinguishable from the state described by .
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In algebraic topology, a peripheral subgroup for a space-subspace pair X ⊃ Y is a certain subgroup of the fundamental group of the complementary space, π1(X − Y). Its conjugacy class is an invariant of the pair (X,Y). That is, any homeomorphism (X, Y) → (X′, Y′) induces an isomorphism π1(X − Y) → π1(X′ − Y′) taking peripheral subgroups to peripheral subgroups. A peripheral subgroup consists of loops in X − Y which are peripheral to Y, that is, which stay "close to" Y (except when passing to and from the basepoint). When an ordered set of generators for a peripheral subgroup is specified, the subgroup and generators are collectively called a peripheral system for the pair (X, Y). Peripheral systems are used in knot theory as a complete algebraic invariant of knots.
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While on his mission to locate the Xindi, Archer is briefly transformed into a member of the extinct Loque'eque by a mutagenic virus. Later, he is infected by subspace parasites, creating an alternate timeline in which the Enterprise's mission fails and the Xindi succeed in destroying Earth (although the nature of these parasites mean that this timeline is erased when Phlox's cure for Archer erases the parasites themselves from history). With the help of Daniels, Archer, along with T'Pol, travels back in time to the year 2004 to prevent the release of a Xindi- Reptilian bio-weapon. By the end of the season, Archer is presumed dead when the Xindi superweapon is destroyed, after having convinced three of the five Xindi races that reports of humanity's future conflict with them are wrong.
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Let X be a noetherian scheme. Let C be a subset of the objects of the category of coherent OX- modules which contains the zero sheaf and which has the property that, for any short exact sequence 0 \to A' \to A \to A \to 0 of coherent sheaves, if two of A, A′, and A′ are in C, then so is the third. Let X′ be a closed subspace of the underlying topological space of X. Suppose that for every irreducible closed subset Y of X′, there exists a coherent sheaf G in C whose fiber at the generic point y of Y is a one-dimensional vector space over the residue field k(y). Then every coherent OX-module whose support is contained in X′ is contained in C.EGA III, Théorème 3.1.
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An order unit of a preordered vector space is any element x such that the set [−x, x] is absorbing. The set of all linear functionals on a preordered vector space V that map every order interval into a bounded set is called the order bound dual of V and denoted by Vb If a space is ordered then its order bound dual is a vector subspace of its algebraic dual. A subset A of a vector lattice E is called order complete if for every non-empty subset B ⊆ A such that B is order bounded in A, both \sup B and \inf B exist and are elements of A. We say that a vector lattice E is order complete is E is an order complete subset of E.
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An order unit of a preordered vector space is any element x such that the set [−x, x] is absorbing. The set of all linear functionals on a preordered vector space X that map every order interval into a bounded set is called the order bound dual of X and denoted by Xb. If a space is ordered then its order bound dual is a vector subspace of its algebraic dual. A subset A of an ordered vector space X is called order complete if for every non-empty subset B ⊆ A such that B is order bounded in A, both \sup B and \inf B exist and are elements of A. We say that an ordered vector space X is order complete is X is an order complete subset of X.
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2-dimensional section of Reeb foliation 3-dimensional model of Reeb foliation In mathematics (differential geometry), a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation.Candel and Conlon 2000, Foliations I, p. 5 If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class Cr), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1 (otherwise, C0 is a topological foliation).
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Cat Rapes Dog is a Swedish electropunk, EBM and heavy metal band, formed in 1984 by Joel Rydström and Magnus Fransson. Initially the band self-released a number of albums on cassette, (then a trend followed by many independent acts of their type), plus a cassette album & vinyl singles on Swedish label Front Music Production. Between 1989 and 1991 Cat Rapes Dog released three studio albums on KK Records, a Belgian independent label specialising in new beat, EBM and industrial acts (such as Front Line Assembly or Vomito Negro) that released over 300 titles between 1987 and 1996. Multiple Cat Rapes Dog albums were later released on Swedish independent labels Energy Rekords (Die Krupps, VNV Nation, Front 242, etc.) and Subspace Communications (Covenant, Apoptygma Berzerk, etc.) and on Artoffact Records of Toronto, Canada in 2013.
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In May 2020 it was announced that he would be assuming the title chaire de combinatoire at the College de France beginning in October 2020, though he intends to continue to reside in Cambridge and maintain a part-time affiliation at the University, as well enjoy the privileges of his life Fellowship of Trinity College. Gowers initially worked on Banach spaces. He used combinatorial tools in proving several of Stefan Banach's conjectures in the subject, in particular constructing a Banach space with almost no symmetry, serving as a counterexample to several other conjectures.1998 Fields Medalist William Timothy Gowers from the American Mathematical Society With Bernard Maurey he resolved the "unconditional basic sequence problem" in 1992, showing that not every infinite-dimensional Banach space has an infinite- dimensional subspace that admits an unconditional Schauder basis.
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In general, one cannot rule out "ergodic" flows (which basically means that an orbit is dense in some open set), or "subergodic" flows (which an orbit dense in some submanifold of dimension greater than the orbit's dimension). We can't have self-intersecting orbits. For most "practical" applications of first-class constraints, we do not see such complications: the quotient space of the restricted subspace by the f-flows (in other words, the orbit space) is well behaved enough to act as a differentiable manifold, which can be turned into a symplectic manifold by projecting the symplectic form of M onto it (this can be shown to be well defined). In light of the observation about physical observables mentioned earlier, we can work with this more "physical" smaller symplectic manifold, but with 2n fewer dimensions.
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Similarly to algebraic subvarieties, finite unions and intersections of semialgebraic sets are still semialgebraic sets. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski–Seidenberg theorem says that they are also closed under the projection operation: in other words a semialgebraic set projected onto a linear subspace yields another such (as case of elimination of quantifiers). These properties together mean that semialgebraic sets form an o-minimal structure on R. A semialgebraic set (or function) is said to be defined over a subring A of R if there is some description as in the definition, where the polynomials can be chosen to have coefficients in A. On a dense open subset of the semialgebraic set S, it is (locally) a submanifold.
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Scientists believe an explosion of sufficient magnitude will cause this node to collapse, as evidenced by the destruction of the Lucifer 32 years ago. The detonation of the Lucifers reactors sealed off the Sol jump node in Delta Serpentis and severed all contact with Earth. The plan works but it is pyrrhic victory, as the GTVA loses the Colossus, their only match for Shivan juggernauts, in a diversionary engagement at the other end of the Capella system. With not much left to do but escort the remaining evacuation convoys to the Capella–Vega jump node while a second payload is sent to the jump node, the GTVA soon begins to detect activity from the system's star, which is being bombarded with an intense subspace field by numerous Sathanas- class ships.
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In mathematics, the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty, on the set of all closed subgroups of a locally compact group G. The intuitive idea may be seen in the case of the set of all lattices in a Euclidean space E. There these are only certain of the closed subgroups: others can be found by in a sense taking limiting cases or degenerating a certain sequence of lattices. One can find linear subspaces or discrete groups that are lattices in a subspace, depending on how one takes a limit. This phenomenon suggests that the set of all closed subgroups carries a useful topology. This topology can be derived from the Vietoris topology construction, a topological structure on all non-empty subsets of a space.
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Metamodel of Optimal Prognosis (MOP): The prediction quality of an approximation model may be improved if unimportant variables are removed from the model. This idea is adopted in the Metamodel of Optimal Prognosis (MOP) which is based on the search for the optimal input variable set and the most appropriate approximation model (polynomial or Moving Least Squares with linear or quadratic basis). Due to the model independence and objectivity of the CoP measure, it is well suited to compare the different models in the different subspaces. Multi-disciplinary optimization: The optimal variable subspace and approximation model found by a CoP/MOP procedure can also be used for a pre-optimization before global optimizers (evolutionary algorithms, Adaptive Response Surface Methods, Gradient-based methods, biological-based methods) are used for a direct single-objective optimization.
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The League of Evil Exes was founded when Ramona left Gideon for Toronto. When he finds her six other exes online, he manipulates them into believing that Ramona was responsible for their break-ups. He uses their anger to mold them in the image of evil. They believe this to be a plan to get revenge on her, but Gideon's secret intention is to use Ramona's ability of subspace travel to manipulate the minds of everyone on Earth. The team was founded two weeks prior to the events of Vol. 1. Each time Scott defeats one of them they become (or drop) coins according to their place in the list in similar fashion to the enemies from the River City Ransom NES video game from the '80s. 1\.
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The term reflection is loose, and considered by some an abuse of language, with inversion preferred; however, point reflection is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the identity map – which is also true of other maps called reflections. More narrowly, a reflection refers to a reflection in a hyperplane (n-1 dimensional affine subspace – a point on the line, a line in the plane, a plane in 3-space), with the hyperplane being fixed, but more broadly reflection is applied to any involution of Euclidean space, and the fixed set (an affine space of dimension k, where 1 \leq k \leq n-1) is called the mirror. In dimension 1 these coincide, as a point is a hyperplane in the line.
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In theoretical physics, a twisted sector is a subspace of the full Hilbert space of closed string states in a particular theory over a (good) orbifold. In the first quantized formalism of string theory (or in two-dimensional conformal field theory) the target space is an orbifold M/G if the observables of the string are only defined modulo G. Consequently, the value of the field after one cycle around the closed string need only be the same as its original value modulo some G transformation. i.e. there exists some g\in G such that :X(\sigma+2\pi,\tau)=g[X(\sigma,\tau)] For each conjugacy class of G, we have a different superselection sector (wrt the worldsheet). The conjugacy class consisting of the identity gives rise to the untwisted sector and all the other conjugacy classes give rise to twisted sectors.
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A topological space X is reducible if it can be written as a union X = X_1 \cup X_2 of two closed proper subsets X_1, X_2 of X. A topological space is irreducible (or hyperconnected) if it is not reducible. Equivalently, all non empty open subsets of X are dense or any two nonempty open sets have nonempty intersection. A subset F of a topological space X is called irreducible or reducible, if F considered as a topological space via the subspace topology has the corresponding property in the above sense. That is, F is reducible if it can be written as a union F = (G_1\cap F)\cup(G_2\cap F), where G_1,G_2 are closed subsets of X, neither of which contains F. An irreducible component of a topological space is a maximal irreducible subset.
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According to the Ascended Master Teachings of Anne Bellringer of Rapid City, South Dakota, who began teaching in 1990, Hatonn, an android Pleiadean Master who flies aboard the flying saucer Phoenix (one of the flying saucers of the Ashtar Galactic Command Flying Saucer Fleet, piloted by Hatonn’s partner the Master Soltec), monitors events on Earth for the Galactic Hall of Records at the galactic core. He feeds information about events on Earth via subspace relay to the supercomputers at the "Galactic Hall of Records". However, Bellringer says that this activity is secondary to Hatonn's primary task, which is functioning as the liaison officer between Sanat Kumara and the Pleiadeans in order for him to be able to help Earth safely navigate in 2012 through the approaching so-called "photon belt", said to emanate from the Pleiades.Hatonn and the Photon Belt:Four Winds10.
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The disjoint union space X, together with the canonical injections, can be characterized by the following universal property: If Y is a topological space, and fi : Xi → Y is a continuous map for each i ∈ I, then there exists precisely one continuous map f : X → Y such that the following set of diagrams commute: Characteristic property of disjoint unions This shows that the disjoint union is the coproduct in the category of topological spaces. It follows from the above universal property that a map f : X → Y is continuous iff fi = f o φi is continuous for all i in I. In addition to being continuous, the canonical injections φi : Xi → X are open and closed maps. It follows that the injections are topological embeddings so that each Xi may be canonically thought of as a subspace of X.
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M. R. Brown and B. Hiley showed that, as alternative to its formulation terms of configuration space (x-space), the quantum potential can also be formulated in terms of momentum space (p-space).M. R. Brown: The quantum potential: the breakdown of classical symplectic symmetry and the energy of localisation and dispersion, arXiv.org (submitted on 6 Mar 1997, version of 5 Feb 2002, retrieved 24 July 2011) (abstract)M. R. Brown, B. J. Hiley: Schrodinger revisited: an algebraic approach, arXiv.org (submitted 4 May 2000, version of 19 July 2004, retrieved June 3, 2011) (abstract) In line with David Bohm's approach, Basil Hiley and mathematician Maurice de Gosson showed that the quantum potential can be seen as a consequence of a projection of an underlying structure, more specifically of a non-commutative algebraic structure, onto a subspace such as ordinary space (x-space).
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During 1975-1982 Sergei had published a series of impressive papers on the arithmetic of Siegel modular forms. His PhD thesis contains very fine arithmetic constructions related to the ray classes of ideals of imaginary quadratic fields. Continuing his research on the theory of modular forms, he found an elegant analytical description of the Maass subspace of Siegel modular forms of genus 2, an explicit formula for the generating Hecke series of the symplectic group of genus 3, and the first explicit formulas for the action of degenerate Hecke operators on the space of theta-series In the mid-1980s, switching to the computational complexity of algorithms in algebra and number theory, he found a delicate and simple algorithm for factorization of polynomials over finite fields. The algorithm has a quasi-polynomial complexity under the assumption of generalized Riemann's hypothesis.
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In mathematics, a hyperplane section of a subset X of projective space Pn is the intersection of X with some hyperplane H. In other words, we look at the subset XH of those elements x of X that satisfy the single linear condition L = 0 defining H as a linear subspace. Here L or H can range over the dual projective space of non-zero linear forms in the homogeneous coordinates, up to scalar multiplication. From a geometrical point of view, the most interesting case is when X is an algebraic subvariety; for more general cases, in mathematical analysis, some analogue of the Radon transform applies. In algebraic geometry, assuming therefore that X is V, a subvariety not lying completely in any H, the hyperplane sections are algebraic sets with irreducible components all of dimension dim(V) − 1\.
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An affine hyperplane is an affine subspace of codimension 1 in an affine space. In Cartesian coordinates, such a hyperplane can be described with a single linear equation of the following form (where at least one of the a_i's is non-zero and b is an arbitrary constant): :a_1x_1 + a_2x_2 + \cdots + a_nx_n = b.\ In the case of a real affine space, in other words when the coordinates are real numbers, this affine space separates the space into two half-spaces, which are the connected components of the complement of the hyperplane, and are given by the inequalities :a_1x_1 + a_2x_2 + \cdots + a_nx_n < b\ and :a_1x_1 + a_2x_2 + \cdots + a_nx_n > b.\ As an example, a point is a hyperplane in 1-dimensional space, a line is a hyperplane in 2-dimensional space, and a plane is a hyperplane in 3-dimensional space.
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Thirteen years ago in the calendar, the computer controlling the newly discovered energy source called was infected by a virus that caused it to create the evil energy being known as Messiah who wishes to take over mankind and create a world made for machines. Though sent into subspace by the scientists' sacrifice, Messiah's actions established the formation of the Energy Management Center's Special Ops Unit from three children who were caught in the crossfire, the Go-Busters, and their Buddyloids. Tokumei Sentai Go-Busters T-Shirt Buddyloid Design In the present, 2012 NCE, a mysterious figure named Enter leads a group called Vaglass on incursions to gather enough Enetron to bring Messiah back. However, training for this day, the Go-Busters and their Buddyloids are deployed to combat Vaglass's Metaloids and Megazords to protect the city's Enetron from them.
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In any topological space X, the empty set and the whole space X are both clopen. (regarding the real numbers and the empty set in R) (regarding topological spaces) Now consider the space X which consists of the union of the two open intervals (0,1) and (2,3) of R. The topology on X is inherited as the subspace topology from the ordinary topology on the real line R. In X, the set (0,1) is clopen, as is the set (2,3). This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen. Now let X be an infinite set under the discrete metricthat is, two points p, q in X have distance 1 if they're not the same point, and 0 otherwise.
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In 1985, Saban Entertainment combined footage from GoShogun and Akū Dai Sakusen Srungle (Great Military Operation in Subspace Srungle or Mission Outer Space Srungle), a similar show produced by Kokusai Eiga-sha, to form Macron 1. Taking two (or more) unrelated series and re-editing them to appear as one storyline was common practice in adapting anime series to American television, as the number of episodes in a typical anime frequently fell short of the minimum number required for five-days-a-week syndication in the US market (65). Aside from Macron 1, Voltron: Defender of the Universe, Robotech, and Captain Harlock and the Queen of a Thousand Years were also stitched together in this manner. The combined series Macron 1 was produced and released in the United States, using the same voice cast as Carl Macek's Robotech adaptation.
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The first two steps of the Gram–Schmidt process In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product. The Gram–Schmidt process takes a finite, linearly independent set S = {v1, ..., vk} for and generates an orthogonal set that spans the same k-dimensional subspace of Rn as S. The method is named after Jørgen Pedersen Gram and Erhard Schmidt, but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. In the theory of Lie group decompositions it is generalized by the Iwasawa decomposition. The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix).
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In this geometric description, physical four-dimensional spacetime, M, is considered as a sheaf of gauge-related subspaces of G̃. For the case in which the curvature vanishes, F = 0, there is no excitation of the Lie group, G, and the Higgs field has a vacuum expectation value, Φ=Φ0, corresponding to a positive cosmological constant, Λ = − 12 Φ02, with the vacuum spacetime, as a subspace of G, identified as de Sitter spacetime, satisfying R = −6Λee. Within a Lie group, the Maurer–Cartan form, θ, is the natural frame and determines the Haar measure for integration over the group manifold. With the Killing form of the Lie algebra, this also determines a natural metric and Hodge duality operator on the group manifold. For a deforming Lie group, the Maurer–Cartan form is replaced by the superconnection, G, defined over the entire deforming Lie group manifold via gauge transformation.
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Let K be a closed convex subset of a real vector space V and ∂K be the boundary of K. The solid tangent cone to K at a point x ∈ ∂K is the closure of the cone formed by all half-lines (or rays) emanating from x and intersecting K in at least one point y distinct from x. It is a convex cone in V and can also be defined as the intersection of the closed half-spaces of V containing K and bounded by the supporting hyperplanes of K at x. The boundary TK of the solid tangent cone is the tangent cone to K and ∂K at x. If this is an affine subspace of V then the point x is called a smooth point of ∂K and ∂K is said to be differentiable at x and TK is the ordinary tangent space to ∂K at x.
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In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras. It states that if G is a connected and solvable linear algebraic group defined over an algebraically closed field and :\rho\colon G \to GL(V) a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L of V such that : \rho(G)(L) = L. That is, ρ(G) has an invariant line L, on which G therefore acts through a one- dimensional representation. This is equivalent to the statement that V contains a nonzero vector v that is a common (simultaneous) eigenvector for all \rho(g), \,\, g \in G . It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one.
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Quite generally, the group of homographies with coefficients in K acts on the projective line P1(K). This group action is transitive, so that P1(K) is a homogeneous space for the group, often written PGL2(K) to emphasise the projective nature of these transformations. Transitivity says that there exists a homography that will transform any point Q to any other point R. The point at infinity on P1(K) is therefore an artifact of choice of coordinates: homogeneous coordinates :[X : Y] \sim [\lambda X : \lambda Y] express a one- dimensional subspace by a single non-zero point lying in it, but the symmetries of the projective line can move the point to any other, and it is in no way distinguished. Much more is true, in that some transformation can take any given distinct points Qi for to any other 3-tuple Ri of distinct points (triple transitivity).
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Space folding permits nearly-instantaneous ultra-long distance travel: a space-fold transports a spacecraft in a very short amount of time by first swapping the location of the spacecraft with super dimension space or subspace, and then swapping the Super Dimension space with the space at the destination. According to U.N. Spacy First Lieutenant Hayase Misa during Space War I (2009–2012) an hour passes in super dimension space as approximately ten days passes in normal space. One of the latest Macross TV series, Macross Frontier, further expands on that concept by introducing fold faults or dislocations, which further retard fold travel and interfere with fold communications. Also explained in Macross Frontier are the limitations of space folding, such as the geometric increase in energy requirement with the mass of the object to be folded, which prevents very large objects from being folded with ease across vast distances.
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Another seminal contribution to nonlinear learning models was to enable a nonparametric basis function pursuit via sparse kernel-based learning, what led to the first approach to tensor completion and extrapolation with applications to spectrum cartography, network flow prediction, and imputation of gene expression data. A cornerstone of data science is learning from big data, where the latter refers to the volume (dimensionality and number) of data, their velocity (of streaming data), and variety (multimodality). To extract the sought information that often resides in small subspaces, and cope with subsampled or missing data, Giannakis and collaborators put forth an online censoring approach for large- scale regressions and trackers, where only informative data are retained for learning. Instead of censoring, they also adopted a limited number of random data projections (sketches) and validated whether they contain informative data, before employing them for (subspace) clustering to obtain desirable performance-complexity tradeoffs.
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Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner. One can thus ask whether a given topological ring R is complete. If it is not, then it can be completed: one can find an essentially unique complete topological ring S that contains R as a dense subring such that the given topology on R equals the subspace topology arising from S. If the starting ring R is metric, the ring S can be constructed as a set of equivalence classes of Cauchy sequences in R, this equivalence relation makes the ring S Hausdorff and using constant sequences (which are Cauchy) one realises a (uniformly) continuous morphism (CM in the sequel) such that, for all CM where is Hausdorff and complete, there exists a unique CM such that f=g\circ c. If R is not metric (as, for instance, the ring of all real-variable rational valued functions i.e.
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In the same paper, Dehn also constructed an example of a non-Legendrian geometry where there are infinitely many lines through a point not meeting another line, but the sum of the angles in a triangle exceeds . Riemann's elliptic geometry over Ω(t) consists of the projective plane over Ω(t), which can be identified with the affine plane of points (x:y:1) together with the "line at infinity", and has the property that the sum of the angles of any triangle is greater than The non-Legendrian geometry consists of the points (x:y:1) of this affine subspace such that tx and ty are finite (where as above t is the element of Ω(t) represented by the identity function). Legendre's theorem states that the sum of the angles of a triangle is at most , but assumes Archimedes's axiom, and Dehn's example shows that Legendre's theorem need not hold if Archimedes' axiom is dropped.
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There are many biclustering algorithms developed for bioinformatics, including: block clustering, CTWC (Coupled Two-Way Clustering), ITWC (Interrelated Two-Way Clustering), δ-bicluster, δ-pCluster, δ-pattern, FLOC, OPC, Plaid Model, OPSMs (Order-preserving submatrixes), Gibbs, SAMBA (Statistical-Algorithmic Method for Bicluster Analysis), Robust Biclustering Algorithm (RoBA), Crossing Minimization, cMonkey, PRMs, DCC, LEB (Localize and Extract Biclusters), QUBIC (QUalitative BIClustering), BCCA (Bi- Correlation Clustering Algorithm) BIMAX, ISA and FABIA (Factor Analysis for Bicluster Acquisition), runibic, and recently proposed hybrid method EBIC (Evolutionary-based biclustering), which was shown to detect multiple patterns with very high accuracy. Biclustering algorithms have also been proposed and used in other application fields under the names coclustering, bidimensional clustering, and subspace clustering. Given the known importance of discovering local patterns in time-series data, recent proposals have addressed the biclustering problem in the specific case of time series gene expression data. In this case, the interesting biclusters can be restricted to those with contiguous columns.
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The problem of finding sets of n points minimizing the number of convex quadrilaterals is equivalent to minimizing the crossing number in a straight-line drawing of a complete graph. The number of quadrilaterals must be proportional to the fourth power of n, but the precise constant is not known. It is straightforward to show that, in higher- dimensional Euclidean spaces, sufficiently large sets of points will have a subset of k points that forms the vertices of a convex polytope, for any k greater than the dimension: this follows immediately from existence of convex k-gons in sufficiently large planar point sets, by projecting the higher- dimensional point set into an arbitrary two-dimensional subspace. However, the number of points necessary to find k points in convex position may be smaller in higher dimensions than it is in the plane, and it is possible to find subsets that are more highly constrained.
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Let V be a real topological vector space and let S be a Borel- measurable subset of V. S is said to be prevalent if there exists a finite- dimensional subspace P of V, called the probe set, such that for all v ∈ V we have v + p ∈ S for λP-almost all p ∈ P, where λP denotes the dim(P)-dimensional Lebesgue measure on P. Put another way, for every v ∈ V, Lebesgue-almost every point of the hyperplane v + P lies in S. A non-Borel subset of V is said to be prevalent if it contains a prevalent Borel subset. A Borel subset of V is said to be shy if its complement is prevalent; a non- Borel subset of V is said to be shy if it is contained within a shy Borel subset. An alternative, and slightly more general, definition is to define a set S to be shy if there exists a transverse measure for S (other than the trivial measure).
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An archetypical irreducible reductive dual pair of type II consists of a pair of general linear groups and arises as follows. Let U and V be two vector spaces over F, X = U ⊗F V be their tensor product, and Y = HomF(X, F) its dual. Then the direct sum W = X ⊕ Y can be endowed with a symplectic form such that X and Y are lagrangian subspaces, and the restriction of the symplectic form to X × Y ⊂ W × W coincides with the pairing between the vector space X and its dual Y. If G = GL(U) and G′ = GL(V), then both these groups act linearly on X and Y, the actions preserve the symplectic form on W, and (G, G′) is an irreducible reductive dual pair. Note that X is an invariant lagrangian subspace, hence this dual pair is of type II. An archetypical irreducible reductive dual pair of type I consists of an orthogonal group and a symplectic group and is constructed analogously.
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As Janeway explains, Omega is unstable and even the explosion of one particle out in space can nullify subspace for many light years around it, rendering faster-than-light travel impossible within that area. Moving to the coordinates of the explosion they encounter the planet and its resident alien race that created it. The society is on the brink of economic failure and is making Omega particles to “give their children a chance at a future.” Seven of Nine displays an interest in the scientists' methods, however, hoping to save the Omega particles and harness them because she believes them to be perfection—infinite parts working together as one (like the Borg)—despite ample Starfleet and Borg evidence of their incredible danger: The Borg, referring to the Omega particle as “Particle 010,” are expected to assimilate it at all costs, even though they have experienced the loss of a large quantity of Borg vessels to Omega particle explosions while trying to harness the power of the substance.
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The game begins 32 years after the events in Descent: FreeSpace. Following the end of the Great War, both the GTA and PVE cemented their alliance by combining together to form the Galactic Terran–Vasudan Alliance (GTVA)—a single entity formed to cement the alliance between the Terran and Vasudan races after the destruction of Vasuda Prime by the Lucifer and the subsequent collapse of all subspace nodes to the Sol system as a result of the superdestroyer's destruction inside the Sol–Delta Serpentis jump node. Despite this alliance, however, opposition still exists to this union in the form of a faction of Terrans led by Great War veteran, Admiral Aken Bosch, who leads the rebel group under the banner of the Neo-Terran Front (NTF). The NTF's rebellion led to the faction gaining control over the Sirius, Polaris and Regulus star systems, while engaging the GTVA for 18 months, before launching attacks on the Vasudan systems of Deneb and Alpha Centauri.
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The key point in this definition is that if and only if the difference of v1 and v2 lies in W.Some authors (such as ) choose to start with this equivalence relation and derive the concrete shape of V/W from this. This way, the quotient space "forgets" information that is contained in the subspace W. The kernel ker(f) of a linear map consists of vectors v that are mapped to 0 in W. The kernel and the image are subspaces of V and W, respectively. The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field F) is an abelian category, that is, a corpus of mathematical objects and structure-preserving maps between them (a category) that behaves much like the category of abelian groups. Because of this, many statements such as the first isomorphism theorem (also called rank–nullity theorem in matrix-related terms) :V / ker(f) ≡ im(f).
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The number of independent pieces of information that go into the estimate of a parameter are called the degrees of freedom. In general, the degrees of freedom of an estimate of a parameter are equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself (most of the time the sample variance has N − 1 degrees of freedom, since it is computed from N random scores minus the only 1 parameter estimated as intermediate step, which is the sample mean). Mathematically, degrees of freedom is the number of dimensions of the domain of a random vector, or essentially the number of "free" components (how many components need to be known before the vector is fully determined). The term is most often used in the context of linear models (linear regression, analysis of variance), where certain random vectors are constrained to lie in linear subspaces, and the number of degrees of freedom is the dimension of the subspace.
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Sakurai chose to use these trailers, which benefit from Internet sharing, as opposed to including a story campaign similar to the Subspace Emissary mode featured in Brawl, as he believed the impact of seeing the mode's cinematic cutscenes for the first time was ruined by people uploading said scenes to video sharing websites. At E3 2013, Sakurai stated that the tripping mechanic introduced in Brawl was removed, with him also stating that the gameplay was between the fast-paced and competitive style of Melee and the slower and more casual style of Brawl. While the games didn't feature cross-platform play between the Wii U and 3DS, due to each version featuring certain exclusive stages and gamemodes, there is an option to transfer customized characters and items between the two versions. The game builds upon the previous game's third-party involvement with the addition of third-party characters such as Capcom's Mega Man and Bandai Namco's Pac-Man, as well as the return of Sega's Sonic the Hedgehog.
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This is the standard concept. Heuristically, if we have a space M for which each point m ∊ M corresponds to an algebro-geometric object Um, then we can assemble these objects into a tautological family U over M. (For example, the Grassmannian G(k, V) carries a rank k bundle whose fiber at any point [L] ∊ G(k, V) is simply the linear subspace L ⊂ V.) M is called a base space of the family U. We say that such a family is universal if any family of algebro-geometric objects T over any base space B is the pullback of U along a unique map B → M. A fine moduli space is a space M which is the base of a universal family. More precisely, suppose that we have a functor F from schemes to sets, which assigns to a scheme B the set of all suitable families of objects with base B. A space M is a fine moduli space for the functor F if M represents F, i.e., there is a natural isomorphism τ : F → Hom(−, M), where Hom(−, M) is the functor of points.
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