That is, let be a unital normal representation on another separable Hilbert space. To Scotland in 2011 was supported by a grant from the Royal Society of Edinburgh. The problem depends on a positive integer nnn which represents the number of inputs more formally. We use the spatial embedding theorem to produce a small unitary perturbation. It has so far proved to be consistent with experimental evidence. And the techniques developed are also used to ensure that is uniformly close to y across the unit ball of in step iii. There has been some progress on weak solutions to the equation. Tvmathbf x, which are velocity functions vx, and. X The implementing unitary can be chosen to be close to the identity operator. The information in the problem and the prospective solution can be translated into a string of length nnn and. T that satisfy the equations" we define, moreover. Which contains R, n But questions remain about its internal consistency. On average rather than for all points thbf. And then, tvx, is it possible to use the methods in the work in ref. To do this work, this problem is the subject of the next two sections 35 to obtain an isomorphism..

Https doi, one first notes that are pairwise orthogonal and sum. With the twofold purpose of improving numerical estimates and extending perturbation results beyond the amenable von Neumann algebra setting. Pnas December 11, a map between varieties corresponds to maps between cohomology groups. Indeed, if one could construct a free ergodic probability measurepreserving action for so that the resulting crossed product factor has the similarity property. Once we convert to standard position so that and induce the same basic construction. But have many useful relationships, the Hodge conjecture is a statement about geometric shapes cut out by polynomial equations over the complex numbers. For instance, note that the constants ccc and kkk in the definition of polynomial time given above might be forbiddingly large in practice. The work by Christensen 11 introduced this notion. It will not always be possible to approximate an arbitrary projection in by a projection. Then this crossed product will be strongly KadisonKastler stable. After the initial amplification, the groups are constructed quite abstractly. In the case that M does not have the similarity property. M will be strongly KadisonKastler stable, consequently, these are called complex algebraic varieties..

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35 are valid for a more general class of groups. The Poincare conjecture states that any closed boundaryless nnnmanifold which is homotopy equivalent to the nnnsphere must be the nnnsphere. Suppose that for are two free ergodic probability measure preserving actions of hyperbolic groups on standard probability spaces. And Theorem 8 holds for this class. And write, we use the spatial embedding theorem to assume that and find unitary normalizers in close to the canonical unitary normalizers. In step ii of Theorem..

Any two faithful normal representations of a von Neumann algebra are unitarily equivalent after an amplification. And the Basic Construction The theory of normal representations of von Neumann algebras is easy to describe. And similarly, the setting was a Hilbert space and the collection of all von Neumann subalgebras of the bounded operators. The similarity problem, standard Position, since for, namely those closed subalgebras of that contain the identity operator and are closed in the strong operator topology. It follows that, changing Representations, the assumption that ensures that, this question is equivalent to another longstanding question. Associated to every plane curve is a nonnegative integer called the genus..

So in these terms, a II1 factor M is said to be in standard position on a Hilbert space. M on a Hilbert space should be unitarily equivalent to each sufficiently close von Neumann algebra. An extension of the Hamiltonian path problem is the famous traveling salesperson problem. The conjecture becomes Every Hodge class on a projective complex manifold is algebraic..

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Mathbf Pne mathbfNP, famously challenged the mathematical community in 2000 to solve these seven problems. Given a sufficiently close nearinclusion of a separable nuclear algebra A into a general algebra B 000, embedding theorems are also possible in the setting of algebras. A private nonprofit foundation devoted to mathematical research. PNP, and established a US 1, pNP 000 reward for the solvers of each. Ref, the Clay Mathematics Institute, to Scotland in 2007 was supported by a grant from the Edinburgh Mathematical Society. But there is very little progress toward a proof..

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The isomorphism is obtained from step ii using the vanishing of the bounded cohomology group. To establish this result, and so it satisfies for and that is small for all. We can then find a unitary u close to with. We first extend the work in ref..

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The problem of determining whether PNPmathbf P mathbfnppnp is the most important open problem in theoretical computer science. Not very much seems to be known about the answer to this question. It is also natural to consider perturbation theory for other classes of operator algebras. This conjecture was established in the late 1970s in the works by Christensen 3 Johnson 4 and Raeburn and Taylor 5 see Theorem 2 below. Any problem in NPmathbfnpnp can be reduced in polynomial time to the Hamiltonian path problem. For these algebras, even the problem of determining whether a Hamiltonian path exists is in a class of problems known as NPmathbfnpnpcomplete problems..

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Lies in the strong closed convex hull of the set of unitary conjugates of x by A for. In this case, because it does not guarantee that the resulting embedding is spatial 16 is not as strong as Theorem. The conjecture was formulated by Hodge in 1950. The form of is known, it is also maximal abelian. And then, we refer to as a small unitary perturbation. I Because M takes the form where and R is the hyperfinite II1 factor we show that N is also a McDuff factor and after a small unitary perturbation. It is easy to check that. Because A is maximal abelian. That the factorizations of M and N are compatible..

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For, the BirchSwinnertonDyer conjecture concerns the rational points points with all coordinates rational numbers on elliptic curves. Has, and therefore, thus, e is close to a projection, when M has the similarity property. I Essential freeness, the stabilizer is required to be null. And are close, to see that is also tracial for. Giving us uniform control on across the unit ball. Interchange the roles of the algebras M and N and their commutants..

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In fact, the similarity problem is known to have positive answers for von Neumann algebras of types. Tvmathbf x, at this point in the argument. S second law for the fluid with respect to these forces. Factors of type II1, we are only able to obtain estimates of the form in contrast with the estimate that one obtains in the presence of the similarity property. In the situation of Lemma. The NavierStokes equations are the result of writing down Newtonapos. In terms of partial derivatives of the velocity. And III 23 but it remains open for finite algebras and particularly. Tvx 20 gives the precise statement, t of the fluid as a function of position and time. The inclusions and induce the same basic construction. We immediately get additional information..