Real Embedding Research Articles

Zoll metrics, branched covers, and holomorphic disks

We strengthen our previous results [12] regarding the moduli spaces of Zoll metrics and Zoll projective structures on S. In particular, we describe a concrete, open condition which suffices to guarantee that a totally real embedding RP ↪→ CP2 arises from a unique Zoll projective structure on the two-sphere. Our methods ultimately reflect the special role such structures play in the initial value problem for the three-dimensional Lorentzian Einstein–Weyl equations.

Fast Dimension Reduction Using Rademacher Series on Dual BCH Codes

The Fast Johnson-Lindenstrauss Transform (FJLT) was recently discovered by Ailon and Chazelle as a novel technique for performing fast dimension reduction with small distortion from ed2 to ed2 in time O(max{d log d,k3}). For k in [Ω(log d), O(d1/2)] this beats time O(dk) achieved by naive multiplication by random dense matrices, an approach followed by several authors as a variant of the seminal result by Johnson and Lindenstrauss (JL) from the mid 80's. In this work we show how to significantly improve the running time to O(d log k) for k = O(d1/2−Δ), for any arbitrary small fixed Δ. This beats the better of FJLT and JL. Our analysis uses a powerful measure concentration bound due to Talagrand applied to Rademacher series in Banach spaces (sums of vectors in Banach spaces with random signs). The set of vectors used is a real embedding of dual BCH code vectors over GF(2). We also discuss the number of random bits used and reduction to e1 space. The connection between geometry and discrete coding theory discussed here is interesting in its own right and may be useful in other algorithmic applications as well.

Nonlinear gravitons, null geodesics, and holomorphic disks

We develop a global twistor correspondence for pseudo-Riemannian conformal structures of signature (++--) with self-dual Weyl curvature. Near the conformal class of the standard indefinite product metric on S2×S2, there is an infinite-dimensional moduli space of such conformal structures, and each of these has the surprising global property that its null geodesics are all periodic. Each such conformal structure arises from a family of holomorphic disks in CP3 with boundary on some totally real embedding of RP3 into CP3. Some of these conformal classes are represented by scalar-flat indefinite Kähler metrics, and our methods give particularly sharp results in connection with this special case

REAL SURFACES IN ELLIPTIC SURFACES

We study the structure of complex points on real surfaces, embedded into complex elliptic surfaces. We show, for example, that any compact connected surface has a totally real embedding into a blow-up of a K3 surface. We also exhibit smooth disc bundles over compact connected orientable surfaces that have a Stein structure as Stein domains inside elliptic surfaces.

Linking Class of Lagrangian or Totally Real Embeddings

Given a totally real embedding j of the 2-torus \(\mathbb{T}^{\text{2}}\) into ℂ2, one defines a 1-class σ1 – its linking class – which is a tool to detect arcwise connected components of the space of totally real embeddings EmbTr(\(\mathbb{T}^{\text{2}}\), ℂ2). We generalize the construction of the linking class to any totally real embedding j of a connected, oriented, compact manifold without boundary Mn into ℂn. We obtain an (n − 1)-class σn− 1 which is still an invariant for isotopy classes of totally real embeddings. We show that this class is nontrivial by computing it for some families of totally real embeddings. We then study the relationship between isotopy classes of ordinary embeddings and the linking class. With additional assumptions on Mn (n ≥ 4 and Mn parallelizable) we obtain the following: two totally real embeddings of Mn into ℂn which belong to the same isotopy class of totally real immersion, belong to the same isotopy class of ordinary embedding if and only if (1) their linking classes are the same (if n odd); (2) the images of their linking classes by the coefficient homomorphism μ: Hn− 1 (Mn, ℤ) → Hn− 1 (Mn, ℤ2) are the same (if n even).

On totally real spheres in complex space

Let M ,N be two totally real and real analytic submanifolds in Cn . We say that M and N are biholomorphically equivalent if there is a biholomorphic mapping F defined in a neighborhood of M such that F (M ) = N . As a standard fact of complexification, one knows that all totally real and real analytic embeddings of M in C n are biholomorphically equivalent if M is of maximal dimension n . However, the topology of the manifold plays a major role in the existence of totally real immersions or embeddings. For instance, R.O. Wells [23] proved that if an n-dimensional compact and orientable manifold M admits a totally real embedding in Cn , then its Euler number must vanish. It was also observed by Wells that if M is a manifold of dimension n and it admits a totally real immersion in Cn , then its complexified tangent bundle T cM = TM⊗C is trivial. Conversely, the triviality of T cM also implies the existence of totally real immersions of M in Cn . This was obtained by M.L. Gromov in [11] through the method of convex integration. A stronger result due to J.A. Lees [17] says that M also admits Lagrangian immersions in Cn . The sphere S k : x 2 1 + . . . + x 2 k+1 = 1 in R k+1 gives us a trivial totally real embedding of S k in Ck+1. On the other hand, the works of Gromov [11], AhernRudin [1] and Stout-Zame [21] tell us that S k admits a totally real and real analytic embedding in Ck if and only if k = 1, 3. Our main result is the following. Theorem 1.1. If k ≤ 4, all totally real and real analytic embeddings of S k in Cn are biholomorphically equivalent. If k ≥ 5 and nk = k + 2[ k−1 4 ], there exist totally

On-board and real-time expert control

Development of an on board real time expert system for controlling engineering systems requires tradeoffs between AI and real time approaches. The article presents an alternative that lies between embedding AI in real time and embedding real time in AI. We used this framework to develop an expert system development environment for maintaining and updating information about an evolving system as its state changes because of abnormal events such as faults, or as a consequence of external actions taken upon the system. We've successfully applied it in several engineering domains such as process control, medical monitoring, and onboard diagnosis. The author explains how it was applied to a system that controls automobile manoeuvring.

ON THE ARITHMETIC 2-BRIDGE KNOTS AND LINK ORBIFOLDS AND A NEW KNOT INVARIANT

Let (p/q, n) denote the orbifold with singular set the two bridge knot or link p/q and isotropy group cyclic of orden n. An algebraic curve [Formula: see text] (set of zeroes of a polynomial r(x, z)) is associated to p/q parametrizing the representations of [Formula: see text] in PSL [Formula: see text]. The coordinates x, z, are trace(A2)=x, trace(AB)=z where A and B[Formula: see text] are the images of canonical generators a, b of [Formula: see text]. Let (xn, zn) be the point of [Formula: see text] corresponding to the hyperbolic orbifold (p/q, n). We prove the following result: The (orbifold) fundamental group of (p/q, n) is arithmetic if and only if the field Q(xn, zn) has exactly one complex place and ϕ(xn)<ϕ(zn)<2 for every real embedding [Formula: see text]. Consider the angle α for which the cone-manifold (p/q, α) is euclidean. We prove that 2cosα is an algebraic number. Its minimal polynomial (called the h-polynomial) is then a knot invariant. We indicate how to generalize this h-polynomial invariant for any hyperbolic knot. Finally, we compute h-polynomials and arithmeticity of (p/q, n) with p≦40, and (p/q, n) with p≦99q2≡1 mod p. We finish the paper with some open problems.

Minimal Torsion in Isogeny Classes of Elliptic Curves

Let K be a field finitely generated over its prime field, and let $w(K)$ denote the number of roots of unity in K. If K is of characteristic 0, then there is an integer D, divisible only by those primes dividing $w(K)$, such that for any elliptic curve $E/K$ without complex multiplication over K, there is an elliptic curve $E\prime /K$ isogenous to E such that $E\prime {(K)_{{\text {tors}}}}$ is of order dividing D. In case K admits a real embedding, we show $D = 2$, and a nonuniform result is proved in positive characteristic.

Minimal torsion in isogeny classes of elliptic curves

Let K be a field finitely generated over its prime field, and let w ( K ) w(K) denote the number of roots of unity in K. If K is of characteristic 0, then there is an integer D, divisible only by those primes dividing w ( K ) w(K) , such that for any elliptic curve E / K E/K without complex multiplication over K, there is an elliptic curve E ′ / K E\prime /K isogenous to E such that E ′ ( K ) tors E\prime {(K)_{{\text {tors}}}} is of order dividing D. In case K admits a real embedding, we show D = 2 D = 2 , and a nonuniform result is proved in positive characteristic.

The Bass-Milnor-Serre theorem for nonstandard models in Peano arithmetic

The aim of this paper is to extend the Bass-Milnor-Serre theorem to the nonstandard rings associated with nonstandard models of Peano arithmetic, in brief to Peano rings.First, we recall the classical setting. Let k be an algebraïc number field, and let θ be its ring of integers. Let n be an integer ≥ 3, and let G be the group Sln(θ) of (n, n) matrices of determinant 1 with coefficients in θ.The profinite topology in G is the topology having as fundamental system of open subgroups the subgroups of finite index.Congruence subgroups of finite index of G are the kernels of the maps Sln(θ) → Sln(θ/I) for which all ideals I of θ are of finite index. By taking these subgroups as a fundamental system of open subgroups, one obtains the congruence topology on G. Every open set for this topology is open in the profinite topology.We denote by Ḡ (resp., Ĝ) the completion of G for the congruence (resp., profinite) topology.The Bass-Milnor-Serre theorem [1] consists of the two following statements:(A) If k admits a real embedding, then we have an exact sequenceThat is, Ĝ and Ḡ are isomorphic.(B) If k is totally imaginary, then one has an exact sequencewhere μ(k)is the group of the roots of unity of k.

An Etale Cohomology Duality Theorem for Number Fields with A Real Embedding

An Etale Cohomology Duality Theorem for Number Fields with A Real Embedding

An étale cohomology duality theorem for number fields with a real embedding

The restriction on 2 2 -primary components in the Artin-Verdier duality theorem [2] has been eliminated by Zink [9], who has shown that the sheaf of units for the étale topology over the ring of integers of any number field acts as a dualizing sheaf for a modified cohomology of sheaves. The present paper provides an alternate means of removing the 2 2 -primary restriction. Like Zink’s, it involves a topology which includes infinite primes, but it avoids modified cohomology and will be more directly applicable in the proof of a theorem of Lichtenbaum regarding zeta- and L L -functions [4, 5]. Related results—including the cohomology of units sheaves, the norm theorem, and punctual duality theorem of Mazur [6]—are also affected by the use of a topology including the infinite primes. The corresponding results in the new setting are included here.

Some totally real embeddings of three-manifolds

LetV be a complex hypersurface in an open subset of ℂ3, and letM be a smooth compact real hypersurface inV. Using a theorem of Gromov we prove that there exist small C1 perturbations\(\tilde M\) ofM in ℂ3 such that\(\tilde M\) is a totally real submanifold of ℂ3. As a consequence we show that certain quotients of the three-sphere admit totally real embeddings into ℂ3. In some special cases including the real projective three-space we find explicit totally real embeddings into ℂ3. Our construction is similar to that of Ahern and Rudin who found a totally real embedding of the three-sphere into ℂ3.

Totally real embeddings of 𝑆³ in 𝐶³

An explicit totally real embedding of S 3 {S^3} in C 3 {{\text {C}}^3} is exhibited. It has the form F ( z , w ) = ( z , w , P ( z , w ) ) F(z,w) = (z,w,P(z,w)) where P P is a (nonholomorphic) polynomial of degree 4, and ( z , w ) (z,w) ranges over the unit sphere in C 2 {{\text {C}}^2} .