Real Embedding Research Articles

The Frobenius problem over number fields with a real embedding

The Frobenius problem over number fields with a real embedding

Width of SL(n,𝒪 S ,I)

We give an estimate for the width of the congruence subgroup in Tits–Vaserstein generators, where is a localization of the ring of integers in a number field K. We assume that either K has a real embedding, or the ideal I is prime to the number of roots of unity in K.

Quantum-Inspired Complex-Valued Language Models for Aspect-Based Sentiment Classification.

Aiming at classifying the polarities over aspects, aspect-based sentiment analysis (ABSA) is a fine-grained task of sentiment analysis. The vector representations of current models are generally constrained to real values. Based on mathematical formulations of quantum theory, quantum language models have drawn increasing attention. Words in such models can be projected as physical particles in quantum systems, and naturally represented by representation-rich complex-valued vectors in a Hilbert Space, rather than real-valued ones. In this paper, the Hilbert Space representation for ABSA models is investigated and the complexification of three strong real-valued baselines are constructed. Experimental results demonstrate the effectiveness of complexification and the outperformance of our complex-valued models, illustrating that the complex-valued embedding can carry additional information beyond the real embedding. Especially, a complex-valued RoBERTa model outperforms or approaches the previous state-of-the-art on three standard benchmarking datasets.

CR regular embeddings of 𝑆⁴ⁿ⁻¹ in ℂ²ⁿ⁺¹

Ahern and Rudin have given an explicit construction of a totally real embedding of S 3 S^3 in C 3 \mathbb {C}^3 . As a generalization of their example, we give an explicit example of a CR regular embedding of S 4 n − 1 S^{4n-1} in C 2 n + 1 \mathbb {C}^{2n+1} . Consequently, we show that the odd dimensional sphere S 2 m − 1 S^{2m-1} with m > 1 m>1 admits a CR regular embedding in C m + 1 \mathbb {C}^{m+1} if and only if m m is even.

Aspects of Enumerative Geometry with Quadratic Forms

Using the motivic stable homotopy category over a field k , a smooth variety X over k has an Euler characteristic \chi(X/k) in the Grothendieck-Witt ring \operatorname{GW}(k) . The rank of \chi(X/k) is the classical \mathbb{Z} -valued Euler characteristic, defined using singular cohomology or étale cohomology, and the signature of \chi(X/k) under a real embedding \sigma:k\to \mathbb{R} gives the topological Euler characteristic of the real points X^\sigma(\mathbb{R}) . We develop tools to compute \chi(X/k) , assuming k has characteristic \neq 2 and apply these to refine some classical formulas in enumerative geometry, such as formulas for the top Chern class of the dual, symmetric powers and tensor products of bundles, to identities for the Euler classes in Chow-Witt groups. We also refine the classical Riemann-Hurwitz formula to an identity in \operatorname{GW}(k) and compute \chi(X/k) for hypersurfaces in \mathbb{P}^{n+1}_k defined by a polynomial of the form \sum_{i=0}^{n+1}a_iX_i^m ; this latter includes the case of an arbitrary quadric hypersurface. This paper is a revision of [ M. Levine ,"Toward an enumerative geometry with quadratic forms", Preprint, ].

On equivariant and motivic slices

Let $k$ be a field with a real embedding. We compare the motivic slice filtration of a motivic spectrum over $Spec(k)$ with the $C_2$-equivariant slice filtration of its equivariant Betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. In particular, we show that, up to reindexing, the towers agree for all spectra obtained from localized quotients of $MGL$ and $MR$, and for motivic Landweber exact spectra and their realizations. As a consequence, we deduce that equivariant spectra obtained from localized quotients of $MR$ are even in the sense of Hill--Meier, and give a computation of the slice spectral sequence converging to $\pi_{*,*}BP\langle n \rangle/2$ for $1 \le n \le \infty$.

Canonical complex extensions of Kähler manifolds

Given a complex manifold $X$, any K\"ahler class defines an affine bundle over $X$, and any K\"ahler form in the given class defines a totally real embedding of $X$ into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of $X$. For compact K\"ahler manifolds of non-negative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of Lempert--Sz\H{o}ke and to the existence question for good complexifications in the sense of Totaro. Moreover, we study projective manifolds for which the induced affine bundle is not just Stein but affine and prove that these must have big tangent bundle. In the course of our investigation, we also obtain a simpler proof of a result of Yang on manifolds having non-negative holomorphic bisectional curvature and big tangent bundle.

On the bounded generation of arithmetic SL2

Let K be a number field and S be a finite set of primes of K containing the archimedean valuations. Let 𝒪 be the ring of S-integers in K Morgan, Rapinchuck, and Sury [A. V. Morgan et al, Algebra Number Theory 12, 1949-1974 (2018)] have proved that if the group of units [Formula: see text] is infinite, then every matrix in SL2(𝒪) is a product of at most 9 elementary matrices. We prove that under the additional hypothesis that K has at least 1 real embedding or S contains a finite place we can get a product of at most 8 elementary matrices. If we assume a suitable generalized Riemann hypothesis, then every matrix in SL2(𝒪) is the product of at most 5 elementary matrices if K has at least 1 real embedding, the product of at most 6 elementary matrices if S contains a finite place, and the product of at most 7 elementary matrices in general.

Hasse Principle for Rost Motives

AbstractWe prove a Hasse principle for binary direct summands of the Chow motive of a smooth projective quadric $Q$ over a number field $F$. Besides, we show that such summands are twists of Rost motives. In the case when $F$ has at most one real embedding we describe a complete motivic decomposition of $Q$.

The Chinese are taking over: Chinese small entrepreneurs in the Cayo District of Belize

The transnational entrepreneurship debate discusses the economic and entrepreneurial consequences of transnational relations and trade for ethnic entrepreneurs. The multi-ethnic society of Belize is an example where transnationalism is an important factor because of its implications for the history, roots, and future of ethnic entrepreneurs. Our case study of ethnically Chinese entrepreneurs points to the shadow of ethnic entrepreneurship that is in danger of being forgotten in the more business and politically oriented discussions about transnationalism. In a national context, where a discourse of Chinese transnational influence has developed, locally-based ethnically Chinese entrepreneurs appear to be subject to stereotyping and stigmatisation. Chinese entrepreneurs' real embedding in transnational networks may be unclear and may be easy targets for gossip and unwarranted generalisation. We thus argue that the investigating a local, rural context may reveal the ambiguous social consequences of economic prosperity brought about by transnationalism.

Totally real embeddings with prescribed polynomial hulls

We embed compact $C^\infty$ manifolds into $\mathbb C^n$ as totally real manifolds with prescribed polynomial hulls. As a consequence we show that any compact $C^\infty$ manifold of dimension $d$ admits a totally real embedding into $\mathbb C^{\lfloor \frac{3d}{2}\rfloor}$ with non-trivial polynomial hull without complex structure.

Ground states of groupoid [formula omitted]-algebras, phase transitions and arithmetic subalgebras for Hecke algebras

We consider the Hecke pair consisting of the group PK+ of affine transformations of a number field K that preserve the orientation in every real embedding and the subgroup PO+ consisting of transformations with algebraic integer coefficients. The associated Hecke algebra Cr∗(PK+,PO+) has a natural time evolution σ, and we describe the corresponding phase transition for KMSβ-states and for ground states. From work of Yalkinoglu and Neshveyev it is known that a Bost–Connes type system associated to K has an essentially unique arithmetic subalgebra. When we import this subalgebra through the isomorphism of Cr∗(PK+,PO+) to a corner in the Bost–Connes system established by Laca, Neshveyev and Trifković, we obtain an arithmetic subalgebra of Cr∗(PK+,PO+) on which ground states exhibit the ‘fabulous’ property with respect to an action of the Galois group G(Kab∕H+(K)), where H+(K) is the narrow Hilbert class field.In order to characterize the ground states of the C∗-dynamical system (Cr∗(PK+,PO+),σ), we obtain first a characterization of the ground states of a groupoid C∗-algebra, refining earlier work of Renault. This is independent from number theoretic considerations, and may be of interest by itself in other situations.

Generic immersions and totally real embeddings

We show that, for a closed orientable [Formula: see text]-manifold, with [Formula: see text] not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex [Formula: see text]-space ensures the existence of a totally real embedding into complex [Formula: see text]-space. This implies that a closed orientable [Formula: see text]-manifold with non-vanishing Kervaire semi-characteristic possesses no CR-regular embedding into complex [Formula: see text]-space. We also pay special attention to the cases of CR-regular embeddings of spheres and of simply-connected 5-manifolds.

Existence results of totally real immersions and embeddings into ℂ^{ℕ}

We prove that the existence of totally real immersions of manifolds is a closed property under cut-and-paste constructions along submanifolds including connected sums. We study the existence of totally real embeddings for simply connected 5 5 -manifolds and orientable 6 6 -manifolds and determine the diffeomorphism and homotopy types. We show that the fundamental group is not an obstruction for the existence of a totally real embedding for high-dimensional manifolds in contrast with the situation in dimension four.

A remark about weak fillings

Let L be a closed manifold of dimension n≥2 which admits a totally real embedding into Cn. Let ST*L be the space of rays of the cotangent bundle T*L of L, and let DT*L be the unit disk bundle of T*L defined by any Riemannian metric on L. We observe that ST*L endowed with its standard contact structure admits weak symplectic fillings W which are diffeomorphic to DT*L and for which any closed Lagrangian submanifold N⊂W has the property that the map H1(N,R)→H1(W,R) has a nontrivial kernel. This relies on a variation on a theorem by Laudenbach and Sikorav.

Torsion points on CM elliptic curves over real number fields

We study torsion subgroups of elliptic curves with complex multiplication (CM) defined over number fields which admit a real embedding. We give a complete classification of the groups which arise up to isomorphism as the torsion subgroup of a CM elliptic curve defined over a number field of odd degree: there are infinitely many. Restricting to the case of prime degree, we show that there are only finitely many isomorphism classes. More precisely, there are six “Olson groups” which arise as torsion subgroups of CM elliptic curves over number fields of every degree, and there are precisely 17 “non-Olson” CM elliptic curves defined over a prime degree number field.

Research on Multi Frame Synchronous Operation Transfer Technology of Cloud Computing Platform Based on Real Time Grid Embedding Algorithm

Research on Multi Frame Synchronous Operation Transfer Technology of Cloud Computing Platform Based on Real Time Grid Embedding Algorithm

On homogeneous hypersurfaces in $${\mathbb {C}}^3$$ C 3

We consider a family \(M_t^n\), with \(n\geqslant 2\), \(t>1\), of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in \({\mathbb {C}}^n\) due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of \(M_t^n\) in \({\mathbb {C}}^n\) for \(n=3,7\). In our earlier article we showed that \(M_t^7\) is not embeddable in \({\mathbb {C}}^7\) for every t and that \(M_t^3\) is embeddable in \({\mathbb {C}}^3\) for all \(1<t<1+10^{-6}\). In the present paper, we improve on the latter result by showing that the embeddability of \(M_t^3\) in fact takes place for \(1<t<\sqrt{(2+\sqrt{2})/3}\). This is achieved by analyzing the explicit totally real embedding of the sphere \(S^3\) in \({\mathbb {C}}^3\) constructed by Ahern and Rudin. For \(t\geqslant {\sqrt{(2+\sqrt{2})/3}}\), the problem of the embeddability of \(M_t^3\) remains open.

Estimation method of real embedding of working organs under the testing of soil cultivating machinery

A method is suggested, plan and equipment for the estimation of working organs’ real embedding in technological process under soil cultivating machinery testing are developed.

The hull of Rudin’s Klein bottle

In 1981 Walter Rudin exhibited a totally real embedding of the Klein bottle into C 2 \mathbb {C}^2 . We show that the polynomially convex hull of Rudin’s Klein bottle contains an open subset of C 2 \mathbb {C}^2 . We also describe another totally real Klein bottle in C 2 \mathbb {C}^2 whose hull has topological dimension equal to three.