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For certain bordered submanifolds M ⊂ ℂ² we show that M can be embedded properly and holomorphically into ℂ². An application is that any subset of a torus with two boundary components can be embedded properly into ℂ².

We prove that a closed co-oriented contact (2m + 1)-manifold (M^{2m + 1}, ξ) can be a contact submanifold of the standard contact structure on ℝ^{4m + 1}, if it satisfies one of the following conditions: (1) m is odd (m ≥ 3) and H₁(M^{2m + 1}; ℤ) = 0, (2) m is even (m ≥ 4) and M^{2m + 1} is 2-connected, (3) m = 2 and M⁵ is simply-connected.

Embedding task graphs onto hypercubes is a difficult problem. When the embedding is one-to-one, schedule length is strongly influenced by dilation. Therefore, it is desirable to find low dilation embeddings. This paper describes a heuristic embedding technique based upon evolutionary strategies. The technique has been extensively investigated using task graphs which are trees, forests, and butterflies. In all cases the technique has found low dilation embeddings.

According to the Campbell–Magaard theorem, any analytical spacetime can be locally and analytically embedded into a five-dimensional pseudo-Riemannian Ricci-flat manifold. This embedding for Gödel's universe can be found. The embedding space is Ricci-flat and has a non-Lorentzian signature of type (+ + - - -). We also show that the embedding found is global.

Many Einstein spaces can be embedded globally in pseudo-Euclidean spaces of dimension N > 4. The geometrical quantities characterizing the embedded manifold can be expressed by means of derivatives of the embedding functions z^A (x^µ), A, B, = 1, 2, …N, µ, ν, …= 0, 1, 2, 3. An infinitesimal deformation of embedding can be expanded into a series , giving rise to a similar expansion of geometrical quantities of the embedded Einstein space, and the Einstein equations in vacuo, too. We show how gravitational wave solutions appear naturally in this context.

Knowledge Graphs (KGs) are gaining popularity and are being widely used in a plethora of applications. They owe their popularity to the fact that KGs are an ideal form to integrate and retrieve data originating from various sources. Using KGs as input for Machine Learning (ML) tasks allows to perform predictions on these popular graph structures. However, KGs cannot directly be used as ML input in their graph representation, they first require to be transformed to a vector space representation through an embedding technique. As ML techniques are data-driven, they can generalize over unseen input data that deviates to some extent from the data they were trained upon. To fully exploit the generalization capabilities of ML algorithms when using embedded KGs as input, small changes in the KGs should also result in small changes in the embedding space. Various embedding techniques for graphs in general exist, however, they have not been tailored towards embedding whole KGs, while KGs can be considered a special kind of graph that adheres to a certain KG schema. This paper evaluates if these existing embedding techniques that embed the whole graphs can represent the similarity between KGs in their embedding space, allowing ML algorithms to generalize over their input. We compare the similarities between KGs in terms of changes in size, entity labels, and KG schema. We found that most techniques were able to represent the similarities in terms of size and entity labels in their embedding space, however, none of the techniques were able to capture the similarities in KG schema.

Let S be a set of m polygons in the plane with a total of n vertices. A translation order for S in direction is an order on the polygons such that no collisions occur if the polygons are moved one by one to infinity in direction according to this order. We show that S can be preprocessed in O(n log n) time into a data structure of size O(m) such that a translation order for a query direction can be computed in O(m) time, if it exists. It is also possible to test in O(log n) time whether a translation order exists, with a structure that uses O(n) storage. These results are achieved using new results on relative convex hulls and on embeddings with few vertices. Translation orders correspond to valid displaying orders for hidden surface removal with the painter’s algorithm. Our technique can be used to generate displaying orders for polyhedral terrains, for parallel as well as perspective views.

In this paper, we show that every spinal open book decomposition of a closed oriented $3$-manifold $M$ spinal open book embeds into a certain spinal open book decomposition of ${\#}_k{S}^2\tilde{\times }{S}^3$, the connected sum of $k$ copies of the twisted $S^3$-bundle over $S^2$, where $k$ depends on the spinal open book decomposition of $M$. We also discuss spinal open book embeddings of a huge class of spinal open books of closed oriented $3$-manifolds into the trivial spinal open book of the $5$-sphere $S^5$. Finally, we show that given a closed oriented $3$-manifold $M$, there exists a spinal open book for $M$ such that $M$ spinal open book embeds into the trivial spinal open book of $S^5.$ In particular, this gives another proof of Hirsch’s theorem which states that every closed orientable $3$-manifold embeds in $S^5.$

The main theorem of this article is that every countable model of set theory 〈M, ∈^M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈L^M, ∈^M〉 by means of an embedding j : M → L^M. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈^M〉 and 〈N, ∈^N〉 are countable models of set theory, then either M is isomorphic to a submodel of N or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω₁ + 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most Ord^M; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈HF^M, ∈^M〉 of M. Indeed, 〈HF^M, ∈^M〉 is universal for all countable acyclic binary relations.

We derive a sequential algorithm Find-Ham-Cycle with the following property. On input: k and n (specifying the k-ary n-cube ); F, a set of at most 2n − 2 faulty links; and v, a node of , the algorithm outputs nodes v⁺ and v⁻ such that if Find-Ham-Cycle is executed once for every node v of then the node v⁺ (resp. v⁻) denotes the successor (resp. predecessor) node of v on a fixed Hamiltonian cycle in in which no link is in F. Moreover, the algorithm Find-Ham-Cycle runs in time polynomial in n and log k. We also obtain a similar algorithm for an n-dimensional hypercube with at most n − 2 faulty links. We use our algorithms to obtain distributed algorithms to embed Hamiltonian cycles k-ary n-cubes and hypercubes with faulty links; our hypercube algorithm improves on a recently-derived algorithm due to Leu and Kuo, and our k-ary n-cube algorithm is the first distributed algorithm for embedding a Hamiltonian cycle in a k-ary n-cube with faulty links.

We calculate the invariant subspaces in the linear representation of the group of algebraic automorphisms of ℂⁿ on the vector space of algebraic vector fields on ℂⁿ and more generally we do this in a setting with parameter. As an application to the field of Several Complex Variables we get a new proof of the Andersén–Lempert observation and a parametric version of the Andersén–Lempert theorem. Further applications to the question of embeddings of ℂ^k into ℂⁿ are announced.

The observable representation provides an embedding of a discrete space in a low dimensional continuous space. Typically, the discrete space is a model of a complex system. This graphical representation is known to highlight significant properties of the original space and can serendipitously reveal unanticipated relationships. We report on the current status of this technique and give examples of its applications and rationale.

We investigate the existence of embeddings of shearlet coorbit spaces associated to weighted mixed L^p-spaces into classical Sobolev spaces in dimension three by using the description of coorbit spaces as decomposition spaces. This different perspective on these spaces enables the application of embedding results that allow the complete characterization of embeddings for certain integrability exponents, and thus provides access to a deeper understanding of the smoothness properties of coorbit spaces, and of the influence of the choice of shearlet groups on these properties. We give a detailed analysis, identifying which features of the dilation groups have an influence on the embedding behavior, and which do not. Our results also allow to comment on the validity of the interpretation of shearlet coorbit spaces as smoothness spaces.

The coherent state representation of the Jacobi group is indexed with two parameters, , describing the part coming from the Heisenberg group, and k, characterizing the positive discrete series representation of SU(1,1). The Ricci form, the scalar curvature and the geodesics of the Siegel–Jacobi disk are investigated. The significance in the language of coherent states of the transform which realizes the fundamental conjecture on the Siegel–Jacobi disk is emphasized. The Berezin kernel, Calabi's diastasis, the Kobayashi embedding and the Cauchy formula for the Siegel–Jacobi disk are presented.

Good communication is indubitably the foundation of effective teamwork. Over time teams develop their own communication styles and often exhibit entrainment, a conversational phenomena in which humans synchronize their linguistic choices. Conversely, teams may experience conflict due to either personal incompatibility or differing viewpoints. We tackle the problem of predicting team conflict from embeddings learned from multiparty dialogues such that teams with similar post-task conflict scores lie close to one another in vector space. Embeddings were extracted from three types of features: (1) dialogue acts, (2) sentiment polarity, and (3) syntactic entrainment. Machine learning models often suffer domain shift; one advantage of encoding the semantic features is their adaptability across multiple domains. To provide intuition on the generalizability of different embeddings to other goal-oriented teamwork dialogues, we test the effectiveness of learned models trained on the Teams corpus on two other datasets. Unlike syntactic entrainment, both dialogue act and sentiment embeddings are effective for identifying team conflict. Our results show that dialogue act-based embeddings have the potential to generalize better than sentiment and entrainment-based embeddings. These findings have potential ramifications for the development of conversational agents that facilitate teaming.