Regular Embeddings Research Articles

Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces

AbstractLet A denote the cylinder $${\mathbb {R}} \times S^1$$ R × S 1 or the band $${\mathbb {R}} \times I$$ R × I , where I stands for the closed interval. We consider 2- of closed curves (“”) and compact surfaces (“”) in A, up to cobordisms that also are 2-moderate immersions in $$A \times [0, 1]$$ A × [ 0 , 1 ] of surfaces and solids. By definition, the 2-moderate immersions of curves and surfaces do not have tangencies of order $$\ge 3$$ ≥ 3 to the fibers of the obvious projections $$A \rightarrow S^1$$ A → S 1 , $$A \times [0, 1] \rightarrow S^1 \times [0, 1]$$ A × [ 0 , 1 ] → S 1 × [ 0 , 1 ] or $$A \rightarrow I$$ A → I , $$A \times [0, 1] \rightarrow I \times [0, 1]$$ A × [ 0 , 1 ] → I × [ 0 , 1 ] . These bordisms come in different flavors: in particular, we consider one flavor based on of doodles and blobs in A. We compute the bordisms of regular embeddings and construct many invariants that distinguish between the bordisms of immersions and embeddings. In the case of oriented doodles on $$A= {\mathbb {R}} \times I$$ A = R × I , our computations of 2-moderate immersion bordisms $$\textbf{OC}^{\textsf{imm}}_{\mathsf {moderate \le 2}}(A)$$ OC moderate ≤ 2 imm ( A ) are near complete: we show that they can be described by an exact sequence of abelian groups $$\begin{aligned} 0 \rightarrow {\textbf{K}} \rightarrow \textbf{OC}^{\textsf{imm}}_{\mathsf {moderate \le 2}}(A)\big /\textbf{OC}^{\textsf{emb}}_{\mathsf {moderate \le 2}}(A) {\mathop {\longrightarrow }\limits ^{{\mathcal {I}} \rho }} {\mathbb {Z}} \times {\mathbb {Z}} \rightarrow 0, \end{aligned}$$ 0 → K → OC moderate ≤ 2 imm ( A ) / OC moderate ≤ 2 emb ( A ) ⟶ I ρ Z × Z → 0 , where $$\textbf{OC}^{\textsf{emb}}_{\mathsf {moderate \le 2}}(A) \approx {\mathbb {Z}} \times {\mathbb {Z}}$$ OC moderate ≤ 2 emb ( A ) ≈ Z × Z , the epimorphism $${\mathcal {I}} \rho $$ I ρ counts different types of crossings of immersed doodles, and the kernel $${\textbf{K}}$$ K contains the group $$({\mathbb {Z}})^\infty $$ ( Z ) ∞ whose generators are described explicitly.

Deep Learning Model for Tamil Part-of-Speech Tagging

Abstract Part-of-Speech (POS) tagging is one of the popular Natural Language Processing (NLP) tasks. It is also considered to be a preliminary task for other NLP applications such as speech recognition, machine translation, and sentiment analysis. A few works have been published on POS tagging for the Tamil language. However, the performance of the POS tagger with unknown words is not explored in the literature. The appearance of unknown words is a frequently occurring problem in POS tagging and makes it a challenging task. In this paper, we propose a deep learning-based POS tagger for Tamil using Bi-directional Long Short Term Memory (BLSTM). The performance of the POS tagger was evaluated using known and unknown words. The POS tagger with regular word-level embeddings produces 99.83 and 92.46% accuracies for all known and 63.21% unknown words. It clearly shows that the accuracy decreases when the number of unknown words increases. To improve the performance of the POS tagger with unknown words, the proposed BLSTM model that uses word-level, character-level and pre-trained word embeddings. Test results of this model show a 2.57% improvement for 63.21% of unknown words, with an accuracy of 95.03%.

Automated Assessment of Initial Answers to Questions in Conversational Intelligent Tutoring Systems: Are Contextual Embedding Models Really Better?

This paper assesses the ability of semantic text models to assess student responses to electronics questions compared with that of expert human judges. Recent interest in text similarity has led to a proliferation of models that can potentially be used for assessing student responses. However, it is unclear whether these models perform as well as early models of distributional semantics. We assessed 5166 response pairings of 219 participants across 118 electronics questions and scored each with 13 different computational text models, including models that use Regular Expressions, distributional semantics, embeddings, contextual embeddings, and combinations of these features. Regular Expressions performed the best out of the stand-alone models. We show other semantic text models performing comparably to the Latent Semantic Analysis model that was originally used for the current task, and in a small number of cases outperforming the model. Models trained on a domain-specific electronics corpus for the task performed better than models trained on general language or Newtonian physics. Furthermore, semantic text models combined with RegEx outperformed stand-alone models in agreement with human judges. Tuning the performance of these recent models in Automatic Short Answer Grading tasks for conversational intelligent tutoring systems requires empirical analysis, especially in domain-specific areas such as electronics. Therefore, the question arises as to how well recent contextual embedding models compare with earlier distributional semantic language models on this task of answering questions about electronics. These results shed light on the selection of appropriate computational techniques for text modeling to improve the accuracy, recall, weighted agreement, and ultimately the effectiveness of automatic scoring in conversational ITSs.

A classification of nonorientable regular embeddings of simple graphs of order $$p^3$$

A classification of nonorientable regular embeddings of simple graphs of order $$p^3$$

From the Nash–Kuiper theorem of isometric embeddings to the Euler equations for steady fluid motions: Analogues, examples, and extensions

Direct linkages between regular or irregular isometric embeddings of surfaces and steady compressible or incompressible fluid dynamics are investigated in this paper. For a surface (M, g) isometrically embedded in R3, we construct a mapping that sends the second fundamental form of the embedding to the density, velocity, and pressure of steady fluid flows on (M, g). From a Partial Differential Equations perspective, this mapping sends solutions to the Gauss–Codazzi equations to the steady Euler equations. Several families of special solutions of physical or geometrical significance are studied in detail, including the Chaplygin gas on standard and flat tori as well as the irregular isometric embeddings of the flat torus. We also discuss tentative extensions to multiple dimensions.

Adversarial Auto-encoder Domain Adaptation for Cold-start Recommendation with Positive and Negative Hypergraphs

This article presents a novel model named Adversarial Auto-encoder Domain Adaptation to handle the recommendation problem under cold-start settings. Specifically, we divide the hypergraph into two hypergraphs, i.e., a positive hypergraph and a negative one. Below, we adopt the cold-start user recommendation for illustration. After achieving positive and negative hypergraphs, we apply hypergraph auto-encoders to them to obtain positive and negative embeddings of warm users and items. Additionally, we employ a multi-layer perceptron to get warm and cold-start user embeddings called regular embeddings. Subsequently, for warm users, we assign positive and negative pseudo-labels to their positive and negative embeddings, respectively, and treat their positive and regular embeddings as the source and target domain data, respectively. Then, we develop a matching discriminator to jointly minimize the classification loss of the positive and negative warm user embeddings and the distribution gap between the positive and regular warm user embeddings. In this way, warm users’ positive and regular embeddings are connected. Since the positive hypergraph maintains the relations between positive warm user and item embeddings, and the regular warm and cold-start user embeddings follow a similar distribution, the regular cold-start user embedding and positive item embedding are bridged to discover their relationship. The proposed model can be easily extended to handle the cold-start item recommendation by changing inputs. We perform extensive experiments on real-world datasets for both cold-start user and cold-start item recommendations. Promising results in terms of precision, recall, normalized discounted cumulative gain, and hit rate verify the effectiveness of the proposed method.

PLFace: Progressive Learning for Face Recognition with Mask Bias

The outbreak of the COVID-19 coronavirus epidemic has promoted the development of masked face recognition (MFR). Nevertheless, the performance of regular face recognition is severely compromised when the MFR accuracy is blindly pursued. More facts indicate that MFR should be regarded as a mask bias of face recognition rather than an independent task. To mitigate mask bias, we propose a novel Progressive Learning Loss (PLFace) that achieves a progressive training strategy for deep face recognition to learn balanced performance for masked/mask-free faces recognition based on margin losses. Particularly, our PLFace adaptively adjusts the relative importance of masked and mask-free samples during different training stages. In the early stage of training, PLFace mainly learns the feature representations of mask-free samples. At this time, the regular sample embeddings shrink to the corresponding prototype, which represents the center of each class while being stored in the last linear layer. In the later stage of training, PLFace converges on mask-free samples and further focuses on masked samples until the masked sample embeddings are also gathered in the center of the class. The entire training process emphasizes the paradigm that normal samples shrink first and masked samples gather afterward. Extensive experimental results on popular regular and masked face benchmarks demonstrate that our proposed PLFace can effectively eliminate mask bias in face recognition. Compared to state-of-the-art competitors, PLFace significantly improves the accuracy of MFR while maintaining the performance of normal face recognition.

Nonorientable regular embeddings of graphs of order $$p^3$$

Nonorientable regular embeddings of graphs of order $$p^3$$

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.

Global representation of Segre numbers by Monge\u2013Amp\xe8re products

On a reduced analytic space X we introduce the concept of a generalized cycle, which extends the notion of a formal sum of analytic subspaces to include also a form part. We then consider a suitable equivalence relation and corresponding quotient mathcal {B}(X) that we think of as an analogue of the Chow group and a refinement of de Rham cohomology. This group allows us to study both global and local intersection theoretic properties. We provide many mathcal {B}-analogues of classical intersection theoretic constructions: For an analytic subspace Vsubset X we define a mathcal {B}-Segre class, which is an element of mathcal {B}(X) with support in V. It satisfies a global King formula and, in particular, its multiplicities at each point coincide with the Segre numbers of V. When V is cut out by a section of a vector bundle we interpret this class as a Monge–Ampère-type product. For regular embeddings we construct a mathcal {B}-analogue of the Gysin morphism.

The Structure of Root Data and Smooth Regular Embeddings of Reductive Groups

AbstractWe investigate the structure of root data by considering their decomposition as a product of a semisimple root datum and a torus. Using this decomposition, we obtain a parametrization of the isomorphism classes of all root data. By working at the level of root data, we introduce the notion of a smooth regular embedding of a connected reductive algebraic group, which is a refinement of the commonly used regular embeddings introduced by Lusztig. In the absence of Steinberg endomorphisms, such embeddings were constructed by Benjamin Martin. In an unpublished manuscript, Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped with a Frobenius endomorphism, to those whose derived subgroup is simple and simply connected. Using our investigations into root data we give new proofs of Asai's results and generalize them so that they are compatible with Steinberg endomorphisms. As an illustration of these ideas, we answer a question posed to us by Olivier Dudas concerning unipotent supports.

Nonlinear supervised dimensionality reduction via smooth regular embeddings

The recovery of the intrinsic geometric structures of data collections is an important problem in data analysis. Supervised extensions of several manifold learning approaches have been proposed in the recent years. Meanwhile, existing methods primarily focus on the embedding of the training data, and the generalization of the embedding to initially unseen test data is rather ignored. In this work, we build on recent theoretical results on the generalization performance of supervised manifold learning algorithms. Motivated by these performance bounds, we propose a supervised manifold learning method that computes a nonlinear embedding while constructing a smooth and regular interpolation function that extends the embedding to the whole data space in order to achieve satisfactory generalization. The embedding and the interpolator are jointly learnt such that the Lipschitz regularity of the interpolator is imposed while ensuring the separation between different classes. Experimental results on several image data sets show that the proposed method outperforms traditional classifiers and the supervised dimensionality reduction algorithms in comparison in terms of classification accuracy in most settings.

On even-closed regular embeddings of graphs

A map is said to be even-closed if all of its automorphisms act like even permutations on the vertex set. In this paper the study of even-closed regular maps is approached by analysing two distinguished families. The first family consists of embeddings of a well-known family of graphs on distinct orientable surfaces, whereas in the second family we consider all graphs having orientable-regular embeddings on a particular surface. In particular, the classification of even-closed orientable-regular embeddings of the complete bipartite graphs Kn,n and classification of even-closed orientable-regular maps on the torus are given.

Edge-girth-regular graphs

We consider a new type of regularity we call edge-girth-regularity. An edge-girth-regular (v,k,g,λ)-graph Γ is a k-regular graph of order v and girth g in which every edge is contained in λ distinct g-cycles. This concept is a generalization of the well-known concept of (v,k,λ)-edge-regular graphs (that count the number of triangles) and appears in several related problems such as Moore graphs and Cage and Degree/Diameter Problems. All edge- and arc-transitive graphs are edge-girth-regular as well. We derive a number of basic properties of edge-girth-regular graphs, systematically consider cubic and tetravalent graphs from this class, and introduce several constructions that produce infinite families of edge-girth-regular graphs. We also exhibit several surprising connections to regular embeddings of graphs in orientable surfaces.

An equivalence between pseudo-holomorphic embeddings into almost-complex Euclidean space and CR regular embeddings into complex space

We show that a pseudo-holomorphic embedding of an almost-complex 2n -manifold into almost-complex (2n + 2) -Euclidean space exists if and only if there is a CR regular embedding of the 2n -manifold into complex (n + 1) -space. We remark that the fundamental group does not place any restriction on the existence of either kind of embedding when n is at least three. We give necessary and sufficient conditions in terms of characteristic classes for a closed almost-complex 6-manifold to admit a pseudo-holomorphic embedding into \mathbb R^8 equipped with an almost-complex structure that need not be integrable.

Classification of nonorientable regular embeddings of cartesian products of graphs

In this paper, we consider the classification of nonorientable regular embeddings of cartesian products Gd of a graph G. We show that if G is a bipartite graph and d≥3, then there is no nonorientable regular embedding. This is a generalization of the result that there is no nonorientable regular embeddings of Qn for n≥3 shown by R. Nedela and the author in 2007. When G is non-bipartite and d≥3, we show that if there is a nonorientable regular embedding of Gd, then there is a nonorientable regular embedding of G. Furthermore, we show that any nonorientable regular embedding of Gd with d≥3 is isomorphic to a Petrie dual of some orientable regular embedding of Gd.Using these results, we classify the nonorientable regular embeddings of cartesian products Cnd of a cycle Cn; for even n there is no nonorientable regular embedding except when d=1, and for odd n there is a unique nonorientable regular embedding for each d.

Erratum to: A note on deformations of regular embeddings

Erratum to: A note on deformations of regular embeddings

Classification of Regular Embeddings of Complete Multipartite Graphs

AbstractA 2‐cell embedding of a graph Γ into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags. In this article, we classify the regular embeddings of the complete multipartite graph . We show that if the number of partite sets is greater than 3, there exists no such embedding; and if the number of partite sets is 3, for any n, there exist one orientable regular embedding and one nonorientable regular embedding of up to isomorphism.

A note on deformations of regular embeddings

In this paper we give a description of the first order deformation space of a regular embedding of reduced algebraic schemes. We compare our result with results of Ran (in particular [Ran, Prop. 1.3]).

Strong regular embeddings of Deligne-Mumford stacks and Hypertoric geometry

We introduce the notion of strong regular embeddings of Deligne-Mumford stacks. These morphisms naturally arise in the related contexts of generalized Euler sequences and hypertoric geometry.