Regular Embedding Research Articles

The generalized Harer conjecture for the homology triviality

AbstractThe classical Harer conjecture states the stable homology triviality of the canonical embedding , which was proved by Song and Tillmann. The main part of the proof is to show that , induced from is a double‐loop space map. In this paper, we give a proof of the generalized Harer conjecture concerning the homology triviality for every regular embedding . The main strategy of the proof is to remove all the interchangeable subsurfaces from and collapse the new boundary components. Then, we obtain (the union of) covering spaces over a disk with marked points that we can analyze. The final goal is to show that the map induced by preserves the actions of the framed little 2‐disks operad.

Color Tuning in Bovine Rhodopsin through Polarizable Embedding.

Bovine rhodopsin is among the most studied proteins in the rhodopsin family. Its primary activation mechanism is the photoisomerization of 11-cis retinal, triggered by the absorption of a UV-visible photon. Different mutants of the same rhodopsin show different absorption wavelengths due to the influence of the specific amino acid residues forming the cavity in which the retinal chromophore is embedded, and rhodopsins activated at different wavelengths are, for example, exploited in the field of optogenetics. In this letter, we present a procedure for systematically investigating color tuning in models of bovine rhodopsin and a set of its mutants embedded in a membrane bilayer. Vertical excitation energy calculations were carried out with the polarizable embedding potential for describing the environment surrounding the chromophore. We show that polarizable embedding outperformed regular electrostatic embedding in determining both the vertical excitation energies and associated oscillator strengths of the systems studied.

ESA-FedGNN: Efficient secure aggregation for federated graph neural networks.

Graph Neural Network (GNN) architecture is a state-of-the-art model, which can obtain complete node embedding features and rich data information by aggregating the information of nodes and neighbors. Therefore, GNNs are widely used in electronic shopping, drug discovery (especially for the treatment of COVID-19) and other fields, promoting the explosive development of machine learning. However, user interaction, data sharing and circulation are highly sensitive to privacy, and centralized storage can lead to data isolation. Therefore, Federated Learning with high efficiency and strong security and privacy enhancement technology based on secure aggregation can improve the security dilemma faced by GNN. In this paper, we propose an Efficient Secure Aggregation for Federated Graph Neural Network(ESA-FedGNN), which can efficiently reduce the cost of communication and avoid computational redundancy while ensuring data privacy. Firstly, a novel secret sharing scheme based on numerical analysis is proposed, which employs Fast Fourier Transform to improve the computational power of the neural network in sharing phase, and leverages Newton Interpolation method to deal with the disconnection and loss of the client in reconstruction phase. Secondly, a regular graph embedding based on geometric distribution is proposed, which optimizes the aggregation speed by using data parallelism. Finally, a double mask is adopted to ensure privacy and prevent malicious adversaries from stealing model parameters. We achieve improvements compared to in state-of-the-art works. This research helps to provide security solutions related to the practical development and application of privacy-preserving graph neural network technology.

Projective Embeddings of Ball Quotients, Birational to a Bi-elliptic Surface

For a neat lattice $$\Gamma < SU(1,2)$$, whose quotient $${\mathbb B} / \Gamma$$ is birational to a bi-elliptic surface, we compute the dimensions of the cuspidal $$\Gamma$$-modular forms $$[ \Gamma,n]_o$$ and all modular forms $$[ \Gamma, n]$$ of weight $$n \geq 2. $$ The work provides a sufficient condition for a subspace $$V \subset [ \Gamma, n]$$ to determine a regular projective embedding of the Baily-Borel compactification $$\widehat{ {\mathbb B} / \Gamma}$$ and applies this criterion to a specific example.

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.

Bounded birationality and isomorphism problems are computable

Let X, Y be two irreducible subvarieties of the projective space Pn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^n$$\\end{document}, and d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 1$$\\end{document} an integer number. The main result of this paper is an algorithm to construct explicitly, in terms of d and the ideals defining X and Y, a quasi-affine algebraic variety parametrising the set of all birational maps f from X onto Y which can be extended to a self-rational map of Pn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^n$$\\end{document} of algebraic degree ≤d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le d$$\\end{document}. We also prove similar results for the case f is a dominant rational map, regular morphism, isomorphism or regular embedding. Similar results are valid for varieties over an arbitrary algebraically closed field, and also for maps on non-projective varieties.

A class of orientably‐regular embeddings of graphs of order [formula omitted

Let G be the automorphism group of an orientably regular embedding of a connected simple graph of order p3, where p is a prime. It was shown in [19] that except for the case when p=2 and G=S4, G contains a normal Sylow p-subgroup P of order pk where k∈{3,4,5}, while such embeddings with k=5 were determined. In this paper, a further classification is achieved for all such embeddings with k=3.

ON EXTENDED WEIGHT MONOIDS OF SPHERICAL HOMOGENEOUS SPACES

Given a connected reductive complex algebraic group $G$ and a spherical subgroup $H \subset G$, the extended weight monoid $\widehat \Lambda^+_G(G/H)$ encodes the $G$-module structures on spaces of global sections of all $G$-linearized line bundles on $G/H$. Assuming that $G$ is semisimple and simply connected and $H$ is specified by a regular embedding in a parabolic subgroup $P \subset G$, in this paper we obtain a description of $\widehat \Lambda^+_G(G/H)$ via the set of simple spherical roots of $G/H$ together with certain combinatorial data explicitly computed from the pair $(P,H)$. As an application, we deduce a new proof of a result of Avdeev and Gorfinkel describing $\widehat \Lambda^+_G(G/H)$ in the case where $H$ is strongly solvable.

OFNE: a framework of opinion features regulated network embedding

Network embedding technologies that transform the nodes of a network into a low-dimensional vector space have many various potential applications such as node classification, community detection and Internet public opinion analysis and so forth. Most existing approaches for network embedding are calculated by utilizing the topology structure of a target network which simply describes the social relations among nodes. However, another factor derived from the significant “opinions” is usually neglected by those works. In particular, there exist dramatic cases of Internet Public Affair, where many users with the same opinion have no social connections. On top of that, the social network can be very sparse, which is unsuitable for network embedding. Therefore, this paper proposes an efficient approach opinion feature network embedding (OFNE) that combines both social relations and opinion features within a network. OFNE defines an opinion feature edge to preserve the opinion features. Experimental results on real-world datasets across different domains demonstrate that the proposed approach OFNE outperforms the regular social network embedding approaches, especially when the opinions have explicit sentiment orientation.

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.

Parentheticality, assertion strength, and polarity

Sentences with slifting parentheticals (such as The dean greeted the secretary, Jill said; Ross, in: Gross, Schutzenberger (eds) The formal analysis of natural language, Mouton, The Hague, 1973) grammaticalize an intriguing interaction between truth-conditional meaning and speech act function. In such sentences, the assertion strength of the slifted clause (the non-parenthetical part of the sentence) is modulated by the parenthetical, which provides evidential support (Urmson in Mind 61(244):480–496, 1952; Asher in J Semant 17:31–50, 2000; Rooryck in Glot Int 5(4):125–133, 2001; Jayez and Rossari in: Corblin, de Swart (eds) Handbook of French semantics, CSLI, Stanford, 2004; Davis et al. in Proc Semant Linguist Theory 17:71–88, 2007; Simons in Lingua 117:1034–1056, 2007; Murray in Semant Pragmat 7(2):1–53, 2014; Maier and Bary in: Brochhagen et al. (eds) Proceedings of the 20th Amsterdam colloquium, 2015; AnderBois in Semant Pragmat 9(19):1–55, 2016; Hunter in Dialogue Discourse 7(4):1–35, 2016). Starting with the idea that assertability comes in degrees (Lewis in Philos Rev 85:297–315, 1976; Davis et al. in Proc Semant Linguist Theory 17:71–88, 2007), this paper develops a probabilistic updatemodel that captures the role of parentheticality as a language tool for qualifying commitments. A crucial role here is played by the rule of Jeffrey conditionalization (Jeffrey in The logic of decision, The University of Chicago Press, Chicago, 1990), which factors in the uncertainty of the parenthetical information itself. The model also derives certain effects of parenthetical modification not found in regular embedding constructions, including the fact that slifting parentheticals are limited to creating upward-entailing environments.

Audio watermarking based on synergy between Lucas regular sequence and Fast Fourier Transform

The growth of digital technology over the recent years has led to increased sending and saving of electronic media. Taking advantage of digital works without observing copyright is always considered as an important threat for publishers. One of the techniques proposed for overcoming this threat is watermarking. Audio watermarking is one of the most challenging fields of watermarking, because detection of watermark in audio is much easier than in image. In this paper, a new approach is proposed for audio watermarking based on the values of energy levels in FFT domain. Despite preservation of imperceptibility and robustness, this method has also a very high capacity. The main idea of the proposed method is employing a regular embedding procedure on the basis of closing the energy of the sub-frames to Lucas math sequence, which is capable of embedding two bits in each audio frame. In addition, appropriate parameters values of the proposed approach such as frame-size and frequency-band are calculated by an intelligent recursive tuning process. Extraction process is done in a blind manner and without any need to the host original signal. In order to evaluate the proposed method, five types of audio files including Electronic, Classic, Jazz, Blues, and Rock audio files have been used in wave format. The results show that despite having a high watermarking capacity, the proposed method has a high imperceptibility and an acceptable robustness against the common attacks such as Noise Addition, 64 kbps MPEG Compression (MP3), FFT-Invert, Re-sampling etc., compared with the previous methods.

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.

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.

The complete bipartite graphs with a unique edge‐transitive embedding

AbstractJones, Nedela, and Škoviera (2008) showed that a complete bipartite graph has a unique orientably regular embedding if and only if . In this article, we extend this result by proving that a complete bipartite graph has a unique orientably edge‐transitive embedding if and only if .

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.

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]).

Graded polynomial identities for matrices with the transpose involution

Let G be a group of order k. We consider the algebra Mk(C) of k by k matrices over the complex numbers and view it as a crossed product with respect to G by embedding G in the symmetric group Sk via the regular representation and embedding Sk in Mk(C) in the usual way. This induces a natural G-grading on Mk(C) which we call a crossed-product grading. We study the graded ⁎-identities for Mk(C) equipped with such a crossed-product grading and the transpose involution. To each multilinear monomial in the free graded algebra with involution we associate a directed labeled graph. Use of these graphs allows us to produce a set of generators for the (T,⁎)-ideal of identities. It also leads to new proofs of the results of Kostant and Rowen on the standard identities satisfied by skew matrices. Finally we determine an asymptotic formula for the ⁎-graded codimension of Mk(C).

Embeddability of homeomorphisms of the circle in set-valued iteration groups

Let F:S1→S1 be a homeomorphism without periodic points. It is known that F is embeddable in a continuous iteration group if and only if F is minimal. We deal with F which is not minimal. In this case, F satisfying some additional assumptions can be embedded but only in a nonmeasurable iteration groups. There are infinitely many such nonmeasurable groups. We propose here a new approach to the problem of embeddability. For a given homeomorphism F without periodic points we construct some substitute of an iteration group, namely the unique special set-valued iteration group {Ft:S1→cc[S1],t∈R}, which is regular in a sense and in which F can be embedded i.e. F(x)∈F1(x). We also determine a maximal countable and dense subgroup T⊂R such that {Ft:S1→cc[S1],t∈T} has a continuous selection {ft:S1→S1,t∈T} being the best regular embedding of F. If there exists a nonmeasurable embedding {ft:S1→S1,t∈R} of F, then there exists an additive function γ:R→T such that ft(z)∈Fγ(t)(z),t∈R. We determine a unique maximal subgroup T with this property.