arXiv research

This paper answers a question about discrete embeddings to maximal surfaces.

New embeddings show answer to Baker-Laidacker question can be yes or no.

Paper explores coarse embeddings between symmetric spaces and Euclidean buildings, answering open questions.

Text embedding representing natural language documents in a semantic vector space can be used for document retrieval using nearest neighbor lookup. In order to study the feasibility of neural models specialized for retrieval in a semantically meaningful way, we suggest the use of the Stanford Question Answering Dataset…

Tubes in manifolds require wide spaces.

Study extends Nirenberg-Spencer's question to families of submanifolds.

In the field of Natural Language Processing (NLP), we revisit the well-known word embedding algorithm word2vec. Word embeddings identify words by vectors such that the words' distributional similarity is captured. Unexpectedly, besides semantic similarity even relational similarity has been shown to be captured in word…

3028 obstructions found for embedding without knots.

The AI2 Reasoning Challenge (ARC), a new benchmark dataset for question answering (QA) has been recently released. ARC only contains natural science questions authored for human exams, which are hard to answer and require advanced logic reasoning. On the ARC Challenge Set, existing state-of-the-art QA systems fail to s…

We show that any embedded minimal torus in S^3 is congruent to the Clifford torus. This answers a question posed by H.B. Lawson, Jr., in 1970.

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A *-graph is a graph endowed with a formal adjacency structure on the half-edges around each vertex, and an embedding of a *-graph is…

e-Yantra Robotics Competition (eYRC) is a unique Robotics Competition hosted by IIT Bombay that is actually an Embedded Systems and Robotics MOOC. Registrations have been growing exponentially in each year from 4500 in 2012 to over 34000 in 2019. In this 5-month long competition students learn complex skills under seve…

Discrete maximal surfaces identified from s-embeddings.

We review classical approaches to the problem of isometrically embedding a Riemannian surface into Euclidean 3-space, including coordinate-based approaches exposited by Darboux and Eisenhart, as well as the moving-frames based approaches advocated by Cartan. In particular, the first approach involves reducing the probl…

Quantum kernels can be efficiently embedded into classical feature spaces.

A conjecture due to Gompf asserts that no nontrivial Brieskorn homology sphere admits a pseudoconvex embedding in ${\mathbb C}^2$, with either orientation. A related question asks whether every compact contractible 4-manifold admits the structure of a Stein domain. We verify Gompf's conjecture, with one orientation, fo…

Ribbonness proven for slice knots, solving an old question.

Researchers solve a question about embedding knots into Legendrian structures.

A new query embedding method improves KB performance on complex queries.

We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $\mathcal{C}^2$-boundary in $\mathbb{C}^n$ into the unit ball of $\mathbb{C}^N$, for $N$ large enough, thereby answering a question of Alarcon and Forstneric.

We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution to the mean curvature flow with genus $0$. More generally, we show that the only properly embedded self-similar shrinkers in $\mathbb{R}^3$ with vanishing intersection form are the sphere, the cylinder…

For a polarized Kähler manifold $(X, L)$, we show the equivalence between relative balanced embeddings introduced by Mabuchi and $σ$-balanced embeddings introduced by Sano, answering a question of Hashimoto. We give a GIT characterization of the existence of a $σ$-balanced embedding, and relate the optimal weight $σ$ t…

New compacta with unique embedding properties found.

Spaces over BO are equivalent to thickened manifolds.

New examples show immersions not homologous to embeddings.

We investigate the operation of torus surgery on tori embedded in $S^4$. Key questions include which 4-manifolds can be obtained in this way, and the uniqueness of such descriptions. As an application we construct embeddings of 3-manifolds into 4-manifolds by viewing Dehn surgery as a cross section of a surgery on a su…

In this note we prove certain necessary and sufficient conditions for the existence of an embedding of statistical manifolds. In particular, we prove that any compact smooth ($C^1$ resp.) statistical manifold can be embedded into the space of probability measures on a finite set. As a result, we get an answer to the La…

The objective of ordinal embedding is to find a Euclidean representation of a set of abstract items, using only answers to triplet comparisons of the form "Is item $i$ closer to the item $j$ or item $k$?". In recent years, numerous algorithms have been proposed to solve this problem. However, there does not exist a fai…

If one tries to embed a metric space uniformly in Hilbert space, how close to quasi-isometric could the embedding be? We answer this question for finite dimensional CAT(0) cube complexes and for hyperbolic groups. In particular, we show that the Hilbert space compression of any hyperbolic group is 1.

SE-KGE embeds spatial data into KGs for better spatial reasoning.

Minimal sphere dimension for equivariant embedding of circles.

Paper introduces RiskEmbed, a finetuned model for financial risk management.

We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's result concerning asymmetric complete minimal submanifolds of Euclidean space wi…

We study the question of how many embedded symplectic or Lagrangian tori can represent the same homology class in a simply connected symplectic 4-manifold.

Question-answering systems and voice assistants are becoming major part of client service departments of many organizations, helping them to reduce the labor costs of staff. In many such systems, there is always natural language understanding module that solves intent classification task. This task is complicated becau…

In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose co…

We show that any infinite order element $g$ of a virtually cyclic hyperbolically embedded subgroup of a group $G$ is Morse, that is to say any quasi-geodesic connecting points in the cyclic group $C$ generated by $g$ stays close to $C$. This answers a question of Dahmani-Guirardel-Osin. What is more, we show that hyper…

The paper shows examples of 2-complexes that can't be embedded in R^4, hiding obstructions in higher Milnor invariants.

We consider the question of existence of embedded doubly periodic minimal surfaces in Euclidean 3-space with Scherk-type ends, surfaces that topologically are Scherk's doubly periodic surface with handles added in various ways. We extend the existence results of H. Karcher and F. Wei to more cases, and we find other ca…

This paper investigates the space of codimension zero embeddings of a Poincare duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincare immersions to a certain space of "unlinked" Poincare embeddings. The layers of this tower are described in terms of the coefficien…

In this paper, we provide a theoretical understanding of word embedding and its dimensionality. Motivated by the unitary-invariance of word embedding, we propose the Pairwise Inner Product (PIP) loss, a novel metric on the dissimilarity between word embeddings. Using techniques from matrix perturbation theory, we revea…

The rate at which medical questions are asked online far exceeds the capacity of qualified people to answer them, and many of these questions are not unique. Identifying same-question pairs could enable questions to be answered more effectively. While many research efforts have focused on the problem of general questio…

Kernel mean embeddings have recently attracted the attention of the machine learning community. They map measures $μ$ from some set $M$ to functions in a reproducing kernel Hilbert space (RKHS) with kernel $k$. The RKHS distance of two mapped measures is a semi-metric $d_k$ over $M$. We study three questions. (I) For a…

The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest pa…

Solves embedding problem for 5D manifolds into Calabi-Yau 3-folds.

The paper characterizes Pfaffian embeddings from 2,3,5-manifolds to 7-dimensional isotropic spaces.

Let X be a simply-connected closed oriented 4-manifold and A an embedded surface of genus g and negative self-intersection -N. We show that for fixed genus g there is an upper bound on N if the homology class of A is divisible or characteristic. In particular, for genus zero, there is a lower bound on the self-intersec…

We introduce an invariant linked to some foundational questions in geometric measure theory and provide bounds on this invariant by decomposing an arbitrary cycle into uniformly rectifiable pieces. Our invariant measures the difficulty of cutting a nonorientable closed manifold or mod-2 cycle in $\mathbb{R}^n$ into ori…