arXiv research

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, …

It is well-known that any pair of closed orientable 3-manifolds are related by a finite sequence of Dehn surgeries on knots. Furthermore Kawauchi showed that such knots can be taken to be hyperbolic. In this article, we consider the minimal length of such sequences connecting a pair of 3-manifolds, in particular, a pai…

Characterizes sequences from two-component link diagrams.

For any quiver mutation sequence, we define a pair of matrices that describe a fixed point equation of a cluster transformation determined from the mutation sequence. We give an explicit relationship between this pair of matrices and the Jacobian matrix of the cluster transformation. Furthermore, we show that this rela…

We submit a new way to detect pairs of non-cobordant surface-links. We find a new example of a pair of non-cobordant surface-links with the following properties: Orr invariant, Cochran sequence, Sato-Levine invariant, the alinking number and one of Stallings's theorems cannot distinguish them. However our new way can d…

Sequence probability predicts correctness in LLMs, but not for repeated prompts

A framework infers causal direction from symbolic sequences using compression measures.

KERMIT models sequences and pairs using a single neural network.

New approach speeds up DNA sequence alignment.

Given a link in the three-sphere, Ozsváth and Szabó showed that there is a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double cover. The aim of this paper is to explicitly calculate this spectral sequence in terms of bordered Floer homolo…

Optimal Farey sequence for $Γ_0(2^n)$ with upper bound $2^{n-1}$.

The paper extends sequences while preserving statistical properties using a mixture model.

Sequence distortion measures large-scale distances in metric spaces.

We refine Matveev's result asserting that any two closed oriented 3-manifolds can be related by a sequence of borromean surgeries if and only if they have isomorphic first homology groups and linking pairings. Indeed, a borromean surgery induces a canonical isomorphism between the first homology groups of the involved …

The Knight Move Conjecture claims that the Khovanov homology of any knot decomposes as direct sums of some "knight move" pairs and a single "pawn move" pair. This is true for instance whenever the Lee spectral sequence from Khovanov homology to Q^2 converges on the second page, as it does for all alternating knots and …

The availability of large amounts of time series data, paired with the performance of deep-learning algorithms on a broad class of problems, has recently led to significant interest in the use of sequence-to-sequence models for time series forecasting. We provide the first theoretical analysis of this time series forec…

Model learns association between text and speech without paired data.

Single individual haplotyping is an NP-hard problem that emerges when attempting to reconstruct an organism's inherited genetic variations using data typically generated by high-throughput DNA sequencing platforms. Genomes of diploid organisms, including humans, are organized into homologous pairs of chromosomes that d…

Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygon…

Established equivalence of Atiyah classes for generalized holomorphic vector bundles.

We construct a sequence of pairs of 3-manifolds each with torus boundary and with the following two properties: 1) For the result of a carefully chosen glueing of the nth pair of 3-manifolds along their boundary tori, the ratio of the genus of the resulting 3-manifold to the sum of the genera of the pair of 3-manifolds…

A measure called relative cluster entropy distinguishes between correlated and uncorrelated sequences.

New sequences prove some link diagrams can't be transformed by specific moves.

In this article we construct a canonical Kähler-Einstein current on a LC (log canonical) pairs of log general type as the limit of a sequence of canonical Kähler-Einstein currents on KLT(Kawamata log terminal) pairs of log general type. We call the volume form associated with the canonical Kähler-Einstein current the c…

Given a link in the three-sphere, Z. Szabó and the second author constructed a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double-cover. The aim of this paper and its sequel is to explicitly calculate this spectral sequence, using bordere…

Programmatic Motion Concepts learn human actions from paired videos.

The paper examines Yang-Mills-Higgs pairs on vector bundles and proves stability and energy identity.

We develop an epsilon-controlled algebraic L-theory, extending our earlier work on epsilon-controlled algebraic K-theory. The controlled L-theory is very close to being a generalized homology theory; we study analogues of the homology exact sequence of a pair, excision properties, and the Mayer--Vietoris exact sequence…

We introduce Courant algebroids, providing definitions, some historical notes, and some elementary properties. Next, we summarize basic properties of graded manifolds. Then, drawing on the work of Roytenberg and others, we introduce the graded or supergraded language demonstrating a cochain complex / cohomology for (ge…

Let $Γ$ be a f.g. discrete group and let $\tilde M$ be a Galois $Γ$-covering of a smooth closed manifold $M$. Let $S_*^Γ(\tilde{M})$ be the analytic structure group, appearing in the Higson-Roe analytic surgery sequence $\to S_*^Γ(\tilde M)\to K_*(M)\to K_*(C_r^*Γ)\to$. We prove that for an arbitrary discrete group $Γ$…

Traditional pairwise sequence alignment is based on matching individual samples from two sequences, under time monotonicity constraints. However, in many application settings matching subsequences (segments) instead of individual samples may bring in additional robustness to noise or local non-causal perturbations. Thi…

A theorem of Kirby states that two framed links in the 3-sphere produce orientation-preserving homeomorphic results of surgery if they are related by a sequence of stabilization and handle-slide moves. The purpose of the present paper is twofold: First, we give a sufficient condition for a sequence of handle-slides on …

Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long exact sequences in homology and further algebraic structure which is then used to derive topological and computational results. Two of our applications include giv…

The paper shows that knot projections without triple chords can be simplified.

The study examines spaces of non-extendable quasimorphisms for group pairs.

The paper introduces a group $LSP$ of obstructions for splitting a homotopy equivalence along a pair of submanifolds. We develop exact sequences relating the $LSP$-groups with various surgery obstruction groups for manifold triple and structure sets arising from triples of manifolds. The natural map from the surgery ob…

Study SGD dynamics in sequence models, revealing training phases and influence of sequence length.

In the article the necessary and sufficient conditions for a representation of Lipschitz function of two variables as a difference of two convex functions are formulated. An algorithm of this representation is given. The outcome of this algorithm is a sequence of pairs of convex functions that converge uniformly to a p…

The paper studies sequences of solutions to Hitchin-Simpson equations on Kähler manifolds.

The paper improves collapsing Alexandrov spaces results using good coverings.

In this paper, we propose a novel neural network model called RNN Encoder-Decoder that consists of two recurrent neural networks (RNN). One RNN encodes a sequence of symbols into a fixed-length vector representation, and the other decodes the representation into another sequence of symbols. The encoder and decoder of t…

New pairing defined from Morse complexes for compact manifolds.

A theory of differential characters is developed for manifolds with boundary. This is done from both the Cheeger-Simons and the deRham-Federer viewpoints. The central result of the paper is the formulation and proof of a Lefschetz-Pontrjagin Duality Theorem, which asserts that the pairing: Ch^k(X,dX) x Ch^{n-k-1}(X) --…

In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on prof…

AFP-CKSAAP predicts antifreeze proteins using k-spaced amino acid pairs with deep neural networks.

Topological model for coloured Alexander invariants from quantum group representations.

In this paper we consider two sequence tagging tasks for medieval Latin: part-of-speech tagging and lemmatization. These are both basic, yet foundational preprocessing steps in applications such as text re-use detection. Nevertheless, they are generally complicated by the considerable orthographic variation which is ty…

This paper studies the behavior of sequences of solutions to Seiberg-Witten like equations for a pair consisting of a Hermitian connection on a line bundle over a 4-dimensional manifold and a section of the self-dual spinor bundle of a complex Clifford module on the manifold. Examples include the cases where the Cliffo…