arXiv research

Study virtualized Delta, Sharp, and Pass moves for oriented virtual knots and links.

We prove that the crossing changes, Delta moves, and sharp moves are unknotting operations on welded knots.

We investigate the behaviour of Rasmussen's invariant $s$ under the sharp operation on knots and obtain a lower bound for the sharp unknotting number. This bound leads us to an interesting move that transforms arbitrary knots into non-alternating knots.

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. R…

Sharp curvature pinching for mean curvature flow in spheres proved.

New formulas limit minimal submanifolds' area in curved spaces.

Dynamic trading strategies, in the spirit of trend-following or mean-reversion, represent an only partly understood but lucrative and pervasive area of modern finance. Assuming Gaussian returns and Gaussian dynamic weights or signals, (e.g., linear filters of past returns, such as simple moving averages, exponential we…

The Margulis constant for Kleinian groups is the smallest constant $c$ such that for each discrete group $G$ and each point $x$ in the upper half space ${\bold H}^3$, the group generated by the elements in $G$ which move $x$ less than distance c is elementary. We take a first step towards determining this constant by p…

DASH improves ensemble generalizability by encouraging diverse, flat loss landscapes.

We present a detailed study of the performance of a trading rule that uses moving average of past returns to predict future returns on stock indexes. Our main goal is to link performance and the stochastic process of the traded asset. Our study reports short, medium and long term effects by looking at the Sharpe ratio …

STORM-PG uses momentum for faster policy gradient updates.

A new method corrects bias in machine learning for trading by filtering out non-executable prices.

This paper improves bounds on how many Delta-moves are needed to trivialize a link.

Sharp analysis of power iteration for tensor PCA, improving convergence and stopping criteria.

Study on hard Legendrian unknots using normal rulings.

We provide sharp stability estimates for the Alexandrov Soap Bubble Theorem in the hyperbolic space. The closeness to a single sphere is quantified in terms of the dimension, the measure of the hypersurface and the radius of the touching ball condition. As consequence we obtain a new pinching result for hypersurfaces i…

The asymptotic behavior of the heat kernel of a Riemannian manifold gives rise to the classical concepts of parabolicity, stochastic completeness (or conservative property) and Feller property (or $C^{0}$-diffusion property). Both parabolicity and stochastic completeness have been the subject of a systematic study whic…

Let $M$ be an $n$-vertex combinatorial triangulation of a $\ZZ_2$-homology $d$-sphere. In this paper we prove that if $n \leq d + 8$ then $M$ must be a combinatorial sphere. Further, if $n = d + 9$ and $M$ is not a combinatorial sphere then $M$ can not admit any proper bistellar move. Existence of a 12-vertex triangula…

A well-interpretable measure of information has been recently proposed based on a partition obtained by intersecting a random sequence with its moving average. The partition yields disjoint sets of the sequence, which are then ranked according to their size to form a probability distribution function and finally fed in…

Improved trading strategy using deep learning and changepoint detection for market changes.

The World Trade Web (WTW) is a weighted network whose nodes correspond to countries with edge weights reflecting the value of imports and/or exports between countries. In this paper we introduce to this macroeconomic system the notion of extinction analysis, a technique often used in the analysis of ecosystems, for the…

We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of $\mathbb{R}^{n+1}$, $n\geq1$, and denote by $osc(H)$ the oscillation of its mean curvature. We prove that there exists a positive $\varepsilon$, depending on $n$ and upper …

A new method uses deep learning to efficiently solve complex physics equations in high dimensions.

The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures $f(k_1,\ldots,k_{n-1})$ satisfying suitable conditions. In this paper…

We study the emergence of instabilities in a stylized model of a financial market, when different market actors calculate prices according to different (local) market measures. We derive typical properties for ensembles of large random markets using techniques borrowed from statistical mechanics of disordered systems. …

A new knot move preserves pass-move equivalence and differs in count.

Behavior cloning training instabilities amplified by SGD noise over long horizons.

It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeister moves O1, O2 and O3. Depending on orientations of fragments involved in the moves, one may distinguish 4 different versions of each of the O1 and O2 moves, and 8 versions of the O3 move. We introd…

In this paper, we introduce an equivalence relation on the set of local moves and classify local moves, called the extended $ST$-moves, up to the equivalence. Moreover, by inducing a binary relation on the set of equivalence classes of local moves, we show that an extended $ST$-move realizes the crossing change or the …

Minimal sets of moves for isotopic knots and trivalent graphs identified.

The H(n)-move simplifies virtual and welded knots and links.

We prove that the classical set of moves for standard spines of 3-manifolds (i.e. the MP-move and the V-move) does not suffice to relate to each other any two standard skeleta of a 3-manifold with marked boundary. We also describe a condition on the 3-manifold with marked boundary that tells whether the generalised set…

Large learning rates cause parameter instability, leading to better generalization.

The (ordinary) unknotting-number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. It is very natural to consider the `unknotting-number' associated with other local-moves on n-dimensional knots, where n is a natural number. In this paper we prove the follow…

New rational band moves simplify knot classification.

Study proposes adaptive RL for dynamic portfolio optimization.

New methods for delta-moves on algebraically split links identified.

Minimal moves for surfaces in 4D discovered, linking planar and spatial moves.

Minimal generating sets of Reidemeister moves identified and classified.

We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main …

New diagonal move simplifies knots and links efficiently.

New moves help untangle complex knots.

Minimal moves for surfaces in 4D identified.

New virtualized Δ-move simplifies virtual knots and links.

Link concordance equals homotopy for high-dimensional spheres.

Algorithm for recognizing and performing Reidemeister moves in Gauss diagrams.

Minimal sets of moves for rotational Reidemeister diagrams are identified.

The $Ξ$-move is a local move generated by forbidden moves in virtual knot theory. This move was introduced by Taniguchi and the second author, who showed that it characterizes the odd writhe of virtual knots, which is a fundamental invariant defined by Kauffman. In this paper, we extend this result by classifying $2$-c…