arXiv research

Study on 2-valued dynamics on complex plane, showing some dynamics can't be group actions.

Formula for Z_2-valued index of symmetric operators on manifolds.

Study shows singular set of certain graphs has codimension 1.

In this paper we consider a generalization of the Faddeev model for the maps from a closed three-manifold into the two-sphere. We give a novel representation of smooth $S^2$-valued maps based on flat connections. This representation allows us to obtain an analytic description of the homotopy classes of $S^2$-valued m…

Study of spectral flow in symmetric Toeplitz operator families.

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary d…

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $S$ be an orientable surface in $\mathbb{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $ω$. Together with Minkowski m…

Compactifies maximal component of surface group representations into a closed ball.

Study predicts coastal water quality using machine learning, identifying salinity as key factor.

The study characterizes homology 4-manifolds with $g_2\leq 5$ combinatorially.

This thesis is concerned with the residues modulo 4 and 8 of the signature of a 4k-dimensional oriented geometric Poincare complex. The Z_8-valued Brown-Kervaire invariant of Z_4-valued quadratic forms is used to prove that if the signature is divisible by 4, the divisibility by 8 is detected by the Arf invariant of a …

Proves singular set of certain integral hypercurrents has measure zero.

Jeffrey and Kirwan suggested expressions for intersection pairings on the reduced space of a Hamiltonian G-space in terms of multiple residues. In this paper we prove a residue formula for symplectic volumes of reduced spaces of a quasi-Hamiltonian SU(2)-space. The definition of quasi-Hamiltonian G-spaces was recently …

We study the fixed point theory of n-valued maps of a space X using the fixed point theory of maps between X and its configuration spaces. We give some general results to decide whether an n-valued map can be deformed to a fixed point free n-valued map. In the case of surfaces, we provide an algebraic criterion in term…

We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior. Applications including a compactness theorem for a class of immersed stable minimal h…

Paper proves Rohlin invariant's uniqueness and extends homology sphere invariants.

We introduce a representation theory for risk operations on locally compact groups in a partition of unity on a topological manifold for Markowitz-Tversky-Kahneman (MTK) reference points. We identify (1) risk torsion induced by the flip rate for risk averse and risk seeking behaviour, and (2) a structure constant or co…

It is shown that if a $C^2$ surface $M\subset\mathbb R^3$ has negative curvature on the complement of a point $q\in M$, then the $\mathbb Z/2$-valued Poincaré-Hopf index at $q$ of either distribution of principal directions on $M-\{q\}$ is non-positive. Conversely, any non-positive half-integer arises in this fashion. …

We define and study Vassiliev invariants for (long) Morse knots. It is shown that there are Vassiliev invariants which can distinguish some topologically equivalent Morse knots. In particular, there is an invariant of order 3 for Morse knots with one maximum that distinguishes two different representations of the figur…

Given a group endowed with a Z/2-valued morphism we associate a Gauss diagram theory, and show that for a particular choice of the group these diagrams encode faithfully virtual knots on a given arbitrary surface. This theory contains all of the earlier attempts to decorate Gauss diagrams, in a way that is made precise…

New invariant detects infinite order cabled knots.

Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.

New solutions found for G2 system on squashed manifolds.

Given a surface F, we are interested in Z/2 valued invariants of immersions of F into R^3, which are constant on each connected component of the complement of the quadruple point discriminant in Imm(F,R^3). Such invariants will be called ``q-invariants.'' Given a regular homotopy class A in Imm(F,R^3), we denote by V_n…

We show that any 2-valued C^{1, α} (α\in (0, 1)) function u = {u_{1}, u_{2}} on an open ball B in {\mathbb R}^{n} with values u_{1}, u_{2} \in {\mathbb R}^{k} whose graph, viewed as a varifold with multiplicity 2 at points where u_{1} = u_{2} and with multiplicity 1 at points where u_{1}, u_{2} are distinct, is station…

Invariant obstructs separating coassociative 4-folds.

In precision agriculture (PA), soil sampling and testing operation is prior to planting any new crop. It is an expensive operation since there are many soil characteristics to take into account. This paper gives an overview of soil characteristics and their relationships with crop yield and soil profiling. We propose a…

New invariant for spin 3-manifolds using super 3-cocycles.

Modern deep learning methods provide effective means to learn good representations. However, is a good representation itself sufficient for sample efficient reinforcement learning? This question has largely been studied only with respect to (worst-case) approximation error, in the more classical approximate dynamic pro…

The study examines stationary integral varifolds near multiplicity 2 planes, proving regularity under specific conditions.

Study compactifies representations space of hyperbolic surfaces.

Optimal liquidation strategy with price impact and signal exploitation.

New deep learning model estimates scattering timescale of FRBs efficiently.

Second order partial differential equations which describe spherical surfaces (ss) or pseudospherical surfaces (pss) are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature $K = 1$ or $K = -1$, respectively, and they can be seen as the compatibility condition of an …

Study uses Perelman and Ricci flow methods to analyze economic inequality.

Third-order PDEs describe spherical and pseudospherical surfaces.

New activation function BrownianReLU improves LSTM network performance on financial time series.

Study reflection symmetry and APS boundary conditions on a warped cylinder.

The paper extends minimal network theory to the sphere, proving local minimality.

Estimates mode of discrete distributions with fewer samples.

LLMs can fail to maximize aligned values even after training, due to irrational reasoning.

A new method for network regression using optimal transport.

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R^3, with values in any Abelian group. We show they are all functions of the universal order 1 invariant that we construct as T \oplus P \oplus Q where T is a Z valued invariant reflecting the number of triple poin…

Study spectral flow on a warped cylinder with special boundary conditions.

Combines RFs and GLMs for better accuracy and interpretable feature importance.

Bernstein theorem proven for 2-valued minimal graphs in 4D.

For knots in S^3, the bi-graded hat version of knot Floer homology is defined over Z; however, for a link L in S^3 with #|L|=l>1, there are 2^{l-1} bi-graded hat versions of link Floer homology defined over Z, the multi-graded hat version of link Floer homology is only defined over F_2 from holomorphic considerations, …

In the semantic segmentation of street scenes the reliability of the prediction and therefore uncertainty measures are of highest interest. We present a method that generates for each input image a hierarchy of nested crops around the image center and presents these, all re-scaled to the same size, to a neural network …