arXiv research

New theory explains how noisy, high-dimensional data can still lead to robust predictions.

TIE framework detects out-of-distribution samples and estimates uncertainty without external datasets.

We propose a bandit algorithm that explores by randomizing its history of rewards. Specifically, it pulls the arm with the highest mean reward in a non-parametric bootstrap sample of its history with pseudo rewards. We design the pseudo rewards such that the bootstrap mean is optimistic with a sufficiently high probabi…

We consider the problem of generating automatic code given sample input-output pairs. We train a neural network to map from the current state and the outputs to the program's next statement. The neural network optimizes multiple tasks concurrently: the next operation out of a set of high level commands, the operands of…

A four-pronged approach to dealing with Social Science Phenomenon is outlined. This methodology is applied to Financial Services, Economic Growth and Well-Being. The four prongs are like the four directions for an army general looking for victory. Just like the four directions, we need to be aware that there is a degre…

In G2 manifolds, 3-dimensional associative submanifolds (instantons) play a role similar to J-holomorphic curves in symplectic geometry. In [21], instantons in G2 manifolds were constructed from regular J-holomorphic curves in coassociative submanifolds. In this exposition paper, after reviewing the background of G2 ge…

Distributivity in algebraic structures appeared in many contexts such as in quasigroup theory, semigroup theory and algebraic knot theory. In this paper we give a survey of distributivity in quasigroup theory and in quandle theory.

We study the behaviour of quasi-geodesics in Out(F_n). Given an element f in Out(F_n) there are several natural paths connecting the origin to f in Out(F_n); for example, paths associated to sequences of Stallings folds and paths induced by the shadow of greedy folding paths in Outer Space. We show that none of these p…

The paper constructs a complex for the Dirac operator in 4 dimensions.

This paper is an updated version of a survey on projective configurations of subspaces in general position. The preceding version was published in Russian in 1989 and in English in 1990 (in Leningrad Math. J.) opening a new section ``Light reading for the professional''. The paper is written in the form of introduction…

Rare data in a large-scale database are called outliers that reveal significant information in the real world. The subspace-based outlier detection is regarded as a feasible approach in very high dimensional space. However, the outliers found in subspaces are only part of the true outliers in high dimensional space, in…

A time-flat condition on spacelike 2-surfaces in spacetime is considered here. This condition is analogous to constant torsion condition for curves in three dimensional space and has been studied in [2, 4, 5, 12, 13]. In particular, any 2-surface in a static slice of a static spacetime is time-flat. In this article, we…

Submanifolds in Lorentz-Minkowski space are investigated from various mathematical viewpoints and are of interest also in relativity theory. We define the hyperbolic surface and the de Sitter surface of a curve in the spacelike hypersurface M in the Minkowski 4-space. These surfaces are respectively located in the hype…

Study shows knots in homology spheres can be topologically equivalent to those in $S^3$.

Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the boundary of M which are nullhomotopic in M. We also show that the set of all simp…

New bounds show triangulated surfaces are evenly distributed in moduli space.

In this article we compute the best Sobolev constants for various Hardy-Sobolev inequalities with sharp Hardy term. This is carried out in three different environments: interior point singularity in Euclidean space, interior point singularity in hyperbolic space and boundary point singularity in Euclidean domains.

In [ES2], the first and the third authors introduced new classes in the Johnson cokernels of the mapping class groups of surfaces by a representation theoretic approach based on some previous results for the Johnson cokernels of the automorphism groups of free groups. On the other hand, in [KK1], Kawazumi and the secon…

The Cornu spirals on plane are the curves whose curvatures are linear. Generalized planar cornu spirals and Euler spirals in E^3, the curves whose curvatures are linear are defined in [1,5]. In this study, these curves are presented as the ratio of two rational linear functions. Also here, generalized Euler spirals in …

The paper tackles multi-player information asymmetry bandits in metric spaces.

In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beiträge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the synthetic approach to jet bundles in terms of infinitesimal spaces Dⁿ's. In our succeeding [Synthetic differential geo…

In contrast with knots, whose properties depend only on their extrinsic topology in $S^3$, there is a rich interplay between the intrinsic structure of a graph and the extrinsic topology of all embeddings of the graph in $S^3$ . For example, it was shown in [2] that every embedding of the complete graph $K_7$ in $S^3$ …

In this paper we prove existence of complete minimal surfaces in some metric semidirect products. These surfaces are similar to the doubly and singly periodic Scherk minimal surfaces in $\mathbb R^3$. In particular, we obtain these surfaces in the Heisenberg space with its canonical metric, and in Sol3 with a one-param…

H-minimal Lagrangian submanifolds in general Kähler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with nega…

Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.

In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic $\mathbb{R}P^{2m}$ in $\mathbb{C}P^n$. Then it minimizes volume among the isotropic submanifolds in the same $\mathbb{Z}/2$ homology class in $\mathbb{C}P^n$ (but not among all submanifolds in this…

Sharp area estimates for minimal submanifolds in curved spaces.

In recent years, there are many progress made in Kähler geometry. In particular, the topics related to the problems of the existence and uniqueness of extremal Kähler metrics, as well as obstructions to the existence of such metrics in general Kähler manifold. In this talk, we will report some recent developments in th…

The paper classifies reversible elements in Seifert-fibered spaces and braid groups.

For this quarter of century, differential operators in a lower dimensional submanifold embedded or immersed in real $n$-dimensional euclidean space $\EE^n$ have been studied as quantum mechanical models, which are realized as restriction of the operators in $\EE^n$ to the submanifold. For this decade, the Dirac operato…

We establish the existence and characterization of a primal and a dual facelift - discontinuity of the value function at the terminal time - for utility-maximization in incomplete semimartingale-driven financial markets. Unlike in the lower- and upper-hedging problems, and somewhat unexpectedly, a facelift turns out to…

Clarifies the various fairness definitions in ML.

Review of MLMC in financial engineering, focusing on option pricing and risk management.

Due to the lack of enough generalization in the state-space, common methods in Reinforcement Learning (RL) suffer from slow learning speed especially in the early learning trials. This paper introduces a model-based method in discrete state-spaces for increasing learning speed in terms of required experience (but not r…

We discuss here researches on econophysics done from India in the last two decades. The term `econophysics' was formally coined in India (Kolkata) in 1995. Since then many research papers, books, reviews, etc. have been written by scientists. Many institutions are now involved in this research field and many conference…

In this study, some characterizations of Euler spirals in E_1^{3} have been presented by using their main property that their curvatures are linear. Moreover, discussing some properties of Bertrand curves and helices, the relationship between these special curves in E_1^{3} have been investigated with different theorem…

In [18], L. R. Pears proved that Bertrand curves in E-n(n > 3) are degenerate curves. This result restate in [16] by Matsuda and Yorozu. They proved that there is no special Bertrand curves in E-n(n > 3) and they define new kind of Bertrand curves called (1, 3)-type Bertrand curves in 4-dimensional Euclidean space. In …

In this paper, we introduce a sequence of invariants of a knot K in S^3: the knot Floer homology groups of the preimage of K in the m-fold cyclic branched cover over K. We exhibit the knot Floer homology in the m-fold branched cover as the categorification of a multiple of the Turaev torsion in the case where the m-fol…

In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar curvature, then along the flow by a positive power $α$ of the mean curvature $H$, t…

MLPs can approximate any function in context, challenging the importance of in-context universality.

In this note we consider the problem of local classification of Bonnet pairs of surfaces in 3-dimensional Minkowski space. We use split quaternions in a way similar to the use of quaternions in, arXiv:dg-ga/9610006, for the solution in Euclidean 3-Space.

Invariances to translation, rotation and other spatial transformations are a hallmark of the laws of motion, and have widespread use in the natural sciences to reduce the dimensionality of systems of equations. In supervised learning, such as in image classification tasks, rotation, translation and scale invariances ar…

Hypersurfaces with constant Ricci eigenvalues in real space forms are classified.

New spheres can split a 4D link in ways not possible in 3D.

In this paper, we define a rectifying spacelike curve in the Minkowski space-time $E_1^4$ as a curve whose position vector always lies in orthogonal complement $N^{\bot}$ of its principal normal vector field $N$. In particular, we study the rectifying spacelike curves in $E_1^4$ and characterize such curves in terms of…

We give a brief review of a research made in the field of differential geometry in Estonia in the period from the beginning of the 19th century to the present time. The biographic data of mathematicians who made a valuable contribution to the development of differential geometry in Estonia in mentioned period are prese…

This work introduces a method to decompose uncertainty in in-context learning for large language models.

In this paper, we compare the performances of FAISS and FENSHSES on nearest neighbor search in Hamming space--a fundamental task with ubiquitous applications in nowadays eCommerce. Comprehensive evaluations are made in terms of indexing speed, search latency and RAM consumption. This comparison is conducted towards a b…