arXiv research

The paper proves the existence of a special Kähler metric on a minimal ruled surface.

In this paper, the Bando-Futaki invariants on hypersurfaces are derived in terms of the degree of the defining polynomials, the dimension of the underlying projective space, and the given holomorphic vector field. In addition, the holomorphic invariant introduced by Tian and Chen (Ricci Flow on Kähler-Einstein surfaces…

In this paper we introduce higher extremal Kahler metrics. We provide an example of the same on a minimal ruled surface. We also prove a perturbation result that implies that there are non-trivial examples of higher constant scalar curvature metrics, which are basically metrics where the top Chern form is harmonic. We …

The paper finds conical higher cscK metrics on minimal ruled surfaces with conical singularities.

Top/O's first two k-invariants are zero.

We use the theory of dual of Fréchet-Schwartz (DFS) spaces to establish a sufficient condition for top-degree solvability for the differential complex associated to a hypocomplex locally integrable structure. As an application, we show that the top-degree cohomology of left-invariant hypocomplex structures on a compact…

A parameter-invariant variational problem with a manifestly covariant Lagrangian function of second order is considered, which covers the case of the free relativistic top at constraint manifold of constant acceleration

Extends homotopical theory to locally compact groups, refining their compactness properties.

Quantum invariants for fibered links determined by genus and Hopf invariant.

The topological fundamental group $π_{1}^{top}$ is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased space $X$, we compute the topological fundamental group of the "hoop earring" spac…

We define the stabilizing number $\operatorname{sn}(K)$ of a knot $K \subset S^3$ as the minimal number $n$ of $S^2 \times S^2$ connected summands required for $K$ to bound a nullhomotopic locally flat disc in $D^4 \# n S^2 \times S^2$. This quantity is defined when the Arf invariant of $K$ is zero. We show that $\oper…

For any compact oriented manifold $M$, we show that that the top degree multi-vector fields transverse to the zero section of $\wedge^{\text{top}}TM$ are classified, up to orientation preserving diffeomorphism, in terms of the topology of the arrangement of its zero locus and a finite number of numerical invariants. Th…

The structure set $\ST^{TOP}(M)$ of an $n$-dimensional topological manifold $M$ for $n \geqslant 5$ has a homotopy invariant functorial abelian group structure, by the algebraic version of the Browder-Novikov-Sullivan-Wall surgery theory. An element $(N,f) \in \ST^{TOP}(M)$ is an equivalence class of $n$-dimensional ma…

We introduce Invariant Risk Minimization (IRM), a learning paradigm to estimate invariant correlations across multiple training distributions. To achieve this goal, IRM learns a data representation such that the optimal classifier, on top of that data representation, matches for all training distributions. Through theo…

We study the stability of symmetric trajectories of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$ invariant metric and an $SO(2)\times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the $SO(2)\times SO(2)$ momentu…

Rotation invariant algorithms fail with hard labels sampled from sparse targets.

The large $k$ asymptotics (perturbation series) for integrals of the form $\int_{\cal F}μe^{i k S}$, where $μ$ is a smooth top form and $S$ is a smooth function on a manifold ${\cal F}$, both of which are invariant under the action of a symmetry group ${\cal G}$, may be computed using the stationary phase approximation…

Two graph homologies help compute embedding space.

The integral of the top dimensional term of the multiplicative sequence of Pontryagin forms associated to an even formal power series is calculated for special Riemannian metrics on the unit ball of a hermitean vector space. Using this result we calculate the generating function of the reduced Dirac and signature eta-i…

A second order variational description of the autoparallel curves of some differential-geometric connection for the third order Mathisson's 'new mechanics' of a relativistic free spinning particle is suggested starting from general requirements of invariance and 'variationality'.

Study curves on a Whitney umbrella using geometric invariants.

We give a generalization of the Nambu mechanics based on vector Hamiltonians theory. It is shown that any divergence-free phase flow in $\mathbb{R}^n$ can be represented as a generalized Nambu mechanics with $n-1$ integral invariants. For the case when the phase flow in $\mathbb{R}^n$ has $n-3$ or less first integrals,…

We prove that a rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three we prove that only multiples of the top Chern number, which is the Euler …

Proves a plumbing-multiplicative property of a Links-Gould invariant.

We give an interpretation of Yetter's Invariant of manifolds $M$ in terms of the homotopy type of the function space $TOP(M,B(G))$, where $G$ is a crossed module and $B(G)$ is its classifying space. From this formulation, there follows that Yetter's invariant depends only on the homotopy type of $M$, and the weak homot…

We study the sutured Floer homology invariants of the sutured manifold obtained by cutting a knot complement along a Seifert surface, R. We show that these invariants are finer than the "top term" of the knot Floer homology, which they contain. In particular, we use sutured Floer homology to distinguish two non-isotopi…

Within its traditional range of perversity parameters, intersection cohomology is a topological invariant of pseudomanifolds. This is no longer true once one allows superperversities, in which case intersection cohomology may depend on the choice of the stratification by which it is defined. Topological invariance also…

New methods reveal colored Jones polynomials from quantum R-matrices and knot invariants.

Paper analyzes trade-offs in top-k classification accuracies and proposes a new loss function.

New conformally invariant forms help identify Einstein metrics.

Paper tackles reinforcement learning generalization through invariant policy optimization.

The paper proves inequalities for closed surfaces involving mean curvature.

We introduce a new weight-decay scaling rule to maintain sublayer gains across different widths in modern scale-invariant architectures.

A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three only multiples of the top Chern number, which is the Euler characteristic, are invarian…

This paper explores a variety of models for frame-based music transcription, with an emphasis on the methods needed to reach state-of-the-art on human recordings. The translation-invariant network discussed in this paper, which combines a traditional filterbank with a convolutional neural network, was the top-performin…

Integrates magnetic geodesic and sub-Riemannian flows on Stefel variety, proving integrability and Lax presentations.

The paper sets genus bounds for twisted quantum invariants.

Expressiveness and generalization of deep models was recently addressed via the connection between neural networks (NNs) and kernel learning, where first-order dynamics of NN during a gradient-descent (GD) optimization were related to gradient similarity kernel, also known as Neural Tangent Kernel (NTK). In the majorit…

Class ambiguity is typical in image classification problems with a large number of classes. When classes are difficult to discriminate, it makes sense to allow k guesses and evaluate classifiers based on the top-k error instead of the standard zero-one loss. We propose top-k multiclass SVM as a direct method to optimiz…

The top-$k$ error is often employed to evaluate performance for challenging classification tasks in computer vision as it is designed to compensate for ambiguity in ground truth labels. This practical success motivates our theoretical analysis of consistent top-$k$ classification. Surprisingly, it is not rigorously und…

Paper bridges quantum and classical mechanics for open systems.

Study links and quivers, proving polynomial equality conjecture.

Paper introduces a new loss function for deep imbalanced classification.

Some intrinsic tools from the formal theory of variational equations are being demonstrated at work in application to one concrete example of the third-order evolution equation of free relativistic top in three-dimensional space-time. The main goal is to introduce a combined approach consisting in the simultaneous util…

Unified model for prediction and deferral selects top-k entities efficiently.

Work on making classifiers robust against adversarial attacks for top-k predictions.

Proposes top-label calibration and M2B framework for multiclass to binary calibration.

This work provides theoretical and empirical evidence that invariance-inducing regularizers can increase predictive accuracy for worst-case spatial transformations (spatial robustness). Evaluated on these adversarially transformed examples, we demonstrate that adding regularization on top of standard or adversarial tra…