arXiv research

We show that thick morphisms (or microformal morphisms) between smooth (super)manifolds, introduced by us before, are classical limits of `quantum thick morphisms' defined here as particular oscillatory integral operators on functions.

Improved bounds for Carleson-Sjölin operators on manifolds with specific curvature conditions.

Geometric approach to Dirac operator evolution on spacetimes.

Global propagator for massless Dirac operator defined and analyzed.

We give an explicit formula, as a formal differential operator, for quantum microformal morphisms of (super)manifolds that we introduced earlier. Such quantum microformal morphisms are essentially oscillatory integral operators or Fourier integral operators of a particular kind. They act on oscillatory wave functions, …

Study oscillatory integrals with degenerate singular points in multivariable phase functions.

The paper calculates asymptotic expansions for specific types of oscillatory integrals.

The geometry of oscillatory integrals on manifolds with intermediate symmetry.

We introduce a Hilbert $A$-module structure on the higher oscillatory module, where $A$ denotes the $C^*$-algebra of bounded endomorphisms of the basic oscillatory module. We also define the notion of an exterior covariant derivative in an $A$-Hilbert bundle and use it for a construction of an $A$-elliptic complex of d…

In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his re…

The study improves norms of spectral projectors on specific surfaces.

Let $M$ be a closed Riemannian manifold carrying an effective and isometric action of a compact connected Lie group $G$. We derive a refined remainder estimate in the stationary phase approximation of certain oscillatory integrals on $T^\ast M \times G$ with singular critical sets that were examined previously in order…

Mathematical study of learning long-term integration in linear RNNs.

We show that, for odd $d$, the $L^{\frac{d+2}2}$ bounds of Sogge and Xi for the Nikodym maximal function over manifolds of constant sectional curvature, are unstable with respect to metric perturbation, in the spirit of the work of Sogge and Minicozzi. A direct consequence is the instability of the bounds for the corre…

Improved Kuznecov remainder estimates for generic metrics.

Study shows global oscillatory solutions for Yang-Mills heat flow in 4D space.

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(λ) = \exp(i(λ_1ψ_1 + ... + λ_nψ_n)) for λin R^n, gives rise to a manifold M of orthogonal projections for the subspaces u(λ)H^2(R) of L^2(R). For classes of admissible functions ψ_i the strong operator topology cl…

We extend a result of the second author \cite[Theorem 1.1]{soggekaknik} to dimensions $d \geq 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<\frac{2(d+1)}{d-1}$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee …

This paper proposes a novel kernel-based optimization scheme to handle tasks in the analysis, e.g., signal spectral estimation and single-channel source separation of 1D non-stationary oscillatory data. The key insight of our optimization scheme for reconstructing the time-frequency information is that when a nonparame…

Sparse regression models CMs from oscillatory shear data efficiently.

D-LinOSS models learn to dissipate energy, improving performance on long-range tasks.

Neural networks enjoy widespread use, but many aspects of their training, representation, and operation are poorly understood. In particular, our view into the training process is limited, with a single scalar loss being the most common viewport into this high-dimensional, dynamic process. We propose a new window into …

A spectral approach to building the exterior calculus in manifold learning problems is developed. The spectral approach is shown to converge to the true exterior calculus in the limit of large data. Simultaneously, the spectral approach decouples the memory requirements from the amount of data points and ambient space …

This work is a contribution to the area of Strict Quantization (in the sense of Rieffel) in the presence of curvature and non-Abelian group actions. More precisely, we use geometry to obtain explicit oscillatory integral formulae for strongly invariant strict deformation quantizations of a class of solvable symplectic …

In this paper we study the existence of a first zero and the oscillatory behavior of solutions of the ordinary differential equation $(vz')'+Avz = 0$, where $A,v$ are functions arising from geometry. In particular, we introduce a new technique to estimate the distance between two consecutive zeros. These results are ap…

Optimal Strichartz estimates for Schrödinger on Zoll manifolds.

Researchers extend microlocal analysis across event horizons of rotating black holes.

For node level graph encoding, a recent important state-of-art method is the graph convolutional networks (GCN), which nicely integrate local vertex features and graph topology in the spectral domain. However, current studies suffer from several drawbacks: (1) graph CNNs relies on Chebyshev polynomial approximation whi…

We prove an estimate for spherical functions $\varphi_λ(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $λ$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.…

Study measures inequality in social-economic systems using Fokker-Planck equations and Lotka-Volterra dynamics.

The paper develops a new theory to understand deep learning optimization.

We extend neural networks with fractional and mixed activation functions for better function approximation.

Interpretable machine-learning models can be unstable under multicollinearity, leading to oscillatory weights that do not reflect meaningful contributions.

Oscillations lie at the core of many biological processes, from the cell cycle, to circadian oscillations and developmental processes. Time-keeping mechanisms are essential to enable organisms to adapt to varying conditions in environmental cycles, from day/night to seasonal. Transcriptional regulatory networks are one…

Data-driven spatial filtering algorithms optimize scores such as the contrast between two conditions to extract oscillatory brain signal components. Most machine learning approaches for filter estimation, however, disregard within-trial temporal dynamics and are extremely sensitive to changes in training data and invol…

We explore nature of price formation in financial markets and develop a theory of bid and ask price dynamics in which the two prices form due to quantum-chaotic interaction between buy and sell orders. In this model bid and ask prices are represented by eigenvalues of a 2x2 price operator corresponding to 'bid' and 'as…

In this paper we engage in a general study of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants of mapping tori of surface mapping class group elements. We use the geometric construction of the Witten-Reshetikhin-Turaev TQFT via the geometric quantization of moduli spaces of flat connections on surfa…

Improved formulation of spinfoam quantum gravity with cosmological constant, ensuring all amplitudes are finite and providing semiclassical asymptotics.

We show that accelerated gradient descent, averaged gradient descent and the heavy-ball method for non-strongly-convex problems may be reformulated as constant parameter second-order difference equation algorithms, where stability of the system is equivalent to convergence at rate O(1/n 2), where n is the number of ite…

In a recent work the first named author, Levitin and Vassiliev have constructed the wave propagator on a closed Riemannian manifold $M$ as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. In this paper, first we give a natural reinterpretation of the und…

Introduces Exponentially Weighted Signature for better path representation.

A new measure of dependence for various data types.

Study analyzes low-energy behavior of Schrödinger operators with Coulomb potentials.

Neural signals are characterized by rich temporal and spatiotemporal dynamics that reflect the organization of cortical networks. Theoretical research has shown how neural networks can operate at different dynamic ranges that correspond to specific types of information processing. Here we present a data analysis framew…

We study the propagator of the wave equation on a closed Riemannian manifold $M$. We propose a geometric approach to the construction of the propagator as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. This enables us to provide a global invariant defi…

One of the goals of this article is to define a an unified setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure. We first remark that some known examples coming from the theory of metric measured spaces and also from oscill…

We show that one can obtain logarithmic improvements of $L^2$ geodesic restriction estimates for eigenfunctions on 3-dimensional compact Riemannian manifolds with constant negative curvature. We obtain a $(\logλ)^{-\frac12}$ gain for the $L^2$-restriction bounds, which improves the corresponding bounds of Burq, Gérard …

Transforms game optimization dynamics into frequency domain for precise hyperparameter analysis.