Transformers reduce redundancy by focusing on invariant relational quantities.
problem Substantial internal redundancy in Transformer models due to coordinate-dependent representations and continuous symmetries.
method Reformulate representations, attention mechanisms, and optimization dynamics in terms of invariant relational quantities, eliminating redundant degrees of freedom by construction.
result Architectures that operate directly on relational structures, providing a principled geometric framework for reducing parameter redundancy and analyzing optimization.
A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem
problem Defining and analyzing a mass-type invariant for smooth metric measure spaces
method Defining a mass-type quantity and showing its geometric invariance properties
result The mass-type quantity has a close relation with the fractional Yamabe problem and the relevant Green's function
New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.
problem Defining curvature measures for 4D manifolds with corners.
method Defined two new extrinsic curvature quantities, one conformal invariant, and a new conformally invariant operator.
result Gauss-Bonnet theorem reformulated in terms of new curvature measures.
Random walks on cell complexes link to Laplacians and Novikov-Shubin invariants.
problem Computing Novikov-Shubin invariants for complex cell structures.
method Construct random walks on cell complexes, relate to Laplacians, and use return probabilities.
result Novikov-Shubin invariants can be recovered from random walk return probabilities.
We express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the Euler-Lagrange equations of the associated energy functionals. Many solutions are given and discussed. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as …
Let M be a compact 3-manifold with a triangulation τ. We give an inequality relating the Euler characteristic of a surface F normally embedded in M with the number of normal quadrilaterals in F. This gives a relation between a topological invariant of the surface and a quantity derived from its combinatorial …
For a nonconstant holomorphic map between projective Riemann surfaces with conformal metrics, we consider invariant Schwarzian derivatives and projective Schwarzian derivatives of general virtual order. We show that these two quantities are related by the "Schwarzian derivative" of the metrics of the surfaces (at least…
Simple matrix formulas for Grassmannian curvatures.
problem Modeling Grassmannian for curvature calculations.
method Symmetric orthogonal matrices and standard matrix operations.
result Explicit, simple formulas for various curvatures.
There is an elegant relation found by Fabricius-Bjerre [Math. Scand 40 (1977) 20--24] among the double tangent lines, crossings, inflections points, and cusps of a singular curve in the plane. We give a new generalization to singular curves in RP^2. We note that the quantities in the formula are naturally dual to each …
Derives monotonic quantities for p-harmonic functions on manifolds.
problem Understanding p-harmonic functions on manifolds with nonnegative scalar curvature. method Derives local and global monotonic quantities associated with p-harmonic functions. result Establishes inequalities relating mass, capacity, and Willmore functional.
Paper derives invariantised Euler-Lagrange equations for Herglotz problems.
problem Nonconservative Herglotz variational problems.
method Invariant calculus of variations and moving frames.
result Derivation of generalised Euler-Lagrange equations and conserved quantities.
We give a new characterisation of the unparametrised geodesics, or distinguished curves, for affine, pseudo-Riemannian, conformal, and projective geometry. This is a type of moving incidence relation. The characterisation is used to provide a very general theory and construction of quantities that are necessarily conse…
We relate the Gromov norm on homology classes to the harmonic norm on the dual cohomology and obtain double sided bounds in terms of the volume and other geometric quantities of the underlying manifold. Along the way, we provide comparisons to other related norms and quantities as well.
In this paper, we consider two different monotone quantities defined for the Ricci flow and show that their asymptotic limits coincide for any ancient solutions. One of the quantities we consider here is Perelman's reduced volume, while the other is the local quantity discovered by Ecker, Knopf, Ni and Topping. This es…
Study finds conserved quantities for two types of curves on conformal sphere.
problem Identifying conserved quantities for specific types of curves on a conformal sphere.
method Used parallel tractor and Lagrangian formalism to compute conserved quantities.
result Found relation between conserved quantities of two curve types.
This note generalizes the visual angle to convex sets in 3D space.
problem Analyzing geometric properties of convex sets in 3D space.
method Generalizing the visual angle to convex sets in Euclidean space and expressing geometric quantities in terms of integrals of functions related to the solid angle.
result Invariant quantities of the original convex set can be expressed by integrals of functions related to the solid angle.
The set of local gauge invariant quantities for linearized gravity on the Kerr spacetime presented by two of the authors (S.A, T.B.) in (arXiv:1803.05341) is shown to be complete. In particular, any gauge invariant quantity for linearized gravity on Kerr that is local and of finite order in derivatives can be expressed…
The notion of Nonlocal Mean Curvature (NMC) appears recently in the mathematics literature. It is an extrinsic geometric quantity that is invariant under global reparameterization of a surface and provide a natural extension of the classical mean curvature. We describe some properties of the NMC and the quasilinear dif…
In this paper, we study the theory of linearized gravity and prove the linear stability of Schwarzschild black holes as solutions of the vacuum Einstein equations. In particular, we prove that solutions to the linearized vacuum Einstein equations centered at a Schwarzschild metric, with suitably regular initial data, r…
The field of risk theory has traditionally focused on ruin-related quantities. In particular, the socalled Expected Discounted Penalty Function has been the object of a thorough study over the years. Although interesting in their own right, ruin related quantities do not seem to capture path-dependent properties of the…
This is the first of two papers where we address and partially confirm a conjecture of Deser and Schwimmer, originally postulated in high energy physics. The objects of study are scalar Riemannian quantities constructed out of the curvature and its covariant derivatives, whose integrals over compact manifolds are invar…
The paper develops a comprehensive theory of submanifolds in conformal geometries.
problem Understanding submanifolds in conformal geometries of arbitrary dimension.
method Using conformal tractor calculus, the paper provides a new framework for studying submanifolds.
result The theory includes a new notion of distinguished submanifolds and characterizes them in various dimensions.
We give a family of monotone quantities along smooth solutions to the inverse curvature flows in Euclidean spaces. We also derive a related geometric inequality for closed hypersurfaces with positive k-th mean curvature.
In their papers published in 1993 and 1994, by expressing certain physical quantity in two distinct ways, Bershadsky-Cecotti-Ooguri-Vafa discovered a remarkable equivalence between Ray-Singer analytic torsion and elliptic instanton numbers for Calabi-Yau threefolds. After their discovery, in a paper published in 2008, …
We study the connections between link invariants, the chromatic polynomial, geometric representations of models of statistical mechanics, and their common underlying algebraic structure. We establish a relation between several algebras and their associated combinatorial and topological quantities. In particular, we def…
This paper analyzes the quantitative relations between stock prices and quantities of tradable stock shares in Chinese stock markets at six time points by means of Exploratory Data Analysis (EDA) method. It is found the resulting formulae have the same structure but different parameters. This paper also uses these rela…
Study on quantum sl3 invariant for positive links.
problem Characterizing and understanding the quantum sl3 invariant of positive links. method Skein theory of sl3-webs, explicit formulae, diagrammatic quantities, obstructions. result Positive links are fibered if and only if the second coefficient of the polynomial is 1.
New insights on quantifying space deformation.
problem Finding best way to quantify space deformation.
method Analyzing relationships between intrinsic and extrinsic quantities.
result Chen's work provided deeper understanding of Riemannian inequalities.
Bilateral trade relationships in the international level between pairs of countries in the world give rise to the notion of the International Trade Network (ITN). This network has attracted the attention of network researchers as it serves as an excellent example of the weighted networks, the link weight being defined …
Develops Floer cohomology for 4-manifolds with involutions and links.
problem Floer cohomology for 4-manifolds with involutions and links.
method Floer cohomology framework for spin^c 4-manifolds with involution and oriented links.
result Proves Frøyshov-type inequalities relating 4-manifold topological quantities and homology cobordism invariants.
We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in Rn. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypers…
This survey article is about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure …
L-CNNs learn gauge invariant quantities on lattices.
problem Learning gauge invariant quantities on lattices.
method Novel convolutional layer preserving gauge equivariance and forming Wilson loops.
result L-CNNs can approximate any gauge covariant function on the lattice.
In the previous paper [MR2430243] we computed some geometric quantities such as curvature and flag curvature for a general left invariant Finsler metric on a two-step nilpotent group. In the present paper we give a more complete description of the Chern--Rund connection defined by a left invariant Randers metric on the…
Renormalized volume invariant for knots in 3-sphere computed.
problem Computing renormalized volume for knot embeddings in 3-sphere.
method Renormalizing volume associated to singular Yamabe metric.
result Renormalized volume is a global conformal invariant for knots in 3-sphere.
Machine learning predicts topological properties of Calabi-Yau manifolds.
problem Predicting topological quantities of Calabi-Yau manifolds.
method Machine learning approach using neural networks and symbolic regressors.
result High performance scores in predicting Sasakian Hodge numbers and Crowley-Nördstrom invariant.
The class of the Riemannian almost product manifolds with nonintegrable structure is considered. Some identities for curvature tensor as certain invariant tensors and quantities are obtained.
DAT-CGAN improves time series generation for better decision support.
problem Generating accurate time series data for decision support.
method DAT-CGAN uses multi-Wasserstein loss and overlapped block-sampling for improved sample efficiency.
result DAT-CGAN outperforms GAN-based baselines in generating data relevant to decision processes.
New geometric quantities help classify manifolds and relate to entropy.
problem Classifying Riemannian manifolds using geometric quantities.
method Introducing and analyzing asymptotic geometric quantities like p-capacity, eigenvalues, and Maz'ya constant.
result Geometric quantities coincide with entropy in specific conditions, characterizing manifolds.
We prove boundedness and polynomial decay statements for solutions to the spin ±1 Teukolsky-type equation projected to the ℓ=1 spherical harmonic on Reissner-Nordström spacetime. The equation is verified by a gauge-invariant quantity which we identify and which involves the electromagnetic and curvature tensor…
In this paper we study the relation between conserved quantities of nonholonomic systems and the hamiltonization problem employing the geometric methods of [1,3]. We illustrate the theory with classical examples describing the dynamics of solids of revolution rolling without sliding on a plane. In these cases, using th…
Derives new monotone quantities for p-harmonic functions on asymptotically flat 3-manifolds.
problem Estimating the mass of 3-manifolds with non-negative scalar curvature and minimal boundary.
method Derives monotone quantities for p-harmonic functions and applies them to derive a sharp mass-capacity estimate.
result Derives a sharp mass-capacity estimate relating the ADM mass of a 3-manifold to the p-capacity of its boundary.
A new geometric cocycle measures mass of hyperbolic manifolds.
problem Measuring mass of hyperbolic manifolds.
method Constructing a cocycle mapping ALH metrics to a tractor-valued form field.
result Invariant c(h) can be integrated over a sphere boundary. Consider a family of smooth immersions F(⋅,t):Mn→Rn+1 of closed hypersurfaces in Rn+1 moving by the mean curvature flow ∂t∂F(p,t)=−H(p,t)⋅ν(p,t), for t∈[0,T). We show that at the first singular time of the mean curvature flow, certain subcritic…
The paper introduces a method to incorporate expert opinion on observable quantities into statistical models.
problem Tackling the challenge of integrating expert knowledge on observable quantities into statistical models.
method The approach involves updating a prior belief using a loss function that reflects expert opinion on observable quantities.
result The method allows for a flexible specification of expert opinion and is straightforward to implement.
In this note we take some initial steps in the investigation of a fourth order analogue of the Yamabe problem in conformal geometry. The Paneitz constants and the Paneitz invariants considered are believed to be very helpful to understand the topology of the underlined manifolds. We calculate how those quantities chang…
SU(2) flat connection on 2D Riemann surface is shown to relate to the generalized twisted geometry in 3D space with cosmological constant. Various flat connection quantities on Riemann surface are mapped to the geometrical quantities in discrete 3D space. We propose that the moduli space of SU(2) flat connections on Ri…
In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild met- rics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively cor- respond to the closed and co-closed, or polar and…