The classical Schläfli formula, and its ``higher'' analogs given in [SS03], are relations between the variations of the volumes and ``curvatures'' of faces of different dimensions of a polyhedra (which can be Euclidean, spherical or hyperbolic) under a first-order deformation. We describe here analogs of those formulas…
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48 results for “Bonahon-Schläfli formula”
Most animals possess the ability to actuate a vast diversity of movements, ostensibly constrained only by morphology and physics. In practice, however, a frequent assumption in behavioral science is that most of an animal's activities can be described in terms of a small set of stereotyped motifs. Here we introduce a m…
New framework learns interaction rules from animal trajectories.
problem Challenges in extracting interaction rules from animal movement data.
method Augmented behavioral models with neural networks and theory-guided regularization.
result Improved performance over baselines and novel biological insights.
Double descent phenomenon explained in simple terms.
problem Understanding the surprising drop in test error in overparameterized models.
method Informal explanation using linear algebra and probability, visual intuition with polynomial regression, mathematical analysis with ordinary linear regression.
result Three factors create double descent: data undersampling, model size, and parameter count. Ablating any one of these factors prevents double descent.
Co-eye: A Multi-resolution Symbolic Representation to TimeSeries Diversified Ensemble Classificationcs.LG
Co-eye combines multiple symbolic representations to improve time series classification accuracy.
problem Challenges in time series classification due to domain diversity.
method Inspired by compound eyes, Co-eye uses multiple symbolic representations and hyper-parameterised lenses to classify time series data.
result Co-eye outperforms state-of-the-art techniques in accuracy and robustness across various domains.
Proves a general connected sum formula for families Seiberg-Witten invariants.
problem Limited connected sum formulae for families Seiberg-Witten theory.
method Develops a general connected sum formula incorporating previous results.
result Proves a new connected sum formula for Seiberg-Witten families.
Derives an integral formula for G2-structures.
problem Calculating properties of G2-structures.
method Applies an integral formula for G-structures to G2.
result Derives an integral formula relating curvatures and quadratic invariants.
Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
Paper derives trace formula for magnetic Laplacian at zero energy.
problem Trace formula for magnetic Laplacian at zero energy.
method Generalizes Gutzwiller trace formula, focuses on zero energy level.
result Derives trace formula at zero energy level.
The Gauss formula is extended to various Laplacians on submanifolds.
problem Deriving formulas for Laplacians on submanifolds.
method Extending the Gauss formula to different types of Laplacians.
result Formulas for various Laplacians on submanifolds.
Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin classmath.AT
The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based…
We prove two tropical gluing formulae for Gromov-Witten invariants of exploded manifolds, useful for calculating Gromov-Witten invariants of a symplectic manifold using a normal-crossing degeneration. The first formula generalizes the symplectic-sum formula for Gromov-Witten invariants. The second formula is stronger, …
Note on new cancellation formulas for manifolds.
problem Generalizing anomaly cancellation formulas to manifolds.
method Proving new (a, b) type cancellation formulas and using transgression.
result Obtained characteristic forms with modularity properties.
The main result of the present paper is a coincidence formula for foliated manifolds. To prove this we establish Kuenneth formula, Poincare duality and intersection product in the context of tangential de Rham cohomology and homology of tangential currents. We apply the formula to get a dynamical Lefschetz formula for …
Formula calculates volume of two-bridge knots.
problem Calculating the volume of two-bridge knots.
method Derived from Hopf formula and Fox derivatives.
result Closed formula for the volume of two-bridge knots.
Introduces a universal Bochner formula for scalar curvature.
problem None explicitly stated; focuses on a new formula.
method Introduces a universal Bochner formula.
result Contains special cases like stability inequality and Schrödinger-Lichnerowicz-type formula.
Formula connects surgeries to Seiberg-Witten invariants.
problem Understanding how surgeries affect Seiberg-Witten invariants.
method Proves surgery formulas for Seiberg-Witten invariants and families.
result Expresses new invariants in terms of original ones.
It has been shown that the Alvarez-Gaum-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula i…
Proves a formula for a special invariant of 4-manifolds.
problem Calculating the Bauer-Furuta invariant for connected sums of 4-manifolds.
method Uses a finite dimensional approximation of the Seiberg-Witten monopole map to derive a formula for the families Bauer-Furuta invariant of a fibrewise connected sum.
result Derives a general connected sum formula for the families Bauer-Furuta invariant.
Proves a special case of the Gaussian kinematic formula using large sphere limits.
problem Proving a special case of the Gaussian kinematic formula.
method Viewing the GKF as the limit of spherical kinematic formulas for large dimension spheres.
result Proves a special case of the Gaussian kinematic formula.
New Crofton formulae derived from existing ones.
problem Generalizing Crofton formulae for products.
method Calculations in the ring of normal densities.
result Generalizations of Crofton formulae in terms of mixed Riemannian volume.
Formulae for non-symmetric connections derived from covariant derivatives.
problem Deriving commutation formulae for non-symmetric affine connections.
method Covariant derivatives of tensors with respect to symmetric and non-symmetric affine connections.
result Formulae for non-symmetric connections derived from covariant derivatives.
Formula connects curvature to volume in special geometric spaces.
problem Deriving formulas for curvature in specific geometric spaces.
method Used strong locality of Laplacian and eigenfunction approximation.
result Proved integral type Gauss-Green formula linking curvature to volume.
Application of a unified Kenmotsu-type formula for surfaces in Euclidean or Lorentzian three-spacemath.DG
Kenmotsu's formula describes surfaces in Euclidean 3-space by their mean curvature functions and Gauss maps. In Lorentzian 3-space, Akutagawa-Nishikawa's formula and Magid's formula are Kenmotsu-type formulas for spacelike surfaces and for timelike surfaces, respectively. We apply them to a few problems concerning rota…
The paper proves T-duality and Hori formulae for winding loop spaces.
problem Realizing T-duality and Hori formulae for loop spaces.
method Proving T-duality and Hori formulae for winding q-loop spaces.
result T-duality and Hori formulae for winding q-loop spaces are proven.
New formulas for measuring geometric properties of definable sets.
problem Measuring geometric properties of definable sets in a non-standard setting.
method Proved two kinematic formulas integrating on .
result Generalized Cauchy-Crofton and infinitesimal linear kinematic formulas.
Guillemin trace formula adapted for group actions.
problem Distributional trace for proper, cocompact group actions.
method Developing an equivariant version of the distributional trace.
result Equivariant Guillemin trace formula for group actions.
Unified entropy formula for real, complex, and quaternionic DLNs.
problem Deriving a formula for DLNs over different fields.
method Extending Menon and Yu's formula to complex and quaternionic DLNs.
result Unified entropy formula for DLNs over , , and .
Proves an Euler-type formula for Möbius strip partitions.
problem No specific problem stated; focuses on a mathematical formula.
method Analyzes partitions of the Möbius strip.
result Proves an Euler-type formula for Möbius strip partitions.
We prove a quasi-Poisson bracket formula for the space of representations of the fundamental groupoid of a surface with boundary, which generalizes Goldman's Poisson bracket formula. We also deduce a similar formula for quasi-Poisson cross-sections.
Paper derives matrix formulae and proves skein relations for non-orientable surfaces in quasi-cluster algebras.
problem Understanding quasi-cluster algebras on non-orientable surfaces.
method Developed matrix formulae and proved skein relations for quasi-cluster variables.
result Laurent expansion and skein relations for quasi-cluster variables on non-orientable surfaces.
New formulas for coassociative submanifolds' volume variation.
problem Understanding volume changes in coassociative submanifolds.
method Proved new variation formulae using data.
result Highlight the role of ambient torsion and Ricci curvature in volume changes.
New methods derive a generalized Frenkel trace formula for Lie groups.
problem Deriving a generalized Frenkel trace formula for Lie groups.
method Applying supersymmetric localization to quantum mechanical and gauged sigma models.
result Presented two complementary approaches for the derivation of the trace formula.
Alternative closed-form formula for spread call option prices under log-normal models.
problem Valuation of spread call options under log-normal models.
method Developed an alternative closed-form formula for spread call option prices.
result Our formula performs better for certain range of model parameters than existing closed-form formula.
Gauss-Bonnet type formulas for frontal bundles over surfaces with boundary and their applicationsmath.DG
The paper derives new Gauss-Bonnet formulas for frontal bundles over surfaces with boundary.
problem Deriving new formulas for coherent tangent bundles over surfaces with boundary.
method Defining frontal bundles and applying Gauss-Bonnet theorems to derive formulas.
result Four new Gauss-Bonnet type formulas for frontal bundles are derived.
In this paper, we obtain asymptotic formulas with error estimates for the implied volatility associated with a European call pricing function. We show that these formulas imply Lee's moment formulas for the implied volatility and the tail-wing formulas due to Benaim and Friz. In addition, we analyze Pareto-type tails o…
Paper provides seven Gauss diagram formulas for degree three long virtual knots.
problem Tackles the complete list of seven distinct Gauss diagram formulas for degree three long virtual knots.
method Gives seven Gauss diagram formulas for degree three long virtual knots and 23 for classical knots.
result Each Gauss diagram formula for degree three long virtual knots is represented as classical knots formulas, supporting Goussarov-Polyak-Viro conjecture.
Proves a formula in Heegaard Floer homology using combinatorial methods.
problem Proving Lipshitz's Maslov index formula in Heegaard Floer homology.
method Combinatorial proof via Heegaard diagrams.
result Validated Lipshitz's Maslov index formula in Heegaard Floer homology.
We give a formula of the colored Alexander invariant in terms of the homological representation of the braid groups which we call truncated Lawrence's representation. This formula generalizes the famous Burau representation formula of the Alexander polynomial.
We derive formulas for F measures' standard error and confidence intervals.
problem Estimating F measures' accuracy with confidence.
method Analytic formulas based on asymptotic normality.
result Valid formulas for sample size planning.
Derives Selberg trace formula on Riemann surfaces and generalizes to other spaces.
problem Deriving and generalizing the Selberg trace formula.
method Supersymmetric localization principle and path integral derivation.
result Derives Selberg trace formula on arbitrary compact Riemann surfaces and generic compact locally symmetric spaces.
This paper derives an explicit formula for Branson's Q-curvature in even-dimensional conformal geometry. The ingredients in the formula come from the Poincare metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses Q-curva…
Wave trace singularity formula for fibre bundles generalizes Poisson summation.
problem Wave trace singularity formula for fibre bundles.
method Proved a wave trace singularity formula for a family of generalised Laplacians defined by a Riemannian fibre bundle.
result Generalizes Poisson summation formulae for families.
We give an immersion formula, the Sym-Bobenko formula, for minimal surfaces in the 3-dimensional Heisenberg space. Such a formula can be used to give a generalized Weierstrass type representation and construct explicit examples of minimal surfaces.
Proves Juhl formulas for curved Ovsienko--Redou operators, confirming conjectures.
problem Formal self-adjointness of curved Ovsienko--Redou operators and their linear analogues.
method Proves Juhl type formulas for curved Ovsienko--Redou operators and their linear analogues.
result Confirms two conjectures of Case, Lin, and Yuan on formal self-adjointness.
New formula for spherical polygon area via prequantization.
problem Traditional area formula for spherical polygons requires measuring angles.
method Uses prequantization to create a new formula that doesn't require angle measurement.
result New formula applicable to a wider range of degenerate curves and polygons.
The paper develops formulas for hyperbolic simplices based on edge lengths.
problem Understanding the geometry of hyperbolic simplices using only edge lengths.
method Develops geometric formulas for hyperbolic simplices based on edge lengths.
result Distance and projection formulas in hyperbolic simplices.
We derive Verlinde's formula from the fixed point formula for loop groups proved in the companion paper "A fixed point formula for loop group actions", and extend it to compact, connected groups that are not necessarily simply-connected.