Classifies metric symplectic Lie algebras with quadratic extensions.
problem Classifying metric symplectic Lie algebras.
method Standard model, quadratic cohomology sets, isomorphism classes.
result Complete list of metric symplectic Lie algebras in special cases.
We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T*-extension of a nilpotent algebra admitting an invertiblederivation and also as the double extension of…
The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a non-degenerate invariant symmetric bilinear form. We show that any metric Lie algebra without simple ideals has the structure of a so called balanced quadratic extension of a…
The paper defines and studies the first Pontryagin class for quadratic Lie 2-algebroids.
problem Defining and studying the first Pontryagin class for quadratic Lie 2-algebroids.
method Detailed study of transitive Lie 2-algebroids, introduction of quadratic Lie 2-algebroids, definition of first Pontryagin class, construction of quadratic Lie 2-algebroids.
result The first Pontryagin class is the obstruction class for the existence of a CLWX-extension and trivial for certain quadratic Lie 2-algebroids.
The paper studies quadratic Poisson structures on Lie algebras, finding a 10-parametric family.
problem Compatibility of quadratic Poisson structures with linear structures on Lie algebras.
method Developed general theory and studied families of functions in involution.
result Found a 10-parametric family of quadratic Poisson structures on $\gl(3)^*$.
In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable quadratic polynomials and prove that when they satisfy a-priori bounds, the topolog…
Ricci solitons as critical points of quadratic curvature functionals
problem Einstein metrics and Ricci solitons as critical points of quadratic Riemannian functionals
method Study of Ricci solitons as critical points of a special quadratic curvature functional
result Ricci solitons are non-Einstein critical points of these functionals
Study on pseudo-Hermitian quadratic nilpotent Lie algebras with methods and classifications.
problem Characterizing and classifying pseudo-Hermitian quadratic nilpotent Lie algebras.
method Construction methods and double extension by planes.
result Complete classification of nilpotent quadratic Lie algebras and pseudo-Hermitian metrics up to dimension 8.
New connection found between complex polynomials and surface homeomorphisms.
problem Investigating the existence of generalized pseudo-Anosov maps from quadratic polynomials.
method Developed a new connection between dynamics of quadratic polynomials and surface homeomorphisms, focusing on Hubbard trees.
result Identified conditions for constructing generalized pseudo-Anosov maps from quadratic polynomials.
Study on neural network dynamics in high dimensions with quadratic activation.
problem Understanding training dynamics in overparameterized neural networks.
method Derivation of gradient flow equations and analysis under l2-regularization.
result Characterization of estimator performance and spectral properties in the high-dimensional limit.
Developed a theory of local convexity for second order differential equations on Lie algebroids.
problem Analyzing convexity in differential equations on Lie algebroids.
method Theory development for local convexity of SODEs on Lie algebroids.
result Extensive discussion of homogeneous quadratic SODEs on Lie algebroids.
Bayes-optimal learning of a neural network with quadratic activations is achieved with GAMP-RIE.
problem Learning a neural network with quadratic activations from quadratic samples.
method Combining approximate message passing with rotationally invariant matrix denoising.
result Derives a closed-form expression for Bayes-optimal test error.
Classifies symplectic Lie algebras with degenerate center.
problem Understanding symplectic Lie algebras with degenerate center.
method Standard model, quadratic cohomology sets, classification scheme.
result Complete list of 6-dimensional nilpotent symplectic Lie algebras.
Classifies limits of groups of involutions in SL(2,F) over local fields.
problem Classifying limits of groups of involutions in SL(2,F) over local fields.
method Classifying involutions, proving polar decompositions, classifying limits.
result Classification of Chabauty limits of various groups of involutions.
A new QHR model extends HR model with a quadratic variance function.
problem Modeling volatility with greater flexibility and stationarity.
method Introducing a quadratic variance function to the HR model, maintaining Markovian property.
result Stationary distribution of the QHR model is Pearson type IV.
We generalize Conway's approach to integral binary quadratic forms on Q to study integral binary hermitian forms on quadratic imaginary extensions of Q. In Conway's case, an indefinite form that doesn't represent 0 determines a line ("river") in the spine T associated with SL(2,Z) in the hyperbolic plane. In our genera…
Proves curvature of conference graphs and finds local matchings.
problem Proving precise values of curvature in conference graphs.
method Combining parameter relations and combinatorial approach.
result Existence of local perfect matchings in broader classes of graphs.
Proposes a new framework for invariant quadratic P&L predictions in option books.
problem Inconsistent second-order P&L predictions across different factor parameterizations.
method Local, model-agnostic framework using covariant Hessian defined by an affine connection.
result Coordinate-invariant quadratic P&L predictions that match desk targets.
Paper connects MoE and self-attention, proposing active-attention.
problem Improving efficiency and performance of self-attention mechanisms.
method Established connection between MoE and self-attention, analyzed quadratic gating functions, proposed active-attention mechanism.
result Active-attention outperforms standard self-attention in various tasks.
The recent financial crisis has led to so-called multi-curve models for the term structure. Here we study a multi-curve extension of short rate models where, in addition to the short rate itself, we introduce short rate spreads. In particular, we consider a Gaussian factor model where the short rate and the spreads are…
Let M be a finite volume hyperbolic manifold, we show the equidistribution in M of the equidistant hypersurfaces to a finite volume totally geodesic submanifold C. We prove a precise asymptotic on the number of geodesic arcs of lengths at most t, that are perpendicular to C and to the boundary of a cuspidal n…
Group quantization applied to finance models.
problem Constructing functional spaces for financial models.
method Group Quantization formalism applied to dynamical symmetry groups.
result Functional spaces and operators for quadratic potentials in finance.
The main purpose of the paper is twofold: First, to extend a well known theorem of Ruh-Vilms in the Euclidean space to symmetric spaces and, secondly, to apply this result to extend Hoffman-Osserman-Schoen Theorem (HOS Theorem) to 3-dimensional symmetric spaces. Precisely, it is defined a Gauss map of a hypersurface M^…
Novikov initiated the study of the algebraic properties of quadratic forms over polynomial extensions by a far-reaching analogue of the Pontrjagin-Thom transversality construction of a Seifert surface of a knot and the infinite cyclic cover of the knot exterior. In this paper the analogy is applied to explain the relat…
Study derivative-free methods for linear policies in linear-quadratic systems.
problem Optimizing policies in linear-quadratic systems with limited derivative information.
method Derivative-free methods applied to linear policies over various noise and reward feedback settings.
result These methods converge to near-optimal policies with a polynomial number of zero-order evaluations.
We give the first algorithm for Matrix Completion whose running time and sample complexity is polynomial in the rank of the unknown target matrix, linear in the dimension of the matrix, and logarithmic in the condition number of the matrix. To the best of our knowledge, all previous algorithms either incurred a quadrat…
New metrics defined in Finsler geometry with specific properties.
problem Understanding the properties of Finsler metrics and their subclasses.
method Introducing the generalized Berwald projective Weyl metric and proving properties of the class of generalized Douglas metrics.
result All GDW metrics with vanishing Landsberg curvature are of R-quadratic type. The paper tackles denoising of function samples modulo 1.
problem Recover smooth estimates of a function's modulo 1 samples from noisy data.
method Formulates and solves a quadratically constrained quadratic program relaxation.
result Demonstrates robustness of the approach to noise.
We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami's theorem about projectively flat Riemanni…
Quadratic discriminant analysis (QDA) is a standard tool for classification due to its simplicity and flexibility. Because the number of its parameters scales quadratically with the number of the variables, QDA is not practical, however, when the dimensionality is relatively large. To address this, we propose a novel p…
We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Ru…
Extends curves to hemispheres in metric spaces, proving isoperimetric inequalities.
problem Extending curves to hemispheres in metric spaces.
method Proving curves can be extended to hemispheres with Lipschitz condition.
result Metric spaces satisfy quadratic isoperimetric inequalities.
A novel regression method using Kirszbraun extension with improved runtime and performance.
problem Regression between Hilbert spaces.
method Framework based on Kirszbraun's extension theorem, decomposed into training and prediction stages solved via MWU scheme.
result Empirical results show a significant improvement over standard solvers.
The paper proposes a method to select clusters, models, and algorithms based on quadratic discriminant scores.
problem Selecting the number of clusters, models, and algorithms in cluster analysis.
method Develops quadratic scores for cluster quality, uses bootstrap resampling, and compares partitions.
result The proposed method achieves better overall performance compared to other state-of-the-art methods.
Algorithm recovers multiple time series from aggregated data.
problem Recovering multiple nonnegative time series from a few temporal aggregates.
method Extends NMF algorithms to use linear measurements as observations, incorporating individual autocorrelation.
result Effective recovery of multiple time series from aggregated data.
Uniform bounds found for Sierpinski carpet hyperbolic components.
problem Bounding hyperbolic components of Sierpinski carpet type.
method Establishing uniform a priori bounds and analyzing quadratic-like restrictions.
result Sierpinski carpet hyperbolic components of disjoint type are bounded.
A new method automatically and dynamically sets learning rates in deep learning.
problem Determining the appropriate learning rate in deep learning tasks is challenging and often subjective.
method Local Quadratic Approximation (LQA) to automatically and dynamically set learning rates.
result The proposed method leads to nearly optimal learning rates in a computationally efficient way.
A surface in the 4-sphere is trivially embedded, if it bounds a 3-dimensional handle body in the 4-sphere. For a surface trivially embedded in the 4-sphere, a diffeomorphism over this surface is extensible if and only if this preserves the Rokhlin quadratic form of this embedded surface.
The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones,…
Develops MGQDA for multi-group classification with theoretical guarantees and practical applications.
problem Complex multi-group classification problems with nonlinear decision boundaries and group-specific covariance patterns.
method MGQDA, a method based on quadratic discriminant analysis that projects predictors onto a lower-dimensional subspace.
result MGQDA achieves competitive or improved predictive performance compared to existing methods.
Effective field theories with explicit Lorentz violation are intimately linked to Riemann-Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces. An example is presented that is constructed from a …
The paper studies quasifuchsian manifolds and their boundary foliations, providing formulas and extensions.
problem Understanding the boundary behavior of quasifuchsian manifolds and their foliations.
method Variation formula for renormalized volume, upper bound on extremal length, extensions of quadratic differential.
result Upper bound on extremal length of horizontal measured foliation and extensions of quadratic differential.
HAMD optimizes cubic portfolios without quadratization, achieving better results.
problem Optimizing higher-order portfolio models with reduced distortion.
method Hybrid pipeline combining continuous Hamiltonian search, cardinality-preserving projection, and iterated local search.
result HAMD achieves significantly lower native cubic objective values than classical heuristics.
Solves portfolio optimization with cardinality constraints using column generation.
problem Portfolio optimization with cardinality constraints.
method Column generation method applied to a subset of assets in a master convex quadratic problem, using dual information to propose new assets.
result Solves portfolio optimization problems efficiently with cardinality constraints.
Quadratic differentials on Riemann surfaces uniquely determine foliations.
problem Understanding the relationship between quadratic differentials and foliations on Riemann surfaces.
method Extending prior results to arbitrary Fuchsian groups, analyzing measured foliations and their Dirichlet integrals.
result A finite-area holomorphic quadratic differential uniquely determines a horizontal foliation on a Riemann surface.
The paper solves the existence problem of sphere packings in higher dimensions.
problem Existence of crystallographic sphere packings in certain higher dimensions.
method Geometric doubling procedure and computations with Lorentzian quadratic forms.
result Solves the existence problem of crystallographic sphere packings in higher dimensions.
We give sharp sectional curvature estimates for complete immersed cylindrically bounded m-submanifolds φ:M→N×Rℓ, n+ℓ≤2m−1 provided that either φ is proper with the second fundamental form with certain controlled growth or M has scalar curvature with strong quadratic decay. This l…
Transformer models predict financial time series movements accurately.
problem Applying transformer models to financial time series prediction.
method Transformer architecture applied to synthetic and real S&P500 data.
result Transformer models predict financial time series movements accurately.