We demonstrate the use of several tools from Algebraic Combinatorics such as Young tableaux, symmetry operators, the Littlewood-Richardson rule and discrete Fourier transforms of symmetric groups in investigations of algebraic curvature tensors.
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48 results for “discrete algebraic rules”
Open problem: Establishing bounds for Cayley-table completion to discover discrete algorithmic axioms.
problem Discovering discrete algorithmic axioms missing in deep learning.
method Cayley-table completion as a testbed for algorithmic complexity minimization.
result Formal exact recovery bounds for Cayley-table completion.
We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as Markov cubature rules. The polynomial property allows us to study such rules using …
NeuRules learns interpretable rule lists from data without pre-discretization.
problem Interpretable rule lists for high-stakes decisions in machine learning.
method Neuro-symbolic continuous optimization with temperature annealing.
result NeuRules outperforms existing methods in learning interpretable rule lists.
In this work, we consider the hedging error due to discrete trading in models with jumps. Extending an approach developed by Fukasawa [In Stochastic Analysis with Financial Applications (2011) 331-346 Birkhäuser/Springer Basel AG] for continuous processes, we propose a framework enabling us to (asymptotically) optimize…
We analyze quantum Yang-Mills theory on using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather than positive definite. Specifically, we work with Lie-algebra valu…
The study classifies singularities in discrete improper affine spheres.
problem Classifying singularities in discrete improper affine spheres.
method Analysis of discrete improper affine spheres based on asymptotic nets, distinguishing singular edges and vertices.
result First step in classifying singularities of discrete nets.
Paper explores folding patterns of curved creases preserving their geometric properties.
problem Investigating rigid-ruling folding motions of curved crease-rule patterns.
method Deriving conditions for rigid-ruling foldability and analyzing combinations of creases.
result Constant fold-angle creases are only compatible with other constant fold-angle creases.
This work discovers algebraic structures from data using a differentiable measure.
problem Discovering discrete algebraic rules from data.
method Formalizes the problem through Cayley-table completion and uses HyperCube operator-valued tensor factorization.
result Derives an absolute lower bound for the differentiable measure of algebraic complexity, proving it is attained only for group structures.
New STDP rule for spiking neurons solves discrete action reinforcement learning tasks.
problem Applying standard STDP to discrete action reinforcement learning tasks.
method Feedback-modulated TD-STDP learning rule for spiking neuron networks.
result Feedback modulation improves credit assignment in reinforcement learning.
A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus is itself a ruled surface over a curve of genus . In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in case .
Defines CAMC discrete nets and their properties.
problem Understanding CAMC discrete nets and their properties.
method Defining CAMC discrete nets and proving properties.
result Properties of CAMC discrete nets are equivalent to properties of compatible interpolating quadrics.
Study curvature and torsion from cross-ratios in discrete curves.
problem Define curvature and torsion for discrete curves using cross-ratios.
method Use Möbius invariant point-insertion-rule to construct circles and express torsion using cross-ratio.
result Discrete curvature and torsion defined using cross-ratios converge to smooth curvature and torsion as sampling density increases.
Discrete structure rules for validating molecular structures are usually limited to fulfillment of the octet rule or similar simple deterministic heuristics. We propose a model, inspired by language modeling from natural language processing, with the ability to learn from a collection of undirected molecular graphs, en…
We investigate the performance of dynamic portfolios constructed using more than 21,000 technical trading rules on 12 categorical and country-specific markets over the 2004-2015 study period, on rolling forward structures of different lengths. We also introduce a discrete false discovery rate (DFRD+/-) method for contr…
Machine learning accelerates Lie algebra computations.
problem Computing tensor products and branching rules of Lie algebras.
method Machine learning for Lie algebra computations.
result Achieves significant speed-ups in Lie algebra computations.
Rewriting theory applied to diagrammatic algebras for categorification.
problem Finding bases in graded -foams.
method Algorithmic approach combining linear and higher rewriting, modulo rules capturing categorical properties.
result First proof of a basis theorem for graded -foams.
This paper constructs an algebra on a 3-torus with specific properties for fluid dynamics.
problem Constructing an algebraic structure on a 3-torus with specific properties.
method Combining combinatorial graded intersection algebra with Sullivan's and Lawrence-Sullivan-Ranade's subcomplexes.
result The construction of an algebra with specific properties on the 3-torus.
Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.
problem Characterizing dense subgroups of algebraic groups.
method Bi-Lipschitz rigidity theorem for Zariski dense discrete subgroups.
result No -smooth slim limit set for higher rank semisimple algebraic groups.
Any ruled surface in Euclidean 3-space is described as a curve of unit dual vectors in the algebra of dual quaternions (=the even Clifford algebra of type (0,3,1)). Combining this classical framework and Singularity Theory, we characterize local diffeomorphic types of singular ruled surfaces in terms of geometric invar…
Develops deep jump learning for continuous treatment OPE.
problem Estimating mean outcomes under new treatment rules using historical data from different rules.
method Adaptive deep discretization of continuous treatment space using deep learning and multi-scale change point detection.
result Validated method through theoretical results, simulations, and real application to Warfarin Dosing.
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables us to explore Poisson algebras in the realm of non-associative algebras. We stud…
Unifying Linear Algebramath.GT
Unified determinants via a single equation.
problem Defining determinants with all known properties.
method Proposing a single equation implying all known properties of determinants.
result Unified definition of determinants with all properties.
The paper analyzes discrete approximations to minimize curve length in Euclidean space.
problem Minimizing the length of curves between two sets in Euclidean space.
method Finite differences and numerical integration for discrete approximations.
result The squared length of the reconstructed curve converges to the squared minimal length with rate .
Neural networks with binary weights are computation-efficient and hardware-friendly, but their training is challenging because it involves a discrete optimization problem. Surprisingly, ignoring the discrete nature of the problem and using gradient-based methods, such as the Straight-Through Estimator, still works well…
Based on the classical Plücker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in . Algebraically, these are encoded in a discrete integrable system which appears in various guises in the theory of continuous and discrete integrable systems. Geometrically, the e…
A Clifford algebra model for M"obius geometry is presented. The notion of Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced, and the structure equations for adapted frames are derived. These equations are discretized and the geometry of the occuring discrete nets and sphere congruences is disc…
We show that the space of algebraic covariant derivative curvature tensors R' is generated by Young symmetrized tensor products W*U or U*W, where W and U are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2),(3)}, {(2),(2 1)} or {(1 1)…
Based on the Lie theoretical methods of algebraic Fourier transformation, we classify in the case of generic values of inducing parameters the scalar singular vectors corresponding to the diagonal branching rules for scalar generalized Verma modules in the case of orthogonal Lie algebra and its conformal parabolic suba…
New algebraic rules for 5D shapes based on 3D cocycles.
problem Creating rules for 5D shapes.
method Using simplicial 3-cocycles to parameterize heptagon relations.
result Parameterized heptagon relations for 5D shapes.
New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.
problem Solving systems of hydrodynamic-type equations.
method Introducing and analyzing quasi-rectifiable Lie algebras and vector fields.
result New methods for solving hydrodynamic-type equations.
CoLA automates efficient numerical linear algebra for complex matrix structures.
problem Efficiently solving large-scale linear algebra problems with complex matrix structures.
method Combining linear operator abstraction with compositional dispatch rules.
result Automatic and efficient numerical algorithms for various linear algebra operations.
New discrete cobordism category for nested manifolds and relations to algebraic structures.
problem Discrete cobordism category for nested manifolds.
method Stratified Morse theory, Cyl-objects, doubling construction, cylindrical bar construction.
result Relations between Cyl-objects and algebraic structures like Temperley-Lieb algebras.
Research Issues in Mining User Behavioral Rules for Context-Aware Intelligent Mobile Applicationscs.LG
Context-awareness in smart mobile applications is a growing area of study, because of it's intelligence in the applications. In order to build context-aware intelligent applications, mining contextual behavioral rules of individual smartphone users utilizing their phone log data is the key. However, to mine these rules…
Abstraction and realization are bilateral processes that are key in deriving intelligence and creativity. In many domains, the two processes are approached through rules: high-level principles that reveal invariances within similar yet diverse examples. Under a probabilistic setting for discrete input spaces, we focus …
Spatio-temporal Lie-Poisson discretization for incompressible magnetohydrodynamics on the spheremath.NA
New method preserves MHD equations on sphere without costly matrix exponentials.
problem Discretizing MHD equations on sphere for numerical simulations.
method Lie-Poisson discretization, geometric quantization, semi-direct product Lie algebras.
result Preserves Lie-Poisson structure and Casimir functions.
In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two Legendrian isotopy invariants: augmentation number via point-counting over a finite field, for the augmentation variety of the associated Chekanov-Eliashberg differential graded algebra, and ruling polynomial via…
We present the design and implementation of a custom discrete optimization technique for building rule lists over a categorical feature space. Our algorithm produces rule lists with optimal training performance, according to the regularized empirical risk, with a certificate of optimality. By leveraging algorithmic bou…
Para-associative Algebroidsmath.DG
New algebraic structure for vector bundles with special properties.
problem Developing new algebraic structures for vector bundles.
method Introducing para-associative algebroids and showing local triviality conditions.
result Existence of a differential connection is necessary and sufficient for local triviality.
Calibrating a trading rule using a historical simulation (also called backtest) contributes to backtest overfitting, which in turn leads to underperformance. In this paper we propose a procedure for determining the optimal trading rule (OTR) without running alternative model configurations through a backtest engine. We…
The study examines numerical aspects of Karhunen-Loève expansions for stochastic processes.
problem Constructing Karhunen-Loève expansions for second-order stochastic processes.
method Spectral decomposition of covariance operator via Fredholm integral equation, discretization, singular value decomposition of weight-scaled sample matrix.
result Consistent solutions for model-based and data-driven KLE construction, characterized by convergence of SVD-based eigenvalue estimates and KL coefficients distributions.
We investigate the relationship between market efficiency of rice futures transaction in Osaka and the Japanese government intervention in rice distributions by directly buying and selling rice during the interwar period, from the middle 1910s to 1939, considering the context of "discretion versus rules." We use a time…
The paper studies spaces of non-compact real algebraic curves and their uniformisation.
problem Understanding the spaces of non-compact real algebraic curves and their uniformisation.
method Construction of spaces of non-compact real algebraic curves and description of their connected components using Fuchsian groups.
result Any connected component of the spaces of non-compact real algebraic curves is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group.
We classify 4-dimensional austere submanifolds in Euclidean space ruled by 2-planes. The algebraic possibilities for second fundamental forms of an austere 4-fold M were classified by Bryant, falling into three types which we label A, B, and C. We show that if M is 2-ruled of Type A, then the ruling map from M into the…
Discrete Lagrange problems solved with Lie group constraints.
problem Solving discrete Lagrange problems with Lie group constraints.
method Proving critical sections are solutions of unconstrained variational problems, applying Noether theory and multisymplectic forms.
result Critical sections of discrete Lagrange problems are solutions of unconstrained variational problems.
We study satellites of Legendrian knots in R^3 and their relation to the Chekanov-Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in R^3 and augmentations of its DGA by showing that the DGA has finite-dimensional represe…
The Chekanov-Eliashberg differential graded algebra of a Legendrian knot L is a rich source of Legendrian knot invariants, as is the theory of generating families. The set P(L) of homology groups of augmentations of the Chekanov-Eliashberg algebra is an invariant, as is a count of objects from the theory of generating …
We propose and analyze numerical methods for the Heath-Jarrow-Morton (HJM) model. To construct the methods, we first discretize the infinite dimensional HJM equation in maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite dimensional system of stochastic dif…