We prove that under certain linear reciprocal transformation, an evolutionary PDE of hydrodynamic type that admits a bihamiltonian structure is transformed to a system of the same type which is still bihamiltonian.
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48 results for “reciprocal transformations”
Legendre transformations link related integrable hierarchies.
problem Understanding relationships between integrable hierarchies.
method Legendre-type transformations of generalized Frobenius manifolds.
result Linear reciprocal transformations link related hierarchies.
Local classification of quasilinear systems with linearizable characteristic webs.
problem Classifying quasilinear systems with linearizable characteristic webs.
method Reciprocal transformations to uncoupled Hopf equations.
result Every quasilinear system with n>3 components can be transformed to n uncoupled Hopf equations.
We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation . The transformation is related to the special case of the famous Bäcklund transformation of the sine-Gordon equation with the Bäcklund parameter . It is also a nonlocal symmetry…
Transformers use ReLUs to approximate softmax efficiently.
problem Analyzing resource usage in softmax transformer models.
method Translating ReLU approximation results to softmax attention mechanisms.
result Economic resource bounds for softmax attention mechanisms.
In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian evolutionary PDEs. We prove that our class of infinite dimensional Jacobi structu…
For two solutions of the WDVV equations that are related by the inversion symmetry, we show that the associated principal hierarchies of integrable systems are related by a reciprocal transformation, and the tau functions of the hierarchies are related by a Legendre type transformation. We also consider relationships b…
For two solutions of the WDVV equations that are related by two types of symmetries of the equations given by Dubrovin, we show that the associated principal hierarchies of integrable systems are related by certain reciprocal transformation, and the tau functions of the hierarchies are either identical or related by a …
Classifies reciprocal elements in Hecke groups, generalizing Sarnak's work.
problem Classifying reciprocal elements in Hecke groups.
method Classifying and parametrizing reciprocal classes in Hecke groups for .
result Generalizes Sarnak's result on reciprocal elements in the modular group.
This paper introduces belief propagation for Gaussian reciprocal processes and links it to stability theory.
problem Efficient inference algorithms for reciprocal processes, especially those with a single loop network.
method Introduces belief propagation for Gaussian reciprocal processes and links it to stability theory.
result Established a link between convergence analysis of belief propagation for Gaussian reciprocal processes and stability theory for differentially positive systems.
Estimates growth of reciprocal classes in Hecke groups.
problem Estimating the growth of reciprocal conjugacy classes in Hecke groups.
method Using free product structure and word lengths of reciprocal elements, with tools from basic probability theory.
result Estimates the asymptotic growth of reciprocal conjugacy classes in Hecke groups.
Modeling and Estimation of Discrete-Time Reciprocal Processes via Probabilistic Graphical Modelsstat.ML
Paper models reciprocal processes using probabilistic graphical models.
problem Modeling acausal generalizations of Markov processes.
method Probabilistic graphical models for reciprocal processes, message passing algorithms for smoothing.
result Provides a principled solution to the smoothing problem.
The paper models social networks with varying levels of reciprocity.
problem Understanding diverse reciprocal behavior in social networks.
method Developed a preferential attachment model with heterogeneous reciprocity.
result Captures the heavy-tailed nature of empirical degree distributions and identifies multiple user groups.
The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.
problem Counting reciprocal hyperbolic elements in Hecke groups.
method Analyzes conjugacy classes of hyperbolic elements associated with reciprocal geodesics.
result Determines the asymptotic growth rate and limiting constant of primitive conjugacy classes of reciprocal hyperbolic elements.
Reformulates Lehmer and Schinzel-Zassenhaus questions on polynomial measures.
problem Understanding the difference between reciprocal and skew-reciprocal polynomials.
method Comparing Mahler measures and houses of polynomials.
result The complexity of orientation-preserving and orientation-reversing mapping classes.
Formulates Hilbert reciprocity law on 3-manifolds.
problem Developing arithmetic topology on 3-manifolds.
method Formulated an analogue of Hilbert reciprocity law using intersection forms and Kummer extensions.
result Cyclic covers of links are analogues of Kummer extensions.
Minimal spectral radii found for specific matrix types.
problem Finding smallest spectral radii for certain matrix classes.
method Analyzing skew-reciprocal integer matrices of fixed even dimensions.
result Most classes of matrices have smaller spectral radii than their reciprocal counterparts.
Study on geodesics and dihedral groups in lattices.
problem Growth and distribution of conjugacy classes of dihedral subgroups.
method Generalizing earlier work on reciprocal geodesics, proving equidistribution.
result Reciprocal geodesics are equidistributed in the unit tangent bundle.
Reciprocity in Interbank Marketsq-fin.CP
The paper models reciprocity in interbank markets using a statistical null model.
problem Understanding the importance of individual banks in financial networks.
method Developed an exponential random graph model to account for reciprocal links on both topological and weighted levels.
result Weighted reciprocity in interbank markets is more significant than network size and volume before the financial crisis.
Reciprocal Learningstat.ML
Reciprocal learning unifies various machine learning algorithms.
problem Understanding the convergence of machine learning algorithms.
method Introducing reciprocal learning as a generalization of various algorithms using decision theory.
result Reciprocal learning algorithms converge at linear rates to an approximately optimal model under certain conditions.
New method reconstructs interbank networks enforcing reciprocity to improve stability and risk prediction.
problem Lack of public interbank network data and difficulty in replicating cycles.
method Proposes a new network reconstruction method enforcing sparsity and link reciprocity from aggregate data.
result Adding reciprocity improves prediction of network properties, including largest real eigenvalue and eccentricity of eigenvalues.
Study geodesics entering a fixed cusp neighborhood multiple times.
problem Understanding geodesics entering a specific cusp neighborhood multiple times.
method Investigate reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
result Characterized the class of reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
Growth rates of geodesics on modular orbifolds are studied.
problem Understanding growth rates of geodesics on modular orbifolds.
method Exhaustion of modular orbifold by compact subsurfaces, analysis of low lying geodesics and reciprocal geodesics.
result Growth rates of low lying geodesics and reciprocal geodesics converge to the full set's growth rate.
The paper proves generalization bounds and stopping rules for self-selected data in reciprocal learning.
problem Generalization of learning algorithms using self-selected data.
method Proves universal generalization bounds using covering numbers and Wasserstein ambiguity sets.
result Provides stopping rules for reciprocal learning algorithms to ensure out-of-sample performance.
The abstract discusses conjectures about Chern-Simons invariants of 3-manifolds.
problem Conjectures about the reciprocity of Chern-Simons invariants of 3-manifolds.
method Supporting evidence through Galois descent of a -group.
result The conjectures hold under the condition of Galois descent of a -group.
Given a knot and an SL(n,C) representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given of a knot group and SL(3,Z) representation that is not conjugate to its dual for …
Reciprocity laws for line bundles on circle fibrations over complex manifolds.
problem Analytic reciprocity laws for complex line bundles on fibrations in oriented circles.
method Study of Gysin maps and first Chern classes in cohomology.
result Sum of Gysin maps of Chern classes equals zero in under specific conditions.
We show that the characteristic series for the greedy normal form of a Coxeter group is always a rational series, and prove a reciprocity formula for this series when the group is right-angled and the nerve is Eulerian. As corollaries we obtain many of the known rationality and reciprocity results for the growth series…
Study on 3D foliated dynamical systems with Hilbert reciprocity law.
problem Understanding and classifying 3D foliated dynamical systems.
method Decomposition theorem, concrete examples construction, integration theory for smooth Deligne cohomology.
result Introduction of geometric Hilbert reciprocity law for 3D FDS.
On Reciprocitymath.AC
We prove a reciprocity formula between Gauss sums that is used in the computation of certain quantum invariants of 3-manifolds. Our proof uses the discriminant construction applied to the tensor product of lattices.
This paper gives a new definition of the Contou-Carrere symbol in terms of an exponential of a Chen iterated integral and proves the corresponding reciprocity law.
Efficient algorithm uncovers mutual interests in reciprocal recommendation.
problem Learning to recommend users based on mutual interests in a sequential learning framework.
method Formulated assumptions and designed a computationally efficient algorithm.
result Algorithm uncovers mutual likes at a pace comparable to a clearvoyant algorithm.
Modelling sparsity, heterogeneity, reciprocity and community structure in temporal interaction datastat.ML
Novel network models capture social interaction features like reciprocity and community.
problem Modeling temporal social interaction data with reciprocity and community structure.
method Self-exciting Hawkes point processes with conditional intensity function.
result Proposed model outperforms competing approaches for link prediction.
The paper introduces admissible hierarchical clustering methods for asymmetric networks.
problem Characterizing and implementing hierarchical clustering methods for asymmetric networks.
method The paper characterizes admissible hierarchical clustering methods and proposes algorithms for their implementation.
result The paper describes three families of intermediate methods for admissible hierarchical clustering of asymmetric networks.
The study examines the growth of reciprocal classes in Hecke groups, proving an asymptotic formula.
problem Analyzing the growth of reciprocal classes in Hecke groups.
method Utilizes the free product structure of Hecke groups, combinatorial counting, and recurrence relations.
result Proves an asymptotic formula for the number of reciprocal classes in Hecke groups.
Study bihamiltonian structures and Frobenius manifolds for specific Toda hierarchies.
problem Local bihamiltonian structures and Frobenius manifolds for asymmetric rational reductions of 2D-Toda hierarchy.
method Construct three-dimensional generalized Frobenius manifold, relate to other hierarchies via transformations.
result Explicit relation between RR2T and bi-graded Toda and constrained KP hierarchies.
We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade network and show that a strong embedding in …
Paper tackles division difficulty, proposing new methods to improve accuracy.
problem Division is the most challenging arithmetic operation for both humans and computers.
method Proposes two novel approaches: Neural Reciprocal Unit (NRU) and Neural Multiplicative Reciprocal Unit (NMRU), and improves an existing division module.
result Improves division accuracy from 70.2% to 91.6%.
New model accounts for scale variation and noise in pairwise comparisons.
problem Nonreciprocal pairwise comparisons in decision analysis.
method Additive model with structured matrix and random perturbation.
result Explicit estimators and probability assessments of admissible ranking regions.
Elephants or Goldfish? An Empirical Analysis of Carrier Reciprocity in Dynamic Freight Marketsq-fin.GN
Empirical study shows carriers ignore past shippers' behavior, focusing only on current actions.
problem Opportunistic behavior by shippers and carriers in dynamic freight markets.
method Empirical analysis of carrier reciprocity in US truckload transportation sector.
result Carriers do not remember shippers' past behaviors but respond to current actions.
Study how past eon's matter affects present eon in Penrose's cyclic cosmology.
problem Determining present eon's matter content from past eon's matter.
method Use Penrose's reciprocity hypothesis to link past and present eons' matter.
result Perfect fluid matter content of past eon influences present eon's matter content.
This article investigates local properties of the further generalized Weierstrass relations for a spin manifold immersed in a higher dimensional spin manifold from viewpoint of study of submanifold quantum mechanics. We show that kernel of a certain Dirac operator defined over , which we call submanifold Dir…
Stochastic Geometric Mechanics in Nonequilibrium Thermodynamics: Schrödinger meets Onsagercond-mat.stat-mech
Schrödinger and Onsager's ideas linked in nonequilibrium thermodynamics.
problem Linking Schrödinger's variational problem with Onsager's nonequilibrium statistical mechanics.
method Analyzing the historical context and comparing the two approaches.
result Schrödinger's ideas have not yet been fully integrated into the classical context of Onsager's work.
New method learns operators with geometric singularities from few samples.
problem Learning operators with geometric singularities from limited data.
method Double fibration transforms and cross-attention architectures.
result Operators can be learned superalgebraically from few samples.
We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a funct…
The paper develops methods for clustering directed dissimilarity networks.
problem Clustering directed dissimilarity networks.
method Admissible methods based on axioms of value and transformation.
result Unique admissible clustering method exists when modifying the axiom of value.
This paper argues that the fundamental principle of contemporary financial economics is balanced reciprocity, not the principle of utility maximisation that is important in economics more generally. The argument is developed by analysing the mathematical Fundamental Theory of Asset Pricing with reference to the emergen…
Bayesian rLASSO improves model selection and prediction.
problem Improving model selection and prediction in linear regression.
method A Bayesian formulation of the rLASSO problem using inverse Laplace priors and scale mixture distributions.
result Bayesian rLASSO outperforms classical rLASSO in estimation, prediction, and variable selection.