Search results for "Polynomial"
showing 10 items of 566 documents
A cubic defining algebra for the Links-Gould polynomial
2012
We define a finite-dimensional cubic quotient of the group algebra of the braid group, endowed with a (essentially unique) Markov trace which affords the Links-Grould invariant of knots and links. We investigate several of its properties, and state several conjectures about its structure.
Event-Based Trajectory Prediction Using Spiking Neural Networks
2021
International audience; In recent years, event-based sensors have been combined with spiking neural networks (SNNs) to create a new generation of bio-inspired artificial vision systems. These systems can process spatio-temporal data in real time, and are highly energy efficient. In this study, we used a new hybrid event-based camera in conjunction with a multi-layer spiking neural network trained with a spike-timing-dependent plasticity learning rule. We showed that neurons learn from repeated and correlated spatio-temporal patterns in an unsupervised way and become selective to motion features, such as direction and speed. This motion selectivity can then be used to predict ball trajectory…
New applications of graded Lie algebras to Lie algebras, generalized Lie algebras and cohomology
2007
We give new applications of graded Lie algebras to: identities of standard polynomials, deformation theory of quadratic Lie algebras, cyclic cohomology of quadratic Lie algebras, $2k$-Lie algebras, generalized Poisson brackets and so on.
Quantizations from reproducing kernel spaces
2012
Abstract The purpose of this work is to explore the existence and properties of reproducing kernel Hilbert subspaces of L 2 ( C , d 2 z / π ) based on subsets of complex Hermite polynomials. The resulting coherent states (CS) form a family depending on a nonnegative parameter s . We examine some interesting issues, mainly related to CS quantization, like the existence of the usual harmonic oscillator spectrum despite the absence of canonical commutation rules. The question of mathematical and physical equivalences between the s -dependent quantizations is also considered.
18 parameter deformations of the Peregrine breather of order 10 solutions of the NLS equation
2015
We present here new solutions of the focusing one-dimensional nonlinear Schrödinger (NLS) equation which appear as deformations of the Peregrine breather of order 10 with 18 real parameters. With this method, we obtain new families of quasi-rational solutions of the NLS equation, and we obtain explicit quotients of polynomial of degree 110 in x and t by a product of an exponential depending on t. We construct new patterns of different types of rogue waves and recover the triangular configurations as well as rings and concentric rings as found for the lower-orders.
Test methodologies for the unitary root
1998
This paper provides a critical survey on unit root testing procedures frequently used in empirical literature. The importance of the hypothesis of linearity of the deterministic component is stressed because its misspecification may result in the false conclusion that the time series has a unit root. Testing strategies elaborated by Dickey, Bell and Miller (1986), Perron (1988) and Dolado, Jenkinson and Sosvilla-Rivero (1990) are compared and their limit is evaluated. An empirical testing strategy is then proposed and applied to the analysis of the nonstationarity exhibited by real GDP in France. We show that it is possible to find some flexible specifications that enable us to reject the u…
A branch-and-cut algorithm for the Team Orienteering Problem
2017
The Team Orienteering Problem aims at maximizing the total amount of profit collected by a fleet of vehicles while not exceeding a predefined travel time limit on each vehicle. In the last years, several exact methods based on different mathematical formulations were proposed. In this paper, we present a new two-index formulation with a polynomial number of variables and constraints. This compact formulation, reinforced by connectivity constraints, was solved by means of a branch-and-cut algorithm. The total number of instances solved to optimality is 327 of 387 benchmark instances, 26 more than any previous method. Moreover, 24 not previously solved instances were closed to optimality.
Nonlinear data description with Principal Polynomial Analysis
2012
Principal Component Analysis (PCA) has been widely used for manifold description and dimensionality reduction. Performance of PCA is however hampered when data exhibits nonlinear feature relations. In this work, we propose a new framework for manifold learning based on the use of a sequence of Principal Polynomials that capture the eventually nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) is shown to generalize PCA. Unlike recently proposed nonlinear methods (e.g. spectral/kernel methods and projection pursuit techniques, neural networks), PPA features are easily interpretable and the method leads to a fully invertible transform, which is a desirable property…
Fuzzy sigmoid kernel for support vector classifiers
2004
This Letter proposes the use of the fuzzy sigmoid function presented in (IEEE Trans. Neural Networks 14(6) (2003) 1576) as non-positive semi-definite kernel in the support vector machines framework. The fuzzy sigmoid kernel allows lower computational cost, and higher rate of positive eigenvalues of the kernel matrix, which alleviates current limitations of the sigmoid kernel.
Semisupervised Kernel Feature Extraction for Remote Sensing Image Analysis
2014
This paper presents a novel semisupervised kernel partial least squares (KPLS) algorithm for nonlinear feature extraction to tackle both land-cover classification and biophysical parameter retrieval problems. The proposed method finds projections of the original input data that align with the target variable (labels) and incorporates the wealth of unlabeled information to deal with low-sized or underrepresented data sets. The method relies on combining two kernel functions: the standard radial-basis-function kernel based on labeled information and a generative, i.e., probabilistic, kernel directly learned by clustering the data many times and at different scales across the data manifold. Th…