Search results for "Polynomial"

A family of kernel anomaly change detectors

2014

This paper introduces the nonlinear extension of the anomaly change detection algorithms in [1] based on the theory of reproducing kernels. The presented methods generalize their linear counterparts, under both the Gaussian and elliptically-contoured assumptions, and produce both improved detection accuracies and reduced false alarm rates. We study the Gaussianity of the data in Hilbert spaces with kernel dependence estimates, provide low-rank kernel versions to cope with the high computational cost of the methods, and give prescriptions about the selection of the kernel functions and their parameters. We illustrate the performance of the introduced kernel methods in both pervasive and anom…

Investigation of buckling characteristics of cracked variable stiffness composite plates by an eXtended Ritz approach

2021

Abstract Variable Angle Tow (VAT) composite plates are characterized by in-plane variable stiffness properties, which opens to new concepts of stiffness tailoring and optimization to achieve higher structural performance for advanced lightweight structures where damage tolerance consideration are often mandatory. In this paper, a single-domain eXtended Ritz formulation is proposed to study the buckling behaviour of variable stiffness laminated cracked plates. The plate behaviour is described by the first order shear deformation theory whose generalized displacements, namely reference plane translations and rotations, are expressed via suitable admissible trial functions. These consist of a …

Semi-Supervised Remote Sensing Image Classification based on Clustering and the Mean Map Kernel

2008

This paper presents a semi-supervised classifier based on the combination of the expectation-maximization (EM) algorithm for Gaussian mixture models (GMM) and the mean map kernel. The proposed method uses the most reliable samples in terms of maximum likelihood to compute a kernel function that accurately reflects the similarity between clusters in the kernel space. The proposed method improves classification accuracy in situations where the available labeled information does not properly describe the classes in the test image.

The central polynomials of the infinite-dimensional unitary and nonunitary Grassmann algebras

2010

For a fixed field $k$, let $k_0\langle X\rangle$ and $k_1\langle X \rangle$ denote respectively the free nonunitary associative $k$-algebra and the free unitary associative $k$-algebra on the countable set $X=\{x_1, x_2, \ldots\}.$ A polynomial $f\in k_i\langle X\rangle$ is called a central polynomial for an associative algebra $A$ if every evaluation of $f$ on $A$ lies in the center of $A.$ The set of all central polynomials of $A$ is a $T$-space of $k_i\langle X\rangle,$ i.e, a subspace closed under all endomorphisms of $k_i\langle X\rangle.$ In this paper the authors describe the T-space of central polynomials for both the unitary and the nonunitary infinite-dimensional Grassmann algebra…

Graded central polynomials for the matrix algebra of order two

2009

Let K be an infinite integral domain and $A=M_2(K)$ the algebra of $2\times 2$ matrices over $K$. The authors consider the natural $\mathbb{Z}_2$-grading of $A$ obtained by requiring that the diagonal matrices and the off-diagonal matrices are of homogeneous degree $0$ and $1$, respectively. When $K$ is a field, a basis of the graded identities of $A$ was described in [O. M. Di Vincenzo, On the graded identities of $M_{1,1}(E).$ Israel J. Math. 80 (1992), no. 3, 323-–335] in case $\mbox{char}\, K = 0$ and in [P. E. Koshlukov and S. S. de Azevedo, Graded identities for T-prime algebras over fields of positive characteristic. Israel J. Math. 128 (2002), 157-–176] when $K$ is infinite and $\mb…

On the Complexity of the Bernstein Combinatorial Problem

2012

International audience

Exploratory remarks and discussion on a potential program for interlock even more the mathematics and physics

2021

These remarks are endowed with exploratory argumentation for disrupt further discussion and in favor of the in-depth consolidation of a mathematical and physics identification based on 2 key concepts: 1) finite support and 2) a notion of infinite intrinsic to the usage of the complex numbers. General relativity shows up linked to a kind of a Gelfand representation as an approximation of an analog of a hidden Markov Model. This has deep connections with the Stone–Weierstrass theorem and these discussion are an invitation to the physics community to study the physics x mathematics identification in the case of a holding true multiverse hypothesis. Photon in this setup stands to the analog of …

Computational Experiments with the Roots of Fibonacci-like Polynomials as a Window to Mathematics Research

2022

Fibonacci-like polynomials, the roots of which are responsible for a cyclic behavior of orbits of a second-order two-parametric difference equation, are considered. Using Maple and Wolfram Alpha, the location of the largest and the smallest roots responsible for the cycles of period p among the roots responsible for the cycles of periods 2kp (period-doubling) and kp (period-multiplying) has been determined. These purely computational results of experimental mathematics, made possible by the use of modern digital tools, can be used as a motivation for confirmation through not-yet-developed methods of formal mathematics. peerReviewed

On the growth of the identities of algebras

2006

On several notions of complexity of polynomial progressions

2021

For a polynomial progression $$(x,\; x+P_1(y),\; \ldots,\; x+P_{t}(y)),$$ we define four notions of complexity: Host-Kra complexity, Weyl complexity, true complexity and algebraic complexity. The first two describe the smallest characteristic factor of the progression, the third one refers to the smallest-degree Gowers norm controlling the progression, and the fourth one concerns algebraic relations between terms of the progressions. We conjecture that these four notions are equivalent, which would give a purely algebraic criterion for determining the smallest Host-Kra factor or the smallest Gowers norm controlling a given progression. We prove this conjecture for all progressions whose ter…