Lyapunov exponents and extensivity of strongly coupled chaotic maps in regular graphs

Acknowledgements J.G. acknowledges funds from the Agencia Nacional de Investigación e In nonvación (ANII), Uruguay, POS_NAC_2018_1_151185, and the Comisión Academica de Posgrado (CAP), Universidad de la República, Uruguay. Both authors acknowledge funds from the Comisión Sectorial de Investigación Ci475 entifíca (CSIC), Uruguay, group grant “CSIC2018 - FID13 - grupo ID 722”.Peer reviewedPublisher PD

TRACE MATRIKS BERBENTUK KHUSUS BERPANGKAT BILANGAN BULAT POSITIF

DISKA LILY PRATIWI (2022) : TRACE MATRIKS BERBENTUK KHUSUS BERPANGKAT BILANGAN BULAT POSITIF Tugas akhir ini bertujuan untuk menentukan bentuk umumtrace matriks 〖FLDcirc〗_r berbentuk khusus berpangkat dua, tiga dan empat.Sebelum menentukan bentuk umum trace matriks 〖FLDcirc〗_r berbentuk khusus tersebut maka akan ditentukan perpangkatan matriks FLDcircr berbentuk khusus dari (〖A_n)〗^2 sampai (〖A_n)〗^4. Selanjutnya pembuktiantrace matriks FLDcircr berbentuk khusus berpangkat dua, tiga dan empat dengan pembuktian langsung menggunakan definisi tracematriks. Diberikan juga contoh aplikasi dari trace matriks FLDcircr berbentuk khusus dari (〖A_n)〗^2 sampai(〖A_n)〗^4. Kata Kunci : matriks〖FLDcirc〗_r,matriks 〖FLDcirc〗_rberbentuk khusus, perpangkatan matriks, pembuktian langsung, trace matriks

On the norm of normal matrices (Research on preserver problems on Banach algebras and related topics)

In this article, we present some recent results related to the calculation of the induced p-norm of n × n circulant matrices A(n, a, b) with diagonal entries equal to a ∈ ℝ and off-diagonal entries equal to b ∈ ℝ. For circulant matrices with nonnegative entries, an explicit formula for the induced p-norm (1 ≤ p ≤ ∞) is given, whereas for A(n, −a, b), a > 0 the situation is no longer so simple and calls for a more subtle analysis. As a matter of fact, while the 2-norm of A(n, −a, b) is precisely determined, the exact value of the induced p-norm for 1 < p < ∞, p ≠ 2, still remains elusive. Nevertheless, we provide a lower bound as well as two different categories of upper bounds. As an indication of not being far from the exact values, our estimates coincide at both ends points (i.e., p = 1 and p = ∞) as well as at p = 2 with the precise values. As an abstract approach, we also introduce the ∗-algebra generated by a normal matrix A accompanied by an axis-oriented norm, and obtain some estimations of the norm of elements of the ∗-algebra. We then exhibit the connection between the new generalized estimates and the previously obtained estimates in the special case where A is a circulant matrix. Finally, using an optimization-oriented approach, we provide insight on the nature of the maximizing vectors for ∥Ax∥p/∥x∥p . This leads us to formulate a conjecture that, if proven valid, would make it possible to derive an exact formula for the induced p-norm of A(n, a, b) whenever a = 1−n/n and b = 1/n

Utilization and experimental evaluation of occlusion aware kernel correlation filter tracker using RGB-D

Unlike deep-learning which requires large training datasets, correlation filter-based trackers like Kernelized Correlation Filter (KCF) uses implicit properties of tracked images (circulant matrices) for training in real-time. Despite their practical application in tracking, a need for a better understanding of the fundamentals associated with KCF in terms of theoretically, mathematically, and experimentally exists. This thesis first details the workings prototype of the tracker and investigates its effectiveness in real-time applications and supporting visualizations. We further address some of the drawbacks of the tracker in cases of occlusions, scale changes, object rotation, out-of-view and model drift with our novel RGB-D Kernel Correlation tracker. We also study the use of particle filter to improve trackers\u27 accuracy. Our results are experimentally evaluated using a) standard dataset and b) real-time using Microsoft Kinect V2 sensor. We believe this work will set the basis for better understanding the effectiveness of kernel-based correlation filter trackers and to further define some of its possible advantages in tracking

From spline wavelet to sampling theory on circulant graphs and beyond– conceiving sparsity in graph signal processing

Graph Signal Processing (GSP), as the field concerned with the extension of classical signal processing concepts to the graph domain, is still at the beginning on the path toward providing a generalized theory of signal processing. As such, this thesis aspires to conceive the theory of sparse representations on graphs by traversing the cornerstones of wavelet and sampling theory on graphs. Beginning with the novel topic of graph spline wavelet theory, we introduce families of spline and e-spline wavelets, and associated filterbanks on circulant graphs, which lever- age an inherent vanishing moment property of circulant graph Laplacian matrices (and their parameterized generalizations), for the reproduction and annihilation of (exponen- tial) polynomial signals. Further, these families are shown to provide a stepping stone to generalized graph wavelet designs with adaptive (annihilation) properties. Circulant graphs, which serve as building blocks, facilitate intuitively equivalent signal processing concepts and operations, such that insights can be leveraged for and extended to more complex scenarios, including arbitrary undirected graphs, time-varying graphs, as well as associated signals with space- and time-variant properties, all the while retaining the focus on inducing sparse representations. Further, we shift from sparsity-inducing to sparsity-leveraging theory and present a novel sampling and graph coarsening framework for (wavelet-)sparse graph signals, inspired by Finite Rate of Innovation (FRI) theory and directly building upon (graph) spline wavelet theory. At its core, the introduced Graph-FRI-framework states that any K-sparse signal residing on the vertices of a circulant graph can be sampled and perfectly reconstructed from its dimensionality-reduced graph spectral representation of minimum size 2K, while the structure of an associated coarsened graph is simultaneously inferred. Extensions to arbitrary graphs can be enforced via suitable approximation schemes. Eventually, gained insights are unified in a graph-based image approximation framework which further leverages graph partitioning and re-labelling techniques for a maximally sparse graph wavelet representation.Open Acces

On polynomial equations and circulant matrices

This thesis is an exposition of the article Polynomial Equations and Circulant Matrices by Dan Kalman and James E. White. It uses circulant matrices as an alternative way for solving polynomial equations of degree less than or equal to four. The circulant matrix technique provides a unified treatment for the problem solving these equations