Related papers: A Dynamical System-based Key Equation for Decoding…
Computation biology helps to understand all processes in organisms from interaction of molecules to complex functions of whole organs. Therefore, there is a need for mathematical methods and models that deliver logical explanations in a…
Dynamical systems are ubiquitous in science and engineering as models of phenomena that evolve over time. Although complex dynamical systems tend to have important modular structure, conventional modeling approaches suppress this structure.…
An existing dialogue between number theory and dynamical systems is advanced. A combinatorial device gives necessary and sufficient conditions for a sequence of non-negative integers to count the periodic points in a dynamical system. This…
We consider generalizations of Reed-Muller codes, toric codes, and codes from certain plane curves, such as those defined by norm and trace functions on finite fields. In each case we are interested in codes defined by evaluating arbitrary…
Identification of local structure in intensive data -- such as time series, images, and higher dimensional processes -- is an important problem in astronomy. Since the data are typically generated by an inhomogeneous Poisson process, an…
Encoding time-series with Linear Dynamical Systems (LDSs) leads to rich models with applications ranging from dynamical texture recognition to video segmentation to name a few. In this paper, we propose to represent LDSs with…
The dynamic cavity method provides the most efficient way to evaluate probabilities of dynamic trajectories in systems of stochastic units with unidirectional sparse interactions. It is closely related to sum-product algorithms widely used…
Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of…
Recently, a novel method based on coding partitions [1]-[4] has been used to derive power series expansions to previously intractable problems. In this method the coefficients at $k$ are determined by summing the contributions made by each…
In this article, we study systems of $n \geq 1$, not necessarily linear, discrete differential equations (DDEs) of order $k \geq 1$ with one catalytic variable. We provide a constructive and elementary proof of algebraicity of the solutions…
The vector space of the multi-indexed sequences over a field and the vector space of the sequences with finite support are dual to each other, with respect to a \textit{scalar product}, which we used to define \textit{orthogonals} in these…
In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…
We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of…
This work redefines the framework of chaos in dynamical systems by extending Devaney's definition to multiple mappings, emphasizing the pivotal role of nonlinearity. We propose a novel theorem demonstrating how nonlinear dynamics within a…
A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as `machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special…
In this article, an efficient sequential linear programming algorithm (SLP) for uncertainty analysis-based data-driven computational mechanics (UA-DDCM) is presented. By assuming that the uncertain constitutive relationship embedded behind…
The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering.…
We illustrate relationships between classical kernel-based dimensionality reduction techniques and eigendecompositions of empirical estimates of reproducing kernel Hilbert space (RKHS) operators associated with dynamical systems. In…
Discrete models have a long tradition in engineering, including finite state machines, Boolean networks, Petri nets, and agent-based models. Of particular importance is the question of how the model structure constrains its dynamics. This…
Polynomial dynamical systems describing interacting particles in the plane are studied. A method replacing integration of a polynomial multi--particle dynamical system by finding polynomial solutions of a partial differential equations is…