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Infinitesimal contraction analysis provides exponential convergence rates between arbitrary pairs of trajectories of a system by studying the system's linearization. An essentially equivalent viewpoint arises through stability analysis of a…
Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm…
An invertible dynamical system with some hyperbolic structure is considered. Upper estimates for the correlations of continuous observables is given in terms of modulus of continuity. The result is applied to certain H\'enon maps and…
We consider here a generalization of a well known discrete dynamical system produced by the bisection of reflection angles that are constructed recursively between two lines in the Euclidean plane. It is shown that similar properties of…
A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the…
An extension of the CCS-method [Chem. Phys. 2004, 304, p. 103-120] for simulating non-adiabatic dynamics with quantum effects of the nuclei is put forward. The time-dependent Schr\"{o}dinger equation for the motion of the nuclei is solved…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
This paper defines and develops cycle indices for the finite classical groups. These tools are then applied to study properties of a random matrix chosen uniformly from one of these groups. Properties studied by this technique will include…
This work examines the problem of topology inference over discrete-time nonlinear stochastic networked dynamical systems. The goal is to recover the underlying digraph linking the network agents, from observations of their state-evolution.…
Let X --> B be a proper submersion with a Riemannian structure. Given a differential K-theory class on X, we define its analytic and topological indices as differential K-theory classes on B. We prove that the two indices are the same.
This is a chapter (planned to appear in Wiley's upcoming Encyclopedia of Operations Research and Management Science) describing parts of the theory of convex polyhedra that are particularly important for optimization. The topics include…
Many "good" topologies for interconnection networks are based on line digraphs of regular digraphs. These digraphs support unitary matrices. We propose the property "being the digraph of a unitary matrix" as additional criterion for the…
New criteria for which Cayley graphs of cyclic groups of any order can be completely determined--up to isomorphism--by the eigenvalues of their adjacency matrices is presented. Secondly, a new construction for pairs of nonisomorphic Cayley…
We demonstrate when and how an entire left-infinite orbit of an underlying dynamical system or observations from such left-infinite orbits can be uniquely represented by a pair of elements in a different space, a phenomenon which we call…
Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral…
A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played…
The Cayley-Hamilton theorem is used to implement an iterative process for the efficient numerical computation of matrix power series and their differentials. In addition to straight-forward applications in lattice gauge theory simulations…
Despite the importance of non-equilibrium statistical mechanics in modern physics and related fields, the topic is often omitted from undergraduate and core-graduate curricula. Key aspects of non-equilibrium physics, however, can be…
We develop the relative Morse index theory for linear self-adjoint operator equation without compactness assumption and give the relationship between the index defined in [44] and [45]. Then we generalize the method of saddle point…
We give a new formula for the rotation number (or Whitney index) of a smooth closed plane curve. This formula is obtained from the winding numbers associated with the regions and the crossing points of the curve. One difference with the…