Related papers: An Extended Small-Gain Theorem
The duality between controllability and observability enables methods developed for full-state control to be applied to full-state estimation, and vice versa. In applications in which control or estimation of all state variables is…
It is widely believed that engineering a model to be invariant/equivariant improves generalisation. Despite the growing popularity of this approach, a precise characterisation of the generalisation benefit is lacking. By considering the…
The results of training a neural network are heavily dependent on the architecture chosen; and even a modification of only its size, however small, typically involves restarting the training process. In contrast to this, we begin training…
In this article, two types of methods from different perspectives based on spectral normalization are described for ensuring the stability of the system controlled by a neural network. The first one is that the L2 gain of the feedback…
We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of…
The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given…
In this paper, we introduce the method of adding additional factors and a parameter to multiple zeta values and prove some generalizations of the duality theorem and several relations among multiple zeta values. In particular, we are able…
We present an extension of Martin-L\"of Type Theory that contains a tiny object; a type for which there is a right adjoint to the formation of function types as well as the expected left adjoint. We demonstrate the practicality of this type…
We give improved bounds for the equidistribution of (multiparameter) nilsequences subject to any degree filtration. The bounds we obtain are single exponential in dimension, improving on double exponential bounds of Green and Tao. To obtain…
One of the two basic theorems in [5] on the existence of solutions of PDEs is improved with the use of a group analysis type argument.
This note provides a simple result showing, under suitable technical assumptions, that if a system S adapts to a class of external signals U, then S must necessarily contain a subsystem which is capable of generating all the signals in U.…
Centrality metrics are a popular tool in Network Science to identify important nodes within a graph. We introduce the Potential Gain as a centrality measure that unifies many walk-based centrality metrics in graphs and captures the notion…
Inspired and underpinned by the idea of integral feedback, a distributed constant gain algorithm is proposed for multi-agent networks to solve convex optimization problems with local linear constraints. Assuming agent interactions are…
This paper develops new limit theory for data that are generated by networks or more generally display cross-sectional dependence structures that are governed by observable and unobservable characteristics. Strategic network formation…
Given an additive network of input-output systems where each node of the network is modeled by a locally convergent Chen-Fliess series, two basic properties of the network are established. First, it is shown that every input-output map…
We introduce a minimal extended evolving model for small-world networks which is controlled by a parameter. In this model the network growth is determined by the attachment of new nodes to already existing nodes that are geographically…
Green, Tao and Ziegler prove ``Dense Model Theorems'' of the following form: if R is a (possibly very sparse) pseudorandom subset of set X, and D is a dense subset of R, then D may be modeled by a set M whose density inside X is…
The study proposes new results on the set input-to-state stability (ISS) subject to a small input time delay for compact, invariant sets that contains the origin. First, using the nonlinear small-gain theory, we prove a Razumikhin-type…
By using nonstandard analysis, we prove embeddability properties of difference sets $A-B$ of sets of integers. (A set $A$ is "embeddable" into $B$ if every finite configuration of $A$ has shifted copies in $B$.) As corollaries of our main…
The small-world phenomenon has been already the subject of a huge variety of papers, showing its appeareance in a variety of systems. However, some big holes still remain to be filled, as the commonly adopted mathematical formulation…