Lecture 10 on Differential Equations & Differential Invariants and Lecture 11 on Differential Equations & Proofs equipped us with powerful tools for proving properties of differential equations without having to solve them. Differential invariants (DI) [Pla10a] prove properties of differential equations by induction based on the right-hand side of the differential equation, rather than its much...

in this paper we introduce the concept of entropy operator for continuous systems of finite topological entropy. it is shown that it generates the kolmogorov entropy as a special case. if $phi$ is invertible then the entropy operator is bounded with the topological entropy of $phi$ as its norm.

A bstract We study four-derivative corrections to 5d $$ \mathcal{N} N = 2 minimal gauged supergravity using tools from conformal supergravity. There are two supersymmetric invariants at the order and we show explicitly how write down action in Poincaré frame terms of their coefficients. apply our results cont...

Acoustic and entropy waves interacting in a duct with steady heat source mean flow are analysed using an asymptotic expansion (AE) for low frequencies. The analytical AE solutions obtained by taking advantage of invariants applying multi-step strategy. proposed provide first-order corrections to the compact model form integrals variables. An eigenvalue system is then built predict thermoacousti...

The literature contains several extensions of the standard definitions of topological entropy for a continuous self-map f : X→X from the case when X is a compact metric space to the case when X is allowed to be noncompact. These extensions all require the space X to be totally bounded, or equivalently to have a compact completion, and are invariants of uniform conjugacy. When themap f is unifor...

This thesis describes techniques for static analysis of polynomial equalities in interprocedural programs. It elaborates on approaches for analysing polynomial equalities over different domains as well as techniques to apply polynomial analysis to infer interprocedurally valid equalities of uninterpreted terms. This work is organised in three major theoretical parts, followed by a practical par...

Recently, based on elementary geometry, Gutman proposed several geometry-based invariants (i.e., $SO$, $SO_{1}$, $SO_{2}$, $SO_{3}$, $SO_{4}$, $SO_{5}$, $SO_{6}$). The Sombor index was defined as $SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, the first $SO_{1}(G)= \frac{1}{2}\sum\limits_{uv\in E(G)}|d_{u}^{2}-d_{v}^{2}|$, where $d_{u}$ denotes degree of vertex $u$. In this paper, w...

Abstract We show that well-known invariants like Lusternik–Schnirelmann category and topological complexity are particular cases of a more general notion, we call homotopic distance between two maps. As consequence, several properties those can be proved in unified way new results arise.