In [J. Graph Theory 13 (1989) 749–762], McCuaig and Shepherd gave an upper bound of the domination number for connected graphs with minimum degree at least two. In this paper, we propose a simple strategy which, together with the McCuaig-Shepherd theorem, gives a sharp upper bound of the domination number via the number of leaves. We also apply the same strategy to other domination-like invaria...

Glossary 1 1. Definition of the subject and its importance 2 2. Basic Results 2 3. Panorama of Examples 8 4. Mixing notions and multiple recurrence 10 5. Topological group Aut(X,μ) 13 6. Orbit theory 15 7. Smooth nonsingular transformations 21 8. Spectral theory for nonsingular systems 22 9. Entropy and other invariants 25 10. Nonsingular Joinings and Factors 27 11. Applications. Connections wi...

We generalize the topological entanglement entropy to a family of topological Rényi entropies parametrized by a parameter alpha, in an attempt to find new invariants for distinguishing topologically ordered phases. We show that, surprisingly, all topological Rényi entropies are the same, independent of alpha for all nonchiral topological phases. This independence shows that topologically ordere...

We discuss metric, algorithmic and geometric issues related to broadly understood complexity of high dimensional convex sets. The specific topics we bring up include metric entropy and its duality, derandomization of constructions of normed spaces or of convex bodies, and different fundamental questions related to geometric diversity of such bodies, as measured by various isomorphic (as opposed...

We investigate linear and additive codes in partially ordered Hamming-like spaces that satisfy the extension property, meaning that automorphisms of ideals extend to automorphisms of the poset. The codes are naturally described in terms of translation association schemes that originate from the groups of linear isometries of the space. We address questions of duality and invariants of codes, es...

Using a special form of Ulam’s method, we estimate the measure-theoretic entropy of a triple (M, T , μ), where M is a smooth manifold, T is a C1+γ uniformly hyperbolic map, and μ is the unique physical measure of T . With a few additional calculations, we also obtain numerical estimates of (i) the physical measureμ, (ii) the Lyapunov exponents of T with respect toμ, (iii) the rate of decay of c...

In this article, we calculate various topological invariants such as symmetric division degree index, redefined Zagreb VL first and second exponential multiplicative indices, entropy, entropies, entropy. We take the chemical compound named Proanthocyanidins, which is a very useful polyphenol in human’s diet. They are beneficial for one’s health. These compounds extracted from grape seeds. treme...

Although content-based image retrieval (CBIR) is not a new subject, it keeps attracting more and more attention, as the amount of images grow tremendously due to internet, inexpensive hardware and automation of image acquisition. One of the applications of CBIR is fetching images from a database. This paper presents a new method for automatic image retrieval using moment invariants and image en...

Real finite type invariants have diagrammatic descriptions and relate to Lie Algebras. Analogues of the corresponding results for virtual finite type invariants exist, but are less well understood. This article collects computational results about the dimensions of the Polyak algebra and other spaces related to virtual finite type invariants. The code describes the corresponding Gauss diagram f...