We count how many ”different” Morse functions exist on the 2-sphere. There are several ways of declaring that two Morse functions f and g are ”indistinguishable” but we concentrate only two natural equivalence relations: homological (when the regular sublevel sets f and g have identical Betti numbers), and geometric (when f is obtained from g via global, orientation preserving changes of coordi...

The approximate analytic bound state solutions of the Klein-Gordon equation with equal scalar and vector exponential-type potentials including the centrifugal potential term are obtained for any arbitrary orbital quantum number l and dimensional space D. The relativistic/non-relativistic energy spectrum formula and the corresponding un-normalized radial wave functions, expressed in terms of the...

In this paper we study topology of the variety of closed pla-nar n-gons with given side lengths l1,. .. , ln. The moduli space M ℓ where ℓ = (l1,. .. , ln), encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces M ℓ as functions of the length vector ℓ = (l1,. .. , ln). We also find sharp upper bounds on the sum of Betti numbers of M ℓ depending only on the num...

Reeb graphs provide a method for studying the shape of a manifold by encoding the evolution and arrangement of level sets of a simple Morse function defined on the manifold. Since their introduction in computer graphics they have been gaining popularity as an effective tool for shape analysis and matching. In this context one question deserving attention is whether Reeb graphs are robust agains...

One of the challenges of Volume Modelling is the definition of theoretical frameworks to support object manipulation and representation. Opposite to what is seen in the field of surface modelling, few satisfactory tools have been presented that ensure topological robustness to volumetric models. In this paper we offer one such mathematical framework for volumetric model definition based on tetr...

In this paper, we describe, analyze and compare techniques for extracting spatial knowledge from a terrain model. Specifically, we investigate techniques for extracting a morphological representation from a terrain model based on an approximation of a Morse-Smale complex. A Morse-Smale complex defines a decomposition of a topographic surface into regions with vertices at the critical points and...

As service-based Internet systems get increasingly complex they become harder to manage at design time as well as at runtime. Nowadays, many systems are described in terms of precisely specified models, e.g., in the context of model-driven development. By making the information in these models accessible at runtime, we provide better means for analyzing and monitoring the service-based systems....

Last year the Metropolitan Museum received a portrait by Samuel F. B. Morse of his daughter Susan, the bequest of Herbert Lee Pratt. This important addition to the collection of American paintings provides an opportunity to review a group of pictures by Morse and other objects associated in one way or another with this most interesting man. Morse the inventor is so firmly embedded in the popula...