This article surveys the Euler calculus — an integral calculus based on Euler characteristic — and its applications to data, sensing, networks, and imaging.

We prove a comparison theorem of Faber-Krahn type and a sharp bound for the compact surfaces with negative Euler characteristic via the Ricci flow.

Contractible transformations of graphs consist of contractible gluing and deleting of vertices and edges of graphs. They partition all graphs into the family of homotopy classes. Contractible transformations do not change the Euler characteristic and the homology groups of graphs. In this paper we describe the minimal representatives of some homotopy classes and find the formula for computing t...

We empirically analyze the impact of transaction costs on the performance of essentially affine interest rate models. We test the implied Euler restrictions and calculate the specification error bound of Hansen and Jagannathan to measure model misspecification. Using both short-maturity and long-maturity bond return data we find, under the assumption of frictionless markets, strong evidence of ...

We give a classification of toric anti-self-dual conformal structures on compact 4-orbifolds with positive Euler characteristic. Our proof is twistor theoretic: the interaction between the complex torus orbits in the twistor space and the twistor lines induces meromorphic data, which we use to recover the conformal structure. A compact anti-self-dual orbifold can also be constructed by adding a...

By gluing together copies of an all-right angled Coxeter polytope a number of open hyperbolic 6-manifolds with Euler characteristic −1 are constructed. They are the first known examples of hyperbolic 6-manifolds having the smallest possible volume.

The purpose of this article is to give a brief and elementary approach to the classification of mapping-classes for a surface with negative Euler characteristic. Work of Thurston ([9], [10], [7]) has brought out a deep analogy with the classical structure of the modular group SL2(Z) acting on the upper half plane H by constructing a completion of L, the space of simple loops in a base surface v...

We define reduced zeta functions of Lie algebras, which can be derived from motivic zeta functions using the Euler characteristic. We show that reduced zeta functions of Lie algebras possessing a suitably well-behaved basis are easy to analyse. We prove that reduced zeta functions are multiplicative under certain conditions and investigate which reduced zeta functions have functional equations.

We develop a model predictive control (MPC) design for systems with discrete-time dynamics evolving on smooth manifolds. We show that the properties of conventional MPC for dynamics evolving on R are preserved and we develop a design procedure for achieving similar properties. We also demonstrate that for discrete-time dynamics on manifolds with Euler characteristic not equal to 1, there do not...

2 Euler characteristic analysis 14 2.1 Formulation of the result and the nature of the generating function . . . . . 14 2.2 Standard Borel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Arbitrary Borel in gl(N |N). . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Arbitrary Borel in gl(M |N). . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Arb...