A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set) near a parameter reflects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system. We establish a new instance of this phenomenon in terms of entropy. Indeed, we prove an “entropy formula” relating the entropy of a polynomial...

Geometry perturbations affect the eigenmodes of a resonant cavity and thereby can improve but also impair the performance characteristics of the cavity. To investigate the effects of both, intentional and inevitable geometry variations parameter studies are to be undertaken. Using common eigenmode solvers involves to perform a full eigenmode computation for each variation step, even if the geom...

The generation of design parameter sensitivity derivatives is required for gradient-based optimization. Such sensitivity derivatives are elusive at best when working with geometry defined within the solid modeling context of Computer-Aided Design (CAD) systems. Solid modeling CAD systems are often proprietary and always complex, thereby necessitating ad hoc procedures to infer parameter sensiti...

Large-scale learning problems require algorithms that scale benignly with respect to the size of the dataset and the number of parameters to be trained; leading numerous practitioners to favor the classic stochastic gradient descent (SGD [1, 2, 3]) over more sophisticated methods. Besides its fast convergence, SGD has been observed to sometimes lead to signi cantly better generalization perform...

In general, a learning machine will behave better as the number of training examples increases. It is important to know how fast and how well the behavior is improved. The average prediction error, the average of the probability that the trained machine mispredicts the output signal, is one of the most popular criteria to see the behavior. However, it is not easy to evaluate the average predict...

Differential geometry (DG) based solvation models are a new class of variational implicit solvent approaches that are able to avoid unphysical solvent-solute boundary definitions and associated geometric singularities, and dynamically couple polar and non-polar interactions in a self-consistent framework. Our earlier study indicates that DG based non-polar solvation model outperforms other meth...

We consider the estimation of hidden Markovian process by using information geometry with respect to transition matrices. We consider the case when we use only the histogram of k-memory data. Firstly, we focus on a partial observation model with Markovian process and we show that the asymptotic estimation error of this model is given as the inverse of projective Fisher information of transition...

Three-dimensional protein structures usually contain regions of local order, called secondary structure, such as α-helices and β-sheets. Secondary structure is characterized by the local rotational state of the protein backbone, quantified by two dihedral angles called ϕ and ψ. Particular types of secondary structure can generally be described by a single (diffuse) location on a two-dimensional...

A single regular analytic arc in the plane has no conformai differential invariants. The conformai theory of curvilinear angles was initiated by Kasner, and has been elaborated by him and others. The present paper is concerned with conformai differential invariants of a real one-parameter family of regular analytic arcs in the plane. We assume that the family is defined in some region R of the ...