A relation between edit distance for ordered trees and edit distance for Euler strings

We consider a relationship between the unit cost edit distance for two rooted ordered trees and the unit cost edit distance for the corresponding Euler strings. We show that the edit distance between trees is at least half of the edit distance between the Euler strings and is at most 2h+1 times the edit distance between the Euler strings, where h is the minimum height of two trees. The result can be extended for more general cost functions.     

From Euler and Navier–Stokes equations to shallow waters by asymptotic analysis

In this paper, we study the Navier–Stokes and Euler equations in a domain with small depth. With this aim, we introduce a small adimensional parameter ε related to the depth. First we make a change of variable to a domain independent of ε and then we use asymptotic analysis to study what happens when ε becomes small. This way we obtain two new models for ε small that, after coming back to the original domain and without making a priori assumptions about velocity or pressure behaviour, give us a shallow water model including a new diffusion term (obtained from Navier–Stokes equations) and a shallow water model without viscosity and explicit dependence on depth (obtained from Euler equations).     

On one asymptotic formula for the Euler constant

We derive a new asymptotic representation for the Euler constant γ.     

On the equivalence of the k-step iterative euler methods and successive overrelaxation (SOR) methods for k-cyclic matrices

For the solution of the linear system x = Tx + c (1), where T is weakly cyclic of index k ≥ 2, the block SOR method together with two classes of monoparametric k-step iterative Euler methods, whose (optimum) convergence properties were studied in earlier papers, are considered. By establishing the existence of the matrix analog of the Varga's relation, connecting the eigenvalues of the SOR and the Jacobi matrices associated with (1), it is proved that the aforementioned SOR method is equivalent to a certain monoparametric k-step iterative Euler method derived from (1). By suitably modifying the existing theory, one can then determine (optimum) relaxation factors for which the SOR method in question converges, (optimum) regions of convergence etc., so that one can obtain, what is known, several new results. Finally, a number of theoretical applications of practical importance is also presented.     

Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces

Unlike the connected sum in classical topology, its digital version is shown to have some intrinsic feature. In this paper, we study both the digital fundamental group and the Euler characteristic of a connected sum of digital closed k_i-surfaces, i∈{0,1}.     

Euler polygonal method for metric dynamical systems

Following some ideas developed for equations with a nonlinear differential, we consider the existence and uniqueness of solution for the metric dynamical system (F,x₀) and we study the approximation, under appropriate hypotheses, of the unique solution using the Euler polygonal approximation method in relation with function F.     

The Multiple Zeta Value data mine

We provide a data mine of proven results for Multiple Zeta Values (MZVs) of the form with weight and depth k and for Euler sums of the form with signs ?i=±1. Notably, we achieve explicit proven reductions of all MZVs with weights w?22, and all Euler sums with weights w?12, to bases whose dimensions, bigraded by weight and depth, have sizes in precise agreement with the Broadhurst-Kreimer and Broadhurst conjectures. Moreover, we lend further support to these conjectures by studying even greater weights (w?30), using modular arithmetic. To obtain these results we derive a new type of relation for Euler sums, the Generalized Doubling Relations. We elucidate the “pushdown” mechanism, whereby the ornate enumeration of primitive MZVs, by weight and depth, is reconciled with the far simpler enumeration of primitive Euler sums. There is some evidence that this pushdown mechanism finds its origin in doubling relations. We hope that our data mine, obtained by exploiting the unique power of the computer algebra language form, will enable the study of many more such consequences of the double-shuffle algebra of MZVs, and their Euler cousins, which are already the subject of keen interest, to practitioners of Quantum Field Theory, and to mathematicians alike.     

Winding and Euler numbers for 2D and 3D digital images

New algorithms for computing the Euler number of a 3D digital image S are given, based on smoothing the image to a differentiable object and applying theorems of differential geometry and algebraic topology. They run in O(n) time, where n is the number of object elements of S with neighbors not in S. The basic idea is general and easily extended to images defined by other means, such as a hierarchical data structure or a union of isothetic (hyper) rectangles.     

A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured triangular grids

The convergence of high-order methods, such as recently developed spectral difference (SD) method, can be accelerated using both implicit temporal advancement and a p-multigrid (p = polynomial degree) approach. A p-multigrid method is investigated in this paper for solving SD formulations of the scalar wave and Euler equations on unstructured grids. A fast preconditioned lower-upper symmetric Gauss-Seidel (LU-SGS) relaxation method is implemented as an iterative smoother. Meanwhile, a Runge-Kutta explicit method is employed for comparison. The multigrid method considered here is nonlinear and utilizes full approximation storage (FAS) [Ta’asan S. Multigrid one-shot methods and design strategy, Von Karman Institute Lecture Note, 1997 [28]] scheme. For some p-multigrid calculations, blending implicit and explicit smoothers for different p-levels is also studied. The p-multigrid method is firstly validated by solving both linear and nonlinear 2D wave equations. Then the same idea is extended to 2D nonlinear Euler equations. Generally speaking, we are able to achieve speedups of up to two orders using the p-multigrid method with the implicit smoother.     

Simulation of microsatellite attitude using Kalman filtering in orbit results

This paper describes the various steps of the design and the implementation of the new reduced model of the extended Kalman filter (EKF) for small Euler angles on the Algerian microsatellite (Alsat-1). A specific implementation of the EKF is proposed to extract attitude and body rate in a circular low earth orbit (LEO) with a boom deployed for passive pitch and roll stabilisation and active Y-momentum wheel plus Z-reaction wheel and magnetorquer control. Numerical results show the effectiveness of the implementation. The simulation results have been validated using Monte–Carlo method. In-orbit test results are presented to evaluate the performance of the new reduced model of the EKF for small Euler angles during accurate Nadir-pointing control.     

eulerForce: Force-directed layout for Euler diagrams

Euler diagrams use closed curves to represent sets and their relationships. They facilitate set analysis, as humans tend to perceive distinct regions when closed curves are drawn on a plane. However, current automatic methods often produce diagrams with irregular, non-smooth curves that are not easily distinguishable. Other methods restrict the shape of the curve to for instance a circle, but such methods cannot draw an Euler diagram with exactly the required curve intersections for any set relations. In this paper, we present eulerForce, as the first method to adopt a force-directed approach to improve the layout and the curves of Euler diagrams generated by current methods. The layouts are improved in quick time. Our evaluation of eulerForce indicates the benefits of a force-directed approach to generate comprehensible Euler diagrams for any set relations in relatively fast time.     

Topology Preservation Within Digital Surfaces

Given two connected subsets Y⊂X of the set of the surfels of a connected digital surface, we propose three equivalent ways to express Y being homotopic to X. The first characterization is based on sequential deletion of simple surfels. This characterization enables us to define thinning algorithms within a digital Jordan surface. The second characterization is based on the Euler characteristics of sets of surfels. This characterization enables us, given two connected sets Y⊂X of surfels, to decide whether Y is n-homotopic to X. The third characterization is based on the (digital) fundamental group.     

A note on generalized Euler numbers and polynomials

In this paper we construct new generalized Euler polynomials and generalized Euler numbers attached to χ. We investigate some of the properties that are related to generalized Euler polynomials. We also derive the existence of a specific interpolation function that interpolates generalized Euler polynomials at negative integers. Finally, we investigate computationally the roots of the generalized Euler polynomials E _{n, χ}(x) for values of the index n.     

Stochastic Backward Euler: An Implicit Gradient Descent Algorithm for <Emphasis Type="Italic">k</Emphasis>-Means Clustering

In this paper, we propose an implicit gradient descent algorithm for the classic k-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch gradient in every iteration. It is the average of the fixed-point trajectory that is carried over to the next gradient step. We draw connections between the proposed stochastic backward Euler and the recent entropy stochastic gradient descent for improving the training of deep neural networks. Numerical experiments on various synthetic and real datasets show that the proposed algorithm provides better clustering results compared to k-means algorithms in the sense that it decreased the objective function (the cluster) and is much more robust to initialization.     

Die Weiterentwicklung der Aireyschen Konvergenzfaktoren für das Exponentialintegral zu einer Darstellung mit Restglied

In this paper Aireys converging factors of the exponential integral are transformed into an asymptotic expansion with remainder. This expansion is obtained by an Euler transformation of the remaining divergent series in analogy to the formal procedure due to Airey. The remainder termR _N (z) is developed by Abels asymptotic principle. Transforming the terms of this series we get Aireys converging factors. Finally we improve the estimation of the remainder term which has been given by J. B. Rosser.     

Optimal evaluation of generalized Euler angles with applications to control

Given two linearly independent matrices in so(3), Z₁ and Z₂, every rotation matrix, X_f∈SO(3), can be written as the product of alternate elements from the one-dimensional subgroups corresponding to Z₁ and Z₂, namely X_f=e^Z₁t₁e^Z₂t₂e^Z₁t₃?e^Z₁t_s. The parameters t_i, i=1,…,s are called Generalized Euler Angles. In this paper, the minimum number of factors required for the factorization of X_f∈SO(3), as a function of X_f, is evaluated. An algorithm is given to determine the generalized Euler angles, in the optimal factorization. The results can be applied to the bang-bang control, with minimum number of switches, of some classical and quantum systems.     

Holes and Genus of 2D and 3D Digital Images

"Hole" has been a confusing idea in the 3D digital literature. We replace counting holes by the clear geometrical idea of counting non-separating cuts, and show that this gives the Betti number b₁, while b₀ counts components and b₂ cavities. Connected sets with equal b₁ and b₂ must match topologically when b₁ = 0 (implying simple connectedness). When b₁ ≠ 0, contrary to digital folklore, they need not. This paper is a conceptually self-contained introduction for computer scientists to these numbers of 2D and 3D images, and to other topological features such as Euler and linking numbers.     

A conservative pressure-correction method for flow at all speeds

A new fully conservative Mach-uniform staggered scheme is discussed. With this scheme one can compute flow with a Mach number ranging from the incompressible limit M↓0 up to supersonic flow M>1, with nearly uniform efficiency and accuracy. Earlier methods are based on a nonconservative discretisation of the energy equation. This results in small discrepancies in the computed shock speed for the Euler equations. The new method has similar Mach-uniform properties as the earlier methods, but is found to converge to the correct weak solution.     

What Makes an Effective Representation of Information: A Formal Account of Observational Advantages

In order to effectively communicate information, the choice of representation is important. Ideally, a chosen representation will aid readers in making desired inferences. In this paper, we develop the theory of observation: what it means for one statement to be observable from another. Using observability, we give a formal characterization of the observational advantages of one representation of information over another. By considering observational advantages, people will be able to make better informed choices of representations of information. To demonstrate the benefit of observation and observational advantages, we apply these concepts to set theory and Euler diagrams. In particular, we can show that Euler diagrams have significant observational advantages over set theory. This formally justifies Larkin and Simon’s claim that “a diagram is (sometimes) worth ten thousand words”.