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Hydrogen presents a promising opportunity for the reduction of CO 2 emissions in combustion processes. Due to its wide ignition limits, operation in lean mode is possible, which significantly reduces NO x emissions. However, this lean operation also leads to a reduction in the resulting torque. In contrast, stoichiometric operation increases maximum power output but leads to increased NO x emissions. In particular, a cost-effective three-way catalyst can be used in stoichiometric operation, enabling effective emission control. This investigation proposes an innovative approach that involves lean-burn operation at part load conditions and switching to stoichiometric operation at full load. The transition between these two modes has a considerable impact on overall NO x emissions. To optimize this process, new functions were developed that implement countermeasures such as lambda control, ignition timing adjustment, catalyst purging, and shortening the switching range through the use of variable valve timing and variable turbine geometry. The results show that nitrogen oxide (NO x ) emissions downstream of the three-way catalyst are kept below in the lean operating range and below in the stoichiometric operating range. By optimizing the transition between the two operating modes and using advanced emission control technologies, it is possible to reduce NO x emissions by 84% while maintaining power efficiency under different load conditions. In addition, the almost torque-neutral switching between the two operational modes ensures that the vehicle’s drivability is not impaired. By incorporating additional dosing of a urea-water solution in an active SCR system, a significant improvment in NO x reduction is attained, achieving levels comparable to those of diesel internal combustion engines. This dual-mode operation strategy improves the feasibility of hydrogen as a viable fuel alternative in future energy systems.

We provide explicit asymptotically optimal quadrature rules for uniform C k -splines, k = 0,1, over the real line. The nodes of these quadrature rules are given in terms of the zeros of ultraspherical polynomials (Gegenbauer polynomials) and related polynomials. We conjecture that our derived rules are the only possible periodic asymptotically optimal quadrature rules for these spline spaces.

3 families of 4-dimensional lattices \\(L_k, M_k, M_k / 2 \\subset \\mathbb{R}^2\\) are defined. Each lattice is defined by 2 quadratic extensions and has a \\emph{finite} number of unit vectors, but the number of unit vectors in each of the 3 familes is \\emph{unbounded}. \\(L_3\\) is the Moser lattice.

In a recently published article by G. Ambrus et al. a new upper bound for the density of an unit avoiding, periodic set is given as \\(0.2470\\), the first upper bound \\(< 1/4\\). A construction of Croft 1967 gave a lower bound \\(\\delta_C = 0.22936\\) for the density. To this date, no better construction with a higher lower bound has been given. In this article I give a construction planar sets with a higher density than Croft's tortoises. No explicit value for this density is given, it's just shown that Croft's density is a local minima of the density of a here constructed 1-parameter family of planar sets. So the densities are \\(> \\delta_C\\).

The conditions for cubic equations, to have 3 real roots and 2 of the roots lie in the closed interval \\([-1, 1]\\) are given. These conditions are visualized. This question arises in physics in e.g. the theory of tops.

The article shows how two choices are possible whenever computing the geometric mean, and the repetition of this process can in general yield 2-to-the power N different values when the choices are compounded in the first N steps of evaluation of the arithmetic-geometric mean. This happens not only in the simple AGM involved in the computation of the complete elliptic integral of the first kind, but also in analogous methods for the computation of the complete and incomplete elliptic integrals of the first and second kind.

The Somos-4 equation defines the sequences with this name. Looking at these sequences with an additional property we get a quartic polynomial in 4 variables. This polynomial defines a rational, projective surface in \\(\\mathbb{RP}^{3}\\). Here some generators of the subgroup of \\(Cr_3 (\\mathbb{R})\\) are determined, whose birational maps are automorphisms of the quartic surface.

Two infinite families of Cremona maps depending on one real parameter are given. For all integers \\(n \\ge 1\\) the first family of Cremona maps consists of group elements in \\(Bir \\left( \\mathbb{P}^{n} \\right)\\) with bidegree \\((n, n)\\), the second family of Cremona maps consists of group elements in \\(Bir \\left( \\mathbb{P}^{2n} \\right)\\) with bidegree \\((2 n, 2 n)\\). For the first family and \\(n \\ge 5\\), for the second family and \\(n \\ge 3\\) the existence of this group elements and the properties depend on a conjecture. But computational results suggest that the conjecture is true for all \\(n\\).

A theory of spline quadrature rules for arbitrary continuity class in a closed interval \\([a, b]\\) with arbitrary nonuniform subintervals based on semi-classical orthogonal Jacobi polynomials is proposed. For continuity class \\(c \\ge 2\\) this theory depends on a conjecture.

Closed formulae for all Gaussian or optimal, 1-parameter quadrature rules in a compact interval [a, b] with non uniform, asymmetric subintervals, arbitrary number of nodes per subinterval for the spline classes \\(S_{2N, 0}\\) and \\(S_{2N+1, 1}\\), i.e. even and odd degree are presented. Also rules for the 2 missing spline classes \\(S_{2N-1, 0}\\) and \\(S_{2N, 1}\\) (the so called 1/2-rules), i.e. odd and even degree are presented. These quadrature rules are explicit in the sense, that they compute the nodes and their weights in the first/last boundary subinterval and, via a recursion the other nodes/weights, parsing from the first/last subinterval to the middle of the interval. These closed formulae are based on the semi-classical Jacobi type orthogonal polynomials.