On the non-equivalence of maximum polynomial degree nordsieck-gear and classical methods

Author(s):  
Roy Danchick
Keyword(s):  
2021 ◽  
Vol 105 (563) ◽  
pp. 253-262
Author(s):  
R. W. D. Nickalls

This Article explores how root multiplicity and polynomial degree influence the structure of the roots of a univariant polynomial. After setting up the notation, we draw upon a result derived in [1], and show that all polynomial roots have a common underlying structure comprising just five parameters. Finally we present some examples involving the lower polynomials.


2014 ◽  
Vol 7 (12) ◽  
pp. 4133-4150 ◽  
Author(s):  
L. M. A. Alvarado ◽  
A. Richter ◽  
M. Vrekoussis ◽  
F. Wittrock ◽  
A. Hilboll ◽  
...  

Abstract. Satellite observations from the SCIAMACHY, GOME-2 and OMI spectrometers have been used to retrieve atmospheric columns of glyoxal (CHOCHO) with the DOAS method. High CHOCHO levels were found over regions with large biogenic and pyrogenic emissions, and hot-spots have been identified over areas of anthropogenic activities. This study focuses on the development of an improved retrieval for CHOCHO from measurements by the OMI instrument. From sensitivity tests, a fitting window and a polynomial degree are determined. Two different approaches to reduce the interference of liquid water absorption over oceanic regions are evaluated, achieving significant reduction of the number of negative columns over clear water regions. The impact of using different absorption cross-sections for water vapour is evaluated and only small differences are found. Finally, a high-temperature (boundary layer ambient: 294 K) absorption cross-section of nitrogen dioxide (NO2) is introduced in the DOAS retrieval to account for potential interferences of NO2 over regions with large anthropogenic emissions, leading to improved fit quality over these areas. A comparison with vertical CHOCHO columns retrieved from GOME-2 and SCIAMACHY measurements over continental regions is performed, showing overall good consistency. However, SCIAMACHY CHOCHO columns are systematically higher than those obtained from the other instruments. Using the new OMI CHOCHO data set, the link between fires and glyoxal columns is investigated for two selected regions in Africa. In addition, mapped averages are computed for a fire event in Russia between mid-July and mid-August 2010. In both cases, enhanced CHOCHO levels are found in close spatial and temporal proximity to elevated levels of MODIS fire radiative power, demonstrating that pyrogenic emissions can be clearly identified in the new OMI CHOCHO product.


2021 ◽  
Vol 2 (4) ◽  
pp. 1-9
Author(s):  
Scott Aaronson

I offer a case that quantum query complexity still has loads of enticing and fundamental open problems—from relativized QMA versus QCMA and BQP versus IP , to time/space tradeoffs for collision and element distinctness, to polynomial degree versus quantum query complexity for partial functions, to the Unitary Synthesis Problem and more.


2019 ◽  
Vol 53 (3) ◽  
pp. 987-1003 ◽  
Author(s):  
Claudio Canuto ◽  
Ricardo H. Nochetto ◽  
Rob P. Stevenson ◽  
Marco Verani

Both practice and analysis of p-FEMs and adaptive hp-FEMs raise the question what increment in the current polynomial degree p guarantees a p-independent reduction of the Galerkin error. We answer this question for the p-FEM in the simplified context of homogeneous Dirichlet problems for the Poisson equation in the two dimensional unit square with polynomial data of degree p. We show that an increment proportional to p yields a p-robust error reduction and provide computational evidence that a constant increment does not.


Author(s):  
Dominik Scholz ◽  
Alexander Du¨ster ◽  
Ernst Rank

Our approach is to use fully three-dimensional models for both the fluid and the structure. For thin-walled structures, which are typically sensitive to loads resulting from the surrounding fluid, it will be shown that the use of high-order hexahedral elements with high aspect ratios is feasible. Furthermore, it will be demonstrated that three-dimensional elements of high order can be used very efficiently by choosing a high polynomial degree in in-plane direction and a low polynomial degree in thickness direction. By varying the polynomial degrees in the local directions of the elements, the choice of an appropriate structural model can be achieved in an adaptive way. This will be demonstrated by means of a numerical example.


Sign in / Sign up

Export Citation Format

Share Document