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

1974 ◽  
pp. 92-106
Author(s):  
Roy Danchick
Keyword(s):  
Polynomial Degree

On the structure of polynomial roots

2021 ◽  
Vol 105 (563) ◽  
pp. 253-262
Author(s):  
R. W. D. Nickalls
Keyword(s):  
Underlying Structure ◽  
Polynomial Roots ◽  
Polynomial Degree

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.

scholarly journals An improved glyoxal retrieval from OMI measurements

2014 ◽  
Vol 7 (12) ◽  
pp. 4133-4150 ◽  
Author(s):  
L. M. A. Alvarado ◽  
A. Richter ◽  
M. Vrekoussis ◽  
F. Wittrock ◽  
A. Hilboll ◽  
...  
Keyword(s):  
Cross Sections ◽  
Anthropogenic Activities ◽  
Absorption Cross ◽  
Fire Event ◽  
Data Set ◽  
Polynomial Degree ◽  
Temperature Boundary ◽  
Sensitivity Tests ◽  
Radiative Power ◽  
The Impact

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.

Open Problems Related to Quantum Query Complexity

2021 ◽  
Vol 2 (4) ◽  
pp. 1-9
Author(s):  
Scott Aaronson
Keyword(s):  
Synthesis Problem ◽  
Query Complexity ◽  
Partial Functions ◽  
Open Problems ◽  
Polynomial Degree ◽  
Quantum Query Complexity ◽  
Time Space ◽  
Quantum Query

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.

scholarly journals A saturation property for the spectral-Galerkin approximation of a Dirichlet problem in a square

2019 ◽  
Vol 53 (3) ◽  
pp. 987-1003 ◽  
Author(s):  
Claudio Canuto ◽  
Ricardo H. Nochetto ◽  
Rob P. Stevenson ◽  
Marco Verani
Keyword(s):  
Dirichlet Problem ◽  
Poisson Equation ◽  
Galerkin Approximation ◽  
Error Reduction ◽  
Two Dimensional ◽  
Dirichlet Problems ◽  
Polynomial Degree ◽  
Unit Square ◽  
The Poisson Equation ◽  
Dimensional Unit

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.

Fully Three Dimensional Modeling of Fluid-Structure Interaction Problems Using High Order Finite Elements for the Structural Simulation

2004 ◽  
Author(s):  
Dominik Scholz ◽  
Alexander Du¨ster ◽  
Ernst Rank
Keyword(s):  
Structural Model ◽  
Three Dimensional ◽  
High Order ◽  
Polynomial Degree ◽  
Three Dimensional Modeling ◽  
Thin Walled Structures ◽  
Aspect Ratios ◽  
Hexahedral Elements ◽  
Plane Direction ◽  
Three Dimensional Models

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.

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