The translate and line properties for 2-primitive elements in quadratic extensions

Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^{n}-1$ . We say that the extension $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$ -primitive elements property if, for every $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}\in \mathbb{F}_{q^{n}}^{\ast }$ such that $\mathbb{F}_{q^{n}}=\mathbb{F}_{q}(\unicode[STIX]{x1D703})$ , there exists some $x\in \mathbb{F}_{q}$ such that $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D703}+x)$ has multiplicative order $(q^{n}-1)/r$ . We prove that, for sufficiently large prime powers $q$ , $\mathbb{F}_{q^{n}}/\mathbb{F}_{q}$ possesses the line property for $r$ -primitive elements. We also discuss the (weaker) translate property for extensions.

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Generalized cyclotomic numbers and cyclic codes of prime power length over Z4

Asian-European Journal of Mathematics ◽

10.1142/s1793557119500852 ◽

2019 ◽

Vol 12 (05) ◽

pp. 1950085 ◽

Cited By ~ 1

Author(s):

Sudhir Batra ◽

Sonal Jain

Keyword(s):

Prime Power ◽

Cyclic Codes ◽

Self Duality ◽

Primitive Idempotents ◽

Multiplicative Order ◽

Cyclotomic Numbers ◽

Explicit Expressions

Generalized cyclotomic numbers of order [Formula: see text] with respect to an odd prime power are obtained. Hence, explicit expressions for primitive idempotents in the ring [Formula: see text] are obtained in two cases, when the multiplicative order of 2 modulo [Formula: see text] is [Formula: see text] and [Formula: see text], where [Formula: see text] is an odd prime. Orthogonality and self-duality of some [Formula: see text] cyclic codes are also discussed. Further, a method for obtaining cyclic self-dual/isodual codes of length [Formula: see text] over [Formula: see text] is given.

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Structure Information From Electron Diffraction Intensities

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100136180 ◽

1990 ◽

Vol 48 (2) ◽

pp. 516-517

Author(s):

J. Gjønnes ◽

N. Bøe ◽

K. Gjønnes

Keyword(s):

Computational Effort ◽

Unit Cell ◽

Small Unit ◽

Structure Information ◽

Dispersion Surface ◽

Integrated Intensity ◽

The One ◽

Image Position ◽

Convergent Beam ◽

Beam Patterns

Structure information of high precision can be extracted from intentsity details in convergent beam patterns like the one reproduced in Fig 1. From low order reflections for small unit cell crystals,bonding charges, ionicities and atomic parameters can be derived, (Zuo, Spence and O’Keefe, 1988; Zuo, Spence and Høier 1989; Gjønnes, Matsuhata and Taftø, 1989) , but extension to larger unit cell ma seem difficult. The disks must then be reduced in order to avoid overlap calculations will become more complex and intensity features often less distinct Several avenues may be then explored: increased computational effort in order to handle the necessary many-parameter dynamical calculations; use of zone axis intensities at symmetry positions within the CBED disks, as in Figure 2 measurement of integrated intensity across K-line segments. In the last case measurable quantities which are well defined also from a theoretical viewpoint can be related to a two-beam like expression for the intensity profile:With as an effective Fourier potential equated to a gap at the dispersion surface, this intensity can be integrated across the line, with kinematical and dynamical limits proportional to and at low and high thickness respctively (Blackman, 1939).

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Computational Effort vs. Accuracy Tradeoff in Uncertainty Quantification

16th AIAA Non-Deterministic Approaches Conference ◽

10.2514/6.2014-1178 ◽

2014 ◽

Author(s):

Joshua G. Mullins ◽

Sankaran Mahadevan

Keyword(s):

Uncertainty Quantification ◽

Computational Effort ◽

Accuracy Tradeoff

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A Combined Spin-Flip and IP/EA Approach for Handling Spin and Spatial Degeneracies: Application to Double Exchange Systems

10.26434/chemrxiv.7445939.v1 ◽

2018 ◽

Author(s):

Shannon Houck ◽

Nicholas Mayhall

Keyword(s):

Single Molecule ◽

Double Exchange ◽

Computational Effort ◽

Strongly Correlated ◽

Single Molecule Magnets ◽

New Approach ◽

Spin Flip ◽

Model Hamiltonian ◽

Simultaneous Existence ◽

Exchange Parameters

<div>Many multiconfigurational systems, such as single-molecule magnets, are difficult to study using traditional computational methods due to the simultaneous existence of both spin and spatial degeneracies. In this work, a new approach termed n-spin-flip Ionization Potential/Electron Affinity (<i>n</i>SF-IP or <i>n</i>SF-EA) is introduced which combines the spin-flip method of Anna Krylov with particle-number changing IP/EA methods. We demonstrate the efficacy of the approach by applying it to the strongly-correlated N<sub>2</sub><sup>+</sup> as well as several double exchange systems. We also demonstrate that when these systems are well-described by a double exchange model Hamiltonian, only 1SF-IP/EA is required to extract the double exchange parameters and accurately predict energies for the low-spin states. This significantly reduces the computational effort for studying such systems. The effects of including additional excitations (using a RAS-<i>n</i>SF-IP/EA scheme) are also examined, with particular emphasis on hole and particle excitations.</div>

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A Combined Spin-Flip and IP/EA Approach for Handling Spin and Spatial Degeneracies: Application to Double Exchange Systems

10.26434/chemrxiv.7445939 ◽

2018 ◽

Author(s):

Shannon Houck ◽

Nicholas Mayhall

Keyword(s):

Single Molecule ◽

Double Exchange ◽

Computational Effort ◽

Strongly Correlated ◽

Single Molecule Magnets ◽

New Approach ◽

Spin Flip ◽

Model Hamiltonian ◽

Simultaneous Existence ◽

Exchange Parameters

<div>Many multiconfigurational systems, such as single-molecule magnets, are difficult to study using traditional computational methods due to the simultaneous existence of both spin and spatial degeneracies. In this work, a new approach termed n-spin-flip Ionization Potential/Electron Affinity (<i>n</i>SF-IP or <i>n</i>SF-EA) is introduced which combines the spin-flip method of Anna Krylov with particle-number changing IP/EA methods. We demonstrate the efficacy of the approach by applying it to the strongly-correlated N<sub>2</sub><sup>+</sup> as well as several double exchange systems. We also demonstrate that when these systems are well-described by a double exchange model Hamiltonian, only 1SF-IP/EA is required to extract the double exchange parameters and accurately predict energies for the low-spin states. This significantly reduces the computational effort for studying such systems. The effects of including additional excitations (using a RAS-<i>n</i>SF-IP/EA scheme) are also examined, with particular emphasis on hole and particle excitations.</div>

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Molecular Dynamics Using Non-Variational Polarizable Force Fields: Theory, Periodic Boundary Conditions Implementation and Application to the Bond Capacity Model

10.26434/chemrxiv.8343614.v2 ◽

2019 ◽

Author(s):

Pier Paolo Poier ◽

Louis Lagardere ◽

Jean-Philip Piquemal ◽

Frank Jensen

Keyword(s):

Boundary Conditions ◽

Periodic Boundary ◽

Force Fields ◽

Periodic Boundary Conditions ◽

Computational Effort ◽

Polarizable Force Fields ◽

Energy Models ◽

Capacity Model ◽

Energy Gradient ◽

Polarization Models

<div> <div> <div> <p>We extend the framework for polarizable force fields to include the case where the electrostatic multipoles are not determined by a variational minimization of the electrostatic energy. Such models formally require that the polarization response is calculated for all possible geometrical perturbations in order to obtain the energy gradient required for performing molecular dynamics simulations. </p><div> <div> <div> <p>By making use of a Lagrange formalism, however, this computational demanding task can be re- placed by solving a single equation similar to that for determining the electrostatic variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p><div><div><div> </div> </div> </div> <p> </p><div> <div> <div> <p>variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p> </div> </div> </div> </div> </div> </div> </div> </div> </div>

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Review for "Efficient MILP formulations for AC optimal power flow to reduce computational effort"

10.1002/2050-7038.12434/v2/review3 ◽

2020 ◽

Keyword(s):

Power Flow ◽

Optimal Power Flow ◽

Computational Effort ◽

Optimal Power

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Review for "Efficient MILP formulations for AC optimal power flow to reduce computational effort"

10.1002/2050-7038.12434/v3/review1 ◽

2020 ◽

Keyword(s):

Power Flow ◽

Optimal Power Flow ◽

Computational Effort ◽

Optimal Power

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Double-smoothing in Kernel Hazard Rate Estimation

Methods of Information in Medicine ◽

10.3414/me0447 ◽

2008 ◽

Vol 47 (02) ◽

pp. 167-173 ◽

Cited By ~ 5

Author(s):

A. Pfahlberg ◽

O. Gefeller ◽

R. Weißbach

Keyword(s):

Hazard Rate ◽

Nearest Neighbor ◽

Bandwidth Selection ◽

Study Data ◽

Computational Effort ◽

Finite Sample ◽

Adaptive Smoothing ◽

Rate Estimation ◽

Data Adaptive ◽

Double Smoothing

Summary Objectives: In oncological studies, the hazard rate can be used to differentiate subgroups of the study population according to their patterns of survival risk over time. Nonparametric curve estimation has been suggested as an exploratory means of revealing such patterns. The decision about the type of smoothing parameter is critical for performance in practice. In this paper, we study data-adaptive smoothing. Methods: A decade ago, the nearest-neighbor bandwidth was introduced for censored data in survival analysis. It is specified by one parameter, namely the number of nearest neighbors. Bandwidth selection in this setting has rarely been investigated, although the heuristical advantages over the frequently-studied fixed bandwidth are quite obvious. The asymptotical relationship between the fixed and the nearest-neighbor bandwidth can be used to generate novel approaches. Results: We develop a new selection algorithm termed double-smoothing for the nearest-neighbor bandwidth in hazard rate estimation. Our approach uses a finite sample approximation of the asymptotical relationship between the fixed and nearest-neighbor bandwidth. By so doing, we identify the nearest-neighbor bandwidth as an additional smoothing step and achieve further data-adaption after fixed bandwidth smoothing. We illustrate the application of the new algorithm in a clinical study and compare the outcome to the traditional fixed bandwidth result, thus demonstrating the practical performance of the technique. Conclusion: The double-smoothing approach enlarges the methodological repertoire for selecting smoothing parameters in nonparametric hazard rate estimation. The slight increase in computational effort is rewarded with a substantial amount of estimation stability, thus demonstrating the benefit of the technique for biostatistical applications.

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