scholarly journals The Remez inequality for linear combinations of shifted Gaussians

2008 ◽  
Vol 146 (03) ◽  
pp. 523 ◽  
Author(s):  
TAMÁS ERDÉLYI
Keyword(s):  
Linear Combinations ◽  
Remez Inequality

Descriptive statistics

2017 ◽  
Author(s):  
Andrew Gelman ◽  
Deborah Nolan
Keyword(s):  
Descriptive Statistics ◽  
Metabolic Rates ◽  
World Record ◽  
Linear Combinations ◽  
Record Times ◽  
Starting Point ◽  
Statistics Course ◽  
Scatter Plots ◽  
Economic Indexes ◽  
The Mean

Descriptive statistics is the typical starting point for a statistics course, and it can be tricky to teach because the material is more difficult than it first appears. The activities in this chapter focus more on the topics of data displays and transformations, rather than the mean, median, and standard deviation, which are covered easily in a textbook and on homework assignments. Specific topics include: distributions and handedness scores; extrapolation of time series and world record times for the mile run; linear combinations and economic indexes; scatter plots and exam scores; and logarithmic transformations and metabolic rates.

scholarly journals Some inequalities of linear combinations of independent random variables: II

Bernoulli ◽  
2013 ◽  
Vol 19 (5A) ◽  
pp. 1776-1789 ◽  
Author(s):  
Xiaoqing Pan ◽  
Maochao Xu ◽  
Taizhong Hu
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Linear Combinations

scholarly journals Reconstruction of Relative Phase of Self-Transmitting Devices by Using Multiprobe Solutions and Non-Convex Optimization

Sensors ◽  
2021 ◽  
Vol 21 (7) ◽  
pp. 2459
Author(s):  
Rubén Tena Sánchez ◽  
Fernando Rodríguez Varela ◽  
Lars J. Foged ◽  
Manuel Sierra Castañer
Keyword(s):  
Iterative Algorithm ◽  
Degrees Of Freedom ◽  
Relative Phase ◽  
Low Cost ◽  
Initial Guess ◽  
Noise Model ◽  
Medium Size ◽  
Phase Reconstruction ◽  
Linear Combinations ◽  
Iterative Optimization Algorithm

Phase reconstruction is in general a non-trivial problem when it comes to devices where the reference is not accessible. A non-convex iterative optimization algorithm is proposed in this paper in order to reconstruct the phase in reference-less spherical multiprobe measurement systems based on a rotating arch of probes. The algorithm is based on the reconstruction of the phases of self-transmitting devices in multiprobe systems by taking advantage of the on-axis top probe of the arch. One of the limitations of the top probe solution is that when rotating the measurement system arch, the relative phase between probes is lost. This paper proposes a solution to this problem by developing an optimization iterative algorithm that uses partial knowledge of relative phase between probes. The iterative algorithm is based on linear combinations of signals when the relative phase is known. Phase substitution and modal filtering are implemented in order to avoid local minima and make the algorithm converge. Several noise-free examples are presented and the results of the iterative algorithm analyzed. The number of linear combinations used is far below the square of the degrees of freedom of the non-linear problem, which is compensated by a proper initial guess. With respect to noisy measurements, the top probe method will introduce uncertainties for different azimuth and elevation positions of the arch. This is modelled by considering the real noise model of a low-cost receiver and the results demonstrate the good accuracy of the method. Numerical results on antenna measurements are also presented. Due to the numerical complexity of the algorithm, it is limited to electrically small- or medium-size problems.

scholarly journals New ideas for handling of loop and angular integrals in D-dimensions in QCD

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Valery E. Lyubovitskij ◽  
Fabian Wunder ◽  
Alexey S. Zhevlakov
Keyword(s):  
Computer Algebra ◽  
Cross Sections ◽  
Orthogonal Basis ◽  
Computer Algebra Systems ◽  
Covariant Formalism ◽  
Linear Combinations ◽  
Recursion Relations ◽  
New Ideas ◽  
Rotational Invariant ◽  
Geometric Picture

Abstract We discuss new ideas for consideration of loop diagrams and angular integrals in D-dimensions in QCD. In case of loop diagrams, we propose the covariant formalism of expansion of tensorial loop integrals into the orthogonal basis of linear combinations of external momenta. It gives a very simple representation for the final results and is more convenient for calculations on computer algebra systems. In case of angular integrals we demonstrate how to simplify the integration of differential cross sections over polar angles. Also we derive the recursion relations, which allow to reduce all occurring angular integrals to a short set of basic scalar integrals. All order ε-expansion is given for all angular integrals with up to two denominators based on the expansion of the basic integrals and using recursion relations. A geometric picture for partial fractioning is developed which provides a new rotational invariant algorithm to reduce the number of denominators.

scholarly journals Linear Combinations of the Complex Degrees of Coherence

Photonics ◽  
2021 ◽  
Vol 8 (5) ◽  
pp. 146
Author(s):  
Zhangrong Mei ◽  
Olga Korotkova
Keyword(s):  
Spatial Coherence ◽  
Light Fields ◽  
Linear Combinations

We propose a method for structuring the spatial coherence state of light via mixed linear combinations of N complex degrees of coherence (CDC) and specify the conditions under which such combinations represent a valid CDC. Several examples demonstrate that this method opens previously unknown avenues for modeling random sources, radiating to light fields with unique features.

scholarly journals A Novel S-Box Dynamic Design Based on Nonlinear-Transform of 1D Chaotic Maps

Electronics ◽  
2021 ◽  
Vol 10 (11) ◽  
pp. 1313
Author(s):  
Wenhao Yan ◽  
Qun Ding
Keyword(s):  
Chaotic Systems ◽  
Chaotic Maps ◽  
Chaotic Map ◽  
Logistic Map ◽  
Sample Entropy ◽  
One Dimension ◽  
Linear Combinations ◽  
Dynamic Design ◽  
Construction Methods ◽  
Low Dimensional

In this paper, a method to enhance the dynamic characteristics of one-dimension (1D) chaotic maps is first presented. Linear combinations and nonlinear transform based on existing chaotic systems (LNECS) are introduced. Then, a numerical chaotic map (LCLS), based on Logistic map and Sine map, is given. Through the analysis of a bifurcation diagram, Lyapunov exponent (LE), and Sample entropy (SE), we can see that CLS has overcome the shortcomings of a low-dimensional chaotic system and can be used in the field of cryptology. In addition, the construction of eight functions is designed to obtain an S-box. Finally, five security criteria of the S-box are shown, which indicate the S-box based on the proposed in this paper has strong encryption characteristics. The research of this paper is helpful for the development of cryptography study such as dynamic construction methods based on chaotic systems.

scholarly journals Cache-Aided General Linear Function Retrieval

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 25
Author(s):  
Kai Wan ◽  
Hua Sun ◽  
Mingyue Ji ◽  
Daniela Tuninetti ◽  
Giuseppe Caire
Keyword(s):  
Linear Function ◽  
Linear Functions ◽  
General Linear ◽  
Linear Combinations ◽  
Past Work ◽  
Single File ◽  
Caching Scheme ◽  
Retrieval Scheme ◽  
Data Points ◽  
Coded Caching

Coded Caching, proposed by Maddah-Ali and Niesen (MAN), has the potential to reduce network traffic by pre-storing content in the users’ local memories when the network is underutilized and transmitting coded multicast messages that simultaneously benefit many users at once during peak-hour times. This paper considers the linear function retrieval version of the original coded caching setting, where users are interested in retrieving a number of linear combinations of the data points stored at the server, as opposed to a single file. This extends the scope of the authors’ past work that only considered the class of linear functions that operate element-wise over the files. On observing that the existing cache-aided scalar linear function retrieval scheme does not work in the proposed setting, this paper designs a novel coded caching scheme that outperforms uncoded caching schemes that either use unicast transmissions or let each user recover all files in the library.

A pairwise additivity scheme for conformational energies of substituted ethanes

1975 ◽  
Vol 343 (1632) ◽  
pp. 1-10 ◽  
Keyword(s):  
Large Deviations ◽  
Ab Initio ◽  
Dihedral Angle ◽  
Molecular Orbital ◽  
Methyl Groups ◽  
Molecular Orbital Calculations ◽  
Linear Combinations ◽  
Strongly Interacting ◽  
Conformational Energies

Ab initio molecular orbital calculations are used to explore additivity in the conformational energies of poly-substituted ethanes in terms of conformational energies of ethane and appropriate mono- and 1,2-di-substituted derivatives. Such relations would allow complex calculations for poly-substituted ethanes to be replaced by much simpler ones on a small number of parent molecules. General expressions for the linear combinations are derived from the assumption that interactions between vicinal substituents are pairwise additive and depend only on the vicinal dihedral angle. The additivity scheme is tested for 15 ethanes, di-, tri- or tetrasubstituted by cyano and methyl groups and for a smaller number of fluoroethanes. Additivity applies to within 0.1- 0.3 k J mol -1 in the methylethanes and mostly to within about 0.7- 0.8 kJ mol -1 in cyanoethanes. Large deviations are found among the geminally substituted fluoroethanes. It is suggested that the additivity approximation is most successful in the absence of strongly interacting geminal groups. Predictions are made of conformational energies of ten hexa(cyano- and methyl-) substituted ethanes.

Sign in / Sign up

Export Citation Format

Share Document