On sine series with positive coefficients

1962 ◽  
Vol 80 (1) ◽  
pp. 382-383
Author(s):  
R. P. Boas
Keyword(s):  
Sine Series ◽  
Positive Coefficients

scholarly journals Removing the Correlation Term in Option Pricing Heston Model: Numerical Analysis and Computing

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
R. Company ◽  
L. Jódar ◽  
M. Fakharany ◽  
M.-C. Casabán
Keyword(s):  
Option Pricing ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Diffusion Reaction ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Volatility Model ◽  
Classical Technique ◽  
Domain Consistency ◽  
Positive Coefficients

This paper deals with the numerical solution of option pricing stochastic volatility model described by a time-dependent, two-dimensional convection-diffusion reaction equation. Firstly, the mixed spatial derivative of the partial differential equation (PDE) is removed by means of the classical technique for reduction of second-order linear partial differential equations to canonical form. An explicit difference scheme with positive coefficients and only five-point computational stencil is constructed. The boundary conditions are adapted to the boundaries of the rhomboid transformed numerical domain. Consistency of the scheme with the PDE is shown and stepsize discretization conditions in order to guarantee stability are established. Illustrative numerical examples are included.

A federated interpretable scorecard and its application in credit scoring

2021 ◽  
pp. 2142009
Author(s):  
Fanglan Zheng ◽  
Erihe ◽  
Kun Li ◽  
Jiang Tian ◽  
Xiaojia Xiang
Keyword(s):  
Privacy Protection ◽  
Data Privacy ◽  
Credit Scoring ◽  
Training Session ◽  
Model Performance ◽  
Parameter Tuning ◽  
Kolmogorov Smirnov ◽  
Data Privacy Protection ◽  
Tuning Process ◽  
Positive Coefficients

In this paper, we propose a vertical federated learning (VFL) structure for logistic regression with bounded constraint for the traditional scorecard, namely FL-LRBC. Under the premise of data privacy protection, FL-LRBC enables multiple agencies to jointly obtain an optimized scorecard model in a single training session. It leads to the formation of scorecard model with positive coefficients to guarantee its desirable characteristics (e.g., interpretability and robustness), while the time-consuming parameter-tuning process can be avoided. Moreover, model performance in terms of both AUC and the Kolmogorov–Smirnov (KS) statistics is significantly improved by FL-LRBC, due to the feature enrichment in our algorithm architecture. Currently, FL-LRBC has already been applied to credit business in a China nation-wide financial holdings group.

scholarly journals Fractal fits to Riemann zeros

2007 ◽  
Vol 85 (4) ◽  
pp. 345-357 ◽  
Author(s):  
P B Slater
Keyword(s):  
Potential Model ◽  
Sine Series ◽  
Dominant Term ◽  
One Dimensional ◽  
Smooth Potential ◽  
Smooth Part ◽  
Least Squares Fit ◽  
The One ◽  
Higher Order Corrections ◽  
03.65 Sq

Wu and Sprung (Phys. Rev. E, 48, 2595 (1993)) reproduced the first 500 nontrivial Riemann zeros, using a one-dimensional local potential model. They concluded — as did van Zyl and Hutchinson (Phys. Rev. E, 67, 066211 (2003)) — that the potential possesses a fractal structure of dimension d = 3/2. We model the nonsmooth fluctuating part of the potential by the alternating-sign sine series fractal of Berry and Lewis A(x,γ). Setting d = 3/2, we estimate the frequency parameter (γ), plus an overall scaling parameter (σ) that we introduce. We search for that pair of parameters (γ,σ) that minimizes the least-squares fit Sn(γ,σ) of the lowest n eigenvalues — obtained by solving the one-dimensional stationary (nonfractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) — to the lowest n Riemann zeros for n = 25. For the additional cases, we study, n = 50 and 75, we simply set σ = 1. The fits obtained are compared to those found by using just the smooth part of the Wu–Sprung potential without any fractal supplementation. Some limited improvement — 5.7261 versus 6.392 07 (n = 25), 11.2672 versus 11.7002 (n = 50), and 16.3119 versus 16.6809 (n = 75) — is found in our (nonoptimized, computationally bound) search procedures. The improvements are relatively strong in the vicinities of γ = 3 and (its square) 9. Further, we extend the Wu-Sprung semiclassical framework to include higher order corrections from the Riemann–von Mangoldt formula (beyond the leading, dominant term) into the smooth potential. PACS Nos.: 02.10.De, 03.65.Sq, 05.45.Df, 05.45.Mt

A Solution Method for an Optimal Controlled Vibrating Circle Shell by Measure and Classical Trajectory

2011 ◽  
Vol 52-54 ◽  
pp. 1855-1860
Author(s):  
J.A. Fakharzadeh ◽  
F.N. Jafarpoor
Keyword(s):  
Wave Equations ◽  
Classical Trajectory ◽  
Polar Coordinates ◽  
Approximation Schemes ◽  
Solution Technique ◽  
Underlying Space ◽  
The Mean ◽  
Positive Coefficients ◽  
And Control ◽  
Finite Linear

The mean idea of this paper is to present a new combinatorial solution technique for the controlled vibrating circle shell systems. Based on the classical results of the wave equations on circle domains, the trajectory is considered as a finite trigonometric series with unknown coefficients in polar coordinates. Then, the problem is transferred to one in which its unknowns are a positive Radon measure and some positive coefficients. Extending the underlying space helps us to prove the existence of the solution. By using the density properties and some approximation schemes, the problem is deformed into a finite linear programming and the nearly optimal trajectory and control are identified simultaneously. A numerical example is also given.

scholarly journals On the uniform convergence of double sine series

2009 ◽  
Vol 193 (1) ◽  
pp. 79-97 ◽  
Author(s):  
Péter Kórus ◽  
Ferenc Móricz
Keyword(s):  
Uniform Convergence ◽  
Sine Series
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