On the peripheral spectrum of monic operator polynomials with positive coefficients

1992 ◽  
Vol 15 (3) ◽  
pp. 479-495 ◽  
Author(s):  
R. T. Rau
Keyword(s):  
Peripheral Spectrum ◽  
Monic Operator ◽  
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.

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 Formulae and Asymptotics for Coefficients of Algebraic Functions

2014 ◽  
Vol 24 (1) ◽  
pp. 1-53 ◽  
Author(s):  
CYRIL BANDERIER ◽  
MICHAEL DRMOTA
Keyword(s):  
Generating Functions ◽  
Entire Functions ◽  
Limit Laws ◽  
Context Free Grammar ◽  
Algebraic Functions ◽  
Strongly Connected ◽  
Non Gaussian ◽  
Positive Coefficients ◽  
Dyadic Numbers ◽  
Context Free

We study the coefficients of algebraic functions ∑n≥0fnzn. First, we recall the too-little-known fact that these coefficientsfnalways admit a closed form. Then we study their asymptotics, known to be of the typefn~CAnnα. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the critical exponents α cannot be 1/3 or −5/2; they in fact belong to a proper subset of the dyadic numbers. We initiate the study of the set of possible values forA. We extend what Philippe Flajolet called the Drmota–Lalley–Woods theorem (which states that α=−3/2 when the dependency graph associated to the algebraic system defining the function is strongly connected). We fully characterize the possible singular behaviours in the non-strongly connected case. As a corollary, the generating functions of certain lattice paths and planar maps are not determined by a context-free grammar (i.e., their generating functions are not ℕ-algebraic). We give examples of Gaussian limit laws (beyond the case of the Drmota–Lalley–Woods theorem), and examples of non-Gaussian limit laws. We then extend our work to systems involving non-polynomial entire functions (non-strongly connected systems, fixed points of entire functions with positive coefficients). We give several closure properties for ℕ-algebraic functions. We end by discussing a few extensions of our results (infinite systems of equations, algorithmic aspects).

Does symmetry Imply PPT Property?

2015 ◽  
Vol 15 (9&10) ◽  
pp. 812-824
Author(s):  
Daniel Cariello
Keyword(s):  
Density Matrix ◽  
Tensor Rank ◽  
Natural Question ◽  
Density Matrices ◽  
Partial Transposition ◽  
Class Of Matrices ◽  
Positive Coefficients

Recently it was proved that many results that are true for density matrices which are positive under partial transposition (or simply PPT), also hold for another class of matrices with a certain symmetry in their Hermitian Schmidt decompositions. These matrices were called symmetric with positive coefficients (or simply SPC). A natural question appeared: What is the connection between SPC matrices and PPT matrices? Is every SPC matrix PPT? Here we show that every SPC matrix is PPT in $M_2\otimes M_2$. This theorem is a consequence of the fact that every density matrix in $M_2\otimes M_m$, with tensor rank smaller or equal to 3, is separable. Although, in $M_3\otimes M_3$, we present an example of SPC matrix with tensor rank 3 that is not PPT. We shall also provide a non trivial example of a family of matrices in $M_k\otimes M_k$, in which both, the SPC and PPT properties, are equivalent. Within this family, there exists a non trivial subfamily in which the SPC property is equivalent to separability.

Asymptotic analysis and local lattice central limit theorems

1979 ◽  
Vol 16 (03) ◽  
pp. 541-553 ◽  
Author(s):  
P.A.P. Moran
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Stirling Numbers ◽  
Central Limit Theorems ◽  
Discrete Distributions ◽  
Asymptotic Formulae ◽  
Local Lattice ◽  
Exact Bounds ◽  
Direct Numerical Integration ◽  
Positive Coefficients

Methods of evaluating the coefficients of high powers of functions defined by power series with positive coefficients are considered. Such methods, which were originally used by Laplace, can, for example, be used to obtain asymptotic formulae for Stirling numbers. They are equivalent to using local lattice central limit theorems. An alternative method using direct numerical integration on a contour integral giving the required coefficient is described. Exact bounds for the accuracy of this method can often be obtained by considerations of the unimodality of discrete distributions. The results are illustrated using convolutions of the rectangular and logarithmic distributions.

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