Dynamical Field Inference and Supersymmetry

Knowledge on evolving physical fields is of paramount importance in science, technology, and economics. Dynamical field inference (DFI) addresses the problem of reconstructing a stochastically-driven, dynamically-evolving field from finite data. It relies on information field theory (IFT), the information theory for fields. Here, the relations of DFI, IFT, and the recently developed supersymmetric theory of stochastics (STS) are established in a pedagogical discussion. In IFT, field expectation values can be calculated from the partition function of the full space-time inference problem. The partition function of the inference problem invokes a functional Dirac function to guarantee the dynamics, as well as a field-dependent functional determinant, to establish proper normalization, both impeding the necessary evaluation of the path integral over all field configurations. STS replaces these problematic expressions via the introduction of fermionic ghost and bosonic Lagrange fields, respectively. The action of these fields has a supersymmetry, which means there exists an exchange operation between bosons and fermions that leaves the system invariant. In contrast to this, measurements of the dynamical fields do not adhere to this supersymmetry. The supersymmetry can also be broken spontaneously, in which case the system evolves chaotically. This affects the predictability of the system and thereby makes DFI more challenging. We investigate the interplay of measurement constraints with the non-linear chaotic dynamics of a simplified, illustrative system with the help of Feynman diagrams and show that the Fermionic corrections are essential to obtain the correct posterior statistics over system trajectories.

A New Application of the $${\bar{\partial }}$$-Method

By constructing a new calculating rule of Lie bracket, we construct a new nonlinear Schrödinger hierarchy and its reduction equations via using the $${\bar{\partial }}$$ ∂ ¯ -method. Furthermore, some soliton solutions of such the equation are obtained by making use of Dirac function.

Calculation of the Fermi–Dirac Function Distribution in Two-Dimensional Semiconductor Materials at High Temperatures and Weak Magnetic Fields

This article investigated the effects of a quantizing magnetic field and temperature on Fermi energy oscillations in nanoscale semiconductor materials. It is shown that the Fermi energy of a nanoscale semiconductor material in a quantizing magnetic field is quantized. The distribution of the Fermi–Dirac function is calculated in low-dimensional semiconductors at weak magnetic fields and high temperatures. The proposed theory explains the experimental results in two-dimensional semiconductor structures with a parabolic dispersion law.

A Phenomenological Analysis of COVID-19

A phenomenological analysis of the time evolution of some COVID-19 data in terms of a Fermi-Dirac function is presented. In spite of its simplicity, the approach appears to describe the data well and allows to correlate the information in a universal plot in terms of non-dimensional or reduced variables Nr = N(t)/Nmax, and tr = t/ΔT, with N(t) being the total number of cases as a function of time, Nmax the number of total infected cases, and ΔT the diffuseness of the Fermi/Dirac function associated with the rate of infection. The analysis of the reported data for the first outbreak in some selected countries and the results are presented and discussed. The approach is also applicable to subsequent waves. Support of our framework is provided by the SIS limit of the SIR model, and simulations carried out with the SEICRD extension.

Optimal Execution considering Trading Signal and Execution Risk Simultaneously

In this paper, we study the optimal execution problem by considering the trading signal and the transaction risk simultaneously. We propose an optimal execution problem by taking into account the trading signal and the execution risk with the associated decay kernel function and the transient price impact function being of generalized forms. In particular, we solve the stochastic optimal control problems under the assumptions that the decay kernel function is the Dirac function and the transient price function is a linear function. We give the optimal executing strategies in state-feedback form and the Hamilton‐Jacobi‐Bellman equations that the corresponding value functions satisfy in the cases of a constant execution risk and a linear execution risk. We also demonstrate that our results can recover previous results when the process of the trading signal degenerates.

From the Fermi–Dirac distribution to PD curves

Purpose In machine learning applications, and in credit risk modeling in particular, model performance is usually measured by using cumulative accuracy profile (CAP) and receiving operating characteristic curves. The purpose of this paper is to use the statistics of the CAP curve to provide a new method for credit PD curves calibration that are not based on arbitrary choices as the ones that are used in the industry. Design/methodology/approach The author maps CAP curves to a ball–box problem and uses statistical physics techniques to compute the statistics of the CAP curve from which the author derives the shape of PD curves. Findings This approach leads to a new type of shape for PD curves that have not been considered in the literature yet, namely, the Fermi–Dirac function which is a two-parameter function depending on the target default rate of the portfolio and the target accuracy ratio of the scoring model. The author shows that this type of PD curve shape is likely to outperform the logistic PD curve that practitioners often use. Practical implications This paper has some practical implications for practitioners in banks. The author shows that the logistic function which is widely used, in particular in the field of retail banking, should be replaced by the Fermi–Dirac function. This has an impact on pricing, the granting policy and risk management. Social implications Measuring credit risk accurately benefits the bank of course and the customers as well. Indeed, granting is based on a fair evaluation of risk, and pricing is done accordingly. Additionally, it provides better tools to supervisors to assess the risk of the bank and the financial system as a whole through the stress testing exercises. Originality/value The author suggests that practitioners should stop using logistic PD curves and should adopt the Fermi–Dirac function to improve the accuracy of their credit risk measurement.

On the Adaptive Quadrature of Fermi-Dirac Functions and their Derivatives

In this paper, using the Python SciPy module “quad”, a fast auto-adaptive quadrature solver based on the pre-compiled QUADPACK Fortran package, computational research is undertaken to accurately integrate the generalised Fermi-Dirac function and all its partial derivatives up to the third order. The numerical results obtained with quad method when combined with optimised break points achieve an excellent accuracy comparable to that obtained by other publications using fixed-order quadratures.

The band theory of solids

The solution of Schrodinger’s equation is discussed for a model in which atoms are represented by potential wells, from which the band structure follows. Three further models are discussed, the Ziman model (which is based on the effect of Bragg reflection upon the wave functions), and the Feynman model (based on coupled equations), and the tight binding model (based on a more realistic solution of the Schrödinger equation). The concept of effective mass is introduced, followed by the effective number of electrons. The difference between metals and insulators based on their band structure is discussed. The concept of holes is introduced. The band structure of divalent metals is explained. For finite temperatures the Fermi–Dirac function is combined with band theory whence the distinction between insulators and semiconductors is derived.

Upon Element-Free Galerkin Method and its Using in Static and Dynamic Structure Analysis

This paper presents, in a synthetically way, the fundamentals of the element-free Galerkin (EFG) method - a meshfree method - under development but with many capabilities for solving complex problems in mechanical engineering, like impact problems etc. For interpolation, the EFG method uses moving least-squares (MLS) interpolants in curve and surface fitting. Unlike other interpolants, the MLS interpolants do not pass through the data because the Dirac function properties are not available. This aspect could be a disadvantage of the EFG method but next to it, there are many advantages. Upon these issues a discussion exists in this paper. Finally, some applications of the EFG method are presented referring to static and dynamic analysis of structures. The examples and conclusions can be useful for knowing and using of the EFG method