Article
Mathematics
Kaizhi Wang, Lin Wang, Jun Yan
Summary: The paper provides necessary and sufficient conditions for the existence of viscosity solutions of nonlinear first order PDEs, proving compactness of the set of solutions. Furthermore, it explores the long-term behavior of viscosity solutions for Cauchy problems using weak KAM theory and dynamic methods.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Abed Bounemoura, Jacques Fejoz
Summary: Some scales of spaces of ultra-differentiable functions are introduced, characterized by a real sequence M bounding the growth of derivatives. Fundamental results of Hamiltonian perturbation theory are proven, including the invariant torus theorem and Nekhoroshev's theorem. The competition between stability properties of nearly integrable systems and the distance to integrability is illuminated by formulas relating the growth M of derivatives to arithmetic conditions and stability time.
MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Applied
Jean Bernard Lasserre, Victor Magron, Swann Marx, Olivier Zahm
Summary: This paper discusses a method for minimizing a sum of rational functions over a compact set of high dimension, utilizing the pushforward measure and semidefinite programming. It also points out the potential of exploring related problems regarding Lebesgue or Haar measure integrals.
SIAM JOURNAL ON OPTIMIZATION
(2021)
Article
Chemistry, Physical
Vibin Abraham, Nicholas J. Mayhall
Summary: Size extensivity is an important property for many-body methods, but traditional configuration interaction (CI) methods lack this property. Coupled electron pair approximation (CEPA) methods can ensure size extensivity, but they face singularity issues. In this study, we extend the CEPA methods to a new formulation based on tensor product states (TPS) and demonstrate their application in various systems. The results show that the TPS-CEPA method can eliminate singularities and provide improved numerical results.
JOURNAL OF CHEMICAL THEORY AND COMPUTATION
(2022)
Article
Automation & Control Systems
Karl Kunisch, Donato Vasquez-Varas, Daniel Walter
Summary: A learning-based method is proposed to obtain feedback laws for nonlinear optimal control problems. The method approximates the infinite dimensional problem using a polynomial ansatz and employs a penalty term combined with the proximal point method to find sparse solutions. The proposed methodology provides a promising approach for mitigating the curse of dimensionality.
JOURNAL OF MACHINE LEARNING RESEARCH
(2023)
Article
Acoustics
D. Brizard, E. Jacquelin
Summary: This paper examines two polynomial approximations for calculating the characteristic equation of wave dispersion in solid or hollow circular cylinders. The accuracy and programming difficulty of these approximations are compared. The results show that both approximations can accurately compute dispersion curves for solid cylinders with a sufficient order. However, for hollow cylinders, the accuracy of dispersion curves is limited due to the numerical limits of thinner cylinders. It is found that Widehammar's approximation is more accurate and easier to program compared to the Jacobi mode approximation.
JOURNAL OF SOUND AND VIBRATION
(2022)
Article
Mathematics, Applied
Panrui Ni, Kaizhi Wang, Jun Yan
Summary: This paper studies Hamilton Jacobi equations satisfying certain conditions in smooth manifolds. The author finds a compact interval C such that solutions to the equation exist if and only if c belongs to C. In addition, the author also investigates the long-time behavior of the unique viscosity solution and obtains some results regarding the properties of the solution.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2022)
Article
Mathematics, Applied
Ran Ran Zhang, Chuang Xin Chen, Zhi Bo Huang
Summary: This study investigates the uniqueness results of meromorphic function f(z) and linear difference polynomial L(z, f) sharing small functions, advancing existing research on Bruck conjecture in both differential and difference cases. Some sufficient conditions are also presented to show that f(z) and L(z, f) cannot share certain small functions.
Article
Mathematics, Applied
Yusuke Okuyama
Summary: A geometric description of the parabolic bifurcation locus in the space Rat(d) of rational functions on P-1 of degree d > 1 is provided, generalizing the study by Morton and Vivaldi in the case of monic polynomials. The results are new even for quadratic rational functions.
Article
Mathematics
Vladimir A. Kulyukin
Summary: We prove that for a primitive recursive function h(x, t), there exists a feedforward artificial neural network N(x, t) that can generate a sequence corresponding to a given n-tuple z and a positive natural number m.
Article
Astronomy & Astrophysics
Carl M. Bender, C. Karapoulitidis, S. P. Klevansky
Summary: The Dyson-Schwinger (DS) equations for a quantum field theory are discussed, which are satisfied exactly by the Green's functions. The problem of underdetermination in the finite sequence of DS equations is addressed by setting the highest Green's function(s) to zero. The accuracy of this approach is examined in the special case of D = 0, where the DS equations reduce to coupled polynomial equations. The study finds that the roots of the polynomial approximants converge to limits that differ slightly from the exact answers. Sophisticated asymptotic techniques are developed to increase the accuracy.
Article
Statistics & Probability
Luis A. Gil-Alana, OlaOluwa S. Yaya
Summary: This paper introduces a testing procedure for fractional orders of integration in the context of non-linear terms approximated by Fourier functions. The test statistic shows asymptotic standard normal distribution and performs well in finite samples as demonstrated by Monte Carlo experiments. Various applications using real life time series data are presented, such as US unemployment rates, US GNP, and Purchasing Power Parity (PPP) of G7 countries.
JOURNAL OF APPLIED STATISTICS
(2021)
Article
Mathematics, Applied
Matvey Smirnov
Summary: This paper proves that a holomorphic function on a neighborhood of a compact convex set in Cn can be uniformly approximated by polynomials, with an error that decreases exponentially fast with the growth of the polynomial degree. The method presented in this paper is based on the vanishing of certain cohomology groups and involves Cech cohomology. It is more elementary compared to previous methods, but it does not provide effective estimates.
COMPLEX ANALYSIS AND OPERATOR THEORY
(2023)
Article
Computer Science, Artificial Intelligence
Mihai Prunescu
Summary: The study investigates the existence of rational-valued approximation processes by continuous functions of two variables, proving the aleph(0)-categoricity of the theory of densely ordered sets with generic predicates. A particular model and continuous choice-function are constructed, with computable processes under certain common-sense conditions, and the potential for constructing other functions with surprising properties as a by-product.
Article
Business
E. A. de Groot, R. Segers, D. Prins
Summary: This paper builds on Schumpeter's theory and uses a mathematical model and empirical research to examine the relationship between the stability of subcycles in the economy and their lengths. The study concludes that for subcycles to remain stable, their lengths should not be close multiples, and the cycles should be non-resonant. This finding is supported by recent empirical evidence showing that the ratios between subcycle lengths in GDP align with the golden ratio.
TECHNOLOGICAL FORECASTING AND SOCIAL CHANGE
(2022)