Article
Chemistry, Physical
Mamta Yadav, Yashwant Singh
Summary: We have developed a theory to trace the solvent degrees of freedom in colloid-solvent mixtures based on the grand partition function. Our approach uses density functional formalism and considers the solvent-induced interactions expressed in terms of two functionals. The theory provides insights into the nature of the potential and its dependence on thermodynamic state, packing fractions, and size ratio. We applied the theory to additive and nonadditive binary hard-sphere mixtures and observed interesting features such as non-monotonic dependence of the attractive part of the potential on the packing fraction.
JOURNAL OF MOLECULAR LIQUIDS
(2022)
Article
Physics, Condensed Matter
Gerhard Jung
Summary: Generalized Langevin equations (GLEs) can be systematically derived through dimensional reduction from high-dimensional microscopic systems. The derivation for linear models can be based on either projection operator techniques, such as the Mori-Zwanzig (MZ) formalism, or by integrating out the bath degrees of freedom. However, exact analytical results show that these two routes can lead to fundamentally different GLEs, with the differences originating from the non-equilibrium nature of the microscopic stochastic model. The most important conceptual difference between the two routes is that the MZ result inherently satisfies the generalized second fluctuation-dissipation theorem, while the integration result can violate it. Numerical and simulation results for time-delayed feedback control and the active Ornstein-Uhlenbeck process are provided to supplement the theoretical findings.
JOURNAL OF PHYSICS-CONDENSED MATTER
(2022)
Article
Mathematics
Arup Chattopadhyay, Guixiang Hong, Avijit Pal, Chandan Pradhan, Samya Kumar Ray
Summary: This paper investigates the existence of isometric embedding of S-q(m) into S-p(n), with conclusions regarding the value of q and extending classical results. It demonstrates specific conditions under which isometric embedding must have q=2. The study complements existing research on isometric and almost isometric embedding theory on non-commutative L-p-spaces.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Physics, Atomic, Molecular & Chemical
Zhen Zhang, Yao Kun Lei, Jun Zhang, Yi Qin Gao
Summary: In this study, a reasonable and powerful one-dimensional reaction coordinate that effectively describes the reaction progression is obtained using isometric mapping and locally linear embedding methods, while considering the nonlinearity of the transition state. The contribution of intrinsic molecular properties and solvent-solute interactions to the nonlinear reaction coordinate is analyzed to explore the reaction mechanism. Additionally, another coordinate is identified to characterize the heterogeneity of reaction mechanisms.
CHINESE JOURNAL OF CHEMICAL PHYSICS
(2022)
Article
Mathematics
Xiaohuan Wang, Jihui Zhang
Summary: This paper considers the existence and regularity of positive solutions to a degenerate fourth order elliptic problem. Firstly, a new Caffarelli-Kohn-Nirenberg type inequality for the fourth order case is established. Then, by using the corresponding embedding, the existence of positive solutions to the degenerate fourth order elliptic problem is obtained. Finally, the regularity of the positive solutions is also studied.
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS
(2022)
Article
Mathematics
Lijia Ding, Kai Wang
Summary: In this article, we investigate Bergman type operators on the complex unit ball and explore their properties, including boundedness, compactness, and norm estimates. The results provide new insights into the Hardy-Littlewood-Sobolev type theorem on the unit ball and give upper bounds for the optimal constants.
TAIWANESE JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics, Applied
Arup Chattopadhyay, Guixiang Hong, Chandan Pradhan, Samya Kumar Ray
Summary: In this article, the study of isometric embeddability of S-q(m) into S-p(n) is extended beyond the previous range of p and q. It is shown that there is no isometric embedding of l(q)(m) into l(p)(n), and also no isometric embedding of S-q(m) into S-p(n) under certain conditions. The non-commutative Clarkson's inequality, Kato-Rellich theorem, and multiple operator integrals are utilized in the analysis.
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
(2023)
Proceedings Paper
Computer Science, Software Engineering
Jerome Feret, Albin Salazar
Summary: In recent decades, logical or discrete models have become a successful paradigm in capturing and predicting the behaviors of molecular interaction systems. This paper presents a sound coarse-graining approach for stochastic reaction networks, which introduces overlapped intervals to reduce fictitious oscillations in the coarse-grained models and computes bounds for the probabilities of transition pairs in the coarse-grained model. The approach provides a framework for designing more precise discretization methods and investigating the underlying structure of logical and discrete models.
VERIFICATION, MODEL CHECKING, AND ABSTRACT INTERPRETATION, VMCAI 2023
(2023)
Article
Mathematics
Michiya Mori, Peter Semrl
Summary: Loewner's theorem states that operator monotone functions on real intervals can be described by holomorphic functions on the upper half-plane. We characterize local order isomorphisms on operator domains using biholomorphic automorphisms of the generalized upper half-plane. We provide an explicit description of such maps and examine the properties of maximal local order isomorphisms. Furthermore, in the finite-dimensional case, we prove that every order embedding of a matrix domain is a homeomorphic order isomorphism onto another matrix domain.
CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES
(2023)
Article
Mathematics, Applied
Hitoshi Tanaka
Summary: In the open range 1 < p < q < infinity, under a certain geometrical condition on weights, a two-weight norm inequality for the product fractional integral operator with singular kernel on each coordinate subspace is shown to be derived from the Fefferman-Phong type characteristic for product cubes. This geometrical condition is revealed to be the testing condition of Carleson-type embedding for product dyadic cubes.
BULLETIN DES SCIENCES MATHEMATIQUES
(2021)
Article
Mathematics, Applied
Gustavo Hoepfner, Paulo Liboni, Dorina Mitrea, Irina Mitrea, Marius Mitrea
Summary: This study explores the theory of multilayer potential operators associated with any given homogeneous constant-coefficient higher-order elliptic system L in an open set Omega satisfying geometric measure theoretic assumptions. The results indicate that singular integral operators are bounded on Lebesgue spaces in the case of a uniformly rectifiable domain Omega, which is significant for boundary value problems of the higher-order system L in such a domain.
Article
Mathematics, Applied
Stefan Klus, Feliks Nuske, Sebastian Peitz, Jan-Hendrik Niemann, Cecilia Clementi, Christof Schuette
PHYSICA D-NONLINEAR PHENOMENA
(2020)
Article
Physics, Multidisciplinary
Stefan Klus, Feliks Nueske, Boumediene Hamzi
Article
Physics, Multidisciplinary
Feliks Nueske, Peter Koltai, Lorenzo Boninsegna, Cecilia Clementi
Summary: This paper investigates the reduction of high-dimensional systems to effective models through the conditioning approach, comparing the spectrum of the generator of the resulting effective dynamics to the full generator. A new relative error bound for reversible systems is proven in terms of eigenfunction approximation error. Numerical examples suggest that Kramers-Moyal type approximations for computing the spectrum of the reduced generator are largely insensitive to the time window used for the estimators, with implications for underdamped Langevin dynamics.
Article
Mathematics, Applied
Andreas Bittracher, Stefan Klus, Boumediene Hamzi, Peter Koltai, Christof Schuette
Summary: In this article, a novel kernel-based machine learning algorithm is presented for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. By embedding and learning the transition manifold in a reproducing kernel Hilbert space, the approach is enhanced and shown to preserve the manifold structure under the embedding with distortion bounds. This results in a more robust and efficient algorithm compared to previous parameterization approaches.
JOURNAL OF NONLINEAR SCIENCE
(2021)
Article
Meteorology & Atmospheric Sciences
Antonio Navarra, Joe Tribbia, Stefan Klus
Summary: Ensemble methods have become popular tools in atmospheric, climate, and ocean dynamics studies for obtaining statistical information on systems and approximating probability distributions. This paper introduces a method to directly estimate probability distribution evolution and create predictor systems based on nonlinear formulations, showcasing its application in two examples. The objective is to demonstrate the utility of these methods in complex, multidimensional atmosphere and ocean environments.
JOURNAL OF THE ATMOSPHERIC SCIENCES
(2021)
Article
Multidisciplinary Sciences
Jan-Hendrik Niemann, Stefan Klus, Christof Schuette
Summary: This paper demonstrates how Koopman operator theory can be used to derive reduced models of agent-based systems using only simulation data. The goal is to learn coarse-grained models and represent the reduced dynamics using ordinary or stochastic differential equations. Experimental results show that the obtained reduced systems are in good agreement with analytical results when the number of agents is sufficiently large.
Article
Computer Science, Artificial Intelligence
Patrick Gelss, Stefan Klus, Ingmar Schuster, Christof Schuette
Summary: The proposed method aims to approximate high- or even infinite-dimensional feature vectors to reduce training data size, storage consumption, and computational complexity, while enhancing the generalizability of learned target functions. Significant improvements have been demonstrated in classification and regression problems across various application areas.
KNOWLEDGE-BASED SYSTEMS
(2021)
Article
Mathematics, Applied
Feliks Nueske, Patrick Gelss, Stefan Klus, Cecilia Clementi
Summary: Recent years have witnessed significant progress in data-driven analysis of dynamical systems based on Koopman operator theory and the TT format for solving large-scale problems. Combining these two approaches enables handling high-dimensional problems with rich basis functions or feature sets, and deriving efficient algorithms for reducing the system's evolution operator representation.
PHYSICA D-NONLINEAR PHENOMENA
(2021)
Article
Physics, Multidisciplinary
Stefan Klus, Feliks Nueske, Sebastian Peitz
Summary: Koopman operator theory has wide applications in various research areas and this paper demonstrates the use of data-driven methods to analyze quantum physics problems and solve the Schrödinger equation, opening up a new avenue for research.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Mathematics, Applied
Stefan Klus, Natasa Djurdjevac Conrad
Summary: This paper discusses the challenges of clustering directed graphs and proposes a clustering algorithm based on the concept of metastable sets.
JOURNAL OF NONLINEAR SCIENCE
(2023)
Article
Mathematics
Christof Schuette, Stefan Klus, Carsten Hartmann
Summary: This article discusses how to overcome the 'timescale barrier' in molecular dynamics through transfer operator-based techniques. It focuses on the approximation of dynamical behavior on long timescales, the theory and algorithmic development of transfer operators, and their relation to rare event simulation techniques. The article takes a mathematical perspective and refers to existing research articles for applications in real-world molecular systems.
Article
Chemistry, Physical
Feliks Nueske, Stefan Klus
Summary: The use of random Fourier features as a stochastic approximation method allows for more efficient estimation of slow kinetic processes.
JOURNAL OF CHEMICAL PHYSICS
(2023)
Article
Computer Science, Artificial Intelligence
Moritz Hoffmann, Martin Scherer, Tim Hempel, Andreas Mardt, Brian de Silva, Brooke E. Husic, Stefan Klus, Hao Wu, Nathan Kutz, Steven L. Brunton, Frank Noe
Summary: Deeptime is a Python library that offers various tools for estimating dynamical models based on time-series data and provides analysis methods to compute important properties of the system. It is easy to use and extend, and compatible with scikit-learn.
MACHINE LEARNING-SCIENCE AND TECHNOLOGY
(2022)
Article
Computer Science, Artificial Intelligence
Stefan Klus, Patrick Gelss, Feliks Nueske, Frank Noe
Summary: This paper explores symmetric and antisymmetric kernels by analyzing their properties and computing the feature space dimensions of resulting polynomial kernels. It proves that the reproducing kernel Hilbert spaces induced by symmetric and antisymmetric Gaussian kernels are dense in the space of symmetric and antisymmetric functions. Additionally, a Slater determinant representation of the antisymmetric Gaussian kernel is proposed for efficient evaluation in high-dimensional state spaces, and shows that the size of the training data set can be significantly reduced by exploiting symmetries or antisymmetries.
MACHINE LEARNING-SCIENCE AND TECHNOLOGY
(2021)
Article
Computer Science, Theory & Methods
Kateryna Melnyk, Stefan Klus, Gregoire Montavon, Tim O. F. Conrad
APPLIED NETWORK SCIENCE
(2020)