Efficient Multi-Configurational Wavefunction Method with Dynamical Correlation Using Non-Orthogonal Configuration Interaction Singles and Doubles (NOCISD)

**equations**defining the NOCISD wavefunction commonly require the solution a poorly condition generalized eigenvalue problem, which we avoid by projecting the

**equation**s to a small dimension space defined by the CISD eigenvectors of each reference determinant. Expand abstract.

**equations**defining the NOCISD wavefunction commonly require the solution a poorly condition generalized eigenvalue problem, which we avoid by projecting the

**equations**to a small dimension space defined by the CISD eigenvectors of each reference determinant. We show that NOCISD results are in good qualitative agreement with other state-of-the-art method for challenging problems such as the electron transfer in the ethylene dimer radical cation and LiF, as well as the description of the Jahn-Teller distortion in the cyclopentadienyl and nitrogen trioxide radicals.

4/10 relevant

chemRxiv

Effective Hamiltonians Derived from **Equation**-of-Motion Coupled-Cluster Wave-Functions: Theory and Application to the Hubbard and Heisenberg Hamiltonians

**equations**and illustrate the theory by examples of systems with electronic states of covalent and ionic characters. Expand abstract.

**equation**-of-motion coupled-cluster (EOM-CC) wave functions to rigorously derive effective Hamiltonians in the Bloch’s and des Cloizeaux’s forms. We report the key

**equations**and illustrate the theory by examples of systems with electronic states of covalent and ionic characters. We show that the Hubbard and Heisenberg Hamiltonians are extracted directly from the so-obtained effective Hamiltonians. By making quantitative connections between many-body states and simple models, the approach also facilitates the analysis of the correlated wave functions. Artifacts affecting the quality of electronic structure calculations such as spin contamination are also discussed.

6/10 relevant

chemRxiv

Stability Analysis of Delayed Age-structured Resource Consumer Model of Population Dynamics with Saturated Intake Rate

**equation**with delay, resource dynamics are described by ordinary differential

**equations**. Expand abstract.

**equation**with delay, resource dynamics are described by ordinary differential

**equations**. The delay models the lagged effect of food intake on consumer fertility and mortality. The model is studied by combination of analytical and numerical methods. Using an explicit recurrent algorithm, the existence and uniqueness of a traveling wave for consumer population is proved under the assumptions that guarantee continuity and smoothness of solutions. Two foraging regimes are considered. The first assumes fixed interaction strength between consumers and resources. The second assumes adaptive interaction strength where consumers forage as to maximize their fitness measured by the reproductive number. Sufficient conditions for existence of solutions for the corresponding optimal control are given. Conditions for existence of trivial, semi-trivial and non-trivial equilibria and their local asymptotic stability are obtained. Numerical experiments illustrate these theoretical results.

4/10 relevant

bioRxiv

Residual Compressive Strength of Short Tubular Steel Columns with Local Corrosion Damage

**equation**was found to be within 11.4%. Expand abstract.

**equation**is presented to evaluate the residual compressive strength of short columns with local corrosion wherein the volume of the corrosion damage was used as a reduction factor in calculating the compressive strength. The percentage error using the presented

**equation**was found to be within 11.4%.

4/10 relevant

Preprints.org

SelectNet: Self-paced Learning for High-dimensional Partial Differential
**Equations**

**equations**(PDEs) successfully; however, its convergence is slow and might not be guaranteed even within a simple class of PDEs. Expand abstract.

**equations**(PDEs) successfully; however, its convergence is slow and might not be guaranteed even within a simple class of PDEs. To improve the convergence of the network-based residual model, we introduce a novel self-paced learning framework, SelectNet, which quantifies the difficulty of training samples, chooses simpler samples in the early stage of training, and slowly explores more challenging samples, e.g., samples with larger residual errors, mimicking the human cognitive process for more efficient learning. In particular, a selection network and the PDE solution network are trained simultaneously; the selection network adaptively weighting the training samples of the solution network achieving the goal of self-paced learning. Numerical examples indicate that the proposed SelectNet model outperforms existing models on the convergence speed and the convergence robustness, especially for low-regularity solutions.

6/10 relevant

arXiv

Primary ideals and their differential **equations**

**equations**with constant coefficients. Expand abstract.

**equations**with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in this context. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, and the join construction, and we present an explicit algorithm for computing Noetherian operators.

9/10 relevant

arXiv

On a class of Lebesgue-Ljunggren-Nagell type **equations**

**equation**$ax^2+b^{2l}=4y^n$, $x, y\in\Bbb Z$, $l \in \Bbb N$, $n$ odd prime, $\gcd(x,y)=1$. We completely solve the above Diophantine

**equation**for $a\in\{7,11,19,43,67,163\}$, and $b$ a power of an odd prime, under the conditions $2^{n-1}b^l\not\equiv \pm 1(\mod a)$ and $\gcd(n,b)=1$. For other square-free integers $a>3$ and $b$ a power of an odd prime, we prove that the above Diophantine

**equation**has no solutions for all integers $x$, $y$ with ($\gcd(x,y)=1$), $l\in\mathbb{N}$ and all odd primes $n>3$, satisfying $2^{n-1}b^l\not\equiv \pm 1(\mod a)$, $\gcd(n,b)=1$, and $\gcd(n,h(-a))=1$, where $h(-a)$ denotes the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-a})$.

6/10 relevant

arXiv

New mathematical modelling tools for co-culture experiments: when do we need to explicitly account for signalling molecules?

**equation**descriptions of both models. Expand abstract.

**equation**. However, the spatial and temporal distributions of such signalling molecules are not often reported or observed experimentally. This leads to a mismatch between the amount of experimental data available and the complexity of the mathematical model used to simulate the experiment. To address this mismatch, we develop a discrete model of cell migration that can be used to describe a new suite of co-culture cell migration assays involving two interacting subpopulations of cells. In this model, the migration of cells from one subpopulation is regulated by the presence of signalling molecules that are secreted by the other subpopulation of cells. The spatial and temporal distribution of the signalling molecules is governed by a discrete conservation statement that is related to a reaction-diffusion

**equation**. We simplify the model by invoking a steady state assumption for the diffusible molecules, leading to a reduced discrete model allowing us to describe how one subpopulation of cells stimulates the migration of the other subpopulation of cells without explicitly dealing with the diffusible molecules. We provide additional mathematical insight into these two stochastic models by deriving continuum limit partial differential

**equation**descriptions of both models. To understand the conditions under which the reduced model is a good approximation of the full model, we apply both models to mimic a set of novel co-culture assays and we systematically explore how well the reduced model approximates the full model as a function of the model parameters.

4/10 relevant

bioRxiv

Universal Differential **Equations** for Scientific Machine Learning

**equations**, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. Expand abstract.

**equations**(UDEs), which augments scientific models with machine-learnable structures for scientifically-based learning. We show how UDEs can be utilized to discover previously unknown governing equations, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. This advance is coupled with open-source software that allows for training UDEs which incorporate physical constraints, delayed interactions, implicitly-defined events, and intrinsic stochasticity in the model. Our examples show how a diverse set of computationally-difficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating climate simulations by 15,000x, can be handled by training UDEs.

7/10 relevant

arXiv

An adaptive discontinuous Petrov-Galerkin method for the Grad-Shafranov
**equation**

**equation**. Expand abstract.

**equation**. An ultraweak formulation of the DPG scheme for the

**equation**is given based on a minimal residual method. The DPG scheme has the advantage of providing more accurate gradients compared to conventional finite element methods, which is desired for numerical solutions to the Grad-Shafranov

**equation**. The numerical scheme is augmented with an adaptive mesh refinement approach, and a criterion based on the residual norm in the minimal residual method is developed to achieve dynamic refinement. Nonlinear solvers for the resulting system are explored and a Picard iteration with Anderson acceleration is found to be efficient to solve the system. Finally, the proposed algorithm is implemented in parallel on MFEM using a domain-decomposition approach, and our implementation is general, supporting arbitrary order of accuracy and general meshes. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithm.

6/10 relevant

arXiv