General Solution Research Articles

Translational motion of a spheroidal drop in a viscous fluid

The problem of translational motion of a spheroidal drop along its axis of revolution in a viscous incompressible fluid is investigated semi-analytically. The flow fields in the exterior and interior of the drop are governed by the Stokes equations. Stream function formulation is adopted to solve the hydrodynamic equations in both regions. The general solution for the stream function in prolate and oblate spheroidal coordinates is expressed in an infinite-series form of semi-separation of variables. The leading order coefficients in the stream function are obtained using suitable boundary conditions. The hydrodynamic drag force experienced by the spheroidal drop is numerically evaluated with adequate convergence behavior for various values of the internal-to-external viscosity ratio and axial-to-radial aspect ratio of the drop. The numerical values of the drag force for the infinite and infinitesimal viscosity ratios agree with the available corresponding results for the slow translation of a slip spheroidal particle in the limiting conditions of no slip and full slip, respectively. At intermediate values of the viscosity ratio, the hydrodynamic force may not be a monotonic function of the aspect ratio. For a spheroidal drop with a fixed aspect ratio, its drag force increases monotonically with an increase in the viscosity ratio.

Sample-analysis based adversarial attack with saliency map

With the widespread application of deep learning, the vulnerability of neural networks has attracted considerable attention, raising reliability and security concerns. Therefore, research on the robustness of neural networks has become increasingly critical. In this paper, we propose a novel sample-analysis based robustness evaluation method that overcomes the drawbacks of existing techniques, such as solving difficulty, single strategy, and loose radius. Our algorithm comprises two parts: robustness evaluation and adversarial attacks. Specifically, we introduce formal definitions of multiple sample types and a general solution to the problem of adversarial samples. We formulate a disturbance model-based description of adversarial samples in the adversarial attack algorithm and utilize saliency map to solve them. Our experimental results demonstrate that our adversarial attack algorithm not only achieves a high attack success rate in a relatively small disturbance range but also generates multiple adversarial examples for each clean example. Our algorithm can evaluate the robustness of complex datasets and models, overcome the lack of a single strategy in solving adversarial examples, and provide a more accurate radius of robustness.

Legal proof: why knowledge matters and knowing does not

I discuss the knowledge account of legal proof in Moss (2023) and develop an alternative. The unifying thread throughout this article are reflections on the beyond reasonable doubt (BRD) standard of proof. In Section 1, I will introduce the details of Moss’s account and how she motivates it via the BRD standard. In Section 2, I will argue that there are important disanalogies between BRD and knowledge that undermine Moss’s argument. There is however another motivation for the knowledge account: combined with auxiliary claims, that is probabilistic knowledge and moral encroachment, it can provide a general solution to the puzzle of statistical evidence. Section 3 spells out the details. In Section 4, I suggest to combine the knowledge account with pragmatic encroachment, instead of moral encroachment, in order to stay clear of the thorny issues whether corporations have moral rights. In Section 5, I argue that the verdicts of Moss’s account in cases of false justified beliefs and non-luminous knowledge conflict with the BRD standard and thus call for abandoning the account. Based on the social function of the BRD standard, I suggest a replacement for the knowledge account that is also just as potent as a general solution for the puzzle of statistical evidence. While I will grant that knowledge is neither always necessary nor always sufficient for convictions, I will argue that the concept of knowledge nonetheless plays a significant and ineliminable role in legal decision-making.

Breaking the Base Barrier: Cu(II)-Mediated C-H Heteroarylation of o-Carboranes with Base-Sensitive Heteroaryl Halides.

While cage C-arylation reactions using strong bases are among the most frequently used transformations in carborane chemistry, there has been no general solution to allow for the use of weak bases in the reaction. Moreover, base-metal-promoted C-H heteroarylation with base-sensitive heteroaryl halides remained elusive. Herein, copper-mediated cage C-H (hetero)arylation has been achieved without the need for strong bases, leading to the facile synthesis of a wide range of C-(hetero)arylated carboranes in good to excellent yields with a broad substrate scope and good functional group compatibility.

General Solution to a Nonlocal Linear Differential Equation of First-Order

General Solution to a Nonlocal Linear Differential Equation of First-Order

Numerical modeling of underground flow through porous media using volumes scheme

The Richards equation attracts the attention of several researchers due to its importance in the hydrogeology field especially the flow through porous soil. It is a non linear partial differential equation that has no general analytic solution. Thus, the need to use numerical methods to solve it. In this work, a finite volumes scheme is used to simulate the pressure form and the mixed form of Richards equation in one dimension. Euler explicit and implicit schemes are used for the time discretization and a condition of scheme’s stability is given. Different test cases are done to validate the accuracy and the efficiency of the proposed numerical model

Modified state space method for long-term behavior of viscoelastic laminated angle-ply plate

This work proposes a modified state space method (SSM) to develop the analytical solution for viscoelastic laminated angle-ply plates subjected to long-term loading situations. Within the analytical model, the plate’s stresses and displacements are governed by orthotropic elasticity theory combined with Boltzmann superposition. The convolution within the governing equation is resolved through the application of Laplace transform. The modified SSM is proposed to provide a general solution for each lamina by extracting the complex variable of the Laplace transform from the state space equation. This modification addresses a limitation of the traditional SSM where the exponential function of the state space matrix is computationally infeasible to calculate. The analytical solution of the laminated angle-ply plate is ultimately derived using the transfer matrix method followed by the inverse Laplace transform. Through comparative analysis, it is observed that the proposed solution exhibits greater accuracy than simplified solutions and higher efficiency compared to the finite element method. In addition, the work explores the bending creep and recovery behaviors of viscoelastic laminated angle-ply plate under various parameters.

Unifying Graph Neural Networks with a Generalized Optimization Framework

Graph Neural Networks (GNNs) have received considerable attention on graph-structured data learning for a wide variety of tasks. The well-designed propagation mechanism, which has been demonstrated effective, is the most fundamental part of GNNs. Although most of the GNNs basically follow a message passing manner, little effort has been made to discover and analyze their essential relations. In this paper, we establish a surprising connection between different propagation mechanisms with an optimization problem. We show that despite the proliferation of various GNNs, in fact, their proposed propagation mechanisms are the optimal solutions of a generalized optimization framework with a flexible feature fitting function and a generalized graph regularization term. Actually, the optimization framework can not only help understand the propagation mechanisms of GNNs, but also open up opportunities for flexibly designing new GNNs. Through analyzing the general solutions of the optimization framework, we provide a more convenient way for deriving corresponding propagation results of GNNs. We further discover that existing works usually utilize naïve graph convolutional kernels for feature fitting function, or just utilize one-hop structural information (original topology graph) for graph regularization term. Correspondingly, we develop two novel objective functions considering adjustable graph kernels showing low-pass or high-pass filtering capabilities and one novel objective function considering high-order structural information during propagation respectively. Extensive experiments on benchmark datasets clearly show that the newly proposed GNNs not only outperform the state-of-the-art methods but also have good ability to alleviate over-smoothing, and further verify the feasibility for designing GNNs with the generalized unified optimization framework.

Photothermal effects induced by laser radiation in a 2D axi-symmetric generalized thermoelastic semiconducting half space using MGL model

In this paper, a novel model based on the strain-temperature rate-dependent theory is investigated to study the coupling influences of thermal, elastic, and plasma waves during the photothermal process in a 2D axi-symmetric semiconducting half space. The surface absorption technique is used to illuminate the surface of the medium by a laser beam whose temporal and spatial profiles are Gaussian, moreover, the surface is also considered traction free. Integral transform method is applied using Laplace and Hankel transformations to obtain a general solution to the considered problem. Inverse Laplace transform is obtained computationally using Rimann-Sum approximation. Silicon element is chosen as an application to show the compatibility of the results of the MGL model with GL and CTE models, moreover the relaxation times effects of the new model were studied.

Conditions for the existence of power solutions to a linear difference equation with constant coefficients

With the help of the formula for the general solution of a difference equation with constant coefficients, it is shown that the set of solutions to this equation contains classical solutions of the type km\λk. We present necessary and sufficient conditions on the coefficients of the equation and the initial parameters under which such solutions are obtained.

Fractal Schrödinger equation: implications for fractal sets

This paper delves into the world of fractal calculus, investigating its implications for fractal sets. It introduces the Fractal Schrödinger equation and provides insights into its consequences. The study presents a general solution for the time-dependent Schrödinger equation, unveiling its core aspects. Exploring quantum mechanics in the context of fractals, the paper analyzes the probability density of the radial hydrogen atom, unveiling its behavior within fractal dimensions. The investigation extends to deciphering the intricate energy levels of the hydrogen atom, uncovering the interplay of quantum mechanics and fractal geometry. Innovatively, the research applies the Fractal Schrödinger equation to simple harmonic motion, leading to the introduction of the fractal probability density function for the harmonic oscillator. The paper employs a series of illustrative figures that enhance the comprehension of the findings. By intertwining quantum mechanics and fractal mathematics, this research paves the way for deeper insights into their relationship.

More holographic M5 branes in AdS7 × S4

We study classical M5 brane solutions in the probe limit in the AdS7×S4 spacetime geometry with worldvolume 3-form flux. These solutions describe the holography of codimension-4 defects in the 6d boundary dual N=(0,2) supersymmetric gauge theories. Starting from some half-BPS solutions which follow stricter BPS conditions, we find a general 116-BPS solution. We show how some of the giant-like M5 brane solutions here may be related to the codimension-2 surface operators in the 4d gauge theory.

The General Solution to a Classical Matrix Equation AXB = C over the Dual Split Quaternion Algebra

The General Solution to a Classical Matrix Equation AXB = C over the Dual Split Quaternion Algebra

A general analytical solution for axisymmetric electro‐osmotic consolidation of unsaturated soil with semi‐permeable boundary

A general analytical solution for axisymmetric electro‐osmotic consolidation of unsaturated soil with semi‐permeable boundary

Particle motion around traversable wormholes: Possibility of closed timelike geodesics

The present work investigates the general wormhole solution in Einstein gravity with an exponential shape function around an ultrastatic and a finite redshift geometry. The geodesic motion around the wormholes is studied in which the deflection angle of the orbiting photon sphere is found to be negative after a certain region, indicating the presence of repulsive effect of gravity in both the ultrastatic and finite redshift wormholes. Various unbounded and bounded timelike trajectories are presented on the wormhole embedding diagrams, in which some of the bound orbits involve intersection points that may lead to causality violating geodesics. Another class of closed timelike geodesics are obtained in the unstable circular trajectory that appeared at the wormhole throat. Finally, the trajectories are classified in terms of the family of CTG orbits.

When outpatient appointment meets online consultation: A joint scheduling optimization framework

While the provision of Online Consultation Services (OCS) has brought convenience to patients, the original offline outpatient workflow may struggle to maintain efficiency. Hospital managers’ concerns lie on how to reconcile the operations of offline service with the provision of OCS and what are the latent impacts of OCS on the operational performances of healthcare system. The outpatient appointment scheduling is a crucial part of the workflow decision that affects both quality of offline services and the fragmented idle time that doctors can reply to their Follow-up patients (FUP) during working hours. To smooth the fluctuations in workload of OCS, we introduce a novel decision variable—the recommended maximum number of FUP that a doctor should serve during overtime hours (FUP-MN)—and propose an inter-day service shifting strategy that any requests for OCS exceeding the FUP-MN will be deferred to be served in the fragmented idle time of the next session. We formulate a stochastic programming model to search for the optimal combination of the appointment schedule and FUP-MN. Propositions are conducted to demonstrate the model is valid not only in coordinating the dual demand for online and offline services, but also in restricting the potential overflow of OCS requests. We provide a general solution approach and also develop a heuristic algorithm to quickly obtain optimal workflows at different OCS demand levels. The numerical experimental results indicate that an appropriate introduction of OCS can give the healthcare system a strong ability to resist the adverse effects of uncertainty.

Multi-task collaborative method based on manifold optimization for automated test case generation based on path coverage

Automated test case generation based on path coverage is not only a large-scale black-box optimization problem, but also a key scientific problem in software automatic testing technology. Evolutionary algorithms and other search-based algorithms are representative methods for this problem. However, existing research mainly focuses on the multi-function case, where generating test cases for multiple functions is difficult due to the combinatorial explosion of the path number. In this paper, we propose a multi-task collaborative method based on manifold optimization by considering the topological manifold relationship between the test case space (decision space) and the program path space (target space). This method achieves the goal of collaborative optimization for different function coverage tasks by coordinating the allocation of computing resources and knowledge transfer mechanisms among them. To verify the effectiveness of the proposed method, we compare it with general solution methods for single-function automated test case generation based on path coverage, such as the manifold-inspired search-based algorithm. The experimental results show that the proposed method outperforms the compared single-function optimization algorithms especially on programs with strong coding similarities. This study verifies the feasibility of collaborative optimization methods for solving large-scale black-box optimization problems, such as the automated test case generation based on path coverage, and expands their application scenarios.

General solution to a difference equation and the long-term behavior of some of its solutions

Closed-from formulas for the general solution to a difference equation are given, generalizing some special cases in the literature. We also analyze and give some comments on the results on the long-term behaviour of some solutions of the special cases.

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

Economic operation of an agent-based virtual storage aggregated residential electric-heating loads in multiple electricity markets

The increasing penetration of intermittent renewable energy sources leads to a higher demand for flexible balancing service in the power system. Consequently, demand-side management techniques, such as utilizing flexible resources from residential electric-heating loads, provide a promising solution. However, due to various uncertainties such as stochastic weather and market prices, it is challenging to develop an economically efficient strategy. Hence, this paper proposes an economic operation optimization approach for residential electric-heating loads to develop optimal bidding and scheduling strategy in the multi-stage market. First, the dynamic characteristics of aggregated electric-heating loads are modeled as the virtual energy storage systems (VESS) to quantify the flexibility potential arising from the combination of building thermal inertia, user tolerance and thermal energy storage (TES). Subsequently, the joint energy and balancing service market is introduced to exploit the operating flexibility of VESS. A multi-agent deep reinforcement learning (MADRL) optimization framework is proposed to address the multi-objective problem at different decision frequency, which involves achieving lower economic costs while maintaining indoor comfort. Finally, we combine the DRL with imitation learning (IL) by introducing the behavior cloning mechanism to mitigate training process instability. It reduces the computational costs and eliminates the need for redundant additional structure. Comparative simulation experiments demonstrate the efficiency of data-driven optimization approach, which provides an adaptive, economic cost-effective and general solution for multi-type aggregators to activate their flexibility in dynamic markets.