General Solution Research Articles

Interior solution of azimuthally symmetric case of Laplace equation in orthogonal similar oblate spheroidal coordinates

Curvilinear coordinate systems distinct from the rectangular Cartesian coordinate system are particularly valuable in the field calculations as they facilitate the expression of boundary conditions of differential equations in a reasonably simple way when the coordinate surfaces fit the physical boundaries of the problem. The recently finalized orthogonal similar oblate spheroidal (SOS) coordinate system can be particularly useful for a physical processes description inside or in the vicinity of the bodies or particles with the geometry of an oblate spheroid. The solution of the azimuthally symmetric case of the Laplace equation was found for the interior space in the orthogonal SOS coordinates. In the frame of the derivation of the harmonic functions, the Laplace equation was separated by a special separation procedure. A generalized Legendre equation was introduced as the equation for the angular part of the separated Laplace equation. The harmonic functions were determined as relations involving generalized Legendre functions of the first and of the second kind. Several lower-degree functions are reported. Recursion formula facilitating determination of the higher-degree harmonic functions was found. The general solution of the azimuthally symmetric Laplace equation for the interior space in the SOS coordinates is reported.

Organ- and Cell-Selective Delivery of mRNA In Vivo Using Guanidinylated Serinol Charge-Altering Releasable Transporters.

Selective RNA delivery is required for the broad implementation of RNA clinical applications, including prophylactic and therapeutic vaccinations, immunotherapies for cancer, and genome editing. Current polyanion delivery relies heavily on cationic amines, while cationic guanidinium systems have received limited attention due in part to their strong polyanion association, which impedes intracellular polyanion release. Here, we disclose a general solution to this problem in which cationic guanidinium groups are used to form stable RNA complexes upon formulation but at physiological pH undergo a novel charge-neutralization process, resulting in RNA release. This new delivery system consists of guanidinylated serinol moieties incorporated into a charge-altering releasable transporter (GSer-CARTs). Significantly, systematic variations in structure and formulation resulted in GSer-CARTs that exhibit highly selective mRNA delivery to the lung (∼97%) and spleen (∼98%) without targeting ligands. Illustrative of their breadth and translational potential, GSer-CARTs deliver circRNA, providing the basis for a cancer vaccination strategy, which in a murine model resulted in antigen-specific immune responses and effective suppression of established tumors.

TyBox: An Automatic Design and Code Generation Toolbox for TinyML Incremental On-Device Learning

Incremental on-device learning is one of the most relevant and interesting challenges in the field of Tiny Machine Learning (TinyML). Indeed, differently from traditional TinyML solutions where the training is typically carried out on the Cloud and inference only occurs on the tiny devices (e.g., embedded systems or Internet-of-Things units), incremental on-device TinyML allows both the inference and the training of TinyML models directly on tiny devices. This ability paves the way for TinyML-enabled intelligent devices that can learn directly on the field and adapt to evolving environments, different working conditions, or specific users. The literature in this field is quite limited with very few solutions focusing only on the incremental fine-tuning of machine learning models, whereas a general solution encompassing algorithms and code generation for incremental on-device TinyML is still perceived as missing. The aim of this article is to introduce, to the best of our knowledge for the first time in the literature, a toolbox called TyBox for the automatic design and code generation of incremental on-device TinyML classification models. In more detail, starting from a “static” TinyML model, TyBox is able to (i) automatically design the “incremental” on-device version of the TinyML model that has been suitably designed to take into account the technological constraint on the RAM memory of the target tiny device, and (ii) autonomously provide the C++ codes and libraries to support the inference and learning of the incremental on-device TinyML model directly on the tiny devices. TyBox has been extensively compared with a state-of-the-art incremental learning solution for TinyML and tested on an off-the-shelf tiny device (i.e., the Arduino Nano 33 BLE) in three relevant TinyML application tasks and scenarios: binary image classification, multi-class image classification, and ultra-wide-band human activity recognition. In addition, TyBox is released to the scientific community as a public repository.

A comprehensive AI model development framework for consistent Gleason grading

BackgroundArtificial Intelligence(AI)-based solutions for Gleason grading hold promise for pathologists, while image quality inconsistency, continuous data integration needs, and limited generalizability hinder their adoption and scalability.MethodsWe present a comprehensive digital pathology workflow for AI-assisted Gleason grading. It incorporates A!MagQC (image quality control), A!HistoClouds (cloud-based annotation), Pathologist-AI Interaction (PAI) for continuous model improvement, Trained on Akoya-scanned images only, the model utilizes color augmentation and image appearance migration to address scanner variations. We evaluate it on Whole Slide Images (WSI) from another five scanners and conduct validations with pathologists to assess AI efficacy and PAI.ResultsOur model achieves an average F1 score of 0.80 on annotations and 0.71 Quadratic Weighted Kappa on WSIs for Akoya-scanned images. Applying our generalization solution increases the average F1 score for Gleason pattern detection from 0.73 to 0.88 on images from other scanners. The model accelerates Gleason scoring time by 43% while maintaining accuracy. Additionally, PAI improve annotation efficiency by 2.5 times and led to further improvements in model performance.ConclusionsThis pipeline represents a notable advancement in AI-assisted Gleason grading for improved consistency, accuracy, and efficiency. Unlike previous methods limited by scanner specificity, our model achieves outstanding performance across diverse scanners. This improvement paves the way for its seamless integration into clinical workflows.

A Solution Matrix by IEVP under the Central Principle Submatrix Constraints

The n×n real matrix P is called centrosymmetric matrix if P=RPR, where R is permutation matrix with ones on cross diagonal (bottom left to top right) and zeroes elsewhere. In this article, the solvability conditions for left and right inverse eigenvalue problem (which is special case of inverse eigenvalue problem) under the submatrix constraint for generalized centrosymmetric matrices are derived, and the general solution is also given. In addition, we provide a feasible algorithm for computing the general solution, which is proved by a numerical example.

The general algebraic solution of dual fuzzy linear systems and fuzzy Stein matrix equations

In this paper, we present a new method for solving a dual fuzzy linear system (DFLS), AX˜+C˜=BX˜+D˜, where the coefficient matrices A and B are arbitrary real m×n matrices and C˜ and D˜ are given fuzzy number vectors. A necessary and sufficient condition for the R-consistency of the associated system of linear equations is obtained. The straightforward method for solving m×n DFLS based on an arbitrary {1}-inverse of A−B is introduced. Also, as an application, we present the first algorithm for solving the fuzzy Stein matrix equations, based on {1}-inverses. Finally, these results are illustrated by examples.

New exact optical soliton solutions of the derivative nonlinear Schrödinger equation family

In this study, we use a systematic approach named the generalized unified method (GUM) to construct the general exact solutions of the derivative nonlinear Schrödinger (DNLS) family that also includes perturbed terms, which are the Kaup–Newell equation, the Chen–Lee–Liu equation, and the Gerdjikov–Ivanov equation. The GUM provides more general exact solutions with free parameters for nonlinear partial differential equations such that some solutions obtained by different exact solution methods, including the hyperbolic function solutions, the trigonometric function solutions, and the exponential solutions, are derived from these solutions by giving special values to these free parameters. Additionally, the used method reduces a large number of calculations compared to other exact solution methods, enabling computations to be made in a short, effortless, and elegant way. We investigate the DNLS family in this work because of its extensive applications in nonlinear optics. Particularly, the obtained optical soliton solutions of the DNLS family are useful for describing waves in optics and facilitating the interpretation of the propagation of solitons through optical fibers. Furthermore, this work not only contributes significantly to the advancement of soliton dynamics and their applications in photonic systems but also be productively used for more equations that occur in mathematical physics and engineering problems. Finally, 2D and 3D graphs of some derived solutions are plotted to illustrate behaviors of optical soliton.

An investigation on the torsional vibration of a FG strain gradient nanotube

AbstractIn this work, an attention is paid to the prediction of torsional vibration frequencies of functionally graded porous nanotubes based on the Lam strain gradient elasticity theory. The nanotubes are formed of functionally graded porous nanomaterials that vary in the radial direction. This study also aims to obtain the analytical solution of the strain gradient model presented by Lam for torsional vibration response, in a simple manner, for different rigid or restrained boundary conditions. The torsion angle of a functionally graded nanotube is defined by an infinite Fourier series. Then, the Stokes’ transformation is applied to force the boundary conditions to the desired state. An eigenvalue problem is established with the help of the two systems of equations obtained. This eigenvalue problem, which includes deformable springs at both ends of the nanotube, appears as a general analytical solution that can find torsional vibration frequencies. It is shown that the vibrational responses can be significantly influenced by the through‐radius gradings of material, material length scale parameters and deformable springs of the functionally graded nanotubes and consequently can be predicted by giving proper values to torsional spring parameters.

Soliton molecules and breather positon solutions for the coupled modified nonlinear Schrödinger equation

The coupled modified nonlinear Schrödinger equation, which can be regarded as a combination of nonlinear Schrödinger and derivative nonlinear Schrödinger equations, is investigated by Darboux transformation (DT) method. Based on the vector modified derivative nonlinear Schrödinger equation spectral problem, the related Lax pair and DT in compact determinant form are all successfully constructed to ensure integrability of the coupled modified nonlinear Schrödinger equation. According to DT method and the limiting technique, two main types of solutions that soliton molecules (SMs) and breather positon (B-P) solutions are systematically discussed. Beginning form zero plane wave backgrounds, the general M−SM-(N−M)-soliton solutions (0≤M≤N), including M SMs and N−M solitons, are subtly derived by the received DT. In particular, two degenerate cases can be reduced from the above general solutions, i.e., N-SM solutions (M=N) and N-soliton solutions (M=0). From the nonzero plane wave backgrounds, the higher-order B-P solutions are constructed via both DT and the limiting technique. It is interestingly shown that the central region of B-P solutions exhibit the patterns of rogue waves, and thus they are suggested to explain the generating mechanism of rogue waves. Finally, the corresponding dynamics of these received solutions are discussed in detail.

Nonlinear gravitational waves in Horndeski gravity: scalar pulse and memories

We present and analyze a new non-perturbative radiative solution of Horndeski gravity. This exact solution is constructed by a disformal mapping of a seed solution of the shift-symmetric Einstein-Scalar system belonging to the Robinson-Trautman geometry describing the gravitational radiation emitted by a time-dependent scalar monopole. After analyzing in detail the properties of the seed, we show that while the general relativity solution allows for shear-free parallel transported null frames, the disformed solution can only admit parallel transported null frames with a non-vanishing shear. This result shows that, at the nonlinear level, the scalar-tensor mixing descending from the higher-order terms in Horndeski dynamics can generate shear out of a pure scalar monopole. We further confirm this analysis by identifying the spin-0 and spin-2 polarizations in the disformed solution using the Penrose limit of our radiative solution. Finally, we compute the geodesic motion and the memory effects experienced by two null test particles with vanishing initial relative velocity after the passage of the pulse. This exact radiative solution offers a simple framework to witness nonlinear consequences of the scalar-tensor mixing in higher-order scalar-tensor theories.

Set-valued state estimators for a class of uncertain discrete-time nonlinear systems

Set-membership state estimation problem for a certain class of discrete-time nonlinear systems is tackled in this work. First, this problem is formulated as a reachability analysis problem. Then, to avoid the wrapping effect, explicit expression of the general solution of this class of systems is used to design new set-valued state estimation schemes. Moreover, for practical considerations, two periodically re-initialized algorithms based on both interval analysis and zonotope computation are introduced. The merit of the proposed approach is illustrated on a numerical example and its performance is compared to that of other methods borrowed from the literature.

Ultra-flexible inorganic coating realized by scale-like nanosheets

Rapid development of foldable devices poses a challenging requirement on functional inorganic coatings with high flexibility. However, the inherent brittleness often renders inorganic coatings incompatible with flexibility. How to make flexible inorganic coatings remains challenging. Here, inspired by the scale structure of animal skins, we realize ultra-flexible scale-like coatings on polyimide (PI) film with layered double hydroxide (LDH) nanosheets. The adhesion is enhanced without peeling due to the bonding of polyimide with amino group attached to nanosheets. LDH nanosheets parallel to the substrate form scale-like structure that absorbs and releases stress to prevent coatings from cracking. And the coatings exhibit ultra-flexibility by surviving from 20,000 cycles of bending fatigue test. We apply the LDH-coated PI for atomic oxygen protection, which has important applications in aerospace industry. The coatings are designed as this special scale-like structure to release stress while possessing protection ability. Due to the barrier and adsorption of this flexible LDH coating, the erosion yield of the PI under an exposure of ~5 × 1021 atoms/cm2 is decreased by about 4 orders of magnitude to 0.015 % (~5 × 10−28 cm3/atom), which can satisfy the requirement for applications in harsh space environment of low earth orbit. Our work provides a general solution for simultaneously realizing high mechanical strength and flexibility in the coating technology, which may find applications in other areas, such as flexible electronics.

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.

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.