General Expression Research Articles

Two-dimensional P1 approximation (P1-2D) for the Description of the Radiant Field on Cylindrical Solar Photocatalytic Reactors

The local volumetric rate of photon absorption (LVRPA) was formulated by solving the radiative transfer equation (RTE) in polar coordinates with the P1 approximation approach (P1-2D) for the description of the radiant field in cylindrical solar photocatalytic reactors. A general expression of the LVRPA was formulated that can be employed on cylindrical photocatalytic reactors with an incident radiation constant along the reactor length. CPC and tubular photocatalytic reactors were used as reactor models and Lambert's cosine law (irradiance) was considered when using the boundary conditions. Simulations were carried out using the commercial TiO2-P25, its optical properties taken from the literature. The LVRPA was found to decrease exponentially from the reactor wall to its center. literature rate of photon absorption per unit of reactor length (VRPA/H) increased exponentially with the catalyst loading until a value where no significant increase was observed and was found to increase with reactor radius, information that agrees with the literature. The optimum catalyst loading with the CPC reactor was about 0.364 g/L with a reactor radius equal to 1.65 cm similar to that found in the literature when using the six-flux model in two dimensions (SFM-2D). The apparent optical thickness τ_App1 newly formulated with the P1 approximation was introduced for optimization purposes and was found more reliable than the optical thickness τ. This parameter not only removes the dependence of the optimum catalyst loading on the reactor's radius but also its dependence on catalyst albedo. τ_App1 was found about 9.73 and 14.6 for CPC and tubular reactors respectively and provides the optimum catalyst loading and the reactor radius that optimize the radiation absorption inside both reactors.

Finite-frequency noise, Fano factor, ΔT-noise and cross-correlations in double quantum dots.

A theoretical study on electrical current fluctuations in a double quantum dot connected to electronic reservoirs is presented, with the aim of deriving the finite-frequency noise, the Fano factor and the ΔT-noise. We establish a general expression for the noise in terms of Green functions in the double quantum dot and self-energies in the reservoirs. This result is then applied to model double quantum dots in various situations. For a non-interacting double quantum dot, we have highlighted several interesting features in the physical properties of this system. In particular, we have demonstrated the possibility of obtaining a significant reduction in zero-frequency noise and Fano factor either when the system is placed in a given operating regime, or when a temperature gradient is applied between the two reservoirs, resulting in a negative ΔT-noise being generated. In addition, in the vicinity of honeycomb vertices, a sign change is observed in the finite-frequency cross-correlator between the two reservoirs, in contrast to what is obtained for the zero-frequency cross-correlator, which remains negative throughout the (ε1,ε2)-plane, ε1, ε2 being the level energies in each of the two dots. By using an approximate first-level numerical approach, we finally study how the finite-frequency noise in a double quantum dot evolves under the influence of Coulomb interactions.

An infralimbic cortex neuronal ensemble encoded during learning attenuates fear generalization expression.

Generalization allows previous experience to adaptively guide behavior when conditions change. The infralimbic (IL) subregion of the ventral medial prefrontal cortex plays a known role in generalization processes, although mechanisms remain unclear. A basic physical unit of memory storage and expression in the brain are sparse, distributed groups of neurons known as ensembles (i.e., the engram). Here, we set out to determine whether neuronal ensembles established in the IL during learning contribute to generalized responses. Generalization was tested in male and female mice by presenting a novel, ambiguous, tone generalization stimulus following Pavlovian defensive (fear) conditioning. The first experiment was designed to test a role for IL in generalization using chemogenetic manipulations. Results show IL regulates defensive behavior in response to ambiguous stimuli. IL silencing led to a switch in defensive state, from vigilant scanning to generalized freezing, while IL stimulation reduced freezing in favor of scanning. Leveraging activity-dependent "tagging" technology (ArcCreER T2 x eYFP system), a neuronal ensemble, preferentially located in IL Layer 2/3, was associated with the generalization stimulus. Remarkably, in the identical discrete location, fewer reactivated neurons were associated with the generalization stimulus at the remote timepoint (30 days) following learning. When an IL neuronal ensemble established during learning was selectively chemogenetically silenced, generalization increased. Conversely, IL neuronal ensemble stimulation reduced generalization. Overall, these data identify a crucial role for IL in suppressing generalized responses. Further, an IL neuronal ensemble, formed during learning, functions to later attenuate the expression of generalization in the presence of ambiguous threat stimuli.

Goos-Hänchen shift for an Airyprime beam.

We propose a comprehensive theory to analyze the Goos-Hänchen (GH) shift for an arbitrarily polarized Airyprime beam reflected at the air-dielectric interface. We derive general expressions for the spatial and angular GH shifts and establish a close relationship between the GH shift of the Airyprime beam and the GH shift of the Airy beam. We also predict the novel optical effects of a significantly enhanced spatial GH shift and an almost disappeared angular GH shift when the horizontally polarized Airyprime beam is reflected near Brewster's angle.

Leading two-point resistances from transfer matrices in cylindrical, spider web, axial and grid resistor networks

Resistor networks are increasingly being considered in heuristic research as models for natural or artificial matter. The equivalent resistance between two nodes, the Two-Point Resistance (TPR), can be calculated using a variety of methods. The transfer matrix (TM) method was originally considered as a numerical tool for estimating percolation thresholds in random networks with a repeating pattern. The TM method is revisited here as an efficient tool to obtain, in a fast and elegant way, iteration relations and exact explicit expressions for leading TPRs that include a node in the last repeated pattern. Several rotationally invariant networks are studied, such as simple cylindrical networks, spider web networks and cylindrical networks with a central resistive axis, in which case the TM matrices are circulant matrices. Examples of explicit expressions are given for orders of rotation ≤4 or 5, depending on the case. The method can be applied in a similar way to networks with less symmetry, such as grids. The general expressions of TPRs obtained using the TM method can provide quantitative guidelines for resistor networks developed in materials science, environmental issues or industrial applications.

A novel mapping method for equivalent stiffness and equivalent damping of [formula omitted]-DOF series hydraulic robot

Impedance control is a commonly used approach for robots to interact with environments. At present, the impedance control is mostly implemented at the foot end, while the impedance control at the joint space offers faster response speed and is a more favorable option. However, due to the influence of installation position and moment arm of hydraulic drive unit (HDU), it is more complicated to establish the mapping relationship of equivalent stiffness and equivalent damping between the foot end and joint HDU. To address this challenge, this paper provides a general calculation expression of mapping matrices between the foot end and joint HDU, which also considers the influence of above nonlinear factors. Additionally, the accuracy of mapping matrices is verified through the impedance control, and the “finite difference method” (FDM) is used to enhance the accuracy of validation. Finally, the accuracy of mapping matrices is validated through the single leg of a 2-DOF series hydraulic robot.

Weak field deflection angle and analytical parameter estimation of the Lorentz-violating Bumblebee parameter through the black hole shadow using EHT data

Abstract We explored how the Lorentz symmetry breaking parameter $\ell$ affects the Reissner-Nordst"{o}m BH solution in the context of weak field deflection angle, and the black hole shadow. We aim to derive the general expression for the weak deflection angle using the non-asymptotic version of the Gauss-Bonnet theorem, and we presented a way to simplify the calculations under the assumption that the distance of the source and the receiver are the same. \textcolor{blue}{Through the Solar System test, $\ell$ is constrained from around $-10^{-9}$ orders of magnitude to $0$, implying challenging detection of $\ell$ through the deflection of light rays from the Sun.} We also studied the black hole shadow in an analytic way, where we applied the EHT results under the far approximation in obtaining an estimate expression for $\ell$. Using the realistic values of the black hole mass and observer distance for Sgr. A* and M87*, it was shown that $M/r_{\rm o} \neq 1$ is satisfied, implying the relevance and potential promise of the spontaneous Lorentz symmetry breaking parameter's role on the shadow radius uncertainties as measured by the EHT. We find constraints for $\ell$ to be negatively valued, where the upper and lower bounds are $\sim -1.94$ and $\sim -2.04$, respectively.

Analysing Flexural Response in RC Beams: A Closed-Form Solution Designer Perspective from Detailed to Simplified Modelling

This paper presents a detailed analytical approach for the bending analysis of reinforced concrete beams, integrating both structural mechanics principles and Eurocode 2 provisions. The general analytical expressions derived for the curvature were applied for the transverse displacement analysis of a simply supported reinforced concrete beam under four-point loading, focusing on key limit states: the initiation of cracking, the yielding of tensile reinforcement and the compressive failure of concrete. The displacement’s results were validated through experimental testing, showing a high degree of accuracy in the elastic and crack propagation phases. Deviations in the yielding phase were attributed to the conservative material assumptions within the Eurocode 2 framework, though the analytical model remained reliable overall. To streamline the computational process for more complex structures, a simplified model utilising a non-linear rotational spring was further developed. This model effectively captures the influence of cracking with significantly reduced computational effort, making it suitable for serviceability limit state analyses in complex loading scenarios, such as seismic impacts. The results demonstrate that combining detailed analytical methods with this simplified model provides an efficient and practical solution for the analysis of reinforced concrete beams, balancing precision with computational efficiency.

Solidification process of hollow metal droplets impacting a substrate

Recently, there has been a renewed interest in hollow droplet impact, as the internal bubble makes a difference in the spreading morphology during thermal spraying and three-dimensional printing. This paper is dedicated to the solidification process of a hollow metal droplet impacting a cold substrate. The influence of initial impact velocity and substrate temperature on the spreading process is numerically investigated by employing a phase-field method coupled with a solidification model. Solidification was shown to strongly alter droplet spreading and the flow state of the center liquid column. To describe the flow state of the center liquid column, three patterns, namely ‘no-jet’, ‘transition region’, and ‘counter-jet’ are defined. The simulation results show that the formation of the center counter-jet is inhibited by the solidification layer when Péclet number Pe<357, where Pe is the ratio of convection rate to diffusion rate. In addition, the critical Pe at which the phenomenon of counter-jet occurs gradually rises with the substrate temperature falling. Finally, a quantitative analysis of the maximum spreading diameter is proposed by establishing an energy conservation equation. The predicted diameters are in good agreement with the simulated results. It allows us to provide a general expression for the hollow droplets' maximum spreading diameter, using the impact and temperature parameters.

Transcription factor 4 expression in the developing non-human primate brain: a comparative analysis with the mouse brain.

Transcription factor 4 (TCF4) has been implicated in a range of neuropsychiatric disorders, including major depressive disorder, bipolar disorder, and schizophrenia. Mutations or deletions in TCF4 cause Pitt-Hopkins syndrome (PTHS), a rare neurodevelopmental disorder. A detailed understanding of its spatial expression across the developing brain is necessary for comprehending TCF4 biology and, by extension, to develop effective treatments for TCF4-associated disorders. However, most current knowledge is derived from mouse models, which are invaluable for preclinical studies but may not fully capture the complexities of human neuropsychiatric phenotypes. This study compared TCF4 expression in the developing mouse brain to its regional and cellular expression patterns in normal prenatal, neonatal, and young adult rhesus macaque brains, a species more relevant to human neurodevelopment. While the general developmental expression of TCF4 is largely conserved between macaques and mice, we saw several interspecies differences. Most notably, a distinct layered pattern of TCF4 expression was clear in the developing macaque neocortex but largely absent in the mouse brain. High TCF4 expression was seen in the inner dentate gyrus of adult mice but not in macaques. Conversely, TCF4 expression was higher in the adult macaque striatum compared to the mouse striatum. Further research is needed to show the significance of these interspecies differences. Still, they underscore the importance of integrating rodent and primate studies to comprehensively understand TCF4 function and its implications for human disorders. Moreover, the primate-specific expression patterns of TCF4 will inform genetic and other therapeutic strategies to treat TCF4-associated disorders.

Generalized Itô’s lemma and the stochastic thermodynamics of diffusion with resetting

Abstract Methods from the theory of stochastic processes are increasingly being used to extend classical thermodynamics to mesoscopic non-equilibrium systems. One characteristic feature of these systems is that averaging the stochastic entropy with respect to an ensemble of stochastic trajectories leads to a second law of thermodynamics that quantifies the degree of departure from thermodynamic equilibrium. A well known mechanism for maintaining a diffusing particle out of thermodynamic equilibrium is stochastic resetting. In its simplest form, the position of the particle instantaneously resets to a fixed position x 0 at a sequence of times generated from a Poisson process of constant rate r. Within the context of stochastic thermodynamics, instantaneous resetting to a single point is a unidirectional process that has no time-reversed equivalent. Hence, the average rate of entropy production calculated using the Gibbs–Shannon entropy cannot be related to the degree of time-reversal symmetry breaking. The problem of unidirectionality can be avoided by considering resetting to a random position or diffusion in an intermittent confining potential. In this paper we show how stochastic entropy production along sample paths of diffusion processes with resetting can be analyzed in terms of extensions of Itô’s formula for stochastic differential equations (SDEs) that include both continuous and discrete processes. First, we use the stochastic calculus of jump-diffusion processes to calculate the rate of stochastic entropy production for instantaneous resetting, and show how previous results are recovered upon averaging over sample trajectories. Second, we formulate single-particle diffusion in a switching potential as a hybrid SDE and develop a hybrid extension of Itô’s stochastic calculus to derive a general expression for the rate of stochastic entropy production. We illustrate the theory by considering overdamped Brownian motion in an intermittent harmonic potential. Finally, we calculate the average rate of entropy production for a population of non-interacting Brownian particles moving in a common switching potential. In particular, we show that the latter induces statistical correlations between the particles, which means that the total entropy is not given by the sum of the 1-particle entropies.

Magnetohydrodynamic Motions of Oldroyd-B Fluids in Infinite Circular Cylinder That Applies Longitudinal Shear Stresses to the Fluid or Rotates Around Its Axis

Two classes of magnetohydrodynamic (MHD) motions of the incompressible Oldroyd-B fluids through an infinite cylinder are analytically investigated. General expressions are firstly established for shear stress and velocity fields corresponding to the motion induced by longitudinal shear stress on the boundary. For validation, the expression of the shear stress is determined by two different methods. Using an important remark regarding the governing equations for shear stress and fluid velocity corresponding to the two different motions, this expression is then used to provide the dimensionless velocity field of the MHD motion of the same fluids generated by a cylinder that rotates around its symmetry axis. Obtained results can generate exact solutions for any motion of this kind of Oldroyd-B fluids. Consequently, both types of motions are completely solved. For illustration, some case studies are considered, and adequate velocity fields are provided. The steady-state components of these velocities are presented in different forms whose equivalence is graphically proved. The influence of the magnetic field on the fluid behavior is graphically investigated. It was found that the fluid flows slower, and a steady state is earlier reached in the presence of a magnetic field. The fluid behavior when shear stress is given on the boundary is also investigated.

Solving the Dual Generalized Commutative Quaternion Matrix Equation AXB = C

Dual generalized commutative quaternions have broad application prospects in many fields. Additionally, the matrix equation AXB=C has important applications in mathematics and engineering, especially in control systems, economics, computer science, and other disciplines. However, research on the matrix equation AXB=C over the dual generalized commutative quaternions remains relatively insufficient. In this paper, we derive the necessary and sufficient conditions for the solvability of the dual generalized commutative quaternion matrix equation AXB=C. Furthermore, we provide the general solution expression for this matrix equation, when it is solvable. Finally, a numerical algorithm and an example are provided to confirm the reliability of the main conclusions.

Scaling Relationships of the Structural and Rheological Behavior of Tadpole Polymer Chains in Dilute Solution Systems Using Brownian Dynamics Simulations.

Tadpole polymers, also known as lasso polymers, feature molecular structures that combine a single ring with a single linear side branch, leading to distinct conformational, dynamical, and rheological characteristics compared to their corresponding counterparts, particularly pure linear and pure ring polymers. To elucidate the mechanisms underlying these distinctive behaviors, comprehensive mesoscopic Brownian dynamics (BD) simulations of dilute solution systems of tadpole polymers were conducted using a bead-rod chain model under both equilibrium and flow conditions. Three types of tadpole polymer chains were prepared by varying the ring-to-linear ratio within the tadpole chain and comparing them with the corresponding linear and ring chains. Depending on this ratio, tadpole polymer chains exhibit entirely different structural properties and rotational dynamics, both in equilibrium and under shear flow. As the linear proportion within the tadpole chain increased, the structural, dynamic, and rheological properties of the tadpole polymer chains became more similar to those of pure linear polymers. Conversely, with an increasing ring proportion, these properties began to resemble those of pure ring polymers. Based on these observed tendencies, a simple general scaling expression is proposed for tadpole polymer properties that integrates scaling expressions for both pure linear and pure ring polymers. Our results indicate that the conformational, dynamic, and rheological properties of tadpole polymers, as predicted by these simple scaling expressions, are in good agreement with the simulated values, a result we consider statistically significant.

Statistical adiabatic channel model for termolecular reactions.

In this work, we present a statistical adiabatic channel model for termolecular reactions, A + B + C → Products. Our approach relies on hyperspherical coordinates, where the adiabatic channels are readily defined in the hyper-radius after averaging the hyperangular degrees of freedom. In this way, we find a general expression for termolecular rate constants. We focus on ion-neutral association reactions to test our approach's accuracy and predictive power, finding a good agreement between theory and experiment, especially in those reactions' temperature dependence.

Low and moderate x gluon contribution to exclusive Compton scattering processes

Abstract We revisit the high energy semi-classical description of the exclusive processes DVCS, TCS, and Double DVCS by explicitly keeping track of the Feynman x dependence in both the hard and the hadronic matrix elements. This is achieved by a modification of the standard shock wave approximation to derive the effective Feynman rules, which leads to a generic expression on which we then perform a partial twist expansion to get rid of quantities suppressed by the proper physical scales. We obtain a compact factorized master formula that can be used to investigate the Bjorken limit at leading twist. In particular, we recover the full one-loop result in the collinear limit for pure gluon exchange with the target. Finally, we discuss the subtleties in taking the simultaneous collinear and small x limit.

Schwinger–Keldysh Path Integral Formalism for a Quenched Quantum Inverted Oscillator

In this work, we study the time-dependent behavior of quantum correlations of a system of an inverted oscillator governed by out-of-equilibrium dynamics using the well-known Schwinger–Keldysh formalism in the presence of quantum mechanical quench. Considering a generalized structure of a time-dependent Hamiltonian for an inverted oscillator system, we use the invariant operator method to obtain its eigenstate and continuous energy eigenvalues. Using the expression for the eigenstate, we further derive the most general expression for the generating function as well as the out-of-time-ordered correlators (OTOCs) for the given system using this formalism. Further, considering the time-dependent coupling and frequency of the quantum inverted oscillator characterized by quench parameters, we comment on the dynamical behavior, specifically the early, intermediate and late time-dependent features of the OTOC for the quenched quantum inverted oscillator. Next, we study a specific case, where the system of an inverted oscillator exhibits chaotic behavior by computing the quantum Lyapunov exponent from the time-dependent behavior of OTOCs in the presence of the given quench profile.

Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries

We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.

2D Skyrmion topological charge of spin textures with arbitrary boundary conditions: a two-component spinorial BEC as a case study

Abstract We derive the most general expression for the Skyrmion topological charge for a two-dimensional spin texture, valid for any type of boundary conditions or for any arbitrary spatial region within the texture. It reduces to the usual one for the appropriate boundary conditions. The general expression resembles the Gauss-Bonet theorem for the Euler-Poincaré characteristic of a 2D surface, but it has definite differences, responsible for the assignment of the proper signs of the Skyrmion charges. Additionally, we show that the charge of a single Skyrmion is the product of the value of the normal component of the spin texture at the singularity times the Index or winding number of the transverse texture, a generalization of a Poincaré theorem. We illustrate our general results analyzing in detail a two-component spinor Bose–Einstein condensate (BEC) in 2D in the presence of an external magnetic field, via the Gross-Pitaevskii equation. The condensate spin textures present Skyrmions singularities at the spatial locations where the transverse magnetic field vanishes. We show that the ensuing superfluid vortices and Skyrmions have the same value for their corresponding topological charges, in turn due to the structure of the magnetic field.

A nonequilibrium statistical mechanics derivation of the hydrodynamic equations of simple fluids using a noncanonical form of the Poisson bracket

The continuum level hydrodynamic equations of a simple fluid were derived by Irving and Kirkwood directly from discrete particle dynamics using statistical mechanics almost 75 years ago. Their elegant derivation demonstrated the fundamental molecular basis of macroscopic fluid flow and culminated in molecular expressions for the stress tensor and heat current density that have since been employed in countless molecular simulations to date. In this article, an alternative derivation is presented, which leads to more general expressions for the fundamental transport relationships and which arrives at them in a more straightforward chain of consistency that ensues directly from the Principle of Least Action. The main point of departure from the Irving–Kirkwood derivation is the application of a transformation mapping of the total momentum of each individual particle onto the sum of its peculiar momentum and its momentum relative to the local velocity field. This mapping provides a phase-space distribution function applicable in the space of particle positions and peculiar momentum, from which a noncanonical Poisson bracket can be derived in terms of the same set of microscopic variables. For a given dynamic variable, expressed in terms of particle positions and peculiar momenta, the expectation value of the noncanonical Poisson bracket of the dynamic variable is shown to correspond to the evolution equation of the expectation value of the dynamic variable. This allows for a direct derivation of all macroscopic density evolution equations (mass, momentum, and energy density fields) using a systematic procedure free of assumptions concerning the macroscopic state of the system. Furthermore, an explicit expression of the time evolution of the entropy density at the hydrodynamic level is derived following the same procedure. Finally, in the limit of short-range interparticle interactions, a molecular-based expression for the local stress tensor as properly defined from continuum mechanics is developed at the hydrodynamic level that elucidates the continuum mechanics connection of the general stress expression of Irving and Kirkwood.