Symmetric Part Research Articles

Assembly of nanoparticles at symmetric coffee stain locations

The precise picture of the nanoparticle array deposits on the substrate surface is crucial since many uses for nanostructures are dependent on how the assembled array's collective behavior responds to various physicochemical properties. The well-known seed-mediated approach was used to synthesize gold nanorods stabilized by CTAB molecules. Due to the Helium Ion Microscope’s (HIM) remarkable ability to reveal the real nature of the deposited entities, gold nanorod arrays at symmetrical parts of the coffee-stain ring have been studied using this technique. The object of this study is to evaluate three different positions within the coffee-stain ring at the convective symmetric locations. In the context of the self-assembled mechanism, the formation of various liquid crystalline phases induced by the evaporating symmetric convective flow inside the sessile droplet have been examined and compared at the symmetric positions within the ring.

Invariance embedded physics-infused deep neural network-based sub-grid scale models for turbulent flows

In this paper, we present two novel physics-infused neural network (NN) architectures that satisfy various invariance conditions for constructing efficient and robust Deep Learning (DL)-based sub-grid scale (SGS) turbulence models for use in Large Eddy Simulation (LES) procedures widely used in various fluid engineering applications. The first architecture, called tensor basis neural networks (TBNN), recently proposed in the context of Reynolds-averaged Navier–Stokes (RANS) modeling, and introduced herein for SGS modeling of wall-bounded turbulence, embeds the analytical expansion of the SGS turbulence stresses into integrity basis tensors composed of the symmetric and anti-symmetric parts of the resolved velocity gradient tensor, thus incorporating the Galilean, rotational and reflectional invariances. In our second approach, a relatively simple yet powerful architecture, called Galilean Invariance embedded Neural Network (GINN) incorporates the Galilean invariance, and takes in as inputs the independent components of the integrity basis tensors in addition to the invariant inputs in a single input layer. Explicit filtering of data from direct simulations of the canonical channel flow at two friction Reynolds numbers Reτ≈395 and 590 provided accurate data for training and testing these models. Both sets of models are used to predict the SGS stresses for feature datasets generated with different filter widths, and at different Reynolds numbers. The GINN model yields less prediction error on test datasets (mean squared error ∼0.4) compared to the TBNN model (mean squared error ∼0.5). Upon comparison, it is revealed that GINN has better feature extraction capacity owing to its ability to establish relations between and extract information from cross-components of the integrity basis tensors and the SGS stresses. Based on their predictive performance, both models proposed herein have shown great promise for use in a posteriori actual LESs. The present work illustrates the significance of physics infusion as well as invariance embedding into NN architecture for constructing efficient and robust DL-based SGS turbulent models in achieving superior predictive efficacy.

A two-relaxation-time lattice Boltzmann model for simulating incompressible thermal flows in porous media

In this work, a two-relaxation-time (TRT) lattice Boltzmann (LB) model for simulating incompressible thermal flows in porous media at the representative elementary volume (REV) scale is developed. In the REV-scale TRT-LB model, the framework of the TRT-LB equations with forcing and source terms is described clearly as simply as possible. Specifically, the forcing (or source) term is divided into two parts: the symmetric part and the anti-symmetric part, which is consistent with the philosophy of the TRT model. Through the Chapman-Enskog expansion procedure, the macroscopic governing equations can be derived from the present model. Numerical simulations of several benchmark problems are carried out to validate the present model, and the present results match well with the analytical solutions and/or those numerical results in the literature.

Comparative evaluation of the efficacy of BlueM active oxygen gel and Coe-pak dressing on post-operative healing of gingival depigmentation: a study protocol

Introduction Dark colored gingiva is an esthetic concern that cannot be ignored, especially when accompanied by a high lip line or gummy smile. Gingival depigmentation is a periodontal plastic surgical procedure wherein the hyperpigmentation is removed or reduced by various techniques. Protecting the wound from mechanical trauma and stability of the surgical site during the healing process are among the most important advantages of periodontal dressing application after surgery. Aim This study is aimed to comparatively evaluate the effectiveness of reactive oxygen gel species (BlueM gel) and the traditional Coe-Pack dressing on gingival healing and pain after surgical depigmentation. Protocol This split-mouth randomized clinical trial will be conducted on 20 patients aged 18–45 years with maxillary physiologic gingival pigmentation classes (II) (III) and (IV) according to the Dummett–Gupta Oral Pigmentation Index (40 treated sites). The maxilla will be randomly divided into two symmetrical parts—from the right second premolar to the midline and from the midline to the left second premolar—to receive either BlueM gel or Coe- Pack as a dressing after surgical depigmentation with a scalpel. Various indices will be assessed, such as plaque index, papillary bleeding index, pain index, healing index and reepithelization index with toluidine blue, and the follow-up period will be one month. Expected results- A total of 20 patients will be included in this study. Statistically significant differences are expected to show in the pain index and healing index after one, two, three, four, and five days. The BlueM gel group is expected to show a higher significant difference after one, two, three and four weeks in the reepithelization index.

Matrix splitting preconditioning based on sine transform for solving two-dimensional space-fractional diffusion equations

Finite difference discretization of the two-dimensional space-fractional diffusion equations derives a complicated linear system consisting of identity matrix and four scaled block-Toeplitz with Toeplitz block (BTTB) matrices resulted from the left and right Riemann–Liouville fractional derivatives in different directions. Incorporating with the diffusion coefficients and the symmetric parts of the BTTB matrices, we construct a diagonal and symmetric splitting (DSS) iteration method, which is demonstrated to be convergent conditionally when the considered space-fractional diffusion equations have sufficiently close diffusion coefficients. By further replacing the symmetric Toeplitz matrices involved in the BTTB matrices with τ matrices, an approximated DSS (ADSS) preconditioner based on two-dimensional fast sine transform is designed to accelerate the convergence rates of the Krylov subspace iteration methods. In this way, the total computational complexity of the ADSS-preconditioned GMRES method will be of O(n2logn), where n2 represents the dimension of the corresponding discrete linear system. In addition, theoretical analysis demonstrates that the eigenvalues of the ADSS-preconditioned matrix are weakly clustered around a complex disk centered at 1 with the radius less than 1. Numerical experiments show that the ADSS-preconditioned GMRES method is much more efficient than the other existing methods, and can show h-independent convergence behavior.

Novel SOFC Repeating Unit Design for the Uniform Distribution of Temperature and Thermo-Electrochemical Reaction Zone in a Commercial Scale Stack

Despite the substantial benefits provided by solid oxide fuel cell (SOFC), its low durability at high-temperature hydrocarbon-fueled operations has inhibited widespread market deployment. Thanks to the tremendous efforts of researchers, SOFC has exhibited sufficient performance to compete with other power generation technologies in the market. However, the lifespan problem due to long-term degradation remains the final obstacle for large-scale deployment. Especially when the hydrocarbon is used as fuel to take an advantage of fuel flexibility, carbon deposition can worsen the problem.A conventional crossflow type stack has the advantage of small volume, which can increase the power and energy density, but suffereing from large nonuniformities of temperature and thermo-electrochemical reactions. However, it has nonuniform temperature and species distribution, where thermo-electrochemical reactions are concentrated at the fuel inlet region. These imbalanced distributions of temperature and species are the major limitation of conventional design since certain regions of the stack are inevitably exposed to a much harsher environment than the remaining regions. For example, nickel reoxidation may occur where fuel is depleted, interfacial separation due to high polarization may appear where electrochemical reactions are concentrated, and carbon deposition may be developed where thermochemical reactions extremely occur. These specific regions might be degraded much faster than the remaining, which can threaten the entire stability of the stack.In this study, we present the novel SOFC stack design that can solve the instability problem of the conventional crossflow type design. First, the vertical gas manifold holes are rearranged to be placed in all perimeters, symmetrical in both diagonals [1]. Since this arrangement makes the stack consist of four symmetric parts, all thermodynamic variables and chemical reactions are distributed symmetrically, and the temperature gradient in the planar direction can be reduced. Second, the guide rib pattern and flow resistance field in the interconnect are optimized to have uniform gas distribution [2]. Finally, a new component, called a ‘cover’, is added in between the interconnect and cell to control the mass transport in a vertical direction [3]. Through the control of mass transport in the vertical direction, chemical reactions can occur in much larger regions rather than concentrated in the fuel inlet regions. To verify the advantages of the novel design, we compare it with the conventional design through the three-dimensional numerical model [4]. As shown in Figure 1, the nonuniformities of temperature, fuel concentration, and chemical reaction are improved through the design modification. Compared to the conventional design, the standard deviations of each variable in the novel design are decreased by 65.07%, 25.14%, and 20.28%, respectively. Entire design processes, various simulation results, and detailed analysis will be provided.

Analysis of the Curative Effect of Diffusion Tensor Imaging-Guided Percutaneous Endoscopic Lumbar Discectomy

Diffusion tensor imaging (DTI), a novel method of describing nerve structure, is a special form of magnetic resonance imaging (MRI). This new imaging method can be used to locate the diseased nerve roots in lumbar disc herniation. The objective of this study is to compare patient outcomes between single-level and doublesection percutaneous endoscopic lumbar discectomy (PELD) in the treatment of lumbar intervertebral disc herniation with single nerve root compression, where single-sided PELD is guided by magnetic resonance diffusion tensor imaging (DTI). The clinical data of patients with lumbar intervertebral disc herniation with double compression of single nerve root symptoms in the Affiliated Hospital of Weifang Medical University from January 2019 to May 2021 were retrospectively summarized and divided into single-level percutaneous endoscopic discectomy (PELD) group after DTI localization and double-section PELD group. The operation time, intraoperative bleeding, VAS score and JOA score of the two groups were compared, as well as the preoperative and postoperative fractional anisotropy (FA) values in the DTI group. The operation time and intraoperative bleeding volume of patients in the DTI group were significantly lower than those in the double segment group, and there was no significant difference between VAS scores and JOA scores in the two groups. After the operation, the nerve root FA value of the responsible compression site of patients in the DTI group increased significantly, but it was still lower than the healthy symmetrical part. The single-level PELD based on DTI has achieved a similar effect to that of the doublesegment PELD in 3 months after the operation, which can improve the FA value of the lesion nerve root. Its bleeding amount is less, and the operation time is shorter, but the efficacy of this technology still requires long-term follow-up of large samples.

On Unsteady Internal Flows of Incompressible Fluids Characterized by Implicit Constitutive Equations in the Bulk and on the Boundary

Long-time and large-data existence of weak solutions for initial- and boundary-value problems concerning three-dimensional flows of incompressible fluids is nowadays available not only for Navier–Stokes fluids but also for various fluid models where the relation between the Cauchy stress tensor and the symmetric part of the velocity gradient is nonlinear. The majority of such studies however concerns models where such a dependence is explicit (the stress is a function of the velocity gradient), which makes the class of studied models unduly restrictive. The same concerns boundary conditions, or more precisely the slipping mechanisms on the boundary, where the no-slip is still the most preferred condition considered in the literature. Our main objective is to develop a robust mathematical theory for unsteady internal flows of implicitly constituted incompressible fluids with implicit relations between the tangential projections of the velocity and the normal traction on the boundary. The theory covers numerous rheological models used in chemistry, biorheology, polymer and food industry as well as in geomechanics. It also includes, as special cases, nonlinear slip as well as stick–slip boundary conditions. Unlike earlier studies, the conditions characterizing admissible classes of constitutive equations are expressed by means of tools of elementary calculus. In addition, a fully constructive proof (approximation scheme) is incorporated. Finally, we focus on the question of uniqueness of such weak solutions.

On the consistency of the matrix equation X T A X = B when B is skew-symmetric: improving the previous characterization

We provide a necessary and sufficient condition for the matrix equation to be consistent, when A is an arbitrary complex square matrix and B is skew-symmetric. This problem is equivalent to find the largest dimension of a subspace in which the bilinear form A is symplectic. The necessity is valid for any A and B as above, whereas the sufficiency is proved to be valid for any skew-symmetric matrix B and for all complex square matrices A whose canonical form for congruence (CFC) does not contain blocks . The provided condition improves the one in [Borobia A, Canogar R, De Terán F. Lin Multilin Algebra, 2022. DOI:10.1080/03081087.2022.2093825], because it includes the case where CFC(A) includes symmetric blocks, and it is given in terms of the size of A and the rank of its symmetric and skew-symmetric parts. More precisely, if A is , we prove that the equation is consistent if and only if , where is the dimension of .

Multi-Person Pose Estimation in the Wild: Using Adversarial Method to Train a Top-Down Pose Estimation Network

Recent studies estimate human anatomical key points through the single monocular image, in which multichannel heatmaps are the key factor in determining the quality of human pose estimation. Multichannel heatmaps can efficiently handle the image-to-coordinate mapping task and the processing of semantic features. Most methods ignore physical constraints and internal relationships of human body parts, which easily misclassify left and right symmetrical parts as similar features. Some studies use RNNs on the top to incorporate priors about the structure of pose components and body configuration. Therefore, a novel top-down convolutional network is proposed to consider these priors during training, which can improve the robustness under complex field conditions in the wild. In order to learn the prior knowledge of human pose configuration, the hierarchy of fully convolutional networks (discriminator) is used to distinguish real poses from fake ones. Consequently, the pose network is inclined to make a pose estimation that the discriminator misjudges as true, which is reasonable in complex situations. The performance of the method is experimentally validated by pose estimation on the MS COCO human key point detection task. The proposed approach outperforms the original method and generates robust pose predictions, demonstrating efficiency by using adversarial learning.

Spontaneous vacuum decay in low-energy collisions of heavy nuclei beyond the monopole approximation

The problem of spontaneous vacuum decay in low-energy collisions of heavy nuclei is considered beyond the scope of the monopole approximation. The time-dependent Dirac equation is solved in a rotating coordinate system with $z$-axis directed along the internuclear line and the origin placed at the center of mass. The probabilities of electron-positron pair creation and the positron energy spectra are calculated in the approximation neglecting the rotational coupling. The two-center potential is expanded over spherical harmonics and the convergence with respect to the number of terms in this expansion is studied. The results show that taking into account the two-center potential instead of its spherically symmetric part preserves all the signatures of the transition to the supercritical regime that have been found in the framework of the monopole approximation and even enhances some of them.

Suppression of Wave Instability in a Liquid Film Flow Down a Non-Uniformly Heated Slippery Inclined Plane Using Odd Viscosity

Abstract We study the effects of odd viscosity on the stability of a thin Newtonian liquid film flowing down a nonuniformly heated plane under a slip boundary condition. The effect of odd viscosity arises in classical fluids when the time-reversal symmetry breaks down. Due to the odd viscosity, the odd part of the Cauchy stress tensor consists of symmetric and antisymmetric parts and shows several striking effects. We apply the Navier slip boundary condition for the slippery inclined plane at the solid–liquid interface. For our problem, we first derive an evolution equation whose solution describes the film thickness. The equation contains parameters considering the effect of inertia, thermocapillarity, slip length, and odd viscosity. We then perform the linear stability analysis and find that odd viscosity can significantly suppress the combined destabilizing effects of the thermocapillarity and slip length. Next, we analyze the dynamics using the weakly nonlinear approach, which provides details of different subregions of the instability zone. We observe that as the influence of the odd viscosity increases, the supercritical stable and explosive zones shrink while the unconditional stable and subcritical unstable zones expand. We also perform numerical investigation and observe that linear analysis, weakly nonlinear theory, and numerical results are consistent.

The average conformation tensor of inter-atomic bonds as an alternative state variable to the strain tensor: definition and first application — the case of nanoelasticity

Most of the mechanical models for solid state materials are in a methodological framework where a strain tensor, whatever it is, is considered as a thermodynamic state variable. As a consequence, the Cauchy stress tensor is expressed as a function of a strain tensor—and, in many cases, of one or more other state variables, such as the temperature. Such a choice for the kinematic state variable is clearly relevant in the case of infinitesimal or finite elasticity. However, one can ask whether an alternative state variable could not be considered. In the case of finite elastoplasticity, the choice of a strain tensor as the basic, kinematic state variable is not totally without issue, in particular in relation to the physical meaning of the internal state variable describing the permanent strains. In any case, this paper proposes an alternative to the strain tensor as a state variable, which is not based on the deformation (Lagrangian) gradient: the average conformation tensor of inter-atomic bonds. The purpose, however, is restricted to (1) a particular type of materials, namely the pure substances (copper or aluminum, for instance), (2) the nanoscale, and (3) the case of elasticity. The very simple case of two atoms of a pure substance in the solid state is first considered. It is shown that the kinematics of the inter-atomic bond can be characterized by a so called ``conformation'' tensor, and that the tensorial internal force acting on it can be immediately deduced from a single scalar function, depending only on the conformation tensor: the state potential of free energy (or interaction potential). Using an averaging procedure, these notions are then extended to a finite set of atoms, namely an atom and its first neighbours, which can be seen as the ``unit cell'' of a pure substance in the solid state considered as a discrete medium. They are also transposed to the Continuum case, where an expression of the Cauchy stress tensor is proposed as the first derivative of a state potential of density (per unit mass) of average free energy of inter-atomic bonds, which is an explicit function of the average conformation tensor of inter-atomic bonds. By applying a standard procedure in Continuum Thermodynamics, it is then shown that the objective part of the material derivative of this new state variable, at least in the case when the pure substance can be considered as an elastic medium, is equal to the symmetric part of the Eulerian velocity gradient, that is the rate of deformation tensor. In the case of uniaxial tension, a simple relationship is eventually set out between the average conformation tensor and a strain tensor, which is correctly approximated by the usual infinitesimal strain tensor as long as the conformation variations (from an initial state of conformation) are ``small''. From this latter result, and assuming an elastic behavior, a simple expression for the state potential of density of average free energy is inferred, showing great similarities with—but not equivalent to—the classical model of isotropic, linear elasticity (Hooke's law).

Analytical Study on the Random Seismic Responses of an Asymmetrical Suspension Structure

An asymmetrical suspension structure, without vertical column support and without supplying the flexibility of spatial arrangement, is more sensitive to ground movement. The structural responses of an asymmetrical suspension structure subjected to Clough–Penzien spectrum excitation were analytically investigated in this study. First, the governing equation was decoupled into an independent state equation in generalized coordinates through the real mode decomposition method and by creatively combining it with finite element methods to acquire modal coefficients. Through the pseudo excitation method (PEM), the frequent domain solution of the dynamic response was acquired, and its power spectrum density function was then quadratically decomposed to obtain its corresponding 0–2-order spectral moments. A practical case study was performed to verify the high accuracy and computational efficiency of the proposed closed-form solution comparative to the traditional PEM. Finally, an extended analysis of the effect of the suspended span and comparisons to a normal framed structure and symmetrical suspension structures were carried out. The analysis results indicate that the larger suspended span could consume more seismic energy and result in smaller horizontal displacement and acceleration. Moreover, the comparison results also point out that the existence of the suspension part showed better seismic energy dissipation capacity compared to the normal framed structure, and two symmetrical suspension parts also performed better than a single asymmetrical part in seismic energy dissipation.

Weak-strong uniqueness and vanishing viscosity for incompressible Euler equations in exponential spaces

In the class of admissible weak solutions, we prove a weak-strong uniqueness result for the incompressible Euler equations assuming that the symmetric part of the gradient belongs to Lloc1([0,+∞);Lexp(Rd;Rd×d)), where Lexp denotes the Orlicz space of exponentially integrable functions. Moreover, under the same assumptions on the limit solution to the Euler system, we obtain the convergence of vanishing-viscosity Leray–Hopf weak solutions of the Navier–Stokes equations.

On the Kelvin–Voigt model in anisotropic viscoelasticity

We propose an anisotropic and nonlinear generalization of the Kelvin–Voigt viscoelastic model obtained considering the additive splitting of the Cauchy stress tensor in an elastic and a dissipative part. The former one corresponds to a fiber-reinforced hyperelastic material while the dissipative effect is described by the most general linear transverse-isotropic tensorial function of symmetric part of the velocity gradient. In a such a way we characterize the dissipative contribution via three viscoelastic moduli. We then show, by a detailed analysis of the simple shear quasistatic motion and the corresponding creep phenomena, that this motion may be used to determine experimentally the viscoelastic parameters.

Designing Highly Precise Overlay Targets for Asymmetric Sidewall Structures Using Quasi-Periodic Line Widths and Finite-Difference Time-Domain Simulation.

The present study introduces an optimized overlay target design to minimize the overlay error caused by asymmetric sidewall structures in semiconductor manufacturing. To achieve this goal, the overlay error formula was derived by separating the asymmetric bottom grating structure into symmetric and asymmetric parts. Based on this formula, it was found that the overlay target design with the linewidth of the bottom grating closed to the grating period could effectively reduce the overlay error caused by the sidewall asymmetry structure. Simulation results demonstrate that the proposed design can effectively control the measurement error of different wavelengths within ±0.3 nm, even under varying sidewall angles and film thicknesses. Overall, the proposed overlay target design can significantly improve the overlay accuracy in semiconductor manufacturing processes.

Affleck-Dine cogenesis of baryon and dark matter

We propose a mechanism for cogenesis of baryon and dark matter (DM) in the universe via the Affleck-Dine (AD) route. An AD field which breaks the lepton number symmetry, leads to the generation of lepton asymmetry by virtue of its cosmic evolution, which then gets transferred into lepton and dark sectors. While the lepton asymmetry gets converted into baryon asymmetry via sphalerons, the dark sector asymmetry leads to the final DM abundance with the symmetric part being annihilated away due to resonantly enhanced annihilation, which we choose to be provided by a gauged B − L portal. Stringent constraints from DM direct detection forces DM and B − L gauge boson masses to be light, in the few GeV ballpark. While a large portion of the model parameter space is already ruled out, the remaining parameter space is within sensitivity of laboratory as well as cosmology based experiments. The AD field also plays the role of inflaton with the required dynamics by virtue of its non-minimal coupling to gravity, consistent with observations.

Fractional Higher Differentiability for Solutions of Stationary Stokes and Navier-Stokes Systems with Orlicz Growth

We consider weak solutions (u,π):Ω→ℝn×ℝ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(u,\\pi ):{\\Omega }\ o \\mathbb {R}^{n}\ imes \\mathbb {R}$\\end{document} to stationary ϕ-Navier-Stokes systems of the type −diva(x,Eu)+∇π+[Du]u=fdivu=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$ \\left \\{ \\begin {array}{ll} -\\mathrm {div~} a(x,\\mathcal {E} u)+\ abla \\pi +[Du]u=f \\\\ \\mathrm {div~} u=0 \\end {array} \\right . $\\end{document} in Ω⊂ℝn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\Omega }\\subset \\mathbb {R}^{n}$\\end{document}, and to the corresponding ϕ-Stokes systems, in which the convective term [Du]u does not appear. In the above system, the function a(x,ξ) depends Hölder continuously on x and satisfies growth conditions with respect to the second variable expressed through a Young function ϕ. The notation Eu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {E} u$\\end{document} is used for the symmetric part of the gradient Du. We prove results on the fractional higher differentiability of both the symmetric part of the gradient Eu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {E} u$\\end{document} and of the pressure π.

Spin-lattice interaction parameters from first principles: Theory and implementation

A scheme is presented to calculate on a first-principles level the spin-lattice coupling (SLC) parameters needed to perform combined molecular-spin dynamics (MSD) simulations. By treating changes to the spin configuration and atomic positions on the same level, closed expressions for the atomic SLC parameters could be derived in a coherent way up to any order. The properties of the SLC parameters are discussed considering separately the symmetric and antisymmetric parts of the SLC tensor. The changes due to atomic displacements of the spin-spin exchange coupling (SSC) parameters estimated using the SLC parameters are compared with the SSC parameters calculated for an embedded cluster with the central atom displaced, demonstrating good agreement of these results. Moreover, this allows to study the impact of different SLC contributions, linear and quadratic with respect to displacements, on the properties of the modified SSC parameters. In addition, we represent an approach to calculate the site-diagonal SLC parameters characterizing local magnetic anisotropy induced by a lattice distortion, which is a counterpart of the approach based on magnetic torque used for the investigations of magneto-crystalline anisotropy (MCA) as well as for calculations of the MCA constants. In particular, the dependence of the induced magnetic torque on different types of atomic displacements is analyzed.