Comparison of different symmetry indices for the quantification of dynamic joint angles

Background Symmetry is a sign of physiological and healthy movements, as pathologies are often described by increased asymmetries. Nevertheless, based on precisely measured data, even healthy individuals will show small asymmetries in their movements. However, so far there do not exist commonly accepted methods and reference values for gait symmetry in a healthy collective. Therefore, a comparison and presentation of reference values calculated by 3 different methods of symmetry indices for lower limb joint angles during walking, ascending, and descending stairs were shown. Methods Thirty-five healthy participants were analyzed during walking, ascending, and descending stairs with the help of the inertial measurement system MyoMotion. Using the normalized symmetry index (SInorm), the symmetry index (SI) as the integral of the symmetry function, and another normalized symmetry index (NSI), the symmetry of joint angles was evaluated. For statistical evaluation of differences, repeated measurement models and Bland–Altman-Plots were used. Results Apart from a bias between the symmetry indices, they were comparable in the predefined limits of 5%. For all parameters, significantly higher asymmetry was found for ankle dorsi/-plantarflexion, compared with the hip and knee flexion. Moreover, the interaction effect of the joint and movement factors was significant, with an increased asymmetry of the hip and knee during descending stairs greater than while ascending stairs or walking, but a reduced symmetry of the ankle during walking when compared to descending. The movement only showed significant effects when analyzing the SInorm. Conclusion Even for healthy individuals, small asymmetries of movements were found and presented as reference values using 3 different symmetry indices for dynamic lower limb joint angles during 3 different movements. For the quantification of symmetrical movements differences between the joints, movements, and especially their interaction, are necessary to be taken into account. Moreover, a bias between the methods should be noted. The potential for each presented symmetry index to identify pathological movements or track a rehabilitation process was shown but has to be proven in further research. Trial registration: DRKS00025878.

Midpoint Inequalities in Fractional Calculus Defined Using Positive Weighted Symmetry Function Kernels

The aim of our study is to establish, for convex functions on an interval, a midpoint version of the fractional HHF type inequality. The corresponding fractional integral has a symmetric weight function composed with an increasing function as integral kernel. We also consider a midpoint identity and establish some related inequalities based on this identity. Some special cases can be considered from our main results. These results confirm the generality of our attempt.

Symmetry function in gait pattern analysis in patients after unilateral transfemoral amputation using a mechanical or microprocessor prosthetic knee

Background Above-knee amputations (AKAs) contribute to gait asymmetry. The level of asymmetry is affected by the type of knee prosthetic module. There is limited evidence suggesting that more technically advanced solutions (microprocessor modules; MicPK) are superior to less advanced ones (mechanical modules; MechPK). The study aimed to evaluate the variable range of hip and pelvic joint movements during gait and look for differentiating areas with an increased level of asymmetry of the gait cycle in individuals who underwent an AKA and are equipped with MicPK or MechPK. Methods Twenty-eight individuals participated in the study; 14 were assigned to a study group of individuals who underwent a unilateral AKA, and the other 14 were healthy participants as a control group. The movement task was recorded using the optoelectronic SMART-E system following the standard Davis protocol (the Newington model). A new method of quantifying gait symmetry using the symmetry function (SF) is proposed. SF is an integral measure of absolute differences in time-standardized signals between sides throughout the whole cycle of motion variability. Results In the frontal plane, there were significant differences between groups in the asymmetry of the range of movement in the hip joint of the intact limb. In the middle of the support phase, the intact limb was more adducted in individuals with MicPK and less abducted in people with MechPK (differences in mean 180%, p < 0.000; max 63%, p < 0.000; min 65%, p < 0.000). In the sagittal plane, the range of asymmetry of the flexion and thigh extension of the intact limb was similar to and only slightly different from the physiological gait. In the transverse plane, higher asymmetry values were noted for individuals with MicPK. In the final stage of the swing phase, the thigh was more rotated both externally and internally. The size of the asymmetry, when compared to gait of healthy individuals, reached 50% (differences in mean 115%, p < 0.232; max 62% p < 0.26; min 50, p < 0.154). Conclusions In the study group, the assessed ranges of pelvic and thigh movement in the hip joint differed only in the frontal plane. Individuals who underwent a unilateral above-knee amputation may gain less from using MicPK than anticipated.

FUNGSI SIMETRI TERHADAP GARIS x = a DAN SIFAT-SIFATNYA

An even function is a function with a graph that is symmetric with respect to the y-axis or the line x = 0. In this paper, we will introduce a more general function of the even function, we call it as symmetry function with respect to the line x = a, which is a function whose graph is symmetric with respect to the line x = a. This paper discusses the properties of the symmetry function with respect to the line x = a, which is derived from the pre-existing properties of the even function. Some of the results obtained above, the linear combination of the symmetry functions with respect to the line x = a is a symmetry function with respect to the line x = a and the composition of any function with a symmety function with respect to the line x = a is a symmetry function with respect to the line x = a. Keywords: Even function, a symmetry function with respect to the line x=a, symmetric graph

Symmetry function as a new tool for evaluating the symmetry of gait in transfemoral amputees

Background: Numerous studies have demonstrated significant asymmetries in unilateral amputee gait. The underlying dissimilarities between prosthetic and intact limbs have not yet been widely examined. To gain more insight into the functionality of asymmetries, we propose a new tool, the symmetry function (SF), to evaluate the symmetry of walking in terms of kinematic and dynamic variables of patients after unilateral transfemoral amputation and to identify areas with the largest side deviations in the movement cycle. Methods: An instrumented motion analysis system was used to register the gait of fourteen patients after unilateral trans-femoral amputation (TFA). Measurements involved evaluating the time series of gait variables characterizing a range of motion and the time series of the ground reaction force components. Comparison of the involved limb with the uninvolved limb in TFA patients was carried out on the basis of the SF values.Results: The symmetry function proved to be an excellent tool to localize the regions of asymmetry and their positive or negative directions in the full gait cycle. The difference between sides revealed by the symmetry function was the highest for the pelvis and the hip. In the sagittal plane, the pelvis was asymmetrically tilted, reaching the highest SF value of more than 25% at 60% cycle time. In the transverse plane, the pelvis was even more asymmetrically positioned throughout the entire gait cycle (50% difference on average). The hip in the frontal plane reached a 60% difference in SF throughout the single support phase for the prosthetic and then for the intact limb. Conclusions: The symmetry function allows for the detection of gait asymmetries and shifts in the center of gravity and may assess the precise in time adaptation of prostheses and rehabilitation monitoring, especially in unilateral impairments.Trial registration: The trial registration number (TRN): 379991 issued by the Australian New Zealand Clinical Trials Registry (ANZCTR) on 07.05.2020 (retrospectively registered).

Schur Convexity and Inequalities for a Multivariate Symmetric Function

In the article, we provide the Schur, Schur multiplicative, and Schur harmonic convexities properties for the symmetry function Fnx,r=Fnx1,x2,⋯,xn;r=∏1≤i1<i2<⋯<ir≤n ∑j=1r xij/1−xij1/r on 0,1n and find several new analytical inequalities by use of the majorization theory, where x=x1,⋯,xn∈0,1n, r=1,2,⋯,n and i1,i2,⋯,in are positive integers.

Applying Axial Symmetries to Historical Silk Fabrics: SILKNOW’s Virtual Loom

Symmetry is part of textile art in patterns and motifs that decorate fabrics, which are made by the interlacement of warp and wefts. Moreover, the 3D representation of fabrics have already been studied by some authors; however, they have not specifically dealt with preserving historical weaving techniques. In this paper, we present the SILKNOW’s Virtual Loom, a tool intended to document, preserve and reproduce silk historical weaving techniques from the 15th to the 19th centuries. We focus on the symmetry function and its contribution to art history, textile conservation, and modern design. We analyzed 2028 records from Garin 1820 datasets—a historical industry that still weaves with these techniques—and we reconstructed some historical designs that presented different types of defects. For those images (including fabrics and drawings) that had a symmetrical axis, we applied the symmetry functionality allowing to reconstruct missing parts. Thanks to these results, we were able to verify the usefulness of the Virtual Loom for conservation, analysis and new interpretative advantages, thanks to symmetry analysis applied to historical fabrics.

Atomic partial charge predictions for furanoses by random forest regression with atom type symmetry function

Atom type symmetry function that utilizes atom types defined in traditional force fields demonstrated improvements for describing structures of furanoses, and the capability of predicting their conformational adaptive charges with random forest regression models.