Invariant Conditions Research Articles

From Evidence to Practice: Denosumab in Osteoporosis and Joint Health

Osteoporosis and osteopenia are prevalent conditions in India, particularly among postmenopausal women and the elderly,leading to increased fracture risks and morbidity. Denosumab, a monoclonal antibody, demonstrates superior efficacyover other treatments in reducing fracture risk and enhancing bone mineral density (BMD). Clinical trials highlight itseffectiveness in preventing periprosthetic bone loss, improving implant stability, and mitigating femoral head collapse invarious conditions; combination therapies involving denosumab further amplify BMD gains and reduce fracture risk. Thearticle reviews the efficacy and safety of denosumab in managing osteoporosis, joint replacement therapy, and avascularnecrosis based on a review of clinical evidence and studies. Denosumab emerges as a cornerstone therapy for osteoporosismanagement, offering multifaceted benefits in fracture prevention, joint replacement therapy, and avascular necrosis.

The anti-self-dual deformation complex and a conjecture of Singer

Abstract Let ( M 4 , g ) (M^{4},g) be a smooth, closed, oriented anti-self-dual (ASD) four-manifold. ( M 4 , g ) (M^{4},g) is said to be unobstructed if the cokernel of the linearisation of the self-dual Weyl tensor is trivial. This condition can also be characterised as the vanishing of the second cohomology group of the ASD deformation complex, and is central to understanding the local structure of the moduli space of ASD conformal structures. It also arises in construction of ASD manifolds by twistor and gluing methods. In this article, we give conformally invariant conditions which imply an ASD manifold of positive Yamabe type is unobstructed.

Recognizing Sign Language using Machine Learning and Deep Learning Models

Individuals with hearing impairments communicate mostly through sign language. Our goal was to create an American Sign Language recognition dataset and utilize it in a neural network-based machine learning model that can interpret hand gestures and positions into natural language. In our study, we incorporated the SVM, CNN and Resnet-18 models to enhance predictability when interpreting ASL signs through this new dataset, which includes provisions such as lighting and distance limitations. Our research also features comparison results between all the other models implemented under invariant conditions versus those using our proposed CNN model. As demonstrated by its high levels of precision at 95.10% despite changes encountered during testing procedures like varying data sets or scene configurations where losses are minimal (0.545), there exists great potential for future applications in image recognition systems requiring deep learning techniques. Furthermore, these advancements may lead to significant improvements within various fields related explicitly to speech-language therapy sessions designed specifically around helping people overcome challenges associated with deafness while building bridges towards improved social integration opportunities.

Stochastic Partial Differential Equations and Invariant Manifolds in Embedded Hilbert Spaces

AbstractWe provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional Itô diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs.

Balance function generated by limiting conditions

This article conducts an analysis of the inherent constraints governing the formation of the price function that describes the interaction between two markets. The research not only identifies these constraints but also obtains an explicit form of the specified function.The key factors considered in constructing the price function are defined in the article. Through analyzing these constraints and their impact on market interaction, a formula for the price function is provided. This approach not only reveals the essence of natural constraints in forming the price function but also provides a contextual foundation for negotiations shaping a fair exchange price for the interaction process between two markets. This offers a theoretical basis for modeling and solving similar problems arising during practical economic activities.Two economies, Economy 1 and Economy 2, producing goods X and Y with linear Production Possibility Curve (PPC) graphs, are under consideration. The cost of producing one unit of good X relative to Y is denoted as R1 for Economy 1 and R2 for Economy 2. Exchange between economies occurs in a market, where the possible exchange is Δx units of X for Δy = Rmarket · Δx units of Y, and vice versa.If R1 is less than R2, Economy 1 specializes in the production of X, and Economy 2 specializes in Y, fostering mutually beneficial trade. For mutually beneficial exchange on the market with a price Rmarket, it is necessary and sufficient that R1 ≤ Rmarket ≤ R2.The article also explores the concept of a fair exchange price, specifying conditions for symmetry, reciprocity, and scale invariance. Notably, it indicates that the unique solution satisfying these conditions is f(R1,R2) = √ R1 · R2.In the context of balanced exchange, where economies gain equal profit per unit of the acquired good, the balanced exchange price Rmarket[balance] is determined as Rmarket = √ R1 · R2. This serves as a fair price, meeting the aforementioned conditions of symmetry, reciprocity, and scale invariance.In the provided example with R1 = 2 and R2 = 8, the article examines the mutually beneficial interval for Rmarket and computes the balanced and fair exchange price.

Small-Scale Cosmology Independent of the Standard Model

‘Small-scale cosmology’ is a theory designed to incorporate the linear redshift versus distance relation, which is inferred from observations, into the theoretical framework independent of the global Robertson–Walker–Friedman (RWF)-type models. The motivation behind this is that the RWF cosmological models, based on the assumptions of homogeneity and a constant matter density, as well as the concept of expanding space inherent to them are not applicable on the scales of observations from which the linear Hubble law is inferred. Therefore, explaining the Hubble law as the small redshift limit of the RWF model or as an effect of expanding space is inconsistent. Thus, the Hubble linear relation between the redshift of an extragalactic object and its distance should be considered an independent law of nature valid in the range of the distances where the RWF cosmology is not valid. In general, the theory, based on that concept, can be developed in different ways. In the present paper, ‘small-scale cosmology’ is formulated as a theory operating in the (redshift–object coordinates) space, which allows developing a conceptual and computational basis of the theory along the lines of that of special relativity. In such a theory, the condition of invariance of the Hubble law with respect to a change in the observer acceleration plays a central role. In pursuing this approach, the effectiveness of group theoretical methods is exploited. Applying the Lie group method yields transformations of the variables (the redshift and space coordinates of a cosmological object) between the reference frames of the accelerated observers. In this paper, the transformations are applied to studying the effects of the solar system observer acceleration on the observed shape, distribution and rotation curves of galaxy clusters.

MHD free convection with Joule heating and entropy generation inside an H-shaped hollow structure

In this research, a free convective flow of water inside an H-shaped hollow structure which is subjected to the existence of an exterior magnetic field and Joule heating is computationally investigated. The structure's right and left upright surfaces are maintained at invariant ambient thermal condition, while the top and bottom-most surfaces of the structure are in adiabatic condition. The rest of the inner walls are heated isothermally. Computational analysis is carried out for different configurations of the chamber by solving Navier-Stokes and heat energy equations via the finite element approach. Parametric computations are conducted by varying Hartmann numbers (0 ≤ Ha ≤ 20), Rayleigh numbers (103 ≤ Ra ≤ 106), width of the vertical sections (0.2 ≤ d/L ≤ 0.4, where L denotes the structure's reference dimension), and thickness of the horizontal middle section (0.2 ≤ t/L ≤ 0.4). To find out the impact of the governing parameters on thermal performance for different configurations, the mean Nusselt number along the hot walls, mean temperature of fluid, overall entropy generation, and thermal performance criterion are assessed. In addition, the variations in fluid motion and thermal patterns are reported in terms of streamlines, isotherms, and heatlines. With a larger mean Nusselt number and smaller thermal performance criterion, better heat transmission performance is found for thicker horizontal middle section and wider vertical sections. The maximum reduction in thermal performance criterion is found to be 87.8 % for increasing the width of the vertical sections. However, in the cases of Ha and d/L, there is an interesting transition in Nusselt number noticed for different Rayleigh numbers.

Two characterizations of the dense rank

In this paper, we have considered the dense rank for assigning positions to alternatives in weak orders. If we arrange the alternatives in tiers (i.e., indifference classes), the dense rank assigns position 1 to all the alternatives in the top tier, 2 to all the alternatives in the second tier, and so on. We have proposed a formal framework to analyze the dense rank when compared to other well-known position operators, such as the standard, modified and fractional ranks. As the main results, we have provided two different axiomatic characterizations which determine the dense rank by considering position invariance conditions along horizontal extensions (duplication), as well as through vertical reductions and movements (truncation, and upward or downward independency).

Delay-dependent set-invariance for linear difference equations with multiple delays: a polyhedral approach

This paper revisits the set-invariance for linear delay-difference equations and proposes novel delay-dependent notions in contrast with the existing delay-independent constructions available in the literature. The main tool in this endeavour is the model transformation by means of a matrix parametrization, which opens the way to the elaboration of delay-dependent conditions for set invariance. For the class of polyhedral sets, it will be shown how these transformed models enable particularly structured invariance conditions. Aside from the mathematical developments, a discussion of the delay-dependent conditions with respect to the confinement of the state trajectories of the original system will be presented. The relationship will be established by means of constraints involving the initial states for the time-delay systems. The characterization of this set of admissible initial conditions can be analysed in terms of complexity and conservativeness with respect to the classical delay-independent set-invariance. An illustrative example is provided to underline that confinement of state trajectories in a set can be achieved even though this is not invariant according to the classical definition.

The generalized sectorial decompositions of semi-sectorial operators

This paper considers the generalized sectorial decompositions of semi-sectorial operators and quasi-sectorial operators on the complex Hilbert space. Under an invariant subspace condition, a necessary and sufficient condition for the existence of the generalized sectorial decomposition of a semi-sectorial operator is given. The generalized sectorial decomposition of a quasi-sectorial operator is given without any conditions. Some sufficient conditions for a block upper triangular operator matrix to be a semi-sectorial operator or a quasi-sectorial operator are given.

Synthesis of ℋ∞ Control for Descriptor Hybrid Systems with Actuator Saturation

This paper addresses a mode-dependent state-feedback H∞ control for stochastic descriptor hybrid systems, considering both the absence and presence of actuator saturation. Firstly, the necessary and sufficient conditions for the stochastic admissibility criterion with H∞ performance γ of the closed-loop system are proposed. Given the proposed non-convex condition, the author reformulates it into linear matrix inequalities (LMIs). Then, to extend the result to the systems with actuator saturation, the actuator-saturated control input is expressed as a linear combination of a given state-feedback control input and a virtual control input that always remains under the saturation level. To verify this expression, the set invariant condition is also suggested by using the singular mode-dependent Lyapunov function candidate. Therefore, the conditions for the existence of both the mode-dependent state-feedback H∞ control and the ellipsoidal shape invariant sets are successfully derived in terms of LMIs. Two numerical examples demonstrate the effectiveness of the proposed method by solving optimization problems subject to the proposed LMIs that minimize H∞ performance γ and maximize the invariant set, respectively.

Measurement invariance of the satisfaction with leisure satisfaction scale by gender, marital status, and age

BackgroundLeisure satisfaction is the degree of positive perception and emotions that an individual acquires as a result of participating in leisure time activities, and it has an important function in maintaining and increasing leisure time participation. Some studies on leisure satisfaction address the comparisons between groups. These studies are based on the premise that the measurement tool used to reveal the between-group differences measures the same feature in subgroups.ObjectiveIn this study, we investigated whether the differences between the groups were due to the measurement tool by examining the psychometric properties of the leisure satisfaction scale.MethodsThe study sample comprised 2344 exercising individuals, including 1228 (52.3%) women and 1116 (47.6%) men. The structural invariance of the leisure satisfaction scale, developed by Beard and Ragheb (Journal of Leisure Research 12:20-33, 1980) and adapted into Turkish by Gökçe and Orhan (Spor Bilimleri Dergisi 22:139–145, 2011), was tested through multiple-group confirmatory factor analysis.ResultsThe results indicated that the structural and metric invariance conditions were fulfilled across gender, marital status, and age in all subscales of the leisure satisfaction scale. Scalar invariance was obtained in educational and social satisfaction subscales across gender and in physical satisfaction subscale across marital status.ConclusionThe study provides evidence for the future comparisons according to these three variables, indicating that the differences obtained will result from the real differences between groups rather than the measurement tool properties.

Shifted model spaces and their orthogonal decompositions

We generalize certain well known orthogonal decompositions of model spaces and obtain similar decompositions for the wider class of shifted model spaces, allowing us to establish conditions for near invariance of the latter with respect to certain operators which include, as a particular case, the backward shift \(S^*\). In doing so, we illustrate the usefulness of obtaining appropriate decompositions and, in connection with this, we prove some results on model spaces which are of independent interest. We show moreover how the invariance properties of the kernel of an operator \(T\), with respect to another operator, follow from certain commutation relations between the two operators involved.

Necessary and sufficient Conditions of Forward Invariance in Hybrid Dnamical Systems: An LMI Approach*

This study focuses on the characterization of the forward invariance property in hybrid dynamical systems, considering state-dependent conditions for both differential and difference inclusions. The first step is to establish necessary and sufficient conditions for (controlled) forward invariance, using cones and set inclusions. For linear hybrid systems, an explicit characterization of the invariance property of polyhedral sets is given in terms of linear matrix inequalities. The result is illustrated by a practical example involving a bouncing ball.

A comparison of finiteness conditions in quadratic form theory

We discuss and relate finiteness conditions for certain field invariants which are studied in quadratic form theory. This includes the [Formula: see text]-invariant, the reduced stability index and the symbol lengths for Galois cohomology groups with coefficients in [Formula: see text], as well as a new invariant called the splitting height.

Non-Classical Symmetry Analysis of a Class of Nonlinear Lattice Equations

In this paper, a non-classical symmetry method for obtaining the symmetries of differential–difference equations is proposed. The non-classical symmetry method introduces an additional constraint known as the invariant surface condition, which is applied after the infinitesimal transformation. By solving the governing equations that satisfy this condition, we can obtain the corresponding reduced equation. This allows us to determine the non-classical symmetry of the differential–difference equation. This method avoids the complicated calculation involved in extending the infinitesimal generator and allows for a wider range of symmetry forms. As a result, it enables the derivation of a greater number of differential–difference equations. In this paper, two kinds of (2+1)-dimensional Toda-like lattice equations are taken as examples, and their corresponding symmetric and reduced equations are obtained using the non-classical symmetry method.

Elucidating deformation pathways and interface characteristic of self-accommodated dual γ/ε phase microstructure in Fe–Mn–Si–Al alloy

There has been an intense scientific interest in investigating the phenomenon of deformation-induced ε-ε martensite (hcp) interaction owing to the thermodynamically paradoxical reverse transformation and mechanical twinning. In this study, detailed transmission electron microscopy has been employed to examine the crystallographic orientation relationship, phase stability and boundaries between the intersection phases on the ε-ε intersections in a 10% tensile deformed Fe–30Mn–4Si–2Al alloy. The transformation/twinning scheme is systematically summarized initiating from the austenite matrix (γ) and two deformation-induced ε-martensite variants. It involves diverse intersection reactions: mechanical ε-twins, a 90°-rotating γ-phase from the γ matrix (γR), and 90°-rotate ε-phase re-transformed from the intersection γ. All these phases share a common 〈101〉γ || <21¯1¯0>ε axis that is equivalent to the intersection axis of the crossing {111}<12¯1>γ shears and interrelated with rotational angles with respect to the axis. Electron diffraction, coupled with stereographic analysis, revealed the distinct orientation relationships between γ–ε and γR–ε phases. The boundaries between the intersection γR and neighboring ε-phase are inclined from the corresponding {111}γ||{101¯1}ε planes. By means of the phenomenological theory of martensite crystallography (PTMC), the inclination of the boundary is rationalized by considering the lattice-invariant shear on the double {111}γ plane inside the intersection γ to satisfy the invariant plane condition of the boundary.

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.

Lie symmetry analysis for the Cargo–Leroux model with isentropic perturbation pressure equation of state

In this article, the well-known Cargo–Leroux model with isentropic perturbation equation of state is analyzed using the Lie symmetry method. By using invariant conditions of system of partial differential equations, six dimensional Lie algebra is obtained. The optimal system for system of partial differential equations is constructed using adjoint representation and the invariants of associated Lie algebras of the system. Further, with the help of one-dimensional optimal system invariant solutions are constructed. Also, physically significant solutions such as traveling wave solutions, specifically the kink-type solitons and peakon-type solitons are obtained by using traveling wave transformations and all the solutions are graphically demonstrated. Finally, the hyperbolic nature of system of partial differential equations is examined by studying the evolutionary behavior of a discontinuity wave.

MOG as symmetry breaking in scalar–vector–tensor gravity

ABSTRACT The Modified Gravity Model (MOG) has been proposed as a solution to the dark matter problem, but it does not meet the gauge invariant condition. The aim of this work is to propose a gauge-invariant theory, which suggests that symmetry can break at a low temperature in the Universe, leading to the MOG theory. This theory has the potential to alter the dynamics of the early and late Universe and naturally produce cosmological inflation.