Torsion Function Research Articles

Analyzing Torsion Functions under Robin and Dirichlet Boundary Conditions

In this paper, the author introduces the mathematical concepts regarding torsion functions and their derivations under the Robin and Dirichlet boundary conditions which are integral in mechanical engineering, theoretical physics, and other disciplines. Functions that satisfy given or required partial differential equations contain a wealth of information about materials or structures when subjected to stress. We examine these functions under two types of boundary conditions: Robin, which contains Dirichlet and Neumann conditions, Dirichlet that provides conditions on the boundary. Also, in this section, we formally describe the mathematical model used in the study and provide some definitions, notations, and basic propositions that form the foundation for subsequent analysis. In answer to such concerns, the torsion function problems solved herein utilize analytical and numerical computations to reveal the differences and similarities in behavior between the two boundary conditions. Strong empirical evidence further supports our findings, proving the real-life relevance of the revealed theoretical concepts. Lastly, the implications for these findings in the real-world application are also presented and analysed with regards to advantages and disadvantages of various boundary conditions in the engineering of systems. We end this paper with a presentation of the general implications of the results attained from the work done and possible expansions and directions for future studies as a way of continuing the discourse on the theoretical analysis and actual applications. In summary, this paper provides a useful perspective in the analysis of torsion functions solution under mixed boundary conditions which will be of interest for further development of the theory and application into practice.

Two-sided Lieb–Thirring bounds

We prove upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. As a corollary, we obtain two-sided estimates for the sum of the negative eigenvalues of atomic Hamiltonians with Kato potentials. Instead of being in terms of the potential itself, as in the usual Lieb–Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of (−Δ + V + M)uM = 1 in Rd; here M∈R is chosen so that the operator is positive. We further prove that the infimum of (uM−1−M) is a lower bound for the ground state energy E0 and derive a simple iteration scheme converging to E0.

Ahlfors regularity of continua that minimize maxitive set functions

The primary objective of this paper is to establish the Ahlfors regularity of minimizers of set functions that satisfy a suitable maxitive condition on disjoint unions of sets. Our analysis focuses on minimizers within continua of the plane with finite 1-dimensional Hausdorff measure. Through quantitative estimates, we prove that the length of a minimizer inside the ball centered at one of its points is comparable to the radius of the ball. By operating within an abstract framework, we are able to encompass a diverse range of entities, including the inradius of a set, the maximum of the torsion function, and spectral functionals defined in terms of the eigenvalues of elliptic operators. These entities are of interest for several applications, such as structural engineering, urban planning, and quantum mechanics.

On Location of Maximum of Gradient of Torsion Function

On Location of Maximum of Gradient of Torsion Function

Torsional vibration behavior of a restrained non-circular nanowire in an elastic matrix

This study introduces an approach to analyze torsional vibration in non-circular nanowires within magnetic fields, considering various boundary conditions on an elastic foundation. Analytical formulas for natural angular frequencies are obtained using nonlocal strain gradient theory. The analysis covers three non-circular cross-sections, incorporating the warping effect. Elastic springs at the wire ends simulate support conditions, restricting rotation around the wire’s axis. The torsion function around the axis is represented by a Fourier series, discretized at the spring points and linked using the Stokes’ transform alongside the boundary values. This leads to an eigenvalue problem that includes higher-order material size parameters (strain gradient, nonlocal), spring parameters, and the warping function. The study’s novelty lies in effectively solving torsional vibration for non-circular sections, addressing warping function, elastic medium, and size effects under both deformable and non-deformable boundary conditions. The presented solution is capable of determining vibration frequencies for both rigid and deformable boundary conditions. This is accomplished by specifying the torsional spring stiffness values, thereby obviating the necessity for additional recalculations. In order to verify the obtained results and compare them with the existing literature, nanowires with free and clamped boundary conditions are solved numerically by changing the spring parameters in the eigenvalue problem. In the formulation of natural angular frequency; length scale parameters, warping, magnetic field, elastic medium effects are included. In addition, since the support conditions are modeled with elastic springs, the resulting formulas are quite general and can be used to solve different types of torsional vibration problems.

Theoretical and observational prescription of warm-inflation in FLRW universe with torsion

The paper deals with Warm Inflationary scenario in FLRW with torsion both from theoretical and observational point of view. In the background of flat FLRW model the Hubble parameter is found to be proportional to the torsion function and warm inflation is examined both in weak and strong dissipation regimes for the power law choice of potential using slow-roll approximation with quasi-stable criteria for radiation. The slow-roll parameters, no.of e-folds, scalar spectral index, and tensor-to-scalar ratio are determined in the present model for mainly three choices of the dissipation coefficient Gamma using the Planck data set. Finally, we focus on single-field chaotic quartic potential with the above choices of the dissipation coefficient to confront the warm inflation observational predictions directly with the latest observational data set.

Polynomial Entropy and Polynomial Torsion for Fibered Systems

Given a continuous fibered dynamical system, we first introduce the notion of polynomial torsion of a fiber,which measures the “infinitesimal variation” of the dynamics between the fiber and the neighboring ones.This gives rise to an (upper semicontinous) torsion function,defined on the base of the system, which is a newC^{0} (fiber) conjugacy invariant. We prove that the polynomial entropy of the system is the supremum ofthe torsion of its fibers, which yields a new insight into the creation of polynomial entropy in fibered systems.We examine the relevance of these results in the context of integrable Hamiltoniansystems or diffeomorphisms, with the particular cases of C^{0}-integrable twist maps on the annulus and geodesic flows.Finally, we bound from below the polynomial entropy of ell-modal interval maps in terms of their lap number and answer a question by Gomes and Carneiro.

Saint-venant Torsion of Functional Graded Orthotropic Piezoelectric Hollow Circular Cylinder

This paper gives an analytical solution to the Saint-Venant torsion of the hollow and solid circular cylinder made of orthotropic functionally graded piezoelectric material. The elastic flexibility coefficients and piezoelectric constants and permittivities have only radial dependence. The considered material non-homogeneity is described by power function. The solution of the Saint-Venant torsion problem is presented for Prandtl’s stress function, shearing stresses, electric displacement potential function, torsion function and electric potential function. An example illustrates the application of the formulated analytical method.

A Rigidity Result for the Robin Torsion Problem

Let Omega subset mathbb {R}^2 be an open, bounded and Lipschitz set. We consider the torsion problem for the Laplace operator associated to Omega with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison, proved in Alvino et al. (Commun Pure Appl Math 76:585–603, 2023).. We prove that the equality is achieved only if Omega is a disk and the torsion function u is radial.

Symmetry and quantitative stability for the parallel surface fractional torsion problem

We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded open set Ω ⊂ R n \Omega \subset \mathbb R^n . More precisely, we prove that if the fractional torsion function has a C 1 C^1 level surface which is parallel to the boundary ∂ Ω \partial \Omega then Ω \Omega is a ball. If instead we assume that the solution is close to a constant on a parallel surface to the boundary, then we quantitatively prove that Ω \Omega is close to a ball. Our results use techniques which are peculiar of the nonlocal case as, for instance, quantitative versions of fractional Hopf boundary point lemma and boundary Harnack estimates for antisymmetric functions. We also provide an application to the study of rural-urban fringes in population settlements.

Testicular torsion and subsequent testicular function in young men from the general population

Is prior testicular torsion associated with testicular function (semen quality and reproductive hormones) in young men from the general population? In young men from the general population, no differences in semen parameters were observed in those who had experienced testicular torsion compared to controls and observations of higher FSH and lower inhibin B were subtle. Testicular function may be impaired after testicular torsion, but knowledge is sparse and based on studies with small sample sizes and no control group or a less than ideal control group. A cross-sectional population-based study was carried out including 7876 young Danish men with unknown fertility potential, examined from 1996 to 2020. All men (median age 19.0 years) had a physical examination, provided a blood and semen sample, and filled in a questionnaire including information about prior testicular torsion, birth, lifestyle and current and previous diseases. Markers of testicular function, including testis volume, semen parameters and reproductive hormones, were compared between men operated for testicular torsion and controls, using multiple linear regression analyses. The average participation rate was 24% for the entire study period. In total, 57 men (0.72%) were previously operated for testicular torsion (median age at surgery 13.4 years) of which five had only one remaining testicle. Men with prior testicular torsion were more often born preterm (25% versus 9.5% among controls), and they had significantly higher FSH and lower inhibin B levels, and a lower inhibin B/FSH ratio than controls in crude and adjusted models. The association was mainly driven by the subgroup of men who had undergone unilateral orchiectomy. No differences in semen parameters were observed. A limitation is the retrospective self-reported information on testicular torsion. Also, results should be interpreted with caution owing to the high uncertainty of the observed differences. Overall, the results of our study are reassuring for men who have experienced testicular torsion, especially when treated with orchiopexy, for whom reproductive hormone alterations were subtle and without obvious clinical relevance. Our study found no differences in semen parameters, but follow-up studies are needed to assess any long-term consequences for fertility. Financial support was received from the Danish Ministry of Health; the Danish Environmental Protection Agency; the Research fund of Rigshospitalet, Copenhagen University Hospital; the European Union (Contract numbers BMH4-CT96-0314, QLK4-CT-1999-01422, QLK4-CT-2002-00603, FP7/2007-2013, DEER Grant agreement no. 212844); A.P. Møller and wife Chastine Mckinney Møllers Foundation; Svend Andersens Foundation; the Research Fund of the Capital Region of Denmark; and ReproUnion (EU/Interreg). The authors have nothing to declare. N/A.

L-Form Switching in Escherichia coli as a Common β-Lactam Resistance Mechanism.

ABSTRACTCell wall deficient bacterial L-forms are induced by exposure to cell wall-targeting antibiotics and immune effectors such as lysozyme. L-forms of different bacteria (including Escherichia coli) have been reported in human infections, but whether this is a normal adaptive strategy or simply an artifact of antibiotic treatment in certain bacterial species remains unclear. Here we show that members of a representative, diverse set of pathogenic E. coli readily proliferate as L-forms in supratherapeutic concentrations of the broad-spectrum antibiotic meropenem. We report that they are completely resistant to antibiotics targeting any penicillin-binding proteins in this state, including PBP1A/1B, PBP2, PBP3, PBP4, and PBP5/6. Importantly, we observed that reversion to the cell-walled state occurs efficiently, less than 20 h after antibiotic cessation, with few or no changes in DNA sequence. We defined for the first time a logarithmic L-form growth phase with a doubling time of 80 to 190 min, followed by a stationary phase in late cultures. We further demonstrated that L-forms are metabolically active and remain normally susceptible to antibiotics that affect DNA torsion and ribosomal function. Our findings provide insights into the biology of L-forms and help us understand the risk of β-lactam failure in persistent infections in which L-forms may be common.IMPORTANCE Bacterial L-forms require specialized culture techniques and are neither widely reported nor well understood in human infections. To date, most of the studies have been conducted on Gram-positive and stable L-form bacteria, which usually require mutagenesis or long-term passages for their generation. Here, using an adapted osmoprotective growth media, we provide evidence that pathogenic E. coli can efficiently switch to L-forms and back to a cell-walled state, proliferating aerobically in supratherapeutic concentrations of antibiotics targeting cell walls with few or no changes in their DNA sequences. Our work demonstrates that L-form switching is an effective adaptive strategy in stressful environments and can be expected to limit the efficacy of β-lactam for many important infections.

Dexterity Analysis and Motion Optimization of In-Situ Torsionally-Steerable Flexible Surgical Robots

Flexible robots with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in-situ</i> torsion can be used in laryngeal endoscopic surgery which can maintain the position and approach vector of the end-effector during the operation. However, the inherent errors would be produced by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in-situ</i> torsional motion which are different due to the various configuration of serpentine module in robot. In this letter, the kinematic model is established according to the structure of serpentine module. The dexterity analysis shows that the singular position is reduced and the angular velocity of dexterity is improved comparing with the robot without <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in-situ</i> torsion function. The theoretical position errors caused by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in-situ</i> torsion is quantitatively analyzed by simulation. It is found that the maximum error is 5.19 mm at the bending angle of 120 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${^\circ }$</tex-math></inline-formula> . In addition, the existence of joints in the robot arm also leads to the occurrence of rotation errors. The configuration and number of the joints are optimized to improve the accuracy. Finally, the experiments are carried out to verify the effectiveness of the proposed design and the model. The results indicate that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in-situ</i> torsionally steerable flexible robots have higher motion dexterity, its inherent error during the in-situ torsion motion can be eliminated by structural optimization.

Dark matter from torsion in Friedmann cosmology

A cosmological model in an Einstein–Cartan framework endowed with torsion is studied. For a torsion function assumed to be proportional to Hubble expansion function, namely phi =-alpha H, the contribution of torsion function as a dark matter component is studied in two different approaches. In the first one, the total matter energy density is altered by torsion coupling alpha , giving rise to an effective dark matter and cosmological constant terms that reproduce quite well the flat cosmic concordance model. In the second approach, starting with just standard baryonic matter plus a cosmological constant term, it is obtained that the coupling of torsion with baryons and cosmological constant term naturally gives rise to a dark matter contribution, together a modified cosmological term. In this model the dark matter sector can be interpreted as an effective coupling of the torsion function with the ordinary baryonic matter and cosmological constant. Finally, it is shown that both models are totally compatible with recent cosmological data from Supernovae and Hubble parameter measurements.

Some properties of the torsion function with Robin boundary conditions

In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the L^{\infty} and L^p norms, for p\ge 1 , of the torsion function, seen as a functional on a bounded simply connected open set \Omega\subset\mathbb{R}^n , and prove that the balls are critical shapes for these functionals, when the volume of \Omega is preserved.

Subtrochanteric osteotomy in the management of femoral maltorsion results in anteroposterior malcorrection of the greater trochanter: computed simulations of 3D surface models of 100 cadavers.

The purpose of this study was to investigate the greater trochanter's (GT) behaviour in simulated subtrochanteric osteotomy. Measurement of functional and anatomical femoral torsion, and position of the GT and lesser trochanter was performed using 3-dimensional (3D) surface models of 100 cadaveric femora. Femoral torsion between 2° and 22° was defined as normal, femora with <2° and >22° of femoral torsion were assigned to the low- and high-torsion group. Subtrochanteric osteotomy was simulated to normalise torsional deformities to 12°. With subtrochanteric osteotomy, functional torsion was simultaneously corrected while adjusting anatomical torsion (R2 = 0.866, p < 0.001). Compared to the normal-torsion group, an anteroposterior (AP) overcorrection of ±0.5 centimetres (range 0.02-1.1 cm) of the GT resulted in the high- and low-torsion group, respectively (p < 0.001): Mean AP GT distance to a standardised coronal plane was 2.1 ± 0.3 cm (range 12-30 cm) in the normal-torsion group compared to 1.61 ± 0.1 cm (range 1.4-1.71 cm) and 2.6 ± 0.6 cm (range 1.8-3.6 cm) for the corrected high and low-torsion groups, respectively. The extent of the GT shift in AP direction correlated strongly with the extent to which anatomical femoral torsion was corrected (R2 = 0.946; p < 0.001). Subtrochanteric osteotomy for femoral maltorsion reliably adjusts anatomical and functional torsion, but also results in a ±1 cm AP shift of the GT per 10° of torsional correction. However, this effect of the procedure is most likely not clinically relevant in relation to hip abductor performance.

Spectral upper bound for the torsion function of symmetric stable processes

We prove a spectral upper bound for the torsion function of symmetric stable processes that holds for convex domains in R d \mathbb {R}^d . Our bound is explicit and captures the correct order of growth in d d , improving upon the existing results of Giorgi and Smits [Indiana Univ. Math. J. 59 (2010), pp. 987–1011] and Biswas and Lőrinczi [J. Differential Equations 267 (2019), pp. 267–306]. Along the way, we make progress towards a torsion analogue of Chen and Song’s [J. Funct. Anal. 226 (2005), pp. 90–113] two-sided eigenvalue estimates for subordinate Brownian motion.

Saint-Venant torsion of thin-walled nonhomogeneous elliptical tube

This paper deals with the uniform torsion of thin-walled elliptical tube. The material of the tube is nonhomogeneous and it depends on one of the curvilinear coordinates which defines the cross section of thin-walled bar with closed profile. The approximate solution for the stresses, torsion function and torsional rigidity are obtained by the application of two extreme value theorems of linearized elasticity.

Complex Valued Analytic Torsion and Dynamical Zeta Function on Locally Symmetric Spaces

AbstractWe show that the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an arbitrary flat vector bundle has a meromorphic extension to the whole complex plane and that its leading term in the Laurent series at the zero point is related to the regularised determinant of the flat Laplacian of Cappell–Miller. When the flat vector bundle is close to an acyclic and unitary one, we show that the dynamical zeta function is regular at the zero point and that its value is equal to the complex valued analytic torsion of Cappell–Miller. This generalises the author’s previous results for unitarily flat vector bundles as well as Müller and Spilioti’s results on hyperbolic manifolds.

Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds

Let M be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets D in M for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in L^2(D), and the supremum of the torsion function for D are comparable with the square of the capacitary width for D if the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale.