Curvature Terms Research Articles

Second Curvature Factor and Term Premium in U.S. Yield Curve: Insights from Deep Learning

Second Curvature Factor and Term Premium in U.S. Yield Curve: Insights from Deep Learning

Second Curvature Factor and Term Premium in the U.S. Yield Curve: Insights from Deep Learning

Second Curvature Factor and Term Premium in the U.S. Yield Curve: Insights from Deep Learning

A Homogenized Bending Theory for Prestrained Plates

The presence of prestrain can have a tremendous effect on the mechanical behavior of slender structures. Prestrained elastic plates show spontaneous bending in equilibrium—a property that makes such objects relevant for the fabrication of active and functional materials. In this paper we study microheterogeneous, prestrained plates that feature non-flat equilibrium shapes. Our goal is to understand the relation between the properties of the prestrained microstructure and the global shape of the plate in mechanical equilibrium. To this end, we consider a three-dimensional, nonlinear elasticity model that describes a periodic material that occupies a domain with small thickness. We consider a spatially periodic prestrain described in the form of a multiplicative decomposition of the deformation gradient. By simultaneous homogenization and dimension reduction, we rigorously derive an effective plate model as a Gamma -limit for vanishing thickness and period. That limit has the form of a nonlinear bending energy with an emergent spontaneous curvature term. The homogenized properties of the bending model (bending stiffness and spontaneous curvature) are characterized by corrector problems. For a model composite—a prestrained laminate composed of isotropic materials—we investigate the dependence of the homogenized properties on the parameters of the model composite. Secondly, we investigate the relation between the parameters of the model composite and the set of shapes with minimal bending energy. Our study reveals a rather complex dependence of these shapes on the composite parameters. For instance, the curvature and principal directions of these shapes depend on the parameters in a nonlinear and discontinuous way; for certain parameter regions we observe uniqueness and non-uniqueness of the shapes. We also observe size effects: The geometries of the shapes depend on the aspect ratio between the plate thickness and the composite period. As a second application of our theory, we study a problem of shape programming: We prove that any target shape (parametrized by a bending deformation) can be obtained (up to a small tolerance) as an energy minimizer of a composite plate, which is simple in the sense that the plate consists of only finitely many grains that are filled with a parametrized composite with a single degree of freedom.

On quasi focal curves with quasi frame in space

In this study, we firstly characterize focal curves by considering quasi frame in the ordinary space. Then, we obtain the relation of each quasi curvatures of curve in terms of focal curvatures. Finally, we give some new conditions with constant quasi curvatures in the ordinary space.

The branch‐cut cosmology: A topological canonical quantum‐mechanics approach

AbstractIn this contribution, we sketch a branch‐cut quantum formulation of the Wheeler‐DeWitt equation analytically continued to the complex plane. As a starting point, we base our approach on the Hořava‐Lifshitz formulation of gravity, which employs higher spatial‐derivative terms of the spacetime curvature for renormalization reasons. Following standard procedures, the quantization of the Lagrangian density is achieved by raising the Hamiltonian, the dynamical variable , which represents the the branch‐cut complex scale factor, and the conjugate momentum pln to the category of operators. We arrive at a Schrödinger‐type equation with a non‐linear potential. Solutions are then obtained and discussed for different potential parameterizations. The results reinforce the conception of a quantum leap between the contraction and expansion phases of the branch‐cut universe, in good agreement with the Bekenstein criterion.

A computationally efficient implementation of continuum dislocation dynamics: Formulation and application to ultrafine-grained Mg polycrystals

Continuum dislocation dynamics (CDD) represents the evolution of systems of curved and connected dislocation lines in terms of density-like field variables which include the volume density of loops (or ’curvature density’) as an additional field. Since dislocation curvature represents a spatial derivative of the underlying discrete dislocation density tensor, the curvature field evolution equation of necessity contains numerically inconvenient higher-order derivatives of the density fields. We propose a simple approximation to express curvature in terms of density fields, and demonstrate its application to a benchmark problem in deformation of Mg polycrystals.

Surrogate model for a mechanical metamaterial undergoing microstructure instabilities and phase transformations

This paper addresses the modeling of periodic metamaterials with microstructure instabilities. Such instabilities are produced by bistable elastic mechanisms, i.e., snap-through deformation mechanisms, inducing phase transformations. Metamaterials with these features can be used for energy dissipation in elastic regimes between other applications.A new surrogate model of the bistable mechanism to replace High-Fidelity Finite Element simulations is developed in two sequential stages. In the first stage, a semi-analytical model of the bistable structural element is presented. Following an approach taken from the literature, the analysis of the bistable element is generalized to include load cases that had not been considered in the original study. These loading conditions naturally happen in functionally 2D and 3D metamaterials with complex microarchitectures. In a second stage, a frame element with linear kinematics and small deformations is described. The constitutive relations of the frame element, i.e., axial stress and moment in terms of axial strain and curvature, follow the same equations of the semi-analytical model derived in the first stage. A Variational Principle of Energetic Equivalence supplies the link between the semi-analytical model of the curved beam and the frame element. In this way, the response of this frame element, of only six degrees of freedom, reproduces the non-linear geometric behavior of the bistable element.The surrogate model is particularly efficient for simulating a large number of unit cells of the metamaterial and with the capacity to reproduce its behavior under general loading conditions. Thus, it is appropriate to assessing the metamaterial limit behavior, typically concerning phenomena such as energy dissipation and hysteresis under closed load cycles.

Two second order instantaneous invariants of axodes

It is well established that generalized spatial motion of a rigid body can be defined in terms of two axodes; a fixed axode and a moving axode. These axodes are ruled surfaces that uniquely roll and slide upon one another. Presented are two instantaneous invariants that define relative motion between two axodes. These invariants are based on the instantaneous screw axis (ISA) or line-tangency shared by two axodes in mesh. One invariant is the relative angular displacement (rolling) about the ISA and the other invariant is the relative axial displacement (sliding) along the ISA. Both invariants are ratios expressed in terms of the normal curvature and geodesic torsion for any point on the ISA. An induced curvature is introduced to quantify the combined rolling and sliding displacement. Two osculating hyperboloids are established in terms of induced curvature, which when combined, create a skew axis gear pair with uniform motion that matches the instantaneous invariants of the axodes. A skew axis gear set with elliptical motion is presented to demonstrate generalized axodes, instantaneous invariants, induced curvature, and osculating hyperboloids.

Cosmological particle creation in Weyl geometry

We investigated the possibility of the homogeneous and isotropic cosmological solution in Weyl geometry, which differs from the Riemannian geometry by adding the so called Weyl vector. The Weyl gravity is obtained by constructing the gravitational Lagrangian both to be quadratic in curvatures and conformal invariant. It is found that such solution may exist provided there exists the direct interaction between the Weyl vector and the matter fields. Assuming the matter Lagrangian is that of the perfect fluid, we found how such an interaction can be implemented. Due to the existence of quadratic curvature terms and the direct interaction the perfect fluid particles may be created straight from the vacuum, and we found the expression for the rate of their production which appeared to be conformal invariant. In the case of creating the Universe ‘from nothing’ in the vacuum state, we investigated the problem, whether this vacuum may persist or not. It is shown that the vacuum may persist with respect to producing the non-dust matter (with positive pressure), but cannot resist to producing the dust particles. These particles, being non-interactive, may be considered as the candidates for dark matter.

Highly nonlinear hyperelastic shells: Statics and dynamics

Investigated in this paper are comprehensive static, dynamic and internal-resonance analyses on a wide range of hyperelastic shell structures, including cylindrical, spherical, doubly-curved, and hyperbolic hyperelastic shallow shells. Donnell's nonlinear shell theory and the Mooney-Rivlin strain energy density model are used to formulate the hyperelastic shell structure. Coupled equations of motion are obtained using Hamilton's principle with highly nonlinear terms due to the curvature in the structure, together with the material nonlinearity and large deformations. The coupled equations of motion are converted to a large set of equations using a two-dimensional Galerkin technique and are solved by employing the Newton-Raphson approach and dynamic equilibrium technique. The strength of the current methodology and model is first verified by comparing the static response of the structure with those obtained by using a finite element software. After verifying the model, a detailed analysis of the bending behaviour of the structure under a time-independent pressure is presented. Moreover, the free and forced vibration responses of the shell structure are presented for different cases, showing that the curvature terms play a significant role in changing the mechanical response of the hyperelastic shell. Furthermore, it is shown that for specific sets of curvatures, internal resonances are present which leads to a complicated, rich nonlinear responses. The nonlinear forced vibration response of the hyperelastic shell is also presented for different shell types and resonances.

"Flexures and bends of the large intestine: Current terminology and a suggestion to simplify it".

There are a number of inconsistencies in the description of the bends of the colon down to the anus. This is historically based on the fact that anatomists saw the colon in its position in the abdominal cavity down to the pelvis and thus from the "outside" and also described it in this way. This view is still useful in clinical practice today (e.g. for the abdominal surgeons). For the greater part of clinicians, however, the view has shifted due to modern endoscopy. This allows examiners to see the terminal section of the intestine and the colon from the "inside". To accommodate both "ways of looking" in terms of modern medicine, we have been guided by today's clinical needs, and here we attempt to reconcile these with the historically evolved anatomical terms to create a nomenclature that meets all the needs of students, anatomists and clinicians looking at the large intestine from the inside and outside. With this in mind, we propose to speak of colic flexures (right colic flexure=RCF=hepatic flexure, flexura coli sinistra; left colic flexure=LCF=splenic flexure, flexura coli dextra; descending-sigmoid flexure=DSF; sigmoid-rectum flexure=SRF) for the colon (colon). For the rectum (rectum), we suggest the term bend (superior, intermediate and inferior) when viewed in the frontal plane, the term curvature (sacral curvature; anorectal curvature=perineal curvature) when viewed in the sagittal plane.

Bouncing Cosmology in Modified Gravity with Higher-Order Gauss–Bonnet Curvature Term

In this paper, we studied the bouncing behavior of the cosmological models formulated in the background of the Hubble function in the F(R,G) theory of gravity, where R and G, respectively, denote the Ricci scalar and Gauss–Bonnet invariant. The actions of the bouncing cosmology are studied with a consideration of the different viable models that can resolve the difficulty of singularity in standard Big Bang cosmology. Both models show bouncing behavior and satisfy the bouncing cosmological properties. Models based on dynamical, deceleration, and energy conditions indicate the accelerating behavior at the late evolution time. The phantom at the bounce epoch is analogous to quintessence behavior. Finally, we formulate the perturbed evolution equations and investigate the stability of the two bouncing solutions.

Internal resonance and bending analysis of thick visco-hyper-elastic arches

In this study, a comprehensive analysis of visco-hyper-elastic thick soft arches under an external time-independent as well as time-dependent loads is presented from bending and internal resonance phenomenon perspectives. Axial, transverse and rotation motions are considered for modelling the thick and soft arch in the framework of the Mooney–Rivlin and Kelvin–Voigt visco-hyper-elastic schemes and third-order shear deformable models. The arch is assumed to be incompressible and is modelled using von Kármán geometric nonlinearity in the strain–displacement relationship. Using a virtual work method, the bending equations are derived. For the vibration analysis, three, coupled, highly nonlinear equations of motions are obtained using force-moment balance method. The Newton–Raphson method together with the dynamic equilibrium technique is used for the bending and vibration analyses. A detailed study on the influence of having visco-hyper-elasticity and arch curvature in the frequency response of the system is given in detail, and the bending deformation due to the applied static load is presented. The influence of having thick, soft arches with different slenderness ratios is shown, and the forced vibration response is discussed. Moreover, internal resonance in the system is studied showing that the curvature term in the structure can lead to three-to-one internal resonances, showing a rich nonlinear frequency response. The results of this study are a step forward in studying the visco-hyper-elastic behaviour of biological structures and soft tissues.Graphic abstract

Ultrahigh-Dose-Rate Proton Irradiation Elicits Reduced Toxicity in Zebrafish Embryos

PurposeRecently, ultrahigh-dose-rate radiation therapy (UHDR-RT) has emerged as a promising strategy to increase the benefit/risk ratio of external RT. Extensive work is on the way to characterize the physical and biological parameters that control the so-called “Flash” effect. However, this healthy/tumor differential effect is observable in in vivo models, which thereby drastically limits the amount of work that is achievable in a timely manner. Methods and MaterialsIn this study, zebrafish embryos were used to compare the effect of UHDR irradiation (8-9 kGy/s) to conventional RT dose rate (0.2 Gy/s) with a 68 MeV proton beam. Viability, body length, spine curvature, and pericardial edema were measured 4 days postirradiation. ResultsWe show that body length is significantly greater after UHDR-RT compared with conventional RT by 180 µm at 30 Gy and 90 µm at 40 Gy, while pericardial edema is only reduced at 30 Gy. No differences were obtained in terms of survival or spine curvature. ConclusionsZebrafish embryo length appears as a robust endpoint, and we anticipate that this model will substantially fasten the study of UHDR proton-beam parameters necessary for “Flash.”

Nonlinear dynamic modeling of bistable variable stiffness composite laminates

The room-temperature equilibrium stable states of cured unsymmetric composite laminates have been the focus of recent research, with a particular emphasis on shape morphing applications. It has been shown that changing the fiber orientation of unsymmetrical laminates using curvilinear fiber path description can results in a plethora of bistable configurations with an enriched design space. In bistable structures, snap-through involves transition from one stable shape to another, which is a non-linear phenomenon exhibiting rich dynamics during the shape transition. Past works involving such dynamic characteristics show encouraging potential in designing efficient morphing strategies. In this work, a novel semi-analytical model using Föppl von Kármán kinematics has been formulated to predict the non-linear dynamic characteristic of bistable variable stiffness (VS) laminates. An efficient energy formulation is adopted where the membrane and bending energies are decoupled using the semi-inverse constitutive equation. The in-plane stress resultants and the energy components are expressed in terms of curvatures using the in-plane equilibrium equations and compatibility conditions. Using Hamilton’s principle in conjunction with the Rayleigh–Ritz approach, a set of non-linear equations are generated, which is solved to obtain the dynamics of the snap-through process. The accuracy of the predicted non-linear vibration results of bistable plates from the semi-analytical model is verified using a fully non-linear finite element framework and validated exemplarily by tests on a straight fiber laminate configuration. Finally, a parametric study is performed by tailoring the VS parameters to identify the effect of different curvilinear fiber alignments on the dynamic characteristics of bistable VS laminates.

Pasta Phases in Neutron Star Mantle: Extended Thomas–Fermi vs. Compressible Liquid Drop Approaches

Nuclear pasta phases in the neutron stars mantle can affect the mechanical and transport properties of superdense matter, thus playing an important role in the dynamics and evolution of neutron stars. In this paper, we compare results obtained by the Extended Thomas–Fermi (ETF) method with the compressible liquid drop model (CLDM), based on the thermodynamically consistent description of the surface properties calculated for the two-phase plane interface and the same energy-density functional (for numerical illustration, we applied the Skyrme-type functional SLy4). Our ETF calculations found that pasta phases in cylindrical form cover a significant crustal region (both normal and inverse phases, aka spaghetti and bucatini are presented). Meanwhile, within the applied CLDM framework, which includes the thermodynamically required effect of neutron adsorption on the cluster’s surface but neglects curvature corrections, only the spaghetti phase was found to be energetically favorable in the small density range prior to crust–core transition. On the other hand, the recent CLDM of Dinh Thi et al., 2021, which, on the contrary, accounts for curvature term but neglects neutron adsorption, predicts pasta phase onset in better agreement with the ETF. This fact highlights the importance of the curvature effects and allows counting on the potential validity of the CLDMs as a convenient, transparent and accurate tool for investigation of the pasta-phase properties.

Wald’s entropy in Coincident General Relativity

The equivalence principle and its universality enables the geometrical formulation of gravity. In the standard formulation of General Relativity (GR) á la Einstein, the gravitational interaction is geometrized in terms of the spacetime curvature. However, if we embrace the geometrical character of gravity, two alternative, though equivalent, formulations of GR emerge in flat spacetimes, in which gravity is fully ascribed either to torsion or to non-metricity. The latter allows a much simpler formulation of GR oblivious to the affine spacetime structure, the Coincident General Relativity (CGR). The entropy of a black hole can be computed using the Euclidean path integral approach, which strongly relies on the addition of boundary or regulating terms in the standard formulation of GR. A more fundamental derivation can be performed using Wald’s formula, in which the entropy is directly related to Noether charges and is applicable to general theories with diffeomorphism invariance. In this work we extend Wald’s Noether charge method for calculating black hole entropy to spacetimes endowed with non-metricity. Using this method, we show that CGR with an improved action principle gives the same entropy as the well-known entropy in standard GR. Furthermore the first law of black hole thermodynamics holds and an explicit expression for the energy appearing in the first law is obtained.

Multi-step prediction in linearized latent state spaces for representation learning

In this paper, we derive a novel method as a generalization over LCEs such as E2C. The method develops the idea of learning a locally linear state space by adding a multi-step prediction, thus allowing for more explicit control over the curvature. We show that the method outperforms E2C without drastic model changes which come with other works, such as PCC and P3C. We discuss the relation between E2C and the presented method and derive update equations. We provide empirical evidence, which suggests that by considering the multi-step prediction, our method – ms-E2C – allows learning much better latent state spaces in terms of curvature and next state predictability. Finally, we also discuss certain stability challenges we encounter with multi-step predictions and how to mitigate them.

Inflation from dynamical projective connections

We show how the recently developed string-inspired, projectively-invariant gravitational model Thomas-Whitehead gravity (TW gravity) naturally gives rise to a field acting as the inflaton. In the formulation of TW gravity, a field $\mathcal{D}_{ab}$ is introduced into the projective connection components and is related to a rank-two tensor field $\mathcal{P}_{ab}$. Through the dynamical action of TW gravity, in terms of projective curvature, the tensor field $\mathcal{P}_{ab}$ acquires dynamics. By decomposing $\mathcal{P}_{ab}$ into its trace and traceless degrees of freedom, and choosing the connection to be Levi-Civita, we demonstrate that TW gravity contains a non-minimally coupled scalar field with a specific potential. Considering only the trace degrees of freedom, we demonstrate that the scalar field acts as an inflaton in the slow roll approximation. We find a range of values for the parameters introduced by TW gravity that fit the experimental constraints of the most recent cosmological data.

D’Atri spaces and the total scalar curvature of hemispheres, tubes and cylinders

Csikós and Horváth proved in J Geom Anal 28(4): 3458-3476, (2018) that if a connected Riemannian manifold of dimension at least 4 is harmonic, then the total scalar curvatures of tubes of small radius about an arbitrary regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about geodesic segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D’Atri space, furthermore, if the space has bounded sectional curvature, then it is enough to require the total scalar curvature condition just for cylinders to imply that the space is D’Atri. This result gives a negative answer to a question posed by Gheysens and Vanhecke. To prove these statements, we give a characterization of D’Atri spaces in terms of the total scalar curvature of geodesic hemispheres in any dimension.