Degree Of Singularity Research Articles

Solving gyrokinetic systems with higher-order time dependence

We discuss theoretical and numerical aspects of gyrokinetics as a Lagrangian field theory when the field perturbation is introduced into the symplectic part. A consequence is that the field equations and particle equations of motion in general depend on the time derivatives of the field. The most well-known example is when the parallel vector potential is introduced as a perturbation, where a time derivative of the field arises only in the equations of motion, so an explicit equation for the fields may still be written. We will consider the conceptually more problematic case where the time-dependent fields appear in both the field equations and equations of motion, but where the additional term in the field equations is formally small. The conceptual issues were described by Burby (J. Plasma Phys., vol. 82 (3), 2016, 905820304): these terms lead to apparent additional degrees of freedom to the problem, so that the electric field now requires an initial condition, which is not required in low-frequency (Darwin) Vlasov–Maxwell equations. Also, the small terms in the Euler–Lagrange equations are a singular perturbation, and these two issues are interlinked. For well-behaved problems the apparent additional degrees of freedom are spurious, and the physically relevant solution may be directly identified. Because we needed to assume that the system is well behaved for small perturbations when deriving gyrokinetic theory, we must continue to assume that when solving it, and the physical solutions are thus the regular ones. The spurious nature of the singular degrees of freedom may also be seen by changing coordinate systems so the varying field appears only in the Hamiltonian. We then describe how methods appropriate for singular perturbation theory may be used to solve these asymptotic equations numerically. We then describe a proof-of-principle implementation of these methods for an electrostatic strong-flow gyrokinetic system; two basic test cases are presented to illustrate code functionality.

Shape optimization of heterogeneous materials based on isogeometric boundary element method

In this paper, the isogeometric boundary element method (IGABEM) is used to optimize the shape of heterogeneous materials. In contrast to the isogeometric finite element method (IGAFEM), the iterative optimization algorithm based on IGABEM can be implemented directly from Computer-Aided Design (CAD) without returning the optimization results to CAD designers. The discontinuous element method is extended to IGABEM which deals with corner point problems of inhomogeneous materials. After the singularity of sensitivity analysis of boundary integral equations is demonstrated, the power series expansion method (PSEM) is applied to IGABEM to evaluate various degrees of singularity about sensitivity analysis, which shows more accuracy and efficiency than the element sub-division method (ESDM). A set of control points on the geometric boundary are chosen as design variables, which can be passed from the design model to the analysis model, and the objective function is the elastic energy increment. Finally, several numerical examples in 2D and 3D problems are presented to demonstrate the validity and robustness of the present method.

Scaling Effects in the Mechanical System of the Flexible Epidermal Electronics and the Human Skin

Abstract The “island-bridge” mesh structure is widely adopted for flexible epidermal electronics to simultaneously achieve the electronic functions and mechanical flexibility. Mechanical intuition tells that the small size of the “island” is beneficial to the flexibility of the structure and the adaptability to complex geometric targets. Here, a plane-strain model and an axisymmetric model are established for square “island” and cycle “island,” respectively, to analyze the mechanical system consisting of the flexible epidermal electronics and the human skin. It is found that the pressure between the “island” and the human skin is positive at the inner region and reaches a peak value at the center, while is negative at the outer region and approaches infinite at the boundary of the contact region. With the increase in the size a/R0, the amplitude of the pressure significantly increases, as well as the singular degree of the pressure at the boundary. The reduction of the “island” size is beneficial for the optimization of the “comfort level” of the flexible epidermal electronics. The models degenerate into the famous Johnson-Kendall-Roberts (JKR) model for the limit case with extremely hard and thick “island.”

Interfacial Strength of Plasma-sprayed Hydroxyapatite Coatings

Both tensile and shear adhesion strength tests were performed to evaluate interfacial strength between hydroxyapatite coatings on titanium alloy substrates subjected to anodization and heat treatment prior to deposition as well as post-deposition heat treatment. Results of the tensile adhesion strength tests were influenced by the size of the blasting media and the processing of pre- and post-heat treatments but not influenced by anodization. The finite element method (FEM) analysis of stress distribution during the shear adhesion strength test was also performed to evaluate the degree of stress singularity. The results show that the stress singularity parameters were dominant factors of stress distribution in the stress singularity fields, and they were also expected to influence the “essential” interfacial strength. In addition, the size of the blasting media had the same influence on the shear and tensile adhesion strength tests. This suggests the possibility of estimating tensile adhesion strength using the results of shear adhesion strength.

От подхода 2Д – к подходу 4Д

От подхода 2Д – к подходу 4Д

Amenable cones: error bounds without constraint qualifications

We provide a framework for obtaining error bounds for linear conic problems without assuming constraint qualifications or regularity conditions. The key aspects of our approach are the notions of amenable cones and facial residual functions. For amenable cones, it is shown that error bounds can be expressed as a composition of facial residual functions. The number of compositions is related to the facial reduction technique and the singularity degree of the problem. In particular, we show that symmetric cones are amenable and compute facial residual functions. From that, we are able to furnish a new H\"olderian error bound, thus extending and shedding new light on an earlier result by Sturm on semidefinite matrices. We also provide error bounds for the intersection of amenable cones, this will be used to provided error bounds for the doubly nonnegative cone.

Chitooligosaccharide supplementation prevents the development of high fat diet-induced non-alcoholic fatty liver disease (NAFLD) in mice via the inhibition of cluster of differentiation 36 (CD36)

Abstract The effects of (GlcN)2-3 on the development of high fat diet-induced non-alcoholic fatty liver disease in C57BL/6J mice and the potential structural-functional relationship between different singular degrees of polymerization (DPs) COSs and CD36 activity were investigated. (GlcN)2-3 was found to significantly inhibit the levels of triglyceride, low density lipid protein and total cholesterol in the serum and liver, thus reducing hepatic steatosis and, ultimately, altering lipid accumulation. This phenomenon was associated with a decrease in the mRNA and protein expressions of CD36, PXR, DGAT2, LXRα and PPARγ, which subsequently decreased the uptake of FFAs and triglyceride synthesis. Using structural analysis, (GlcN)2-3 blocked the core cavity and inhibited the translocation of FFAs in CD36. Furthermore, the molecular size and steric hindrance effect play crucial roles in the deactivation of CD36. These findings will provide a better understanding of the modulating actions of specific singular-DPs COS in high fat diet-induced hepatic steatosis.

Position, Jacobian, decoupling and workspace analysis of a novel parallel manipulator with four pneumatic artificial muscles

This paper firstly presents a novel 4-SPS/S (active chain/passive chain) parallel manipulator (PPM) driven by four pneumatic artificial muscles (PAMs). SPS denotes a spherical pair–prismatic pair–spherical pair chain and S denotes a spherical pair chain. The PPM proposed can be utilized as shoulder, wrist, waist, hip and ankle simulators or ankle rehabilitation robot. And we present a comprehensive analysis on the PPM proposed, including degrees of freedom (DOF), position, kinematic, Jacobian, singularity, decoupling and workspace analysis. In order to decrease the influence of the maximum angle of spherical pairs in four active chains on the workspace, a novel connector for PAM is proposed. The structure of the PPM is explicitly described, and DOF of the PPM are analyzed based on constraint screw theory. The posture of the moving platform (MP) of the PPM is obtained though Z–Y–X (α–β–γ) type Euler angles and the inverse position solution is obtained. When the MP moves toward any a single Euler angle direction, the closed-form direct position solution is presented by geometric analysis. A back propagation (BP) neural network model for the whole direct position solution is established. Two methods for the Jacobian matrix, one velocity composition, one differentiation, are addressed, and the acceleration inversion is obtained. Based on the Jacobian matrix, the singularity and kinematic decoupling of the PPM are both analyzed. The geometric model of expanded PAM is introduced, and the change of diameters of four PAMs in active chains is taken into consideration when chains of the PPM interfere. Based on length ranges of four active chains, the maximum angle of spherical pairs and possible interferences between chains of the PPM, the configuration workspace is presented. Finally, the characteristic of the workspace and the influence of the maximum angle of spherical pairs in active chains on the configuration workspace are both analyzed.

Analytical investigation of a two-layered elastic half-space containing a cylindrical elastic inhomogeneity under forced torsional rotation

This paper presents a rigorous investigation for a two-layered linear elastic half-space containing a cylindrical elastic inhomogeneity with length equal to the thickness of the top layer undergoing forced torsional rotation applied on a rigid circular disc with the same radius as the cylinder, which is welded on the top of the cylinder. To investigate this complicated system analytically, a combination of Fourier sine and cosine integral transforms for depth, respectively in the domain of inhomogeneity and its matrix at the top layer, and Hankel integral transforms for radial distance in the lower half-space as the remaining part of the matrix are used, which translate the related boundary value problem to a system of coupled singular integral equations for the non-dimensional shear stresses resulted from the continuity and boundary conditions. The system of coupled singular integral equations is solved for some collocation points with a smoothed variable of distance, which is adapted with the use of a free parameter. The validities of both the analytical procedure and numerical evaluations are verified with its degenerated case for the homogeneous case, which is the Reissner–Sagoci problem. In addition, the degree of singularities of the shear stresses in between different regions are estimated in terms of material properties of the layer, inhomogeneity, and the half-space. Moreover, some new graphical results are presented for more understanding of related engineering behavior.

Quasilinear SPDEs via Rough Paths

We are interested in (uniformly) parabolic PDEs with a nonlinear dependence of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable:∂2u-P(a(u)∂12u+σ(u)f)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\partial_2u -P( a(u)\\partial_1^2u + \\sigma(u)f ) =0,$$\\end{document}where P is the projection on mean-zero functions, and f is a distribution which is only controlled in the low regularity norm of { C^{alpha-2}} for {alpha > frac{2}{3}} on the parabolic Hölder scale. The example we have in mind is a random forcing f and our assumptions allow, for example, for an f which is white in the time variable x2 and only mildly coloured in the space variable x1; any spatial covariance operator {(1 + |partial_1|)^{-lambda_1 }} with {lambda_1 > frac13} is admissible. On the deterministic side we obtain a {C^alpha}-estimate for u, assuming that we control products of the form {vpartial_1^2v} and vf with v solving the constant-coefficient equation {partial_2 v-a_0partial_1^2v=f}. As a consequence, we obtain existence, uniqueness and stability with respect to {(f, vf, v partial_1^2v)} of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing f using stochastic arguments. For this we extend the treatment of the singular product {sigma(u)f} via a space-time version of Gubinelli’s notion of controlled rough paths to the product {a(u)partial_1^2u}, which has the same degree of singularity but is more nonlinear since the solution u appears in both factors. In fact, we develop a theory for the linear equation {partial_t u - P(apartial_1^2 u +sigma f)=0} with rough but given coefficient fields a and {sigma} and then apply a fixed point argument. The PDE ingredient mimics the (kernel-free) Safonov approach to ordinary Schauder theory.

On elliptic equations with singular potentials and nonlinear boundary conditions

We consider the Laplace equation in the half-space satisfying a nonlinear Neumann condition with boundary potential. This class of problems appears in a number of mathematical and physics contexts and is linked to fractional dissipation problems. Here the boundary potential and nonlinearity are singular and of power-type, respectively. Depending on the degree of singularity of potentials, first we show a nonexistence result of positive solutions in D 1 , 2 ( R + n ) \mathcal {D}^{1,2}(\mathbb {R}^n_+) with a L p L^p -type integrability condition on ∂ R + n \partial \mathbb {R}^n_{+} . After, considering critical nonlinearities and conditions on the size and sign of potentials, we obtain the existence of positive solutions by means of minimization techniques and perturbation methods.

Mobility properties analyses of a wall climbing hexapod robot

In this paper, we investigate the Degree of freedom (DOF), workspace and singularity of a wall climbing hexapod robot. The robot has two typical working modes, which are the six or three legs attaching on the wall, so robot can be regarded as 6SRRR or 3SRRR parallel mechanism, respectively. First, the DOF of the robot is analyzed by the screw theory. The result indicates that two configurations of the robot possess 6-DOF, and the screw theory makes the calculation of the DOF become extremely simple. Moreover, the workspace of the robot body is studied with constraint equations, which obtains the influence of structural parameters on workspace. After that, a new simple Jacobian matrix is proposed to analyze the singularity, and obtain the singular configurations of the robot, which greatly simplifies the calculation of Jacobian matrix of the robot. Finally, by experiments to verify that the singularity analysis method is correct. The singularity analysis of this paper could be applied for effective control of the robot to avoid singular configurations.

Minimal regular models of quadratic twists of genus two curves

Let $K$ be a complete discrete valuation field with ring of integers $R$ and residue field $k$ of characteristic $p>2$. We assume moreover that $k$ is algebraically closed. Let $C$ be a smooth projective geometrically connected curve of genus $2$. If $K(\sqrt{D})/K$ is a quadratic field extension of $K$ with associated character $\chi$, then $C^{\chi}$ will denote the quadratic twist of $C$ by $\chi$. Given the minimal regular model $\mathcal X$ of $C$ over $R$, we determine the minimal regular model of the quadratic twist $C^{\chi}$. This is accomplished by obtaining the stable model $\mathcal{C}^{\chi}$ of $C^{\chi}$ from the stable model $\mathcal{C}$ of $C$ via analyzing the Igusa and affine invariants of the curves $C$ and $C^{\chi}$, and calculating the degrees of singularity of the singular points of $\mathcal{C}^{\chi}$.

Level‐ limit linear series

AbstractWe consider all one‐parameter families of smooth curves degenerating to a singular curve X and describe limits of linear series along such families. We treat here only the simplest case where X is the union of two smooth components meeting transversely at a point P. We introduce the notion of level‐δ limit linear series on X to describe these limits, where δ is the singularity degree of the total space of the degeneration at P. If the total space is regular, that is, , we recover the limit linear series introduced by Osserman in . So we extend his treatment to a more general setup. In particular, we construct a projective moduli space parameterizing level‐δ limit linear series of rank r and degree d on X, and show that it is a new compactification, for each δ, of the moduli space of Osserman exact limit linear series. Finally, we generalize by associating with each exact level‐δ limit linear series on X a closed subscheme of the dth symmetric product of X, and showing that, if is a limit of linear series on the smooth curves degenerating to X, then is the limit of the corresponding spaces of divisors. In short, we describe completely limits of divisors along degenerations to such a curve X.

Singularity Degree of the Positive Semidefinite Matrix Completion Problem

The singularity degree of a semidefinite programming problem is the smallest number of facial reduction steps to make the problem strictly feasible. We introduce two new graph parameters, called the singularity degree and the nondegenerate singularity degree, based on the singularity degree of the positive semidefinite matrix completion problem. We give a characterization of the class of graphs whose parameter value is at most one for each parameter. Specifically, we show that the singularity degree of a graph is at most one if and only if the graph is chordal, and the nondegenerate singularity degree of a graph is at most one if and only if the graph is the clique sum of chordal graphs and $K_4$-minor free graphs. We also show that the singularity degree is bounded by two if the treewidth is bounded by two and exhibit a family of graphs with treewidth three, whose singularity degree grows linearly in the number of vertices.

On the use of the Peak Stress Method for the calculation of Residual Notch Stress Intensity Factors: a preliminary investigation

Abstract: Residual stresses induced by welding processes significantly affect the engineering properties of structural components. If the toe region of a butt-welded joint is modeled as a sharp V-notch, the distribution of the residual stresses in that zone is asymptotic with a singularity degree which follows either the linear-elastic or the elastic-plastic solution, depending on aspects such as clamping conditions, welding parameters, material and dimension of plates. The intensity of the local residual stress fields is quantified by the Residual Notch Stress Intensity Factors (R-NSIFs), which can be used in principle to include the residual stress effect in the fatigue assessment of welded joints. Due to the need of extremely refined meshes and to the high computational resources required by non-linear transient analyses, the R-NSIFs have been calculated in literature only by means of 2D models. It is of interest to propose new coarse-mesh-based approaches which allow residual stresses to be calculated with less computational effort. This work is aimed to investigate the level of accuracy of the Peak Stress Method in the R-NSIFs evaluation.

Music-Specific Emotion: An Elusive Quarry

Expressive music, almost everyone agrees, evokes an emotional response of some kind in receptive listeners, at least some of the time, in at least some conditions of listening. But is such an emotional response distinctive of or unique to the music that evokes it? In other words, is there such a thing as music-specific emotion? This essay is devoted to an exploration of that question and others related to it. In the main part of the essay a sixpart component model of a standard emotion is set out, and a case is made for the music-specificity of an emotional response to music of a mirroring sort, with respect to four of the six components of such emotion. In the latter part of the essay aspects of the phenomenology of the music-listening experience, and especially that of the inner sounding of music being heard, are explored further, and issues of the value, communicability, and degree of singularity of the emotional response to expressive music are addressed.

A note on alternating projections for ill-posed semidefinite feasibility problems

We observe that Sturm's error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of $$\mathcal {O}\Big (k^{-\frac{1}{2^{d+1}-2}}\Big )$$O(k-12d+1-2), where d is the singularity degree of the problem--the minimal number of facial reduction iterations needed to induce Slater's condition. Consequently, for almost all such problems (in the sense of Lebesgue measure), alternating projections converge at a worst-case rate of $$\mathcal {O}\Big (\frac{1}{\sqrt{k}}\Big )$$O(1k).

Nullity of b-Bridge Coalescence Graphs

The nullity η(G) (degree of singularity) of a graph G is the algebraic multiplicity of thenumber zero in the spectrum of G. If G is a graph containing a vertex of degree one andH be the subgraph obtained from G, by deleting this vertex together with the vertexadjacent to it then, η(G) = η(H). In this paper, we proved that nullity of a graph is themaximum number of independent variables in a high zero-sum weighting for it. Theabove procedures are applied to evaluate the nullity of b-bridge coalescence graphs.They are also applied to determine the nullity of edge introducing between t-tuplecoalescence graphs and nullity of paths introducing between (n, m)-tuples of coalescencegraphs.

Three-dimensional cracked discs under anti-plane loading and effects of the boundary conditions

Purpose– Accordingly to the recent multi-scale model proposed by Sih and Tang, different orders of stress singularities are related to different material dependent boundary conditions associated with the interaction between the V-notch tip and the material under the remotely applied loading conditions. This induces complex three-dimensional stress and displacement fields in the proximity of the notch tip, which are worthy of investigation. The paper aims to discuss these issues.Design/methodology/approach– Starting from Sih and Tang’s model, in the present contribution the authors propose some analytical expressions for the calculation of the strain energy density (SED) averaged over a control volume embracing the V-notch tip. The expressions vary as a function of the different boundary conditions. Dealing with the specific crack case, the results from the analytical frame are compared with those determined numerically under linear-elastic hypotheses, by applying different constraints to the through-the-thickness crack edges in three-dimensional discs subjected to Mode III loading. Free-free and free-clamped cases are considered.Findings– Due to three-dimensional effects, the application of a nominal Mode III loading condition automatically provokes coupled Modes (I and II). Not only the intensity of the induced modes but also their degree of singularity depend on the applied conditions on the crack flanks. The variability of local SED through the thickness of the disc is analysed by numerical analyses and compared with the theoretical trend.Originality/value– The capability of the SED to capture the combined three-dimensional effects is discussed in detail showing that this parameter is particularly useful when the definition of the stress intensity factors (SIFs) is ambiguous or the direct comparison between SIFs with odd dimensionalities is not possible.