Singular Part Research Articles

Bulk viscosity of hard sphere fluids by equilibrium and nonequilibrium molecular dynamics simulations.

The bulk viscosity, ηb, of the hard sphere (HS) fluid is computed by equilibrium and nonequilibrium molecular dynamics (NEMD) simulations, the latter using an adaptation of the time-stepping method for continuous potential systems invented by Hoover et al. [Phys. Rev. A 21, 1756 (1980)], which employs an imposed cyclic density variation on the system by affine scaling of the particle coordinates. The time-stepping method employed for HS is validated against exact event-driven hard sphere methodology for a series of equilibrium quantities over a wide density range, including the pressure, singular parts of the hard sphere viscosities, and the nonsingular parts of the shear viscosity time correlation functions. The time steps used are typically only a little smaller than those employed in continuous potential simulations. Exact pressure tensor fluctuation expressions are derived for the singular (or infinite limiting frequency) equilibrium parts of the viscosities, which were employed in the simulations. The values obtained agree well with the predictions of the Enskog theory for all densities considered. The bulk viscosity obtained by NEMD is shown to be noticeably frequency dependent for densities in excess of ∼0.8, decaying approximately exponentially to the Enskog and equilibrium simulation values at all densities considered for frequencies in excess of ∼5 in hard sphere units. Temperature profiles during the cycle and the effects of strain amplitude on the computed frequency dependent bulk viscosity are presented. The bulk viscosity increases with the maximum density amplitude.

The Neumann problem for one-dimensional parabolic equations with linear growth Lagrangian: evolution of singularities

In this paper, we obtain existence and uniqueness of strong solutions to the inhomogenous Neumann initial-boundary problem for a parabolic PDE which arises as a generalization of the time-dependent minimal surface equation. Existence and regularity in time of the solution are proved by means of a suitable pseudoparabolic relaxed approximation of the equation and the corresponding passage to the limit. Our main result is monotonicity in time of the positive and negative singular parts of the distributional space derivative for bounded variation initial data. Sufficient conditions for instantaneous L^1-W^{1,1}_{{mathrm{loc}}} or BV-W^{1,1}_{{mathrm{loc}}} regularizing effects are also discussed.

Lower and upper bounds of condition number for Vandermonde‐wise matrices and method of fundamental solutions using pseudo radial‐lines

AbstractConsider the method of fundamental solutions (MFS) for 2D Laplace's equation in a bounded simply connected domain . In the standard MFS, the source nodes are located on a closed contour outside the domain boundary , which is called pseudo‐boundary. For circular, elliptic, and general closed pseudo‐boundaries, analysis and computation have been studied extensively. New locations of source nodes are proposed along two pseudo radial‐lines outside . Numerical results are very encouraging and promising. Since the success of the MFS mainly depends on stability, our efforts are focused on deriving the lower and upper bounds of condition number (Cond). The study finds stability properties of new Vandermonde‐wise matrices on nodes with . The Vandermonde‐wise matrix is called in this article if it can be decomposed into the standard Vandermonde matrix. New lower and upper bounds of Cond are first derived for the standard Vandermonde matrix, and then for new algorithms of the MFS using two pseudo radial‐lines. Both lower and upper bounds of Cond are intriguing in the stability study for the MFS. Numerical experiments are carried out to verify the stability analysis made. Since the fundamental solutions (as ) are the basis functions of the MFS, new Vandermonde‐wise matrices are found. Since the nodes with may come from approximations and interpolations by the Laurent polynomials with singular part, the conclusions in this article are important not only to the MFS but also to matrix analysis.

Total positivity of copulas from a Markov kernel perspective

The underlying dependence structure between two random variables can be described in manifold ways. This includes the examination of certain dependence properties such as lower tail decreasingness (LTD), stochastic increasingness (SI) or total positivity of order 2, the latter usually considered for a copula (TP2) or (if existent) its density (d-TP2). In the present paper we investigate total positivity of order 2 for a copula's Markov kernel (MK-TP2 for short), a positive dependence property that is stronger than TP2 and SI, weaker than d-TP2 but, unlike d-TP2, is not restricted to absolutely continuous copulas, making it presumably the strongest dependence property defined for any copula (including those with a singular part such as Marshall-Olkin copulas). We examine the MK-TP2 property for different copula families, among them the class of Archimedean copulas and the class of extreme value copulas. In particular we show that, within the class of Archimedean copulas, the dependence properties SI and MK-TP2 are equivalent.

Higher integrability of the gradient for the thermal insulation problem

We prove the higher integrability of the gradient for local minimizers of the thermal insulation problem: an analogue of De Giorgi’s conjecture for the Mumford–Shah functional. We deduce that the singular part of the free boundary has Hausdorff dimension strictly less than $n - 1$.

Mathematical justification of the point vortex dynamics in background fields on surfaces as an Euler–Arnold flow

The point vortex dynamics in background fields on surfaces is justified as an Euler–Arnold flow in the sense of de Rham currents. We formulate a current-valued solution of the Euler–Arnold equation with a regular-singular decomposition. For the solution, we first prove that, if the singular part of the vorticity is given by a linear combination of delta functions centered at \(q_n(t)\) for \(n=1,\ldots ,N\), \(q_n(t)\) is a solution of the point vortex equation. Conversely, we next prove that, if \(q_n(t)\) is a solution of the point vortex equation for \(n=1,\ldots ,N\), there exists a current-valued solution of the Euler–Arnold equation with a regular-singular decomposition such that the singular part of the vorticity is given by a linear combination of delta functions centered at \(q_n(t)\). As a corollary, we generalize the Bernoulli law to the case where the flow field is a curved surface and where the presence of point vortices is taken into account. From the viewpoint of the application, the mathematical justification is of significance since the point vortex dynamics in the rotational vector field on the unit sphere is regarded as a mathematical model of geophysical flow in order to take effect of the Coriolis force on inviscid flows into consideration.

Lebesgue type decompositions and Radon–Nikodym derivatives for pairs of bounded linear operators

For a pair of bounded linear Hilbert space operators A and Bone considers the Lebesgue type decompositions of B with respect toA into an almost dominated part and a singular part, analogous to theLebesgue decomposition for a pair of measures in which case one speaks ofan absolutely continuous and a singular part. A complete parametrizationof all Lebesgue type decompositions will be given, and the uniqueness ofsuch decompositions will be characterized. In addition, it will be shownthat the almost dominated part of B in a Lebesgue type decomposition hasan abstract Radon–Nikodym derivative with respect to the operator A.

On Non-Hermitian Positive (Semi)Definite Linear Algebraic Systems Arising from Dissipative Hamiltonian DAEs

We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An important feature of the linear algebraic systems is that the (non-Hermitian) system matrix has a positive definite or semidefinite Hermitian part. In the positive definite case we can solve the linear algebraic systems iteratively by Krylov subspace methods based on efficient three-term recurrences. We illustrate the performance of these iterative methods on several examples. The semidefinite case can be challenging and requires additional techniques to deal with the “singular part,” while the “positive definite part” can still be treated with the three-term recurrence methods.

Planar potential flow on Cartesian grids

Potential flow has many applications, including the modelling of unsteady flows in aerodynamics. For these models to work efficiently, it is best to avoid Biot–Savart interactions. This work presents a grid-based treatment of potential flows in two dimensions and its use in a vortex model for simulating unsteady aerodynamic flows. For flows consisting of vortex elements, the treatment follows the vortex-in-cell approach and solves the streamfunction–vorticity Poisson equation on a Cartesian grid after transferring the circulation from the vortices onto the grid. For sources and sinks, an analogous approach can be followed using the scalar potential. The combined velocity field due to vortices, sinks and sources can then be obtained using the Helmholtz decomposition. In this work, we use several key tools that ensure the approach works on arbitrary geometries, with and without sharp edges. First, the immersed boundary projection method is used to account for bodies in the flow and the resulting body-forcing Lagrange multiplier is identified as the bound vortex sheet strength. Second, sharp edges are treated by decomposing the vortex sheet strength into a singular and non-singular part. To enforce the Kutta condition, the non-singular part can then be constrained to remove the singularity introduced by the sharp edge. These constraints and the Poisson equation are formulated as a saddle-point system and solved using the Schur complement method. The lattice Green's function is used to efficiently solve the discrete Poisson equation with unbounded boundary conditions. The method and its accuracy are demonstrated for several problems.

Causality condition relevant functions-orientated analytical treatment on singularities in 3D TD-BEM

A direct analytical treatment on singularities in the 3D TD-BEM formulation is proposed, where the wavefront singularity and the dual singularity are analytically expressed. In the process of the solution of the spatial singularity, the integration domain is divided into the regular part and the singular part. The singularities in the singular part are analytically eliminated by the direct method of the concept of the finite part of an integral (Hadamard principal integral), while the singular integrals in the regular part are solved by the convenient Gaussian integration. Due to the increase of the dimension and the additional causality relevant function, the 3D TD-BEM formulation is further more intricate than the 2D one. In order to reduce the complexity, in the process of the solution of the boundary integral equation in the 3D TD-BEM formulation, a new coordinate transformation method is proposed to analytically transform the coefficient integrals on the spatial surface element into the 2D plane element. The 3D TD-BEM formulation based on the proposed analytical treatment on singularities is verified to be correct by three examples.

Interpolation based formulation of the oscillatory finite Hilbert transforms

A novel quadrature based on interpolation is proposed for approximation of the finite Hilbert transforms with high-frequency parameter. The transforms are arduous to evaluate due to the existence of Cauchy-type singularity and oscillatory kernel. Accordingly, we fixate the Lagrange interpolation technique with asymptotic-meshfree collocation method to tackle the two challenges. Applying Lagrange’s interpolation and root-coefficient relation of a polynomial to avoid Cauchy-type singularity in the domain. After a large manipulation, the finite Hilbert transform is reduced to a sum of finite Fourier transforms and a singular part integral. The meshfree collocation method is implemented to evaluate the oscillatory Fourier transforms, while the singular part integral is evaluated analytically. Numerical examples confirm the accuracy and efficiency of the proposed algorithm. Results of the new procedure are compared with some state of the art methods and assure better performance of the proposed procedure.

Non-commutative measure theory: Henkin and analytic functionals on $$\mathrm {C}^*$$-algebras

Henkin functionals on non-commutative $$\mathrm {C}^*$$ -algebras have recently emerged as a pivotal link between operator theory and complex function theory in several variables. Our aim in this paper is characterize these functionals through a notion of absolute continuity, inspired by a seminal theorem of Cole and Range. To do this, we recast the problem as a question in non-commutative measure theory. We develop a Glicksberg–König–Seever decomposition of the dual space of a $$\mathrm {C}^*$$ -algebra into an absolutely continuous part and a singular part, relative to a fixed convex subset of states. Leveraging this tool, we show that Henkin functionals are absolutely continuous with respect to the so-called analytic functionals if and only if a certain compatibility condition is satisfied by the ambient weak- $$*$$ topology. In contrast with the classical setting, the issue of stability under absolute continuity is not automatic in this non-commutative framework, and we illustrate its key role in sharpening our description of Henkin functionals. Our machinery yields new insight when specialized to the multiplier algebras of the Drury–Arveson space and of the Dirichlet space, and to Popescu’s noncommutative disc algebra. As another application, we make a contribution to the theory of non-commutative peak and interpolation sets.

Dynamics and stability of non-smooth dynamical systems with two switches

One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. In this paper, we leverage a methodology that, through blow-ups and singular perturbation, allows to extend Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, nonlinear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto-like theorems characterizing semi-local structural stability.

Eigenvalues of singular measures and Connes’ noncommutative integration

In a domain \Omega\subset \mathbb{R}^{\operatorname{N}} we consider compact, Birman–Schwinger type operators of the form \operatorname{T}_{P,\mathfrak{A}}=\mathfrak{A}^* P \mathfrak{A} with P being a Borel measure in \Omega, containing a singular part, and \mathfrak{A} being an order -\operatorname{N}/2 pseudodifferential operator. Operators are defined by means of quadratic forms. For a class of such operators, we obtain a proper version of H. Weyl's law for eigenvalues, with order not depending on dimensional characteristics of the measure. These results lead to establishing measurability, in the sense of Dixmier–Connes, of such operators and the noncommutative version of integration over Lipschitz surfaces and rectifiable sets.

Self-interaction of an arbitrary moving dislocation

The action functional for a linear elastic medium with dislocations is given. The equations of motion following from this action reproduce the Peach–Köhler and Lorentzian forces experienced by dislocations. The explicit expressions for singular and finite parts of the self-force acting on a curved dislocation are derived in the framework of linear theory of elasticity of an isotropic medium. The velocity of dislocation is assumed to be arbitrary but less than the shear wave velocity. The slow speed and near-sonic motions of a dislocation are investigated. In the near-sonic limit, the explicit expression for the leading contribution to the self-force is obtained. In the case of slowly moving dislocations, the effective equations of motion derived in the present paper reproduce the known results.

Geometric structure of mass concentration sets for pressureless Euler alignment systems

We study the limiting dynamics of the Euler Alignment system with a smooth, heavy-tailed interaction kernel ϕ and unidirectional velocity u=(u,0,…,0). We demonstrate a striking correspondence between the entropy function e0=∂1u0+ϕ⁎ρ0 and the limiting ‘concentration set’, i.e., the support of the singular part of the limiting density measure. In a typical scenario, a flock experiences aggregation toward a union of C1 hypersurfaces: the image of the zero set of e0 under the limiting flow map. This correspondence also allows us to make statements about the fine properties associated to the limiting dynamics, including a sharp upper bound on the dimension of the concentration set, depending only on the smoothness of e0. In order to facilitate and contextualize our analysis of the limiting density measure, we also include an expository discussion of the wellposedness, flocking, and stability of the Euler Alignment system, most of which is new.

Scattering amplitudes in the Regge limit and the soft anomalous dimension through four loops

Using rapidity evolution equations we study two-to-two gauge-theory scattering amplitudes in the Regge limit. We carry out explicit computations at next-to-next-to-leading logarithmic accuracy through four loops and present new results for both infrared-singular and finite contributions to the amplitude. New techniques are devised in order to derive the colour structure stemming from three-Reggeon exchange diagrams in terms of commutators of channel operators, obtaining results that are valid for any gauge group, and apply to scattered particles in any colour representation. We also elucidate the separation between contributions to the Regge cut and Regge pole in the real part of the amplitude to all loop orders. We show that planar contributions due to multiple-Reggeon exchange diagrams can be factorised as a Regge pole along with the single-Reggeon exchange, and when this is done, the singular part of the gluon Regge trajectory is directly determined by the cusp anomalous dimension. We explicitly compute the Regge cut component of the amplitude through four loops and show that it is non-planar. From a different perspective, the new results provide important information on soft singularities in general kinematics beyond the planar limit: by comparing the computed corrections to the general form of the four-loop soft anomalous dimension we derive powerful constraints on its kinematic dependence, opening the way for a bootstrap-based determination.

Determination of αs value from tau decays with a renormalon-motivated approach

We apply Borel-Laplace sum rules to the data of the semihadronic tau decay rate. For the higher order terms of the Adler function in the leadingtwist (D = 0) contribution we use a renormalon-motivated model, where the correct leading anomalous dimensions are taken into account in the IR u = 3 (and u = 2) renormalon contributions. In the evaluation of D = 0 contribution of the sum rules we apply two methods: (a) fixed order perturbation theory (FO) and (b) Borel resummation of the singular part with the Principal Value prescription (PV). We use as data the ALEPH data for the (V+A)-channel, and a combined set of data for the V-channel. In the D = 6 OPE term of the Adler function we account for the leading nonzero (and noninteger) anomalous dimension. In the OPE for the Adler function we include terms with dimension up to D = 10 for the (V+A)-channel, and up to D = 14 for the V-channel. In such cases, the extracted values of the coupling αs and of the condensates show a reasonably good convergence under the increase of OPE terms. In order to suppress the quark-hadron duality violations, our sum rules are doubly-pinched in the Minkowskian point. We obtain the averaged extracted values of the coupling αs(m2τ) = 0.3169+0.0070-0.0096, corresponding to αs(M2Z) = 0.1183+0.0009-0.0012.

Математичні моделі температурного поля в термочутливих елементах електронних пристроїв

Nonlinear mathematical models for the analysis of temperature regimes in a thermosensitive isotropic plate heated by locally concentrated heat sources have been developed. For this purpose, the heat-active zones of the plate are described using the theory of generalized functions. Given this equation of thermal conductivity and boundary conditions contain discontinuous and singular right parts. The original nonlinear equations of thermal conductivity and nonlinear boundary conditions are linearized by Kirchhoff transformation. To solve the obtained boundary value problems, the integral Fourier transform was used and, as a result, their analytical solutions in the images were determined. The inverse integral Fourier transform was applied to these solutions, which made it possible to obtain analytical expressions for determining the Kirchhoff variable. As an example, the linear dependence of the thermal conductivity on temperature is chosen, which is often used in many practical problems. As a result, analytical relations were obtained to determine the temperature in the heat-sensitive plate. The given analytical solutions are presented in the form of improper convergent integrals. According to Newton’s method (three eighths), numerical values of these integrals are obtained with a certain accuracy for given values of plate thickness, spatial coordinates, specific power of heat sources, thermal conductivity of structural materials of the plate and geometric parameters of the heat-active zone. The material of the plate is silicon and germanium. To determine the numerical values of temperature in the structure, as well as the analysis of heat transfer processes in the middle of the plate due to local heating, developed software, using which geometric mapping of temperature distribution depending on spatial coordinates, thermal conductivity, specific heat flux density. The obtained numerical values of temperature testify to the correspondence of the developed mathematical models of the analysis of heat exchange processes in the thermosensitive plate with local heating to the real physical process.

Stability estimation of the generalized solution to the direct problem for the acoustic equation

The initial-boundary value problem for the acoustic equation with data on a timelike surface is considered in this paper. Such a problem arises, for example, if it is required to determine the acoustic pressure inside the region from a fixed response to part of the boundary from the source involved at the same boundary. It is assumed that the medium is at rest up to a certain instant of time and the parameters of the medium, for example, acoustic density, are known. The problem is considered in a triangular domain. The advisability of this was shown in the second half of the last century in the works of Romanov V.G. (for example, [1]), where it was proved that the solution to the direct problem of acoustic is representable as the sum of a singular and a continuous terms. The author has written out the form of the singular part, investigated the problem in an integral statement, and also proved conditional well-posedness theorems for three cases: for a small parameter of the domain, for small data, and for the source representability of the sought solution. It is known that the initial-boundary value problem for the acoustic equation with data on a timelike surface is ill-posed. In this paper, the original ill-posed problem is reduced to an inverse problem with respect to some direct (well-posed) problem. The theorem is proved and a stability estimate of the generalized solution to the direct problem is obtained.