Finite Terms Research Articles

Derive Lovelock gravity from string theory in cosmological background

It was proved more than three decades ago, that the first order α′ correction of string effective theory could be written as the Gauss-Bonnet term, which is the quadratic term of Lovelock gravity. In cosmological background, with an appropriate field redefinition, we reorganize the infinite α′ corrections of string effective action into a finite term expression for any specific dimension. This finite term expression matches Lovelock gravity exactly and thus fix the couplings of Lovelock gravity by the coefficients of string effective action. This result thus provides a strong support to string theory.

Antiferromagnets with random vacancies and substitutional spins on the triangular lattice

We discuss theoretically static and dynamical properties of $XY$ and Heisenberg antiferromagnets on triangular lattice with random vacancies and substitutional spins. It is shown that the distortion of $120^\circ$ magnetic order produced by a single defect is described by electrostatic equations for a field of an electrically neutral complex of six charges located around the impurity. The first finite term in the multipole expansion of this field is the octupole moment which decays as $1/r^3$ with the distance $r$. The linearity of equations allows to describe analytically the distortion of the long-range magnetic order at a small concentration $c$ of defects. We obtain analytically renormalization of the elastic neutron scattering cross section and the magnon spectrum $\epsilon_{\bf k}$ in the leading order in $c$. We find that the scattering on impurities renormalizes weakly the bare spectrum $\epsilon_{\bf k}\propto k$ at $k\gg\sqrt c$. However the renormalization is substantial of the long-wavelength magnon spectrum at $k\ll\sqrt c$: $\epsilon_{\bf k}\propto \sqrt{c /\ln(1/k)}$ at $k\to0$ and there is a parametrically large region in which magnons with not too small momenta are overdamped and localized. This strong modification of the long-wavelength spectrum leads to the stabilization of the slightly distorted magnetic long-range order at $T<T_N\sim S^2J/\ln(1/c)$ and to the considerable change in the density of states and in the specific heat. The overdamped modes arise also in quasi-2D spin systems on a stacked triangular lattice.

Mixed boundary value problems in power-law functionally graded circular annulus

The present work proposes a semi-analytical technique for solution of mixed boundary value problem in functionally graded circular annulus wherein shear modulus varies radially in power-law form while Poisson’s ratio is constant. The technique relies on two main steps. In the first step, corresponding to terms in periodic Fourier series applied individually as traction along the annulus surface, stress and displacement field in the annulus is computed harnessing Airy stress functions approach. In the second step, leveraging the strain–displacement relations in polar co-ordinates, mixed boundary conditions are rendered in terms of displacement all along the annulus surface. Assuming the unknown traction along the annulus surface in terms of periodic Fourier series with finite terms, the modified displacement boundary condition, family of solution from the first step and orthogonality of sine and cosine functions is used to generate a system of simultaneous linear equations for the series coefficients. Knowing the coefficients of periodic Fourier series, stress and displacement field can be computed everywhere in the annulus. The first step is digitized in terms of MAPLE functions, exhaustively validated through traction distribution comprising of normal, shear traction on part of the boundary and pair of equal and diametrically opposite point load along the boundary. The second step is corroborated through a problem where the inner surface is subjected to a specified traction distribution and mixed boundary conditions exist along the outer surface in the form of complete constraint along a part and traction-free condition elsewhere.

Numerical Investigation on Motion Responses of a Floating Hemisphere Over a Sloping Bottom

Abstract In the last several decades, some numerical approaches have been proposed to deal with three-dimensional wave-body interaction problems in sloping bottom environment. Most of them either adopt the finite depth Green function or add numerical damping terms into the free surface condition to treat far field radiation condition, which certainly give rise to numerical errors. The hybrid model (Belibassakis, “A Boundary Element Method for the Hydrodynamic Analysis of Floating Bodies in Variable Bathymetry Regions,” Eng. Anal. Boundary Elements 32, pp. 796–810) adopting the consistent coupled-mode system for incident wave propagation problem combining with the three-dimensional bottom-dependent Green function to treat the diffraction and radiation problem is a complete formulation, as the latter function appropriately characterize the far field radiation wave pattern over a smoothly sloping bottom. However, this model has not been validated after its publication. In this connection, comparisons with computational fluid dynamic (CFD) results are presented to verify its accuracy. Application of this hybrid model is also performed to investigate the effects on the floating hemisphere by the sloping bottom.

Dimensional reduction of higher-point conformal blocks

Recently, with the help of Parisi-Sourlas supersymmetry an intriguing relation was found expressing the four-point scalar conformal block of a (d − 2)-dimensional CFT in terms of a five-term linear combination of blocks of a d-dimensional CFT, with constant coefficients. We extend this dimensional reduction relation to all higher-point scalar conformal blocks of arbitrary topology restricted to scalar exchanges. We show that the constant coefficients appearing in the finite term higher-point dimensional reduction obey an interesting factorization property allowing them to be determined in terms of certain graphical Feynman-like rules and the associated finite set of vertex and edge factors. Notably, these rules can be fully determined by considering the explicit power-series representation of just three particular conformal blocks: the four-point block, the five-point block and the six-point block of the so-called OPE/snowflake topology. In principle, this method can be applied to obtain the arbitrary-point dimensional reduction of conformal blocks with spinning exchanges as well. We also show how to systematically extend the dimensional reduction relation of conformal partial waves to higher-points.

Supersymmetry anomalies and the Wess–Zumino model in a supergravity background

We briefly recall the procedure for computing the Ward identities in the presence of a regulator which violates the symmetry being considered. We compute the first nontrivial correction to the supersymmetry Ward identity of the Wess–Zumino model in the presence of background supergravity using dimensional regularization. We find that the result can be removed using a finite local counter term and so there is no supersymmetry anomaly.

On The Vibrations of Pyramidal Beams With Rectangular Cross-Section and Application to Unswept Wings

Summary The bending frequencies of an unswept wing are calculated based on the model of a beam clamped at the root and free at the tip. For a tapered wing with straight leading- and trailing-edges, the chord is a linear function of the span; the same linear function of the span applies to thickness, in the case of constant thickness-to-chord ratio. The latter is usually small, so that the beam differs from the more frequent cases of a conical beam with a circular cross-section or a prismatic beam with a square cross-section. Thus, the bending modes of a non-uniform beam are considered, with mass and area moment of inertia which are respectively quadratic and quartic functions of the span. There is no exact solution expressible in finite terms using elementary functions, and thus power series expansions are used. The bending frequencies are calculated for a delta wing and compared with a rectangular wing, with the same span, mean chord and thickness, mass density and Young’s modulus. It is shown that the fundamental frequency is higher by a factor 4.96 for the delta wing; it is also shown that the general case of the tapered wing is intermediate between the delta and the rectangular wing. Lastly, the analytical results obtained for the bending modes are compared with numerical modal analyses of general tapered wing beams using high-fidelity finite-element model software.

Anti-Plane Dynamics Analysis of a Circular Lined Tunnel in the Ground under Covering Layer

Wave diffusion in the composite soil layer with the lined tunnel structure is often encountered in the field of seismic engineering. The wave function expansion method is an effective method for solving the wave diffusion problem. In this paper, the wave function expansion method is used to present a semi-analytical solution to the shear horizontal (SH) wave scattering problem of a circular lined tunnel under the covering soil layer. Considering the existence of the covering soil layer, the great arc assumption (that is, the curved boundary instead of the straight-line boundary) is used to construct the wavefield in the composite soil layer. Based on the wave field and boundary conditions, an infinite linear equation system is established by adding the application of complex variable functions. The finite term is intercepted and solved, and the accuracy of the solution is analyzed. Although truncation is inevitable, due to the Bessel function has better convergence, a smaller truncation coefficient can achieve mechanical accuracy. Based on numerical examples, the influence of SH wave incident frequency, soil parameters, and lining thickness on the dynamic stress concentration factor of lining is analyzed. Compared with the SH wave scattering problem by lining in a single medium half-space, due to the existence of the cover layer and the influence of its stiffness, the dynamic stress of the lining can be increased or inhibited. In addition, the lining thickness has obvious different effects on the dynamic stress concentration coefficient of the inner and outer walls of different materials.

Supersymmetry anomaly in the superconformal Wess-Zumino model

We present a comprehensive analysis of supersymmetry anomalies in the free and massless Wess-Zumino (WZ) model in perturbation theory. At the classical level the model possesses mathcal{N} = 1 superconformal symmetry, which is partially broken by quantum anomalies. The form of the anomalies and the part of the symmetry they break depend on the multiplet of conserved currents used. It was previously shown that the R-symmetry anomaly of the conformal current multiplet induces an anomaly in Q-supersymmetry, which appears first in 4-point functions. Here we confirm this result by an explicit 1-loop computation using a supersymmetric Pauli-Villars regulator.The conformal current multiplet does not exist in the regulated theory because the regulator breaks conformal invariance, R-symmetry and S-supersymmetry explicitly. The minimal massive multiplet is the Ferrara-Zumino (FZ) one and the supersymmetry preserved by the regulator is a specific field dependent combination of Q- and S- supersymmetry of the conformal multiplet. While this supersymmetry is non anomalous, conformal invariance, R-symmetry and the original Q- and S-supersymmetries are explicitly broken by finite contact terms, both in the regulated and renormalized theories.A conformal current multiplet does exist for the renormalized theory and may be obtained from the FZ multiplet by a set of finite local counterterms that eliminate the explicit symmetry breaking, thus restoring superconformal invariance up to anomalies. However, this necessarily renders both Q- and S-supersymmetries anomalous, as is manifest starting at 4-point functions of conformal multiplet currents. The paper contains a detailed discussion of a number of issues and subtleties related to Ward identities that may be useful in a wider context.

Static properties of skyrmions and nucleons at finite isospin chemical potential

We use the Skyrme Lagrangian with a finite pion mass term to study the stability of the skyrmion at finite isospin chemical potential [Formula: see text]. It is found that there is a critical value [Formula: see text] MeV, and there is no stable solution above the [Formula: see text]. We also explore the behavior of different skyrmion parameters as a function of [Formula: see text], in particular the isovector electric root-mean-square (rms) charge radius in this work. Then, using the Hamiltonian formulation, in terms of collective variables, we study the behavior of nucleons mass splitting at finite isospin chemical potential. We find the proton mass decreases and the neutron mass increases when [Formula: see text]. Finally, we also discuss the magnetic moments and the pion-nucleon sigma term [Formula: see text] as a function of isospin chemical potential for interest.

Barrierless melt nucleation at solid-solid interface in energetic nitramine octahydro-1, 3, 5, 7-tetranitro-1, 3, 5, 7-tetrazocine

The present work aims to investigate the barrierless nucleation of a molten interface at solid-solid phase boundary, significantly below melting temperature during solid-solid phase transformations (PTs) in energetic nitramine octahydro-1, 3, 5, 7-tetranitro-1, 3, 5, 7-tetrazocine by utilizing a thermodynamically consistent multiphase phase-field model. The instability conditions for barrierless PTs have been presented for each phase and the effect of finite barrier term characterizing the interaction of two solid-melt interfaces on the different instability conditions are analyzed for some critical conditions. Additionally, barrierless melt nucleation has been studied for the different range of nanoscale material parameters and temperature which reveals nontrivial interaction of interfaces. Similar phenomena are expected to occur during PTs in pharmaceutical, ferroelectric, colloidal, and superhard materials.

Dynamic analysis for a vertical interface crack and the nearby circular cavity located at the piezoelectric bi-material half-space under SH-waves

The present problem aims to study the scattering behavior of SH-waves by a circular cavity near two symmetrically permeable interface cracks in the piezoelectric bi-material half-space. The steady-state response of the problem is obtained, with the aid of the Green’s function method and the complex function method. Above all, the essential expression of Green’s function is constructed by the mirror method. This expression satisfies the conditions of being stress-free and electric insulation on the horizontal boundary of the orthogonal space where the circular cavity is located, and the condition of bearing a harmonic out-plane line source force on the vertical boundary. Next, on the basis of dividing the bi-material medium into two parts along the vertical boundary, the first kind of Fredholm integral equation with uncertain anti-plane forces is established by using the conjunction method and the crack-division technology. Then, the solution is obtained by solving an algebraic equation with finite terms, which is an effective truncation of the integral equation. Finally, the dynamic stress concentration factor around the edge of the circular cavity and the dynamic stress intensity factor at the crack tip are calculated numerically. On this basis, the effects of incident wave frequency, crack length, crack location and circular cavity position on the dynamic stress concentration factor and dynamic stress intensity factor are discussed.

Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains

In the setting of reversible continuous-time Markov chains, the CDΥ condition has been shown recently to be a consistent analogue to the Bakry-Émery condition in the diffusive setting in terms of proving Li-Yau inequalities under a finite dimension term and proving the modified logarithmic Sobolev inequality under a positive curvature bound. In this article we examine the case where both is given, a finite dimension term and a positive curvature bound. For this purpose we introduce the CDΥ(κ,F) condition, where the dimension term is expressed by a so called CD-function F. We derive functional inequalities relating the entropy to the Fisher information, which we will call entropy-information inequalities. Further, we deduce applications of entropy-information inequalities such as ultracontractivity bounds, exponential integrability of Lipschitz functions, finite diameter bounds and a modified version of the celebrated Nash inequality.

Effect of arc cracks in composite rock strata on ground vibration under SH wave

The existence of cracks complicates the scattering wave field in composite rock strata. The cracks not only affect the dynamic features of the site but also influence the displacement amplitude of the ground vibration. Therefore, it is of engineering significance to study the crack scattering in composite rock strata. Based on the function of complex variables (CVF) and big circle method, this paper constructs a series solution to the scattering wave field of the arc crack in composite rock strata. Specifically, a series of equations were established according to the continuous conditions of stress and displacement at the boundaries of the rock layer and the crack. Then, the model was solved by Fourier series and truncating finite terms, followed by the discussion of the calculation error. Through programming calculations, the authors analyzed the impact of two arc cracks in the “upper-hard and lower-soft” composite rock strata on the ground vibration under incident SH wave. The results show that the magnitude of ground displacement depends on the following factors: the medium parameters of the rock layer, the frequency of the incident wave, and the depth and direction of the cracks; compared with the lower rock layer, the surface rock layer can obviously amplify the amplitude of the ground displacement; with the growing frequency of the incident wave, the influence of horizontal and vertical cracks on the amplitude of the ground displacement first increased and then decreased; the shallower the crack, the greater the impact on ground displacement amplitude; the horizontal crack had a greater influence on the ground displacement amplitude than the vertical crack.

The Many-Body Expansion for Aqueous Systems Revisited: I. Water-Water Interactions.

We revisit the many-body expansion (MBE) for water-water interactions by examining the effects of the basis set, including those resulting from the basis set superposition error (BSSE) correction, and electron correlation on the various terms for selected sizes of water clusters up to n = 21. The analysis is performed at the second-order Møller-Plesset (MP2) perturbation theory with the family of augmented correlation consistent basis sets up to five zeta quality (aug-cc-pVxZ, x = D, T, Q, 5) for the (H2O)n, n = 7, 10, 13, 16, and 21, clusters for which we report either the complete MBE (for n = 7, 10) or the ones through the 6-body (for n = 13) and the 5-body terms (for n = 16, 21). For the n = 3 and 7 clusters, we also report the analysis at the coupled cluster with single, double, and perturbative triple replacements in order to assess the effects of a higher correlation on the magnitude and percentage of the various MBE terms. Our results suggest that the oscillatory behavior around zero found for the 5-body and larger terms is solely an artifact of the (small) size of the basis set. Indeed, all terms above the 4-body converge monotonically to practically zero upon increasing the size of the basis set toward the complete basis set (CBS) limit. In that respect, the BSSE-corrected 5-body and above terms do not exhibit the oscillatory behavior on either side of zero with the basis set observed for the BSSE-uncorrected terms. In addition, the magnitudes of the 5-body and above terms are accurately reproduced even with the smaller basis set of the series (aug-cc-pVDZ) once the BSSE correction is taken into account. The same level of theory (MP2/aug-cc-pVDZ, BSSE-corrected) also accurately reproduces the MP2/CBS values of the 3- and 4-body terms. The contribution of electron correlation to the 3- and 4-body terms is quite small so that neglecting the correlation contribution in all terms above the 3-body results in an error of the order of 0.1%. The BSSE correction to the largest 2-body term in the MBE was accurately estimated from the function a[1 + erf( - b·R)], which is proportional to the common (overlapping) area between two Gaussian distributions whose centers are separated by R with the constants a and b fitted to the calculated BSSE corrections for the individual 2-body terms of the clusters with each basis set and R is the distance between oxygen atoms. Our results demonstrate that the MBE for water-water interactions converges by the 4-body term since any finite terms above the 4-body are artifacts of the size of the basis set. The MBE can thus be safely truncated at the 4-body term when either a very large basis set is used or BSSE corrections are taken into account even with the smaller aug-cc-pVDZ basis set. We expect these findings to have important consequences in the pursuit of accurate ab initio based many-body molecular dynamics simulations for aqueous systems.

The effect of considering Pasternak elastic foundation on acoustic insulation of the finite doubly curved composite structures

This approach focuses on the acoustic characteristic of a simply supported doubly curved composite shell subjected to the Pasternak-type elastic foundation. To carry out the sound analysis of a finite shell, displacements and rotation terms and acoustic pressures are developed based on the infinite longitudinal and transversal modes. According to Hamilton's principle, the dynamic equations of the structure equipped with an elastic foundation are extracted. Subsequently, the construction is stimulated using an acoustic wave. Considering infinite modes, the necessity of terminating this process by applying a large number of modes is concerned. Therefore, in addition to the design of the convergence algorithm, some configurations against variations of dimensions and frequencies are proposed. Before presenting the numerical outcomes, the accuracy of the formulation is checked by either natural frequencies or sound transmission spectra. It is realized that although Winkler spring stiffness is impressive to improve the noise property at the low-frequency domain, the positive effects of Shear layer one are specified in the whole region. The outcomes also contain some 3D new shapes for variations of the Winkler spring and Shear layer stiffness.

One-loop jet functions by geometric subtraction

In factorization formulae for cross sections of scattering processes, final-state jets are described by jet functions, which are a crucial ingredient in the resummation of large logarithms. We present an approach to calculate generic one-loop jet functions, by using the geometric subtraction scheme. This method leads to local counterterms generated from a slicing procedure; and whose analytic integration is particularly simple. The poles are obtained analytically, up to an integration over the azimuthal angle for the observable- dependent soft counterterm. The poles depend only on the soft limit of the observable, characterized by a power law, and the finite term is written as a numerical integral. We illustrate our method by reproducing the known expressions for the jet function for angularities, the jet shape, and jets defined through a cone or kT algorithm. As a new result, we obtain the one-loop jet function for an angularity measurement in e+e− collisions, that accounts for the formally power-suppressed but potentially large effect of recoil. An implementation of our approach is made available as the GOJet Mathematica package accompanying this paper.

On the fourth order Schrödinger equation in three dimensions: Dispersive estimates and zero energy resonances

We study the fourth order Schrödinger operator H=(−Δ)2+V for a short range potential in three space dimensions. We provide a full classification of zero energy resonances and study the dynamic effect of each on the L1→L∞ dispersive bounds. In all cases, we show that the natural |t|−34 decay rate may be attained, though for some resonances this requires subtracting off a finite rank term, which we construct and analyze. The classification of these resonances, as well as their dynamical consequences differ from the Schrödinger operator −Δ+V.

A conservative gyrofluid model: Effect of closure on energetics

We develop a conservative gyrofluid model that is suitable for global flux-driven simulations of electrostatic tokamak plasma turbulence. On the basis of the general gyrofluid moment equations, we derive energy equations which enable us to manipulate the impact of a gyrofluid closure on energetics. We demonstrate that an artificial manipulation of high order moment contributions to gyrofluid moment equations via a closure model can lead to the violation of the energy conservation. A fluid closure is also found to restrict the maximum attainable order of finite Larmor radius terms, implying the loss of dynamical information by the closure.

Dedekind sums arising from newform Eisenstein series

For primitive nontrivial Dirichlet characters [Formula: see text] and [Formula: see text], we study the weight zero newform Eisenstein series [Formula: see text] at [Formula: see text]. The holomorphic part of this function has a transformation rule that we express in finite terms as a generalized Dedekind sum. This gives rise to the explicit construction (in finite terms) of elements of [Formula: see text]. We also give a short proof of the reciprocity formula for this Dedekind sum.