Investigation of tanpura string vibrations using a two-dimensional time-domain model incorporating coupling and bridge friction

Tanpura string vibrations have been investigated previously using numerical models based on energy conserving schemes derived from a Hamiltonian description in one-dimensional form. Such time-domain models have the property that, for the lossless case, the numerical Hamiltonian (representing total energy of the system) can be proven to be constant from one time step
to the next, irrespective of any of the system parameters; in practice the Hamiltonian can be shown to be conserved within machine precision. Models of this kind can reproduce a jvari effect, which results from the bridge-string interaction. However the one-dimensional formulation has recently been shown to fail to replicate the jvaris strong dependence on the thread placement. As a first step towards simulations which accurately emulate this sensitivity to the thread placement, a twodimensional model is proposed, incorporating coupling of controllable level between the two string polarisations at the string termination opposite from the barrier. In addition, a friction force acting when the string slides across the bridge in horizontal direction is introduced, thus effecting a further damping mechanism. In this preliminary study, the string is terminated at the position of the thread. As in the one-dimensional model, an implicit scheme has to be used to solve the system, employing Newton's method to calculate the updated positions and momentums of each string segment. The two-dimensional model is proven to be energy conserving when the loss parameters are set to zero, irrespective of the coupling constant. Both frequency-dependent and independent losses are then added to the string, so that the model can be compared to analogous instruments. The influence of coupling and the bridge friction are investigated.

Violation of Bell inequalities in larger Hilbert spaces: robustness and challenges

We explore the challenges posed by the violation of Bell-like inequalities by d-dimensional systems exposed to imperfect state-preparation and measurement settings. We address, in particular, the limit of high-dimensional systems, naturally arising when exploring the quantum-to-classical transition. We show that, although suitable Bell inequalities can be violated, in principle, for any dimension of given subsystems, it is in practice increasingly challenging to detect such violations, even if the system is prepared in a maximally entangled state. We characterize the effects of random perturbations on the state or on the measurement settings, also quantifying the efforts needed to certify the possible violations in case of complete ignorance on the system state at hand.

Catena-Poly[bis(ethane-1,2-diammonium) [manganese(II)-di-mu-phosphato-kappa O-4 : O ']]: a one-dimensional manganese phosphate

The title compound,{(C2H10N2)(2)[Mn(PO4)(2)]}(n), contains anionic square-twisted chains of formula [Mn(PO4)(2)](4-) constructed from corner-sharing four-membered rings of alternating MnO4 and PO4 units. The Mn and P atoms have distorted tetrahedral coordination and the Mn atom lies on a twofold axis. The linear manganese-phosphate chains are held together by hydrogen-bonding interactions involving the framework O atoms and the H atoms of the ethane-1,2-diammonium cations, which lie in the interchain spaces.

Full-dimensional vibrational calculations for H5O2+ using an ab initio potential energy surface

We report quantum diffusion Monte Carlo (DMC) and variational calculations in full dimensionality for selected vibrational states of H5O2+ using a new ab initio potential energy surface [X. Huang, B. Braams, and J. M. Bowman, J. Chem. Phys. 122, 044308 (2005)]. The energy and properties of the zero-point state are focused on in the rigorous DMC calculations. OH-stretch fundamentals are also calculated using "fixed-node" DMC calculations and variationally using two versions of the code MULTIMODE. These results are compared with infrared multiphoton dissociation measurements of Yeh [L. I. Yeh, M. Okumura, J. D. Myers, J. M. Price, and Y. T. Lee, J. Chem. Phys. 91, 7319 (1989)]. Some preliminary results for the energies of several modes of the shared hydrogen are also reported.

Full-dimensional quantum calculations of ground-state tunneling splitting of malonaldehyde using an accurate ab initio potential energy surface

Quantum calculations of the ground vibrational state tunneling splitting of H-atom and D-atom transfer in malonaldehyde are performed on a full-dimensional ab initio potential energy surface (PES). The PES is a fit to 11 147 near basis-set-limit frozen-core CCSD(T) electronic energies. This surface properly describes the invariance of the potential with respect to all permutations of identical atoms. The saddle-point barrier for the H-atom transfer on the PES is 4.1 kcal/mol, in excellent agreement with the reported ab initio value. Model one-dimensional and "exact" full-dimensional calculations of the splitting for H- and D-atom transfer are done using this PES. The tunneling splittings in full dimensionality are calculated using the unbiased "fixed-node" diffusion Monte Carlo (DMC) method in Cartesian and saddle-point normal coordinates. The ground-state tunneling splitting is found to be 21.6 cm(-1) in Cartesian coordinates and 22.6 cm(-1) in normal coordinates, with an uncertainty of 2-3 cm(-1). This splitting is also calculated based on a model which makes use of the exact single-well zero-point energy (ZPE) obtained with the MULTIMODE code and DMC ZPE and this calculation gives a tunneling splitting of 21-22 cm(-1). The corresponding computed splittings for the D-atom transfer are 3.0, 3.1, and 2-3 cm(-1). These calculated tunneling splittings agree with each other to within less than the standard uncertainties obtained with the DMC method used, which are between 2 and 3 cm(-1), and agree well with the experimental values of 21.6 and 2.9 cm(-1) for the H and D transfer, respectively. (C) 2008 American Institute of Physics.

Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces

We consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface which is assumed to lie within a finite distance of some plane. The paper is concerned with the study of an equivalent variational formulation of this problem set in a scale of weighted Sobolev spaces. We prove well-posedness of this variational formulation in an energy space with weights which extends previous results in the unweighted setting [S. Chandler-Wilde and P. Monk, SIAM J. Math. Anal., 37 (2005), pp. 598–618] to more general inhomogeneous terms in the Helmholtz equation. In particular, in the two-dimensional case, our approach covers the problem of plane wave incidence, whereas in the three-dimensional case, incident spherical and cylindrical waves can be treated. As a further application of our results, we analyze a finite section type approximation, whereby the variational problem posed on an infinite layer is approximated by a variational problem on a bounded region.

On the operator space structure of Hilbert spaces

Operator spaces of Hilbertian JC∗ -triples E are considered in the light of the universal ternary ring of operators (TRO) introduced in recent work. For these operator spaces, it is shown that their triple envelope (in the sense of Hamana) is the TRO they generate, that a complete isometry between any two of them is always the restriction of a TRO isomorphism and that distinct operator space structures on a fixed E are never completely isometric. In the infinite-dimensional cases, operator space structure is shown to be characterized by severe and definite restrictions upon finite-dimensional subspaces. Injective envelopes are explicitly computed.

Toeplitz operators on Bergman spaces with locally integrable symbols

We study the boundedness of Toeplitz operators $T_a$ with locally integrable symbols on Bergman spaces $A^p(\mathbb{D})$, $1 < p < \infty$. Our main result gives a sufficient condition for the boundedness of $T_a$ in terms of some ``averages'' (related to hyperbolic rectangles) of its symbol. If the averages satisfy an ${o}$-type condition on the boundary of $\mathbb{D}$, we show that the corresponding Toeplitz operator is compact on $A^p$. Both conditions coincide with the known necessary conditions in the case of nonnegative symbols and $p=2$. We also show that Toeplitz operators with symbols of vanishing mean oscillation are Fredholm on $A^p$ provided that the averages are bounded away from zero, and derive an index formula for these operators.

A socio-technical network approach to the adoption of low and zero carbon technologies in new housing

Housing in the UK accounts for 30.5% of all energy consumed and is responsible for 25% of all carbon emissions. The UK Government’s Code for Sustainable Homes requires all new homes to be zero carbon by 2016. The development and widespread diffusion of low and zero carbon (LZC) technologies is recognised as being a key solution for housing developers to deliver against this zero-carbon agenda. The innovation challenge to design and incorporate these technologies into housing developers’ standard design and production templates will usher in significant technical and commercial risks. In this paper we report early results from an ongoing Engineering and Physical Sciences Research Council project looking at the innovation logic and trajectory of LZC technologies in new housing. The principal theoretical lens for the research is the socio-technical network approach which considers actors’ interests and interpretative flexibilities of technologies and how they negotiate and reproduce ‘acting spaces’ to shape, in this case, the selection and adoption of LZC technologies. The initial findings are revealing the form and operation of the technology networks around new housing developments as being very complex, involving a range of actors and viewpoints that vary for each housing development.

Rigid body trajectories in different 6D spaces

The objective of this paper is to show that the group SE(3) with an imposed Lie-Poisson structure can be used to determine the trajectory in a spatial frame of a rigid body in Euclidean space. Identical results for the trajectory are obtained in spherical and hyperbolic space by scaling the linear displacements appropriately since the influence of the moments of inertia on the trajectories tends to zero as the scaling factor increases. The semidirect product of the linear and rotational motions gives the trajectory from a body frame perspective. It is shown that this cannot be used to determine the trajectory in the spatial frame. The body frame trajectory is thus independent of the velocity coupling. In addition, it is shown that the analysis can be greatly simplified by aligning the axes of the spatial frame with the axis of symmetry which is unchanging for a natural system with no forces and rotation about an axis of symmetry.

A uniformly valid far field asymptotic expansion of the green function for two-dimensional propagation above a homogeneous impedance plane

A generalized asymptotic expansion in the far field for the problem of cylindrical wave reflection at a homogeneous impedance plane is derived. The expansion is shown to be uniformly valid over all angles of incidence and values of surface impedance, including the limiting cases of zero and infinite impedance. The technique used is a rigorous application of the modified steepest descent method of Ot

On the existence of two-dimensional Euler flows satisfying energy-Casimir stability criteria

The energy-Casimir stability method, also known as the Arnold stability method, has been widely used in fluid dynamical applications to derive sufficient conditions for nonlinear stability. The most commonly studied system is two-dimensional Euler flow. It is shown that the set of two-dimensional Euler flows satisfying the energy-Casimir stability criteria is empty for two important cases: (i) domains having the topology of the sphere, and (ii) simply-connected bounded domains with zero net vorticity. The results apply to both the first and the second of Arnold’s stability theorems. In the spirit of Andrews’ theorem, this puts a further limitation on the applicability of the method. © 2000 American Institute of Physics.

On Arnol'd's second nonlinear stability theorem for two-dimensional quasi-geostrophic flow

Arnol'd's second hydrodynamical stability theorem, proven originally for the two-dimensional Euler equations, can establish nonlinear stability of steady flows that are maxima of a suitably chosen energy-Casimir invariant. The usual derivations of this theorem require an assumption of zero disturbance circulation. In the present work an analogue of Arnol'd's second theorem is developed in the more general case of two-dimensional quasi-geostrophic flow, with the important feature that the disturbances are allowed to have non-zero circulation. New nonlinear stability criteria are derived, and explicit bounds are obtained on both the disturbance energy and potential enstrophy which are expressed in terms of the initial disturbance fields. While Arnol'd's stability method relies on the second variation of the energy-Casimir invariant being sign-definite, the new criteria can be applied to cases where the second variation is sign-indefinite because of the disturbance circulations. A version of Andrews' theorem is also established for this problem.

Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading

In the present paper we study the approximation of functions with bounded mixed derivatives by sparse tensor product polynomials in positive order tensor product Sobolev spaces. We introduce a new sparse polynomial approximation operator which exhibits optimal convergence properties in L2 and tensorized View the MathML source simultaneously on a standard k-dimensional cube. In the special case k=2 the suggested approximation operator is also optimal in L2 and tensorized H1 (without essential boundary conditions). This allows to construct an optimal sparse p-version FEM with sparse piecewise continuous polynomial splines, reducing the number of unknowns from O(p2), needed for the full tensor product computation, to View the MathML source, required for the suggested sparse technique, preserving the same optimal convergence rate in terms of p. We apply this result to an elliptic differential equation and an elliptic integral equation with random loading and compute the covariances of the solutions with View the MathML source unknowns. Several numerical examples support the theoretical estimates.

Eigenvalues of a one-dimensional Dirac operator pencil

We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.