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The Jacobi linear iterative method and weight Jacobi method (WJM) are introduced for solving the large-scale linear problem in the proximity effect correction (PEC) of electron beam lithography. Based on the discussion of PEC physics, a symmetrical and positive defined proximity interaction matrix is constructed to ensure the convergence of the methods. It shows that zeroing the center exposure fraction in the point spread function matrix is equal to the operation of splitting the proximity interaction matrix. Then, the Jacobi method is ready for the PEC calculation. The iterative can be performed in the Fourier domain due to the inherent parallelization of the Jacobi method. The convergent property of the Jacobi method is discussed and then testified by the PEC simulation. Compared with the classical Jacobi method, an improvement of 100% in convergence efficiency can be achieved by introducing the optimized relaxation parameter quasi-ωopt in the WJM. By combining the WJM and Gold nonlinear iteration method, a new method that shows an order of magnitude superior in accuracy to the WJM is proposed. Results indicate the methods introduced here could be used to calculate the PEC problem efficiently. [ABSTRACT FROM AUTHOR]

Full quantum mechanical calculations of vibrational energies of methane and fluoromethane are carried out using a polyspherical description combining Radau and Jacobi coordinates. The Hamiltonian is built in a potential-optimized discrete variable representation, and vibrational energies are solved using an iterative eigensolver. This new approach can be applied to a large variety of molecules. In particular, we show that it is able to accurately and efficiently compute eigenstates for four different molecules : CH4, CHD3, CH2D2, and CH3F. Very good agreement is obtained with the results reported previously in the literature with different approaches and with experimental data. [ABSTRACT FROM AUTHOR]

The global ground and first three excited electronic state adiabatic as well as the corresponding quasidiabatic potential energy surfaces is reported as a function of nuclear geometries in the Jacobi coordinates (..., ..., γ) using Dunning's cc-pVTZ basis set at the internally contracted multi-reference (single and double) configuration interaction level of accuracy. Nonadiabatic couplings, arising out of relative motion of proton and the vibrational motion of CO, are also reported in terms of coupling potentials. The quasidiabatic potential energy surfaces and the coupling potentials have been obtained using the ab initio procedure [Simah et al., J. Chem. Phys. 111, 4523 (1999)] for the purpose of dynamics studies. [ABSTRACT FROM AUTHOR]

We extend the periodic von Neumann basis to non-Cartesian coordinates. The bound states of two isomerizing triatomic molecules, LiCN/LiNC and HCN/HNC, are calculated using the vibrational Hamiltonian in Jacobi coordinates. The phase space localization of the basis functions leads to a flexible and accurate representation of the Hamiltonian. This results in significant savings compared to a basis localized just in coordinate space. The favorable scaling of the method with dimensionality makes it promising for applications to larger systems. [ABSTRACT FROM AUTHOR]

This research paper employs two different computational schemes to the couple Boiti–Leon–Pempinelli system and the (3+1)-dimensional Kadomtsev–Petviashvili equation to find novel explicit wave solutions for these models. Both models depict a generalized form of the dispersive long wave equation. The complex, exponential, hyperbolic, and trigonometric function solutions are some of the obtained solutions by using the modified Khater method and the Jacobi elliptical function method. Moreover, their stability properties are also analyzed, and for more interpretation of the physical features of the obtained solutions, some sketches are plotted. Additionally, the novelty of our paper is explained by displaying the similarity and difference between the obtained solutions and those obtained in a different research paper. The performance of both methods is tested to show their ability to be applied to several nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

A quantum scattering dynamics, time-dependent wavepacket propagation method is applied to study the reaction of H2D++H2→H3++HD on the Xie–Braams–Bowman potential energy surface. The reduced-dimensional, seven-degree-of-freedom approach is employed in this calculation by fixing one Jacobi and one torsion angle related to H2D+ at the lowest saddle point geometry of D2d on the potential energy surface. Initial state selected reaction probabilities are presented for various initial rovibrational states. The ground state reaction probability shows no threshold for this reaction, in other words, this reaction can occur without an activation barrier. The vibrational excitation shows that the stretching motion of H+–HD only has a small effect on the reaction probability; the vibrational excitation of HD in H2D+ hinders the reactivity. By contrast, rotational excitation of H+–HD greatly enhances the reactivity with the reaction probability increased double or triple at high rotational states compared to the ground state. Reactive resonances, seen in all the initial state selected reaction probabilities, are also found in the integral cross section for the ground state of H2D+ and H2. The thermal rate coefficient is also calculated and is found to be in semiquantitative agreement with experiment; however, quantum scattering approaches including more degrees of freedom, especially including all the angles, are necessary to study this reaction in the future. [ABSTRACT FROM AUTHOR]

The S-matrix for a scattering system provides the most detailed information about the dynamics. In this work, we discuss the calculation of S-matrix elements for the A+BC→AB+C, AC+B type reaction. Two methods for extracting S-matrix elements from a single wave packet in reactant Jacobi coordinates are reviewed and compared. Both methods are capable of extracting the state-to-state attributes for both product channels from a single wave packet propagation. It is shown through the examples of H+HD, Cl+H2, and H+HCl reactions that such reactant coordinate based methods are easy to implement, numerically efficient, and accurate. Additional efficiency can be gained by the use of a L-shaped grid with two-dimensional fast Fourier transform. [ABSTRACT FROM AUTHOR]

A new Laplace transform based multistate local CC2 response method for calculating excitation energies of extended molecular systems is presented. By virtue of the Laplace transform trick, the eigenvalue problem involving the local CC2 Jacobian is partitioned along the doubles-doubles block (which is diagonal in the parental canonical method) without losing the sparsity in the integral, amplitude, and amplitude response supermatrices. Hence, only an effective eigenvalue problem involving singles vectors has to be solved, while the doubles part can be computed on-the-fly. Within this framework, a multistate treatment of excited states with state specific and adaptive local approximations imposed on the doubles part is straightforwardly possible. Furthermore, in the context of the density fitting approximation of the two-electron integrals, a procedure to specify the local approximation, i.e., the restricted pair lists and domains, on the basis of an analysis of the object to be approximated itself is proposed. Performance and accuracy of the new Laplace transformed density fitted local CC2 (LT-DF-LCC2) response method are tested for set of different test molecules and states. It turns out that LT-DF-LCC2 response is much more robust than the earlier local CC2 response method proposed before, which failed to find some excited states in difficult cases. [ABSTRACT FROM AUTHOR]

Quantum dynamical calculations are reported for the zero point energy, several low-lying vibrational states, and the infrared spectrum of the H5O2+ cation. The calculations are performed by the multiconfiguration time-dependent Hartree (MCTDH) method. A new vector parametrization based on a mixed Jacobi-valence description of the system is presented. With this parametrization the potential energy surface coupling is reduced with respect to a full Jacobi description, providing a better convergence of the n-mode representation of the potential. However, new coupling terms appear in the kinetic energy operator. These terms are derived and discussed. A mode-combination scheme based on six combined coordinates is used, and the representation of the 15-dimensional potential in terms of a six-combined mode cluster expansion including up to some 7-dimensional grids is discussed. A statistical analysis of the accuracy of the n-mode representation of the potential at all orders is performed. Benchmark, fully converged results are reported for the zero point energy, which lie within the statistical uncertainty of the reference diffusion Monte Carlo result for this system. Some low-lying vibrationally excited eigenstates are computed by block improved relaxation, illustrating the applicability of the approach to large systems. Benchmark calculations of the linear infrared spectrum are provided, and convergence with increasing size of the time-dependent basis and as a function of the order of the n-mode representation is studied. The calculations presented here make use of recent developments in the parallel version of the MCTDH code, which are briefly discussed. We also show that the infrared spectrum can be computed, to a very good approximation, within D2d symmetry, instead of the G16 symmetry used before, in which the complete rotation of one water molecule with respect to the other is allowed, thus simplifying the dynamical problem. [ABSTRACT FROM AUTHOR]

We study the problem of finding a path that joins a given initial state with a final one, where the evolution is governed by classical (Hamiltonian) dynamics. A new algorithm for the computation of long time transition trajectories connecting two configurations is presented. In particular, a strategy for finding transition paths between two stable basins is established. The starting point is the formulation of the equation of motion of classical mechanics in the framework of Jacobi’s principle; a shortening procedure inspired by Birkhoff’s method is then applied to find geodesic solutions. Numerical examples are given for Müller’s potential and the collinear reaction H2+H→H+H2. [ABSTRACT FROM AUTHOR]

In the hidden variable theory, Bohm proved a connection between the Schrödinger and Hamilton–Jacobi equations and showed the existence of classical paths, for which the generalized Bohr quantization condition is valid. In this paper we prove similar properties, starting from the equivalence between the Schrödinger and wave equations in the case of the conservative bound systems. Our approach is based on the equations and postulates of quantum mechanics without using any additional postulate. Like in the hidden variable theory, the above properties are proven without using the approximation of geometrical optics or the semiclassical approximation. Since the classical paths have only a mathematical significance in our analysis, our approach is consistent with the postulates of quantum mechanics. [ABSTRACT FROM AUTHOR]

The application of the concept of body-fixed reference frames, proposed by C. Eckart [Phys. Rev. 47, 552 (1935)], to the problem of the separation of three collective angles in quantum N-body problems is analyzed based on the technique recently developed by Meremianin and Briggs [Phys. Rep. 384, 121 (2003)]. Special attention is paid to the body frame defined by the “second Eckart condition” which minimizes vibro-rotational couplings near the equilibrium position. The important case of the Eckart frame for three-body systems is considered in detail. The connection of the basis vectors of the Eckart frame with Jacobi vectors is derived. All results of this work are valid for an arbitrary choice of internal (body-frame) coordinates. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Presents theoretical and computational details of implementing the reactant-product decoupling (RPD) method. Solution of psi[sub r] in the reactant Jacobi coordinates; Propagation of psi[sub r] in the product arrangement; Transformations of Jacobi coordinates between arrangements; General collocation-quadrature scheme.

In this paper, we study the probability density function, P (c , α , β , n) d c , of the center of mass of the finite n Jacobi unitary ensembles with parameters α > −1 and β > −1; that is the probability that trMn ∈ (c, c + dc), where Mn are n × n matrices drawn from the unitary Jacobi ensembles. We compute the exponential moment generating function of the linear statistics ∑ j = 1 n f ( x j ) ≔ ∑ j = 1 n x j , denoted by M f (λ , α , β , n). The weight function associated with the Jacobi unitary ensembles reads xα(1 − x)β, x ∈ [0, 1]. The moment generating function is the n × n Hankel determinant Dn(λ, α, β) generated by the time-evolved Jacobi weight, namely, w (x; λ, α, β) = xα(1 − x)β e−λx, x ∈ [0, 1], α > −1, β > −1. We think of λ as the time variable in the resulting Toda equations. The non-classical polynomials defined by the monomial expansion, Pn(x, λ) = xn + p(n, λ) xn−1 + ⋯ + Pn(0, λ), orthogonal with respect to w (x, λ, α, β) over [0, 1] play an important role. Taking the time evolution problem studied in Basor et al. [J. Phys. A: Math. Theor. 43, 015204 (2010)], with some change of variables, we obtain a certain auxiliary variable rn(λ), defined by integral over [0, 1] of the product of the unconventional orthogonal polynomials of degree n and n − 1 and w (x; λ, α, β)/x. It is shown that rn(2iez) satisfies a Chazy II equation. There is another auxiliary variable, denoted as Rn(λ), defined by an integral over [0, 1] of the product of two polynomials of degree n multiplied by w (x; λ, α, β)/x. Then Yn(−λ) = 1 − λ/Rn(λ) satisfies a particular Painlevé V: PV(α2/2, − β2/2, 2n + α + β + 1, 1/2). The σn function defined in terms of the λp(n, −λ) plus a translation in λ is the Jimbo–Miwa–Okamoto σ-form of Painlevé V. The continuum approximation, treating the collection of eigenvalues as a charged fluid as in the Dyson Coulomb Fluid, gives an approximation for the moment generating function M f (λ , α , β , n) when n is sufficiently large. Furthermore, we deduce a new expression of M f (λ , α , β , n) when n is finite, in terms the σ function of this is a particular case of Painlevé V. An estimate shows that the moment generating function is a function of exponential type and of order n. From the Paley-Wiener theorem, one deduces that P (c , α , β , n) has compact support [0, n]. This result is easily extended to the β ensembles, as long as the weight w is positive and continuous over [0, 1]. [ABSTRACT FROM AUTHOR]

The effect of hawkmoth-like flexibility on the aerodynamic hovering performance of wings at a Reynolds number of 400 has been assessed by conducting fluid structure interaction simulations incorporating a finite difference based immersed boundary method coupled with a finite-element based structure solver. The stiffness distribution of a hawkmoth forewing was mapped onto three wing shapes ( r ¯ 1 = 0.43, 0.53, and 0.63) defined by the radius of the first moment of wing area each with aspect ratios, AR = 1.5, 2.96, 4.5, and 6.0 using elliptic mesh generation, the Jacobi method for iterations, and the concept of the barycentric coordinate system. The results show that there is a dominant chordwise deformation at AR = 1.5, and the wings also deform in the spanwise direction and their tips deviate from the horizontal stroke plane as AR increases. At AR = 1.5, 2.96, and 4.5, flexibility increases the mean lift (up to 39%, 18%, and 17.6%, respectively) for all wing shapes. At AR = 6.0, the r 1 ¯ = 0.53 and 0.63 flexible wings give lesser lift than the rigid equivalents because of negative lift or small positive lift during the early stroke as the vortical structures remain on the bottom surface. This is attributed to the rapid pitch-down rotation, lesser stroke angular velocity than the rigid wing, and upward motion of the wingtip, away from the horizontal stroke plane. From the design perspective, the anisotropic flexible wings (except r 1 ¯ = 0.53 and 0.63 with AR = 6.0) can be used in micro aerial vehicles for high lift requirements, such as for a high payload. Results here show that in nature, the hawkmoth wings with r 1 ¯ and AR of 0.43-0.44 and 2.73-2.92, respectively, appear to have a combination of the shape, AR, and flexibility that optimizes power economy. [ABSTRACT FROM AUTHOR]

The differential equation that defines the Zernike system, originally proposed to classify wavefront aberrations of the wavefields in the disk of a circular pupil, had been shown to separate in three distinct coordinate systems obtained from polar coordinates on a half-sphere. Here we find and examine the separation in the generic elliptic coordinate system on the half-sphere and its projected disk, where the solutions, separated in Jacobi coordinates, contain Heun polynomials. [ABSTRACT FROM AUTHOR]

We discuss several new bi-Hamiltonian integrable systems on the plane with integrals of motion of third, fourth, and sixth orders in momenta. The corresponding variables of separation, separated relations, compatible Poisson brackets, and recursion operators are also presented in the framework of the Jacobi method. [ABSTRACT FROM AUTHOR]

The Jacobi metric derived from the line element by one of the authors is shown to reduce to the standard formulation in the non-relativistic approximation. We obtain the Jacobi metric for various stationary metrics. Finally, the Jacobi-Maupertuis metric is formulated for time-dependent metrics by including the Eisenhart-Duval lift, known as the Jacobi-Eisenhart metric. [ABSTRACT FROM AUTHOR]

The Bessel process with parameter D > 1 and the Dyson model of interacting Brownian motions with coupling constant ß > 0 are extended to the processes in which the drift term and the interaction terms are given by the logarithmic derivatives of Jacobi's theta functions. They are called the elliptic Bessel process, eBES(D), and the elliptic Dyson model, eDYS(ß), respectively. Both are realized on the circumference of a circle [0,2pr) with radius r > 0 as temporally inhomogeneous processes defined in a finite time interval [0, t*), t* < 8. Transformations of them to Schrödinger-type equations with time-dependent potentials lead us to proving that eBES(D) and eDYS(ß) can be constructed as the time-dependent Girsanov transformations of Brownian motions. In the special cases where D = 3 and ß = 2, observables of the processes are defined and the processes are represented for them using the Brownian paths winding round a circle and pinned at time t*. We show that eDYS(2) has the determinantal martingale representation for any observable. Then it is proved that eDYS(2) is determinantal for all observables for any finite initial configuration without multiple points. Determinantal processes are stochastic integrable systems in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single continuous function called the spatio-temporal correlation kernel. [ABSTRACT FROM AUTHOR]

Through the use of suitable variable transformations, the commonality of all extended magnetohydrodynamics (MHD) models is established. Remarkable correspondences between the Poisson brackets of inertialess Hall MHD and inertial MHD (which has electron inertia, but not the Hall drift) and extended MHD (which has both effects) are established. The helicities (two in all) for each of these models are obtained through these correspondences. The commonality of all the extended MHD models is traced to the existence of two Lie-dragged 2-forms, which are closely associated with the canonical momenta of the two underlying species. The Lie-dragging of these 2-forms by suitable velocities also leads to the correct equations of motion. The Hall MHD Poisson bracket is analyzed in detail, the Jacobi identity is verified through a detailed proof, and this proof ensures the Jacobi identity for the Poisson brackets of all the models. [ABSTRACT FROM AUTHOR]

Constants of motion, Lagrangians, and Hamiltonians admitted by a family of relevant nonlinear oscillators are derived using a geometric formalism. The theory of the Jacobi last multiplier allows us to find Lagrangian descriptions and constants of the motion. An application of the jet bundle formulation of symmetries of differential equations is presented in the second part of the paper. After a short review of the general formalism, the particular case of non-local symmetries is studied in detail by making use of an extended formalism. The theory is related to some results previously obtained by Krasil'shchik, Vinogradov, and coworkers. Finally, the existence of non-local symmetries for such two nonlinear oscillators is proved. [ABSTRACT FROM AUTHOR]

In a previous paper, we presented 3 + 2M term recurrence relations with variable dependent coefficients for M-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. In this paper, we present (conjectures of) the recurrence relations with constant coefficients for these multi-indexed orthogonal polynomials. The simplest recurrence relations have 3 + 2l terms, where l (= M) is the degree of the lowest member of the multi-indexed orthogonal polynomials. [ABSTRACT FROM AUTHOR]

We construct the bosonization of the Fock space F⊗½ of a single neutral fermion by using a 2-point local Heisenberg field. We decompose F⊗½ as a direct sum of irre-ducible highest weight modules for the Heisenberg algebra HZ, and thus we show that under the Heisenberg HZ action the Fock space F⊗½ of the single neutral fermion is isomorphic to the Fock space F⊗1 of a pair of charged free fermions, thereby con-structing the boson-fermion correspondence of type D-A. As a corollary we obtain the Jacobi identity equating the graded dimension formulas utilizing both the Heisenberg and the Virasoro gradings on F⊗³. We construct a family of 2-point-local Virasoro fields with central charge -2 + 12λ - 12λ², λ ∊ C, on FⅫ&#;179;. We construct a W1+∞ representation on F⊗1 and show that under this W1+X action F⊗³ is again isomorphic to F⊗1. [ABSTRACT FROM AUTHOR]

We construct the bosonization of the Fock space F⊗½ of a single neutral fermion by using a 2-point local Heisenberg field. We decompose F⊗½ as a direct sum of irre-ducible highest weight modules for the Heisenberg algebra HZ, and thus we show that under the Heisenberg HZ action the Fock space F⊗½ of the single neutral fermion is isomorphic to the Fock space F⊗1 of a pair of charged free fermions, thereby con-structing the boson-fermion correspondence of type D-A. As a corollary we obtain the Jacobi identity equating the graded dimension formulas utilizing both the Heisenberg and the Virasoro gradings on F⊗³. We construct a family of 2-point-local Virasoro fields with central charge -2 + 12λ - 12λ², λ ∊ C, on FⅫ&#;179;. We construct a W1+∞ representation on F⊗1 and show that under this W1+X action F⊗³ is again isomorphic to F⊗1. [ABSTRACT FROM AUTHOR]

Linear operators R are introduced on tensor products of evaluation modules of U'q（sl(2)） obtained from the complementary and strange series representations. The operators R satisfy the intertwining condition on finite linear combinations of the canonical basis elements of the tensor products. Infinite sums associated with the action of R on six pairs of tensor products are evaluated. For two pairs, the sums are related to the vector-valued big q-Jacobi transform of the matrix elements defining the operator R. In one case, the sums specify the action of R on the irreducible representations present in the decomposition of the underlying indivisible sum of U'q（sl(2)）-tensor products. In both cases, bilinear summation formulae for the matrix elements of R provide a generalization of the unitarity property. Corresponding results are given for the remaining pairs. [ABSTRACT FROM AUTHOR]

The partition function of the Chern-Simons theory on the three-sphere with the unitary group U(N) provides a one-matrix model. The corresponding N-particle system can be mapped to the determinantal point process whose correlation kernel is expressed by using the Stieltjes-Wigert orthogonal polynomials. The matrix model and the point process are regarded as q-extensions of the random matrix model in the Gaussian unitary ensemble and its eigenvalue point process, respectively. We prove the convergence of the N-particle system to an infinite-dimensional determinantal point process in N → ∞, in which the correlation kernel is expressed by Jacobi's theta functions. We show that the matrix model obtained by this limit realizes the oscillatory matrix model in Chern-Simons theory discussed by de Haro and Tierz. [ABSTRACT FROM AUTHOR]

We give a detailed study of the traveling wave solutions of (l = 2) Kaup-Boussinesq type of coupled KdV equations. Depending upon the zeros of a fourth degree polynomial, we have cases where there exist no nontrivial real solutions, cases where asymptotically decaying to a constant solitary wave solutions, and cases where there are periodic solutions. All such possible solutions are given explicitly in the form of Jacobi elliptic functions. Graphs of some exact solutions in solitary wave and periodic shapes are exhibited. Extension of our study to the cases l = 3 and l = 4 are also mentioned. [ABSTRACT FROM AUTHOR]