Parabolic Representation Research Articles

Review of Rydberg Spectral Line Formation in Plasmas

The present review is dedicated to the problem of an array of transitions between highly-excited atomic levels. Hydrogen atoms and hydrogen-like ions in plasmas are considered here. The presented methods focus on calculation of spectral line shapes. Fast and simple methods of universal ionic profile calculation for the Hnα (Δn=1) and Hnβ (Δn=2) spectral lines are demonstrated. The universal dipole matrix elements formulas for the Hnα and Hnβ transitions are presented. A fast method for spectral line shape calculations in the presence of an external magnetic field using the formulas for universal dipole matrix elements is proposed. This approach accounts for the Doppler and Stark–Zeeman broadening mechanisms. Ion dynamics effects are treated via the frequency fluctuation model. The accuracy of the presented model is discussed. A comparison of this approach with experimental data and the results of molecular dynamics simulation is demonstrated. The kinetics equation for the populations of highly-excited ionic states is solved in the parabolic representation. The population source associated with dielectronic recombination is considered.

From quantum groups to Liouville and dilaton quantum gravity

We investigate the underlying quantum group symmetry of 2d Liouville and dilaton gravity models, both consolidating known results and extending them to the cases with mathcal{N} = 1 supersymmetry. We first calculate the mixed parabolic representation matrix element (or Whittaker function) of Uq( mathfrak{sl} (2, ℝ)) and review its applications to Liouville gravity. We then derive the corresponding matrix element for Uq( mathfrak{osp} (1|2, ℝ)) and apply it to explain structural features of mathcal{N} = 1 Liouville supergravity. We show that this matrix element has the following properties: (1) its q → 1 limit is the classical OSp+(1|2, ℝ) Whittaker function, (2) it yields the Plancherel measure as the density of black hole states in mathcal{N} = 1 Liouville supergravity, and (3) it leads to 3j-symbols that match with the coupling of boundary vertex operators to the gravitational states as appropriate for mathcal{N} = 1 Liouville supergravity. This object should likewise be of interest in the context of integrability of supersymmetric relativistic Toda chains. We furthermore relate Liouville (super)gravity to dilaton (super)gravity with a hyperbolic sine (pre)potential. We do so by showing that the quantization of the target space Poisson structure in the (graded) Poisson sigma model description leads directly to the quantum group Uq( mathfrak{sl} (2, ℝ)) or the quantum supergroup Uq( mathfrak{osp} (1|2, ℝ)).

A STUDY ON LENGTH WEIGHT RELATIONSHIP OF FISH Channa punctatus (BLOCH) FROM THE RIVER GADADHAR AT COOCHBEHAR DISTRICT, WEST BENGAL, INDIA

The present study was aimed to describe the length weight relationship and condition factor of a study on Length weight Relationship of Channa punctatus (Bloch) [1] from the Gadadhar river at Coochbehar District, West Bengal, India. In this study, a total of 25 fish specimen comprises of Channa punctatus ranged from 9.3 cm to 14.2 cm in length and 5.12 gm to 19.3 gm in weight were studied. The entire length weight data were analyzed by least square method. The corresponding parabolic representation is w=0.0086413L2.90374 The computed b value (b=2.90374) of the species is not significantly different from the expected value '3' and hence the cube law W=aLb hold good in the case of Channa punctatus. In the species, there is a strong positive relationship between length and weight. The sampling areas were chosen to be spatially and geographically diverse, as well as to have varying environmental conditions. A true relationship was observed between an animal's length and weight. In general, fishes' growth patterns follow the cube laws. When the growth of the fish is isometric, such a relationship will be valid. The value of condition factor (K) was found to be more than 1* in the population of the fish indicated good condition of fish in the Gadadhar river, WestBengal, India (K=1.00308).

Nondiscrete parabolic characters of the free group F2: supergroup density and Nielsen classes in the complement of the Riley slice

A parabolic representation of the free group $F_2$ is one in which the images of both generators are parabolic elements of $PSL(2,\IC)$. The Riley slice is a closed subset ${\cal R}\subset \IC$ which is a model for the parabolic, discrete and faithful characters of $F_2$. The complement of the Riley slice is a bounded Jordan domain within which there are isolated points, accumulating only at the boundary, corresponding to parabolic discrete and faithful representations of rigid subgroups of $PSL(2,\IC)$. Recent work of Aimi, Akiyoshi, Lee, Oshika, Parker, Lee, Sakai, Sakuma \& Yoshida, have topologically identified all these groups. Here we give the first identified substantive properties of the nondiscrete representations and prove a supergroup density theorem: given any irreducible parabolic representation $\rho_*:F_2\to PSL(2,\IC)$ whatsoever, any non-discrete parabolic representation $\rho_0$ has an arbitrarily small perturbation $\rho_\epsilon$ so that $\rho_\epsilon(F_2)$ contains a conjugate of $\rho_*(F_2)$ as a proper subgroup. This implies that if $\Gamma_*$ is any nonelementary group generated by two parabolic elements (discrete or otherwise) and $\gamma_0$ is any point in the complement of the Riley slice, then in any neighbourhood of $\gamma$ there is a point corresponding to a nonelementary group generated by two parabolics with a conjugate of $\Gamma_*$ as a proper subgroup. Using these ideas we then show that there are nondiscrete parabolic representations with an arbitrarily large number of distinct Nielsen classes of parabolic generators.

Spectra of a Rydberg Atom in Crossed Electric and Magnetic Fields

Contemporary spectroscopic studies of astrophysical and laboratory plasmas frequently deal with extremely large values of principle quantum numbers of atomic systems. These atomic states are very sensitive to electric and magnetic fields of the surrounding medium. While interpreting the spectra of such excited atomic systems, one faces the problem of a huge array of radiative transitions between highly excited atomic levels. Moreover, external electric and magnetic fields significantly complicate the problem because of the absence of standard selection rules typical for the spherical quantization. The analytical expression in the parabolic representation for dipole matrix elements obtained by Gordon contains hyper-geometric series and it has a very complex structure. The matrix elements that involve the presence of electric and magnetic fields are calculated while using a representation closely related to the parabolic quantization on two different axes. This matrix element depends in a complex way on the transition probabilities in the parabolic coordinate system (Gordon’s formulas) and the Wigner d-functions. This circumstance leads to even greater computational difficulties. A method of simplification of these complicated expressions for transition probabilities is demonstrated. The semiclassical approximation for coordinate matrix elements (Gulayev) and recurrence properties of the Wigner d-functions are used. The Hnβ line is under consideration. Specific calculations for the transition 10–8 in the case of parallel and perpendicular fields are presented.

Symplectic quandles and parabolic representations of 2-bridge knots and links

In this paper, we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial [Formula: see text] for each 2-bridge link [Formula: see text], and each zero of this polynomial gives a set of arc coloring vectors on the diagram of [Formula: see text] satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of [Formula: see text]. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.

The volume and Chern–Simons invariant of a Dehn-filled manifold

For a compact 3-manifold N with non-empty boundary, Zickert [12] gave a combinatorial formula for computing the volume and Chern–Simons invariant of a boundary parabolic representation π1(N)→PSL(2,C). In this paper, we introduce a notion of deformed Ptolemy varieties and extend the formula of Zickert to a representation that is not necessarily boundary parabolic. This allows us to compute the volume and Chern–Simons invariant of a PSL(2,C)-representation of a closed 3-manifold.

Development of a freely available simulator with graphical interface for modeling nonlinear focused ultrasound fields with shocks

Measurement-based modeling is gaining acceptance as a standard tool for characterizing the nonlinear fields of existing therapeutic ultrasound devices and designing new ones. Here, a freely available simulation tool is presented for modeling axially symmetric, strongly nonlinear HIFU beams with shocks in a layered propagation medium such as water and different types of tissue. Two nonlinear wave equations are included in the simulator: the KZK equation generalized to include an equivalent source boundary condition for strongly focused beams and the Westervelt equation in a nonlinear wide-angle parabolic representation. Both equations are solved in the frequency domain and permit definition of the HIFU transducer as an annular array. The geometrical parameters and power output of the array, electronic focus steering along the beam axis, and acoustic properties of the layered propagation medium can be configured via graphical interface. Visualization and output of various acoustic field parameters such as peak positive and negative pressures, shock amplitude, intensity, heat deposition rate, and total power of a beam are also provided. The simulator can be used for transducers without ideal symmetry through definition of an equivalent radially symmetric source. Widespread availability of such simulation tools will help advance standardized utilization of measurement-based modeling and facilitate the adoption of such approaches for HIFU treatment planning. [Work supported by NIH R01EB7643, R01EB025187, and RSF №14-12-00974.]

On the Concept of Representations of Prime Numbers and Prime Products

The author used to think that the only representation of prime numbers, etc. was the prime number double helix representation. However, in 2011, the author published a paper which puzzled him and it eventually dawned upon him that there was more than one representation of prime numbers. The second representation will be referred to as the hyperbolic representation. The third representation will be called the parabolic representation and is related to the hyperbolic representation. The fourth representation is the triangular representation and is related to the hyperbolic representation. Both of the primary representations, the double helix and the hyperbolic both reject that 2 and 3 are prime numbers. The hyperbolic representation is shown to be related to Lorentz-like transformations.

Twisted Alexander polynomials of 2-bridge knots for parabolic representations

In this paper we show that the twisted Alexander polynomial associated to a parabolic representation determines fiberedness and genus of a wide class of 2-bridge knots. As a corollary we give an affirmative answer to a conjecture of Dunfield, Friedl and Jackson for infinitely many hyperbolic knots.

A cell-local finite difference discretization of the low-order quasidiffusion equations for neutral particle transport on unstructured quadrilateral meshes

We present a quasidiffusion (QD) method for solving neutral particle transport problems in Cartesian XY geometry on unstructured quadrilateral meshes, including local refinement capability. Neutral particle transport problems are central to many applications including nuclear reactor design, radiation safety, astrophysics, medical imaging, radiotherapy, nuclear fuel transport/storage, shielding design, and oil well-logging. The primary development is a new discretization of the low-order QD (LOQD) equations based on cell-local finite differences. The accuracy of the LOQD equations depends on proper calculation of special non-linear QD (Eddington) factors from a transport solution. In order to completely define the new QD method, a proper discretization of the transport problem is also presented. The transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic, parabolic representation of the angular flux on incoming faces. Analytic and numerical tests are used to test the accuracy and spatial convergence of the non-linear method. All tests exhibit O(h2) convergence of the scalar flux on orthogonal, random, and multi-level meshes.

Length-weight Relationship and Condition Factor of Liza subviridis (Valenciennes, 1836) of Parangipettai Waters, Southeast Coast of India

Determination of Length-weight Relationship (LWR) of any commercially important fish is crucial to validate the wild stock level, to predict their wellbeing in the natural habitat and for various sustainable fishery management practices. Liza subviridis (Valenciennes) is noted to be highly abundant along the coast of Parangipettai, South east coast of India. Hence, the present study was aimed to establish Length-weight relationship and condition factor of Greenback mullet, Liza subviridis (Valenciennes) occurring in Vellar estuary, Parangipettai (lat. 11 degrees 30' N, long. 79 degrees 46' E) using least square method. To determine the actual relationship between length and weight of L. subviridis exponent coefficient or equilibrium constant (b) and relative condition factor (Kn) analysis were adopted. The females were found to be heavier than males at similar length. The equilibrium constant 'b' was found to be 2.7106 in males and 2.8927 in females. The corresponding parabolic representation for male was W = 0.0462L(2.7106) and for female W = 0.0382L(2.8927). The equilibrium constant did not obey the cube law as it deviated significantly from 3 in the case of males. The relative condition factor around 1 and little over it revealed the well-being of L. subviridis in Parangipettai waters.

Parabolic sturmians approach to the three-body continuum Coulomb problem

The three-body continuum Coulomb problem is treated in terms of the generalized parabolic coordinates. Approximate solutions are expressed in the form of a Lippmann-Schwinger-type equation, where the Green’s function includes the leading term of the kinetic energy and the total potential energy, whereas the potential contains the non-orthogonal part of the kinetic energy operator. As a test of this approach, the integral equation for the (e −, e −, He++) system has been solved numerically by using the parabolic Sturmian basis representation of the (approximate) potential. Convergence of the expansion coefficients of the solution has been obtained as the basis set used to describe the potential is enlarged.

EVALUATIONS OF THE TWISTED ALEXANDER POLYNOMIALS OF 2-BRIDGE KNOTS AT ±1

Let H(p) be the set of 2-bridge knots K(r), 0<r<1, such that there is a meridian-preserving epimorphism from G(K(r)), the knot group, to G(K(1/p)) with p odd. Then there is an algebraic integer s0such that for any K(r) in H(p), G(K(r)) has a parabolic representation ρ into SL(2, ℤ[s0]) ⊂SL(2, ℂ). Let [Formula: see text] be the twisted Alexander polynomial associated to ρ. Then we prove that for any K(r) in H(p), [Formula: see text] and [Formula: see text], where [Formula: see text], μ ∈ ℤ[s0]. The number μ can be recursively evaluated.

Goya: "to every story there belongs another"

Hofmann places Goya's paintings, drawings and prints in a biographical context, revealing the specific character of each phase of the artist's life and work. He discusses the glory and the pain of evinced by Goya's early work, the artist's parabolic representation of the threat posed by the French Revolution, his dramatic documentation of the French occupation of Spain, his variations on cruelty in the Disasters of War etchings, and the religious faith apparent in his late work. Hofmann also relates the artist and his work to contemporary intellectual developments, drawing comparison with writers, critics and philosophers from Goethe to William Blake to the Marquis de Sade.

Size quantization effects in atomic level broadening near thin metallic films

The broadening of atomic levels near thin metallic films is studied theoretically within the fixed-atom approximation. First-order level widths are calculated by using a Jennings-type jellium potential to describe the electronic states of the film, and hydrogenic wave functions in parabolic (Stark) representation for the atomic orbitals. In the parabolic representation, hybridization effects due to the long-range image-charge interactions are taken into account. Size quantization in the growth direction of the film gives rise to characteristic structures in level widths, atomic occupation probabilities, and transition distances as a function of the film thickness. Details of this structure depend on the orientation of the Stark orbitals with respect to the film and can be related to the dependence of transition matrix elements on the active electron's wave vector component parallel to the surface for the case of a semi-infinite metal. The large variation of the calculated transition distances with the film thickness may result in observable effects in atomic interactions with thin films. (c) 2000 The American Physical Society.

Atomic level broadening near metal surfaces: Inclusion of hybridization effects in first-order calculations

The interaction of slow atoms and ions with metal surfaces results in a hybridization of the atomic orbitals as well as a shift and a broadening of the associated energy levels. In perturbative treatments of the atom-surface interaction, hybridization effects can be taken into account approximately through a proper choice of the atomic wave functions. We examine the inclusion of hybridization effects in a first-order treatment by employing hydrogenic atomic wave functions in the parabolic (Stark) representation as well as in the spheroidal (two-center) representation. These two representations diagonalize different parts of the image-charge interaction. Level widths calculated from the parabolic and the spheroidal representation are compared to nonperturbative level widths. The parabolic representation appears to be a favorable choice for describing hybridization effects in first-order calculations.

Resonant electron transfer between highly charged ions and metal surfaces: first-order vs. nonperturbative transition rates

Transition rates for resonant electron transfer between highly charged ions and metal surfaces are calculated within a simple first-order model and within the nonperturbative coupled-angular-mode (CAM) method. In the first-order model, image potentials are disregarded and hybridization effects are simulated by describing the ionic orbitals in terms of hydrogenic wave functions in the parabolic (Stark) representation. The CAM method employs a fairly realistic electronic potential that includes the classical image potentials. For typical cases, the deviations between first-order and CAM transition rates are such that the associated transition distances agree, on average, within one atomic unit. The reason for this agreement is indicated.

Broadening of atomic levels near metal surfaces: first-order model versus coupled-angular-mode method

The reliability of a first-order model that is in use for calculating the widths of resonantly broadened atomic levels near metal surfaces is examined, in particular for large atomic quantum numbers and core charges. The model employs a simplified electronic potential in which image-charge effects are disregarded. Orbital hybridization is simulated by describing the atomic states in terms of hydrogenic wave functions in the parabolic (Stark) representation. For typical cases, first-order resonance widths are compared to widths calculated from the nonperturbative coupled-angular-mode method which uses a fairly realistic electronic potential including the classical image potentials. Good overall agreement is achieved between first-order and nonperturbative results. A plausible explanation for this agreement is given.

Resonant electron transfer in the interaction of atoms and ions with metal surfaces

Various aspects of the theoretical description of resonant electron transfer in the interaction of atoms and ions with metal surfaces are examined. Transition rates are calculated in first-order adiabatic approximation by applying a novel method for evaluating transition matrix elements for arbitrary parameter values of the ion-metal system. Results for hydrogenic states in the parabolic (Stark) representation are compared to rates obtained from the (nonperturbative) complex-scaling method. The behavior of the first-order rates at asymptotically large ion-surface distances is studied and compared to results of a nonperturbative tunneling theory. Scaling properties of the transition rates are briefly discussed.