Solvable Cases Research Articles

Optical Properties of Modified Pöschl-Teller Quantum Well with Position-Dependent Mass

Optical properties of modified Poschl-Teller quantum well are studied with considering a position- dependent mass of an electron in the confinement potential. Analytical and numerical calculation are investigated based on the point canonical transformation method. Author to whom correspondence should be addressed. Researches shows that there are several theoretical analysis of the optical properties is associated with inter- subband optical transitions in nanostructures which are presented in scientific literatures. 18-21 In this paper, we attempt to investigate the analytical and numerical calcu- lation of the optical properties of modified Poschl-Teller potential (MPTP) with considering position-dependent mass effect. At first, we choose the exact solvable case and use PCTM to solve the wave equation and obtain the wave function and corresponding energy eigenstates. In addi- tion, the numerical solutions are introduced for compari- son and checked the accuracy of our calculations. Then, we use the same method to investigate optical properties of MPTP with an arbitrary position-dependent mass func- tion numerically. Model and discussion are brought in the following section, including the optical properties. Finally, some conclusions are highlighted.

Counter-ions at single charged wall: Sum rules

For inhomogeneous classical Coulomb fluids in thermal equilibrium, like the jellium or the two-component Coulomb gas, there exists a variety of exact sum rules which relate the particle one-body and two-body densities. The necessary condition for these sum rules is that the Coulomb fluid possesses good screening properties, i.e. the particle correlation functions or the averaged charge inhomogeneity, say close to a wall, exhibit a short-range (usually exponential) decay. In this work, we study equilibrium statistical mechanics of an electric double layer with counter-ions only, i.e. a globally neutral system of equally charged point-like particles in the vicinity of a plain hard wall carrying a fixed uniform surface charge density of opposite sign. At large distances from the wall, the one-body and two-body counter-ion densities go to zero slowly according to the inverse-power law. In spite of the absence of screening, all known sum rules are shown to hold for two exactly solvable cases of the present system: in the weak-coupling Poisson-Boltzmann limit (in any spatial dimension larger than one) and at a special free-fermion coupling constant in two dimensions. This fact indicates an extended validity of the sum rules and provides a consistency check for reasonable theoretical approaches.

Testing viscous and anisotropic hydrodynamics in an exactly solvable case

We exactly solve the one-dimensional boost-invariant Boltzmann equation in the relaxation time approximation for arbitrary shear viscosity. The results are compared with the predictions of viscous and anisotropic hydrodynamics. Studying different non-equilibrium cases and comparing the exact kinetic-theory results to the second-order viscous hydrodynamics results we find that recent formulations of second-order viscous hydrodynamics agree better with the exact solution than the standard Israel-Stewart approach. Additionally, we find that, given the appropriate connection between the kinetic and anisotropic hydrodynamics relaxation times, anisotropic hydrodynamics provides a very good approximation to the exact relaxation time approximation solution.

Bosonic transport through a chain of quantum dots

The particle transport through a chain of quantum dots coupled to two bosonic reservoirs is studied. For the case of reservoirs of non-interacting bosonic particles, we derive an exact set of stochastic differential equations, whose memory kernels and driving noise are characterised entirely by the properties of the reservoirs. Going to the Markovian limit an analytically solvable case is presented. The effect of interparticle interactions on the transient behaviour of the system, when both reservoirs are instantaneously coupled to an empty chain of quantum dots, is approximated by a semiclassical method, known as the Truncated Wigner approximation. The steady-state particle flow through the chain and the mean particle occupations are explained via the spectral properties of the interacting system.

Funding Value Adjustment for General Financial Instruments: Theory and Practice

In the first part of this paper (Antonov-Bianchetti, 2013) we developed the theoretical framework for pricing financial instruments under multiple sources of funding, leading to a non-linear pricing PDE and to Funding Value Adjustment (FVA).In this second part we develop the numerical framework for computing FVA for general financial instruments including callable features. Our main technical result is an efficient approximation of the FVA in a fully universal way. Usage of non-linear effective discounting rates permits an exact handling of all solvable special cases (the collateral as linear function of the value) by a single formula. Furthermore, the formula delivers a very accurate approximation for general instruments (barriers, Bermudans, etc.). The proof is based on our second technical result: exact calculation of prices of automatically exercisable instruments (e.g. European-style or Barrier) having different stochastic discount rates before and after exercise.We also address the implementation workflow of the FVA calculation, allowing parallel deal-by-deal computation, and we provide a concrete example of FVA calculation for a Bermudan Swaption with partial collateralization, proving that the quality of the numerical approximation is excellent.

Dipole-excited surface plasmons in metallic nanoparticles: Engineering decay dynamics within the discrete-dipole approximation

A theoretical control of the electromagnetic coupling between localized surface plasmons and pointlike sources of radiation is a relevant topic in nanoscience and nanophotonics. In this paper a numerical approach based on the discrete dipole approximation is presented as a practical and reliable computational tool to study the decay dynamics of a dipole when it is located in the near proximities of metallic nanoparticles whose shapes do not allow a fully analytical treatment. The method is first applied to Ag nanospheres and nanoshells, which represent two analytically solvable cases, and it is shown to lead to a very good agreement with exact results. The approach is then used to consider the response, in terms of perturbations induced on the radiative and nonradiative decay rates, of elongated nanoparticles, like Ag prolate spheroids and nanocones. Results demonstrate how the optical response of conically shaped nanoparticles can be affected by the distance and the orientation of the emitter of radiation, as well as by other geometrical parameters. The particular symmetry of these plasmonic objects results in peculiar features: the absorption efficiencies of the modes depend on the distance of the source of radiation in a counterintuitive way, and this is explained in terms of the excited charge density distributions. The possibility to simulate arbitrary-shaped nanostructures and several dipole-metal configurations presented here, could thus open new avenues for an aware use of surface plasmons in fluorescence spectroscopy applications or single photon emission studies.

JOVIAN STRATOSPHERE AS A CHEMICAL TRANSPORT SYSTEM: BENCHMARK ANALYTICAL SOLUTIONS

We systematically investigated the solvable analytical benchmark cases in both one- and two-dimensional (1D and 2D) chemical-advective-diffusive systems. We use the stratosphere of Jupiter as an example but the results can be applied to other planetary atmospheres and exoplanetary atmospheres. In the 1D system, we show that CH_4 and C_2H_6 are mainly in diffusive equilibrium, and the C_2H_2 profile can be approximated by modified Bessel functions. In the 2D system in the meridional plane, analytical solutions for two typical circulation patterns are derived. Simple tracer transport modeling demonstrates that the distribution of a short-lived species (such as C_2H_2) is dominated by the local chemical sources and sinks, while that of a long-lived species (such as C_2H_6) is significantly influenced by the circulation pattern. We find that an equator-to-pole circulation could qualitatively explain the Cassini observations, but a pure diffusive transport process could not. For slowly rotating planets like the close-in extrasolar planets, the interaction between the advection by the zonal wind and chemistry might cause a phase lag between the final tracer distribution and the original source distribution. The numerical simulation results from the 2D Caltech/JPL chemistry-transport model agree well with the analytical solutions for various cases.

On some polynomially solvable cases and approximate algorithms in the optimal communication tree construction problem

Considering an arbitrary undirected n-vertex graph with nonnegative edge weights, we seek to construct a spanning tree minimizing the sum over all vertices of the maximal weights of the incident edges. We find some particular cases of polynomial solvability and show that the minimal span whose edge weights lie in the closed interval [a, b] is a \(\left( {2 - \frac{{2a}} {{a + b + 2b/(n - 2)}}} \right) \)-approximate solution, and the problem of constructing a 1.00048-approximate solution is NP-hard. We propose a heuristic polynomial algorithm and perform its a posteriori analysis.

On Some Difference Equations of First Order

Abstract One considers two boundary value problems for the Laplacian and biharmonic operator in a plane sector with boundary conditions of the Dirichlet and the Neumann type on the angle sides. The system of difference equations of first order to which this problem is reduced, is explicitly written out. Certain cases of solvability for such difference equations of first order are described; it will be useful for studying similar equations.

Inverse method for viscous flow design using stream-function coordinates

In the paper, a new inverse method for viscous 2D laminar flows is developed. The method is based on incompressible Navier–Stokes equations transformed to the stream-function coordinate system (von Mises coordinates). The flow design problem with appropriate boundary conditions is formulated and solved numerically. The geometrical shape of the boundary is obtained through the integration along streamlines. The method may be coupled with a flow analysis solver to model the influence of known parts of geometry. Results for two analytically solvable cases (the Poiseuille and the Jeffery–Hamel flows) are presented. Then, the foil design problem is considered as an example. Potential applications and developments towards axisymmetric and 3D flows are discussed.

Dependence of objective function on several variables in layout problem and its solution by the method of structural-alphabetic search

The problem of the layout of objects of different sizes is analyzed. The objective function is shown to depend on several variables, which are various combinatorial configurations. According to this principle, the problem is divided into several subproblems; therefore, a self-adjusting algorithm is developed to solve it. A location problem for objects of the same size is solved by the structural-alphabetic search, which is based on the known solvable case.

Maximum matching in multi-interface networks

In heterogeneous networks, devices can communicate by means of multiple wireless interfaces. By choosing which interfaces to switch on at each device, several connections might be established. That is, the devices at the endpoints of each connection share at least one active interface.In this paper, we consider the standard matching problem in the context of multi-interface wireless networks. The aim is to maximize the number of parallel connections without incurring interferences. Given a network G=(V,E), nodes V represent the devices, and edges E represent the connections that can be established. If node x participates in the communication with one of its neighbors by means of interface i, then another neighboring node of x can establish a connection (but not with x) only if it makes use of interface j≠i. The size of a solution for an instance of the outcoming matching problem, which we call Maximum Matching in Multi-Interface networks (MMMI for short), is always in between the sizes of the solutions for the same instance with respect to the standard matching and its induced version problems. However, we prove that MMMI is NP-hard even for proper interval graphs and for bipartite graphs of maximum degree Δ≥3. We also show polynomially solvable cases of MMMI with respect to different assumptions.

A NOTE ON PRIMITIVE SUBGROUPS OF FINITE SOLVABLE GROUPS

In [5], Johnson introduced the primitivity of subgroups and proved that a finite group G is supersolvable if every primitive subgroup of G has a prime power index in G. In that paper, he also posed an interesting problem: what a group looks like if all of its primitive subgroups are maximal. In this note, we give the detail structure of such groups in solvable case. Finally, we use the primitivity of some subgroups to characterize T-group and the solvable <TEX>$PST_0$</TEX>-groups.

Motion Planning with Pulley, Rope, and Baskets

We study a motion planning problem where items have to be transported from the top room of a tower to the bottom of the tower, while simultaneously other items have to be transported in the opposite direction. Item sets are moved in two baskets hanging on a rope and pulley. To guarantee stability of the system, the weight difference between the contents of the two baskets must always stay below a given threshold.We prove that it is \(\varPi_{2}^{p}\)-complete to decide whether some given initial situation of the underlying discrete system can lead to a given goal situation. Furthermore we identify several polynomially solvable special cases of this reachability problem, and we also settle the computational complexity of a number of related questions.

Optimized basis expansion as an extremely accurate technique for solving time-independent Schrödinger equation

We use the optimized trigonometric finite basis method to find energy eigenvalues and eigenfunctions of the time-independent Schrodinger equation with high accuracy. We apply this method to the quartic anharmonic oscillator and the harmonic oscillator perturbed by a trigonometric anharmonic term as not exactly solvable cases and obtain the nearly exact solutions.

Some results on local fields

Let K be a local field with finite residue field of characteristic p. This paper is devoted to the study of the maximal abelian extension of K of exponent p-1 and its maximal p-abelian extension, especially the description of their Galois groups in solvable case. Then some properties of local fields in general case are studied too.

Coloring vertices of claw-free graphs in three colors

We study the computational complexity of the vertex 3-colorability problem in the class of claw-free graphs. Both the problem and the class received much attention in the literature, separately of each other. However, very little is known about the 3-colorability problem restricted to the class of claw-free graphs beyond the fact the problem is NP-complete under this restriction. In this paper we first strengthen this negative fact by revealing various further restrictions under which the problem remains NP-complete. Then we derive a number of positive results that deal with polynomially solvable cases of the problem in the class of claw-free graphs.

Forced Convection Heat Transfer from a Heated Cylinder in an Axial Background Flow in a Porous Medium: Three Exactly Solvable Cases

Assuming a background flow of velocity U = U(x) in the axial direction x of a circular cylinder with surface temperature distribution T w = T w (x) in a saturated porous medium, for the temperature boundary layer occurring on the cylinder three exactly solvable cases are identified. The functions {U(x), T w (x)} associated with these cases are given explicitly, and the corresponding exact solutions are expressed in terms of the modified Bessel function K 0 (z), the incomplete Gamma function Γ (a, z) and the confluent hypergeometric function U(a, b, z), respectively. The correlation between the Nusselt number and the Peclet number as well as the curvature effects on the heat transfer are discussed in all these cases in detail. Some “universal” features of the exponential surface temperature distribution are also pointed out.

The minimum cost perfect matching problem with conflict pair constraints

In this paper we address the minimum cost perfect matching problem with conflict pair constraints (MCPMPC). Given an undirected graph G with a cost associated with each edge and a conflict set of pairs of edges, the MCPMPC is to find a perfect matching with the lowest total cost such that no more than one edge is selected from each pair in the conflict set. MCPMPC is known to be strongly NP-hard. We present additional complexity results and identify new polynomially solvable cases for the general MCPMPC. Several heuristic algorithms and lower bounding schemes are presented. The proposed algorithms are tested on randomly generated instances. Encouraging experimental results are also reported.

Dynamic fluctuations in unfrustrated systems: random walks, scalar fields and the Kosterlitz–Thouless phase

We study analytically the distribution of fluctuations of the quantities whose average yields the usual two-point correlation and linear response functions in three unfrustrated models: the random walk, the d dimensional scalar field and the 2d XY model. In particular we consider the time dependence of ratios between composite operators formed with these fluctuating quantities which generalize the largely studied fluctuation-dissipation ratio, allowing us to discuss the relevance of the notion of effective temperature beyond linear order. The behavior of fluctuations in the aforementioned solvable cases is compared to numerical simulations of the 2d clock model with p = 6,12 states.