Analytical Evaluation of Non-Elementary Integrals Involving Some Exponential, Hyperbolic and Trigonometric Elementary Functions and Derivation of New Probability Measures Generalizing the Gamma-Type and Gaussian-Type Distributions

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, and where α,η and β are real or complex constants are evaluated in terms of the confluent hypergeometric function 1F1 and the hypergeometric function 1F2. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions 1F1 and 1F2. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and Gaussian distributions are also obtained. The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (x2) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations (or methods) which are in general expensive and associated with numerical errors.

Exact solutions of the Schrodinger equation for a class of hyperbolic potential well

We propose a new scheme to study the exact solutions of a class of hyperbolic potential well.We first apply different forms of function transformation and variable substitution to transform the Schrodinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate.And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant,we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well.Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function.The linearly dependent relation between two eigenfunctions is also studied.

FLOOD ROUTING MODELS IN CONFLUENT AND DIVIDING CHANNELS

By introducing a water depth connecting formula, the hydraulic equations in the dividing channel system were coupled and the relation of discharge distribution between the branches of the dividing channels can be yielded. In this manner, a numerical model for the confluent channels was established to study the variation of backwater effects with the parameters in the channel junction. The meeting of flood peaks in the mainstream and tributary can be analyzed with this model.The flood peak meeting is found to be a major factor for the extremely high water level in the mainstream during the 1998 Yangtze River flood. Subsequently the variations of discharge distribution and water level with channel parameters between each branch in this system were studied as well. As a result, flood evolution caused by Jingjiang River shortcut and sediment deposition in the entrance of dividing channels of the Yangtze River may be qualitatively elucidated.It is suggested to be an effective measure for flood mitigation to enhance regulation capability of reservoirs available upstream of the tributaries and harness branch entrance channels.

Solution of the Dirac equation with the ring-shaped oscillator potential

The ring-shaped oscillator potential, obtained by replacing the Coulomb part of the Hartmann potential by a harmonic oscillator term, was investigated. Under the equal vector potential and scalar potential, the Dirac equation was solved in spherical coordinate. The exact energy spectrum of the bound states was presented as a solution to the confluent hypergeometric equation by boundary conditions. Furthermore, the normalized angular and radial wave functions were presented.

Solution of Some Integral Equations Involving Confluent <i>k</i>-Hypergeometric Functions

The principle aim of this research article is to investigate the properties of k-fractional integration introduced and defined by Mubeen and Habibullah [1],and secondly to solve the integral equation of the form , for?k > 0, β > 0, y > 0, 0 is the confluent k-hypergeometric functions, by using k-fractional integration.

Prurigo Pigmentosa and Confluent and Reticulated Papillomatosis a Spectrum of One Disease: A New Case Report

Background: Prurigo Pigmentosa (PP) is a rare inflammatory dermatitis first discovered in 1971. Characterized by a sudden eruption of pruritic reticulated, pink-brown papules coalescing into plaques distributed symmetrically over shoulders, neck, chest, and back. Various triggers have been identified, including the ketogenic diet. Clinicopathological presentation looks similar to confluent and reticulated papillomatosis (CARP) which is a rare dermatosis of unknown etiology characterized by hyperkeratotic pigmented papules & peripheral reticulation involving seborrheic areas. Aim: To document a new case presentation of PP caused by a low-carbohydrate restricted diet and discuss the comparison with CARP clinically, pathologically, and treatment modalities. Case report: A 15-year-old childhood male developed PP 3 weeks after self-initiating a low carbohydrate-restricted ketogenic diet for weight management. Clinically and histopathologically the lesion looks similar to CARP, treated successfully with re-introduction of high carbohydrates in his food, a short course of systemic steroids in combination with oral doxycycline capsules for the one-month duration. Conclusion: PP & CARP have been considered a spectrum of one disease, and PP is a pruritic variant from CARP caused by a low carbohydrate-restricted diet.

A Pair of Pseudo-differential Operators Involving Index Whittaker Transform in L_(2)^(a) (R_(+);m_(a)(x)dx)

Pseudo-differential operators(PDO)Q(x,L_(a,x))and Q(x,L_(a,x))involving the index Whittaker transform are defined.Estimates for these operators in Hilbert space L_(2)^(a)(R+;m_(a)(x)dx)are obtained.A symbol classΩis introduced.Later product and commutators for the PDO are investigated and their boundedness results are discussed.

Commuting Toeplitz Operators on Fock-Sobolev Spaces of Negative Orders

In the setting of Fock-Sobolev spaces of positive orders over the complex plane,Choe and Yang showed that if the one of the symbols of two commuting Toeplitz operators with bounded symbols is non-trivially radial,then the other must also be radial.In this paper,we extend this result to the Fock-Sobolev space of negative order using the Fock-type space with a confluent hyper geometric function.

Abdominal acupuncture plus the Confluent Points of the Eight Extraordinary Meridian for chronic pharyngitis

Objective：To evaluate the effectiveness of abdominal acupuncture plus the Confluent Points of the Eight Extraordinary Meridian in treating chronic pharyngitis. Methods：Thirty eligible subjects with chronic pharyngitis were given abdominal acupuncture plus needling Lieque （LU 7） and Zhaohai （KI 6）, two out of the Confluent Points of the Eight Extraordinary Meridian, once a day, 10 sessions as a course, and the therapeutic efficacy was evaluated after a treatment course. Results：After a treatment course, of the 30 subjects, 21 were recovered, 5 showed markedly effective, 2 were improved, and 2 failed,the total effective rate was 93.3%. Conclusion：Abdominal acupuncture plus the Confluent Points of the Eight Extraordinary Meridian is effective in treating chronic pharyngitis, worth promoting in clinic.

Existence and Regularity of Solution to the Generalized Tricomi Equation with a Singular Initial Datum at a Point

In this paper, we study the existence and regularity of a solution to the initial datum problem of a semilinear generalized Tricomi equation in mixed-type initial datum on the degenerate plane is smooth away from the origin, domain. We suppose that an and has a conormal singularity at this point, then we show that in some mixed-type domain, the solution exists and is conormal with respect to the characteristic conic surface which is issued from the origin and has a cusp singularity.

On Optimality of the Barrier Strategy for the Classical Risk Model with Interest

In this paper, we consider the optimal dividend problem for a classical risk model with a constant force of interest. For such a risk model, a sufficient condition under which a barrier strategy is the optimal strategy is presented for general claim distributions. When claim sizes are exponentially distributed, it is shown that the optimal dividend policy is a barrier strategy and the maximal dividend-value function is a concave function. Finally, some known results relating to the distribution of aggregate dividends before ruin are extended.

Study of the thermal pain threshold latency of acupoints based on Fei Teng Ba Fa

Objective： By applying moxibustion to the eight confluent points in different periods of time, to observe the changes in thermal pain threshold latency of acupoints based on Fei Teng Ba Fa. Methods： A total of 468 healthy college student volunteers received moxibustion at the eight confluent points in three different periods of time, i.e. Chen （7：00-9：00）, Wu （11：00-13：00） and Xu （19：00-21：00）. The thermal pain threshold latency was adopted to measure the changes in pain threshold of the eight confluent points under different conditions （different periods of time, different genders, different acupoints and different states of the acupoints） based on Fei Teng Ba Fa. Results： Finally, thirty subjects dropped out and 438 subjects were included. The comparison of thermal pain threshold latencies of the eight confluent points in the same opening or closing state based on Fei Teng Ba Fa： latencies of the closing points and adjunct points were significantly different in different periods of time （P〈0.05）, the latencies of the males were significantly longer than those of the females （P〈0.05）; there was no significant difference in the latency between the left and right sides （P〉0.05）; in the female group, there was a significant difference in the latency between the lower-limb points and the upper-limb points （P〈0.05）. The comparison of thermal point threshold latencies of the eight confluent points in different opening or closing state： in the period of Wu （11：00-13：00）, the latencies of the opening points were significantly longer than those of the closing points and adjunct points （P〈0.05）; for men, their opening and closing points had significantly longer thermal pain threshold latencies than their adjunct points （P〈0.05）; despite the gender, the latencies of the upper limb opening and closing points were significantly longer than the latency of the adjunct points （P〈0.05）; in the female group, the latencies of the lower-limb opening points were significantly shorter than those of the lower-limb closing and adjunct points （P〈0.05）. Conclusion： Based on Fei Teng Ba Fa, the pain thresholds of the eight confluent points vary in different periods of time, gender, acupoint location and opening/closing state, which can be taken as the evidence of making time-based acu puncture-moxibustion prescriptions.

Semi-exact Solutions of Konwent Potential

In this work we study the quantum system with the symmetric Konwent potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun function. The eigenvalues have to be calculated numerically because series expansion method does not work due to the variable z ≥ 1. The properties of the wave functions depending on the potential parameter A are illustrated for given potential parameters V_0 and a. The wave functions are shrunk towards the origin with the increasing |A|. In particular, the amplitude of wave function of the second excited state moves towards the origin when the positive parameter A decreases. We notice that the energy levels ε_i increase with the increasing potential parameter |A| ≥ 1, but the variation of the energy levels becomes complicated for |A| ∈(0, 1), which possesses a double well. It is seen that the energy levels ε_i increase with |A| for the parameter interval A ∈(-1, 0), while they decrease with |A| for the parameter interval A ∈(0, 1).

Charged scalar fields in a Kerr-Sen black hole:exact solutions, Hawkingradiation, and resonant frequencies

In this study, we consider charged massive scalar fields around a Kerr-Sen spacetime. The radial and angular parts of the covariant Klein-Gordon equation are solved in terms of the confluent Heun function. From the exact radial solution, we obtain the Hawking radiation spectrum and discuss its resonant frequencies. The massless case of the resonant frequencies is also examined.

Analytic solutions in the acoustic black hole analogue of the conical Kerr metric

We study the sound perturbation of a rotating acoustic black hole in the presence of a disclination. The radial part of the massless Klein-Gordon equation is written into a Heun form, and its analytical solution is obtained. These solutions have an explicit dependence on the parameter of the disclination. We obtain the exact Hawking-Unruh radiation spectrum.

Some Representations of Unified Voigt Functions

The authors derive a set of unified representations of the Voigt functions in terms of familiar special functions of Mathematical Physics. Some deductions from these representations are also considered.

Total Duration of Negative Surplus for a Brownian Motion Risk Model with Interest

In this paper,we consider the Brownian motion risk model with interest.The Laplace transform of the first exit time from the upper barrier before hitting the lower barrier is obtained.Using the obtained result and exploiting the limitation idea,we derive the Laplace transform of total duration of negative surplus.