Some microwave applications of the Kummer confluent hypergeometric function

The real negative zeros equation and equation — a finite positive integer) in the first parameter k, resp. â of the real Kummer confluent hypergeometric function equation with â, ĉ and x — all real, â 0 and x > 0, â = ĉ/2 + k (k = â – ĉ/2 — real, negative, −∞ < k < −ĉ/2 ), resp., â — arbitrary real negative, are investigated numerically for selected values of n, assuming x and ĉ as parameters. The results are presented in the form of Tables and graphs. Analyzing them, certain properties of the zeros are revealed and summarized in three theorems which are substantiated numerically and analytically. Theorem 1 states that when x → +∞, then equation and equation tend to the real positive numbers equation, relevantly equation, and in case x → 0, both quantities of interest go to −∞. Theorem 2 determines the number n of zeros studied. It ascertains that it is finite and depends on their values. According to it n = 1; 2; …, p, where equation, correspondingly equation, (equation is the greatest integer less than or equal to equation). Theorem 3 specifies the value of the first zero at x = ĉ to equation, accordingly to equation. An application of the quantities considered for ĉ = 3 is given in the theory of azimuthally magnetized circular ferrite waveguide under slow equation modes excitation.

ABSTRACTGeneralized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.

1. Slow waves recorded from the circular smooth muscle layer of guinea-pig antrum consisted of two components, an initial component and a secondary regenerative component. Whereas both components persisted in the presence of nifedipine, the secondary component was abolished by a low concentration of caffeine. 2. Short segments of single bundles of circular muscle were isolated and impaled with two microelectrodes. Depolarizing currents initiated regenerative responses which resembled those initiated during normal slow waves. These responses had partial refractory periods of 20-30 s and were initiated about 1 s after the onset of membrane depolarization. 3. The regenerative responses persisted in the presence of either nifedipine or cobalt ions but were abolished by caffeine, BAPTA or cyclopiazonic acid. 4. The observations suggest that depolarizing membrane potential changes trigger the release of Ca2+ from intracellular stores and this causes a depolarization by activating sets of unidentified ion channels in the membranes of smooth muscle cells of the circular layer of guinea-pig antrum.

This is an overview paper. This paper is an attempt to show that fractional calculus can be reached through statistical distribution theory. This paper brings together results on fractional integrals and fractional derivatives of the first and second kinds in the real and complex domains in the scalar, vector, and matrix-variate cases, and shows that all these results can be reached through statistical distribution theory. It is shown that the whole area of fractional integrals can be reached through distributions of products and ratios in the scalar variable case and distributions of symmetric products and symmetric ratios in the matrix-variate cases. While summarizing the materials, the real domain results are also listed side by side with the complex domain results so that a comparative study is possible. Fractional integrals and derivatives in the real domain mean that the parameters involved could be real or complex with appropriate conditions, the arbitrary function is real-valued, and the variables involved are all real. These in the complex domain mean that the parameters could be real or complex and the arbitrary function is still real-valued but the variables involved are in the complex domain. Fully complex domain means the variables as well as the arbitrary function are in the complex domain. Most of the materials on fractional integrals and fractional derivatives involving a single matrix or a number of matrices in the real or complex domain are of this author. Slight modifications of the results, compared with the published works in various papers, are there in various sections. In the paragraph on notations, the lemmas that are taken from this author’s own book on Jacobians are common with published works and hence the similarity index with this author’s works will be high. Section Matrix-Variate Joint Distributions and Fractional Integrals in Many Matrix-Variate Cases material on a statistical approach to Kiryakova’s multi-index fractional integral and its extension to the real scalar case of second kind integrals as well as extensions of first and second kind integrals to real and complex matrix-variate cases are believed to be new. Matrix differential operators are introduced in Section Fractional Derivatives and, with the help of these operators, fractional derivatives are constructed from the corresponding fractional integrals. These operators are applicable in a large variety of functions. Applicability is shown through identities created from scale transformed gamma random variables. Some concluding remarks are given and some open problems are pointed out in Section Concluding Remarks.

Fast and slow simple waves are studied in the framework of the anisotropic magnetohydrodynamics of Chew, Goldberger, and Low [1]. Baranov [2] has constructed fields of integral curves for fast and slow waves and in two special cases has shown that such waves break in the compression section. The possibility of breaking of a slow wave in a rarefaction section was noted by Akhiezer et al. [3]. However, their general relations in simple waves [3] have been shown to be incorrect [2, 4]. In the present paper the nature of the variation of the longitudinal and transverse plasma pressures is determined, and the problem of the breaking of fast and slow waves is completely solved. Conditions under which a slow wave breaks in a rarefaction section are found. A fast wave always breaks in a rarefaction section.

Although hypo- and hyperthyroid patients have different symptoms in the gastrointestinal tract, the mechanism of thyroid action on the gut remains poorly understood. Thus the aim of this study was to investigate the effect of hypo- and hyperthyroidism on gastric myoelectrical activity, gastric emptying, dyspeptic symptoms. Twenty-two hyperthyroid (median age 45, 15 females) and 11 hypothyroid (median age 42, 10 females) patients were included into the study. Dyspepsia score, hypo- and hyperthyroid symptom scale, abdominal ultrasonography and upper gastrointestinal endoscopy were performed. Gastric myoelectrical activity was measured by electrogastrograpy (EGG) before and after therapy both preprandially and postprandially and compared with age, gender, and body-matched controls (12 for hypothyroid, 15 for hyperthyroid patients). Radionuclide gastric emptying studies were performed with a solid meal. Hypothyroid patients revealed a significant increase in preprandial tachygastria as compared with controls (12.3% vs 4.8%). The percentage of preprandial normal slow waves (2.4-3.7 cpm) was below 70% (dysmotility) in 7 of 11 hypothyroid patients versus 2 of 12 controls (P < 0.05). Hyperthyroid patients revealed a significantly higher preprandial (3.1 vs 2.8) and postprandial (3.4 vs 3) DF when compared with the controls (P < 0.05). A higher percentage of postprandial taschygastria (7.9 vs 0) was present in hyperthyroid patients than in the controls (P < 0.05). The decrease on postprandial EGG power (power ratio < 1) was observed in 7 patients the in hyperthyroid group and 1 in controls (P < 0.05). The percentage of postprandial normal slow waves was below 70% in 10 of 20 hyperthyroid patients vs 1 of 15 controls (P < 0.05). After therapy these differences disappeared in the euthyroid state. The hypo- and hyperthyroid symptom scale correlated to dyspepsia score. Dyspepsia score in hyperthyroidism correlated to power ratios in hyperthyroid patients. We detected some correlations between serum levels of fT3 or fT4 and some EGG parameters in hypo- and hyperthyroidism. Dyspepsia score and hypo- and hyperthyroid symptom scale were improved significantly after therapy in the euthyroid state. In conclusions, we showed gastric dysrhythmia by EGG in both hypo- and hyperthyroid patients. Dyspeptic symptoms correlated to the activity of thyroid disease. After therapy, these findings and dyspeptic symptoms improved in the euthyroid state. Abnormalities of power ratios may be responsible of dyspeptic symptoms in hyperthyroid patients. EGG may be a useful and noninvasive tool for detecting gastric disturbances during hypo- and hyperthyroidism.

Employing an iterative approach, the differential phase shift provided by the circular waveguide, containing an azimuthally magnetized coaxial ferrite cylinder, surrounded by a dielectric toroid, is computed in normalized form for normal TE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">01</sub> mode. The method uses the positive purely imaginary roots of the complex characteristic equation of the structure, derived in terms of the complex Kummer and the real cylindrical functions, determined, varying the imaginary part of the first complex parameter of the Kummer function. The results are presented in tabular and numerical form and discussed.

Aims. We present an investigation into how the cooling of the background plasma influences the propagation of slow and fast MHD wave modes supported by an unbounded, homogenous plasma. Previous investigations have suggested that the cooling of the plasma and a reduction in density could lead to the damping of fast magneto-acoustic oscillations. We aim to investigate whether cooling of the background plasma at a constant density may be responsible for the damping of slow and fast modes. Methods. The plasma is assumed homogeneous and the background temperature (pressure) is decreasing with time. The temperature change is assumed to be due to optically thin radiation. A special case of the radiative function is chosen to allow an analytical assessment of the effects of cooling on magneto-acoustic MHD modes and ensures the temperature evolution of the background plasma due to this radiation also matches the observed cooling profile of coronal loops. Results. A time-dependent dispersion relation is obtained on the slow timescale of cooling and full time-dependent solutions are found. Leading order equations for the amplitude of the waves are obtained and solved analytically for the slow and fast MHD modes. The cooling of the plasma is found to cause the frequency of the magneto-acoustic modes to decrease with time. The slow modes are found to experience a greater change in frequency than the fast modes. More importantly, the radiative losses also provide a significant damping of the slow mode and a small damping of the component of the fast mode perpendicular to the magnetic field. The damping of the slow mode is found to be strong within typical lifetimes of oscillations observed in coronal structures. Cooling could have important consequences and needs to be assessed when trying to determine what mechanism is responsible for the observed damping of coronal oscillations.

Strongly damped Doppler shift oscillations are observed frequently associated with flarelike events in hot coronal loops. In this paper, a review of the observed properties and the theoretical modeling is presented. Statistical measurements of physical parameters (period, decay time, and amplitude) have been obtained based on a large number of events observed by SOHO/SUMER and Yohkoh/BCS. Several pieces of evidence are found to support their interpretation in terms of the fundamental standing longitudinal slow mode. The high excitation rate of these oscillations in small- or micro-flares suggest that the slow mode waves are a natural response of the coronal plasma to impulsive heating in closed magnetic structure. The strong damping and the rapid excitation of the observed waves are two major aspects of the waves that are poorly understood, and are the main subject of theoretical modelling. The slow waves are found mainly damped by thermal conduction and viscosity in hot coronal loops. The mode coupling seems to play an important role in rapid excitation of the standing slow mode. Several seismology applications such as determination of the magnetic field, temperature, and density in coronal loops are demonstrated. Further, some open issues are discussed.

The real Kummer function, the ordinary and modified difference Bessel functions are applied to derive the characteristic equation of slow TE/spl circ//sub 0n/spl circ// modes in the circular waveguide, containing azimuthally magnetized ferrite cylinder and dielectric toroid. A lemma on a remarkable property of the real zeros of the equation is formulated, based on its numerical solution. Essential features of the slow TE/spl circ//sub 01/ mode are found out, investigating its graphically shown phase curves.

The aims of this study were to evaluate the gastric electrical activity and gastric emptying in preterm and term newborns and to assess the development of gastric motility by comparing newborns of different gestational ages. The cutaneous electrogastrography and the ultrasonographic study of the gastric emptying were performed before and after milk formula in three groups of infants: 12 preterm newborns with a gestational age of 28-32 weeks, 11 preterm newborns with a gestational age of 32-36 weeks, and 10 full-term newborns with a gestational age of 36-40 weeks. All recording sessions were performed 1 week after infants had reached full enteral feeding. The percentage of normal slow waves was similar in the three groups but it was not predominant compared to tachygastria in the earliest premature infants (59.3% (12.7-92.3) vs. 29.6% (3.7-78.8); P < 0.05). In addition, a progressive increase in the normal slow wave percentage (59.3% (17.4-87.4), 60.9% (38.1-89.7), 77.8% (66.4-84.8); P < 0.05) was observed as gestation advanced. As regards gastric emptying parameters, the antral area was greater and T(1/2) was longer in the preterm newborns of 28-32 weeks than preterm newborns of 32-36 weeks and full-term newborns (fasting antral area: 0.96 cm2 (0.6-1.5), 0.63 cm2 (0.4-1.2), 0.55 cm2 (0.1-0.9) respectively, P < 0.05; T(1/2): 83.4 min (76.0-108.5), 70 min (57.5-89.5) and 71.8 min (54.9-81.2), respectively P < 0.05). The comparisons of gastric emptying curves made among the three groups showed a reduced antral dilatation in preterm newborns of 28-32 weeks compared to full-term newborns at 30 and 60 min after a meal. In conclusion, although enteral feeding is important for the development process of gastrointestinal motility, gastric electrical activity and gastric emptying show an intrinsic maturation depending on the gestational age.

On the spatial structure and dispersion of slow magnetosonic modes coupled with Alfvén modes in planetary magnetospheres due to field line curvature

SUMMARY The Lord–Shulman thermoelasticity theory combined with Biot equations of poroelasticity, describes wave dissipation due to fluid and heat flow. This theory avoids an unphysical behaviour of the thermoelastic waves present in the classical theory based on a parabolic heat equation, that is infinite velocity. A plane-wave analysis predicts four propagation modes: the classical P and S waves and two slow waves, namely, the Biot and thermal modes. We obtain the frequency-domain Green's function in homogeneous media as the displacements-temperature solution of the thermo-poroelasticity equations. The numerical examples validate the presence of the wave modes predicted by the plane-wave analysis. The S wave is not affected by heat diffusion, whereas the P wave shows an anelastic behaviour, and the slow modes present a diffusive behaviour depending on the viscosity, frequency and thermoelasticity properties. In heterogeneous media, the P wave undergoes mesoscopic attenuation through energy conversion to the slow modes. The Green's function is useful to study the physics in thermoelastic media and test numerical algorithms.

Power system state estimation is classically formulated as a weighted least squares (WLS) optimization problem over the real domain, with the state variables being the voltage magnitudes and angles or the voltage real and imaginary parts. Although the voltages in ac networks are complex variables, a solution in the complex domain has not been previously reported; this is because a real function of complex variables is nonanalytic in its arguments, i.e., the Taylor series expansion in its arguments alone does not exist. By making use of the Wirtinger calculus, this paper presents a new implementation of the WLS state estimation problem in complex variables. The partial derivatives employed in the method give rise to a structure that is very simple to implement. The method can straightforwardly handle both legacy and phasor measurements; particularly, the processing of phasor current measurements does not require any special provisions, unlike previously reported hybrid state estimator implementations over the real domain. Numerical results are reported on networks with up to 9241 nodes, and they demonstrate that the accuracy of the complex variable implementation is competitive with that of the real variable constrained hybrid state estimator, while being significantly faster.

We focus on the convergence rate of the alternating direction method of multipliers (ADMM) in a complex domain. First, the complex form of variational inequality (VI) is established by using the Wirtinger calculus technique. Second, the \(O(1/K)\) convergence rate of the ADMM in a complex domain is provided. Third, the ADMM in a complex domain is applied to the least absolute shrinkage and selectionator operator (LASSO). Finally, numerical simulations are provided to show that ADMM in a complex domain has the \(O(1/K)\) convergence rate and that it has certain advantages compared with the ADMM in a real domain. doi:10.1017/S1446181118000184