On the Growth Behavior of Hypermonogenic Functions on Upper Half-Space

Using the Clifford algebra C n , with generators $$ {e_{1}}, \ldots ,{e_{n}},e_{i}^{2} = - 1, $$ the Cauchy-Riemann operator in ℝ n+1 is defined as $$ \bar{\partial } = \frac{\partial }{{\partial xo}} + \sum\limits_{{i = 1}}^{n} {\frac{\partial }{{\partial {x_{i}}}}{e_{i}}} , $$ where $$ x = {x_{0}} + \sum\limits_{{i = 1}}^{n} {{x_{i}}{e_{i}}} , $$ is a paravector. We consider Fueter-type paravector functions f = u( x o, p) + I( x) v( x o, p), where $$ {p^{2}} = \sum\limits_{{i = 1}}^{n} {x_{i}^{2}} ,I\left( x \right): = \frac{1}{p} = \sum\limits_{{i = 1}}^{n} {{x_{i}}{e_{i}}} , $$ and u, v are real-valued. The equation $$\bar{\partial }f = 0$$ splits into two parts. One of them depends only on xo,p. This leads to a system of partial differential equations which coincides with the system defining hypermonogenic functions. These functions arise for example as solutions of the Dirac equation in the upper half space ℝ + +1 endowed with the Poincaré metric.

We are studying a function theory of k-hypermonogenic functions connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric $$ds_{k}^{2} = x_{n}^{\frac{2k}{1-n}} \left(dx_{0}^{2} + \cdots + dx_{n}^{2} \right)$$ in the upper half space \({\mathbb{R}_{+}^{n+1} = \left\{\left(x_{0}, \ldots,x_{n}\right)\,|\,x_{i} \in \mathbb{R}, x_{n} > 0\right\}}\). The function theory based on this metric is important, since in case \({k = n-1}\), the metric is the hyperbolic metric of the Poincare upper half space and Leutwiler noticed that the power function \({x^{m}\,(m \in \mathbb{N}_{0})}\), calculated using Clifford algebras, is a conjugate gradient of a hyperbolic harmonic function. We find a fundamental \({k}\)-hyperbolic harmonic function. Using this function we are able to find kernels and integral formulas for k-hypermonogenic functions. Earlier these results have been verified for hypermonogenic functions (\({k = n-1}\)) and for k-hyperbolic harmonic functions in odd dimensional spaces.

A fractional differential equation is studied and its application for describing diffusion on random fractal structures is considered. It represents the simplest generalization of the fractional diffusion equation valid in Euclidean systems. The solution of the fractional equation in one dimension is discussed, and compared with exact results for the fractional Brownian motion and the one-dimensional version of the 'standard' diffusion equation on fractals. In higher dimensions, it correctly describes the asymptotic scaling behaviour of the probability density function on random fractals, as obtained recently by using scaling arguments and exact enumeration calculations for the infinite percolation cluster at criticality.

It has been shown (1) that a two-delayed neutron group model will accurately represent the asymptotic behavior of the reactor transfer function. Correspondingly, the requirement that this asymptotic behavior be accurately represented is sufficient to specify completely the two-delayed neutron group constants. Further, a three-delayed neutron group model has been evolved which is designed to represent both the asymptotic behavior of the transfer function and shutdown behavior. The inhour equation, transfer function, and transient behavior for step inputs of reactivity with and without reactivity feedback loops for the two- and three-delayed neutron group representations are compared with those given by the standard one group and many delayed neutron group treatments. Important consequences of such reduced group representations are savings in machine time in making kinetic studies and increased tractability of the reactor kinetic equations.

The authors study the critical frontiers and the high- and low-temperature asymptotic behavior of the Potts model two-site correlation function for several Sierpinski carpets with fractal dimension lying between one and two. They simulate the Sierpinski carpets by means of appropriate hierarchical lattices, obtaining from their basic cells the two graphs (among others) that arise from the anisotropic bond-moving schemes. They correspond to the asymptotic high- and low-temperature dominant terms for the hierarchical lattice two-site correlation function exact result. By taking the infinite limit size of the basic cell these asymptotic behaviors tends to those of the Sierpinski carpets. It is then possible to compare them with similar results obtained from analytical extension to d<2 of the hypercubic lattice. Their results indicate that these asymptotic behaviors do not always coincide, suggesting that possibly they may not identify the Sierpinski carpets exhibited by Gefen et al. (1984) as a correct implementation of the analytical extension of hypercubic lattices.

In this paper we consider the asymptotic behaviour of the spectral function of an elliptic differential (pseudodifferential) equation or system of equations.For the case of differential operators this problem has been widely studied, and various methods have been developed for its solution (see [1] and [2] for a survey of these methods).We consider in this paper just one of these methods. It is based on a study of the structure of the fundamental solution of the Cauchy problem for a hyperbolic differential (pseudodifferential) equation. The method we use is called “the method of geometrical optics”. For a system of first order differential equations it was originally developed in detail by Lax [8], and for pseudodifferential equations by Hörmander [2] and independently by Eskin [17], [18] and Maslov [19].In [2] Hörmander also investigates the asymptotic behaviour of the spectral function for an elliptic pseudodifferential first order operator. Using some important results of Seeley [5] one can then derive the asymptotic behaviour of the spectral function of an elliptic differential operator of arbitrary order.Similar methods have previously been applied by the author in [3], [4] for second order elliptic differential operators.In this article we give a partial account of the results of [8] and [2]. We also present some new results due to the author, concerning both the structure of the fundamental solution of the Cauchy problem and the asymptotic behaviour of the spectral function.

In this paper, we establish some theorems giving necessary and sufficient conditions for an arbitrary function defined in the unit disc of the complex plane to have boundary values along classes of an equivalence relation over simple curves. Our results generalize the well-known theorems on asymptotic and angular boundary behaviours of meromorphic functions (Lindelölf-, Lehto–Virtanen- and Seidel–Walsh-type theorems). The obtained results are applied to the study of boundary behaviour of meromorphic functions along curves using P-sequences, as well as in the proof of the uniqueness theorem similar to Šaginjan’s one. The constructed examples of functions show that the results cannot be improved.

We present an exact derivation for the asymptotic large distance behavior of the spin two-point correlation function in the XY-model. This allows for the exact obtainment of the critical exponent $\eta=1/4$ at the Kosterlitz-Thouless transition that occurs in this model and in the 2D neutral Coulomb gas and which has been previously obtained by scaling arguments. In order to do that, we use the language of sine-Gordon theory to obtain a Coulomb Gas description of the XY-model spin correlation function, which becomes identified with the soliton correlator of that theory. Using a representation in terms of bipolar coordinates we obtain an exact expression for the asymptotic large distance behavior of the relevant correlator at $\beta^2=8\pi$, which corresponds to the Kosterlitz-Thouless transition. The result is obtained by approaching this point from the plasma (high-temperature) phase of the gas. The vortex correlator of the XY-model is also obtained using the same procedure.

Let G be a wreath product of the form C2 ≀ H, where C2 is the cyclic group of order 2. Under mild conditions for H we determine the asymptotic behavior of the counting functions for number fields K/k with Galois group G and bounded discriminant. Those counting functions grow linearly with the norm of the discriminant and this result coincides with a conjecture of Malle. Up to a constant factor these groups have the same asymptotic behavior as the conjectured one for symmetric groups.

Correlation functions are derived in explicit form for the asymptotic behavior at Τ→∞ for a Brownian oscillator with viscous aftereffects.

We study a matrix model of RNA in which an external perturbation on n nucleotides is introduced in the action of the partition function of the polymer chain. The effect of the perturbation appears in the exponential generating function of the partition function as a factor exp(1-nalpha/L) (where alpha is the ratio of strengths of the original to the perturbed term and L is the length of the chain). The asymptotic behavior of the genus distribution functions as a function of length for the matrix model with interaction is analyzed numerically for all n<or=L. It is found that as nalpha/L is increased from 0 to 1, the term 3L in the number of diagrams a'L,g,alpha at a fixed length L, genus g and alpha, goes to 2L [(3-nalpha/L)L for any nalpha/L] and the total number of diagrams Nalpha' at a fixed length L and alpha but independent of genus g, undergoes a change in the factor exp(sqrt[L]) to 1 (exp[(1-nalpha/L)sqrt[L]] for any nalpha/L). However the exponent L of the dominant length dependent term in a'L,g,alpha stays unchanged. Hence the universality is robust to changes in the interaction (alpha). The distribution functions also exhibit unusual behavior at small lengths.

The main purpose of this paper is to investigate the asymptotic behavior of the reduced probability density, ρp+q+s ({p}, {q}, {s}), of p+q+s molecules when the molecules within each of the three subsets {p}, {q}, and {s}, are close together but there are large separations between the sets of p, q, and s particles. The interest is in the case where p, q, and s are unity or two. One may use the asymptotic long-range correlation function to understand the critical singularity of the thermodynamic quantities at the critical point.

In this paper, we shall obtain some Bernstein-type inequalities for certain class of meromorphic functions with restricted zeros and prescribed poles. Our results extend some well known polynomial inequalities to a class of rational functions.

Asymptotic time behavior of correlation functions and mode-mode coupling theory

An exact analytical solution in closed form is obtained for a two-dimensional initial-boundary-value problem of heat wave propagation in a thick slab of an anisotropic rigid thermal conductor within the dual-phase-lag model. One-sided Fourier transform technique is used to obtain a formal solution. The method requires an essential change of the dependent variable in order to guarantee a suitable asymptotic time behavior of the unknown function. The solution satisfies prescribed boundary temperatures and zero initial conditions. Numerical results are presented to put in evidence the effect of the two thermal relaxation times on the behavior of the heat conduction process. In particular, it is shown that the governing equation for temperature includes both characters of wave propagation and diffusion due to the interplay between relaxation times. The influence of anisotropy is investigated. The present results confirm previous findings about heat propagation within dual-phase-lag. The proposed method can treat other geometries, viz. the half-space, quarter-space and the rectangular domain with arbitrary boundary and initial conditions. The obtained exact solution can be used to check approximate solutions arising from other techniques.