In this paper we have introduced a new geometric mean (F(r))^α which we shall call generalized geometric mean of an entire function f(z) and have obtained a few growth relations of this mean with respect to other known functions such as G(r) and n(r) etc. Some results obtained earlier by Kamthan, Jain and Chugh, Vaish and Srivastava, Kuldeep Kumar and Boas etc. become the particular cases of the results obtained in this paper. We have also considered the generalized geometric mean of the products of two entire functions only and obtained an interesting theorem. This result can easily be extended to any finite number of entire functions.
We use a result of T. K?mura and Y. K?mura [7], which says that every nuclear Fr?chet space is a closed subspace of the space s^@ , to show some properties of nuclear b ? spaces and their ε ? product.
We push further the idea of [4] and show the existence of unique fixed points for mappings defined on complete metric spaces (X,d) which behave like contractions through a positive function G. We call them generalized contractions. Our results extend the results of [4] as well as the classical ones concerning contractive mappings.
In this paper some common fixed point theorems for nonself two point maps and two set maps on subsets of a metrically convex metric space are proved under weaker boundary and commutativity conditions. Our results generalize the results of Dhage [4]
A topological algebra (,)Aτ is said to be ?circle ? exponential? if the circle ? exponential function 1exp(),!nncnωωω?+?Σ ? Α can be defined, satisfying also the property exp()exp()exp(),,:cccωθωθωθωθ+=?Α=o θω (where ).ωθωθωθ=++o The differential equation αωωα=+& can be solved in such an algebra, while each σ ? complete A ? convex algebra [1] is proved to be of the above type. Also we can formulate and solve the differential equation 00,,(TVTLτ V) αααα=+??& for each σ ? complete locally convex space (,)Vτ. It is not quite clear, which type of ? A ? convexity ? would be enough for having a circle ? exponential algebra
Given a * - vector space (, we wish to find suitable families of seminorms on Vand also multiplications *)V,??Γπ on V, in order to obtain a topological * - algebra (,*,,).
In this paper the surfaces of revolution in the 3 ? dimensional Lorentz ? Minkowski space are classified under the condition IIIrArΔ=rr where IIIΔ is the Laplace operator with respect to the third fundamental form and A is a real 33? matrix. More precisely we prove that such surfaces are either minimal or Lorentz hyperbolic cylinders or pseudospheres of real or imaginary radius
An acoustically soft sphere, which is covered by a penetrable eccentric spherical shell, disturbs the propagation of an incident plane wave field. Low ? frequency scattering theory reduces the scattering problem to a sequence of potential problems with respect to the approximation coefficients. It is defined a bispherical coordinate system that describes the given geometry of the obstacle. The low ? frequency coefficients of the zeroth and the first order for the near field as well as the leading two nonvanishing approximations of the normalized scattering amplitude and scattering cross ? section are obtained. The effects of the coating and the eccentricity of the core are analytically expressed.
We obtain a new expression of the maximal value of Banach limits. With the help of this we easily derive G. G. Lorentz?s classical characterization of almost convergent sequences via Banach limits as well as a (known) set having the property that it produces the set of all Banach limits. We also simplify a Theorem of M. Jerison concerning another set with the same property
Two confocal ellipsoids are employed for modeling the human head in the direct Electroencephalography (EEG) problem. The internal ellipsoid represents the surface of the brain tissue, while the space between the two ellipsoids models the membranes, fluid, bone and scalp, which have different conductivity from that of the brain. The electric excitation of the brain is due to an equivalent electric dipole, which is located within the inner ellipsoid. The proposed model is considered to be more realistic than the single ellipsoidal one, since the effect of the substance surrounding the brain is now taken into account. The direct EEG problem consists in finding the electric potential inside the conductive core and the ellipsoidal shell, as well as at the non ? conductive exterior space. The solution of this transmission problem is given analytically in terms of elliptic integrals and ellipsoidal harmonics. Reduction to the single ellipsoidal and the spherical ? shell model recovers corresponding known results. Comparing the analytic expressions obtained for the exterior potential for the cases of the single and of the double ellipsoidal model, we observe that in the second one a more complicated factor appears in every multipole term. This factor is expressed in terms of the conductivity of the shell as well as the difference of the two conductivities and depends on the geometrical parameters of the two ellipsoidal boundaries. In the case of equal conductivities, or when the two ellipsoids coincide this factor reduces to the conductivity of the single model.
Two different complete representations for Stokes flow have been employed for solving a boundary value problem of particle ? in ? shell model, in spherical and in spheroidal geometry. One employs the stream function ψ and the other employs two harmonic potentials. Using anyone of the two representations, a unique solution of the boundary value problem at hand is obtained within the spherical eigenmode environment. In the case of spheroidal geometry though, the eigenspaces have a richer and more complicated structure, which produces a one ? parametric family of solutions and thus causes the loss of uniqueness. An attempt to answer the question of how the change of geometry influences the structure of the eigenspace in both cases is presented in the work at hand.