Derivatives of meromorphic functions and exponential functions

We take up a new method to prove a Picard type theorem. Let f be a meromorphic function in the complex plane, whose zeros are multiple, and let R be a Möbius transformation. If \({\overline {\lim } _{r \to \infty }}\frac{{T\left( {r,f} \right)}}{{{r^2}}} = \infty \) then f′z) = R(e ^z ) has infinitely many solutions in the complex plane.     

Sampling and interpolation of entire functions and exponential systems in convex domains

Research supported by the Norwegian Research Council for Science and Humanities (NAVF).     

Lyapunov functions versus exponential contractions

For nonautonomous linear equations in a Banach space admitting a nonuniform version of exponential contraction, we give an optimal characterization of the exponential behavior in terms of strict Lyapunov sequences. In particular, we construct explicitly strict Lyapunov sequences for each exponential contraction. We also consider the particular case of quadratic Lyapunov functions, and we use the corresponding characterization of the exponential behavior in terms of these functions to show that the stability of an exponential contraction persists under sufficiently small perturbations.     

An approximation property of exponential functions

We solve the inhomogeneous linear first order differential equations of the form y′(x) ? λy(x) = Σ _m=0 ^∞ a _m (x ? c) ^m , and prove an approximation property of exponential functions. More precisely, we prove the local Hyers-Ulam stability of linear first order differential equations of the form y′(x) = λy(x) in a special class of analytic functions.     

On product formulas for exponential functions

A direct and inverse theorem on the generalized Chernoff formula in Banach algebras is proved by using only elementary properties of exponential and logarithm functions. The main result in this paper forms a part of the first author’s Master thesis (1981) at National Central University.     

Nonuniform exponential dichotomies and Lyapunov functions

For the nonautonomous dynamics defined by a sequence of bounded linear operators acting on an arbitrary Hilbert space, we obtain a characterization of the notion of a nonuniform exponential dichotomy in terms of quadratic Lyapunov sequences. We emphasize that, in sharp contrast with previous results, we consider the general case of possibly noninvertible linear operators, thus requiring only the invertibility along the unstable direction. As an application, we give a simple proof of the robustness of a nonuniform exponential dichotomy under sufficiently small linear perturbations.     

Rationality of exponential functions at integer arguments

Whereas exponential equations have been widely studied, we will deal with integer arguments where an exponential function assumes rational values.     

Equisummability in the sense of M. Riesz of expansions in certain systems of exponential functions

We establish the equisummability in the M. Riesz sense of Fourier series expansions in two systems of exponential functions, these latter being the characteristic functions of boundary-value problems for a first-order differential equation.     

Deficits and Excesses of Frames

The excess of a sequence in a Hilbert space is the greatest number of elements that can be removed yet leave a set with the same closed span. We study the excess and the dual concept of the deficit of Bessel sequences and frames. In particular, we characterize those frames for which there exist infinitely many elements that can be removed from the frame yet still leave a frame, and we show that all overcomplete Weyl–Heisenberg and wavelet frames have this property.     

A property of a system of functions close to exponential functions

We consider the system {f _n=xⁿ[l+_n]} in the interval [a,b] (0 a n > 0 and _n(x) such as the condition, we obtain a bound for the coefficients of the polynomial P(x)=#x2211;c_n f _n(x) in terms of P(x)L_p[a,b]. It is found that this bound is not valid without this condition (assuming the other conditions to remain the same).Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 29–36, July, 1972.     

Quadratic Lyapunov functions and nonuniform exponential dichotomies

For nonautonomous linear equations x^′=A(t)x, we show how to characterize completely nonuniform exponential dichotomies using quadratic Lyapunov functions. The characterization can be expressed in terms of inequalities between matrices. In particular, we obtain converse theorems, by constructing explicitly quadratic Lyapunov functions for each nonuniform exponential dichotomy. We note that the nonuniform exponential dichotomies include as a very special case (uniform) exponential dichotomies. In particular, we recover in a very simple manner a complete characterization of uniform exponential dichotomies in terms of quadratic Lyapunov functions. We emphasize that our approach is new even in the uniform case.Furthermore, we show that the instability of a nonuniform exponential dichotomy persists under sufficiently small perturbations. The proof uses quadratic Lyapunov functions, and in particular avoids the use of invariant unstable manifolds which, to the best of our knowledge, are not known to exist in this general setting.     

Scheduling jobs with truncated exponential learning functions

In this paper we consider the single machine scheduling problem with truncated exponential learning functions. By the truncated exponential learning functions, we mean that the actual job processing time is a function which depends not only on the total normal processing times of the jobs already processed but also on a control parameter. The use of the truncated function is to model the phenomenon that the learning of a human activity is limited. We show that even with the introduction of the proposed model to job processing times, several single machine problems remain polynomially solvable. For the following three objective functions, the total weighted completion time, the discounted total weighted completion time, the maximum lateness, we present heuristic algorithms according to the corresponding problems without truncated exponential learning functions. We also analyse the worst-case bound of our heuristic algorithms.