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The following classical characterization of the exponential distribution is well known. Let X 1 ,X 2 , . . . X n be independent and identically distributed random variables. Their common distribution is exponential if and only if random variables X 1 and n min(X 1 , . . .,X n ) have the same distribution. In this note we show that the characterization can be substantially simplified if the exponentiality is characterized within a broad family of distributions that includes, in particular, gamma, Weibull and generalized exponential distributions. Then the necessary and sufficient condition is the equality only expectations of these variables. A similar characterization holds for the maximum.

Defining the logarithmic function as a definite integral with a variable upper limit, an approach used by some popular calculus textbooks, is problematic. We discuss the disadvantages of such a definition and provide a way to fix the problem. We also consider a definition-based, rigorous derivation of the derivative of the exponential function that is easier, more intuitive, and complies with the standard definitions of the number e, the logarithmic, and the exponential functions.

The problem of characterizing a distribution is an important problem which has recently attracted the attention of many researchers. Thus, various characterizations have been established in many different directions. An investigator will be vitally interested to know if their model fits the requirements of a particular distribution. To this end, one will depend on the characterizations of this distribution which provide conditions under which the underlying distribution is indeed that particular distribution. In this work, several characterizations of Malinowska and Szynal (2008) for certain general classes of distributions are revisited and simpler proofs of them are presented. These characterizations are not based on conditional expectation of the kth lower record values (as in Malinowska and Szynal), they are based on: (i) simple truncated moments of the random variable, (ii) hazard function.

For one-parameter natural exponential and power series families we give some general characterizations from an affine relation connecting the mean of member distributions and their length biased versions. Examples subsume many known cases. Other characterizations are explored using random variable relations involving the length biased version of partial sums. Finally, characterizations of the law of X admitting more general weights are obtained by equating the laws of a function H(X) and the weighted version of X.

We extend Picard's theorem on the existence of elliptic solutions of the second kind of linear homogeneous nth_order scalar ordinary differential equations with coefficients being elliptic functions (associated with a common period lattice) to linear homogeneous first-order n x n systems. In particular, the qualitative Floquet-type structure of fundamental systems of solutions in terms of elliptic and exponential functions, polynomials, and Weierstrass zeta functions of the independent variable is determinod. Connections with completely integrable systems are mentioned. In order to set the stage for our extension of Picard's theorem on the existence of elliptic solutions of the second kind of nth_order scalar ordinary differential equations with elliptic coefficients (with a common period lattice) to first-order n x n systems and an explicit description of the corresponding Floquet-type structure of fundamental systems of solutions, we briefly review Floquet theory for singly periodic n x n systems. Denote by M(n), n E N, the set of n x n matrices with entries in C, define GL(n) {A E M(n) I det(A) 7 0}, and consider the linear homogeneous system (1) @'I(z) Q(z)'J'(z), z E C, where @(z) E GL(n), Q(z) E M(n), with Q(z) a continuous periodic matrix of period Q E C\{0}, that is, (2) Q(z+Q) = Q(z), z E C. Concerning (continuously differentiable) fundamental matrices 4>(z) of solutions of (1), one has the following basic Floquet theorem (see, e.g., [5], Ch. 3, [20], Ch. 4, [30], Ch. 5). Theorem 1. Let Q(z) E M(n) with Q(z) a continuous periodic matrix of period Q E C\{0}. Then (1) admits a fundamental matrix 4?(z) E GL(n) of the type (3) 4(z) = P(z) exp (zK), z E C, where P(z) E GL(n), K E M(n), and P(z) is continuously differentiable and periodic of period Q, (4) P(z+Q) = P(z), z E C. Received by the editors September 23, 1996. 1991 Mathematics Subject Classification. Primary 33E05, 34C25; Secondary 58F07. The research was based upon work supported by the National Science Foundation under Grant No. DMS-9623121. (g)1998 American Mathematical Society

We give two characterizations of the exponential function by solving two functional equations related to some classical considerations made by Gregoire de Saint-Vincent in the 17th century.

The present paper is also devoted to an investigation of stability of characterization of the exponential distribution by property (4) for an arbitrary rational k without conditions of type (5).

In this paper we present several axiomatic characterizations of the mapping assigning to the utility function of a consumer its associated indirect utility function as well as characterizations of the inverse correspondence. Some of them refer to the standard case of real-valued functions, while others involve functions taking values in an arbitrary complete chain. In this abstract framework, we also give some simultaneos characterizations of both transformations.

We give some characterizations of the Lagrangian dual objective function λφdefined with respect to a coupling function φ, namely, conditions for the existence of a φ, and characterizations when q> is given. In particular, when φ is the so-called “natural coupling function”, one of these characterizations reduces to the main result of [6].

The simple cosine function used in the formulation of the traditional minimal principle and the related Shake-and-Bake algorithm is here replaced by a function of exponential type and its expected value and variance are derived. These lead to the corresponding exponential minimal principle and its associated Exponential Shake-and-Bake algorithm. Recent applications of the exponential function to several protein structures within the Shake-and-Bake framework suggest that this function leads, in general, to significant improvements in the success rate (percentage of trial structures yielding solution) of the Shake-and-Bake procedure. However, only in space group P1 is it presently possible to assign optimal values a priori for the exponential-function parameters.

Given the data $(p_i, t_i, f_i), $ $i=1, ldots, m, $ we consider the existence problem for the optimal parameters for the exponential function approximating this data in the sense of total least squares. We give sufficient conditions which guarantee the existence of such optimal parameters.

This paper develops a methodology using risk properties to characterize the functional form of a utility measure for decision making under uncertainty. The constant absolute risk property, for example, is known to be necessary and sufficient, with appropriate regularity conditions, for the utility function to have either a linear or an exponential form. A new generalization of this property, called the constant exchange risk property, gives a characterization of six utility functions: the linear function, the exponential function, the quadratic function, the sum of two exponential functions, the sum of a linear and an exponential function, and the product of a linear and an exponential function. Since all of these functional forms have been used previously as approximations, this methodology allows analysts to distinguish beforehand between alternative forms and thus properly specify the utility function in applications.

An example is given of an unbounded operator in a Hilbert space which generates a strongly continuous semigroup and for which the spectral mapping theorem for the exponential function does not hold. The spectra of both the generator and the semigroup are determined explicitly.

Hobby and Rice [8] were apparently the first to notice that the concept of y-polynomials reveals interesting common features of such important problems as approximation by exponential sums or by splines with variable knots. Later, de Boor [l] extended and simplified their results, and gave a unified approach to existence problems of best approximations for many nonlinear approximation problems. While the above-mentioned questions require only knowledge of the topology of the manifold of exponential sums (or splines), an intrinsic knowledge of the dzfirential structure of these manifolds is required to derive necessary or sufficient conditions for a given y-polynomial to be a (local) best approximation. y-Polynomials constitute a manifold with boundary where the respective dimensions of the tangent cones vary [5]. The closed L, unit-balls (0 2n + 1 and X compact. For a y E C(T x X) the y-polynomials c and c,, are defined as in [6]. For 1 Q q ]]f-g]], for all n* 294 002 l-9045/82/120294-10$02.00/O

It is proved that any A-valued analytic function in an open subset Ω of Rm may be extended to a monogenic function in a suitable open neighbourhood Ωof Ω in Rm+1. This result is applied to define a product between monogenic functions, which in its turn is used to determine a hypercomplex exponential function

SUMMARY In this article a systematic approach to providing goodness of fit tests is discussed, for the composite goodness of fit problem of testing that the distribution F of a random sample comes from a parametric family Fo. Characterization procedures are emphasized, and it is shown that, at least for the exponential case, invariant characterizations appear to be better than those which are not invariant. A general technique is developed for producing invariant characterizations and for the exponential case it is shown how these are related to characterizations already in the literature. Power studies are given to examine the tests based on both invariant and non-invariant characterizations.

Series expansions and recurrence relations suitable for numerical computation are developed for the generalized exponential integral functions. Tables of these functions are presented in the microfiche section of this issue.