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A classical inventory problem is studied from the perspective of embedded options, reducing inventory-management to the design of optimal contracts for forward delivery of stock (commodity). Financial option techniques a la Black-Scholes are invoked to value the additional ‘option to expand stock’. A simplified approach which ignores distant time effects identifies an optimal ‘time to deliver’ and an optimal ‘amount to deliver’ for a production process run in continuous time modelled by a Cobb-Douglas revenue function. Commodity prices, quoted in initial value terms, are assumed to evolve as a geometric Brownian process with positive (inflationary) drift. Expected revenue maximization identifies an optimal ‘strike price’ for the expansion option to be exercised and reveals the underlying martingale in a truncated (censored) commodity price. The paper establishes comparative statics of the censor, using sensitivity analysis on the related censor functional equation; key here is that the censor, as a function of the drift and volatility of price, is the solution of a functional equation. Asymptotic approximation allows a tractable analysis of the optimal timing.

It is proved that, for an arbitrary field K not of characteristic 2 and arbitrary \( n \ge 2 \), if functions f : K→K and h : K→K satisfy f [x1, . . . , xn] = h (x1 + . . . + xn) whenever x1, . . . , xn are distinct elements of K, then f is equal to a polynomial of degree at most n over K. (Here f [x1, . . . , xn] denotes the divided difference of f at the distinct points x1, . . . , xn.) The case of a field of characteristic 2 is also considered.

A measurable f from I = [ 0 , 1 ] I = [0,1] to R is equivalent to a Baire 2 function but may not be equivalent to any Baire 1 function. Gorman has obtained the following interesting contrasting facts. If f assumes finitely many values there is a homeomorphism h of I such that f ∘ h f \circ h is equivalent to a Baire 1 function, but there is a measurable f which assumes countably many values which does not have this property. However, the example of Gorman is such that for some homeomorphisms h the function f ∘ h f \circ h is not measurable. It is shown here that if f ∘ h f \circ h is measurable, for every homeomorphism h, then there is an h for which f ∘ h f \circ h is equivalent to a Baire 1 function.

In a typical counter-example construction in geometric measure theory, starting from some initial set one obtains by successive reductions a decreasing sequence of sets Fn, whose intersection has some required property; it is desired that ∩ Fn shall have large Hausdorf F dimension. It has long been known that this can often be accomplished by making each Fn+1 sufficiently “dense” in Fn. Our first theorem expresses this intuitive idea in a precise form that we believe to be both new and potentially useful, if only for simplifying the exposition in such cases. Our second theorem uses just such a construction to solve the problem that originally stimulated this work: can a Borel set in ℝk have Hausdorff dimension k and yet for continuum-many directions in every angle have at most one point on each line in that direction? The set of such directions must have measure zero, since in fact in almost all directions there are lines that meet the Borel set (of dimension k) in a set of dimension 1: this can easily be deduced from Theorem 6.6 of Mattila [5], which generalized Marstrand's result [4] for the case k = 2.

The essential boundary of a measurable set is related to the de Giorgi perimeter and was introduced by Vol'pert in his "improvement" of Federer's work. For a totally disconnected compact set of positive measure in n space the essential boundary can be of Hausdorff n 1 dimension but cannot have a finite (n 1)measure. Let E c R'" be a measurable set with respect to Lebesgue measure. It is said to be of finite perimeter if all the partial derivatives t1(A),... I, Pn(A) of its characteristic function X are totally finite measures over the Borel sets A C R . It is well known that ti = pi(Rn)= JvI, where vi is the infinum of the variations in xi of all functions equivalent to X and the integration is over the (n 1)-space orthogonal to the xi axis, Oxi. The value of the perimeter is then the variation measure of the vector valued measure (t1(A), ... ., n(A)), evaluated for Rn. It was shown by Federer [1] that the perimeter is equal to the (n 1)-measure of a set that he called the reduced boundary of E, consisting of those points at which a certain generalized normal exists. Specifically, a point p is in the reduced boundary of E if there is an (n 1)-plane v through p such that the part of E on one side of v has density 0 at p, and the part of CE on the other side of v has density 0 at p. (The k-measure X k(E) of a set in Rn will mean the k-dimensional Hausdorff measure, normalized so that the k-dimensional unit cube has measure 1.) Vol'pert [2] showed that in the result the reduced boundary may be replaced by the essential boundary ae(E), consisting of those points of Rn which are neither points of density 1 nor points of density 0 of E. Clearly, the essential boundary of E contains the restricted boundary. Vol'pert's remarkable theorem asserts that, if E has finite perimeter, then the Hausdorff (n 1)-measure of the restricted and essential boundaries are equal. The Lebesgue theorem guarantees that ae(E) is of n-measure zero, but in its perimeter role ae(E) has more of an (n 1)-dimensional flavor, and evidently coincides with the ordinary boundary a(E) when this is a sufficiently smooth surface. Going to the opposite extreme, in this note we shall discuss the possible nature of Me(F), with respect to (n 1)-measure, when E is a nowhere dense set of positive measure in R'". Received by the editors October 23, 1984. 1980 Mathemnatics Subject Classificcation. Primary 28A75. 1 These authors were supported in part by grants from the National Science Foundation. Part of the work was accomplished while the third author was visiting the University of California at Santa Barbara during a Special Year in Real Analysis. :'?1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

The computation of eigenvalues and eigenvectors of a real symmetric matrix A with distinct eigenvalues can be speeded up at the end of the Jacobi process when the off-diagonal elements have become sufficiently small for A to be regarded as a perturbation of a diagonal matrix. A leading-order approximation to the eigensolution is calculated by formulae particularly suitable for the distributed array processor (DAP). A single application of this direct method reduces A to diagonal form and is asymptotically equivalent to an entire sweep of the Jacobi method.

Ceder(1) proved (assuming the axiom of choice, as we do throughout this paper) that the Euclidean plane can be partitioned into ℵ0 sets none of which contains an equilateral triangle; indeed he proved that given any denumerable set of triangles, the plane can be partitioned into ℵ0 sets, none containing a triangle similar to one of the given triangles. Erdős and Kakutani(2) proved that the continuum hypothesis implies that the real line can be partitioned into ℵ0, rationally independent sete, and that the existence of such a partition implies the continuum hypothesis. Erdős asked (private communication) whether there is a partition of the plane into ℵ0 sets not containing an isosceles triangle, or more generally in which any four points determine six different distances. Assuming the continuum hypothesis, it will be shown here (Theorem 1) that a, partition of the latter kind does exist. (I communicated this result to Erdős and others some years ago, but subsequently noticed that the argument was incomplete.) Conversely (Theorem 2) the existence of such a partitition, even for the line, implies the continuum hypothesis. This strengthening of the converse half of the Erdős-Kakutani theorem is proved by what is essentially their method (actually in a rather simplified form).

Sierpiński ((6), (7)) proved that the continuum hypothesis implies the following proposition:Euclidean space E3 can be decomposed into three sets Si (i = 1, 2, 3) such that, for some three straight lines Di in E3, the intersection of each line parallel to Di with the corresponding set Si is finite.

It is known that a real-valued function f of two real variables which is continuous in each variable separately need not be continuous in (x, y), but must be in the first Baire class (1). Moreover if f is continuous in x for each y and merely measurable in y for each x then it must be Lebesgue-measurable (7), and this result can be extended to more general product spaces (2). However, the continuum hypothesis implies that this result fails if continuity is replaced by approximate continuity, as can be seen from the proof of Theorem 2 of (2). This makes Mišik's question (5) very natural: is a function which is separately approximately continuous in both variables necessarily Lebesgue-measurable? Our main aim is to establish an affirmative answer. It will be shown that such a function must in fact be in the second Baire class, although not necessarily in the first Baire class (unlike approximately continuous functions of one variable (3)). Finally, we show that the existence of a measurable cardinal would imply that a separately continuous real function on a product of two topological finite complete measure spaces need not be product-measurable.

Besicovitch's construction(1) of a set of measure zerot containing an infinite straight line in every direction was subsequently adapted (2, 3, 4) to provide the following answer to Kakeya's problem (5): a unit segment can be continuously turned round, so as to return to its original position with the ends reversed, inside an arbitrarily small area. The last word on Kakeya's problem itself seems to be F. Cunningham Jr.'s remarkable result(6)‡ that this can be done inside a simply connected subset of arbitrarily small measure of a unit circle.

Asymptotic expansions of certain generalized Bernoulli polynomials are obtained, some in terms of elementary functions and others in terms of gamma functions and their derivatives. The latter results can also be written in terms of elementary functions, by using the known asymptotic expansions of the gamma functions, and the leading terms are obtained in this way. The results obtained give the asymptotic form of the coefficients occurring in all the usual central-difference formulae of the calculus of finite differences.

Denote by |E| the cardinal of a set E . The purpose of the present paper is to prove the following result, constituting the solution of an unpublished problem of Erdos, Hajnal and Milner.

From the fact that Hausdorffs-dimensional measure is a regular Carathéodory outer measure follows (see Saks (3), ch. II, §§ 6, 8) the standard result:TheoremA.If {En} is any increasing sequence of sets, then∧sEn→ ∧s(ΣEn)as n→ ∞.Since ∧sXis denned (for every setX) as, the problem arises whether for every positive δ and every increasing sequence of sets one can proveNow there are four possible ways of defining; it is the lower bound oftaken over coverings ΣUrofXby convex sets, but in most applications it is not important whether the restriction on the convex sets is that they shall be (i) open and of diameter less than δ, (ii) closed and of diameter less than δ, (iii) open and of diameter not exceeding δ, or (iv) closed and of diameter not exceeding δ.

Problem I was raised (oral communication) by Goffman some three years ago, and I found the example then. Problem II was raised at about the same time by Topsoe; Christensen has given an affirmative answer for spaces satisfying certain additional conditions.It is easy to see that if the answer to problem I were affirmative then so would be that to problem II; therefore our counter-example for problem II implies the existence of one for problem I. It is also possible that another counter-example for problem I could be found by analysing the construction of Dieudonn6 in [1], which is also concerned (implicitly) with a failure of Vitali's theorem. Nevertheless our construction may be of independent interest.

The smallest (nontrivial) identity graph is known to have six points and the smallest identity tree seven. It is now shown that the smallest cubic identity graphs have 12 points and that exactly two of them are planar, namely those constructed by Frucht in his proof that every finite group is isomorphic to the automorphism group of some cubic graph. Both of these graphs can be obtained from plane trees by joining consecutive endpoints; it is shown that when applied to identity trees this construction leads to identity graphs except in certain specified cases. In appendices all connected cubic graphs with 10 points or fewer, and all cubic nonseparable planar graphs with 12 points, are displayed.

A set is said to be σ-finite (with respect to Λs-measure) if it can be expressed as a countable sum of sets of finite Λs-measure. I have proved(1) that every non σ-finite analytic set in a Euclidean space contains a closed set of infinite measure, and Prof. Besicovitch asked me whether the closed subset could itself be chosen to be non σ-finite. In this paper an affirmative answer is given.

We investigate how the behaviour, especially at ± ∞, of continuous real solutions f(t) to the equation f(t) = a1f(t + h1) + a2f(t − h2), where a1, a2, h1, h2 are positive real constants, depends on the values of these parameters. Definitive answers are given, except in certain cases when h1/h2 is rational.

A friendship graph is a graph in which every two distinct vertices have exactly one common neighbour. Finite friendship graphs were characterized by Erdös, Rényi, and Sós [1] as those for which the vertices can be enumerated as u, υ1,…υk, w1,…wk in such a way that the only edges are uυiuwi and υiwi (i = 1,…,k). Thus finite friendship graphs are rather rare. In contrast, we shall show that there are as many nonisomorphic friendship graphs of given infinite cardinal as there are nonisomorphic graphs of that cardinal altogether. In fact, we do a little more.

Recent notes (1, 2) suggest the problem. Find necessary and sufficient conditions on a set of n(>3) real constants a1 …, anfor the inequality

The problem is to arrange the numbers 1, 1, 2, 2, …, n, n in a sequence (without gaps) in such a way that for r = 1, 2, …, n the two r’s are separated by exactly r places; for example41312432.Priday has shown in the preceding paper that for every n there exists either such a perfect sequence (as he calls it) or else a hooked sequence, with a gap one place from one end, for example345131425*2.

1. The preceding paper considers the most general theorem on change of variable in a Riemann integral: If g(t) is integrable over [a, b] and f (x) is integrable over G ([a, b]), then (G(t))g(t) dt exists and equals (x)dx.

By the measure of a set of directions in the plane we mean the linear measure of the set of points P on a unit circle for which the direction from the centre to P belongs to the set.The ray issuing from a point P in direction θ will be denoted by P(θ).

A functionf:X→ ℝ iscountably decomposable(into continuous functions) if the topological spaceXcan be partitioned into countably many setsAnwith each restrictionf│Ancontinuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric spaceX(see Corollary 1). We deduce that whenXis complete they include all functions possessing the propertyPdefined by D. E. Peek in (3):each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has propertyP.

Theorem. if a ≥ 2, b ≥ 1, and a + b ≥ ℵ0, then every collection of more than ab sets each of cardinal not exceeding b contains a subcollection of more than ab sets every two of which have the same intersection.

Solving a problem of Erdős(5), I recently proved (4) that for n = 1, 2, …, the hypothesis implies that if the lines in the Euclidean space Ek (k ≥ 2) are distributed into n + 2 disjoint classes Li(i = 1, …,n + 2), then there exists a decomposition of Ek into sets Si such that the intersection of each line of Li with the corresponding set Si is finite. Without assuming any hypothesis other than the axiom of choice, I have also recently proved (3) that if di (i = 1,2,…) is any infinite sequence of mutually non-parallel lines in the plane, then the plane can be decomposed into N0 sets Si such that each line parallel to di intersects the corresponding set 8i in at most a single point.