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Brief mention is made of early work in the design of experiments by Galen, Avicenna, and Laplace, after which a seemingly unknown 1815 paper of Gergonne's on polynomial regression is introduced and discussed, and a translation of the paper presented.

Ergodic theory is a relatively new branch of mathematics which from a mathematical point of view may be regarded as generated by the interaction of measure theory and the theory of transformation groups. Its basic concept of "metric transitivity" or "ergodicity" was introduced in 1928 in a paper of Paul Smith and G. D. Birkhoff on dynamical systems. However, the significance of this concept was not appreciated until late 1931 when J. yon Neumann and G. D. Birkhoff proved the celebrated mean and pointwise ergodic theorems, and one may regard the nearly simultaneous appearance of these papers as marking the birth of the subject. Birkhoff's proof of the much more difficult pointwise ergodic theorem was stimulated by yon Neumann's theorem and yon Neumann, in turn, was stimulated by a key observation of B. O. Koopman. Let De be a surface of constant energy E in the phase space D of some Hamiltonian dynamical system. Let V,(~o) denote the point of phase space representing the "state" of the system t time units after it was represented by ~o. Then, for each t, oJ -. Vt(oJ ) is a one-to-one transformation of f2 e onto itself which conserves the natural volume element ~e in f2e induced in £2 e by the Liouville measure dql .." dqn dpl "'" d p n . Moreover, Vq+t~ = VqVt~ for all real numbers t 1 and tz. Koopman's observation (not so obvious 40 years ago as now) was that we may obtain a unitary representation't --+ Ut of the additive group of the real line in the Hilbert space 5°2(f2e, ~e) by defining Ut( f ) (co ) = f (V, ( , -o) ) .

This paper describes the work of mathematicians during the seventeenth and eighteenth centuries on the pressure-height problem of determining the relationship between atmospheric pressure and altitude. Omitting minor contributions by many other mathematicians, the paper describes the work of Pascal (atmospheric pressure decreases with altitude), E. Mariotte (height increases in geometric progression as pressure decreases in arithmetic progression), E. Halley (the use of logarithms), John Wallace, G.W. Leibniz, Jacques Cassini, Daniel Bernoulli, Pierre Bouguer, J.H. Lambert, G. Fontana, J.A. DeLuc, S. Horsley, J. Playfair, and P.S. Laplace whose formula summarized previous results.

In a previous paper, we developed lower and upper bounds for the generalized hypergeometric functions p F q , p = q , p = q + 1, and certain confluent forms under appropriate restrictions on the variable and parameters. In the present paper, we extend these notions and obtain similar inequalities for certain generalized hypergeometric functions of two variables.

In this paper we consider the problem of characterizing those situations under which the best uniform linear approximation to an arbitrary continuous function is unique. The problem has been solved by Haar where the set of approximants is a finite dimensional subspace, but in this paper we generalize this by allowing the set of approximants to be any subset of a finite dimensional space. Some previous work has been done on this problem by Rice [2, p. 87 ff.] for a number of partial results.

The primary objective of this paper is to show that semicomplex structures may be employed to define a local fixed point index on the category whose objects are all finite unions of compact, convex subsets of locally convex topological vector spaces and whose morphisms are all continuous maps between its objects. This is established via the use of the lc∗ spaces introduced by Lefschetz to generalize absolute neighborhood retracts.

This paper is concerned with nonlinear optimization problems in normed linear spaces. Necessary and sufficient conditions for optimal points are given and the range of applicability of these conditions is studied. The results are applied to nonlinear approximation problems.

Let R be a region in n -space and Q a linear quadrature formula for R of the form (f)= ∑ r=1 k r f(x r ) . It is known that if Q(ƒ) = ∝ R ƒ whenever ƒ is a polynomial of degree 3 or lower, then k ⩾ n + 1. It is known that the minimum possible value of k depends on the region R , being 2 n for the n -cube and n + 2 for the n -simplex ( n > 1). In 1956 Hammer and Stroud conjectured that k ⩾ n + 2 for every R , when n > 1. In this paper we construct an R , and a Q with the required property, with k = n + 1.

It has often been of interest in mathematics to consider the problem of embedding a certain algebraic system in other more restricted systems, i.e., in those satisfying additional properties. In the present paper, an opposite problem is considered, namely, the possibility of finding certain more restricted systems within a given one. A new characterization of a free groupoid, in terms of a successor mapping, is also given.

This paper is concerned with Chebyshev approximation by spline functions with free knots. If a zero of a Chebyshev spline function occurs at a knot, the multiplicity of the zero is suitably extended. Theorems on uniqueness on the whole approximation interval and on subintervals are stated in terms of alternation properties.

Leibniz considered the “ars combinatoria” as a science of fundamental significance, much more extensive than the combinatorics of today. His only publications in the field were his youthful Dissertatio de Arte Combinatoria of 1666 and a short article on probability, but he left an extensive (hitherto unpublished and unstudied) Nachlass dealing with five related topics: the basic operations of combinatorics, symmetric functions in connection with theory of equations, partitions (additive theory of numbers), determinants, and theory of probability and related fields. This paper concentrates on the first and third topics as they appear in published sources and the Nachlass. It shows that Leibniz was in possession of many results not published by other mathematicians until many decades later. These include a recursion formula for partitions of n into k parts (first published by Euler in 1751), the Stirling numbers of the second kind (first published in 1730), and several special cases of the general formula for partitions that was published only in 1840 by Stern.

In this paper we give a general expression for operators on a Hilbert space which are similar to an operator of a certain type such as a contraction, a unitary operator, a dissipative operator, an operator with nuclear imaginary component, etc. Theorem 1 covers these types and many others. Theorem 2 is concerned with those operators whose spectrum lies in the open upper half-plane. By an operator we always mean a bounded linear operator on a Hilbert space !$. As usual H > 0 means that H is a positive operator, i.e., (Hx, x) 2 0 f or all x E &. We write H > 0 to indicate that H is positive and invertible. The spectrum of an operator B is denoted bY 4B).

In this paper we shall show that each ƒϵ L p [0,1] (1 ⩽ p ⩽ ∞) has a best L p approximation from the set of exponential sums, V n ( S ), provided S is closed. Here V n ( S ) denotes the set of all solutions of all n -th order linear homogeneous differential equations with constant coefficients for which the roots of the corresponding characteristic polynomial all lie in S . We thus extend the previously known existence theorems which apply only in the special cases where S is compact or where S = R .

This paper proves that some useful commutivity relations exist among semigroup wreath product factors that are either groups or combinatorial “units” U1, U2, or U3. Using these results it then obtains some characterizations of each of the classes of semigroups buildable from U1's, U2's, and groups (“buildable” meaning “dividing a wreath product of”). We show that up to division U1's can be moved to the right and U2's, and groups to the left over other units and groups, if it is allowed that the factors involved be replaced by their direct products, or in the case of U2, even by a wreath product. From this it is deduced that U1's and U2's do not affect group complexity, that any semigroup buildable from U1's, U2's, and groups has group complexity 0 or 1, and that all such semigroups can be represented, up to division, in a canonical form—namely, as a wreath product with all U1's on the right, all U2's on the left, and a group in the middle. This last fact is handy for developing characterizations. An embedding theorem for semigroups with a unique 0-minimal ideal is introduced, and from this and the commutivity results and some constructions proved for RLM semigroups, there is obtained an algebraic characterization for each class of semigroups that is a wreath product-division closure of some combination of U1's, U2's, and the groups. In addition it is shown, for i = 1,2,3, that if the unit Ui does not divide a semigroup S, then S can be built using only groups and units not containing Ui. Thus, it can be deduced that any semigroup which does not contain U3 must have group complexity either 0 or 1. This then establishes that indeed U3 is the determinant of group complexity, since it is already proved that both U1 and U2 are transparent with regard to the group complexity function, and it is known that with U3 (and groups) one can build semigroups with complexities arbitrarily large. Another conclusion is a combinatorial counterpart for the Krohn-Rhodes prime decomposition theorem, saying that any semigroups can be built from the set of units which divide it together with the set of those semigroups not having unit divisors. Further, one can now characterize those semigroups which commute over groups, showing a semigroup commutes to the left over groups iff it is “R1” (i.e., does not contain U1, i.e., is buildable form U2's and groups), and commutes to the right over groups iff it does not contain U2 (i.e., is buildable from groups and U1's). Finally, from the characterizations and their proofs one sees some ways in which groups can do the work of combinatorials in building combinatorial semigroups.

We show that any invertible nonsingular transformation T of a finite measure space ( Ω , Ol , μ ) admits a countable partition of Ω into disjoint measurable sets Ω 0 , Ω 1 , Ω 2 ,… so that (a) Ω 0 and ∪ i ⩾1 Ω i are invariant under T , (b) T restricted to Ω 0 has a finite equivalent invariant measure, (c) each Ω i is an image under an integral power of T of each Ω i ( i, j ⩾ 1). If Ol is countably generated mod μ the sets Ω i ( i ⩾ 1) can be constructed with the additional property of being strongly generating in Ω / Ω 0 . We also give a streamlined introduction to some known results on existence of invariant measures and, thereby, make the paper completely self-contained.

In this paper we establish the existence of a best Lp approximation, 1 ⩽ p ⩽ ∞, to a given function f∈Lp( D , where D ⊂ Rm is a bounded domain, from the family Vn(S) of all nth order exponential sums in m independent variables for which the corresponding exponential parameters lie in the closed set S ⊆ C. In so doing we extend the previously known existence theorem which corresponds to the special case where m = 1 and D is a finite interval.

In “On the Conflict of Bordism of Finite Complexes” [ J. Differential Geometry ], Conner and Smith introduced a homomorphism called the Todd character, relating complex bordism theory to rational homology. Specifically the Todd character consists of a family of homomorphisms th r : MU s (X) → H s→r (X; Q ) . In L. Smith, The Todd character and the integrality theorem for the Chern character, Ill. J. Math. it was shown (note that the indexing of the Todd character is somewhat different here) that there was an integrality theorem for th analogous to the Adams integrality theorem for the Chern character J. F. Adams, On the Chern character and the structure of the unitary group, Proc. Cambridge Philos. Soc. 57 (1961), 189–199; On the Chern character revisted, Ill. J. Math. Now Adams' first paper contains a wealth of information about the Chern character in addition to the integrality theorem already mentioned. Our objective in the present note is to derive analogous results for the Todd character. As in Smith these may then be used to deduce the results of Adams for the Chern character.

The purpose of this paper is twofold. First, we prove the uniqueness of a polynomial of best uniform approximation, in a certain class P of “monotone” polynomials, to a given continuous function. This is the content of Theorem 3.1 which complements the results of Lorentz and Zeller [l]. Secondly, we prove (Theorem 6.1) that a polynomial of best L, approximation in the class 8, to a given continuous function is also unique. This is the analog of Jackson’s theorem for general polynomials. As a preliminary to Theorem 6.1, we give a necessary condition for a polynomial in B to be a polynomial of best L, approximation to an integrable function.

This paper gives some history of the problem, and includes superperiod data for quadratic and cubic nonlinear terms, together with a computation for a prime number of particles in the string. Extension of the problem to a circular array is discussed, and there is a biliography.

Birkhoff and de Boor first posed the question of the existence of a convergent bicubic spline interpolation scheme for non-rectangular domains. In this paper that query is answered affirmatively for L-shaped domains. Specifically, it is shown that ∥sf − f∥= O(hr) where sf is the bicubic spline interpolant associated with a smooth function f, h is the maximum mesh spacing, r − 4 for uniform partitions, and r = 3 for nonuniform partitions.

In this paper a generalized inner product 〈 x | y 〉 is defined as a binary function with complex values which satisfies the following: (i) for any nonzero vector y and any complex number ζ there exists a vector x such that 〈 x | y | = ζ , ( ii ) 〈 x 1 + x 2 | y 1 + y 2 〉 = 〈 x 1 | y 1 〉 + 〈 x 2 | y 1 〉 + 〈 x 1 | y 2 〉 + 〈 x 2 | y 2 〉, ( iii ) 〈 y | x 〉 = f [〈 x | y 〉], where f is a continuous function and, (iv) 〈 x | μy 〉 = g [ μ , 〈 x | y 〉], where g is a continuous function. These conditions induce several functional equations which are then solved. By making a linear combination of 〈 x | y 〉 with its complex conjugate a new function ( x | y ) is obtained which is either symmetric, antisymmetric, or Hermitian. The functions 〈 x | y 〉 and ( x | y ) have the same orthogonal vectors.

This paper deals with the pointwise saturation problem for the Abel means of Fourier series. Most of the known results on pointwise approximation in connection with summation processes of Fourier series are Jackson-type theorems. (See the books of N. Achieser, P. P. Korovkin, and I. P. Natanson on approximation theory. We wish also to refer to the results due to G. Alexits and his school, cf. [I] and [5]). Here we shall study a Bernstein-type problem for the Abel means, i.e., a converse of the Jackson-type problems. But we shall study it only in case of saturated approximation. Although this type of question has been studied for several years, only a few results are available. BajSanski-BojaniC [3] proved a pointwise “0”-theorem for the Bernstein polynomials, and recently V. A. Andrienko [2] studied this problem for the FejCr means of Fourier series. A generalization of his result was given by the author [4]. On the other hand, these problems are closely connected with theorems concerning generalized derivatives of functions which have their origin in Schwas we denote

It is the purpose of this paper to present a method for the computation of best uniform approximation, through the replacement of the uniform norm by another norm designated by ∥ · ∥ n and also by two pseudonorms designated by ps ∥ · ∥ n and ( A ) ∥ · ∥ n . Existence, uniqueness, characterization and computation of best approximations for ∥ · ∥ n , ps ∥ · ∥ n , and ( A ) ∥ · ∥ n are examined.

for which there exists an x E C([O, 11) such that the polynomial of best approximation of degree j to x (in the sense of CebySev) is pj (j = 0, 1, . . ., n 1). What is the answer in the particular case II = 2? In the present paper we shall consider the following more general problem: Let {Gk} be a sequence of linear subspaces of a normed linear space E. Characterize those sequences (gk) in E, with g, E Gk (k = 1,2,. . .), for which there exists an x E E such that

It is shown in this paper (Theorem 1) that if α and β are real numbers such that α gb is irrational and algebraic, then there exist two (necessarily distinct) distributions S and T on R, both with compact supports, such that δ′ = ΔαS + ΔβT. Here ΔαS means S − Sα and Sα denotes the translate of S by α. It is also shown that δ′ has no such representation if α gb has rational or certain transcendental values, and that S and T can be chosen to have order two and no lower order. If ϑ belongs to any of the spaces D, E, S or their duals then ϑ ∗ S and ϑ ∗ T belong to the same space as ϑ, and ϑ′ = Δ α (ϑ ∗ S) + Δ β (ϑ ∗ T) . The formula δ′ = ΔαS + ΔβT is generalized to Rn by means of the tensor product of distributions, and it follows from this formula that there is no discontinuous translation-invariant linear form on any of the spaces D (Rn), E (Rn), S (Rn) or their duals. The same thing is also proved for E (Tn) and its dual where Tn denotes the n-dimensional torus group.

In this paper, we show the existence of nonnegative interpolation formulas for functions which are solutions of second-order uniformly elliptic equations over bounded domains.

The paper describes an approximation formula for sine (x + h) that differs from the first four terms of the Taylor expansion only by having 4 in place of 6 in the denominator of the fourth term. It appears in Sanskrit stanzas quoted in a work of about the fifteenth century and given here with translation and explanation.