By representing integral operators defined by kernels involving Bessel functions as compositions of two operators, we determine their mapping properties and, in one case, derive an inversion process. The operators considered are of the form

where τ(t) is given respectively by and and in the last three cases the integrals converge in norm.

Let C be the homogeneous Cantor set invariant for x→ax and x→1−a+ax. It has been shown by Darst that the Hausdorff dimension of the set of non-differentiability points of the distribution function of uniform measure on C equals (dimHC)2 = (log 2/log a)2. In this paper we generalize the essential ingredient of the proof of this result. Let Ω = {0, 1, …, r}. Let F be a Moran set associated with {0 < ai < 1, i ∈ Ω} and Ωw = Ω×Ω×⃛. Let ø be the associated coding map from Ωe onto F. Fix a non-empty set Γ ⊆ Ω with Γ≠Ø and let z(σ, n) denote the position of the nth occurrence of the elements of Γ in σ ∈ Ωw.

Over a period of many years there has developed an extensive literature on ballot problems. These, in effect, constitute a very special class of random walk problems, and their recent continued development has been justified by the apparent difficulty of reducing expressions given by the general theory down to the very simple ones that it is possible to obtain in an elementary fashion. In this short note we show that the obstacle presented by this reduction problem is actually a rather small one. For background to the above comments, together with a fairly comprehensive list of references to the ballot theory and its attendant applications, the reader is referred to Takács(2).

Y. Morita proved that, for each prime number p, one can define a p-adic continuous function Γp(x) from p to p, interpolating the sequence

where m runs through the integers m prime to p with 1 ≤ m < n. Our aim is to show how this result is related to Dwork's result on the radius of convergence of

We describe a construction which associates to any median metric space a pseudometric satisfying the binary intersection property for closed balls. Under certain conditions, this implies that the resulting space is, in fact, an injective metric space, bilipschitz equivalent to the original metric. In the course of doing this, we derive a few other facts about median metrics, and the geometry of CAT(0) cube complexes. One motivation for the study of such metrics is that they arise as asymptotic cones of certain naturally occurring spaces.

Let X, X1, X2, …, be independent random variables with a common distribution function F(x) = P {X ≤ x}, x∈ℝ, and for each n∈ℕ, let X1, n ≤ … ≤ Xn, n denote the order statistics pertaining to the sample X1, …, Xn. We assume that 1–F(x) = x−1/cl(x), 0 < x < ∞, where l is some function slowly varying at infinity and c > 0 is any fixed number. The class of all such distribution functions will be denoted by .

It has been shown in recent years ((5)–(8), (10)) that it is possible to obtain closed form solutions for the time harmonic wave equation when a linear combination of the wave function and its normal derivative is prescribed on the surface of a wedge. Boundary-value problems of this type occur in the problem of diffraction by a highly conducting wedge or by a wedge whose surfaces are thinly coated with dielectric. In certain circumstances such surfaces can support surface waves and one important aspect of the solution of the boundary-value problem is the determination of the amplitude of the surface wave excited.

Introduction. The space Cp is the class of operators on a Hilbert space for which the norm ∥K∥p = [trace (KK*)P/2]/p is finite. Equivalently, a compact operator is in Cp if

where the μn are the so-called ‘singular values’ of K (characteristic values of the non-negative compact operator [K] ≡ (KK*)½). The case p = 2 gives the familiar class of Hilbert–Schmidt operators, while C1 is the collection of trace-class or nuclear operators considered by Schatten(12), Lidskii(11), and Gohberg and Krein(7), among others.

A Hadamard matrix Hn is an n by n matrix H = [hij], i, j = 1, …, n in which every entry hij is + 1 or − 1, such that

It is well known that possible orders are n = 1, 2 and n = 4m. An automorphism α of H is given by a pair P, Q of monomial ± 1 matrices such that

Here P permutes and changes signs of rows, while Q acts similarly on columns.

1. Introduction. The main result of this paper is that the simple C*-algebra (5) has the Dixmier property. As a corollary of the proof we obtain that the relative commutant of the Choi algebra (4) in is trivial.

We provide, for N ≥ 1, a solution to the word problem for the semigroup S having the presentation {a, b|a = abaNb}. For N = 2, this is example 7 in Howie and Pride [1]. Since solving the word problem for {a, b|a = abaNb} is equivalent to solving the word problem for {a, b|a = baNba}, the result here should be regarded as a special case of a more general result of Oganesyan [2], who solves the word problem for {a, b|a = bA}, A arbitrary.

We show that, for finitely generated groups, the property of admitting a coarse median structure is preserved under relative hyperbolicity.

Let L1, …, Ln be n homogeneous linear forms in n variables u1, …, un, with non-zero determinant Δ. Suppose that L1, …, Lr have real coefficients, that Lr+1, …, Lr+s have complex coefficients, and that the form Lr+s+j is the complex conjugate of the form Lr+j for j = 1, …, s, where r + 2s = n. Let

for integral u1, …, un, not all zero. For any n numbers α1, …, αn of the same ‘type’ as the forms L1, …, Ln (that is, α1, …, αr real, αr+1, …, αr+s complex, αr+s+j = ᾱr+j), let

The paper deals with the problem of the effect of two-dimensional first order disturbances on the linear and parabolic flows of a viscous fluid. If the laminar flow is in the direction x and is bounded by the planes y = ±h, and if the stream function of the disturbance is assumed to be of the form , it is found that, for small Reynolds numbers R (= Uh/v), C can be developed in the power series

The first term in (1) corresponds to a disturbance in still water and is known from Rayleigh's investigation of that problem; the succeeding C's are real and can be obtained by an application of Schrödinger's perturbation theory of wave mechanics. In the case of the parabolic flow, expressions for C0 and C1 were thus derived for both the symmetrical and anti-symmetrical types of disturbance.

The imaginary part of C, as represented by in this approximation, is found to be positive. When α is of the order of unity, the value of R which makes the second term of the first is about 50 for the symmetrical disturbance and 100 for the antisymmetrical disturbance. C0, the first term in the expression for the phase velocity, is less than 1 and increases with α, which implies a group velocity greater than the phase velocity.

In the case of the linear flow it is found that the C's with even subscripts vanish, implying a vanishing phase velocity. This verifies the assumption to that effect made by Southwell and Chitty, and it also suggests that the region of convergence of (A) is limited to the range of values of R covered by these authors. Expressions are derived for C1 and C3 and are evaluated for α=1. The formulae thus obtained for the damping constant give an initial decrease with R, which is not shown in the corresponding curve of Southwell and Chitty.

An alternative method of determining C is outlined in the last section. Use is made of the orthogonal system of the characteristic functions for the vorticity of a disturbance in still water. The equation determining C is reduced to an infinite determinant in which C occurs only in the diagonal elements. When R→0, the determinant can be evaluated and the roots C are found to coincide with the first two terms in (A).

Finally it is shown that the two-dimensional laminar flow is stable for values of R less than the smaller of the quantities 2α and α3.

Exact octagons, i.e. circular eight-term exact sequences, have cropped up recently in a few places in the literature. Papers of the author [11], and implicitly [10], the book of M. Warshauer[14], and the notes [5], all contain exact octagons. The first three references involve octagons of Witt groups of quadratic and other kinds of forms, the last reference extending the octagons to the setting of L-groups, i.e. surgery obstruction groups.

A braided surface of degree m is a compact oriented surface S embedded in a bidisk such that is a branched covering map of degree m and , where is the projection. It was defined L. Rudolph [14, 16] with some applications to knot theory, cf. [13, 14, 15, 16, 17, 18]. A similar notion was defined O. Ya. Viro: A (closed) 2-dimensional braid in R4 is a closed oriented surface F embedded in R4 such that and pr2 │F: F → S2 is a branched covering map, where is the tubular neighbourhood of a standard 2-sphere in R4. It is related to 2-knot theory, cf. [8, 9, 10]. Braided surfaces and 2-dimensional braids are called simple if their associated branched covering maps are simple. Simple braided surfaces and simple 2-dimensional braids are investigated in some articles, [5, 8, 9, 14, 16], etc. This paper treats of non-simple braided surfaces in the piecewise linear category. For braided surfaces a natural weak equivalence relation, called braid ambient isotopy, appears essentially although it is not important for classical dimensionai braids Artin's argument [1].

The influence of a uniform external magnetic field on the propagation of harmonic waves in a perfectly conducting elastic plate extending to infinity in vacuum, with its plane boundaries free of mechanical tractions, has been studied. With the plane of the field and the direction of wave propagation being normal to the plate, it is found that the effects of electromagnetic radiation into the adjoining free space cannot be neglected in the case of waves polarized in the plane. When the field is aligned with the direction of propagation, the maximum influence appears to be on long flexural waves, whereas if the field is normal to the plate then the effect is very much pronounced on long plate waves, and the flexural waves propagate almost with Alfvén velocity.

Cohen's Factorization Theorem says, in its basic form, that if A is a Banach algebra with a bounded left approximate identity, then every element x ∈ A may be written as a product x = ay for some a, y ∈ A. Such is the beauty and importance of this result that much interest attaches to the question of whether the hypothesis of a bounded left approximate identity can be weakened, or whether a converse result exists. This paper contributes to the study of that question.

In a previous paper (3) the writer has proved

where is a Jacobi polynomial defined as ((4) p.255)

Let and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.