This paper investigates the natural problem of performing the multifractal analysis of heterogeneous sums of Dirac masseswhere (xn)n≥0 is a sequence of points in [0, 1]d and (wn)n≥0 is a positive sequence of weights such that Σn≥0wn < ∞. We consider the case where the points xn are roughly uniformly distributed in [0, 1]d, and the weights wn depend on a random self-similar measure μ, a parameter ρ ∈ (0, 1], and a sequence of positive radii (λn)n≥1 converging to 0 in the following wayThe measure ν has a rich multiscale structure. The computation of its multifractal spectrum is related to heterogeneous ubiquity properties of the system {(xn,λn)n with respect to μ.

We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GLd have uniform exponential conjugacy growth and in fact that the number of distinct polynomials arising as characteristic polynomials of the elements of the ball of radius n for the word metric has exponential growth rate bounded away from 0 in terms of the dimension d only.

We consider a random walk on a two-dimensional rectangular lattice in which steps are strictly between nearest neighbour points. The conditions of the walk are that the walker must, at each step, turn either to the right or to the left of his previous step with respective probabilities ½(1+α), ½(1−α), (≤ α ≤ 1). To fix the ideas it is assumed that he starts from the origin and the probability of each of the four possible starting directions is ¼. If Ar denotes the probability of return to the origin after r steps we shall show that

where β = ½(α + α−1) and Pn is the nth Legendre polynomial. It is clear that Ar is zero for r ≢ 0(mod 4).

The part of combinatorial analysis which deals with systems of representatives draws on a variety of different sources. Thus the theory of graphs, the theory of flows in networks, linear programming, and integral linear programming all provide relevant methods and arguments. Nevertheless, it is in all likelihood possible to give a unified set-theoretic account of the subject, based in the main on Hall's criterion for the existence of transversals and on a certain generalization of the Schröder–Bernstein theorem. This approach has the added advantage of greater flexibility since it enables us sometimes to cope with transfinite situations where other methods may be only applicable to finite cases.

In this paper certain Bessel type eigenfunction expansions are developed by considering a non-seif-adjoint problem which involves a radiation type condition

We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits.

Recent studies by Debye (1), Bloch (2), Stoner (3) and Guggenheim (4) expressing somewhat divergent views suggest that there is still some confusion and uncertainty in the method of application of the thermodynamic argument to magnetic phenomena. The first effective contribution to the subject was made by Larmor (5) over fifty years ago, and developing this further he finally proposed in 1929 a sufficient basis for a general theory of the whole range of phenomena. At about the same time (1927), and apparently in ignorance of Larmor's earlier work, Debye(1) also gave a brief outline of the thermodynamic aspects of the subject, starting, however, from much more tentative magnetic ideas and formulae which are now known to be without adequate physical justification. In 1933, Bloch (2) elaborated the argument still further, using, however, basic ideas more akin to those employed by Larmor, but derived by a very specialized argument. Then in 1935, using the same fundamental equation as Debye, Stoner revived the discussion with a systematic survey of the theoretical results bearing on the thermodynamic aspects of the subject, attempting later (6) to justify the basic magnetic ideas by a discussion which he now admits is not entirely satisfactory.

Born partial wave integrals are considered for electron-neutral atom collisions. It is shown that for relatively general atomic wave functions these may be evaluated analytically. These form the Born reactance matrix RB and can be used to calculate the collision strengths in the approximations ωI, ωII.

We discuss the minimal-to-maximal correspondence between surfaces and show that, under this correspondence, a congruence class of minimal surfaces in 3 determines an 2-family of congruence classes of maximal surfaces in 3. It is proved that further identifications among these classes may exist, depending upon the subgroup of automorphisms preserving the Hopf differential on the underlying Riemann surface. The space of maximal congruence classes inherits a quotient topology from 2. In the case of the Scherk minimal surface, the subgroup has order two and the quotient space is topologically a disc with boundary. Other classical examples are discussed: for the Enneper minimal surface, one obtains a non-Hausdorff space; for the minimal catenoid, a closed interval.

In this paper we prove that R7 admits smooth periodic maps with no fixed points for every period that is not a prime power. Results of P. A. Smith show that such examples do not exist in any lower dimensions.

In the contour integral

f and g are regular functions of z in a neighbourhood of the contour C and of the complex parameters (α1, α2, …, αp) = α in a domain of . N is a positive parameter and asymptotic expansions are considered as N → + ∞. The method of steepest descents provides an asymptotic expansion for each fixed value of α, but this is non-uniform with respect to α near critical values α0 at which certain saddle points tend to coincidence. A more complicated expansion, involving Airy functions or generalizations thereof, is valid and uniform for α near α0. This expansion is known to be valid in a neighbourhood of α0 which does not depend on N, in contrast to certain other expansions having regions of validity which contract to a point or surface as N → ∞. It is thereby suggested that the actual region of validity has a definite geometrical shape determined by the functions f and g and the contour C, just as steepest descents expansions are valid in regions bounded by Stokes' lines. In this paper, procedures are derived for the determination of such regions, subject to assumptions concerning f, g and C. The validity of the expansion is established in regions so determined.

Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of such that G: = E/ is a principal groupoid with Haar system λ. The twisted groupoid C*-algebra C*(E; G, λ) is a quotient of the C*-algebra of E obtained by completing the space of -equivariant functions on E. We show that C*(E; G, λ) is postliminal if and only if the orbit space of G is T0 and that C*(E; G, λ) is liminal if and only if the orbit space is T1. We also show that C*(E; G, λ) has bounded trace if and only if G is integrable and that C*(E; G, λ) is a Fell algebra if and only if G is Cartan.

Let be a second-countable, locally compact, Hausdorff groupoid with Haar system λ and continuously varying, abelian isotropy groups. Let be the isotropy groupoid and : = /. Using the results about twisted groupoid C*-algebras, we show that the C*-algebra C*(, λ) has bounded trace if and only if is integrable and that C*(, λ) is a Fell algebra if and only if is Cartan. We illustrate our theorems with examples of groupoids associated to directed graphs.

Approximate analytical expressions for the wave reflexion amplitude for transmission above one-dimensional real smooth and slowly varying potentials in the absence of classical turning points are obtained in two different ways by means of the phase-integral method developed by N. Fröman and P. O. Fröman. The wave reflexion is shown to be governed by the complex transition points for the problem. The great accuracy of the formulae obtained, as well as their respective ranges of applicability, is illustrated by applying them, up to the eleventh order phase-integral approximation, to an exactly solvable model, for which the potential represents a monotonic step.

The main purpose of this paper is to introduce a method to “stabilise” certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group G, namely Hom(ℤn , G). We show that this stabilised space of homomorphisms decomposes after suspending once with “summands” which can be reassembled, in a sense to be made precise below, into the individual spaces Hom(ℤn , G) after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilised space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilised space also allows the description of the additive reduced homology of the individual spaces Hom(ℤn , G), with the order of the Weyl group inverted.

We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

As a mathematical area generalized inversion1 was inaugurated in 1955 by R. Penrose [128]. Since then there have appeared about 2000 articles and 15 books2 on generalized inverses of matrices and linear operators. See the annotated bibliography by Nashed and Rall [121] for the period up to 1976.

Goussarov and the author independently proved that two knots in S3 have the same values of finite type invariants of degree <n if and only if they are Cn-equivalent, which means that they are equivalent up to modification by a kind of geometric commutator of class n. This property does not generalize to links with more than one component.

In this paper, we study the case of Brunnian links, which are links whose proper sublinks are trivial. We prove that if n ≥ 1, then an (n+1)-component Brunnian link L is Cn-equivalent to an unlink. We also prove that if n ≥ 2, then L can not be distinguished from an unlink by any Goussarov–Vassiliev finite type invariant of degree <2n.

In [11], we considered the question of existence of elliptic fibre space structures on a smooth Calabi–Yau threefold X. Necessary conditions are that there exists a nef divisor D on X with D3=0; D2[nequiv ]0 and D·c2[ges ]0. In the case when D·c2[ges ]0, it was observed that a result from [9] implies that there is an elliptic fibre space structure determined by D, and so [11] concerned itself with the case when D·c2=0. In this case, we were not able to deduce in general the existence of an elliptic fibre space structure, but only in the presence of a further condition. In particular, it follows from the Theorem in [11] that, if r denotes the number of rational surfaces E on X with D[mid ]E≡0, and if the Euler characteristic e(X)≠2r, then some positive multiple of D determines an elliptic fibre space structure on X (the theorem in fact proves a slightly stronger result, which gives the elliptic fibre space structure if there is any non-rational surface E on X with D[mid ]E≡0). The purpose of this note is to clarify the meaning of the above rather mysterious condition on the non-vanishing of e(X)/2−r.

The symmetrical Epstein refractive index profile (with free space conditions on each side of the transition) is generalized to a non-symmetrical form including both a barrier and a well by the discovery of a profile that cannot be obtained from the standard Epstein profile merely by adjusting the parameters. The analytical evaluation of the reflexion and transmission properties of the model is more complicated than that which occurs in the study of the Epstein profile.

Let (fn) and (gn) be two sequences of random variables adapted to an increasing sequence of σ-algebras (ℱn) such that the conditional distributions of fn and gn given ℱn coincide. Suppose further that the sequence (gn) is conditionally independent. Then it is known that where the number C is a universal constant. The aim of this paper is to extend this result to certain classes of Orlicz and rearrangement invariant spaces. This paper includes fairly general techniques for obtaining rearrangement invariant inequalities from Orlicz norm inequalities.