Relative Necessity Reformulated with Jessica Leech

This chapter discusses some serious difficulties for what it calls the standard account of various kinds of relative necessity, according to which any given kind of relative necessity may be defined by a strict conditional—necessarily, if C then p—where C is a suitable constant proposition, such as a conjunction of physical laws. It is argued, with the help of Humberstone (1981), that the standard account has several unpalatable consequences. It is argued that Humberstone’s alternative account has certain disadvantages, and another—considerably simpler—solution is offered. The proposed alternative takes seriously the idea that the standard account omits crucial information which, if suitably replaced, allows the problems to be solved.

On the generalized Kesten–McKay distributions

We examine the properties of distributions with the density of the form: [see formula in PDF] where c, a1, …, an are some parameters and An a suitable constant. We find general forms of An, of k-th moment and of k-th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so called Askey–Wilson scheme. On the way we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters a1, …, an forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases n = 2, 4, 6.

The speed of extinction for some generalized jiřina processes

The speed of extinction for some generalized Jiřina processes {Xn} is discussed. We first discuss the geometric speed. Under some mild conditions, the results reveal that the sequence {cn}, where c does not equal the pseudo-drift parameter at x = 0, cannot estimate the speed of extinction accurately. Then the general case is studied. We determine a group of sufficient conditions such that Xn/cn, with a suitable constant cn, converges almost surely as n → ∞ to a proper, nondegenerate random variable. The main tools used in this paper are exponent martingales and stochastic growth models.

The speed of extinction for some generalized jiřina processes

The speed of extinction for some generalized Jiřina processes {X n } is discussed. We first discuss the geometric speed. Under some mild conditions, the results reveal that the sequence {c n }, where c does not equal the pseudo-drift parameter at x = 0, cannot estimate the speed of extinction accurately. Then the general case is studied. We determine a group of sufficient conditions such that X n /c n , with a suitable constant c n , converges almost surely as n → ∞ to a proper, nondegenerate random variable. The main tools used in this paper are exponent martingales and stochastic growth models.

Locally Restricted Compositions I. Restricted Adjacent Differences

We study compositions of the integer $n$ in which the first part, successive differences, and the last part are constrained to lie in prescribed sets ${\cal L,D,R}$, respectively. A simple condition on ${\cal D}$ guarantees that the generating function $f(x,{\cal L,D,R})$ has only a simple pole on its circle of convergence and this at $r({\cal D})$, a function independent of ${\cal L}$ and ${\cal R}$. Thus the number of compositions is asymptotic to $Ar({\cal D})^{-n}$ for a suitable constant $A=A({\cal L,D,R})$. We prove a multivariate central and local limit theorem and apply it to various statistics of random locally restricted compositions of $n$, such as number of parts, numbers of parts of given sizes, and number of rises. The first and last parts are shown to have limiting distributions and to be asymptotically independent. If ${\cal D}$ has only finitely many positive elements ${\cal D}^+$, or finitely many negative elements ${\cal D}^-$, then the largest part and number of distinct part sizes are almost surely $\Theta((\log n)^{1/2})$. On the other hand, when both ${\cal D}^+$ and ${\cal D}^-$ have a positive asymptotic lower "local log-density", we prove that the largest part and number of distinct part sizes are almost surely $\Theta(\log n)$, and we give sufficient conditions for the largest part to be almost surely asymptotic to $\log_{1/r({\cal D})}n$.

Fractional powers of generators of equicontinuous semigroups and fractional derivatives

We analyze fractional powers Hα, α > 0, of the generators H of uniformly bounded locally equicontinuous semigroups S. The Hα are defined as the αth derivative δα of the Dirac measure δ evaluated on S. We demonstrate that the Hα are closed operators with the natural properties of fractional powers, for example, HαHβ = Hα+β for α, β > 0, and (Hα)β = Hαβ for 1 > α > 0 and β > 0. We establish that Hα can be evaluated by the Balakrishnan-Lions-Peetre algorithm where m is an integer larger than α, Cα, m is a suitable constant, and the limit exists in the appropriate topology if, and only if, x ∈ D(Hα). Finally we prove that H∈ is the fractional derivation of S in the sense where the limit again exists if, and only if, x ∈ D(Hα).

Absolute Riesz summability of a Fourier related series, II

This paper is an endeavour to improve upon the work begun in an earlier paper with the same title. We prove a general theorem on the summability |R, exp((log ω)β+1), ρ| of the series ∑ {sn(x)−s}/n, where {sn(x)} is the sequence of partial sums at a point x of the Fourier series of a Lebesgue integrable 2π-periodic function and s is a suitable constant. While the theorem improves upon the main result contained in the previous paper, corollaries to it include recent results due to Chandra and Yadava.

On Gaussian and Geodesic Curvature of Riemannian Manifolds

In [1], S. S. Chern gave a very elegant and simple proof of the Gauss-Bonnet formula for closed (i.e. compact without boundary) oriented Riemannian manifolds of even dimension:Here, c is a suitable constant depending on the dimension of M and Ω is an n-form (n = dim M) which may be calculated from its curvature tensor. W. Greub gave a coordinate-free description of this integrand Ω (cf. [4]).

On classes of summable functions and their Fourier Series

1. Functions which are summable may be such that certain functions of them are themselves summable. When this is the case they will possess certain special properties additional to those which the mere summability involves. A remarkable instance where this has been recognised is in the case of summable functions whose squares also are summable. The—in its formal statement almost self-evident—Theorem of Parseval which asserts that the sum of the squares of the coefficients of a Fourier series of a function f ( x ) is equal to the integral of the square of f ( x ), taken between suitable limits and multiplied by a suitable constant, has been recognised as true for all functions whose squares are summable. Moreover, not only has the converse of this been shown to be true, but writers have been led to develop a whole theory of this class of functions, in connection more especially with what are known as integral equations. That functions whose (1 + p )th power is summable, where p >0, but is not necessarily unity, should next be considered, was, of course, inevitable. As was to be expected, it was rather the integrals of such functions than the functions themselves whose properties were required. Lebesgue had already given the necessary and sufficient condition that a function should be an integral of a summable function. F. Riesz then showed that the necessary and sufficient condition that a function should be the integral of a function whose (1 + p )th power is summable had a form which constituted rather the generalisation of tire expression of the fact that such a function has bounded variation, than one which included the condition of Lebesgue as a particular case.