The existence,uniqueness, and instability of spherically symmetric solutions of a system of reaction—Diffusion equations

The system $\text{?x}\text{?t}\text{= Δx + F(x,y),}\text{?y}\text{?t}\text{= G(x,y)}$ is investigated, where x and y are scalar functions of time (t ? 0), and n space variables $\text{(ξ}{}_1\text{,…, ξ}{}_{\mathrm{n}}\text{), Δx ≡ ∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{?}{}^2\text{x}\text{?ξ}{}_{\mathrm{i}}{}^2$, and F and G are nonlinear functions. Under certain hypotheses on F and G it is proved that there exists a unique spherically symmetric solution $\text{(x(r),y(r)),}\text{where}\text{r = (ξ}{}_1{}^2\text{+ … + ξ}{}_{\mathrm{n}}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, which is bounded for r ? 0 and satisfies x(0) >x₀, y(0) > y₀, x′(0) = 0, y′(0) = 0, and x′ < 0, y′ > 0, ?r > 0. Thus, (x(r), y(r)) represents a time independent equilibrium solution of the system. Further, the linearization of the system restricted to spherically symmetric solutions, around (x(r), y(r)), has a unique positive eigenvalue. This is in contrast to the case n = 1 (i.e., one space dimension) in which zero is an eigenvalue. The uniqueness of the positive eigenvalue is used in the proof that the spherically symmetric solution described is unique.     

On the order of the error function of the (k,r)-integers

Let k and r be fixed integers such that 1 < r < k. Any positive integer n of the form n = a^kb, where b is r-free, is called a (k, r)-integer. In this paper we prove that if Q_k,r(x) denotes the number of (k, r)-integers ≤ x, then $\text{Q}{}_{\mathrm{k,r}}\text{(x) =}\text{xζ(k)}\text{ζ(r)}\text{+ Δ}{}_{\mathrm{k,r}}\text{(x)}$, where $\text{Δ}{}_{\mathrm{k,r}}\text{(x) = O(x}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}\text{exp}\text{[?B}\text{log}{}^{\mathrm{<mtext>3</mtext><mtext>5</mtext>}}\text{x (}\text{log log}\text{x)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>5</mtext>}}\text{])}$, B being a positive constant depending on r and the O-estimate is uniform in k. On the assumption of the Riemann hypothesis, we improve the above order estimate of Δ_k,r(x) and prove that

$\text{1}\text{x}\text{∫}\text{1}\text{α}\text{δ}{}_{\mathrm{k,r}}\text{(t)dt=0(x}\text{1}\text{k}\text{ω(x))}\text{or}\text{0(x}{}^{\mathrm{3/(4r+1)}}\text{ω(x))}$

, according as $\text{k ≤}\text{(4r + 1)}\text{3}$ or $\text{k >}\text{(4r + 1)}\text{3}$, where ω(x) = exp [B log x(log log x)^?1].     

Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

An elementary proof is given of the author's transformation formula for the Lambert series $\text{G}{}_{\mathrm{p}}\text{(x) =}\text{Σ}{}_{\mathrm{n?1}}{}^{\mathrm{<mtext>\infty </mtext>}}\text{n}{}^{\mathrm{?p}}\text{x}{}^{\mathrm{n}}\text{(1?x}{}^{\mathrm{n}}\text{)}$ relating G_p(e^2πiτ) to G_p(e^2πiAτ), where p > 1 is an odd integer and $\text{Aτ =}\text{(aτ + b)}\text{(cτ + d)}$ is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function $\text{log}\text{η(τ) =}\text{πiτ}\text{12}\text{? G}{}_1\text{(e}{}^{\mathrm{2\pi i\tau }}\text{)}$, and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.     

Weak solutions for a nonlinear dispersive equation

In this paper we study the existence, uniqueness, and regularity of the solutions for the Cauchy problem for the evolution equation $\text{u}{}_{\mathrm{t}}\text{+ (f (u))}{}_{\mathrm{x}}\text{? u}{}_{\mathrm{xxt}}\text{= g(x, t), (}{}^1\text{)}\text{where}\text{u = u(x, t), x}$ is in (0, 1), 0 ? t ? T, T is an arbitrary positive real number,f(s)?C¹$\text{R}$, and g(x, t)?L^∞(0, T; L²(0, 1)). We prove the existence and uniqueness of the weak solutions for (¹) using the Galerkin method and a compactness argument such as that of J. L. Lions. We obtain regular solutions using eigenfunctions of the one-dimensional Laplace operator as a basis in the Galerkin method.     

On a conjecture of S. Chowla and of S. Chowla and H. Walum,I

As an extension of the Dirichlet divisor problem, S. Chowla and H. Walum conjectured that, as x → ∞, $\text{Σ}{}_{\mathrm{n\; \le \; \surd x}}\text{n}{}^{\mathrm{a}}\text{B}{}_{\mathrm{r}}\text{({}\text{x}\text{n}\text{}) = O(x}{}^{\mathrm{<mtext>a</mtext><mtext>2</mtext><mtext>\; +\; </mtext><mtext>1</mtext><mtext>4</mtext><mtext>\; +\; \epsilon </mtext>}}\text{)}$ holds for each ε > 0. Here integers a ≥ 0 and r ≥ 1 are given. B_r(x) denotes the rth Bernoulli polynomial and {x} denotes the fractional part of x. The special case a = 0, r = 2 of this conjecture was also mentioned by S. Chowla. In this paper we prove this conjecture for all $\text{e ≥}\text{1}\text{2}$ and r ≥ 2 with ε = 0 (with x^ε replaced by log x in case $\text{a =}\text{1}\text{2}$).     

Numerical solution of a nonlinear integro-differential equation

Galerkin's method with appropriate discretization in time is considered for approximating the solution of the nonlinear integro-differential equation $\text{u}{}_{\mathrm{t}}\text{(x, t) = ∝}{}_0{}^{\mathrm{t}}\text{a(t ? τ)}\text{?}\text{?x}\text{σ(u}{}_{\mathrm{x}}\text{(x, τ)) dτ + f(x, t)}$, 0 < x < 1, 0 < t < T.An error estimate in a suitable norm will be derived for the difference u ? u^h between the exact solution u and the approximant u^h. It turns out that the rate of convergence of u^h to u as h → 0 is optimal. This result was confirmed by the numerical experiments.     

Sublinear perturbations of the differential equation y(n) = 0 and of the analogous difference equation

According to a result of A. Ghizzetti, for any solution y(t) of the differential equation where $\text{y}{}^{\mathrm{(n)}}\text{(t)+}\text{∑}\text{i=0}\text{n?1}\text{g}{}_{\mathrm{i}}\text{(t) y}{}^{\mathrm{i}}\text{(t)=0}\text{(t ? 1)}$, $\text{∝}{}_1{}^{\mathrm{\infty }}\text{¦g}{}_{\mathrm{i}}\text{(x)¦x}{}^{\mathrm{n?I?1}}\text{dx < ∞}$ (0 ?i ? n ?1, either y(t) = 0 for t ? 1 or there is an integer r with 0 ? r ? n ? 1 such that $\text{lim}\text{t → ∞}\text{y(t)}\text{t}{}^{\mathrm{r}}$ exists and ≠0. Related results are obtained for difference and differential inequalities. A special case of the former has interesting applications in the study of orthogonal polynomials.     

Nonlinear “double porosity” type modelUn modèle non linéaire de type double porosité

We consider a variational problem in a bounded domain $\text{Ω=}\text{F}{}^{\mathrm{(\epsilon )}}\text{∪}\text{M}{}^{\mathrm{(\epsilon )}}$ with a microstructure $\text{F}{}^{\mathrm{(\epsilon )}}$ which is not in general periodic; a^ε=a^ε(x) is of order 1 in $\text{F}{}^{\mathrm{(\epsilon )}}$ and as ε→0. A homogenized model is constructed. To cite this article: L. Pankratov, A. Piatnitski, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 435–440.     

Bounce law at the corners of convex billiards

Let C be a convex subset of $\text{R}{}^{\mathrm{n}}$. Given any elastic shock solution x(·) of the differential inclusion$\text{x}\text{̈}\text{(t)+N}{}_{\mathrm{C}}\text{(x(t))∋0,}\text{t>0,}$the bounce of the trajectory at a regular point of the boundary of C follows the Descartes law. The aim of the paper is to exhibit the bounce law at the corners of the boundary. For that purpose, we define a sequence (C_ε) of regular sets tending to C as ε→0, then we consider the approximate differential inclusion , and finally we pass to the limit when ε→0. For approximate sets defined by $\text{C}{}_{\mathrm{\epsilon }}\text{=C+ε}\text{B}$ (where $\text{B}$ is the unit euclidean ball of $\text{R}{}^{\mathrm{n}}$), we recover the bounce law associated with the Moreau–Yosida regularization.     

Hermite series estimates of a probability density and its derivatives

The following estimate of the pth derivative of a probability density function is examined: $\text{Σ}{}_{\mathrm{k\; =\; 0}}{}^{\mathrm{N}}\text{a}\text{?}{}_{\mathrm{k}}\text{h}{}_{\mathrm{k}}\text{(x)}$, where h_k is the kth Hermite function and $\text{a}\text{?}{}_{\mathrm{k}}\text{= (}\text{(?1)}{}^{\mathrm{p}}\text{n}\text{)}$Σ_{i = 1}ⁿh_k^(p)(X_i) is calculated from a sequence X₁,…, X_n of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n^?α) and O(n^?α log n), respectively, where $\text{α =}\text{2(r ? p)}\text{(2r + 1)}$. Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n^?β) and $\text{O(n}{}^{\mathrm{<mtext>?\beta </mtext><mtext>2</mtext>}}\text{log}\text{n)}$, respectively, where $\text{β =}\text{(2(r ? p) ? 1)}\text{(2r + 1)}$.     

The Tjon-Wu equation in Banach space settings

The authors investigate the Tjon-Wu (TW) equation: (TW)

$\text{?u}\text{?t}\text{(t, x) + u(t, x) = ∫}{}_{\mathrm{x}}{}^{\mathrm{\infty }}\text{dy}\text{y}\text{∫}{}_0{}^{\mathrm{y}}\text{u(t, y ? z) u(t, z)dz, u(0, x) = u}{}_0\text{(x)}$

, which has been obtained from a classical Boltzmann equation by applying the Abel transform. (TW) is considered as an ordinary differential equation first in the space L²={u:[0,∞)→R|∫_x^∞|u(x)|²e^xdx < + ∞}The authors establish existence and uniqueness of solutions in disks of codimension 2 around 0 and around e^?x. Asymptotic stability of these latter functions is also established. The basic tool is an unusual eigenvalue property of the nonlinear right-hand side of (TW) which leads to a reformulation of (TW) as a differential equation in l². Similar results are established in L¹ working with (TW) directly.     

Minimax estimators for a multinormal precision matrix

Let S_p×p ～ Wishart (Σ, k), Σ unknown, k > p + 1. Minimax estimators of Σ^?1 are given for L₁, an Empirical Bayes loss function; and L₂, a standard loss function (R_i ≡ E(L_i ∣ Σ), i = 1, 2). The estimators are $\text{Σ}\text{?}{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{+}\text{br}\text{(S)I}{}_{\mathrm{p\times p}}$, a, b ≥ 0, r(·) a functional on $\text{R}{}^{\mathrm{<mtext>p(p+2)</mtext><mtext>2</mtext>}}$. Stein, Efron, and Morris studied the special cases $\text{Σ}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{(E}\text{Σ}\text{?}{}_{\mathrm{<mtext>k?p?1</mtext>}}{}^{\mathrm{?1}}\text{= Σ}{}^{\mathrm{?1}}\text{)}$ and $\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{+ (b/tr S)I}$, for certain, a, b. From their work $\text{R}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) ≤ R}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) (?Σ)}$, a = k ? p ? 1, b = p² + p ? 2; whereas, we prove $\text{R}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) (?Σ)}$. The reversal is surprising because $\text{L}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) →}\text{L}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S)}$ a.e. (for a particular L₂). Assume $\text{R}$ (compact) ? $\text{S}$, $\text{S}$ the set of p × p p.s.d. matrices. A “divergence theorem” on functions F_p×p : $\text{R}$ → $\text{S}$ implies identities for R_i, i = 1, 2. Then, conditions are given for $\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) (?Σ)}$, i = 1, 2. Most of our results concern estimators with r(S) = t(U)/tr(S), U = p ∣S∣^1/p/tr(S).     

Counterexample—An inadmissible estimator which is generalized bayes for a prior with “light” tails

Previous work on the problem of estimating a univariate normal mean under squared error loss suggests that an estimator should be admissible if and only if it is generalized Bayes for a prior measure, F, whose tail is “light” in the sense that $\text{∫}{}_1{}^{\mathrm{\infty }}\text{f}{}^{\mathrm{1?1}}\text{(x) dx = ∞ =}\text{∫}{}_{\mathrm{?\infty }}{}^{\mathrm{?1}}\text{f}{}^{\mathrm{1?1}}\text{(x) dx}$, where $\text{f}{}^1$ denotes the convolution of F with the normal density. (There is also a precise multivariate analog for this condition.) We provide a counterexample which shows that this suggestion is false unless some further regularity conditions are imposed on F.     

Spectral representation for Schrödinger operators with long-range potentials

A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R³, is obtained, where V(x) is a long-range potential: $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(1</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, grad $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(3</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, $\text{ΛV(x) = O(¦ x}\text{s}\text{?}{}^{\mathrm{?\delta }}\text{) (δ > 0), Λ}$ being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator $\text{F}$ from PL₂(R³) onto $\text{L}{}_2\text{((0, ∞); L}{}_2\text{(Ω)), P}$ being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ),

$\text{(α(H)(Pf,g)=}\text{∫}\text{0}\text{∞}\text{(α(λ)(Ff)(λ),(Fg)(λ))L}{}_2\text{(ω) dλ}$

.     

Orthogonal polynomials on the negative multinomial distribution

Orthogonal polynomials on the multivariate negative binomial distribution,

$\text{(1 + Θ)}{}^{\mathrm{?\alpha ?x}}\text{(}\text{π}\text{j=0}\text{p}\text{Θ}{}_{\mathrm{j}}{}^{\mathrm{x}}\text{j}\text{x}{}_{\mathrm{j}}\text{!}\text{)}\text{Γ(α + x)}\text{Γ(α)}$

where α > 0, Θ₁ > 0, x = ΣΘ_i, x₀, x₁, …, x_p = 0,1, … are constructed and their properties studied.     

Compositions of singular integral operators

The composition of two Calderón-Zygmund singular integral operators is given explicitly in terms of the kernels of the operators. For φ?L¹(Rⁿ) and ε = 0 or 1 and ∝ φ = 0 if ε = 0, let Ker(φ) be the unique function on R^{n + 1} homogeneous of degree ?n ? 1 of parity ε that equals φ on the hypersurface x₀ = 1. Let Sing(φ, ε) denote the singular integral operator , which exists under suitable growth conditions on ? and φ. Then Sing(φ, ε₁) Sing(ψ, ε₂)f = ?2π²(∝ φ)(∝ ψ)f + Sing(A, ε₁, + ε₂)f, where

(with notation $\text{¦t¦}{}_0{}^{\mathrm{a}}\text{= ¦t¦}{}^{\mathrm{a}}\text{and}\text{¦t¦}{}_1{}^{\mathrm{a}}\text{= ¦t¦}{}^{\mathrm{a}}\text{sgn}\text{t}$). This result is used to show that the mapping ψ → A is a classical pseudo-differential operator of order zero if φ is smooth, with top-order symbol

$\text{ω}{}_0\text{(x,?)=?πiθ(?)∫θ(x?y)}\text{sgn}\text{y·?dy}\text{if}\text{?}{}_1\text{=1}$

,

$\text{=?2θ(?)∫θ(x?y)}\text{log}\text{|y·?|dy}\text{if}\text{?}{}_1\text{=0}$

where θ(ξ) is a cut-off function. These results are generalized to singular integrals with mixed homogeneity.     

Kolmogorov-Landau inequalities for monotone functions

Presented in this report are two further applications of very elementary formulae of approximate differentiation. The first is a new derivation in a somewhat sharper form of the following theorem of V. M. Olovyani?nikov: LetN_n (n ? 2) be the class of functionsg(x) such thatg(x), g′(x),…, g⁽ⁿ⁾(x) are ? 0, bounded, and nondecreasing on the half-line ?∞ < x ? 0. A special element ofN_nis

$\text{g}{}_1\text{(x) = 0}\text{if}\text{?∞ < x < ?1, g}{}_1\text{(x) = (1 + x)}{}^{\mathrm{n}}\text{if}\text{?1 ? x ? 0}$

. Ifg(x) ∈ N_nis such that

$\text{g(0) ? g}{}_1\text{(0) = 1, g}{}^{\mathrm{(n)}}\text{(0) ? g}{}_1{}^{\mathrm{(n)}}\text{(0) = n!}$

, then

$\text{g}{}^{\mathrm{(v)}}\text{(0) ? g}{}_1{}^{\mathrm{(v)}}\text{(0)}$

for

1$\text{v = 1,…, n ? 1}$

. Moreover, if we have equality in (1) for some value of v, then we have there equality for all v, and this happens only if $\text{g(x) = g}{}_1\text{(x)}$ in (?∞, 0].The second application gives sufficient conditions for the differentiability of asymptotic expansions (Theorem 4).     

The sum of aΩ(n) over integers n ≤ x with all prime factors between a and y

For a > 0 let $\text{ψ}{}_{\mathrm{a}}\text{(x, y) =}\text{Σ}\text{a}{}^{\mathrm{\Omega (n)}}$, the sum taken over all n, 1 ≤ n ≤ x such that if p is prime and p|n then a < p ≤ y. It is shown for u < about ($\text{log log}\text{x}\text{log log log}\text{x}$) that $\text{ψ}{}_{\mathrm{a}}\text{(x, x}{}^{\mathrm{<mtext>1</mtext><mtext>u</mtext>}}\text{) ? x(}\text{log}\text{x)}{}^{\mathrm{a?1}}\text{p}{}_{\mathrm{a}}\text{(u)}$, where p_a(u) solves a delay differential equation much like that for the Dickman function p(u), and the asymptotic behavior of p_a(u) is worked out.     

A remark on the tail probability of a distribution

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let S_n = Σ_j=1ⁿX_j and $\text{E{N}{}_{\mathrm{\infty }}\text{(r, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{r?2}}\text{P{|S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>r</mtext><mtext>t</mtext>}}\text{}}$. In this paper, we prove that (1) lim_?→0+?^α(r?1)E{N_∞(r, t, ?)} = K(r, t) if E(X₁) = 0, Var(X₁) = 1, and E(| X₁ |^t) < ∞, where 2 ≤ t < 2r ≤ 2t, $\text{K(r, t) =}\text{{2}{}^{\mathrm{<mtext>\alpha (r?1)</mtext><mtext>2</mtext>}}\text{Γ(}\text{(1 + α(r ? 1))}\text{2}\text{)}}\text{{(r ? 1) Γ(}\text{1}\text{2}\text{)}}$, and $\text{α =}\text{2t}\text{(2r ? t)}$; (2) $\text{lim}{}_{\mathrm{?\to 0+}}\text{G(t, ?)}\text{H(t, ?)}\text{= 0}$ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(|X₁|^t) < ∞, where G(t, ?) = E{N_∞(t, t, ?)} = Σ_n=1^∞n^t?2P{| S_n | > ?n} → ∞ as ? → 0+ and $\text{H(t, ?) = E{N}{}_{\mathrm{\infty }}\text{(t, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{t?2}}\text{P{| S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>2</mtext><mtext>t</mtext>}}\text{} → ∞}\text{as}\text{? → 0+}$, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(| X₁ |^t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.     

A canonical decomposition of the probability measure of sets of isotropic random points in Rn

The probability measure of X = (x₀,…, x_r), where x₀,…, x_r are independent isotropic random points in $\text{R}$ⁿ (1 ≤ r ≤ n ? 1) with absolutely continuous distributions is, for a certain class of distributions of X, expressed as a product measure involving as factors the joint probability measure of (ω, ?), the probability measure of p, and the probability measure of $\text{Y}{}^1\text{= (y}{}_0{}^1\text{,…, y}{}_{\mathrm{r}}{}^1\text{)}$. Here ω is the r-subspace parallel to the r-flat η determined by X, ? is a unit vector in ω^⊥ with ‘initial’ point at the origin [ω^⊥ is the (n ? r)-subspace orthocomplementary to ω], p is the norm of the vector z from the origin to the orthogonal projection of the origin on η, and $\text{y}{}_{\mathrm{i}}{}^1\text{=}\text{(x}{}_{\mathrm{i}}\text{? z)}\text{α(p}{}^2\text{)}$, where α is a scale factor determined by p. The probability measure for ω is the unique probability measure on the Grassmann manifold of r-subspaces in $\text{R}$ⁿ invariant under the group of rotations in $\text{R}$ⁿ, while the conditional probability measure of ? given ω is uniform on the boundary of the unit (n ? r)-ball in ω^⊥ with centre at the origin. The decomposition allows the evaluation of the moments, for a suitable class of distributions of X, of the r-volume of the simplicial convex hull of {x₀,…, x_r} for 1 ≤ r ≤ n.