Some Weak Forms of the Axiom of Choice Restricted to the Real Line

It is shown that AC(ℝ), the axiom of choice for families of non‐empty subsets of the real line ℝ, does not imply the statement PW(ℝ), the powerset of ℝ can be well ordered. It is also shown that (1) the statement “the set of all denumerable subsets of ℝ has size 2 ${}^{\text{ℵ}{}_{\text{0}}}$ ” is strictly weaker than AC(ℝ) and (2) each of the statements (i) “if every member of an infinite set of cardinality 2 ${}^{\text{ℵ}{}_{\text{0}}}$ has power 2 ${}^{\text{ℵ}{}_{\text{0}}}$ , then the union has power 2 ${}^{\text{ℵ}{}_{\text{0}}}$ ” and (ii) “ℵ(2 ${}^{\text{ℵ}{}_{\text{0}}}$ ) ≠ ℵ_ω” (ℵ(2 ${}^{\text{ℵ}{}_{\text{0}}}$ ) is Hartogs' aleph, the least ℵ not ≤ 2 ${}^{\text{ℵ}{}_{\text{0}}}$ ), is strictly weaker than the full axiom of choice AC.     

A continuity theorem for operators from W1,q(Ω) into Lr(Ω)

A continuity theorem for an operator $\text{T}{}_{\mathrm{?}}\text{:W}{}^{\mathrm{1,q}}\text{(Ω) → L}{}^{\mathrm{r}}\text{(Ω)}$ of the form $\text{T}{}_{\mathrm{?}}\text{(u) = ?(u, Du)}$ when ? is not a continuous function is proven.     

Homogenization in open sets with holes

Let Q^r be a cylindrical bar with r cylindrical cavities having generators parallel to those of Q^r. Let Ω be the cross-section of the bar, Ω1 the cross-section of the domain occupied by the material and $\text{Ω}{}^{\mathrm{i}}\text{(i = 1,…, r)}$ the cross- sections of the cavities:

$\text{Ω}\text{?}{}^{\mathrm{i}}\text{? Ω}\text{Ω}\text{?}{}^{\mathrm{i}}\text{∩}\text{Ω}\text{?}{}^{\mathrm{k}}\text{= φ, i ≠ k}$

. The study of the elastic torsion of this bar leads to the following problem [see 2., 3., 267–320)]:

$\text{Δ?}{}_{\mathrm{r}}\text{+ 2μα = 0 in Ω1}$

$\text{?}{}_{\mathrm{r¦?\Omega }}\text{= 0}$

(1)

$\text{?}{}_{\mathrm{r}}\text{=}\text{constant on}\text{?Ω}{}^{\mathrm{i}}\text{; i = 1,…, r}$

where μ is the shear modulus of the material, α is the angle of twist and $\text{?}{}_{\mathrm{r}}$ represents the stress function. In this paper the problem (1) with an increasing number of holes which are distributed periodically is considered. One would like to know if $\text{?}{}_{\mathrm{r}}$ has a limit $\text{?}{}_{\mathrm{\infty }}\text{as}\text{r → + ∞}$, and if so, the equation satisfied by this limit. This is an “homogenization” problem — the heterogeneous bar Q^r is replaced by a homogeneous one, the response of which under torsion approximates as closely as possible that of Q^r. A more general problem will be studied and the case of elastic torsion will be obtained as an application. The proof uses the energy method [see Lions (Collège de France, 1975–1977), Tartar (Collège de France, 1977)] and extension theorems. A related problem is the homogenization of a perforated plate [cf. Duvaut (to appear)].     

On r-potent linear mappings

Let L(E) be the set of all linear mappings of a vector space E. Let Z₊ be the set of all positive integers. A nonzero element ? in L(E) is called an r-potent if $\text{?}{}^{\mathrm{r}}\text{=?}\text{and}\text{?}{}^{\mathrm{i}}\text{≠?}\text{for}\text{1<i<r (i,r∈Z}{}_{\mathrm{+}}\text{)}$. We prove that $\text{S(E)= {?∈L(E): ?}\text{is singular}\text{}}$ is a semigroup generated by the set of all r-potents in S(E), where r is a fixed positive integer with 2?r?n=dim(E).     

Toeplitz matrices commuting with tridiagonal matrices

we prove that if R is a nonscalar Toeplitz matrix R_{i, j}=r_?i?j? which commutes with a tridiagonal matrix with simple spectrum, then

$\text{r}{}_{\mathrm{k}}\text{r}{}_1\text{=}\text{u}{}_{\mathrm{k-1}}\text{r}{}_2\text{r}{}_1\text{cos p}\text{u}{}_{\mathrm{k-1}}\text{(cos p)}$

, k=4, 5,…, with U_k the Chebychev polynomial of the second kind, where p is determined from

$\text{cos p=}\text{1}\text{2}\text{r}{}^2{}_1\text{?r}{}_1\text{r}{}_3\text{r}{}^2{}_2\text{?r}{}_1\text{r}{}_3$

.     

Stability conditions for linear retarded functional differential equations

The purpose of this note is to study the exponential stability for the linear retarded functional differential equation $\text{x}\text{?}\text{(t) = ∫}{}_{\mathrm{?1}}{}^0\text{[dη(θ)] x(t ? r(θ))}$, where the delay function r(θ) ? 0 is continuous and η(θ) is of bounded variation on the interval [?1, 0]. It is shown that the spectral limit function for the equation above has a continuous dependence on the pair (η, r). The set of all functions of bounded variation η for which the equation above is exponentially stable for every delay function r, the so-called region of stability globally in the delays, is a cone. Therefore for a fixed r, the set of all η which make our equation exponentially stable, that is, the region of stability for the delay function r, contains a cone. A discussion of the characterization of these regions of stability, as well as of the largest cone contained in each region of stability for a fixed delay function r, is given. Some remarks are made with respect to a similar question for the equation $\text{x}\text{?}\text{(t) = Ax(t) + ∫}{}_{\mathrm{?\; 1}}{}^0\text{[dμ(θ)] x(t?r(θ))}$, where A is a real n by n matrix, μ(θ) is bounded variation on [?1, 0] and r(θ) as before. Several examples illustrate the results obtained.     

On Fourier transform of generalized Brownian functionals

Let $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ and $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ be the spaces of generalized Brownian functionals of the white noises ? and ?, respectively. A Fourier transform from $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>B</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ into $\text{(L}{}^2\text{)}{}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}{}^{\mathrm{?}}$ is defined by ??(?) = ∫$\text{S}$¹: exp[?i ∫_$\text{R}$?(t) ?(t) dt]: ${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}\text{?(}\text{B}\text{?}\text{) dμ(}\text{B}\text{?}$), where : :${}_{\mathrm{<mtext>b</mtext><mtext>?</mtext>}}$ denotes the renormalization with respect to ? and μ is the standard Gaussian measure on the space $\text{S}$¹ of tempered distributions. It is proved that the Fourier transform carries ?(t)-differentiation into multiplication by i?(t). The integral representation and the action of?? as a generalized Brownian functional are obtained. Some examples of Fourier transform are given.     

Wr,p(R)-splines

In [3] Golomb describes, for 1 < p < ∞, the H^r,p(R)-extremal extension $\text{F}{}_1$ of a function $\text{?:E → R}$ (i.e., the H^r,p-spline with knots in E) and studies the cone $\text{H}{}_{1E}{}^{\mathrm{r,p}}$ of all such splines. We study the problem of determining when $\text{F}{}_1$ is in W^r,p ≡ H^r,p ∩ L^p. If $\text{F}{}_1\text{? W}{}^{\mathrm{r,p}}$, then $\text{F}{}_1$ is called a W^r,p-spline, and we denote by $\text{W}{}_{1E}{}^{\mathrm{r,p}}$ the cone of all such splines. If E is quasiuniform, then $\text{F}{}_1\text{? W}{}^{\mathrm{r,p}}$ if and only if . The cone $\text{W}{}_{1E}{}^{\mathrm{r,p}}$ with E quasiuniform is shown to be homeomorphic to l^p. Similarly, $\text{H}{}_{1E}{}^{\mathrm{r,p}}$ is homeomorphic to h^r,p. Approximation properties of the W^r,p-splines are studied and error bounds in terms of the mesh size $\text{¦ E ¦}$ are calculated. Restricting ourselves to the case p = 2 and to quasiuniform partitions E, the second integral relation is proved and better error bounds in terms of $\text{¦ E ¦}$ are derived.     

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.     

Votes and a half-binomial

In two party elections with popular vote ratio $\text{p}\text{q}\text{,}\text{1}\text{2}\text{≤p=1 ?q}$, a theoretical model suggests replacing the so-called MacMahon cube law approximation $\text{(}\text{p}\text{q}\text{)}{}^3$, for the ratio $\text{P}\text{Q}$ of candidates elected, by the ratio $\text{?}{}_{\mathrm{k}}\text{(p)}\text{?}{}_{\mathrm{k}}\text{(q)}$ of the two half sums in the binomial expansion of (p+q)^2k+1 for some k. This ratio is nearly $\text{(}\text{p}\text{q}\text{)}{}^3$ when k = 6. The success probability $\text{g}{}_{\mathrm{k}}\text{(p)=(}\text{p}{}^{\mathrm{a}}\text{(p}{}^{\mathrm{a}}\text{+q}{}^{\mathrm{a}}\text{)}$ for the power law $\text{(}\text{p}\text{q}\text{)}{}^{\mathrm{a}}\text{?}\text{P}\text{Q}$ is shown to so closely approximate $\text{?}{}_{\mathrm{k}}\text{(p)=Σ}{}_0{}^{\mathrm{k}}\text{(}{}_{\mathrm{r}}{}^{\mathrm{2k+1}}\text{)p}{}^{\mathrm{2k+1?r}}\text{q}{}^{\mathrm{r}}$, if we choose $\text{a = a}{}_{\mathrm{k}}\text{=}\text{(2k+1)!}\text{4}{}^{\mathrm{k}}\text{k!k!}$, that $\text{1≤}\text{?}{}_{\mathrm{k}}\text{(p)}\text{g}{}_{\mathrm{k}}\text{(p)}\text{≤1.01884086}$ for $\text{k≥1}\text{if}\text{1}\text{2}\text{≤p≤1}$. Computationally, we avoid large binomial coefficients in computing $\text{?}{}_{\mathrm{k}}\text{(p)}$ for k>22 by expressing $\text{2?}{}_{\mathrm{k}}\text{(p)?1}$ as the sum $\text{(p?q) Σ}{}_0{}^{\mathrm{k}}\text{(4pq)}{}^{\mathrm{s}}\text{a}{}_{\mathrm{s}}\text{(2s+1)}$, whose terms decrease by the factors $\text{(4pq)(1?}\text{1}\text{2s}\text{)}$. Setting K = 4k+3, we compute a_k for the large k using a continued fraction $\text{πa}{}_{\mathrm{k}}{}^2\text{=}\text{K+1}{}^2\text{(}\text{2K+3}{}^2\text{(}\text{2K+5}{}^2\text{(2K+…)}\text{)}\text{)}$ derived from the ratio of π to the finite Wallis product approximation.     

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).     

Absolute continuity of positive spectrum for Schrödinger operators with long-range potentials

Absolute continuity in (0, ∞) for Schrödinger operators ? Δ + V(x), with long range potential V = V₁ + V₂ such that $\text{?V}{}_1\text{?}{}_{\mathrm{r}}\text{, V}{}_2\text{= 0(r}{}^{\mathrm{?1??}}\text{)}$, ? > 0, as $\text{¦ x ¦ → ∞}$, is shown by proving estimates on resolvents near the real axis. Completeness of the modified wave operators for a superposition of Coulomb potentials also follows. Singular local behavior of V is allowed.     

Calculating Bessel functions with Padé approximants

The solution of the two-body Schrodinger equation with a $\text{C}\text{r}{}^2$ potential is a Bessel function. The asymptotic series in r^?1, which is generated from Schrodinger equation, has a zero radius of convergence. The Padé approximants to the asymptotic series in $\text{1}\text{z}\text{for}\text{J}{}_{\mathrm{v}}\text{(z)}$ converge rigorously for v real. For v imaginary convergence appears to be the same as for v real.     

A note on non-separable solutions of linear partial differential equations

A method has been presented for constructing non-separable solutions of homogeneous linear partial differential equations of the type F(D, D′)W = 0, where $\text{D =}\text{?}\text{?x}\text{, D′ =}\text{?}\text{?y}$,

$\text{F(D,D′)=}\text{∑}\text{n}\text{r+s=0}\text{C}{}_{\mathrm{rs}}\text{D}{}^{\mathrm{r}}\text{D′}{}^{\mathrm{s}}\text{,}$

where c_rs are constants and n stands for the order of the equation. The method has also been extended for equations of the form Φ(D, D′, D″)W = 0, where $\text{D =}\text{?}\text{?x}\text{, D′ =}\text{?}\text{?y}\text{, D″ =}\text{?}\text{?z}$ and

$\text{Φ(D,D′,D″)W=}\text{∑}\text{n}\text{r+s+t=0}\text{C}{}_{\mathrm{rst}}\text{D}{}^{\mathrm{r}}\text{D′}{}^{\mathrm{s}}\text{D}{}^{\mathrm{t}}\text{D″}{}^{\mathrm{s}}\text{.}$

As illustration, the method has been applied to obtain nonseparable solutions of the two and three dimensional Helmholtz equations.     

Estimées Ck pour les couronnes de convexes de type fini de Cn

C^k estimates for convex domains of finite type in $\text{C}{}^{\mathrm{n}}$ are known from Alexandre (C. R. Acad. Paris, Ser. I 335 (2002) 23–26). We now want to show the same result for annuli. Precisely, we show that for all convex domains D and D′ relatively compact of $\text{C}{}^{\mathrm{n}}$, of finite type m and m′ such that $\text{D}\text{?D′}$, for all q=1,…,n?2, there exists a linear operator $\text{T}{}^1{}_{\mathrm{q}}$ from $\text{C}{}_{\mathrm{0,q}}\text{(}\text{D′?D}\text{)}$ to $\text{C}{}_{\mathrm{0,q?1}}\text{(}\text{D′?D}\text{)}$ such that for all $\text{k∈}\text{N}$ and all (0,q)-form f, $\text{?}\text{?}$-closed of regularity C^k up to the boundary, $\text{T}{}^1{}_{\mathrm{q}}\text{f}$ is of regularity C^{k+1/max(m,m′)} up to the boundary and $\text{?}\text{?}\text{T}{}_{\mathrm{q}}{}^1\text{f=f}$. We fit the method of Diederich, Fisher and Fornaess to the annuli by switching z and ζ. However, the integration kernel will not have the same behavior on the frontier as in the Diederich–Fischer–Fornaess case and we have to alter the Diederich–Fornaess support function which will not be holomorphic anymore. Also, we take care of the so generated residual term in the homotopy formula and show that it is extremely regular so that solve the $\text{?}\text{?}$ problem for it will not be difficult. To cite this article: W. Alexandre, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

Abstract multiparameter theory,I

In this paper we continue our investigation of multiparameter spectral theory. Let H₁,…, H_k be separable Hilbert spaces and H = ?_{r = 1}^kH_r, be their tensor product. In each space H_r we have densely defined self-adjoint operators T_r and continuous Hermitian operators V_rs. The multiparameter eigenvalue problem concerns eigenvalues λ = (λ₁,…, λ_n) ?R^k and eigenvectors $\text{? = ?}{}_1\text{? ··· ? ?}{}_{\mathrm{k}}\text{? H}$ such that $\text{T}{}_{\mathrm{r}}\text{?}{}_{\mathrm{r}}\text{+ ∑}{}_{\mathrm{s\; =\; 1}}{}^{\mathrm{k}}\text{λ}{}_{\mathrm{s}}\text{V}{}_{\mathrm{rs}}\text{?}{}_{\mathrm{r}}\text{= 0}$. We develop a spectral theory for such systems leading to a Parseval equality and generalized eigenvector expansion. The results are applied to a k × k system of linked secondorder differential equations.     

The virial theorem in relativistic quantum mechanics

For a class of potentials including the Coulomb potential q = μr^?1 with ¦ μ ¦ < 1 (1) (i.e., atomic numbers Z ? 137), the virial theorem $\text{(u, α ·}\text{p}\text{u) = (u, r(}\text{?q}\text{?r}\text{)u)}$ is shown to hold, u being an eigenfunction of the operator

$\text{Hu = TU : = (α ·}\text{p}\text{+ β + q)u}$

,

$\text{D(H) = {u ¦ u ∈ [H}{}_{\mathrm{loc}}{}^1\text{(}\text{R}\text{+}{}^3\text{)]}{}^4\text{, r}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{u, TU ∈ [L}{}^2\text{(R)}{}^3\text{]}{}^4\text{}}$

($\text{R}$₊³ := $\text{R}$?{0}). The result implies in particular that H with (1) does not have any eigenvalues embedded in the continuum. The proof uses a scale transformation.     

Subextension of plurisubharmonic functions with bounded Monge–Ampère mass

Let $\text{Ω?}\text{C}{}^{\mathrm{n}}$ be a hyperconvex domain. Denote by $\text{E}{}_0\text{(Ω)}$ the class of negative plurisubharmonic functions ? on $\text{Ω}$ with boundary values 0 and finite Monge–Ampère mass on $\text{Ω.}$ Then denote by $\text{F}\text{(Ω)}$ the class of negative plurisubharmonic functions ? on $\text{Ω}$ for which there exists a decreasing sequence (?)_j of plurisubharmonic functions in $\text{E}{}_0\text{(Ω)}$ converging to ? such that $\text{sup}{}_{\mathrm{j}}\text{∫}{}_{\mathrm{\Omega }}\text{(dd}{}^{\mathrm{c}}\text{?}{}_{\mathrm{j}}\text{)}{}^{\mathrm{n}}\text{+∞.}$It is known that the complex Monge–Ampère operator is well defined on the class $\text{F}\text{(Ω)}$ and that for a function $\text{?∈}\text{F}\text{(Ω)}$ the associated positive Borel measure is of bounded mass on $\text{Ω.}$ A function from the class $\text{F}\text{(Ω)}$ is called a plurisubharmonic function with bounded Monge–Ampère mass on $\text{Ω.}$We prove that if $\text{Ω}$ and $\text{Ω}$ are hyperconvex domains with $\text{Ω?}\text{Ω}\text{?}\text{C}{}^{\mathrm{n}}$ and $\text{?∈}\text{F}\text{(Ω),}$ there exists a plurisubharmonic function $\text{?}\text{?}\text{∈}\text{F}\text{(}\text{Ω}\text{)}$ such that $\text{?}\text{?}\text{??}$ on $\text{Ω}$ and $\text{∫}{}_{\mathrm{<mtext>\Omega </mtext>}}\text{(dd}{}^{\mathrm{c}}\text{?}\text{?}\text{)}{}^{\mathrm{n}}\text{?∫}{}_{\mathrm{\Omega }}\text{(dd}{}^{\mathrm{c}}\text{?)}{}^{\mathrm{n}}\text{.}$ Such a function is called a subextension of ? to $\text{Ω}\text{.}$From this result we deduce a global uniform integrability theorem for the classes of plurisubharmonic functions with uniformly bounded Monge–Ampère masses on $\text{Ω.}$To cite this article: U. Cegrell, A. Zeriahi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A multivariate correlation ratio

A multivariate correlation ratio of a random vector $\text{Y}$ upon a random vector $\text{X}$ is defined by

$\text{η}{}_{\mathrm{\delta }}\text{(Y;X)={}\text{tr}\text{(δ}{}^{\mathrm{<mtext>?1\; </mtext><mtext>Cov</mtext><mtext>E(Y|X))\}</mtext>}}\text{1}\text{2}\text{{}\text{tr}\text{(δ}{}^{\mathrm{?1}}\text{∑}{}_{\mathrm{Y}}\text{)}}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$

where Λ, a fixed positive definite matrix, is related to the relative importance of predictability for the entries of $\text{Y}$. The properties of η_Λ are discussed, with particular attention paid to a ‘correlation-maximizing’ property. Given are applications of η_Λ to the elliptically symmetric family of distributions and the multinomial distribution. Also discussed is the problem of finding those r linear functions of $\text{Y}$ that are most predictable (in a correlation ratio sense) from $\text{X}$.