A method of feasible directions using function approximations,with applications to min max problems

This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gⁱ(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when $\text{φ(x) =}\text{max}\text{{f(x, y) ¦ y ? Ω}{}_{\mathrm{y}}\text{}}$, a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form $\text{min}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{x}}\text{max}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{y}}\text{f(x, y)}$ under the assumption that Ωm_ξ={x|gⁱ(x)?0, j=1,…,s} ∩$\text{R}$ⁿ, Ω_y={y|ζ_i(y)?0, i-1,…,t} ∩ $\text{R}$^m, with f, g^j, ζⁱ continuously differentiable, f(x, ·) concave, ζⁱ convex for $\text{i = 1,…, t,}\text{and}\text{Ω}{}_{\mathrm{x}}\text{, Ω}{}_{\mathrm{y}}$ compact.     

Asymptotic behavior for an almost periodic,strongly dissipative wave equation

This paper deals with asymptotic behavior for (weak) solutions of the equation $\text{u}{}_{\mathrm{tt}}\text{? Δu + β(u}{}_{\mathrm{t}}\text{) ? ?(t, x)}$, on $\text{R}$⁺ × Ω; u(t, x) = 0, on $\text{R}$⁺ × ?Ω. If $\text{?∈L∞(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by G. Prouse in a previous work. If in addition $\text{?}{}_{\mathrm{t}}\text{∈S}{}^2\text{(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and ? is srongly almost-periodic, we prove for strongly monotone β that all solutions are asymptotically almost-periodic in the energy space. The assumptions made on β are much less restrictive than those made by G. Prouse: mainly, we allow β to be multivalued, and in the one-dimensional case β need not be defined everywhere.     

Limiting distributions of two random sequences

For fixed p (0 ≤ p ≤ 1), let {L₀, R₀} = {0, 1} and X₁ be a uniform random variable over {L₀, R₀}. With probability p let {L₁, R₁} = {L₀, X₁} or = {X₁, R₀} according as $\text{X}{}_1\text{≥}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}$; with probability 1 ? p let {L₁, R₁} = {X₁, R₀} or = {L₀, X₁} according as $\text{X}{}_1\text{≥}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_0\text{+ R}{}_0\text{)}$, and let X₂ be a uniform random variable over {L₁, R₁}. For n ≥ 2, with probability p let {L_n, R_n} = {L_{n ? 1}, X_n} or = {X_n, R_{n ? 1}} according as $\text{X}{}_{\mathrm{n}}\text{≥}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}$, with probability 1 ? p let {L_n, R_n} = {X_n, R_{n ? 1}} or = {L_{n ? 1}, X_n} according as $\text{X}{}_{\mathrm{n}}\text{≥}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}\text{or}\text{<}\text{1}\text{2}\text{(L}{}_{\mathrm{n\; ?\; 1}}\text{+ R}{}_{\mathrm{n\; ?\; 1}}\text{)}$, and let X_{n + 1} be a uniform random variable over {L_n, R_n}. By this iterated procedure, a random sequence {X_n}_{n ≥ 1} is constructed, and it is easy to see that X_n converges to a random variable Y_p (say) almost surely as n → ∞. Then what is the distribution of Y_p? It is shown that the Beta, (2, 2) distribution is the distribution of Y₁; that is, the probability density function of Y₁ is g(y) = 6y(1 ? y) I_0,1(y). It is also shown that the distribution of Y₀ is not a known distribution but has some interesting properties (convexity and differentiability).     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

The construction of stationary two-dimensional Markoff fields with an application to quantum field theory

It is shown that if φ(f)  ∝_R^dφ(y) f(y) dy is a Markoff random field and X_α are multiplicative functionals of φ (with E(X_α) = 1) which converge locally in L₁, then there exists a locally Markoff random field $\text{φ}{}_1$ such that $\text{E(exp(iφ}{}_1\text{(f))) =}\text{lim}{}_{\mathrm{\alpha }}\text{E(X}{}_{\mathrm{<mtext>\alpha </mtext>}}\text{exp}\text{(iφ(φ)))}$. We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take X_α proportional to exp(?λ∝_R² : P(φ(y)) : g_α(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of g_α → 1 and small λ, {X_α} does converge locally in L₁ and that the corresponding $\text{φ}{}_1$ is stationary.     

Explicit zero-free regions for Dirichlet L-functions

Let $\text{L}$_k(S) be the product of the ?(k) Dirichlet L-functions formed with characters modulo k. We prove the existence of explicit numerical zero-free regions for $\text{L}$_k(S). The first result is that $\text{L}$_k(S) has at most a single zero in the region {$\text{s: σ > 1 ?}\text{1}\text{(R}\text{log}\text{M)}\text{}}$, where R = 9.645908801 and M = max {k, k |t|, 10}. The only possible zero in this region is a simple real zero arising from an L-function formed with a real non-principal character. The second result is that if χ₁ and χ₂ are distinct real primitive characters modulo k₁ and k₂, respectively, and if β₁ is a zero of L(s, χ_i), i = 1, 2, then min $\text{{β}{}_1\text{, β}{}_2\text{} < 1 ?}\text{1}\text{(R}{}_1\text{log}\text{M}{}_1\text{)}$, where $\text{R}{}_1\text{=}\text{(5 ? √5)}\text{(15 ? 10√2)}$, and $\text{M}{}_1\text{=}\text{max}\text{{}\text{k}{}_1\text{k}{}_2\text{17}\text{, 13}}$.     

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.     

Laws of iterated logarithm of multiparameter wiener processes

Let {$\text{X(}\text{t}\text{) :}\text{t}\text{∈ R}{}_{\mathrm{+}}{}^{\mathrm{N}}$} denote the N-parameter Wiener process on $\text{R}{}_{\mathrm{+}}{}^{\mathrm{N}}\text{= [0, ∞)}{}^{\mathrm{n}}$. For multiple sequences of certain independent random variables the authors find lower bounds for the distributions of maximum of partial sums of these random variables, and as a consequence a useful upper bound for the yet unknown function , c ≥ 0, is obtained where D_N = Π_{k = 1}^N [0, T_k]. The latter bound is used to give three different varieties of N-parameter generalization of the classical law of iterated logarithm for the standard Brownian motion process.     

Exposed points and extremal problems in H1

If φ ∈ L_∞, we denote by T_φ the functional defined on the Hardy space H¹ by

$\text{T}{}_{\mathrm{\phi }}\text{(?) =}\text{∫}\text{?π}\text{π}\text{?(e}{}^{\mathrm{i\theta }}\text{) φ(e}{}^{\mathrm{i\theta }}\text{)}\text{dθ}\text{2π}$

. Let S_φ be the set of functions in H¹ which satisfy $\text{T}{}_{\mathrm{\phi }}\text{(?) = ∥T}{}_{\mathrm{\phi }}\text{∥}\text{and}\text{∥? ∥}{}_1\text{? 1}$. It is known that if φ is continuous, then S_φ is weak-¹ compact and not empty. For many noncontinuous φ each S_φ is weak-¹ compact and not empty. A complete descr ption of S_φ if S_φ is weak-¹ compact and not empty is obtained. S_φ is not empty if and only if $\text{S}{}_{\mathrm{\phi }}\text{= S}{}_{\mathrm{\psi }}\text{and}\text{ψ = ¦ ?¦}\text{?}$ for some nonzero ? in H¹. It is shown that if $\text{φ = ¦? ¦}\text{?}$ and $\text{? = pg}$, where p is an analytic polynomial and g is a strong outer function, then S_φ is weak-¹ compact. As the consequence, if $\text{? = p}$, then S_φ is weak-¹ compact.     

Sur la convergence uniforme presque complète dans l'estimation de la densité spectrale d'un processus à temps continu après échantillonnage du temps

Let $\text{X={X(t)}}{}_{\mathrm{<mtext>t\in </mtext><mtext>R</mtext>}}$ be a continuous-time strictly stationary and strongly mixing process. In this paper, we prove in the setting of spectral density estimation, at first, under some hard conditions on the spectral density φ_X (because of aliasing phenomenon), the uniformly complete convergence of the spectral density estimate from periodic sampling. Afterwards, to overcome aliasing, we consider the sampled process $\text{{X(t}{}_{\mathrm{n}}\text{)}}{}_{\mathrm{<mtext>n\in </mtext><mtext>Z</mtext>}}$, where {t_n} is a stationary point process independent from X. The uniform complete convergence of the spectral estimate based on the discrete time observations {X(t_k),t_k} is also obtained. The convergence rates are also established. To cite this article: M. Rachdi, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A combinatorial perspective on the non-Radon partitions

Let E be a finite set of points in $\text{R}$^d. Then {A, E ? A} is a non-Radon partition of E iff there is a hyperplane H separating A strictly from E?A. Or equivalently iff ${}_{\mathrm{<mtext>A</mtext>}}\text{O}$ is an acyclic reorientation of (M_Aff(E), O), the oriented matroid canonically determined by E. If (M(E), O) is an oriented matroid without loops then the set $\text{NR(E, O) = {(A, E ? A):}{}_{\mathrm{<mtext>A</mtext>}}\text{O}\text{is acyclic}\text{}}$ determines (M(E), O). In particular the matroidal properties of a finite set of points in $\text{R}$^d are precisely the properties which can be formulated in non-Radon partitions terms. The Möbius function of the poset $\text{A = {A: A ? E,}{}_{\mathrm{<mtext>A</mtext>}}\text{O}\text{is acyclic}\text{}}$ and in a special case its homotopy type are computed. This paper generalizes recent results of P. Edelman (A partial order on the regions of $\text{R}$ⁿ dissected by hyperplanes     

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.     

Positive operators on the n-dimensional ice cream cone

Let $\text{K}{}_{\mathrm{n}}\text{= {x ? R}{}^{\mathrm{n}}\text{: (x}{}_1{}^2\text{+ · +x}{}^2{}_{\mathrm{n?1}}\text{)}\text{1}\text{2}\text{? x}{}_{\mathrm{n}}\text{}}$ be the n-dimensional ice cream cone, and let Γ(K_n) be the cone of all matrices in $\text{R}$ⁿⁿ mapping K_n into itself. We determine the structure of Γ(K_n), and in particular characterize the extreme matrices in Γ(K_n).     

Operator properties of Sobolev imbeddings over unbounded domains

For an open set Ω ? $\text{R}$^N, 1 ? p ? ∞ and λ ∈ $\text{R}$⁺, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators $\text{U}$, 1 ? p, q ? ∞ and a quasibounded domain Ω ? $\text{R}$^N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the given Banach ideal $\text{U}$: Assume the quasibounded domain fulfills condition C_k^l for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any $\text{x ? Ω}$ to the boundary ?Ω tends to zero as $\text{O(¦ x ¦}{}^{\mathrm{?l}}\text{)}$ for $\text{¦ x ¦ → ∞}$, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ $\text{N}$, μ > λ _S($\text{U}$; p,q:N) and v > N/l · λ_D($\text{U}$;p,q), one has that $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ belongs to the Banach ideal $\text{U}$. Here λ_D($\text{U}$;p,q;N)∈$\text{R}$⁺ and λ_S($\text{U}$;p,q;N)∈$\text{R}$⁺ are the D-limit order and S-limit order of the ideal $\text{U}$, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l_pⁿ → l_qⁿ for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω fulfills condition C₁^l.For an open set Ω in $\text{R}$^N, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in $\text{R}$^N and give sufficient conditions on λ such that the Sobolev imbedding operator $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded open set in $\text{R}$^N.     

A Feynman-Kac gauge for solvability of the Schrödinger equation

Let {X_t, t ≥ 0} be Brownian motion in $\text{R}$^d (d ≥ 1). Let D be a bounded domain in $\text{R}$^d with C² boundary, ?D, and let q be a continuous (if d = 1), Hölder continuous (if d ≥ 2) function in D?. If the Feynman-Kac “gauge” E_x{exp(∝₀^τ_Dq(X_t)dt)1_A(X_τD)}, where τ_D is the first exit time from D, is finite for some non-empty open set A on ?D and some x?D, then for any $\text{? ? C}{}^0\text{(?D}$), is the unique solution in $\text{C}{}^2\text{(D) ∩ C}{}^0\text{(}\text{D}\text{?}\text{)}$ of the Schrödinger boundary value problem $\text{(}\text{1}\text{2}\text{Δ}\text{+ q)φ = 0}\text{in}\text{D, φ = ?}\text{on}\text{?D}$.     

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).     

On the probability that k positive integers are relatively prime II

Given a set S of positive integers let $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t)}$ denote the number of k-tuples 〈m₁, …, m_k〉 for which $\text{m}{}_{\mathrm{i}}\text{∈ S ? [1, t]}$ and (m₁, …, m_k) = 1. Also let $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n)}$ denote the probability that k integers, chosen at random from $\text{S ? [1, n]}$, are relatively prime. It is shown that if P = {p₁, …, p_r} is a finite set of primes and S = {m : (m, p₁ … p_r) = 1}, then $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t) = (td(S))}{}^{\mathrm{k}}\text{Π}{}_{\mathrm{\nu ?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) + O(t}{}^{\mathrm{k?1}}\text{)}$ if k ≥ 3 and $\text{Z}{}_2{}^{\mathrm{S}}\text{(t) = (td(S))}{}^2\text{Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^2\text{) + O(t}\text{log}\text{t)}$ where d(S) denotes the natural density of S. From this result it follows immediately that $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) = (ζ(k))}{}^{\mathrm{?1}}\text{Π}{}_{\mathrm{p\in P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$ as n → ∞. This result generalizes an earlier result of the author's where $\text{P = ?}$ and S is then the whole set of positive integers. It is also shown that if S = {p₁^x₁ … p_r^x_r : x_i = 0, 1, 2,…}, then $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → 0}$ as n → ∞.     

The one-dimensional diffusion process and unitary representations of R1

Given a cocycle a(t) of a unitary group {U¹}, ?∞ < t < ∞, on a Hilbert space $\text{H}$, such that a(t) is of bounded variation on [O, T] for every T > O, a(t) is decomposed as a(t) = f;^t₀U^sxds + β(t) for a unique x ? $\text{H}$, β(t) yielding a vector measure singular with respect to Lebesgue measure. The variance is defined as $\text{σ}{}^2\text{({rmU}{}^{\mathrm{t}}\text{}, a(t)) =}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{)∥∝}{}^{\mathrm{t}}{}_0\text{U}{}^{\mathrm{s}}\text{x ds∥}{}^2$ if existing. For a stationary diffusion process on $\text{R}$¹, with Ω₁, the space of paths which are natural extensions backwards in time, of paths confined to one nonsingular interval J of positive recurrent type, an information function I(ω) is defined on $\text{Ω}{}_1$, based on the paths restricted to the time interval [0, 1]. It is shown that $\text{I(Ω)}$ is continuous and bounded on $\text{Ω}{}_1$. The shift τ_t, defines a unitary representation {U^t}. Assuming , dm being the stationary measure defined by the transition probabilities and the invariant measure on J, $\text{I(Ω)}$ has a C^∞ spectral density function f;. It is then shown that σ²({U^t}, I) = f;(O).     

Abstract kinetic equations relevant to supercritical media

The abstract Hilbert space equation $\text{(T?)′(x) = ?(A?)(x)}$, x∈$\text{R}$₊, is studied with a partial range boundary condition $\text{(Q}{}_{\mathrm{+}}\text{?)(0) = ?}{}_{\mathrm{+}}\text{?}\text{Ran}\text{Q}{}_{\mathrm{+}}$. Here T is bounded, injective and self-adjoint, A is Fredholm and self-adjoint, with finite-dimensional negative part, and Q₊ is the orthogonal projection onto the maximal T-positive T-invariant subspace. This models half-space stationary transport problems in supercritical media. A complete existence and uniqueness theory is developed.     

Filtrations for the two parameter jump process

A process which has just one jump, and whose time parameter is the positive quadrant [0, ∞] × [0, ∞], is considered. Following Merzbach, related stopping lines are introduced, and the filtration {$\text{F}$_t1,t2³} considered in this paper is such that, modulo completion, the σ-field $\text{F}$_t1,t2³ is the Borel field on the region

$\text{L}{}_{\mathrm{t1,t2}}\text{={(s}{}_1\text{,s}{}_2\text{); 0?s}{}_1\text{?t}{}_1\text{or}\text{0?s}{}_2\text{?t}{}_2\text{}}$

, together with the atom which is the complement in Ω = [0, ∞]² of L_t1,t2. Optional and predictable projections of related processes are defined, together with their dual projections, and an integral representation for martingales is obtained.