The asymptotic distribution of weighted empirical distribution functions

Let G_n denote the empirical distribution based on n independent uniform (0, 1) random variables. The asymptotic distribution of the supremum of weighted discrepancies between G_n(u) and u of the forms 6w_v(u)D_n(u)6 and 6w_v(G_n(u))D_n(u)6, where D_n(u) = G_n(u)?u, w_v(u) = (u(1?u))^?1+v and 0 ? v < $\text{1}\text{2}$ is obtained. Goodness-of-fit tests based on these statistics are shown to be asymptotically sensitive only in the extreme tails of a distribution, which is exactly where such statistics that use a weight function w_v with $\text{1}\text{2}$ ? v ? 1 are insensitive. For this reason weighted discrepancies which use the weight function w_v with 0 ? v < $\text{1}\text{2}$ are potentially applicable in the construction of confidence contours for the extreme tails of a distribution.     

Attainable sets for linear stochastic control systems

Let x_t^u(w) be the solution process of the n-dimensional stochastic differential equation dx_t^u = [A(t)x_t^u + B(t) u(t)] dt + C(t) dW_t, where A(t), B(t), C(t) are matrix functions, W_t is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rⁿ is (?, δ) attainable if there exists an admissable control u such that P{x_t^u?S_?(x)} ? δ, where S_?(x) is the closed ?-ball in Rⁿ centered at x. The set of all (?, δ) attainable points is denoted by $\text{A}$(t). In this paper, we derive various properties of $\text{A}$(t) in terms of K(t), the attainable set of the deterministic control system $\text{x}\text{?}\text{= A(t)x + B(t)u}$. As well a stochastic bang-bang principle is established and three examples presented.     

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).     

Bounds for the uniform deviation of empirical measures

If X₁,…,X_n are independent identically distributed R^d-valued random vectors with probability measure μ and empirical probability measure μ_n, and if $\text{a}$ is a subset of the Borel sets on R^d, then we show that P{sup_{A∈$\text{a}$}|μ_n(A)?μ(A)|≥ε} ≤ cs($\text{a}$, n²)e^?2n∈2, where c is an explicitly given constant, and s($\text{a}$, n) is the maximum over all (x₁,…,x_n) ∈ R^dn of the number of different sets in {{x₁…,x_n}∩A|A ∈$\text{a}$}. The bound strengthens a result due to Vapnik and Chervonenkis.     

A note on k-plane integral transforms

Let Π be a k-dimensional subspace of Rⁿ, n ? 2, and write x = (x′, x″) with x′ in Π and x″ in the orthogonal complement Π^⊥. The k-plane transform of a measurable function ? in the direction Π at the point x″ is defined by $\text{L?(Π, x″) = ∝Π?(x′, x″) dx′}$. In this article certain a priori inequalities are established which show in particular that if $\text{? ? L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{n}}\text{)}$, $\text{1 ? p}\text{\$}\text{?}\text{n}\text{k}$, then ? is integrable over almost every translate of almost every k-space. Mapping properties of the k-plane transform between the spaces L^p(Rⁿ), p ? 2, and certain Lebesgue spaces with mixed norm on a vector bundle over the Grassmann manifold of k-spaces in Rⁿ are also obtained.     

On solutions of “equations in symmetric groups”

Let x?S_n, the symmetric group on n symbols. Let θ? Aut(S_n) and let the automorphim order of x with respect to θ be defined by

$\text{γθ(x)=}\text{min}\text{{k:x xθ xθ}{}^2\text{? xθ}{}^{\mathrm{k?1}}\text{=1}}$

where xθ is the image of x under θ. Let α_g? Aut(S_n) denote conjugation by the element g?S_n. Let where s and k are positive integers and $\text{a}\text{b}$ denotes a divides b. Further h(s, k : n) ≡ b(1; s, k : n), where 1 denotes the identity automorphim. If g?S_n let c = f(g, s) denote the number of symbols in g which are in cycles of length not dividing the integer s, and let g_s denote the product of all cycles in g whose lengths do not divide s. Then g_s moves c symbols. The main results proved are: (1) recursion: if n ? c + 1 and t = n ? c ? 1 then $\text{b(g; s, 1:n)=∑}{}_{\mathrm{<mtext>i</mtext><mtext>s</mtext>}}\text{b(g; s, 1:n?1)(}{}^{\mathrm{t}}{}_{\mathrm{i?1}}\text{(i?1)!}$ (2) reduction: b(g; s, 1 : c)h(s, 1 : i) = b(g; s, 1 : i + c); (3) distribution: let D(θ, n) ≡ {(k, b) : k?Z⁺ and b = b(θ; 1, k : n) ≠ 0}; then D(θ, m) = D(φ, m) ∨ m ? N = N(θ, φ) iff θ is conjugate to φ; (4) evaluation: the number of cycles in g_s^s of any given length is smaller than the smallest prime dividing s iff b(g_s; s, 1 : c) = 1. If g = (12 … p^m)^t and $\text{sk}\text{p}{}^{\mathrm{m}}$ then $\text{b(g;s,k:p}{}^{\mathrm{m}}\text{) {}{}^0{}_{\mathrm{\pm 1}}\text{(}\text{mod}\text{p)}$.     

The number of positive integers ≤x and free of prime factors >y

The number defined by the title is denoted by Ψ(x, y). Let $\text{u =}\text{log}\text{x}\text{log}\text{y}$ and let ?(u) be the function determined by ?(u) = 1, 0 ≤ u ≤ 1, u?′(u) = ? ?(u ? 1), u > 1. We prove the following:Theorem. For x sufficiently large and log y ≥ (log log x)², Ψ(x,y) ? x?(u) while for 1 + log log x ≤ log y ≤ (log log x)², and ε > 0, $\text{Ψ(x, y) ?}{}_{\mathrm{\epsilon }}\text{x?(u)}\text{exp}\text{(?u e}\text{xp}\text{(?(}\text{log}\text{y)}{}^{\mathrm{<mtext>(</mtext><mtext>3</mtext><mtext>5</mtext><mtext>\; ?\; \epsilon )</mtext>}}\text{))}$.The proof uses a weighted lower approximation to Ψ(x, y), a reinterpretation of this sum in probability terminology, and ultimately large-deviation methods plus the Berry-Esseen theorem.     

Existence and comparison theorems for nonlinear diffusion systems

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v¹(x), v²(x),…, v^m(x)) from $\text{Ω ? R}{}^{\mathrm{n}}$ into R^m which satisfy ψⁱ(x, t) ? vⁱ(x) ? θⁱ(x, t) for all $\text{x ? Ω}\text{and}\text{1 ? i ? m}$, where ψⁱand θⁱ are extended real-valued functions on $\text{}\text{?}\text{gW × [0, T)}$. We find conditions which will ensure that a solution U(x, t) ≡ (u¹(x, t), u²(x, t),…, u^m(x, t)) which satisfies U(x, 0) ?K(0) will also satisfy U(x, t) ?K(t) for all 0 ? t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.     

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.     

Entrainment of frequency in evolution equations

The existence of periodic solutions near resonance is discussed using elementary methods for the evolution equation ·u = Au + ?f(t, u) when the linear problem is totally degenerate (e^2πA = I) and the period of f is entrained with ? (T = 2π(1 + ?μ)). The approach is to solve the periodicity equation u(T,p,?) = p for an element p(?) in D, the domain of A, as a perturbation from an approximate solution p₀. p₀ is a solution of the nonlinear boundary value problem 2πμAp + ∝₀^2πe^?Asf(s, e^Asp) ds = 0 obtained from the periodicity equation by dividing by ?, applying the entrainment assumption, and letting ? → 0. Once p₀ is known, the conventional inverse function theorem is applied in a slightly unconventional manner. Two particular cases where results are obtained are u_t = u_x + ?{g(u) ? h(t, x)} with g strongly monotone and

$\text{d}\text{dt}\text{v}\text{w}\text{=}\text{0}\text{d}\text{dx}\text{d}\text{dx}\text{0}\text{v}\text{w}\text{+ ?}\text{v}{}^3\text{h(t,x)}$

, where in both cases D is a certain class of 2π-periodic functions of x.     

The asymptotic behavior of solutions of a system of reaction-diffusion equations which models the Belousov-Zhabotinskii chemical reaction

We investigate the boundary value problem $\text{?u}\text{?t}\text{=}\text{?}{}^2\text{u}\text{?x}{}^2\text{+ u(1 ? u ? rv)}$, $\text{?v}\text{?t}\text{=}\text{?}{}^2\text{v}\text{?x}{}^2\text{? buv}$, u(?∞, t) = v(∞, t) = 0, u(∞, t) = 1, and v(?∞, t) = γ ?t > 0 where r > 0, b > 0, γ > 0 and x?R. This system has been proposed by Murray as a model for the propagation of wave fronts of chemical activity in the Belousov-Zhabotinskii chemical reaction. Here u and v are proportional to the concentrations of bromous acid and bromide ion, respectively. We determine the global stability of the constant solution (u, v) ≡ (1,0). Furthermore we introduce a moving coordinate and for each fixed x?R we investigate the asymptotic behavior of u(x + ct, t) and v(x + ct, t) as t → ∞ for both large and small values of the wave speed c ? 0.     

Isolated singularities of some semilinear elliptic equations

Let Ω be an open subset of $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">N</mi>}}$, N ? 3, containing 0. We consider the solutions of ?Δu(x) + g(u(x)) = f(x) in Ω-{0}, where g is nondecreasing and f is bounded and we study the possible singularities at 0: when u(x) = o(|x|^{1 ? N}) we prove that u is isotropic near 0 and show that either it is a C¹ function in Ω (removable singularity) or |x|^{N ? 2}u(x) → c, c ≠ 0 (weak singularity) or |x|^{N ? 2} |u(x) |→ + ∞ (strong singularity). We also characterize the g's for which solutions with a weak singularity exist and improve a previous removability result of H. Brézis and L. Véron (Arch. Rational Mech. Anal.23 (1979), 153–166).     

Periodic forcing of solutions of a boundary value problem for a second-order differential equation in Hilbert space

We show that if u is a bounded solution on $\text{R}$⁺ of u″(t) ?Au(t) + f(t), where A is a maximal monotone operator on a real Hilbert space H and f∈L_loc²($\text{R}$⁺;H) is periodic, then there exists a periodic solution ω of the differential equation such that u(t) ? ω(t)   0 and u′(t) ? ω′(t) → 0 as t → ∞. We also show that the two-point boundary value problem for this equation has a unique solution for boundary values in $\text{D(A)}$ and that a smoothing effect takes place.     

Spherically symmetric solutions of a reaction-diffusion equation

Two theorems are proved for the spherically symmetric solutions of the “bistable” reaction-diffusion equation $\text{u}{}_{\mathrm{t}}\text{= Δ}{}_{\mathrm{x}}\text{u + ?(u)}$, where ? is cubic-like and x ∈ Rⁿ. The first theorem says that, for a suitable class of initial data, there are only two types of asymptotic behavior, u(x, t) tends to an equilibrium solution as t → + ∞ or u(x, t) → 1 uniformly on compact sets. The second theorem says that in the latter case, if the solution is followed out along any ray, it approaches, in shape, the one-dimensional travelling wave.     

Periodic solutions of nonlinear delay equations

Existence and uniqueness of 2π-periodic solutions of $\text{d}{}^{\mathrm{j}}\text{x(t)}\text{dt}{}^{\mathrm{j}}\text{+}\text{grad}\text{G(x(t ? τ)) = e(t, x(t), x(t ? τ)) (j = 1, 2)}$, where x(t) is in $\text{R}$ⁿ and e(t, u, v) is a given vector function, 2π-periodic in t, are shown under conditions on the spectrum of the Hessian of G. The equation is studied using a fixed point theorem in the space L²(0, 2π). One feature of this approach is that no relationship between the delay and the period is necessary.     

On compactness of commutators of multiplications and convolutions,and boundedness of pseudodifferential operators

Commutators [a(M), b(D)] of a multiplication (a(M)u)(x) = a(x) u(x) and a convolution b(D) = F^?1b(M)F (F = Fourier transform) are L²-compact if only the continuous functions a and b are bounded and for c = a and c = b we have $\text{lim}{}_{\mathrm{¦x¦\to \infty }}\text{sup}\text{{¦ c(x + h) ? c(x)¦ : ¦ h ¦ ? 1} = 0}$. An improvement of a result by Calderon and Vaillancourt of boundedness of pseudodifferential operators is discussed (including an independent proof). Similar results on L^p-compactness and L^p-boundedness, 1 < p < ∞, using the Hoermander-Mihlin boundedness theorem on $\text{R}$ⁿ-Fourier-multipliers, and with conditions and proofs different from the case of L².     

Entire solutions blowing up at infinity for semilinear elliptic systems

We consider the system Δu=p(x)g(v), Δv=q(x)f(u) in $\text{R}{}^{\mathrm{N}}$, where f,g are positive and non-decreasing functions on (0,∞) satisfying the Keller–Osserman condition and we establish the existence of positive solutions that blow-up at infinity.     

A resolution method for multiobjective problems

A resolution method for multiobjective problems, based on a maximin criterion, is developed. Given the multiobjective problem Max_{Ax=b,x?0}{c_ix; i = 1,2,…,k}, we suppose that the decisor can construct, for each i, a function h_i:$\text{R}$ → $\text{R}$ (or h_i:$\text{R}$ⁿ→-$\text{R}$), such that h_i(c_i,x) is his satisfaction degree produced by the value c_ix, and we substitute the original problem by Max_{Ax=b,x?0～min_i{h_i(c_ix)}. We analize its resolution and basic properties.