Coding theory in white Gaussian channel with feedback

The message m = {m(t)} is a Gaussian process that is to be transmitted through the white Gaussian channel with feedback: $\text{Y(t) = ∫}{}_0{}^{\mathrm{t}}\text{F(s, Y}{}_0{}^{\mathrm{s}}\text{, m)ds + W(t)}$. Under the average power constraint, $\text{E[F}{}^2\text{(s, Y}{}_0{}^{\mathrm{s}}\text{, m)] ≤ P}{}_0$, we construct causally the optimal coding, in the sense that the mutual information I_t(m, Y) between the message m and the channel output Y (up to t) is maximized. The optimal coding is presented by $\text{Y(t) = ∫}{}_0{}^{\mathrm{t}}\text{A(s)[m(s) ?}\text{m}\text{?}\text{(s)] ds + W(t)}$, where $\text{m}\text{?}\text{(s) = E[m(s) ¦ Y(u), 0 ≤ u ≤ s]}$ and A(s) is a positive function such that $\text{A}{}^2\text{(s) E |m(s) ?}\text{m}\text{?}\text{(s)|}{}^2\text{= P}{}_0$.     

Strong approximation theorems for density dependent Markov chains

A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form

$\text{X}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lY}{}_1\text{N ∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X}{}_{\mathrm{N}}\text{(s))ds}$

where $\text{l∈}\text{Z}{}^{\mathrm{t}}$, the Y₁ are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution).The corresponding deterministic model, satisfies

$\text{X(t)=x}{}_0\text{+ ∫}{}^{\mathrm{t}}{}_0\text{∑ lf}{}_1\text{(X(s))ds}$

Under very general conditions lim_N→∞X_N(t)=X(t) a.s. The process X_N(t) is compared to the diffusion processes given by

$\text{Z}{}_{\mathrm{N}}\text{(t)=x}{}_0\text{+∑}\text{1}\text{N}\text{lB}{}_1\text{N∫}{}^{\mathrm{t}}{}_0\text{ft(Z}{}_{\mathrm{N}}\text{(s))ds}$

and

$\text{V(t)=∑ l∫}{}^{\mathrm{t}}{}_0\text{f}{}_1\text{(X(s))}\text{d}\text{W}\text{?}{}_1\text{+∫}{}^{\mathrm{t}}{}_0\text{?F(X(s))·V(s)}\text{ds.}$

Under conditions satisfied by most of the applied probability models, it is shown that X_N,Z_N and V can be constructed on the same sample space in such a way that

$\text{X}{}_{\mathrm{N}}\text{(t)=Z}{}_{\mathrm{N}}\text{(t)+O}\text{logN}\text{N}$

and

$\text{N}\text{(X}{}_{\mathrm{N}}\text{(t)?X(t))=V(t)+O}\text{log N}\text{N}$

     

The asymptotic equivalence of Bayes and maximum likelihood estimation

Let (X, $\text{A}$) be a measurable space, Θ ? $\text{R}$ an open interval and P_Ω ∥ $\text{A}$, Ω ? Θ, a family of probability measures fulfilling certain regularity conditions. Let $\text{Ω}{}_{\mathrm{n}}$ be the maximum likelihood estimate for the sample size n. Let λ be a prior distribution on Θ and let $\text{R}{}_{\mathrm{<mtext>n,</mtext><mtext>x</mtext>}}$ be the posterior distribution for the sample size n given $\text{x}\text{? X}{}^{\mathrm{n}}$. $\text{L:}\text{Θ}\text{× Θ →}\text{R}$ denotes a loss function fulfilling certain regularity conditions and T_n denotes the Bayes estimate relative to λ and L for the sample size n. It is proved that for every compact K ? Θ there exists c_K ≥ 0 such that

This theorem improves results of Bickel and Yahav [3], and Ibragimov and Has'minskii [4], as far as the speed of convergence is concerned.     

Representation of linear functionals on a class of normed Köthe spaces

Under the condition that $\text{L}{}_{\mathrm{\theta ,s}}{}^1$, (the set of singular functionals on a normed Köthe space L_θ) is an abstract L-space, it is proved in this paper that there exists a set of purely finitely additive measures $\text{M}$_θ such that $\text{L}{}_{\mathrm{\theta ,s}}{}^1$? holds. It follows that $\text{L}{}_{\mathrm{\theta ,s}}{}^1$ is an abstract L-space if and only if $\text{L}{}_{\mathrm{\theta ,s}}{}^1$ is Riesz isomorphic and isometric with a band in $\text{L}{}_{\mathrm{\infty ,s}}{}^1$.     

A maximization problem and its application to canonical correlation

Let Σ be an n × n positive definite matrix with eigenvalues λ₁ ≥ λ₂ ≥ … ≥ λ_n > 0 and let M = {x, y | x?Rⁿ, y?Rⁿ, x ≠ 0, y ≠ 0, x′y = 0}. Then for x, y in M, we have that $\text{x′Σy}\text{(x′Σxy′Σy)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$ and the inequality is sharp. If

$\text{∑=}\text{∑}{}_{11}\text{∑}{}_{12}\text{∑}{}_{21}\text{∑}{}_{22}$

is a partitioning of Σ, let θ₁ be the largest canonical correlation coefficient. The above result yields $\text{θ}{}_1\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$.     

On the fair coin tossing process

Let Ω = {1, 0} and for each integer n ≥ 1 let $\text{Ω}{}_{\mathrm{n}}\text{= Ω × Ω × … × Ω}$ (n-tuple) and $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}\text{= {(a}{}_1\text{, a}{}_2\text{, …, a}{}_{\mathrm{n}}\text{)|(a}{}_1\text{, a}{}_2\text{, … , a}{}_{\mathrm{n}}\text{) ? Ω}{}_{\mathrm{n}}\text{and}\text{Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{i}}\text{= k}}$ for all k = 0,1,…,n. Let {Y_m}_m≥1 be a sequence of i.i.d. random variables such that $\text{P(Y}{}_1\text{= 0) = P(Y}{}_1\text{= 1) =}\text{1}\text{2}$. For each A in $\text{Ω}{}_{\mathrm{n}}$, let T_A be the first occurrence time of A with respect to the stochastic process {Y_m}_m≥1. R. Chen and A.Zame (1979, J. Multivariate Anal. 9, 150–157) prove that if n ≥ 3, then for each element A in $\text{Ω}{}_{\mathrm{n}}$, there is an element B in $\text{Ω}{}_{\mathrm{n}}$ such that the probability that T_B is less than T_A is greater than $\text{1}\text{2}$. This result is sharpened as follows: (I) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$, there is an element B also in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$ such that the probability that T_B is less than T_A is greater than $\text{1}\text{2}$; (II) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A = (a₁, a₂,…,a_n) in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$, there is an element C also in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$ such that the probability that T_A is less than T_C is greater than $\text{1}\text{2}$ if n ≠ 2m or n = 2m but a_i = a_{i + 1} for some 1 ≤ i ≤ n?1. These new results provide us with a better and deeper understanding of the fair coin tossing process.     

W1,p-quasiconvexity and variational problems for multiple integrals

Variational problems for the multiple integral $\text{I}{}_{\mathrm{\Omega }}\text{(u) = ∝}{}_{\mathrm{\Omega }}\text{g(▽u(x))dx}$, where $\text{Ω?R}{}^{\mathrm{m}}$ and $\text{u:Ω→R}{}^{\mathrm{n}}$ are studied. A new condition on g, called W^1,p-quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of C. B. Morrey, it being shown in particular to be necessary for sequential weak lower semicontinuity of $\text{I}{}_{\mathrm{\Omega }}$ in $\text{W}{}^{\mathrm{1,p}}\text{(Ω;R}{}^{\mathrm{n}}\text{)}$ and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in $\text{W}{}^{\mathrm{1,p}}\text{(Ω;R}{}^{\mathrm{n}}\text{)}$, p ? n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses.     

On the Γ-convergence of matrix fields related to the adjugate Jacobian

Adjugate Jacobians of mappings $\text{f}{}_{\mathrm{j}}\text{:Ω?}\text{R}{}^2\text{→}\text{R}{}^2$ can be represented in terms of Jacobian matrices: $\text{adj}\text{Df}{}_{\mathrm{j}}\text{=}\text{A}{}_{\mathrm{j}}\text{(x)Df}{}^{\mathrm{t}}{}_{\mathrm{j}}$, for $\text{j=1,2,…}$, by mean of symmetric matrix fields $\text{A}{}_{\mathrm{j}}\text{(x)}$ with $\text{det}\text{A}{}_{\mathrm{j}}\text{(x)=1}$ a.e. Under suitable conditions, we prove that $\text{Df}{}_{\mathrm{j}}\text{?Df}$ weakly in $\text{L}{}^1{}_{\mathrm{<mtext>loc</mtext>}}\text{(Ω;}\text{R}{}^2\text{)}$ if and only if $\text{A}{}_{\mathrm{j}}\text{(x)}$Γ-converges to a matrix $\text{A}\text{(x)}$ with $\text{det}\text{A}\text{(x)=1}$ satisfying $\text{adj}\text{Df=}\text{A}\text{(x)Df}{}^{\mathrm{t}}$. To cite this article: C. Sbordone, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians

Abstract connections between integral kernels of positivity preserving semigroups and suitable L^p contractivity properties are established. Then these questions are studied for the semigroups generated by $\text{?Δ + V}\text{and}\text{H}{}_{\mathrm{\Omega }}$, the Dirichlet Laplacian for an open, connected region Ω. As an application under a suitable hypothesis, Sobolev estimates are proved valid up to ?Ω, of the form $\text{¦η(x)¦? c?}{}_0\text{(x) ∥H}{}_{\mathrm{\Omega }}{}^{\mathrm{k}}\text{η∥}{}_2$, where ?₀ is the unique positive L² eigenfunction of $\text{H}{}_{\mathrm{\Omega }}$.     

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).     

Extension of a Riemannian metric with vanishing curvature

Let $\text{Ω}$ be a connected and simply-connected open subset of $\text{R}{}^{\mathrm{n}}$ such that the geodesic distance in $\text{Ω}$ is equivalent to the Euclidean distance. Let there be given a Riemannian metric (g_ij) of class $\text{C}{}^2$ and of vanishing curvature in $\text{Ω}$, such that the functions g_ij and their partial derivatives of order $\text{?}\text{2}$ have continuous extensions to $\text{Ω}$. Then there exists a connected open subset $\text{Ω}$ of $\text{R}{}^{\mathrm{n}}$ containing $\text{Ω}$ and a Riemannian metric $\text{(}\text{g}\text{?}{}_{\mathrm{ij}}\text{)}$ of class $\text{C}{}^2$ and of vanishing curvature in $\text{Ω}$ that extends the metric (g_ij). To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations

General second-order parabolic and hyperbolic equations on a bounded domain are considered. The input is applied in the Neumann or mixed boundary condition and is expressed as a finite-dimensional feedback. In the parabolic case, the feedback acts, in particular, on the Dirichlet trace of the solution: here it is shown that the resulting closed loop system defines a (feedback) C₀-semigroup on $\text{L}{}_2\text{(Ω)}$ (in fact, on $\text{H}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext><mtext>\; ?\; 2?</mtext>}}\text{(Ω), ρ > 0}$), that is both analytic and compact for positive times, and whose generator has compact resolvent. In the hyperbolic case, the feedback acts on the position vector only, or on its Dirichlet trace in a special case: here a similar result is established regarding the existence of a feedback C₀-cosine operator. Moreover, an example is given, which hints that the class of prescribed feedbacks acting on the Dirichlet trace cannot be substantially enlarged. Functional analytic techniques are employed, in particular perturbation theory. However, perturbation theory for the original variable fails on $\text{L}{}_2\text{(Ω)}$, the space in which the final result is sought. Therefore, our approach employs perturbation theory, after a suitable continuous extension, on the larger space $\text{[H}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>\; +\; 2?</mtext>}}\text{(Ω)]′}$.     

Sur l'équation divu=fOn the equation divu=f

The main result is the following. Let $\text{Ω}$ be a bounded Lipschitz domain in $\text{R}{}^{\mathrm{d}}$, d?2. Then for every $\text{f∈}\text{L}{}^{\mathrm{d}}\text{(Ω)}$ with ∫f=0, there exists a solution $\text{u∈}\text{C}{}^0\text{(}\text{Ω}\text{)∩}\text{W}{}^{\mathrm{1,d}}\text{(Ω)}$ of the equation divu=f in $\text{Ω}$, satisfying in addition u=0 on $\text{?Ω}$ and the estimate

where C depends only on $\text{Ω}$. However one cannot choose u depending linearly on f. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 973–976.     

Approximation to complex ξ from within a wedge

It is shown that for any complex ξ ? $\text{Q}$[i] and any angles θ₁ < θ₂ ≤ θ₁ + π there exists a constant C such that $\text{|ξ ?}\text{P}\text{Q}\text{| <}\text{C}\text{|Q|}{}^2$ and $\text{θ}{}_1\text{<}\text{arg}\text{(ξ ?}\text{P}\text{Q}\text{) < θ}{}_2$.     

Global existence for a delay differential equation

An elastic-plastic bar with simply connected cross section Q is clamped at the bottom and given a twist at the top. The stress function u, at a prescribed cross section, is then the solution of the variational inequality (0.1) $\text{min}{}_{\mathrm{<mtext>v</mtext><mtext>?</mtext><mtext>K</mtext>}}\text{{∝}{}_{\mathrm{Q}}\text{¦}\text{▽}\text{v¦}{}^2\text{? 2θ}{}_1\text{∝}{}_{\mathrm{Q}}\text{v} = ∝}{}_{\mathrm{Q}}\text{¦}\text{▽}\text{u¦}{}^2\text{? 2θ}{}_1\text{∝}{}_{\mathrm{Q}}\text{u, u}\text{?}\text{K,}\text{where}\text{(0.2) K = {v}\text{?}\text{H}{}_0{}^1\text{(Q), ¦}\text{▽}\text{v¦ ? 1}\text{a.e.}\text{}}\text{and}\text{θ}{}_1$ is equal to the angle of the twist (after normalizing the units). Introducing the Lagrange multiplier λ_θ1, the unloading problem consists in solving the variational inequality (0.3) is the twisting angle for the unloaded bar; θ₂ < θ₁. Let (0.4) $\text{K}{}^1\text{= {v}\text{?}\text{H}{}_0{}^1\text{(Q), ?d(x) ? v(x) ? d(x)},}\text{where}\text{d(x) =}\text{dist}\text{.(x, ?Q)}$, and denote by $\text{u}{}^1\text{, w}{}^1$ the solutions of (0.1), (0.3), respectively, when K is replaced by $\text{K}{}^1$. The following results are well known for the loading problem (0.1):(0.5) $\text{u = u}{}^1$; (0.6) the plastic set $\text{P = (X}\text{?}\text{Q}\text{?}\text{; ¦}\text{▽}\text{u(x)¦ = 1}}$ is connected to the boundary. In this paper we show that, in general, (0.7) $\text{w ≠ w}{}^1$; (0.8) the plastic set $\text{P}\text{?}\text{= {x}\text{?}\text{Q}\text{?}\text{; ¦}\text{▽}\text{w(x)¦ = 1}}$ is not connected to the boundary. That is, we construct domains Q for which (0.7) and (0.8) hold for a suitable choice of θ₁, θ₂.     

Sobolev inequalities with remainder terms

The usual Sobolev inequality in $\text{R}$ⁿ, n ? 3, asserts that , with S_n being the sharp constant. This paper is concerned, instead, with functions restricted to bounded domains Ω ? $\text{R}$ⁿ. Two kinds of inequalities are established: (i) If ? = 0 on ?Ω, then with $\text{p =}\text{2}{}^1\text{2}$ and with $\text{q =}\text{n}\text{(n ? 1)}$. (ii) If ? ≠ 0 on ?Ω, then with $\text{q =}\text{2(n ? 1)}\text{(n ? 2)}$. Some further results and open problems in this area are also presented.     

On a conjecture by D. Ž. D̄okovic

It is shown that if $\text{A?Ω}{}_{\mathrm{n}}\text{?{J}{}_{\mathrm{n}}\text{}}$ satisfies

$\text{nkσ}{}_{\mathrm{k}}\text{(A)?(n?k+1)}{}^2\text{σ}{}_{\mathrm{k?1}}\text{(A)}$

$\text{(k=1,2,…,n)}$

, where σ_k(A) denotes the sum of all kth order subpermanent of A, then Per[λJ_n+(1?λ)A] is strictly decreasing in the interval 0<λ<1.     

The KMS condition and spectral passivity in group duality

Let (A, G, α) be a C¹-dynamical system, where G is abelian, and let φ be an invariant state. Suppose that there is a neighbourhood Ω of the identity in $\text{G}\text{?}$ and a finite constant κ such that $\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}{}^1\text{x}{}_{\mathrm{i}}\text{) ? κ}\text{Π}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{φ(x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i}}{}^1\text{)}$ whenever x_i lies in a spectral subspace $\text{R}{}^{\mathrm{\alpha }}\text{(Ω}{}_{\mathrm{i}}\text{)}$, where $\text{Ω}{}_1\text{+ … + Ω}{}_{\mathrm{n}}\text{? Ω}$. This condition of complete spectral passivity, together with self-adjointness of the left kernel of φ, ensures that φ satisfies the KMS condition for some one-parameter subgroup of G.     

On a class of stopping times for M-estimators

For a given score function ψ = ψ(x, θ), let θ_n be Huber's M-estimator for an unknown population parameter θ. Under some mild smoothness assumptions it is known that $\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(θ}{}_{\mathrm{n}}\text{? θ)}$ is asymptotically normal. In this paper the stopping times $\text{τ}{}_{\mathrm{c}}\text{(m) =}\text{inf}\text{{n ≥ m: n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{|θ}{}_{\mathrm{n}}\text{? θ | > c }}$ associated with the sequence of confidence intervals for θ are investigated. A useful representation of M-estimators is derived, which is also appropriate for proving laws of the iterated logarithm and Donskertype invariance principles for (π_n)_n.