On the exponential map of classical Lie groups and linear differential systems with periodic coefficients

Let X and Y be Banach spaces, $\text{?: X → Y}{}^1\text{and}\text{P: X → Y}$; P is said to be strongly ?-accretive if $\text{〈Px ? Py, ?(x ? y)〉 ? c ¦|x ? y¦|}{}^2$ for some c > 0 and each x,y?X. These mappings constitute a generalization simultaneously of monotone mappings ($\text{when}\text{Y = X}{}^1$) and accretive mappings (when Y = X). By applying a theorem of 1. Ekeland, it is shown that a localized class of these mappings must be surjective under appropriate geometric assumptions on $\text{Y}{}^1$ and continuity assumptions on P. The results generalize two theorems of F. E. Browder and the proofs further refine the methodology for dealing with such mappings.     

An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

An algorithm for the electromagnetic scattering due to an axially symmetric body with an impedance boundary condition

Let B be a body in R³ and let S denote the boundary of B. The surface S is described by $\text{S = {(x, y, z): (x}{}^2\text{+ y}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= f(z), ?1 ? z ? 1}}$, where f is an analytic function that is real and positive on (?1, 1) and f(±1) = 0. An algorithm is described for computing the scattered field due to a plane wave incident field, under Leontovich boundary conditions. The Galerkin method of solution used here leads to a block diagonal matrix involving 2M + 1 blocks, each block being of order 2(2N + 1). If, e.g., N = O(M²), the computed scattered field is accurate to within an error bounded by $\text{Ce}{}^{\mathrm{<mtext>?cN</mtext><mtext>1</mtext><mtext>2</mtext>}}$, where C and c are positive constants depending only on f.     

Integral transforms of analytic functions on abstract Wiener spaces

Let (H, B) be an abstract Wiener pair and p_t the Wiener measure with variance t. Let $\text{E}$_a be the class of exponential type analytic functions defined on the complexification [B] of B. For each pair of nonzero complex numbers α, β and f ? $\text{E}$_a, we define

$\text{F}{}_{\mathrm{\alpha ,\beta }}\text{f(y)=}\text{∫}\text{B}\text{f(αx+βy)p}{}_1\text{(dx) (y ?[B]).}$

We show that the inverse $\text{F}$_α,β^?1 exists and there exist two nonzero complex numbers α′,β′ such that

. Clearly, the Fourier-Wiener transform, the Fourier-Feynman transform, and the Gauss transform are special cases of $\text{F}$_α,β. Finally, we apply the transform to investigate the existence of solutions for the differential equations associated with the operator $\text{N}$_c, where c is a nonzero complex number and $\text{N}$_c is defined by

$\text{N}{}_{\mathrm{c}}\text{u(x)=?Δu(x)+c(x,Du(x))}$

where Δ is the Laplacian and (·, ·) is the $\text{B-B}{}^1$ pairing. We show that the solutions can be represented as integrals with respect to the Wiener measure.     

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 expected eigenvalue distribution of a large regular graph

Let K₁, K₂,... be a sequence of regular graphs with degree v?2 such that n(X_i)→∞ and ck(X_i)/n(X_i)→0 as i∞ for each k?3, where n(X_i) is the order of X_i, and c_k(X_i) is the number of k- cycles in X₁. We determine the limiting probability density f(x) for the eigenvalues of X>_i as i→∞. It turns out that

$\text{f(x)=}\text{v}\text{4(v?1)?v}{}^2\text{2π(v}{}^2\text{?x}{}^2\text{)}{}_0$

for ?x??2$\text{v-1}$, otherwise It is further shown that f(x) is the expected eigenvalue distribution for every large randomly chosen labeled regular graph with degree v.     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.     

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 $\text{(}\text{1}\text{2}\text{Δ}\text{+ q)φ = 0}\text{in}\text{D, φ = ?}\text{on}\text{?D}$.     

Norm properties of C-numerical radii

Given n×n complex matrices A, C, the C-numerical radius of A is the nonnegative quantity

$\text{r}{}_{\mathrm{c}}\text{(A)≡ma{|tr(CU}{}^1\text{AU)|:U unitary}}$

. For C=diag(1,0,…,0) it reduces to the classical numerical radius $\text{r(A)= max{|x}{}^1\text{Ax|:x}{}^1\text{x=1}}$. We show that r_c is a generalized matrix norm if and only if C is nonscalar and trC≠0. Next, we consider an arbitrary generalized matrix norm and characterize all constants v?0 for which vN is multiplicative. A technique to obtain such v is then applied to C-numerical radii with Hermitian C. In particular we find that vr is a matrix norm if and only if v?4.     

Oscillation theory at a finite singularity

It is well known that every weak solution (with boundary values 0) of a semilinear equation $\text{Au + ?(x, u) = g}$ is a regular solution if ? fulfils the growth condition $\text{(1) ¦?(x, u)¦? c ¦u¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; ?\; ?</mtext>}}$. Here 2m is the order of A. In this paper we weaken this condition to $\text{c ¦u ¦}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1</mtext>}}\text{? ?(x, u)u ? ?c ¦u ?}{}^{\mathrm{<mtext>(n\; +\; 2m)</mtext><mtext>(n\; ?\; 2m)</mtext><mtext>\; +\; 1\; ?\; ?</mtext>}}$. This requires a technique completely different from that which may be applied in case (1).     

Principle of inclusion-exclusion on semilattices

We introduce an enumeration theorem under lattice action. Let L be a finite semilattice and Ω be a nonempty set. Let $\text{f: L → P(Ω)}$ be a map satisfying f(x ? y) ? f(x) ∩ f(y), where ? and $\text{P}$(Ω) mean “join” and the power set of Ω, respectively. Then

$\text{m}\text{∪}\text{x?L}\text{?(x)}\text{=}\text{Σ}\text{c?C}\text{(?1)}{}^{\mathrm{l(c)}}\text{m}\text{∩}\text{x?c}\text{?(x)}$

, where C is the set of all chains in L and l(c) denotes the length of a chain c. Also the theorem can be dualized. Furthermore, we describe two applications of the theorem to a Boolean lattice of sets and a partition lattice of a set.     

Integral inequalities for generalized concave or convex functions

Let ψ be convex with respect to ?, B a convex body in Rⁿ and f a positive concave function on B. A well-known result by Berwald states that $\text{1}\text{¦B¦}\text{∝}{}_{\mathrm{B}}\text{ψ(f(x)) dx ? n ∝}{}_0{}^1\text{ψ(ξt)(1 ? t)}{}^{\mathrm{n\; ?\; 1)}}\text{dt}$ (1) if ξ is chosen such that $\text{1}\text{¦B¦}\text{∝}{}_{\mathrm{B}}\text{?(f(x)) dx = n ∝}{}_0{}^1\text{?(ξt)(1 ? t)}{}^{\mathrm{n\; ?\; 1)}}\text{dt}$.The main purpose in this paper is to characterize those functions f : B → R₊ such that (1) holds.     

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 $\text{b(g;s,k:p}{}^{\mathrm{m}}\text{) {}{}^0{}_{\mathrm{\pm 1}}\text{(}\text{mod}\text{p)}$.     

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.     

Positive definite distributions on K\GK

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space $\text{C}$¹(G). We show that the corresponding property is no longer true for the space of double cosets $\text{K}\text{G}\text{K}$. If G is of real-rank 1, we construct liner functionals $\text{T}{}_{\mathrm{p}}\text{? (C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{))′}$ for each p, 0 < p ? 2, such that $\text{T}{}_{\mathrm{p}}\text{(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$ but T_p does not extend to a continuous functional on $\text{C}{}^{\mathrm{p}}\text{(K}\text{G}\text{K}\text{)}$. In particular, if p ? 1, T_v does not extend to a continuous functional on $\text{C}{}^1\text{(K}\text{G}\text{K}\text{)}$. We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that $\text{T(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$. The main tool used is a theorem of Trombi-Varadarajan.     

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).     

The inverse balayage problem for Markov chains,part II

This paper continues the study of the inverse balayage problem for Markov chains. Let X be a Markov chain with state space $\text{A ? B}{}_2$, let v be a probability measure on B₂ and let M(v) consist of probability measures μ on A whose X-balayage onto B₂ is v. The faces of the compact, convex set M(v) are characterized. For fixed μ?M(v) the set M(μ,v) of the measures ? of the form $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S) ? ·}}$, where S is a randomized stopping time, is analyzed in detail. In particular, its extreme points and edge are explicitly identified. A naturally defined reversed chain X, for which v is an inverse balayage of μ, is introduced and the relation between X and X^ is studied. The question of which $\text{? ? M(μ, v)}$ admit a natural stopping time $\text{S}{}_{\mathrm{?}}$ of X (not involving an independent randomization) such that $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S}{}_{\mathrm{?}}\text{) ? ·}}$, is shown to have rather different answers in discrete and continuous time. Illustrative examples are presented.     

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.     

Générateurs et semi-groupes dans les espaces de Banach uniformement lisses

Let C be a closed convex subset of a uniformly smooth Banach space. Let S(t) : C → C be a semigroup of type ω. Then the generator A₀ of S(t) has a dense domain in C. Moreover there is is an operator A such that: (i) A₀ ? A and $\text{D(A)}\text{= C, (ii) A + gwI}$ accretive, (iii) R(I + λA) ? C for λ > 0 and ωλ < 1, (iv) $\text{S(t)x =}\text{lim}{}_{\mathrm{n\; \to \; \infty }}\text{(I + (}\text{t}\text{n}\text{)A)}{}^{\mathrm{?n}}\text{x}$ for every x?C.