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.     

Asymptotic evaluation of two families of integrals

In this paper the integrals $\text{f}{}_{\mathrm{mv}}\text{(τ) = ∝}{}_0{}^{\mathrm{\infty }}\text{exp}\text{[?(t + τ)]t}{}^{\mathrm{v}}\text{(}\text{ln}\text{t)}{}^{\mathrm{m}}\text{(t + τ)}{}^{\mathrm{?1}}\text{dt}\text{and}\text{g}{}_{\mathrm{mv}}\text{(τ) = ∝}{}_0{}^{\mathrm{\infty }}\text{exp}\text{[? ¦ ? τ ¦]t}{}^{\mathrm{v}}\text{(}\text{ln}\text{t)}{}^{\mathrm{m}}\text{(t ? τ)}{}^{\mathrm{?1}}\text{dt}$ are investigated for positive real values of the variable τ. Here, m is a nonnegative integer, v is a complex variable with Re(v) > ?1. Both integrals are related to the complex integral Φ_mγ(z) = ∝₀^∞exp[?(t ? z)]t^?γ(ln t)^m(t ? z)^?1dt with 0 ? Re(γ) < 1, the behavior of which is analyzed in detail. The results are applied to obtain asymptotic representations for f_mn(τ) and g_mn(τ), m and n both nonnegative integers, near τ = 0. The latter integrals play a role in the study of the equations of neutron transport and radiative transfer.     

A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras

Let M be a von Neumann algebra with separating and cyclic vector ξ₀. The map $\text{xξ}{}_0\text{→ x}{}^1\text{ξ}{}_0$ with x?M has a least closed extension S. Tomita proved that the isometric involution J and the positive self-adjoint operator Δ obtained from the polar decomposition $\text{S = JΔ}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of S satisfy JMJ = M′ and Δ^itMΔ^?it = M for any real t. More generally, he obtained similar results for the left von Neumann algebra of any generalized Hilbert algebra. In this paper a shorter proof of his results is given.     

On exponential bases for the Sobolev spaces over an interval

We suppose that K is a countable index set and that $\text{Λ = {λ}{}_{\mathrm{k}}\text{¦ k ? K}}$ is a sequence of distinct complex numbers such that forms a Riesz (strong) basis for L²[a, b], a < b. Let Σ = {σ₁, σ₂,…, σ_m} consist of m complex numbers not in Λ. Then, with p(λ) = Π_{k = 1}^m (λ ? σ_k), forms a Riesz (strong) bas Sobolev space H^m[a, b]. If we take σ₁, σ₂,…, σ_m to be complex numbers already in Λ, then, defining p(λ) as before, forms a Riesz (strong) basis for the space H^?m[a, b]. We also discuss the extension of these results to “generalized exponentials” tⁿe^λ_kt.     

Concrete model theory for a class of operators

The operator $\text{L?(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∝}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∝}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ acting on H=∝₀^2π⊕L²(v_t), where m and v_t, 0 ? t ? 2π are measures on [0, 2π] with m smooth and e(s, t) = exp[?∝_t^s∝_Tdv_λ(θ) dm(λ)], satisfies $\text{rank}\text{(I ? LL}{}^1\text{) =}\text{rank}\text{(I ? L}{}^1\text{L) = 1}$. It is, therefore, unitarily equivalent to a scalar Sz.-Nagy-Foia? canonical model. The purpose of this paper is to determine the model explicitly and to give a formula for the unitary equivalence.     

Canonical forms for linear systems

This paper considers canonical forms for the similarity action of Gl(n) on $\text{∑}{}_{\mathrm{n,m}}\text{={(A,B)∈}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{n}}\text{×}\text{C}{}^{\mathrm{n}}\text{·}{}^{\mathrm{m}}\text{}}$:

$\text{Gl(n×∑}{}_{\mathrm{n,m}}\text{→∑}{}_{\mathrm{n,m}}$

,

$\text{(H,(A,B))?(HAH}{}^{-1}\text{,HB)}$

Those canonical forms are obtained as an application of a more general method to select canonical elements M_c in the orbits $\text{O}{}_{\mathrm{M}}$ of a matrix group G acting on a set of matrices $\text{M}\text{?}\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$. We define a total order (?) on $\text{C}{}^{\mathrm{l}}\text{·}{}^{\mathrm{p}}$, different from the lexicographic order ^l? [0^l?x ? x <0, but $\text{0?x≠0 for x∈}\text{R}\text{]}$ and consider normalized $\text{O}{}_{\mathrm{M}}$-elements with a minimal number of parameters:

$\text{min{}\text{M}\text{?}\text{∈}\text{O}{}_{\mathrm{M}}\text{:}\text{M}\text{?}\text{normalized}}$

It is shown that the row and column echelon forms, the Jordan canonical form, and “nice” control canonical forms for reachable (A,B)-pairs have a homogeneous interpretation as such (?)-minimal orbit elements. Moreover new canonical forms for the general action (?) are determined via this method.     

Measuring flatness in linear spaces

Let L be a finite-dimensional normed linear space and let M be a compact subset of L lying on one side of a hyperplane through 0. A measure of flatness for M is the number $\text{D(M) =}\text{inf}\text{{}\text{sup}\text{f(x)}\text{f(y)}\text{: x, y ? M}}$, where the infimum is over all f in $\text{L}{}^1$ which are positive on M. Thus D(M) = 1 if M is flat, but otherwise D(M) > 1. On the other hand, let E(M) be a second measure on M defined as follows: If M is linearly independent, E(M) = 1. If M is linearly dependent, then (1) let Z be a minimal, linearly dependent subset of M; (2) partition Z into mutually exclusive subsets U = {u₁, …, u_p} and V = {v₁, …, v_q} such that there exist positive coefficients a_i and b_i for which Σ_{i = 1}^pa_iu_i = Σ_{i = 1}^qb_iv_i; (3) let $\text{r =}\text{max}\text{{}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{p}}\text{a}{}_{\mathrm{i}}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{q}}\text{b}{}_{\mathrm{i}}\text{,}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{p}}\text{b}{}_{\mathrm{i}}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{q}}\text{a}{}_{\mathrm{i}}\text{}}$; (4) let E(M) be the supremum of all ratios r which can be formed by steps (1), (2) and (3). The main result of this paper is that these two measures are the same: D(M) = E(M). This result is then used to obtain results concerning the Banach distance-coefficient between an arbitrary finite-dimensional normed linear space and Hilbert space.     

Diffusion with interactions between two types of particles and Pressureless gas equations

We construct two d-dimensional independent diffusions $\text{X}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{=a+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{a}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{,}\text{X}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{=b+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{b}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{b}}$, with the same viscosity ν≠0 and the same drift u(x,t)=(pρ_t^a(x)v₁+(1?p)ρ_t^b(x)v₂)/(pρ_t^a(x)+(1?p)ρ_t^b(x)), where ρ_t^a,ρ_t^b are respectively the density of X_t^a and X_t^b. Here $\text{a,b,v}{}_1\text{,v}{}_2\text{∈}\text{R}{}^{\mathrm{d}}$ and p∈(0,1) are given. We show that $\text{(ρ}{}_{\mathrm{t}}\text{(x)=pρ}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{(x)+(1?p)ρ}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{(x),u(x,t):}\text{t?0,}\text{x∈}\text{R}{}^{\mathrm{d}}\text{)}$ is the unique weak solution of the following pressureless gas system

such that $\text{ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pδ}{}_{\mathrm{a}}\text{+(1?p)δ}{}_{\mathrm{b}}\text{,}\text{u(x,t)ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pv}{}_1\text{δ}{}_{\mathrm{a}}\text{+(1?p)v}{}_2\text{δ}{}_{\mathrm{b}}$ as t→0+. To cite this article: A. Dermoune, S. Filali, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Two strong limit theorems for processes with independent increments

Two related almost sure limit theorems are obtained in connection with a stochastic process {ξ(t), ?∞ < t < ∞} with independent increments. The first result deals with the existence of a simultaneous stabilizing function H(t) such that $\text{(ξ(t) ? ξ(0))}\text{H(t)}\text{→ 0}$ for almost all sample functions of the process. The second result deals with a wide-sense stationary process whose random spectral distributions is ξ. It addresses the question: Under what conditions does $\text{(2T)}{}^{\mathrm{?1}}\text{∫}{}_{\mathrm{?T}}{}^{\mathrm{T}}\text{X(t)}\text{X(t + τ)}\text{dt}$ converge as T → ∞ for all τ for almost all sample functions?     

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.     

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.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

A central limit theorem for the sojourn times of strongly ergodic Markov chains

Let X_n be an irreducible aperiodic recurrent Markov chain with countable state space I and with the mean recurrence times having second moments. There is proved a global central limit theorem for the properly normalized sojourn times. More precisely, if $\text{t}{}^{\mathrm{(n}}\text{)}{}_{\mathrm{i}}\text{=Σ}{}^{\mathrm{n}}{}_{\mathrm{k=1}}\text{i}\text{?}{}_{\mathrm{i}}\text{(X}{}_{\mathrm{k}}\text{)}$, then the probability measures induced by {t⁽ⁿ⁾_i/√n?√nπ_i}_i?I(π_i being the ergotic distribution) on the Hilbert-space of square summable I-sequences converge weakly in this space to a Gaussian measure determined by a certain weak potential operator.     

Series expansions of means

A mean M(u, v) is defined to be a homogeneous symmetric function of two positive real variables satisfying min(u, v) ? M(u, v) ? max(u, v) for all u and v. Setting M(u, v) = uM(1, vu^?1) = uM(1, 1 ? t), 0 ? t < 1, we determine power series expansions in t of various generalized means, including $\text{μ}{}_{\mathrm{p}}\text{(1, 1 ? t) = [}\text{1}\text{2}\text{+}\text{(1 ? t)}{}^{\mathrm{p}}\text{2}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{, m}{}_{\mathrm{p}}\text{(u, v) = [}\text{(v}{}^{\mathrm{p\; +\; 1}}\text{? u}{}^{\mathrm{p\; +\; 1}}\text{)}\text{(v ? u)(p + 1)}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$ (Stolarsky's mean), $\text{M}{}_{\mathrm{p}}\text{(u, v) =}\text{(u}{}^{\mathrm{p}}\text{+ v}{}^{\mathrm{p}}\text{)}\text{(u}{}^{\mathrm{p?\; 1}}\text{+ v}{}^{\mathrm{p\; ?\; 1}}\text{)}$ (Lehmer's mean), $\text{E(r, s; u, v) = [}\text{r(u}{}^{\mathrm{s}}\text{? v}{}^{\mathrm{s}}\text{)}\text{s(u}{}^{\mathrm{r}}\text{? v}{}^{\mathrm{r}}\text{)}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>(s\; ?\; r)</mtext>}}$ (Leach and Sholander's mean), and $\text{G(r, s; u, v) = [}\text{(u}{}^{\mathrm{s}}\text{+ v}{}^{\mathrm{s}}\text{)}\text{(u}{}^{\mathrm{r}}\text{+ v}{}^{\mathrm{r}}\text{)}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>(s\; ?\; r)</mtext>}}$ (Gini's mean). The explicit power series coefficients and recurrence relations for these coefficients are found. Finally, applications are shown by proving a theorem that generalizes one due to Lehmer.     

On Guerra's broken replica-symmetry bound

Consider a random Hamiltonian $\text{H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{)}$ for $\text{σ}\text{→}\text{∈Σ}{}_{\mathrm{N}}\text{={0,1}}{}^{\mathrm{<mtext>N</mtext>}}\text{.}$ We assume that the family $\text{(H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{))}$ is jointly Gaussian centered and that for $\text{σ}\text{→}{}^1\text{,}\text{σ}\text{→}{}^2\text{∈Σ}{}_{\mathrm{N}}\text{,}$$\text{N}{}^{\mathrm{?1}}\text{EH}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}{}^1\text{)H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}{}^2\text{)}$ =ξ(N^?1∑_i?Nσ¹_iσ²_i) for a certain function ξ on $\text{R}$. F. Guerra proved the remarkable fact that the free energy of the system with Hamiltonian $\text{H}{}_{\mathrm{N}}\text{(}\text{σ}\text{→}\text{)+h∑}{}_{\mathrm{i?N}}\text{σ}{}_{\mathrm{i}}$ is bounded below by the free energy of the Parisi solution provided that ξ is convex on $\text{R}$. We prove that this fact remains (asymptotically) true when the function ξ is only assumed to be convex on $\text{R}{}^{\mathrm{+}}$. This covers in particular the case of the p-spin interaction model for any p. To cite this article: M. Talagrand, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

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.     

A note on the probabilistic approach to Turán's problem

The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let X_s(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(X_l), given that min(X_l) ≥ 1. The following lower bound on the variance of X_s is proved:

$\text{Var(X}{}_{\mathrm{s}}\text{)?mm}\text{n?2k}\text{s?k}\text{n}\text{s}{}^{\mathrm{?1}}\text{n}\text{k}{}^1$

, where m = |E(H)| and $\text{m}\text{= (}{}_{\mathrm{k}}{}^{\mathrm{n}}\text{) ? m}$. This implies the following: putting t(k, l) = lim_n→∞T(n, l, k)(_kⁿ)^?1 then t(k, l) ≥ T(s, l, k)((_k^s) ? 1)^?1, whenever s ≥ l > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned.     

On the length of optimal TSP circuits in sets of bounded diameter

Let V be a set of n points in R^k. Let d(V) denote the diameter of V, and l(V) denote the length of the shortest circuit which passes through all the points of V. (Such a circuit is an “optimal TSP circuit”.) l^k(n) are the extremal values of l(V) defined by l^k(n)=max{l(V)|V∈V_n^k}, where V_n^k={V|V?R^k,|V|=n, d(V)=1}. A set V∈V_n^k is “longest” if l(V)=l^k(n). In this paper, first some geometrical properties of longest sets in R² are studied which are used to obtain l²(n) for small n′s, and then asymptotic bounds on l^k(n) are derived. Let δ(V) denote the minimal distance between a pair of points in V, and let: δ^k(n)=max{δ(V)|V∈V_n^k}. It is easily observed that $\text{δ}{}^{\mathrm{k}}\text{(n)=O(n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{)}$. Hence, $\text{c}{}_{\mathrm{k}}\text{=}\text{lim sup}{}_{\mathrm{n\to \infty }}\text{δ}{}^{\mathrm{k}}\text{(n)n}{}^{\mathrm{<mtext>1</mtext><mtext>k</mtext>}}$ exists. It is shown that for all $\text{n, c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>?1</mtext><mtext>k</mtext>}}\text{≤δ}{}^{\mathrm{k}}\text{(n)}$, and hence, for all $\text{n, l}{}^{\mathrm{k}}\text{(n)≥ c}{}_{\mathrm{k}}\text{n}{}^{\mathrm{<mtext>1?1</mtext><mtext>k</mtext>}}$. For k=2, this implies that $\text{l}{}^2\text{(n)≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, which generalizes an observation of Fejes-Toth that $\text{lim}{}_{\mathrm{n\to \infty }}\text{l}{}^2\text{(n)n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{≥(}\text{π}{}^2\text{12}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$. It is also shown that $\text{l}{}^{\mathrm{k}}\text{(n) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]nδ}{}^{\mathrm{k}}\text{(n) + o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{) ≤ [}\text{(3?√3)k}\text{(k?1)}\text{]n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{+ o(n}{}^{\mathrm{<mtext>1?</mtext><mtext>1</mtext><mtext>k</mtext>}}\text{)}$. The above upper bound is used to improve related results on longest sets in k-dimensional unit cubes obtained by Few (Mathematika2 (1955), 141–144) for almost all k′s. For k=2, Few's technique is used to show that $\text{l}{}^2\text{(n)≤(}\text{πn}\text{2}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ O(1)}$.     

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

Branching brownian motion with absorption

We consider a branching diffusion {Z_t}_t?0 in which particles move during their life time according to a Brownian motion with drift -μ and variance coefficient σ², and in which each particle which enters the negative half line is instantaneously removed from the population. If particles die with probability c dt+o(dt) in [t,t+dt] and if the mean number of offspring per particle is m>1, then Z_t dies out w.p.l. if $\text{μ?μ}{}_0\text{≡{2σ}{}^2\text{c(m?1)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. If μ<μ₀, then itZ_t grows exponentially with positive probability. Our main concern here is with the critical case where μ=μ₀. Even though $\text{E}\text{{}\text{Z}{}_{\mathrm{T}}\text{}∽}\text{const.}\text{T}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>2</mtext>}}$ in this case, we find that $\text{P}\text{{}\text{Z}{}_{\mathrm{T}}\text{>0}}$ is only exp$\text{{–}\text{const.}\text{T}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{+0(}\text{log}\text{T)}{}^2\text{}}$, and conditionally on {Z_T>0} there are with high probability much fewer particles alive at time T than $\text{E}\text{{Z}{}_{\mathrm{T}}\text{|Z}{}_{\mathrm{T}}\text{0}}$.