Transcendental continued fractions

In a previous paper it was proven that given the continued fractions

$\text{A = a}{}_1\text{+}\text{1}\text{a}{}_2\text{+}\text{1}\text{a}{}_3\text{+… and B = b}{}_1\text{+}\text{1}\text{b}{}_2\text{+}\text{1}\text{b}{}_3\text{+…}$

where the a's and b's are positive integers, then A, B, A ± B, $\text{A}\text{B}$ and AB are irrational numbers if $\text{a}{}_{\mathrm{n}}\text{2}\text{> b}{}_{\mathrm{n}}\text{> a}{}_{\mathrm{n?1}}{}^{\mathrm{5n}}$ for all n sufficiently large, and transcendental numbers if for all n sufficiently large. Using a more direct approach it is proven in this paper that A, B, A ± B, $\text{A}\text{B}$ and AB are transcendental numbers if a_n > b_n > a_n?1^(n?1)2 for all n sufficiently large.     

Interpolation and inner maps that preserve measure

It is shown, for n ? m ? 1, that there exist inner maps Φ: Bⁿ → B^m with boundary values $\text{Φ}{}_1\text{: B}{}^{\mathrm{n}}\text{→ B}{}^{\mathrm{m}}$ such that $\text{σ}{}_{\mathrm{m}}\text{(A) = σ}{}_{\mathrm{n}}\text{(Φ}{}_1{}^{\mathrm{?1}}\text{(A))}$. where σ_n and σ_m are the Haar measures on ?Bⁿ and ?B^m, respectively, and A ? Bⁿ is an arbitrary Borel set.     

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.     

On the law of large numbers for stationary sequences

We show that if X_i is a stationary sequence for which $\text{S}{}_{\mathrm{n}}\text{B}{}_{\mathrm{n}}$ converges to a finite non zero random variable of constant sign, where S_n=X₁+X₂+?+X_n and B_n is a sequence of constants, then B_n is regularly varying with index 1. If in addition $\text{Σ}\text{P}\text{(|X}{}_1\text{|>B}{}_{\mathrm{n}}$ is finite, then $\text{E}\text{|X}{}_1\text{|}$ is finite, and if in addition to this X_i satisfies an asymptotic independence condition, $\text{E}\text{X}{}_1\text{≠ 0}$.     

Segmental solutions of Boolean equations

Davio and Deschamps have shown that the solution set, K, of a consistent Boolean equation $\text{?(x}{}_1\text{, …, x}{}_{\mathrm{n}}\text{)=0}$ over a finite Boolean algebra B may be expressed as the union of a collection of subsets of Bⁿ, each of the form {(x₁, …, x_n) | a_i≤x_i≤b_i, a_i?B, b_i?B, i = 1, …, n}. We refer to such subsets of Bⁿ as segments and to the collection as a segmental cover of K. We show that $\text{?(x}{}_1\text{, …, x}{}_{\mathrm{n}}\text{) = 1}$ is consistent if and only if ? can be expressed by one of a class of sum-of-products expressions which we call segmental formulas. The object of this paper is to relate segmental covers of K to segmental formulas for ?.     

Decomposability of symmetric multilinear functions

Let V denote a finite dimensional vector space over a field K of characteristic 0, let T_n(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let S_n(V) denote the subspace of T_n(V) whose members are the fully symmetric members of T_n(V). If $\text{L}$_n denotes the symmetric group on {1,2,…,n} then we define the projection $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{: T}{}_{\mathrm{n}}\text{(V) → S}{}_{\mathrm{n}}\text{(V)}$ by the formula , where P_σ : T_n(V) → T_n(V) is defined so that P_σ(A)(y₁,y₂,…,y_n = A(y_σ(1),y_σ(2),…,y_σ(n)) for each A?T_n(V) and y_i?V, 1 ? i ? n. If $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, then x₁?x₂? … ?x_n denotes the member of T_n(V) such that $\text{(x}{}_1\text{?x}{}_2\text{· ? ? ?x}{}_{\mathrm{n}}\text{)(y}{}_1\text{,y}{}_2\text{,…,y}{}_{\mathrm{n}}\text{) = П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{x}{}_{\mathrm{i}}\text{(y}{}_{\mathrm{i}}\text{)}$ for each y₁ ,₂,…,y_n in V, and x₁·x₂… x_n denotes $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{(x}{}_1\text{?x}{}_2\text{? … ?x}{}_{\mathrm{n}}\text{)}$. If B? S_n(V) and there exists $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, such that B = x₁·x₂…x_n, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of S_n(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.     

Embedding the complement of an oval in a projective plane of even order

A configuration $\text{D}$ with parameters (v, b, r, k) is an incidence structure ($\text{P}$, $\text{B}$, $\text{I}$), where $\text{P}$ is a set of v “points”, $\text{B}$ is a set of b “blocks” and $\text{I}$ is an “incidence relation” between points and blocks such that each point is incident with r blocks, and each blok is incident with k points. A block may be regarded as the subset of those points of $\text{p}$ with which it is incident.A regular two component pairwise balanced design with parameters

$\text{v,}\text{k}{}_1\text{,}\text{k}{}_2\text{r}{}_1\text{,}\text{r}{}_2\text{,1}$

is an incidence structure $\text{D}$ = ($\text{P}$, $\text{B}$₁ ∪ $\text{B}$₂, $\text{I}$), where $\text{D}$_i = ($\text{P}$, $\text{B}$_i,$\text{I}$) is a configuration with parameters (v, b_i, r_i, k_i), i = 1,2, b_i = vr_i/k_i, and such that any two distinct points of $\text{P}$ are contained in exactly one block of $\text{B}$₁ ∪ $\text{B}$₂. The configuration $\text{D}$_i is called the i^th component of $\text{D}$, i = 1,2.Hence by deleting the points of an oval S from a projective plane П of even order q, we obtain a regular two component pairwise balanced design for which

$\text{v=q}{}^2\text{?1, b}{}_1\text{=}\text{1}\text{2}\text{q(q?1), b}{}_2\text{=}\text{1}\text{2}\text{(q+1)(q+2), r}{}_1\text{=}\text{1}\text{2}\text{q, r}{}_2\text{=}\text{1}\text{2}\text{(q+2), k}{}_1\text{=q+1, k}{}_2\text{q?1.}$

In this paper, we investigate the converse question and prove: Given a regular two component pairwise balanced design with the above parameters, then if q ≠ 6, the design can be embedded in a projective plane of order q.     

An extremal problem for sets with applications to graph theory

Let X₁, …, X_n be n disjoint sets. For 1 ? i ? n and 1 ? j ? h let A_ij and B_ij be subsets of X_i that satisfy |A_ij| ? r_i and |B_ij| ? s_i for 1 ? i ? n, 1 ? j ? h, $\text{(∪}{}_{\mathrm{i}}\text{A}{}_{\mathrm{ij}}\text{) ∩ (∪}{}_{\mathrm{i}}\text{B}{}_{\mathrm{ij}}\text{) = ?}$ for 1 ? j ? h, $\text{(∪}{}_{\mathrm{i}}\text{A}{}_{\mathrm{ij}}\text{) ∩ (∪}{}_{\mathrm{i}}\text{B}{}_{\mathrm{il}}\text{) ≠ ?}$ for 1 ? j < l ? h. We prove that $\text{h?}\text{Π}\text{i=1}\text{n}\text{r}{}_{\mathrm{i}}\text{+s}{}_{\mathrm{i}}\text{r}{}_{\mathrm{i}}$. This result is best possible and has some interesting consequences. Its proof uses multilinear techniques (exterior algebra).     

A regularity result for singular nonlinear elliptic systems in inverse-power weighted Sobolev spaces

The compactness method to weighted spaces is extended to prove the following theorem:Let H_2,s¹(B₁) be the weighted Sobolev space on the unit ball in Rⁿ with norm

$\text{6ν6}{}^1{}_{\mathrm{2,s}}\text{=}\text{∫}{}_{\mathrm{B1}}\text{(}\text{1}\text{r}{}^{\mathrm{s}}\text{)|ν|}{}^2\text{dx + ∫}{}_{\mathrm{B1}}\text{(}\text{1}\text{r}{}^{\mathrm{s}}\text{)|Dν|}{}^2\text{dx.}$

Let n ? 2 ? s < n. Let u? [H_2,s¹(B₁) ∩ L^∞(B₁)]^N be a solution of the nonlinear elliptic system

$\text{∫}{}_{\mathrm{B1}}\text{1}\text{r}{}^{\mathrm{s}}\text{,}\text{∑}\text{i,j=1}\text{n}\text{,}\text{∑}\text{h,K=1}\text{N}\text{A}{}^{\mathrm{hK}}{}_{\mathrm{ij}}\text{(x,u) D}{}^{\mathrm{i}}\text{u}{}_{\mathrm{h}}\text{D}{}^{\mathrm{j\psi }}{}_{\mathrm{K}}\text{dx=0}$

, $\text{∨}\text{ψ}\text{?}\text{¦C}{}_0{}^1\text{(B}{}_1\text{)¦}{}^{\mathrm{N}}\text{,}\text{where}\text{¦A}{}_{\mathrm{ij}}{}^{\mathrm{hk}}\text{¦ ? L, A}{}_{\mathrm{ij}}{}^{\mathrm{hk}}$ are uniformly continuous functions of their arguments and satisfy:

$\text{|η|}{}^2\text{=}\text{∑}\text{i=1}\text{n}\text{,}\text{∑}\text{j=1}\text{N}\text{|η}{}^{\mathrm{i}}{}_{\mathrm{j}}\text{|}{}^2\text{?}\text{∑}\text{i,j=1}\text{n}\text{,}\text{1}\text{r}{}^{\mathrm{s}}\text{,}\text{∑}\text{h,K=1}\text{N}\text{A}{}^{\mathrm{hK}}{}_{\mathrm{ij}}\text{η}{}^{\mathrm{i}}{}_{\mathrm{h}}\text{η}{}^{\mathrm{i}}{}_{\mathrm{k}}\text{,}\text{?η?R}{}^{\mathrm{Nn}}$

. Then there exists an R₁, 0 < R₁ < 1, and an α, 0 < α < 1, along with a set $\text{Ω ? B}{}_1$ such that (1) $\text{H}{}_{\mathrm{n\; ?\; 2}}\text{(Ω) = 0}$, (2) Ω does not contain the origin; Ω does not contain B_R1, (3) $\text{B}{}_1\text{? Ω}$ is open, (4) u is $\text{Lip}{}_{\mathrm{\alpha }}\text{(B}{}_1\text{? Ω)}$; u is Lip_αB_R1.     

Corresponding residue systems in nonnormal extensions of prime degree

Let K₁ and K₂ be number fields and $\text{F = K}{}_1\text{? K}{}_2$. Suppose $\text{K}{}_1\text{F}$ and $\text{K}{}_2\text{F}$ are of prime degree p but are not necessarily normal. Let N₁ and N₂ be the normal closures of K₁ and K₂ over F, respectively, L = K₁K₂, N = N₁N₂, and $\text{B}$ be a prime divisor of N which divides p and is totally ramified in $\text{K}{}_1\text{F}$ and $\text{K}{}_2\text{F}$. Let ${}_{\mathrm{<mtext>N</mtext><mtext>L</mtext>}}$ be the ramification index of $\text{B}$ in $\text{N}\text{L}$, $\text{t}{}_{\mathrm{<mtext>L</mtext><mtext>F</mtext>}}$ be the total ramification number of $\text{B}$ in $\text{L}\text{F}$, and . Then $\text{M}$(K₁, K₂) is exactly divisible by $\text{B}$^M, where .     

Bounds for the variance of certain stationary point processes

For the variance of stationary renewal and alternating renewal processes N_n(·) the paper establishes upper and lower bounds of the form

$\text{?B}{}_1\text{?varN}{}_8\text{(0,x–Aλx?B}{}_2\text{(0<x<∞)}$

, where λ=EN₈(0,1), with constants A, B₁ and B₂ that depend on the first three moments of the interval distributions for the processes concerned. These results are consistent with the value of the constant A for a general stationary point process suggested by Cox in 1963 [1].     

Densité des zéros des transformés de Lévy itérés d'un mouvement brownien

The Lévy transform of a Brownian motion B is the Brownian motion $\text{B′}{}_{\mathrm{t}}\text{=∫}{}_0{}^{\mathrm{t}}\text{sgn}\text{B}{}_{\mathrm{s}}\text{d}\text{B}{}_{\mathrm{s}}$; denote by Bⁿ the Brownian motion obtained from B by iterating n times the Lévy transform. We establish that the set of all instants t such that B_tⁿ=0 for some n, is a.s. dense in the time-axis $\text{R}{}_{\mathrm{+}}$. To cite this article: M. Malric, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Some limit results for longest common subsequences

The length l_n of a longest common subsequence before time n sequences (B₁₁, B₁₂, …) (B₂₁, B₂₂, …) is the cardinality of the largest increasing set of pairs of integers {(j_1α, j_2α)}

$\text{l?}\text{j}{}_{11}\text{<}\text{j}{}_{12}\text{<·<}\text{j}{}_{11}{}_{\mathrm{n}}\text{?}\text{n}\text{l?}\text{j}{}_{21}\text{<}\text{j}{}_{22}\text{<·<}\text{j}{}_{21}{}_{\mathrm{n}}\text{?}\text{n}$

such that ?1?α?l_n, (B_1j1α=B_2j2α). If B₁ and B₂ are independent random sequences with co-ordinates i.i.d. uniform on {1, 2, …, k}, it follows from Kingman's subadditive ergodic theorem that the ratio l_n/n converges to a constant c_k a.s. A method of deriving lower bounds for the constants c_k is given, the bounds obtained improving known lower bounds, for k>2. The rate of decrease of c_k with k is shown to be no faster than 1/√k, contrasting with P{B_1i?B_2i}=1/k. Finally, an alternative method of deriving lower bounds is given and used to improve the lower bound for c₂.     

Arithmetic in generalized Artin-Schreier extensions of k(x)

If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions K_n over k(x) such that [K : k(x)] = pⁿ using the concept of Witt vectors. This is accomplished in the following way; if [β₁, β₂,…, β_n] is a Witt vector over k(x) = K₀, then the Witt equation $\text{y}{}^{\mathrm{p}}\text{?}\text{y}\text{= β}$ generates a tower of extensions through $\text{K}{}_{\mathrm{i}}\text{= K}{}_{\mathrm{i?1}}\text{(}\text{y}{}_{\mathrm{i}}\text{)}$ where $\text{y}\text{= [}\text{y}{}_1\text{,}\text{y}{}_2\text{,…,}\text{y}{}_{\mathrm{n}}\text{]}$. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in K_n. This alternate generation has the form K_i = K_i?1(y_i); y_i^p ? y_i = B_i, where, as a divisor in K_i?1, B_i has the form . In this form q is prime to Πp_j^λ_j and each λ_j is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field K_n over an algebraically closed field of constants.     

A presentation for the special automorphism group of a free group

A presentation is given for SA_n, the group of automorphisms of determinant 1 of a free group F_n of rank n. The canonical isomorphisms $\text{H}{}_2\text{(A}{}_{\mathrm{n}}\text{,}\text{Z}\text{)}\text{?}\text{H}{}_2\text{(SA}{}_{\mathrm{n}}\text{,}\text{Z}\text{)}\text{?}\text{K}{}_2\text{(}\text{Z}\text{)}$ are established for n ≥ 5, where A_n is the full group of automorphisms of F_n.     

Hilbert space representations of general discrete time stochastic processes

We show that under mild conditions the joint densities Px₁,…,x_n) of the general discrete time stochastic process X_n on $\text{pH}$ can be computed via

$\text{Px}{}_1\text{,…,x}{}_{\mathrm{n}}\text{(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{) = 6?T(x}{}_1\text{)…T(x}{}_{\mathrm{n}}\text{)6}{}^2$

where ? is in a Hilbert space $\text{pH}$, and T (x), x ? $\text{pH}$ are linear operators on $\text{pH}$. We then show how the Central Limit Theorem can easily be derived from such representations.     

On the van der Waerden conjecture and zeros of polynomials

Let

$\text{F(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{A}{}_{\mathrm{n}}\text{≠ 0,}$

and

$\text{G(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{B}{}_{\mathrm{n}}\text{≠ 0,}$

be polynomials with real zeros satisfying A_n?1 = B_n?1 = 0, and let

$\text{H(x) =}\text{∑}\text{k=o}\text{n-2}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}\text{.}$

Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H(x) are obtained. These results are related to classical results on composite polynomials.     

Some comments on the hazard gradient

For random variables T₁,…,T_n, the gradient of $\text{R(t) = ?log}\text{P}\text{{T}{}_1\text{> t}{}_1\text{,…,T}{}_{\mathrm{n}}\text{> t}{}_{\mathrm{n}}\text{}}$ is called the hazard gradient. Some properties of this multivariate version of the hazard rate are demonstrated, and some examples are given to show the usefulness of the hazard gradient in characterizing distributions or families of distributions.     

Barriers for uniformly elliptic equations and the exterior cone condition

We consider the mixed boundary value problem $\text{Au = f}\text{in}\text{Ω, B}{}_0\text{u = g}{}_0\text{in}\text{Γ}{}^{\mathrm{?}}\text{, B}{}_1\text{u = g}{}_1\text{in}\text{Γ}{}^{\mathrm{+}}$, where Ω is a bounded open subset of $\text{R}$ⁿ whose boundary Γ is divided into disjoint open subsets Γ⁺ and Γ^? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on $\text{Ω}$ and B_j, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on $\text{Γ}{}^{\mathrm{+}}$. The coefficients of the operators and Γ⁺, Γ^? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset $\text{T}$ of the reals such that, if $\text{D}{}^{\mathrm{s}}\text{= {u ? H}{}^{\mathrm{s}}\text{(Ω): Au = 0}}$ then for $\text{s ? =}\text{1}\text{2}\text{(}\text{mod}\text{1), (B}{}_0\text{,B}{}_1\text{): D}{}^{\mathrm{s}}\text{→ H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{?}}\text{) × H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>3</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{+}}\text{)}$ is a Fredholm operator if and only if s ∈$\text{T}$ . Moreover, $\text{T}$ = ?_xew$\text{T}$_x, where the sets $\text{T}$_x are determined algebraically by the coefficients of the operators at x. If n = 2, $\text{T}$_x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, $\text{T}$_x is either an open interval of length 1 or is empty; and finally, if n ? 4, $\text{T}$_x is an open interval of length 1.