Construction of a recurrence relation for modified moments

In this paper we are constructing a recurrence relation of the form

$\text{∑}\text{i=0}\text{r}\text{ω}{}_{\mathrm{i}}\text{(k)m}{}_{\mathrm{k+i}}{}^{\mathrm{\{\lambda \}}}\text{[f] = ω(k)}$

for integrals (called modified moments)

$\text{m}{}_{\mathrm{k}}{}^{\mathrm{\{\lambda \}}}\text{[f]df=}\text{∫}\text{?1}\text{1}\text{f(x)C}{}_{\mathrm{k}}{}^{\mathrm{(\lambda )}}\text{(x)dx (k = 0,1,…)}$

in which C_k^(λ) is the k-th Gegenbauer polynomial of order $\text{λ(λ > ?}\text{1}\text{2}\text{)}$, and f is a function satisfying the differential equation

$\text{∑}\text{i=0}\text{n}\text{P}{}_{\mathrm{i}}\text{(x)f}{}^{\mathrm{(i)}}\text{(x) = p(x) (?1?x?1)}$

of order n, where p₀, p₁, …, p_n ? 0 are polynomials, and m_k^〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense.     

On the probability that k positive integers are relatively prime II

Given a set S of positive integers let $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t)}$ denote the number of k-tuples 〈m₁, …, m_k〉 for which $\text{m}{}_{\mathrm{i}}\text{∈ S ? [1, t]}$ and (m₁, …, m_k) = 1. Also let $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n)}$ denote the probability that k integers, chosen at random from $\text{S ? [1, n]}$, are relatively prime. It is shown that if P = {p₁, …, p_r} is a finite set of primes and S = {m : (m, p₁ … p_r) = 1}, then $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t) = (td(S))}{}^{\mathrm{k}}\text{Π}{}_{\mathrm{\nu ?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) + O(t}{}^{\mathrm{k?1}}\text{)}$ if k ≥ 3 and $\text{Z}{}_2{}^{\mathrm{S}}\text{(t) = (td(S))}{}^2\text{Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^2\text{) + O(t}\text{log}\text{t)}$ where d(S) denotes the natural density of S. From this result it follows immediately that $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) = (ζ(k))}{}^{\mathrm{?1}}\text{Π}{}_{\mathrm{p\in P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$ as n → ∞. This result generalizes an earlier result of the author's where $\text{P = ?}$ and S is then the whole set of positive integers. It is also shown that if S = {p₁^x₁ … p_r^x_r : x_i = 0, 1, 2,…}, then $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → 0}$ as n → ∞.     

Arnold's canonical matrices and the asymptotic simplification of ordinary differential equations

Let A(x,ε) be an n×n matrix function holomorphic for |x|?x₀, 0<ε?ε₀, and possessing, uniformly in x, an asymptotic expansion $\text{A(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{A}{}_{\mathrm{r}}\text{(x) ε}{}^{\mathrm{r}}$, as ε→0⁺. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion $\text{P(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{P}{}_{\mathrm{r}}\text{(x)ε}{}^{\mathrm{r}}$, as ε→0⁺, is constructed, such that the transformation y = P(x,ε)z takes the differential equation $\text{ε}{}^{\mathrm{h}}\text{dy}\text{dx}\text{= A(x,ε)y,h}$ a positive integer, into $\text{ε}{}^{\mathrm{h}}\text{dz}\text{dx}\text{= B(x,ε)z}$, where B(x,ε) is asymptotically equal, to all orders, to a matrix in a canonical form for holomorphic matrices due to V.I. Arnold.     

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

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

     

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 $\text{A}{}^2\text{(s) E |m(s) ?}\text{m}\text{?}\text{(s)|}{}^2\text{= P}{}_0$.     

On matrix functions which commute with their derivative

We improve several results published from 1950 up to 1982 on matrix functions commuting with their derivative, and establish two results of general interest. The first one gives a condition for a finite-dimensional vector subspace E(t) of a normed space not to depend on t, when t varies in a normed space. The second one asserts that if A is a matrix function, defined on a set ?, of the form A(t)= U diag(B₁(t),…,B_p(t)) U^-1, t ∈ ?, and if each matrix function B_k has the polynomial form

$\text{B}{}_{\mathrm{k}}\text{(t)=}\text{∑}\text{i=0}\text{α}{}_{\mathrm{k}}\text{f}{}_{\mathrm{ki}}\text{(t)C}{}_{\mathrm{k}}{}^{\mathrm{i}}\text{, t∈ ?, k∈{1,…,p}}$

then A itself has the polynomial form

$\text{A(t)=}\text{∑}\text{i=0}\text{d?1}\text{f}{}_{\mathrm{i}}\text{(t)C}{}^{\mathrm{i}}\text{,t∈?}$

, where

$\text{d=}\text{∑}\text{k=1}\text{p}\text{d}{}_{\mathrm{k}}$

, d_k being the degree of the minimal polynomial of the matrix C_k, for every k ∈ {1,…,p}.     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.     

Factorization of fredholm operators on analytic functions

Let Ω be a simply connected domain in the complex plane, and $\text{A(Ω}{}^{\mathrm{n}}\text{)}$, the space of functions which are defined and analytic on $\text{Ω}{}^{\mathrm{n}}$, if K is the operator on elements $\text{u(t, a}{}_1\text{, …, a}{}_{\mathrm{n}}\text{)}\text{of}\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of the kernels k_i(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by is the identity operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$, then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of a kernel w(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by Wu = ∝_an^tw(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠW is the operator; ΠWu = ∝_{an ? 1}w(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠK is the operator; ΠKu = ∑_{i = 1}^{n ? 1} ∝_ai^tk_i(t, s, a₁, …, a_n) ds + ∝_{an ? 1}^tk_n(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. The operator M is of the form m(t, a₁, …, a_n)I, where $\text{m ? A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ and maps elements of $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator $\text{K}{}^1$ on functions u in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$, by . A determinant $\text{δ(I ? K}{}^1\text{)}$ of the operator $\text{I ? K}{}^1$ is defined as an element $\text{m}{}^1\text{(t, a}{}_1\text{, …, a}{}_{\mathrm{n\; +\; 1}}\text{)}$ of $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$. This is mapped into $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ by setting a_{n + 1} = t to give m(t, a₁, …, a_n). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels k_i defining K are separable in the sense that k_i(t, s, a₁, …, a_n) = P_i(t, a₁, …, a_n) Q_i(s, a₁, …, a_n), where P_i is a 1 × p_i matrix and Q_i is a p_i × 1 matrix, each with elements in $\text{A(Ω}{}_{\mathrm{n\; +\; 1}}\text{)}$, explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case.     

Some partitions associated with a partially ordered set

For any partially ordered set P, let $\text{d}{}_{\mathrm{k}}\text{(P)(}\text{d}\text{?}{}_{\mathrm{k}}\text{(P))}$ denote the cardinality of the largest subset of P obtained by taking the union of k antichains (chains). Then there exists a partition Δ = {Δ_l ? Δ₂ > … ? Δ_l} of | P | such that d_k(P) = Δ₁ + Δ₂ + … + Δ_k and $\text{d}\text{?}{}_{\mathrm{k}}\text{(P) = Δ}{}_1{}^1\text{+ Δ}{}_2{}^1\text{+ … + Δ}{}_{\mathrm{k}}{}^1$ for each k, where $\text{Δ}{}^1$ denotes the partition conjugate to Δ. This result can be used to prove a general class of “Dilworth-type” theorems for subfamilies of P.     

Votes and a half-binomial

In two party elections with popular vote ratio $\text{p}\text{q}\text{,}\text{1}\text{2}\text{≤p=1 ?q}$, a theoretical model suggests replacing the so-called MacMahon cube law approximation $\text{(}\text{p}\text{q}\text{)}{}^3$, for the ratio $\text{P}\text{Q}$ of candidates elected, by the ratio $\text{?}{}_{\mathrm{k}}\text{(p)}\text{?}{}_{\mathrm{k}}\text{(q)}$ of the two half sums in the binomial expansion of (p+q)^2k+1 for some k. This ratio is nearly $\text{(}\text{p}\text{q}\text{)}{}^3$ when k = 6. The success probability $\text{g}{}_{\mathrm{k}}\text{(p)=(}\text{p}{}^{\mathrm{a}}\text{(p}{}^{\mathrm{a}}\text{+q}{}^{\mathrm{a}}\text{)}$ for the power law $\text{(}\text{p}\text{q}\text{)}{}^{\mathrm{a}}\text{?}\text{P}\text{Q}$ is shown to so closely approximate $\text{?}{}_{\mathrm{k}}\text{(p)=Σ}{}_0{}^{\mathrm{k}}\text{(}{}_{\mathrm{r}}{}^{\mathrm{2k+1}}\text{)p}{}^{\mathrm{2k+1?r}}\text{q}{}^{\mathrm{r}}$, if we choose $\text{a = a}{}_{\mathrm{k}}\text{=}\text{(2k+1)!}\text{4}{}^{\mathrm{k}}\text{k!k!}$, that $\text{1≤}\text{?}{}_{\mathrm{k}}\text{(p)}\text{g}{}_{\mathrm{k}}\text{(p)}\text{≤1.01884086}$ for $\text{k≥1}\text{if}\text{1}\text{2}\text{≤p≤1}$. Computationally, we avoid large binomial coefficients in computing $\text{?}{}_{\mathrm{k}}\text{(p)}$ for k>22 by expressing $\text{2?}{}_{\mathrm{k}}\text{(p)?1}$ as the sum $\text{(p?q) Σ}{}_0{}^{\mathrm{k}}\text{(4pq)}{}^{\mathrm{s}}\text{a}{}_{\mathrm{s}}\text{(2s+1)}$, whose terms decrease by the factors $\text{(4pq)(1?}\text{1}\text{2s}\text{)}$. Setting K = 4k+3, we compute a_k for the large k using a continued fraction $\text{πa}{}_{\mathrm{k}}{}^2\text{=}\text{K+1}{}^2\text{(}\text{2K+3}{}^2\text{(}\text{2K+5}{}^2\text{(2K+…)}\text{)}\text{)}$ derived from the ratio of π to the finite Wallis product approximation.     

Asymptotic behavior of nonoscillatory solutions of nonlinear differential equations with retarded arguments

For nonlinear retarded differential equations $\text{y}{}^{\mathrm{2n}}\text{(t)?}\text{∑}\text{i=1}\text{m}\text{f}{}_{\mathrm{i}}\text{(t,y(t),y(g}{}_{\mathrm{i}}\text{(t)))=0}$ and $\text{y}{}^{\mathrm{n}}\text{(t)?}\text{∑}\text{i=1}\text{m}\text{P}{}_{\mathrm{i}}\text{(t)F}{}_{\mathrm{i}}\text{(y(g}{}_{\mathrm{i}}\text{(t)))=h(t),}$ the sufficient conditions are given on f_i, p_i, F_i, and h under which every bounded nonoscillatory solution of (${}^1$) or (${}^7$) tends to zero as t → ∞.     

On a conjecture of S. Chowla and of S. Chowla and H. Walum,II

In 1965, Chowla and Walum conjectured that, $\text{G}{}_{\mathrm{a,k}}\text{(x):=}\text{Σ}{}_{\mathrm{n\; \le \; \surd x}}\text{n}{}^{\mathrm{a}}\text{P}{}_{\mathrm{k}}\text{(}\text{x}\text{n}\text{) = O(x}{}^{\mathrm{<mtext>a</mtext><mtext>2</mtext><mtext>\; +\; </mtext><mtext>1</mtext><mtext>4</mtext><mtext>\; +\; \epsilon </mtext>}}\text{)}$ holds for each ε > 0 and x → ∞, where integers a ≥ 0 and k ≥ 1 are given and P_k is the periodic Bernoulli function of order k. Recently, the authors established this conjecture in case k ≥ 2 and $\text{a >}\text{1}\text{2}$ with ε = 0 while if $\text{a =}\text{1}\text{2}$, with a log factor. In this paper, it is proven that the conjecture, in case $\text{|a| <}\text{1}\text{2}$, k = 2, is true “on average”.     

On the closability of some positive definite symmetric differential forms on c0∞(ω)

Let $\text{S}$ be a Dirichlet form in $\text{L}{}^2\text{(Ω; m)}$, where Ω is an open subset of $\text{R}$ⁿ, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let $\text{S}$_k be a Dirichlet form on some k-dimensional submanifold $\text{Ω}{}_{\mathrm{k}}$ of Ω. The paper is devoted to the study of the closability of the forms E with domain $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ and defined by: $\text{(?,g)=}$$\text{E}$$\text{(?, g)+}\text{∑}\text{i}\text{p}\text{=1}$$\text{E}$k_i where 1 ? k_p < ? < n, and where , g^k_i denote restrictions of ?, g in $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ to . Conditions are given for E to be closable if, for each i = 1,…, p, one has k_i = n ? i. Other conditions are given for E to be nonclosable if, for some i, k_i < n ? i.     

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).     

On the order of the error function of the (k,r)-integers

Let k and r be fixed integers such that 1 < r < k. Any positive integer n of the form n = a^kb, where b is r-free, is called a (k, r)-integer. In this paper we prove that if Q_k,r(x) denotes the number of (k, r)-integers ≤ x, then $\text{Q}{}_{\mathrm{k,r}}\text{(x) =}\text{xζ(k)}\text{ζ(r)}\text{+ Δ}{}_{\mathrm{k,r}}\text{(x)}$, where $\text{Δ}{}_{\mathrm{k,r}}\text{(x) = O(x}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}\text{exp}\text{[?B}\text{log}{}^{\mathrm{<mtext>3</mtext><mtext>5</mtext>}}\text{x (}\text{log log}\text{x)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>5</mtext>}}\text{])}$, B being a positive constant depending on r and the O-estimate is uniform in k. On the assumption of the Riemann hypothesis, we improve the above order estimate of Δ_k,r(x) and prove that

$\text{1}\text{x}\text{∫}\text{1}\text{α}\text{δ}{}_{\mathrm{k,r}}\text{(t)dt=0(x}\text{1}\text{k}\text{ω(x))}\text{or}\text{0(x}{}^{\mathrm{3/(4r+1)}}\text{ω(x))}$

, according as $\text{k ≤}\text{(4r + 1)}\text{3}$ or $\text{k >}\text{(4r + 1)}\text{3}$, where ω(x) = exp [B log x(log log x)^?1].     

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.     

A completeness theorem for a nonlinear multiparameter eigenvalue problem

In this paper we study the linked nonlinear multiparameter system

$\text{y}{}_{\mathrm{r}}{}^{\mathrm{n}}\text{(X}{}_{\mathrm{r}}\text{) + M}{}_{\mathrm{r}}\text{Y}{}_{\mathrm{r}}\text{+}\text{∑}\text{s=1}\text{k}\text{λ}{}_{\mathrm{s}}\text{(a}{}_{\mathrm{rs}}\text{(X}{}_{\mathrm{r}}\text{) + P}{}_{\mathrm{rs}}\text{) Y}{}_{\mathrm{r}}\text{(X}{}_{\mathrm{r}}\text{) = 0, r = l,…, k}$

, where x_r? [a_r, b_r], y_r is subject to Sturm-Liouville boundary conditions, and the continuous functions a_rs satisfy ¦ $\text{A ¦ (x) =}\text{det}\text{a}{}_{\mathrm{rs}}\text{(x}{}_{\mathrm{r}}\text{) > 0}$. Conditions on the polynomial operators M_r, P_rs are produced which guarantee a sequence of eigenfunctions for this problem yⁿ(x) = Π_r=1^ky_rⁿ(x_r), n ? 1, which form a basis in $\text{L}{}^2\text{([a, b], ¦ A ¦)}$. Here [a, b] = [a₁, b₁ × … × [a_k, b_k].     

Two transformations of series that commute with compositional inversion

It is shown that the compositional inverse of either of two transformations of a given series can be determined from the compositional inverse of the series. Specifically, if t · f(t) and t · g(t) are compositional inverses, then so are t · f_k(t) and $\text{t · g}{}_{\mathrm{k}}{}^1\text{(t)}$, where f_k(t) is the kth Euler transformation of f(t) and $\text{g}{}_{\mathrm{k}}{}^1\text{(t) =}\text{g(t)}\text{(1 ? kt · g(t))}$.