Normal number constructions for Cantor series with slowly growing bases

Let Q = (q_n)_n=1^∞ be a sequence of bases with q_i ≥ 2. In the case when the q_i are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose Q-Cantor series expansion is both Q-normal and Q-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of Q, and from this construction we can provide computable constructions of numbers with atypical normality properties.     

Galois Groups of Generalized Iterates of Generic Vectorial Polynomials

Let q=p^u>1 be a power of a prime p, and let k_q be an overfield of GF(q). Let m>0 be an integer, let J^* be a subset of {1,…,m}, and let E^*_{m, q}(Y)=Y^qm+∑_j∈J^*X_jY^qm−j where the X_j are indeterminates. Let J³ be the set of all m−ν where ν is either 0 or a divisor of m different from m. Let s(T)=∑_0≤i≤ns_iTⁱ be an irreducible polynomial of degree n>0 in T with coefficients s_i in GF(q). Let E^*[s]_{m, q}(Y) be the generalized sth iterate of E^*_{m, q}(Y); i.e., E^*[s]_{m, q}(Y)=∑_0≤i≤ns_iE^*[i]_{m, q}(Y), where E^*[i]_{m, q}(Y), is the ordinary ith iterate. We prove that if J³⊂J^*, m is square-free, and GCD(m,n)=1=GCD(mnu,2p), then Gal(E^*[s]_{m, q},k_q({X_j:j∈j^*})=GL(m, qⁿ). The proof is based on CT (=the Classification Theorem of Finite Simple Groups) in its incarnation as CPT (=the Classification of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors).     

Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators

Let M be a compact Riemannian manifold with or without boundary, and let −Δ be its Laplace-Beltrami operator. For any bounded scalar potential q, we denote by λi(q) the ith eigenvalue of the Schrödinger type operator −Δ+q acting on functions with Dirichlet or Neumann boundary conditions in case ∂M≠∅. We investigate critical potentials of the eigenvalues λi and the eigenvalue gaps Gij=λj−λi considered as functionals on the set of bounded potentials having a given mean value on M. We give necessary and sufficient conditions for a potential q to be critical or to be a local minimizer or a local maximizer of these functionals. For instance, we prove that a potential q∈L^∞(M) is critical for the functional λ₂ if and only if q is smooth, λ₂(q)=λ₃(q) and there exist second eigenfunctions f₁,…,fk of −Δ+q such that . In particular, λ₂ (as well as any λi) admits no critical potentials under Dirichlet boundary conditions. Moreover, the functional λ₂ never admits locally minimizing potentials.     

An unconditional basis in periodic spaces with dominating mixed smoothness properties

В статье построен без условный базис в функ циональных пространствах с доми нирующей смешанной гладкость ю типа Никольского — А манова:S _p,q ^r (Т ₂),p = (р ₁,р ₂,q=(q ₁,q ₂),r=(r ₁,r ₂), 1 <p _i , < ∞, 1 <q _i < ∞, -∞<r _i <∞,i=1, 2;T ₂ — двухмерный тор. С помощью этого ба зиса исследование собств енных чисел некоторых инте гральных операторов редуцируется к соотв етствующей задаче для матричных операт оров.     

Outsourcing and scheduling for two-machine ordered flow shop scheduling problems

This paper considers a two-machine ordered flow shop problem, where each job is processed through the in-house system or outsourced to a subcontractor. For in-house jobs, a schedule is constructed and its performance is measured by the makespan. Jobs processed by subcontractors require paying an outsourcing cost. The objective is to minimize the sum of the makespan and the total outsourcing cost. Since this problem is NP-hard, we present an approximation algorithm. Furthermore, we consider three special cases in which job j has a processing time requirement p_j, and machine i a characteristic q_i. The first case assumes the time job j occupies machine i is equal to the processing requirement divided by a characteristic value of machine i, that is, p_j/q_i. The second (third) case assumes that the time job j occupies machine i is equal to the maximum (minimum) of its processing requirement and a characteristic value of the machine, that is, max{p_j, q_i} (min{p_j, q_i}). We show that the first and the second cases are NP-hard and the third case is polynomially solvable.     

An umbral calculus for polynomials characterizing U(n) tensor operators

After the change of variables Δ_i = γ_i ? δ_i and x_{i,i + 1} = δ_i ? δ_{i + 1} we show that the invariant polynomials _μG⁽ⁿ⁾_q(, Δ_i, ; , x_i,i+1,) characterizing U(n) tensor operators 〈p, q,…, q, 0,…, 0〉 become an integral linear combination of Schur functions S_λ(γ ? δ) in the symbol γ ? δ, where γ ? δ denotes the difference of the two sets of variables {γ₁ ,…, γ_n} and {δ₁ ,…, δ_n}. We obtain a similar result for the yet more general bisymmetric polynomials ^m_μG⁽ⁿ⁾_q(γ₁ ,…, γ_n; δ₁ ,…, δ_m). Making use of properties of skew Schur functions $\text{S}{}_{\mathrm{<mtext>\lambda </mtext><mtext>\rho </mtext>}}$ and S_λ(γ ? δ) we put together an umbral calculus for ^m_μG⁽ⁿ⁾_q(γ; δ). That is, working entirely with polynomials, we uniquely determine ^m_μG⁽ⁿ⁾_q(γ; δ) from ^m_μG⁽ⁿ⁾_{q ? 1}(γ; δ) and combinatorial rules involving Ferrers diagrams (i.e., partitions), provided that n ≥ (μ + 1)q. (This restriction does not interfere with writing the general case of ^m_μG⁽ⁿ⁾_q(γ; δ) as a linear combination of S_λ(γ ? δ).) As an application we deduce “conjugation” symmetry for ⁿ_μG⁽ⁿ⁾_q(γ; δ) from “transposition” symmetry by showing that these two symmetries are equivalent.     

On the Evaluation of Weil Sums of Dembowski-Ostrom Polynomials

Let F_q denote the finite field of q elements, q=p^e odd, let χ₁ denote the canonical additive character of F_q where χ₁(c)=e^2πiTr(c)/p for all c∈F_q, and let Tr represent the trace function from F_q to F_p. We are interested in evaluating Weil sums of the form S=S(a₁, …, a_n)=∑_x∈Fq χ₁(D(x)) where D(x)=∑ⁿ_i=1 a_ix^pα_i+pβ_i, α_i?β_i for each i, is known as a Dembowski-Ostrom polynomial (or as a D-O polynomial). Coulter has determined the value of S when D(x)=ax^pα+1; in this note we show how Coulter's methods can be generalized to determine the absolute value of S for any D-O polynomial. When e is even, we give a subclass of D-O polynomials whose Weil sums are real-valued, and in certain cases we are able to resolve the sign of S. We conclude by showing how Coulter's work for the monomial case can be used to determine a lower bound on the number of F^l_q-solutions to the diagonal-type equation ∑^l_i=1 x^pγ+1_i+(x_i+λ)^pγ+1=0, where l is even, e/gcd(γ, e) is odd, and h (X)=λ^pe−γX^pe−γ+λ^pγX is a permutation polynomial over F_q.     

Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes

P. Masani and the author have previously answered the question, “When is an operator on a Hilbert space $\text{H}$ the integral of a complex-valued function with respect to a given spectral (projection-valued) measure?” In this paper answers are given to the question, “When is a linear operator from $\text{H}$^q to $\text{H}$^p the integral of a spectral measure?”; here the values of the integrand are linear operators from the square-summable q-tuples of complex numbers to the square-summable p-tuples of complex numbers, and our spectral measure for $\text{H}$^q is the “inflation” of a spectral measure for $\text{H}$. In the course of this paper, we make available tools for handling the spectral analysis of q-variate weakly stationary processes, 1 ≤ q ≤ ∞, which should enable researchers to deal in the future with the case q = ∞. We show as one application of our theory that if U = ∫(in0, 2π]e^?iθE(dθ) is a unitary operator on $\text{H}$ and if T is a bounded linear operator from $\text{H}$^q to $\text{H}$^q (1 ≤ q ≤ ∞) which is a prediction operator for each stationary process (Uⁿx)_?∞^∞ ?$\text{H}$^q (for each x = (xⁱ)_i^j ∈ $\text{H}$^q, Uⁿx = (Uⁿxⁱ)_i=1^q), then T is a spectral integral, ∫(_0,2π)]Φ(θ) E(dθ), and the Banach norm of T, |T|_B = ess sup |Φ(θ)|_B.     

When all solutions of x′ = −∑ qi(t) x(t − Ti(t)) oscillate

In this paper the long-term behavior of solutions to the equation in the title are examined, where q_i(t) and T_i(t) are positive. In particular, it is shown that if lim inf_{t → ∝} ∑_{i = 1}ⁿT_i(t) q_i(t) > 1/e, all solutions oscillate about 0 infinitely often.     

On algebraic relations for Ramanujan’s functions

Let P,Q, and R denote the Ramanujan Eisenstein series. We compute algebraic relations in terms of P(q ⁱ ) (i=1,2,3,4), Q(q ⁱ ) (i=1,2,3), and R(q ⁱ ) (i=1,2,3). For complex algebraic numbers q with 0<|q|<1 we prove the algebraic independence over ? of any three-element subset of {P(q),P(q ²),P(q ³),P(q ⁴)} and of any two-element subset of {Q(q),Q(q ²),Q(q ³)} and {R(q),R(q ²),R(q ³)}, respectively. For all the results we use some expressions of $P(q^{i_{1}}), Q(q^{i_{2}}) $ , and $R(q^{i_{3}}) $ in terms of theta constants. Computer-assisted computations of functional determinants and resultants are essential parts of our proofs.     

Special solutions of the equations Pu = 0, Pu = δ for invariant linear differential operators with polynomial coefficients

Explicit formulas are given for certain solutions of the equations $\text{Pu = δ}\text{and}\text{P}\text{u}\text{?}\text{= 0}$ with P being a differential operator with polynomial coefficients preserving the form ∑_{i = 1}ⁿx_i^p ? ∑_{i = 1}^my_i^q for arbitrary even integers p, q. These formulas are a direct consequence of the invariance of P and depend only up to a constant factor upon the operator P.     

Enumeration of non-crossing pairings on bit strings

A non-crossing pairing on a bit string is a matching of 1s and 0s in the string with the property that the pairing diagram has no crossings. For an arbitrary bit-string w=p₁¹q₁⁰…pr¹qr⁰, let φ(w) be the number of such pairings. This enumeration problem arises when calculating moments in the theory of random matrices and free probability, and we are interested in determining useful formulas and asymptotic estimates for φ(w). Our main results include explicit formulas in the “symmetric” case where each pi=qi, as well as upper and lower bounds for φ(w) that are uniform across all words of fixed length and fixed r. In addition, we offer more refined conjectural expressions for the upper bounds. Our proofs follow from the construction of combinatorial mappings from the set of non-crossing pairings into certain generalized “Catalan” structures that include labeled trees and lattice paths.     

Confidence intervals for marginal parameters under fractional linear regression imputation for missing data

Item nonresponse occurs frequently in sample surveys and other applications. Imputation is commonly used to fill in the missing item values in a random sample {Y_i;i=1,…,n}. Fractional linear regression imputation, based on the model with independent zero mean errors ?_i, is used to create one or more imputed values in the data file for each missing item Y_i, where {X_i,i=1,…,n}, is observed completely. Asymptotic normality of the imputed estimators of the mean μ=E(Y), distribution function θ=F(y) for a given y, and qth quantile θ_q=F^-1(q),0<q<1 is established, assuming that Y is missing at random (MAR) given X. This result is used to obtain normal approximation (NA)-based confidence intervals on μ,θ and θ_q. In the case of θ_q, a Bahadur-type representation and Woodruff-type confidence intervals are also obtained. Empirical likelihood (EL) ratios are also obtained and shown to be asymptotically scaled variables. This result is used to obtain asymptotically correct EL-based confidence intervals on μ,θ and θ_q. Results of a simulation study on the finite sample performance of NA-based and EL-based confidence intervals are reported.     

A property of algebraic univoque numbers

Consider the set $ {\mathcal{U}} $ of real numbers q ≧ 1 for which only one sequence (c _i ) of integers 0 ≦ c _i ≦ q satisfies the equality Σ _i=1 ^∞ c _i q ^?i = 1. We show that the set of algebraic numbers in $ {\mathcal{U}} $ is dense in the closure $ \overline {\mathcal{U}} $ of $ {\mathcal{U}} $ .     

A note on a new matrix decomposition

It is shown that if det A=±1, then A=±^q_i=1B_i, where B_i² = I. This decomposition is used to find the Jacobian of the linear matrix transformation: Y=AX.     

Moments of square-integrable functions

This paper is motivated by [2], where we have given necessary and sufficient conditions for a given basis P in the space of polynomials to be orthogonal with respect to the measure ϱdφ for a certain function ϱ ϵ L₂(dφ). Let P = {p_i: i = 0, 1, …}, p₀ = 1. Then the conditions are (1) a multivariate analog of the three-term recurrence relation holds, see Section 4 for details; and (2) {q_i = ∑_{j = 0}^∞ c_ij P_j, i = 0, 1, …} is a φ-orthonormal basis in the space of polynomials for some coefficients c_ij such that ∑_{i = 0}^∞ c_i0² <-∞. This paper provides an algebraic condition (a condition on the coefficients c_i0) such that ϱ satisfies ∥p∥ <B, Bϵ (0, ∞], and has a cone-positivity property. In particular, our results imply that ϱ is nonnegative a.e. if ∑_{i = 0}^∞ c_i0² < ∞ and ∑_{jϵ Sk} c_j0 q_j defines nonnegative polynomials for certain finite sets S₁, S₂, … of integers.     

Incidence graphs and subdesigns of semisymmetric designs

A semisymmetric design is a connected incidence structure satisfying; two points (blocks) are on 0 or λ blocks (points). Every block (point) is incident with k points (blocks). Properties of the incidence graph of these structures are investigated, leading to bounds on its diameter (d?k if λ = 2, d?[2k/(λ + 1)]+ 1 if λ > 2), and the number of points of these structures (υ?2^k-1 if λ = 2, υ?k2^{[2k/(λ + 1)]} if λ > 2). Bounds are also found for semisymmetric designs containing a subdesign. We give characterizations of semisymmetric designs with λ = 2 (semibiplanes) which contain a subdesign and achieve the bounds. This leads to a construction for a semibiplane with parameters υ = 2^r-1 (q²?1), k = q+q₁+?+q_r, where q_r is aprime power, q_i = q²_i+1 and q=q²₁.     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

Proof of a conjecture of Segre and Bartocci on monomial hyperovals in projective planes

The existence of certain monomial hyperovals D(x ^k ) in the finite Desarguesian projective plane PG(2, q), q even, is related to the existence of points on certain projective plane curves g _k (x, y, z). Segre showed that some values of k (k?=?6 and 2 ⁱ ) give rise to hyperovals in PG(2, q) for infinitely many q. Segre and Bartocci conjectured that these are the only values of k with this property. We prove this conjecture through the absolute irreducibility of the curves g _k .     

An approximation technique for nonlinear integral operations

Approximation results for J. S. Mac Nerney's theory of nonlinear integral operations are established. For the nonlinear product integral _xΠ^y (1 + V)P, approximations of the form Π_{i = 1}ⁿ [1 + L_q(x_i?1, x_i)]P are considered, where L₁(u, v)P = ∝_u^vVP and L_q(u, v)P = ∝_u^vV(r, s)[1 + L_q?1(s, v)]P for q = 2, 3,…. Error bounds are obtained for the difference between the product integral and the preceding product.