Nondecreasing Sequence Research Articles

Complete sequences of sets of integer powers

Call S complete if Σ(S) contains all sufficiently large integers. It has been known for some time (see [B]) that if gcd(a, b) = 1 then the (nondecreasing) sequence formed from the values ab with s0 ≤ s, t0 ≤ t ≤ f(s0, t0) is complete, where s0 and t0 are arbitrary, and f(s0, t0) is sufficiently large. In this note we consider the analogous question for sequences formed from pure powers of integers. Specifically, for a sequence A of integers greater than 1, denote by Pow(A; s) the (nondecreasing) sequence formed from all the powers a where a ∈ A and k ≥ s ≥ 1. Although we are currently unable to prove it, we believe the following should hold:

On the Strong Law of Large Numbers for Random Quadratic Forms

The paper establishes strong laws of large numbers for the quadratic forms \[ Q_n (X,X) = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {a_{ij} X_i X_j } } \] and the bilinear forms \[ Q_n (X,Y) = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {a_{ij} X_i Y_j } } , \] where $X = (X_n )$ is a sequence of independent random variables and $Y = (Y_n )$ is an independent copy of it. In the case of independent identically distributed symmetric p-stable random variables $X_n $ we derive necessary and sufficient conditions for the strong laws of $Q_n (X,X)$ and $Q_n (X,Y)$ for a given nondecreasing sequence $(b_n )$ of normalizing constants. For these classes of variables $(X_n )$ the strong laws $\lim b_n^{ - 1} Q_n (X,X) = 0$ a.s. and $\lim b_n^{ - 1} Q_n (X,Y) = 0$ a.s. are shown to be equivalent provided that $a_{ii} = 0$ for all i.

SCATTERED SETS, CHAINS AND THE BAIRE CATEGORY THEOREM

A scattered set of real numbers can be exhibited in terms of endpoints of the components of some nondecreasing sequence of open sets (chain). We study this connection between scattered sets and chains providing characterizations of scattered sets, semiscattered sets, certain splattered sets and sets with countable closure. It is shown that any application of the Baire Category Theorem on the real line leads naturally to a chain of open sets and hence to an exceptional scattered set. Some applications of this fact are given.

On projections in a space with an indefinite metric

We consider G-projectors (orthogonal projections) defined on an indefinite inner product space, and derive in a systematic way the indefinite counterparts of a number of useful results known to hold for ordinary projectors in Hilbert space. Some of the topological considerations encountered in the literature are avoided here, and several results are obtained using quite elementary matrix-type arguments. In particular, the relation between G-projectors and contractions in an indefinite inner product space is studied. For example, a convergence result is given for a nondecreasing sequence of G-contractive G-projectors. We also prove a result characterizing G-projectors within the class of idempotents, generalizing the corresponding result in Hilbert space.

Stability of random sets generated by multivariate samples

Suppose is an iid random sample of random vectors in and let be the closed random set induced by the random sample. When does there exist a nondecreasing sequence of non-random compact sets such that almost surely where ρ is the Hausdorff metric giving a natural distance measure between two compact sets. When such a monotone sequence exists, we say the sequence of random sets is (almost surely) stable. Sufficient conditions are given on the underlying distribution guaranteeing stability of the induced random sets and examples are given to show the possibly bizarre nature of the approximating sets

Elementary sequences, sub-Fibonacci sequences

Recent research in uniqueness of representability for finite measurement structures has identified a number of novel finite integer sequences. One of the simplest, called an elementary sequence, is a nondecreasing integer sequence x 1, x 2,…, x n with x 1= x 2=1 and, for all k> 2, if x k > 1 then x k = i + j for distinct i, j < k. We investigate combinatorial and number-theoretic questions for elementary sequences and identify interesting open problems. The paper also discusses sub-Fibonacci sequences x 1, x 2,…, x n , which are characterized as nondecreasing integer sequences with x 1= x 2=1 and x k ≤ x k−1 + x k−2 for each k > 2.

Inequalities for non-decreasing sequences

SynopsisIn this paper we prove an extension of inequality (1.1) due to A. Meir.

Stability theorems for large order statistics with varying ranks

Let { X n }; be a sequence of i.i.d. random variables with distribution function F. For r ⩾ 1 and n ⩾ r, let Z rn be the rth largest of }; X 1,…, X n };. Let μ n = F −1(1 − n −1), n ⩾ 1, and suppose that { r n }; is a non-decreasing integer sequence such that 1 ⩽ r n } ⩽ n. It is shown that Z r nn /μ n → 1 in probability if and only if (*) r n /( n(1 − F(δμ n ))) → 0 for every δ < 1. Moreover, if Z rn /μ n → 1 a.s. for some r ⩾ 1, then Z r nn /ν n → 1 a.s. if and only if (*) holds. These results generalize a theorem of Hall (1979).

Global multi-parametric optimal value bounds and solution estimates for separable parametric programs

In this tutorial, a strategy is described for calculating parametric piecewise-linear optimal value bounds for nonconvex separable programs containing several parameters restricted to a specified convex set. The methodology is based on first fixing the value of the parameters, then constructing sequences of underestimating and overestimating convex programs whose optimal values respectively increase or decrease to the global optimal value of the original problem. Existing procedures are used for calculating parametric lower bounds on the optimal value of each underestimating problem and parametric upper bounds on the optimal value of each overestimating problem in the sequence, over the given set of parameters. Appropriate updating of the bounds leads to a nondecreasing sequence of lower bounds and a nonincreasing sequence of upper bounds, on the optimal value of the original problem, continuing until the bounds satisfy a specified tolerance at the value of the parameter that was fixed at the outset. If the bounds are also sufficiently tight over the entire set of parameters, according to criteria specified by the user, then the calculation is complete. Otherwise, another parameter value is selected and the procedure is repeated, until the specified criteria are satisfied over the entire set of parameters. A parametric piecewise-linear solution vector approximation is also obtained. Results are expected in the theory, computations, and practical applications. The general idea of developing results for general problems that are limits of results that hold for a sequence of well-behaved (e.g., convex) problems should be quite fruitful.

Optimal parallel construction of prescribed tournaments

A tournament is a digraph in which every pair of vertices is connected by exactly one arc. The score list of a tournament is the sorted list of the out-degrees of its vertices. Given a nondecreasing sequence of nonnegative integers, is it the score list of some tournament? There is a simple test for answering this question. There is also a simple sequential algorithm for constructing a tournament with a given score list. However, this algorithm has a greedy nature, and seems hard to parallelize. We present a simple parallel algorithm for the construction problem. Our algorithm runs in time O(log n) and uses O( n 2/ log n) processors on a CREW PRAM, where n is the number of vertices. Since the size of the output is Θ( n 2), our algorithm achieves optimal speedup. The tournament constructed has the property that it is the closest possible to a transitive tournament in a precise sense.

Almost Sure Limit Set of the Vector Sequence of Partial Sums and Delayed Sums

Let { xn, n ⩾ 1} be a sequence of positive valued stable andon variables with exponent [Formula: see text] where an is non-decreasing sequence of positive integers. Write [Formula: see text] In this paper we obtain the almost sure limit set of the sequence (ξ n) under certain conditions on an.

A generalized law of the iterated logarithm

Let { S n , n ⩾ } denote the partial sums of a sequence of independent random variables, and let ( B n , n ⩾ 1) be a non-decreasing sequence with B n → ∞. Upper and lower bounds for lim sup n → ∞ S n /(2 B 2 n log log B 2 n ) 1 2 are presented.

Van Lier sequences

We study two types of sequences of positive integers which arise from problems in the measurement of comparative judgements of probability. The first type consists of the Van Lier sequences, which are nondecreasing sequences x1, x2,…,xn of positive integers that start with two 1's and have the property that, whenever j<k≤n,xk−xj can be expressed as a sum of terms from the sequence other than xj. The second type consists of the regular sequences, which are nondecreasing sequences of positive integers that start with two 1's and have the property that each subsequent term is a partial sum of preceding terms. We show that every regular sequence without “gaps” is Van Lier and that every regular sequence which satisfies the Fibonacci-like inequality xk≤xk-2+xk-1 is Van Lier. We also study one-term extensions of Van Lier sequences and obtain some asymptotic results on the number of Van Lier sequences.

Stability For a Multidimensional Inverse Spectral Theorem

We consider the eigenvalue problem in Ω Where Ω is a bounded domain in Rd with smooth boundary,a nd q is a bounded, measurable function on Ω The eigenvalue problem has discrete spectrum; we denote by and a nondecreasing sequence of eigenvalue and corresponding (orthonormal) eigenfunctions. It is known ([N–S–U]) that knowledge of the eigenvalues and the boundary values of the normal derivatives of the corresponding eigenfunctions is sufficient to uniquely determine a coefficient, q.

Strong limit theorems for quasi-orthogonal random fields

This is a systematic and unified treatment of a variety of seemingly different strong limit problems. The main emphasis is laid on the study of the a.s. behavior of the rectangular means ζ mn = 1 (λ 1(m) λ 2(n)) Σ i=1 m Σ k=1 n X ik as either max{ m, n} → ∞ or min{ m, n} → ∞. Here { X ik : i, k ≥ 1} is an orthogonal or merely quasi-orthogonal random field, whereas { λ 1( m): m ≥ 1} and { λ 2( n): n ≥ 1} are nondecreasing sequences of positive numbers subject to certain growth conditions. The method applied provides the rate of convergence, as well. The sufficient conditions obtained are shown to be the best possible in general. Results on double subsequences and 1-parameter limit theorems are also included.

The minimization of lower subdifferentiable functions under nonlinear constraints: An all feasible cutting plane algorithm

Nonlinear, possibly nonsmooth, minimization problems are considered with boundedly lower subdifferentiable objective and constraints. An algorithm of the cutting plane type is developed, which has the property that the objective needs to be considered at feasible points only. It generates automatically a nondecreasing sequence of lower bounds converging to the optimal function value, thus admitting a rational rule for stopping the calculations when sufficient precision in the objective value has been obtained. Details are given concerning the efficient implementation of the algorithm. Computational results are reported concerning the algorithm as applied to continuous location problems with distance constraints.

Polynomially moving ergodic averages

Given an increasing sequence of positive integers { m n } \left \{ {{m_n}} \right \} , a non-decreasing sequence of positive integers { b n } \left \{ {{b_n}} \right \} , and a measurable, measure-preserving ergodic transformation τ \tau on a probability space ( Ω , F , μ ) \left ( {\Omega ,\mathcal {F},\mu } \right ) , the a.s. convergence of the moving averages T n ( f ) = b n − 1 ∑ k = m n + 1 m n + b n f ( τ k ) {T_n}\left ( f \right ) = b_n^{ - 1}\sum \nolimits _{k = {m_n} + 1}^{{m_n} + {b_n}} {f\left ( {{\tau ^k}} \right )} is considered, for f ∈ L p ( Ω ) f \in {L_p}\left ( \Omega \right ) . A counterexample is constructed in the case of polynomial-like { m n } \left \{ {{m_n}} \right \} .

Example of a 𝑇₁ topological space without a Noetherian base

A Noetherian base B \mathcal {B} of a topological space X X is a base for the topology of X X which has the following property: If B 1 ⊂ B 2 ⊂ ⋯ {B_1} \subset {B_2} \subset \cdots is a nondecreasing sequence of elements of B \mathcal {B} , then { B n } n ∈ N {\left \{ {{B_n}} \right \}_{n \in {\mathbf {N}}}} is finite. In this article we give an example of a T 1 {T_1} topological space without a Noetherian base.

A note on ranking functions

In this issue, W.J. Walker introduces the lattice L(n, r) as the set of all possible results when n competitors are matched in a series of r races. A result is an r-term nondecreasing sequence of integers selected from {1,2,…, n}. The dimension of L(n, r) is at most r since it is a subposet of R t . Walker conjectures that L(n, r) is at fact the intersection of r consistent linear extensions and verifies this conjecture when r ⩽ 2 as well as for the case ( n, r) = (4, 3). In this note, we show that the general conjecture does not hold by providing that for every r >/ 3 and every t ⩾ r, there exists an integer n 0 so that if n ⩾ n 0, then L(n, r) is not the intersection of t consistent linear extensions.

Classification of possibilistic uncertainty and information functions

A possibility distribution is defined as a sequence ( p i ), 0 ⩽ p i ⩽ 1, such that max i p i = 1. Joint and marginal distributions are defined in analogy with probability theory; however the operations of minimum and maximum replace multiplication and addition. The classical notion of entropy, as a measure of information associated with a probability distribution has as its counterpart an uncertainty measure associated with a possibility distribution. Higashi and Klir proposed several axioms that an uncertainty function should satisfy. They also proposed one such function — given a non-decreasing sequence ( p i ), its value is p 1 log n + (p 2−p 1) log(n − 1) + (p 3−p 2) log(n − 2) + … In this paper we give a complete description of uncertainty functions, satisfying those axioms. They are classified by montonically increasing functions from the unit interval into itself which fix end-points. To any such function z( x) there corresponds an uncertainty function z(p 1) log n + [,z(p 2) − z(p 1)] log(n − 1) + … We also discuss the role of continuity and deduce uniqueness of the measure under and additional linearity property. Because of the interdependency of the axioms, our results assume only the properties of symmetry, expansibility, monotonicity and additivity.