Solutions to enumeration problems for single-transition serial sequences with an adjacent series height increment bounded above

Sets of n-valued finite serial sequences are investigated. Such a sequence consists of two serial subsequences, beginning with an increasing subsequence and ending in a decreasing one (and vice versa). The structure of these sequences is determined by constraints imposed on the number of series, on series lengths, and on series heights. For sets of sequences the difference between adjacent series heights in which does not exceed a certain given value 1 ≤ |h _j+1 ? h _j | ≤ δ, two algorithms are constructed of which one assigns smaller numbers to lexicographically lower sequences and the other assigns smaller numbers to lexicographically higher sequences.     

Numeration of non-decreasing and non-increasing serial sequences

Finite sets of n-valued serial sequences are examined. Their structure is determined not only by restrictions on the number of series and series lengths, but also by restrictions on the series heights, which define the order number of series and their lengths, but also is limited to the series heights, by whose limitations the order of series of different heights is given. Solutions to numeration and generation problems are obtained for the following sets of sequences: non-decreasing and non-increasing sequences where the difference in heights of the neighboring series is either not smaller than a certain value or not greater than a certain value. Algorithms that assign smaller numbers to lexicographically lower sequences and smaller numbers to lexicographically higher sequences are developed.     

Algorithms for enumeration of single-transition serial sequences

Sets of n-valued single-transition serial sequences consisting of two serial subsequences (an increasing one and a decreasing one) determined by constraints on the number of the series and on their lengths and heights are considered. Enumeration problems for sets of finite sequences in which the difference in height between the neighboring series is not less than some given value are solved. Algorithms that assign smaller numbers to lexicographically lower-order sequences and smaller numbers to lexicographically higher-order sequences are obtained.     

Multiple walsh series and Zygmund sets

The classical Zygmund theorem claims that, for any sequence of positive numbers {? _n } monotonically tending to zero and any δ > 0, there exists a set of uniqueness for the class of trigonometric serieswhose coefficients aremajorized by the sequence {? _n } whosemeasure is greater than 2π ?δ. In this paper, we prove the analog of Zygmund’s theorem for multiple series in the Walsh system on whose coefficients rather weak constraints are imposed.     

Lexicographically ordered trees

We characterize trees whose lexicographic ordering produces an order isomorphic copy of some sets of real numbers, or an order isomorphic copy of some set of ordinal numbers. We characterize trees whose lexicographic ordering is order complete, and we investigate lexicographically ordered ω-splitting trees that, under the open-interval topology of their lexicographic orders, are of the first Baire category. Finally we collect together some folklore results about the relation between Aronszajn trees and Aronszajn lines, and use earlier results of the paper to deduce some topological properties of Aronszajn lines.     

Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation

In the case of the boundary value problem for a singularly perturbed convection-diffusion parabolic equation, conditioning of an ε-uniformly convergent finite difference scheme on a piecewise uniform grid is examined. Conditioning of a finite difference scheme on a uniform grid is also examined provided that this scheme is convergent. For the condition number of the scheme on a piecewise uniform grid, an ε-uniform bound O(δ ₁ ^?2 lnδ ₁ ^?1 + δ ₀ ^?1 ) is obtained, where δ₁ and δ₀ are the error components due to the approximation of the derivatives with respect to x and t, respectively. Thus, this scheme is ε-uniformly well-conditioned. For the condition number of the scheme on a uniform grid, we have the estimate O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ); this scheme is not ε-uniformly well-conditioned. In the case of the difference scheme on a uniform grid, there is an additional error due to perturbations of the grid solution; this error grows unboundedly as ε → 0, which reduces the accuracy of the grid solution (the number of correct significant digits in the grid solution is reduced). The condition numbers of the matrices of the schemes under examination are the same; both have an order of O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ). Neither the matrix of the ε-uniformly convergent scheme nor the matrix of the scheme on a uniform grid is ε-uniformly well-conditioned.     

A recursive formula for even order harmonic series

A useful recursive formula for obtaining the infinite sums of even order harmonic series Σ^∞_n=1 (1/n^2k), k = 1, 2, …, is derived by an application of Fourier series expansion of some periodic functions. Since the formula does not contain the Bernoulli numbers, infinite sums of even order harmonic series may be calculated by the formula without the Bernoulli numbers. Infinite sums of a few even order harmonic series, which are calculated using the recursive formula, are tabulated for easy reference.     

On condition numbers of the Shanks transformation

Quasi-linear functions generate sequence transformation methods whose conditioning depends upon the nature of the sequence to be accelerated. These methods are often well conditioned when they are applied to alternating sequences; however, they are relatively ill-conditioned in case of monotonic convergence. The condition numbers of the Shanks transformation e_k(s_n) are given in order to prove that the closely related ε-algorithm to a such transformation is ill-conditioned when performed on the set of totally monotonic sequences. In the same way, we show that this algorithm is well conditioned on the set of totally oscillating sequences.     

Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ?-uniformly well conditioned or ?-uniformly stable to perturbations of the data of the grid problem (here, ? is a perturbation parameter, ? ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ?-uniformly in the maximum norm at an O(N ^?1lnN + N ₀ ^?1 ) rate, where N + 1 and N ₀ + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ?-uniformly well conditioned and ?-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ^?2lnδ^?1 + δ ₀ ^?1 ); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ = N ^?1lnN and δ₀ = N ₀ ^?1 are the accuracies of the discrete solution in x and t, respectively.     

The semicycles of solutions of neutral difference equations

In this paper, we study the semicycles of solutions of neutral delay difference equation Δ(y_n + p_ny_n−τ) + q_ny_n−σ = 0, where {p_n} and {q_n} are sequences of nonnegative real numbers, τ and σ are positive integers. Upper bound of numbers of terms of semicycles are determined.     

A dependent theory with few indiscernibles

We give a full solution to the question of existence of indiscernibles in dependent theories by proving the following theorem: For every θ there is a dependent theory T of size θ such that for all κ and δ, κ → (δ) _T,1 iff κ → (δ) _θ ^<ω . This means that unless there are good set theoretical reasons, there are large sets with no indiscernible sequences.     

Schauder estimates for a class of second order elliptic operators on a cube

We consider a class of second order elliptic operators on a d-dimensional cube S_d. We prove that if the coefficients are of class C^k+δ(S_d), with k=0,1 and δ∈(0,1), then the corresponding elliptic problem admits a unique solution u belonging to C^k+2+δ(S_d) and satisfying non-standard boundary conditions involving only second order derivatives.     

Sequences of Diophantine approximations

We sharpen a technique of Gelfond to show that, in a sense, the only possible gap-free sequences of “good” Diophantine approximations to a fixed α ∈ C are trivial ones. For example, suppose that a > 1 and that (δ_n)_n=1^∞ and (σ_n)_n=1^∞ are two positive, strictly increasing unbounded sequences satisfying δ_n+1 ≤ aδ_n and σ_n+1 ≤ aσ_n. If there is a sequence of nonzero polynomials P_n ∈ Z[x] with deg P_n ≤ δ_n, deg P_n + log height P_n ≤ σ_n, and ∣P_n(α)∣ ≤ e^{?(2a+1)δ_nσ_n}, then each P_n(α) = 0.     

Asymptotic stability and other properties of trajectories and transfer sequences leading to the bargaining sets

The foundation of a dynamic theory for the bargaining sets started withStearns, when he constructed transfer sequences which always converge to appropriate bargaining sets. A continuous analogue was developed byBillera, where sequences where replaced by solutions of systems of differential equations. In this paper we show that the nucleolus is locally asymptotically stable both with respect toStearns' sequences andBillera's solutions if and only if it is an isolated point of the appropriate bargaining set. No other point of the bargaining set can be locally asymptotically stable. Furthermore, it is always stable in these processes. As by-products of the study we derive the results ofBillera andStearns in a different fashion. We also show that along the non-trivial trajectories and sequences, the vector of the excesses of the payoffs, arranged in a non-increasing order, always decreases lexicographically, thus each bargaining set can be viewed as resulting from a certain monotone process operating on the payoff vectors.     

Constructing polynomials of minimal growth

In [2] a cyclic diagonal operator on the space of functions analytic on the unit disk with eigenvalues (λ _n ) is shown to admit spectral synthesis if and only if for each j there is a sequence of polynomials (p _n ) such that lim _n→∞ p _n (λ _k ) = δ _j,k and lim sup _n→∞ sup _k>j |p _n (λ _k )|^1/k ≤ 1. The author also shows, through contradiction, that certain classes of cyclic diagonal operators are synthetic. It is the intent of this paper to use the aforementioned equivalence to constructively produce examples of synthetic diagonal operators. In particular, this paper gives two different constructions for sequences of polynomials that satisfy the required properties for certain sequences to be the eigenvalues of a synthetic operator. Along the way we compare this to other results in the literature connecting polynomial behavior ([4] and [9]) and analytic continuation of Dirichlet series ([1]) to the spectral synthesis of diagonal operators.     

Holomorphic Solutions of a Functional Equation Related to Nonlinear Difference Systems

Here we consider the following functional equation, $$\Psi(X(x,\Psi(x)))=Y(x, \Psi(x)),$$ where X(x, y) and Y(x, y) are holomorphic functions in |x| < δ ₁, |y| < δ ₁. When we consider a nonlinear simultaneous system of two variables difference equations, we can reduce it to a single difference equation of first order by a solution Ψ of the above functional equation. We obtain a matrix by the linear terms of functions X and Y. When the all eigenvalues of the matrix are equal to 1, it is difficult to have a solution of the above functional equation. In the present paper, we derive a formal solution of the above functional equation under the condition. Further we prove the existence of a solution which is holomorphic and have an asymptotically expansion of the formal solution. Moreover, we will show an example of nonlinear difference system such that our results are applicable.     

Strong approximation by Fourier--Laplace series on the unit sphere S n-1

We study the strong approximation properties of the Cesáro means of order δ of the Fourier--Laplace expansion of functions integrable on the unit sphere S ^n-1, where δ ≥λ? (n-2)/2, the latter being the critical index for Cesáro summability of Fourier--Laplace series on S ^n-1. The main purpose of this paper is to extend known results from the unit circle S ¹to the general sphere S ^n-1 with n≥3. We prove six theorems. To prove Theorems 1-3, our machinery is based on the equiconvergent operator E ^δ _N (f) of the Cesáro means σ^δ _N (f) on S ^n-1 introduced by Wang Kunyang for δ>-1. We prove in Theorem 6 that E ^δ _N (f) is also equiconvergent with σ^δ _N (f) for δ>0 in the case of strong approximation. To prove Theorems 4 and 5, we rely on known equivalence relations between K-functionals and moduli of continuity.     

Practical algorithms to rank necklaces,Lyndon words,and de Bruijn sequences

We present practical algorithms for ranking k-ary necklaces and Lyndon words of length n. The algorithms are based on simple counting techniques. By repeatedly applying the ranking algorithms, both necklaces and Lyndon words can be efficiently unranked. Then, explicit details are given to rank and unrank the length n substrings of the lexicographically smallest de Bruijn sequence of order n.     

Traces of operators with a relatively compact perturbation

This is a continuation of the authors’ series of papers on the theory of regularized traces of abstract discrete operators. We prove a theorem in which the perturbing operator B is subordinate to the operator A ₀ in the sense that BA ₀ ^?δ is a compact operator belonging to some Schatten-von Neumann class of finite order. Apart from covering new classes of operators, and in contrast to our preceding papers, we give a unified statement of the theorem regardless of whether the resolvent of the unperturbed operator belongs to the trace class. Two examples are given in which the result is applied to ordinary differential operators as well as to partial differential operators.