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.     

Worms,gaps, and hydras

We define a direct translation from finite rooted trees to finite natural functions which shows that the Worm Principle introduced by Lev Beklemishev is equivalent to a very slight variant of the well‐known Kirby‐Paris' Hydra Game. We further show that the elements in a reduction sequence of the Worm Principle determine a bad sequence in the well‐quasi‐ordering of finite sequences of natural numbers with respect to Friedman's gapembeddability. A characterization of gap‐embeddability in terms of provability logic due to Lev Beklemishev is also presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterization of certain orders using their associated choice functions

Binary relation and choice functions define two different languages for describing the decision maker's (DM's) preferences over a finite set of alternatives. The study of normal choice functions establishes a link between these two languages. The existence of this kind of function demonstrates the compatibility between the DM's local choices (paired comparisons or binary choices) and the DM's global choices. In this paper, we explore this kind of linkage in depth by studying characterizations of special binary relations (total order, weak order, partial order and quasi order) using certain properties of their associated choice functions.     

TOPOLOGICAL DECOMPOSITIONS OF THE DUALS OF LOCALLY CONVEX VECTOR SEQUENCE SPACES

ABSTRACT

Let Λ be a scalar sequence space which is endowed with a normal locally convex topology. For a separated locally convex space E we denote by Λ(E) the vector space of all sequences g in E for which (>g(i),a<) ε Λ for all a ε E'. We define a locally convex topology ζ on Λ(E) and then characterize the dual of the ζ-closure (denoted by Λ_c (E)) of the finite sequences in Λ(E). We demonstrate the existence of a continuous projection from Λ(E)' onto a subspace of Λ(E)' which is isomorphic to Λ_c(E)'. Furthermore, we find a topological decomposition of Λ^α _c (E)”, where one of the factors is isomorphic to Λ;^α(E). These results are then applied to find necessary and sufficient conditions for Λ^α(E) to be semi-reflexive. A parallel development yields the same results for the space Λ(E') of all sequences f in E' for which (>x, f(i)<) ε Λ; for all x ε E, when E is barrelled. We conclude the paper by application of the results on vector sequence spaces to spaces of operators—including for instance, necessary and sufficient conditions for L_b (E,Λ;) and L_b (Λ,E) to be semi-reflexive.     

Chebyshev approximations in subspaces of spline functions

The main purpose of this paper is to consider strict approximations from subspaces of spline functions of degree m-1 with k fixed knots. Rice defines the strict approximation which is a particular unique best Chebyshev approximation for problems defined on a finite set. In order to determine best approximations on an interval I we define a sequence of strict approximations on finite subsets of I where the subsets fill up the interval. It is shown that the sequences always converge if k≤m. In the case k>m the sequences are convergent if we restrict ourselves to problems defined on certain subsets of I. It seems to be natural to denote these limits as strict approximations. To be able to compute these functions we also develop a Remez type algorithm.     

Development and Initial Validation of the Beliefs About Reformed Science Teaching and Learning (BARSTL) Questionnaire

This article describes the development and the initial validation of an instrument that can be used to assess teachers' beliefs about science teaching and learning. The instrument, which is called the Beliefs About Reformed Science Teaching and Learning (BARSTL) questionnaire, draws on the current national science education reform efforts in order to define a traditional‐reformed teaching and learning belief continuum that can be used to map teachers' beliefs. The reliability and validity of the instrument were examined using a multiple perspective approach. The psychometric properties of the BARSTL suggest that it is a valid and reliable instrument for measuring prospective elementary teachers' beliefs about science education.     

Truth‐table Schnorr randomness and truth‐table reducible randomness

Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth‐table Schnorr randomness (defined in 6 too only by martingales) and truth‐table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth‐table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth‐table Schnorr randomness. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation

For a discrete time second-order stationary process, the Levinson-Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k+1, given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson-Durbin sequence. We also define a generalized Levinson-Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson-Durbin sequence. All generalized Levinson-Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson-Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson-Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1/T, where T is the sample length, can be combined to construct a Levinson-Durbin sequence. By contrast, analogous fixed point processes arising from the Yule-Walker estimator do not combine to construct a Levinson-Durbin sequence, although the Yule-Walker estimator itself does determine a Levinson-Durbin sequence. The least squares and Yule-Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares.     

Microlocal defect measures

In order to study weak continuity of quadratic forms on spaces of L² solutions of systems of partial differential equations, we define defect measures on the space of positions and frequencies.A systematic use of these measures leads in particular to a compensated compactness theorem, generalizing MURAT"TARTAR's compensated compactness to variable coefficients and GOLSE"LIONS"PERTHAME"SENTIS's averaging lemma. We also obtain results on homogenization for differential operators of order I with oscillating coefficients.     

Coherent sequences versus Radin sequences

We attempt to make a connection between the sequences of measures used to define Radin forcing and the coherent sequences of extenders which are the basis of modern inner model theory. We show that in certain circumstances we can read off sequences of measures as defined by Radin from coherent sequences of extenders, and that we can define Radin forcing directly from a coherent extender sequence and a sequence of ordinals; this generalises Mitchell's construction of Radin forcing from a coherent sequence of measures.     

Aggregation and emergence in systems of ordinary differential equations

The aim of this article is to present aggregation methods for a system of ordinary differential equations (ODE's) involving two time scales. The system of ODE's is composed of the sum of fast parts and a perturbation. The fast dynamics are assumed to be conservative. The corresponding first integrals define a few global variables. Aggregation corresponds to the reduction of the dimension of the dynamical system which is replaced by an aggregated system governing the global variables at the slow time scale. The centre manifold theorem is used in order to get the slow reduced model as a Taylor expansion of a small parameter. We particularly look for the conditions necessary to get emerging properties in the aggregated model with respect to the nonaggregated one. We define two different types of emergences, functional and dynamical. Functional emergence corresponds to different functions for the two dynamics, aggregated and nonaggregated. Dynamical emergence means that both dynamics are qualitatively different. We also present averaging methods for aggregation when the fast system converges towards a stable limit cycle.     

On Some Operators Associated to Superoscillations

In this paper we study the convergence of some sequences of operators associated to the Aharonov and Berry’s superoscillating functions. The main tool to define the sequences of operators is the spectral theorem. In particular we discuss the case of sequences of unbounded self-adjoint operators on a Hilbert space. We apply our results to the case where T is the self-adjoint extension of the momentum operator with unbounded spectrum.     

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.     

Isosingular Sets and Deflation

This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations constructed by taking the closure of points with a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which, in a different context, are the sequences arising in Thom–Boardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic set that can be investigated with methods from numerical algebraic geometry.     

Discrete Systems and their Characteristic Spectral Functions

We define first-order discrete systems in the matrix-valued case. They are characterized by sequences of pair of matrices, called admissible sequences. We present two important examples of such sequences, called Szeg? and Nehari sequences. We introduce the characteristic spectral functions associated to a first-order system. We define in particular the scattering function, the Weyl function and the reflection coefficient function and we study the relationships between these functions. Daniel Alpay wishes to thank the Earl Katz family for endowing the chair which supported his research.     

Sequences of Ordered Sets and Catalan Numbers

We define the down sets (lower covers, respectively) sequence of an ordered set. We show that the number of down set sequences of an n-ordered set is equal to the n-th Catalan Number. We give a characterization of down sets sequences of an ordered set and another characterization of lower covers sequences of an ordered set.     

Monotone enclosing of solutions of the Steffensen‐type

In this paper we consider two iterative methods of the Steffensen‐type. The iteration sequences which approximate the solution of f(x) = 0 from opposite sides are generated by Steffensen's method and a secant method, respectively. We show two enclosing theorems and establish orders of convergence. The iteration sequences converge with order two and three, respectively. Numerical examples complete the paper.     

Covering Sequences of Boolean Functions and Their Cryptographic Significance

We introduce the notion of covering sequence of a Boolean function, related to the derivatives of the function. We give complete characterizations of balancedness, correlation immunity and resiliency of Boolean functions by means of their covering sequences. By considering particular covering sequences, we define subclasses of (correlation-immune) resilient functions. We derive upper bounds on their algebraic degrees and on their nonlinearities. We give constructions of resilient functions belonging to these classes. We show that they achieve the best known trade-off between order of resiliency, nonlinearity and algebraic degree.     

由有理数Cauchy列定义的实数的无限十进小数展开

讨论了有理数Cauchy列定义的实数系的一种等价形式———无限十进小数展开,定义了其上的算术运算与顺序并证明了它们和已有的定义一致.     

ON α,β DESTRUCTIBLE GRAPHS

ABSTRACT

A connected simple graph G or order v is said to be α,β destructible if α,β are integral factors of v and an α-set of edges, E', exists whose removal from G isolates exactly the vertices in a β-set, V'. Aspects of α,β destructions of graphs considered include associated reconstruction problems, unique α,β destructibility, annihilability and proper division sequences of numbers associated with paths.