Particular rules for the Θ-algorithm

The -algorithm is an extrapolation algorithm which can be very useful in accelerating some slowly convergent sequences. Like the other acceleration algorithms, the -algorithm is quite sensitive to the propagation of rounding errors due to cancellation in the difference between two almost equal quantities.In order to (partially) avoid this drawback, particular rules are given. They have to be used, instead of the usual rules of the algorithm, when two adjacent quantities in a column are nearly equal. Numerical examples show that these rules can improve the numerical stability of the algorithm in some cases while, in other cases, the improvement is non-existent.     

Particular rules for the vector ε-algorithm

Summary The aim of this paper is to give a singular rule for the vector -algorithm. This singular rule avoids a too rapid growth of rounding errors when applying the vector -algorithm. Another rule is given which works when two adjacent vectors are equal.     

Skew Pieri rules for Hall–Littlewood functions

We produce skew Pieri rules for Hall–Littlewood functions in the spirit of Assaf and McNamara (J. Comb. Theory Ser. A 118(1):277–290, 2011). The first two were conjectured by the first author (Konvalinka in J. Algebraic Comb. 35(4):519–545, 2012). The key ingredients in the proofs are a q-binomial identity for skew partitions and a Hopf algebraic identity that expands products of skew elements in terms of the coproduct and the antipode.     

On unification and admissible rules in Gabbay–de Jongh logics

In this paper we study the admissible rules of intermediate logics. We establish some general results on extensions of models and sets of formulas. These general results are then employed to provide a basis for the admissible rules of the Gabbay–de Jongh logics and to show that these logics have finitary unification type.     

Sure things — Dominance and independence rules for choice under uncertainty

The primary purpose of this paper is to show that differences between models of choice under uncertainty may be derived primarily from different assumptions about the appropriate ways in which states of the world may be compared and combined. It considers different concepts of stochastic dominance arising from different permitted transformations on the ordering of prizes during a comparison of two lotteries. These concepts imply various forms of the Independence axiom and correspond to various non-expected utility theories.     

Singular rules for the calculation of non-normal multivariate Padé approximants

Section 1 describes the univariate situation in the case of non-normal Padé approximants and Cordellier's extension of the famous five-star identity of Wynn. Section 2 repeats our definition of multivariate Padé approximants and proves a number of theorems that remain valid when going from the univariate to the multivariate case. These theorems and more new results given in Section 3, will finally also copy Cordellier's extension from the univariate to the multivariate case.     

Distribution of points for convergent interpolatory integration rules on (−∞, ∞)

Letw be a “nice” positive weight function on (?∞, ∞), such asw(x)=exp(??x?^α) α>1. Suppose that, forn≥1, $$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} } f(x_{jn} )$$ is aninterpolatory integration rule for the weightw: that is for polynomialsP of degree ≤n-1, $$I_n [P]: = \int\limits_{ - \infty }^\infty {P(x)w(x)dx.} $$ Moreover, suppose that the sequence of rules {I _n} _n=1 ^t8 isconvergent: $$\mathop {\lim }\limits_{n \to \infty } I_n [f] = \int\limits_{ - \infty }^\infty {f(x)w(x)dx} $$ for all continuousf:R→R satisfying suitable integrability conditions. What then can we say about thedistribution of the points {x _jn} _j=1 ⁿ ,n≥1? Roughly speaking, the conclusion of this paper is thathalf the points are distributed like zeros of orthogonal polynomials forw, and half may bearbitrarily distributed. Thus half the points haveNevai-Ullmann distribution of order α, and the rest are arbitrarily distributed. We also describe the possible distributions of the integration points, when the ruleI _n has precision other thann-1.     

Gaussian quadrature rules with exponential weights on (−1, 1)

We study the behavior of some “truncated” Gaussian rules based on the zeros of Pollaczek-type polynomials. These formulas are stable and converge with the order of the best polynomial approximation in suitable function spaces. Moreover, we apply these results to the related Lagrange interpolation process and to prove the stability and the convergence of a Nyström method for Fredholm integral equations of the second kind. Finally, some numerical examples are shown.     

On generalized Bohr–Sommerfeld quantization rules for operators with PT symmetry

We give Bohr–Sommerfeld rules corresponding to quasi-eigenvalues in the pseudospectrum for a one-dimensional h-pseudodifferential operator verifying PT symmetry.     

Pieri and Littlewood–Richardson rules for two rows and cluster algebra structure

We study an algebra encoding a twice-iterated Pieri rule for the representations of the general linear group and prove that it has the structure of a cluster algebra. We also show that its cluster variables invariant under a unipotent subgroup generate the highest weight vectors of irreducible representations occurring in the decomposition of the tensor product of two irreducible representations of the general linear group one of whom is labeled by a Young diagram with less than or equal to two rows.     

Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group’s relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters in this context.     

On the manipulability of competitive equilibrium rules in many-to-many buyer–seller markets

We analyze the manipulability of competitive equilibrium allocation rules for the simplest many-to-many extension of Shapley and Shubik’s (Int J Game Theory 1:111–130, 1972) assignment game. First, we show that if an agent has a quota of one, then she does not have an incentive to manipulate any competitive equilibrium rule that gives her her most preferred competitive equilibrium payoff when she reports truthfully. In particular, this result extends to the one-to-many (respectively, many-to-one) models the Non-Manipulability Theorem of the buyers (respectively, sellers), proven by Demange (Strategyproofness in the assignment market game. École Polytechnique, Laboratoire d’Économetrie, Paris, 1982), Leonard (J Polit Econ 91:461–479, 1983), and Demange and Gale (Econometrica 55:873–888, 1985) for the assignment game. Second, we prove a “General Manipulability Theorem” that implies and generalizes two “folk theorems” for the assignment game, the Manipulability Theorem and the General Impossibility Theorem, never proven before. For the one-to-one case, this result provides a sort of converse of the Non-Manipulability Theorem.     

What distributions of points are possible for convergent sequences of interpolatory integration rules?

Suppose that, forn1,

Modularity and incremental innovation: the roles of design rules and organizational communication

As little attention has been paid to the relationship between modularity and near decomposability, extant studies have not unveiled the impact of modularity on incremental innovation completely. We argue that the modular structure is a special case of nearly decomposable structure, in which the interdependencies between modules are specified by design rules, and the degree of modularity is defined by the level of near decomposability and the extent to which intermodule dependencies are specified. The results of our simulation experiments show that in the term of near decomposability, the increase of modularity leads to higher innovation advantage in the short term, but effective communication between modules can help systems with moderate and low modularity gain more innovation benefits in the long term; in the aspect of design rules, modularization may restrict the search space of the incremental innovation within each module, but under some conditions the option value of modularity may offset or even exceed the restriction effect of design rules.     

High-precision Gauss–Turán quadrature rules for Laguerre and Hermite weight functions

Procedures and corresponding Matlab software are presented for generating Gauss–Turán quadrature rules for the Laguerre and Hermite weight functions to arbitrarily high accuracy. The focus is on the solution of certain systems of nonlinear equations for implicitly defined recurrence coefficients. This is accomplished by the Newton–Kantorovich method, using initial approximations that are sufficiently accurate to be capable of producing n-point quadrature formulae for n as large as 42 in the case of the Laguerre weight function, and 90 in the case of the Hermite weight function.     

Spline-Gauss rules and the Nyström method for solving integral equations in quantum scattering

This paper is concerned with an n-point Gauss quadrature

where the points {t_n,j}ⁿ_j=1 and weights {w_n,j}ⁿ_j=1 are chosen to be exact when g is defined on a 2n-dimensional space of polynomial splines. The spline-Gauss quadrature is used in a Nyström method for solving integral equations of the second kind. A practical application is provided by solving integral equations that arise in quantum scattering theory.     

Why learning doesn’t add up: equilibrium selection with a composition of learning rules

In this paper, we investigate the aggregate behavior of populations of learning agents. We compare the outcomes in homogenous populations learning in accordance with imitate the best dynamics and with replicator dynamics to outcomes in populations that mix these two learning rules. New outcomes can emerge. In certain games, a linear combination of the two rules almost always attains an equilibrium that homogenous learners almost never locate. Moreover, even when almost all weight is placed on one learning rule, the outcome can differ from homogenous use of that rule. Thus, allowing even an arbitrarily small chance of using an alternative learning style can shift a population to select a different equilibrium.