Formal Power Research Articles

Formal power series approach to nonlinear systems with additive static feedback

The goal of this paper is to compute the generating series of a closed-loop system when the plant is described in terms of a Chen–Fliess series and an additive static output feedback is applied. The first step is to consider the so-called Wiener-Fliess connection consisting of a Chen–Fliess series followed by a memoryless function. To explicitly compute the generating series, two Hopf algebras are needed: the existing output feedback Hopf algebra used to describe dynamic output feedback and the Hopf algebra of the shuffle group. These two combinatorial structures are combined to compute what will be called the Wiener–Fliess feedback product. It will be shown that this product has a natural interpretation as a transformation group acting on the plant and preserves the relative degree of the plant. The convergence of the Wiener–Fliess composition product and the additive static feedback product are characterised.

On analytic structure of weighted shifts on generalized directed semi-trees

Inspired by natural classes of examples, we define generalized directed semi-trees and construct weighted shifts on them. Given an n-tuple of generalized directed semi-trees with certain properties, we associate an n-tuple of multiplication operators on a Hilbert space of formal power series. Under certain conditions, turns out to be a reproducing kernel Hilbert space consisting of holomorphic functions on some domain in and the n-tuple of multiplication operators on is unitarily equivalent to an n-tuple of weighted shifts on generalized directed semi-trees. Finally, we exhibit two classes of examples of n-tuples of operators, which can be intrinsically identified as weighted shifts on generalized directed semi-trees.

On the growth of Artin–Tits monoids and the partial theta function

We present a new procedure to determine the growth function of a homogeneous Garside monoid, with respect to the finite generating set formed by the atoms. In particular, we present a formula for the growth function of each Artin–Tits monoid of spherical type (hence of each braid monoid) with respect to the standard generators, as the inverse of the determinant of a very simple matrix.Using this approach, we show that the exponential growth rates of the Artin–Tits monoids of type An (positive braid monoids) tend to 3.233636… as n tends to infinity. This number is well-known, as it is the growth rate of the coefficients of the only formal power series x0(y)=−(1+y+2y2+4y3+9y4+⋯) which is the leading root of the classical partial theta function.We also describe the sequence 1,1,2,4,9,… formed by the coefficients of −x0(y), by showing that its kth term (the coefficient of yk) is equal to the number of braids of length k, in the positive braid monoid A∞ on an infinite number of strands, whose maximal lexicographic representative starts with the first generator a1. This is an unexpected connection between the partial theta function and the theory of braids.

Continuity of Formal Power Series Products in Nonlinear Control Theory

Formal power series products appear in nonlinear control theory when systems modeled by Chen–Fliess series are interconnected to form new systems. In fields like adaptive control and learning systems, the coefficients of these formal power series are estimated sequentially with real-time data. The main goal is to prove the continuity and analyticity of such products with respect to several natural (locally convex) topologies on spaces of locally convergent formal power series in order to establish foundational properties behind these technologies. In addition, it is shown that a transformation group central to describing the output feedback connection is in fact an analytic Lie group in this setting with certain regularity properties.

Power sources among district health managers in Ghana: a qualitative study

BackgroundIn Ghana district directors of health services and district hospital medical superintendents provide leadership and management within district health systems. A healthy relationship among these managers is dependent on the clarity of formal and informal rules governing their routine duties. These rules translate into the power structures within which district health managers operate. However, detailed nuanced studies of power sources among district health managers are scarce. This paper explores how, why and from where district health directors and medical superintendents derive power in their routine functions.MethodsA multiple case study was conducted in three districts; Bongo, Kintampo North and Juaboso. In each case study site, a cross-sectional design was used to explore the research question. Purposive sampling technique was used to select study sites and 61 participants for interview and focus group discussion. A total of 11 interviews (3 in each district and 2 with deputy regional directors), and 9 focus group discussions (3 in each district) were conducted. Transcriptions of the voice-recordings were done verbatim, cleaned and imported into the Nvivo version 11 software for analysis using the inductive content analysis approach.ResultsThe findings revealed that legitimacy provides formal power source for district health managers since they are formally appointed by the Director General of the Ghana Health Service after going through the appointment processes. These appointments serve as the primary power source for district health managers based on the existing legal and policy framework of the Ghana Health Service. Additionally, resource control especially finances and medical dominance are major informal sources of power that district health managers often employ for the management and administration of their functional areas in the health districts.ConclusionsThe study concludes that district health managers derive powers primarily from their positions within the hierarchical structure (legitimacy) of the district health system. Secondary sources of power stems from resource control (medical dominance and financial dominance), and these power sources inform the way district health managers relate to each other. This paper recommends that district health managers are oriented to understand the power dynamics in the district health system.

Formal Gevrey solutions: in analytic germs—for higher order holomorphic PDEs

We consider a family of holomorphic PDEs whose singular locus is given by the zero set of an analytic map P with P(0)=0. Our goal is to establish conditions for the existence and uniqueness of formal power series solutions and to determine their divergence rate. In fact, we prove that the solution is Gevrey in P, giving new information on divergency while compared to the classical Gevrey classes. If P is not singular at 0, we also provide Poincaré conditions to recover convergent solutions. Our strategy is to extend the dimension and lift the given PDE to a problem where results of singular PDEs can be applied. Finally, examples where the Gevrey class in P is optimal are included.

Automata and finite order elements in the Nottingham group

The Nottingham group at 2 is the group of (formal) power series t+a2t2+a3t3+⋯ in the variable t with coefficients ai from the field with two elements, where the group operation is given by composition of power series. The depth of such a series is the largest d⩾1 for which a2=…=ad=0.Only a handful of power series of finite order (forcedly a power of 2) are explicitly known through a formula for their coefficients. We argue in this paper that it is advantageous to describe such series in closed computational form through automata, based on effective versions of proofs of Christol's theorem identifying algebraic and automatic series.Up to conjugation, there are only finitely many series σ of order 2n with fixed break sequence (i.e. the sequence of depths of σ∘2i). Starting from Witt vector or Carlitz module constructions, we give an explicit automaton-theoretic description of: (a) representatives up to conjugation for all series of order 4 with break sequence (1,m) for m<10; (b) representatives up to conjugation for all series of order 8 with minimal break sequence (1,3,11); and (c) an embedding of the Klein four-group into the Nottingham group at 2.We study the complexity of the new examples from the algebro-geometric properties of the equations they satisfy. For this, we generalise the theory of sparseness of power series to a four-step hierarchy of complexity, for which we give both Galois-theoretic and combinatorial descriptions. We identify where our different series fit into this hierarchy. We construct sparse representatives for the conjugacy class of elements of order two and depth 2μ±1(μ⩾1). Series with small state complexity can end up high in the hierarchy. This is true, for example, for a new automaton we found, representing a series of order 4 with 5 states (the minimal possible number for such a series).

Fixed Point Property of Variable Exponent Cesàro Complex Function Space of Formal Power Series under Premodular

We have defined the variable exponent of the Cesàro complex function space of formal power series. We have constructed the prequasi-ideal generated by s -numbers and this new space of complex functions. We present some topological and geometric structures of this class of ideal. The existence of Caristi’s fixed point is examined. Some geometric properties related to the fixed point theory are presented. Finally, real-world examples and applications show solutions to some nonlinear difference equations.

The Relational Dynamics of Issue-Selling: Enacting Different Genres for Dealing with Discontent

Actors without formal power can initiate and shape organizational change through issue selling as a means to address their concerns about organizational practices. The literature has emphasized the need for perspective taking and found discontent to be associated with inhibiting fear. Based on a longitudinal field study of middle managers trying to instigate strategic renewal in six different manufacturing companies, we find that discontent can also energize issue selling, particularly if sellers and recipients also engage in perspective taking. We identified three different genres of issue selling and enacting discontent that are jointly shaped by sellers and recipients: 1) productive confrontation, 2) avoiding escalation, and 3) collective moaning. In a process model of issue selling as framing, mobilizing and pursuing, we illustrate how the genres form part of an organization’s collective repertoire of initiating and responding to change. Our findings have implications for pursuing change as well as for managers seeking to create a constructive environment for expressing discontent.

Structural and psychological empowerment and its influencing factors among nursing students in Oman.

This study explored the empowerment of nursing students to contribute to nursing education and recommend ways of increasing the engagement of nursing students in their training. The development of empowerment among nursing students leads to the maximum achievement of learning competencies and enables them to eventually become competent nurses. This study examined the relationship between student nurses' characteristics, structural empowerment, and psychological empowerment. A total of 185 nursing students were recruited through systematic sampling at Sultan Qaboos University, with a 92% response rate. Nursing students' level of psychological empowerment, in terms of meaning, competence, self-determination, and impact, and their level of structural empowerment, in terms of access to opportunity, support, information, and resources were investigated using a self-designed questionnaire. The results showed that "access to support" was rated as the highest dimension by the students, followed by "access to information," "informal power," "access to opportunity," "access to resources," and "formal power." The global empowerment mean score was 3.64 (SD=1.01), and the total structural empowerment score was 20.58 (SD=3.62). The multivariate multiple regression analysis revealed that students' year level and involvement in school organizations had multivariate effects on the four dimensions of psychological empowerment. Students who were involved in school organizations had higher scores for the dimensions of "meaning," "competence," and "impact" than students who were not involved in any school organization. Power can be either developed or acquired, and its definition is expressed based on the achievement objective. Age is not a barrier, as it had little or no impact on nursing student experiences and no correlation with structural empowerment.

The combinator M and the Mockingbird lattice

AbstractWe study combinatorial and order theoretic structures arising from the fragment of combinatory logic spanned by the basic combinator ${{\mathbf{M}}}$ . This basic combinator, named as the Mockingbird by Smullyan, is defined by the rewrite rule ${{\mathbf{M}}} \mathsf{x}_1 \to \mathsf{x}_1 \mathsf{x}_1$ . We prove that the reflexive and transitive closure of this rewrite relation is a partial order on terms on ${{\mathbf{M}}}$ and that all connected components of its rewrite graph are Hasse diagram of lattices. This last result is based on the introduction of new lattices on duplicative forests, which are sorts of treelike structures. These lattices are not graded, not self-dual, and not semi-distributive. We present some enumerative properties of these lattices like the enumeration of their elements, of the edges of their Hasse diagrams, and of their intervals. These results are derived from formal power series on terms and on duplicative forests endowed with particular operations.

On the problem of periodicity of continued fraction expansions of for cubic polynomials over algebraic number fields

AbstractWe obtain a complete description of the fieldsthat are extensions ofof degree at mostand the cubic polynomialssuch that the expansion ofinto a continued fraction in the field of formal power seriesis periodic. We prove a finiteness theorem for cubic polynomialswith a periodic expansion offor extensions ofof degree at most. We obtain a description of the periodic elementsfor the cubic polynomialsdefining elliptic curves with points of order,.Bibliography: 19 titles.

Decompositions and eigenvectors of Riordan matrices

Riordan matrices are infinite lower triangular matrices determined by a pair of formal power series over the real or complex field. These matrices have been mainly studied as combinatorial objects with an emphasis placed on the algebraic or combinatorial structure. The present paper contributes to the linear algebraic discussion with an analysis of Riordan matrices by means of the interaction of the properties of formal power series with the linear algebra. Specifically, it is shown that if a Riordan matrix A is an n×n pseudo-involution then the singular values of A must come in reciprocal pairs. Moreover, we give a complete analysis of existence and nonexistence of the eigenvectors of Riordan matrices. This leads to a surprising partition of the group of Riordan matrices into matrices with three different types of sets of eigenvectors. Finally, given a nonzero vector v, we investigate the Riordan matrices A that stabilize the vector v, i.e.Av=v.

On Existence and Uniqueness of Formal Power Series Solutions of Algebraic Ordinary Differential Equations

Given an algebraic ordinary differential equation (AODE), we propose a computational method to determine when a truncated power series can be extended to a formal power series solution. If a certain regularity condition on the given AODE or on the initial values is fulfilled, we compute all of the solutions. Moreover, when the existence is confirmed, we present the algebraic structure of the set of all formal power series solutions.

The planned curriculum—Not just a matter of teachers

AbstractTeachers are responsible for teaching, and when they plan, they are part of a complex non‐linear social practice of curriculum making. When planning, teachers draw on curriculum materials, which are often designed to promote reform; however, previous studies show that this is not always the case. A study on planning for mathematics teaching in Sweden showed that teachers are influenced by those who have formal and informal power when they plan. This article focuses on structural influence on teachers' planning as curriculum work and the possible consequences for teacher agency. The data comes from a focus group discussion about planning with mathematics teachers in primary school. Tools for the analysis were inspired by actor analysis and “three pillars of institutions”. Results show that the structural influence on teachers is at the regulative, the normative, and the cultural‐cognitive levels. Regulative influence seems to be short‐term, while cultural‐cognitive influence seems to last over the long‐term. The most prominent influence was normative and related to the textbook. Results shed light on teachers' curriculum work in the micro site of activity when they are planning for mathematics teaching. Based on the results, the article suggests that teachers—as key actors when it comes to teaching—must have the opportunity to discuss and problematize assumptions, ideas that are taken‐for‐granted, and power relations in curriculum making to achieve agency and be the professional agents of change they are expected to be.

Calculating the Euler Characteristic of the Moduli Space of Curves

The orbifold Euler characteristic of the moduli space \({{\cal M}_{g;1}}\) of genus g smooth curves with one marked point (g ≥ 1) was calculated by Harer and Zagier: \(\chi ({{\cal M}_{g;1}}) = \zeta (1 - 2g) = - {B_{2g}}/(2g)\), where ζ is the Riemann zeta function and B2g is the (2g)th Bernoulli number. We give a shorter proof of this result using only formal power series and classical combinatorics.

On Mahler's Classification of Formal Power Series Over a Finite Field

Abstract Let K be a finite field, K(x) be the field of rational functions in x over K and K be the field of formal power series over K. We show that under certain conditions integral combinations with algebraic formal power series coefficients of a U 1-number in K are Um -numbers in K , where m is the degree of the algebraic extension of K(x), determined by these algebraic formal power series coefficients.

N-point spherical functions and asymptotic boundary KZB equations

Let G be a split real connected Lie group with finite center. In the first part of the paper we define and study formal elementary spherical functions. They are formal power series analogues of elementary spherical functions on G in which the role of the quasi-simple admissible G-representations is replaced by Verma modules. For generic highest weight we express the formal elementary spherical functions in terms of Harish-Chandra series and integrate them to spherical functions on the regular part of G. We show that they produce eigenstates for spin versions of quantum hyperbolic Calogero–Moser systems. In the second part of the paper we define and study special subclasses of global and formal elementary spherical functions, which we call global and formal N-point spherical functions. Formal N-point spherical functions arise as limits of correlation functions for boundary Wess–Zumino–Witten conformal field theory on the cylinder when the position variables tend to infinity. We construct global N-point spherical functions in terms of compositions of equivariant differential intertwiners associated with principal series representations, and express them in terms of Eisenstein integrals. We show that the eigenstates of the quantum spin Calogero–Moser system associated to N-point spherical functions are also common eigenfunctions of a commuting family of first-order differential operators, which we call asymptotic boundary Knizhnik–Zamolodchikov–Bernard operators. These operators are explicitly given in terms of folded classical dynamical r-matrices and associated dynamical k-matrices.

Quantum queer supergroups via υ-differential operators

By using certain quantum differential operators, we construct a super representation for the quantum queer supergroup Uυ(qn). The underlying space of this representation is a deformed polynomial superalgebra in 2n2 variables whose homogeneous components can be used as the underlying spaces of queer q-Schur superalgebras. We then extend the representation to its formal power series algebra which contains a (super) submodule isomorphic to the regular representation of Uυ(qn). A monomial basis M for Uυ(qn) plays a key role in proving the isomorphism. In this way, we may present the quantum queer supergroup Uυ(qn) by another new basis L together with some explicit multiplication formulas by the generators. As an application, similar presentations are obtained for queer q-Schur superalgebras via the above mentioned homogeneous components.The existence of the bases M and L and the new presentation show that the seminal construction of quantum gln established by Beilinson–Lusztig–MacPherson thirty years ago extends to this “queer” quantum supergroup via a completely different approach.

On quotients of formal power series

Quotient is a basic operation of formal languages, which plays a key role in the construction of minimal deterministic finite automata (DFA) and the universal automata. In this paper, we extend this operation to formal power series and systemically investigate its implications in the study of weighted automata. In particular, we define two quotient operations for formal power series which are called (left or right) quotients and (left or right) residuals respectively. Algebraic properties of the quotient operation and closure properties (w.r.t. regular and context-free series) under quotients and residuals are also investigated for formal power series. Using these operations, we define for each formal power series A two weighted automata that accept A. The first is the minimal deterministic weighted automaton of A, and the second is the universal weighted automaton of A. An effective algebraic method to construct the universal automaton is also presented.