Recursive Properties Research Articles

Robust linear matrix inequality‐based model predictive control with recursive estimation of the uncertainty polytope

The present work is concerned with the recursive estimation of the uncertainty polytope in a robust model predictive control (RMPC) framework. For this purpose, the unknown but bounded error method is employed to update the uncertainty polytope on the basis of sensor measurements at each sampling period. The recursive feasibility and asymptotic stability properties of the proposed approach are demonstrated as an extension of previous results concerning the RMPC formulation. For illustration, a simulated example involving an angular positioning system is presented. The results show that the proposed scheme provides a performance improvement, as indicated by the resulting cost function values.

A Generalized Topological Recursion for Arbitrary Ramification

The Eynard-Orantin topological recursion relies on the geometry of a Riemann surface S and two meromorphic functions x and y on S. To formulate the recursion, one must assume that x has only simple ramification points. In this paper we propose a generalized topological recursion that is valid for x with arbitrary ramification. We justify our proposal by studying degenerations of Riemann surfaces. We check in various examples that our generalized recursion is compatible with invariance of the free energies under the transformation (x,y) -> (y,x), where either x or y (or both) have higher order ramification, and that it satisfies some of the most important properties of the original recursion. Along the way, we show that invariance under (x,y) -> (y,x) is in fact more subtle than expected; we show that there exists a number of counter examples, already in the case of the original Eynard-Orantin recursion, that deserve further study.

Solution to Bethe–Salpeter equation via Mellin–Barnes transform

We consider the Mellin–Barnes (MB) transform of the triangle ladder-like scalar diagram in d=4 dimensions. It is shown how the multi-fold MB transform of the momentum integral corresponding to an arbitrary number of rungs is reduced to the two-fold MB transform. For this purpose, we use the Belokurov–Usyukina reduction method for four-dimensional scalar integrals in position space. The result is represented in terms of the Euler ψ function and its derivatives. We derive new formulas for the MB two-fold integration in the complex planes of two complex variables. We demonstrate that these formulas solve the Bethe–Salpeter equation. We comment on further applications of the solution to the Bethe–Salpeter equation for the vertices in N=4 supersymmetric Yang–Mills theory. We show that the recursive property of the MB transforms observed in the present work for that kind of diagrams has nothing to do with quantum field theory, the theory of integral transforms, or the theory of polylogarithms in general, but has its origin in a simple recursive property of smooth functions, which may be shown by using basic methods of mathematical analysis.

Hilbert’s Fourth Problem: Searching for Harmonic Hyperbolic Worlds of Nature

Recently the new unique classes of hyperbolic functions-hyperbolic Fibonacci functions based on the “golden ratio”, and hyperbolic Fibonacci l-functions based on the “metallic proportions” (l is a given natural number), were introduced in mathematics. The principal distinction of the new classes of hyperbolic functions from the classic hyperbolic functions consists in the fact that they have recursive properties like the Fibonacci numbers (or Fibonacci l-numbers), which are “discrete” analogs of these hyperbolic functions. In the classic hyperbolic functions, such relationship with integer numerical sequences does not exist. This unique property of the new hyperbolic functions has been confirmed recently by the new geometric theory of phyllotaxis, created by the Ukrainian researcherOleg Bodnar(“Bodnar’s hyperbolic geometry). These new hyperbolic functions underlie the original solution of Hilbert’s Fourth Problem (Alexey Stakhov and Samuil Aranson). These fundamental scientific results are overturning our views on hyperbolic geometry, extending fields of its applications (“Bodnar’s hyperbolic geometry”) and putting forward the challenge for theoretical natural sciences to search harmonic hyperbolic worlds of Nature. The goal of the present article is to show the uniqueness of these scientific results and their vital importance for theoretical natural sciences and extend the circle of readers. Another objective is to show a deep connection of the new results in hyperbolic geometry with the “harmonic ideas” of Pythagoras, Plato and Euclid.

BFKL approach and2→5maximally helicity violating amplitude inN=4super-Yang-Mills theory

We study MHV amplitude for the 2 -> 5 scattering in the multi-Regge kinematics. The Mandelstam cut correction to the BDS amplitude is calculated in the leading logarithmic approximation (LLA) and the corresponding remainder function is given to any loop order in a closed integral form. We show that the LLA remainder function at two loops for 2 -> 5 amplitude can be written as a sum of two 2 -> 4 remainder functions due to recursive properties of the leading order impact factors. We also make some generalizations for the MHV amplitudes with more external particles. The results of the present study are in agreement with all leg two loop symbol derived by Caron-Huot as shown in a parallel paper of one of the authors with collaborators.

Individual haplotype assembly of Apis mellifera (honeybee) using a practical branch and bound algorithm

A haplotype is a single nucleotide polymorphism (SNP) sequence and a representative genetic marker describing the diversity of biological organs. Haplotypes have a wide range of applications such as pharmacology and medical applications. In particular, as a highly social species, haplotypes of the Apis mellifera (honeybee) benefit human health and medicine in diverse areas, including venom toxicology, infectious disease, and allergic disease. For this reason, assembling a pair of haplotypes from individual SNP fragments drives research and generates various computational models for this problem. The minimum error correction (MEC) model is an important computational model for an individual haplotype assembly problem. However, the MEC model has been proved to be NP-hard; therefore, no efficient algorithm is available to address this problem. In this study, we propose an improved version of a branch and bound algorithm that can assemble a pair of haplotypes with an optimal solution from SNP fragments of a honeybee specimen in practical time bound. First, we designed a local search algorithm to calculate the good initial upper bound of feasible solutions for enhancing the efficiency of the branch and bound algorithm. Furthermore, to accelerate the speed of the algorithm, we made use of the recursive property of the bounding function together with a lookup table. After conducting extensive experiments over honeybee SNP data released by the Human Genome Sequencing Center, we showed that our method is highly accurate and efficient for assembling haplotypes.

GLL parse-tree generation

Backtracking techniques which are often used to extend recursive descent (RD) parsers can have explosive run-times and cannot deal with grammars with left recursion. GLL parsers are fully general, worst-case cubic parsers which have the recursive descent-like property that they are easy to write and to use for grammar debugging. They have the direct relationship with the grammar that an RD parser has. In this paper we give an algorithm for generating GLL parsers which build an SPPF representation of the derivations of the input, complementing our existing GLL recognition algorithm, and we show that such parsers and recognisers are worst-case cubic.

The Electronic Structure of Coupled Semiconductor Quantum Dots Arranged as a Graphene Hexagonal Lattice under a Magnetic Field

We study the electronic spectrum of coupled quantum dots (QDs) arranged as a graphene hexagonal lattice in the presence of an external perpendicular magnetic field. In our tight-binding model, the effect of the magnetic field is included in both the Peierls phase of the Hamiltonian and the tight-binding basis Wannier function. The energy of the system is analyzed when the magnetic flux through the lattice unit cell is a rational fraction of the quantum flux. The calculated spectrum has recursive properties, similar to those of the classical Hofstadter butterfly. However, unlike the ideal Hofstadter butterfly structure, our result is asymmetric since the impacts of the specific material and the magnetic field on the wavefunctions are included, making the results more realistic.

On recursive properties of certain p-adic Whittaker functions

We investigate recursive properties of certain p -adic Whittaker functions (of which representation densities of quadratic forms are special values). The proven relations can be used to compute them explicitly in arbitrary dimensions in terms of orbits under the orthogonal group acting on the representations. These relations have implications for the first and second special derivatives of the Euler product over all p of these Whittaker functions. These Euler products appear as the main part of the Fourier coefficients of Eisenstein series associated with the Weil representation. In case of signature ( l − 2 , 2 ) , we interpret these implications in terms of the theory of Borcherdsʼ products on orthogonal Shimura varieties. This gives some evidence for Kudlaʼs conjectures in higher dimensions.

Optimal replacement policy for safety-related multi-component multi-state systems

This paper investigates replacement scheduling for non-repairable safety-related systems (SRS) with multiple components and states. The aim is to determine the cost-minimizing time for replacing SRS while meeting the required safety. Traditionally, such scheduling decisions are made without considering the interaction between the SRS and the production system under protection, the interaction being essential to formulate the expected cost to be minimized. In this paper, the SRS is represented by a non-homogeneous continuous time Markov model, and its state distribution is evaluated with the aid of the universal generating function. Moreover, a structure function of SRS with recursive property is developed to evaluate the state distribution efficiently. These methods form the basis to derive an explicit expression of the expected system cost per unit time, and to determine the optimal time to replace the SRS. The proposed methodology is demonstrated through an illustrative example.

Recursive properties of branching and BGG resolution

Recurrent relations for branching coefficients are based on a special type of singular element decomposition. We show that this decomposition can be used to construct the parabolic Verma modules and finally to obtain the generalized Weyl-Verma formulas for characters. We demonstrate how branching coefficients can determine the generalized BGG resolution sequence.

Polynomial approximation of rational Bézier curves with constraints

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bezier curve by the polynomial Bezier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm.

On the correlation between primary damage and long-term nanostructural evolution in iron under irradiation

Atomic displacement cascades in solids are complex phenomena, the outcome of which can be statistically characterised by properties such as their spatial extent, morphology and the spatial correlation of defects. Some properties scale in a simple way with parameters such as the cascade energy, others have limited variability with energy, for example point defect cluster size distributions. Taking advantage of the latter invariance, we use object kinetic Monte Carlo simulations to demonstrate that most properties of displacement cascade play no significant role in the evolution of point defect cluster size distributions after long enough time. It is suggested that reliable long-term predictions are possible, when using only the self-interstitial and vacancy cluster size distributions from low energy displacement cascades as building blocks to represent the complete spectrum of cascade energies obtained under neutron irradiation conditions. This is shown on the basis of recursive properties of displacement cascades evidenced for the first time and taking only approximately into account the average volumes in which vacancies and self-interstitial atoms are confined. The model has been successfully used to simulate the evolution of point defect clusters in iron for displacement rates in the range of 10 −6 dpa/s and doses of the order of 0.1 dpa. The applicability beyond this range and to more complex materials is discussed.

Semiparametric inference in mixture models with predictive recursion marginal likelihood

Predictive recursion is an accurate and computationally efficient algorithm for nonparametric estimation of mixing densities in mixture models. In semiparametric mixture models, however, the algorithm fails to account for any uncertainty in the additional unknown structural parameter. As an alternative to existing profile likelihood methods, we treat predictive recursion as a filter approximation to fitting a fully Bayes model, whereby an approximate marginal likelihood of the structural parameter emerges and can be used for inference. We call this the predictive recursion marginal likelihood. Convergence properties of predictive recursion under model mis-specification also lead to an attractive construction of this new procedure. We show pointwise convergence of a normalized version of this marginal likelihood function. Simulations compare the performance of this new marginal likelihood approach that of existing profile likelihood methods as well as Dirichlet process mixtures in density estimation. Mixed-effects models and an empirical Bayes multiple testing application in time series analysis are also considered.

Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs)

We present a filter design formalism for the synthesis of coupled-resonator optical waveguide (CROW) filters. This formalism leads to expressions and a methodology for deriving the coupling coefficients of CROWs for the desired filter responses and is based on coupled-mode theory as well as the recursive properties of the coupling matrix. The coupling coefficients are universal and can be applied to various types of resonators. We describe a method for the conversion of the coupling coefficients to the parameters based on ring resonators and grating defect resonators. The designs of Butterworth and Bessel CROW filters are demonstrated as examples.

The Therapeutic Alliance: Implications for Therapeutic Process and Therapeutic Goals

Interest in the importance of the therapeutic alliance to the psychotherapeutic process has recently grown across theoretical domains. Through a case example the author demonstrates that in some cases a familiar pattern of relating with the therapist may be more important for the client in terms of treatment success than positive regard early in treatment. Also, charting shifts in the quality of the relationship rather than quantitative changes in the alliance may help the therapist navigate the recursive emergent properties of relationship, goals of treatment and choice and timing of interventions. Finally, the author suggests that the therapist’s internalization of the client may represent a significant leap in the relationship, and help reduce some of the “sloppiness” that is inherent in the therapeutic process.

Folded fans and string functions

Folded fans Fψ describe recursion properties of weights of integrable highest weight modules L u . Being considered simultaneously for the set of string functions $$ \sigma_s^u $$ belonging to a fundamental Weyl chamber and corresponding to the same congruence class, the system of recursion relations gives rise to an equation which connects the string functions and the power series depending on multiplicities of weights of a folded fan Fψ. We apply these equations to study properties of string functions $$ \sigma_s^u $$ associated with integrable modules for affine Lie algebras. New important relations for string functions are thus obtained. The set of folded fans provides a compact and effective tool to study them. Bibliography: 7 titles.

Constrained multi-degree reduction of triangular Bézier surfaces using dual Bernstein polynomials

We propose a novel approach to the problem of multi-degree reduction of Bézier triangular patches with prescribed boundary control points. We observe that the solution can be given in terms of bivariate dual discrete Bernstein polynomials. The algorithm is very efficient thanks to using the recursive properties of these polynomials. The complexity of the method is O ( n 2 m 2 ) , n and m being the degrees of the input and output Bézier surfaces, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined triangular Bézier surfaces, the result is a composite surface of global C r continuity with a prescribed order r . Some illustrative examples are given.

Single machine scheduling with precedence constraints and positionally dependent processing times

In many real-life situations the processing conditions in scheduling models cannot be viewed as given constants since they vary over time thereby affecting actual durations of jobs. We consider single machine scheduling problems of minimizing the makespan in which the processing time of a job depends on its position (with either cumulative deterioration or exponential learning). It is often found in practice that some products are manufactured in a certain order implied, for example, by technological, marketing or assembly requirements. This can be modeled by imposing precedence constraints on the set of jobs. We consider scheduling models with positional deterioration or learning under precedence constraints that are built up iteratively from the prime partially ordered sets of a bounded width (this class of precedence constraints includes, in particular, series–parallel precedence constraints). We show that objective functions of the considered problems satisfy the job module property and possess the recursion property. As a result, the problems under consideration are solvable in polynomial time.

Mean first-passage time for random walks on generalized deterministic recursive trees

We describe a technique that allows the exact analytical computation of the mean first passage time (MFPT) for infinite families of trees using their recursive properties. The method is based in the relationship between the MFPT and the eigenvalues of the Laplacian matrix of the trees but avoids their explicit computation. We apply this technique to find the MFPT for a family of generalized deterministic recursive trees. The method, however, can be adapted to other self-similar tree families.