Finite Polynomial Research Articles

Strategic behavior, social optimization, and revenue maximization in queues with infinite servers

Motivated by the impact of emerging technologies on (toll) parks, this paper studies a problem of customers’ strategic behavior, social optimization, and revenue maximization for infinite-server queues. More specifically, we assume that a customer’s utility consists of a positive reward for receiving service minus a cost caused by the other customers in the system. In the observable setting, we show the existence, uniqueness, and expressions of the individual equilibrium threshold, the socially optimal threshold, and the optimal revenue threshold, respectively. Then, we prove that the optimal revenue threshold is smaller than the socially optimal threshold, which is smaller than the individual one. Furthermore, we also extend the cost functions to any finite polynomial function with nonnegative coefficients. In the unobservable setting, we derive the joining probabilities of individual equilibrium and optimal revenue. Finally, using numerical experiments, we complement our results and compare the social welfare and the revenue under these two information levels.

Finite polynomial cohomology with coefficients

We introduce a theory of finite polynomial cohomology with coefficients in this paper. We prove several basic properties and introduce an Abel–Jacobi map with coefficients. As applications, we use this cohomology theory to study the arithmetic of compact Shimura curves over \mathbb{Q} and to simplify the proofs of the works of Darmon–Rotger and Bertolini–Darmon–Prasanna.

Reduced submodules of finite dimensional polynomial modules

Reduced submodules of finite dimensional polynomial modules

Some Applications of Dynamical Belyi Polynomials

We give necessary and sufficient conditions for post-critically finite polynomials to have persistent bad reduction at a given prime. We also answer in the negative a pair of questions posed by Silverman about conservative polynomials. Our proofs rely on conservative dynamical Belyi polynomials as exemplars of PCF (resp. conservative) maps.

In-in correlators and scattering amplitudes on a causal set

Causal set theory is an approach to quantum gravity in which spacetime is fundamentally discrete at the Planck scale and takes the form of an irregular Lorentzian lattice, or “causal set,” from which continuum spacetime emerges in a large-scale (low-energy) approximation. In this work, we present new developments in the framework of interacting quantum field theory on causal sets. We derive a diagrammatic expansion for in-in correlators in local scalar field theories with finite polynomial interactions. We outline how these same correlators can be computed using the double-path integral, which acts as a generating functional for the in-in correlators. We modify the in-in generating functional to obtain a generating functional for in-out correlators. We define a notion of scattering amplitudes on causal sets with noninteracting past and future regions and verify that they are given by S-matrix elements (matrix elements of the time-evolution operator). We describe how these formal developments can be implemented to compute early Universe observables under the assumption that spacetime is fundamentally discrete. Published by the American Physical Society 2024

‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme

A new interpretation and applications of the ‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of 0 , 1 , … , N , constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the ‘Diophantine’ and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm ‘eigenvectors’ of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.

Leveraging Variable Density Honeycomb Structures for Innovative Design in Mission-Critical Embedded Devices

The imperative for lightweighting technologies, paramount in mission-critical cyber-physical systems (CPSs) including aerospace, automotive and allied sectors, hinges upon optimizing energy efficiency and curbing product weight. Honeycomb structures, celebrated for their exceptional strength-to-weight ratio, have indisputably guided the pursuit of lightweight design. This paper expounds upon the versatility of honeycomb structures by scrutinizing their in-plane mechanical attributes. Leveraging finite element simulations and polynomial fitting, we enhance the prevailing equivalent elastic modulus model for uniform honeycomb structures, expanding its domain to encompass a broader spectrum of relative density values. Deliberations ensue concerning the model’s constraints and its inapplicability to nonuniform honeycomb structures. The investigation introduces nodes as pivotal influencers in the mechanical comportment of nonuniform honeycomb structures, delineating the nexus between the equivalent elastic modulus and node dimensions through a fusion of finite element simulations and mechanical experimentation. Furthermore, this research delves into the tenets and constructs of density-based variable density methodologies within the ambit of topology optimization, with an overarching goal of maximizing stiffness. We furnish a holistic design protocol for optimizing honeycomb structures, underscored by a pragmatic instantiation of the density-based variable density approach. Scrutinizing the geometric interplay between honeycomb structures and design spaces, we posit an innovative paradigm employing concentric circles to approximate cellular envelopes, streamlining numerical cartography and the conversion of optimization outputs into variable density honeycomb configurations. Evaluation of the in-plane mechanical attributes of variable density honeycomb structures reveals that TPU material augments the resilience of both uniform and variable density honeycomb structures, whereas topology optimization amplifies specific stiffness and resilience modulus in variable density honeycomb structures relative to their uniform counterparts. This study sheds light on the complexities of honeycomb structures, providing valuable insights for their optimization in lightweight applications.

Real-space spectral simulation of quantum spin models: Application to generalized Kitaev models

The proliferation of quantum fluctuations and long-range entanglement presents an outstanding challenge for the numerical simulation of interacting spin systems with exotic ground states. Here, we present a toolset of Chebyshev polynomial-based iterative methods that provides a unified framework to study the thermodynamical properties, critical behavior and dynamics of frustrated quantum spin models with controlled accuracy. Similar to previous applications of the Chebyshev spectral methods to condensed matter systems, the algorithmic complexity scales linearly with the Hilbert space dimension and the Chebyshev truncation order. Using this approach, we study two paradigmatic quantum spin models on the honeycomb lattice: the Kitaev-Heisenberg (K-H) and the Kitaev-Ising (K-I) models. We start by applying the Chebyshev toolset to compute nearest-neighbor spin correlations, specific heat and entropy of the K-H model on a 24-spin cluster. Our results are benchmarked against exact diagonalization and a popular iterative method based on thermal pure quantum states. The transitions between a variety of magnetic phases, namely ferromagnetic, Néel, zigzag and stripy antiferromagnetic and quantum spin liquid phases are obtained accurately and efficiently. We also determine the temperature dependence of the spin correlations, over more than three decades in temperature, by means of a finite temperature Chebyshev polynomial method introduced here. Finally, we report novel dynamical signatures of the quantum phase transitions in the K-I model. Our findings suggest that the efficiency, versatility and low-temperature stability of the Chebyshev framework developed here could pave the way for previously unattainable studies of quantum spin models in two dimensions.

Levy and Thurston obstructions of finite subdivision rules

AbstractFor a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. To show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and Tan [Combining rational maps and controlling obstructions. Ergod. Th. & Dynam. Sys.18(1) (1998), 221–245] to a graph intersecting obstruction theorem. As a corollary, we prove that for a pair of post-critically finite polynomials, if at least one polynomial has core entropy zero, then their mating has a Levy cycle if and only if the mating has a Thurston obstruction.

Soft analyzer for sulfur content in tail oil of hydrocracking process based on adaptively regularized dynamic polynomial PLS

AbstractHydrocracking process is an essential and widely used means to remove impurities such as sulfur, nitrogen, and oxygen from heavy oil in refineries. The sulfur content in the tail oil of the hydrogenation process is a key quality index that must be strictly monitored and controlled. However, the traditional measurement ways of the sulfur content suffer from either large delays or high investment and maintenance costs. To this end, a predictive model named “adaptively regularized dynamic polynomial partial least squares (ARDPPLS)” is proposed to develop soft analyzer for sulfur content estimation. The ARDPPLS takes complicated characteristics of the hydrogenation process into consideration, including process dynamics, nonlinearities, and transportation delays of materials and energies. Specifically, the dynamics and nonlinearities are modeled by the finite impulse response paradigm and polynomial fitting, respectively, and the delays of explanatory variables are automatically determined using differential evolution. In particular, a Bayesian inference‐based adaptive regularization scheme for the polynomial partial least squares is designed to tackle overfitting that results from high‐order variable augmentation and high‐order polynomials. The performance of the ARDPPLS is evaluated by a real‐world industrial wax oil hydrocracking process, showing the effectiveness of the ARDPPLS.

Totally real algebraic integers in short intervals, Jacobi polynomials, and unicritical families in arithmetic dynamics

AbstractWe classify all postcritically finite unicritical polynomials defined over the maximal totally real algebraic extension of . Two auxiliary results used in the proof of this result may be of some independent interest. The first is a recursion formula for the diameter of an interval, which uses properties of Jacobi polynomials. The second is a numerical criterion that allows one to give a bound on the degree of any algebraic integer having all of its complex embeddings in a real interval of length less than 4.

Some New Families of Finite Orthogonal Polynomials in Two Variables

In this paper, we generalize the study of finite sequences of orthogonal polynomials from one to two variables. In doing so, twenty three new classes of bivariate finite orthogonal polynomials are presented, obtained from the product of a finite and an infinite family of univariate orthogonal polynomials. For these new classes of bivariate finite orthogonal polynomials, we present a bivariate weight function, the domain of orthogonality, the orthogonality relation, the recurrence relations, the second-order partial differential equations, the generating functions, as well as the parameter derivatives. The limit relations among these families are also presented in Labelle’s flavor.

Conditional GrÖbner Basis: GrÖbner Basis Detection with Parameters

For a given finite polynomial set, finding a monomial order such that the given set is already a GrÖbner basis for the ideal generated by the given set with respect to the found monomial order is called GrÖbner basis detection (GBD) problem and there is also its simpler version, called structural GrÖbner basis detection (SGBD) problem. In this short communication, we give algorithms to solve these problems for polynomials with parameters on their coefficients.

Monk rules for type GLn Macdonald polynomials

In this paper we give Monk rules for Macdonald polynomials which are analogous to the Monk rules for Schubert polynomials. These formulas are similar to the formulas given by Baratta [6], but our method of derivation is to use Cherednik's intertwiners. Deriving Monk rules by this technique addresses the relationship between the work of Baratta and the product formulas of Yip [19]. Specializations of the Monk formula's at q=0 and/or t=0 provide Monk rules for Iwahori-spherical polynomials and for finite and affine key polynomials.

New finite-type multi-indexed orthogonal polynomials obtained from state-adding Darboux transformations

Abstract The Hamiltonians of finite-type discrete quantum mechanics with real shifts are real symmetric matrices of order N + 1. We discuss the Darboux transformations with higher-degree (>N) polynomial solutions as seed solutions. They are state-adding and the resulting Hamiltonians after M steps are of order N + M + 1. Based on 12 orthogonal polynomials ((q-)Racah, (dual, q-)Hahn, Krawtchouk, and five types of q-Krawtchouk), new finite-type multi-indexed orthogonal polynomials are obtained, which satisfy second-order difference equations, and all the eigenvectors of the deformed Hamiltonian are described by them. We also present explicit forms of the Krein–Adler-type multi-indexed orthogonal polynomials and their difference equations, which are obtained from the state-deleting Darboux transformations with lower-degree (≤N) polynomial solutions as seed solutions.

A Polynomial Chaos Expansion Method for Mechanical Properties of Flexoelectric Materials Based on the Isogeometric Finite Element Method

The paper proposes a method for analyzing the mechanical properties of flexoelectric materials based on the isogeometric finite element method (IGA-FEM) and polynomial chaos expansion (PCE). The method discretizes the flexoelectric governing equations utilizing the B-spline shape functions that satisfy the continuity requirement to obtain the mechanical properties (electric potential) of the material. To obtain a mechanical property with different input parameters, we choose the truncated pyramid model as the object of study, and use IGA-FEM and PCE to solve different single uncertain parameters, including independent Young’s modulus and uniformly distributed force, and two kinds of flexoelectric constants, respectively. Numerical examples are presented to bear out the accuracy and viability of the proposed methodology.

Matings of cubic polynomials with a fixed critical point. Part II: α-symmetry of limbs

Following Milnor, define to be the space of monic, centred cubic polynomials with a marked fixed critical point. In this article, a sequel to [Thomas Sharland, Matings of cubic polynomials with a fixed critical point, part I: Thurston obstructions, Conform. Geom. Dyn. 23 (12) (2019), pp. 205–220], we provide a combinatorial sufficient (and conjecturally, necessary) condition (called α-symmetry) for the mating of two postcritically finite polynomials in to be obstructed. To do this, we study the rotation sets associated to the parameter limbs in the connectedness locus of , which allows us to determine when there exist ray classes in the formal mating which contain a closed loop. We give a proof of the necessity of α-symmetry for a particular subset of postcritically finite maps in . Many examples are given to illustrate the results of the paper.

A finiteness property of postcritically finite unicritical polynomials

A finiteness property of postcritically finite unicritical polynomials

A Fractional Order Fast Repetitive Control Paradigm of Vienna Rectifier for Power Quality Improvement

Due to attractive features, including high efficiency, low device stress, and ability to boost voltage, a Vienna rectifier is commonly employed as a battery charger in an electric vehicle (EV). However, the 6k ± 1 harmonics in the acside current of the Vienna rectifier deteriorate the THD of the ac current, thus lowering the power factor. Therefore, the current closed-loop for suppressing 6k ± 1 harmonics is essential to meet the desired total harmonic distortion (THD). Fast repetitive control (FRC) is generally adopted; however, the deviation of power grid frequency causes delay link in the six frequency fast repetitive control to become non-integer and the tracking performance to deteriorate. This paper presents the detailed parameter design and calculation of fractional order fast repetitive controller (FOFRC) for the non-integer delay link. The finite polynomial approximates the non-integer delay link through the Lagrange interpolation method. By comparing the frequency characteristics of traditional repetitive control, the effectiveness of the FOFRC strategy is verified. Finally, simulation and experiment validate the steadystate performance and harmonics suppression ability of FOFRC.

Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic L‐functions

AbstractWe introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that the above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two applications of this result. One is that the twisting of coefficients of automorphic L‐function on and polynomial nilsequences has logarithmic decay; the other is that the mean value of the Möbius function, coefficients of automorphic L‐function, and polynomial nilsequences also has logarithmic decay.