Polynomial Configurations Research Articles

Generalized Legendre Polynomial Configuration Method for Solving Numerical Solutions of Fractional Pantograph Delay Differential Equations

This paper develops a numerical approach for solving fractional pantograph delay differential equations using generalized Legendre polynomials. These polynomials are derived from generalized Taylor bases, which facilitate the approximation of the underlying analytical solutions, leading to the formulation of numerical solutions. The fractional pantograph delay differential equation is then transformed into a finite set of nonlinear algebraic equations using collocation points. Following this step, Newton's iterative method is applied to the resultant set of nonlinear algebraic equations to compute their numerical solutions. An error analysis for this methodology is subsequently presented, accompanied by numerical examples demonstrating its accuracy and efficiency. Overall, this study contributes a more streamlined and productive tool for determining the numerical solution of fractional differential equations.

Fsm Driven NON-Linear Transistion Based Lfsr TPG for Improved Fault Coverage in BIST Application

The Test pattern generator (TPGs) is one of the computationally intensive and performance centric computational block in BIST applications and has been investigated in many existing works to maximize its contribution towards testing the digital circuits; however, design constraints associated to the system leads significant improvements in TPG model. This paper proposes energy efficient FSM state transition-based LFSR TPG to formulate test patterns with maximum length and randomness. Here LFSR generated test patterns generated from conventional LFSR reseeding model with fixed polynomial configuration is decomposed into smaller units and follows FSM encoding to convert into multiple independent test vector blocks. The concatenation of these encoded vector blocks offers improved test vectors which includes transitional and non- transitional encoded values to regulate the overall switching activity. According to the power constrains each binary positions of test vectors are weighted to control certain bit transitions which can significantly optimize the dynamic power consumption during testing process. Finally, FSM controller is used to control state transition in addition to the variable clock to maximize the randomness randomization characteristics of LFSR. Here both clock rate and number state in FSM machine can be dynamically changed to modify the LFSR statistics in accordance with BIST requirements.

Synchronized Multi-Rate Test Pattern Generation Using Hybrid Twisted Ring Counter and LFSR for BIST Application

This work presents a new approach for built-in test pattern generation (BIST) based on the variable rate clock-driven linear feedback shift register (LFSR) and twisted-ring counters (TRCs). The proposed technique operates on different operating clocks for generating deterministic test pattern sequence for the circuit under test (CUT). The test vectors generated by LFSR are transformed and reproduced with TRC results. The TRC design constitutes of only flip-flops, which be used as a potential alternative to improved seed storage and complex multi polynomial configurations in LFSR.Moreover, the control logic is simplified with synchronized multi-rate clock generation and can be easily extended for any test cubes among multiple CUTs. The randomness characteristics of LFSR are retained with improved memory efficiency and considerable complexity reduction. Experimental results validate the proposed test pattern generator (TPG) over existing well-known LFSRs models, and its randomness is proved with ISCAS benchmark circuits.

On Rado conditions for nonlinear Diophantine equations

Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado’s characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial is linear. This would provide a natural generalization of Rado’s theorem.We verify that such a conjecture holds for the equations x2−xy+ax+by+cz=0 and x2−y2+ax+by+cz=0 for a,b,c∈Z such that abc=0 or a+b+c=0. To deal with these equations, we establish new results concerning the partition regularity of polynomial configurations in Z such as x,x+y,xy+x+y, building on the recent result on the partition regularity of x,x+y,xy.

Artificial neural network hybridized with a genetic algorithm for optimization of lipase production from Penicillium roqueforti ATCC 10110 in solid-state fermentation

In the present work, an artificial neural network hybridized with a genetic algorithm (ANN-GA) has been applied to optimize Penicillium roqueforti ATCC 10110 lipase production in solid-state fermentation (SSF). For such a purpose, a feed-forward ANN with polynomial configuration 3-49-1 (i.e. 3 neurons in the input layer, 49 neurons in the hidden layer and 1 neuron in the output layer) was used to computationally model the experiment and a GA was used to optimize lipase production through the ANN model. The input variables optimized by the ANN-GA were fermentation time (1 day), incubation temperature (31.2 °C) and percentage moisture content (78%). Validation was performed by considering the optimal and central point conditions, thus obtaining a lipase activity value of 48.00 U g−1, which is three times greater than by using other methodologies. Furthermore, the ANN model was obtained using 28 essays (small dataset) with interpolation and generalization capability based on a significant and precise data choice and justified by mean square error and determination coefficient values. A total of 5.0 × 107 artificial tests were simulated from the small dataset of 28 experiments.

Polynomial configurations in sets of positive upper density over local fields

Let F(x) = (f1(x), …, fm(x)) be such that 1, f1, …, fm are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, de Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of ℝm with respect to the portion of the graph of F defined by a ≤ log åså ≤ T is at most O(1/(T — a)). We conclude that if I ⊆ ℝm has positive upper density, then the difference set I — I contains vectors of the form F(s)for an unbounded set of values s ∈ ℝ. It follows that the Borel chromatic number of the Cayley graph of ℝm with respect to the set {±F(s): s ∈ ℝ} is infinite. Analogous results are also proven when ℝ is replaced by the field of p-adic numbers ℚp. At the end, we will also show the existence of real analytic functions f1, …, fm, for which the analogous statements no longer hold.

Energy efficiency control of pneumatic actuator systems through nonlinear dynamic optimization

Abstract As one of the most widely used power sources in the engineering field, compressed air is especially common in industrial applications. In a traditional pneumatic circuit, a three-position five-way valve is used to control the air inlet and exhaust with a single pressure, which has low efficiency. This paper proposed a novel bridge-type circuit with variable pressure control using four on-off valves to improve the energy efficiency of compressed air for pneumatic actuators. The key idea of the method was to use air expansion energy to do work. Reducing the volume of consumed air was the goal, and a numerical method was presented to calculate the optimal on-off time sequence of the four valves using nonlinear dynamic optimization. The sequence was obtained based on the combination of a finite element polynomial configuration method and a reduced space sequential quadratic programming algorithm. Then, experimental validation was conducted to illustrate the energy efficiency, including pneumatic cylinder motion test under different conditions. Compared with the traditional circuit, the experimental results show that the bridge-type circuit put forward by this paper can significantly improve the energy efficiency of pneumatic driving systems by 50–70%. Additionally, the bridge-type circuit allows the piston to move smoothly to the end of the stroke, which can take the place of the function of cushion component and speed control valve of the traditional circuit.

Sets of large dimension not containing polynomial configurations

The main result of this paper is the following. Given countably many multivariate polynomials with rational coefficients and maximum degree d, we construct a compact set E⊂Rn of Hausdorff dimension n/d which does not contain finite point configurations corresponding to the zero sets of the given polynomials.Given a set E⊂Rn, we study the angles determined by three-point subsets of E. The main result implies the existence of a compact set in Rn of Hausdorff dimension n/2 which does not contain the angle π/2. (This is known to be sharp if n is even.) We show that there is a compact set of Hausdorff dimension n/8 which does not contain an angle in any given countable set. We also construct a compact set E⊂Rn of Hausdorff dimension n/6 for which the set of angles determined by E is Lebesgue null.In the other direction, we present a result that every set of sufficiently large dimension contains an angle ε close to any given angle.The main result can also be applied to distance sets. As a corollary we obtain a compact set E⊂Rn (n≥2) of Hausdorff dimension n/2 which does not contain rational distances nor collinear points, for which the distance set is Lebesgue null, moreover, every distance and direction is realised only at most once by E.

New polynomial and multidimensional extensions of classical partition results

In the 1970s Deuber introduced the notion of (m,p,c)-sets in N and showed that these sets are partition regular and contain all linear partition regular configurations in N. In this paper we obtain enhancements and extensions of classical results on (m,p,c)-sets in two directions. First, we show, with the help of ultrafilter techniques, that Deuber's results extend to polynomial configurations in abelian groups. In particular, we obtain new partition regular polynomial configurations in Zd. Second, we give two proofs of a generalization of Deuber's results to general commutative semigroups.We also obtain a polynomial version of the central sets theorem of Furstenberg, extend the theory of (m,p,c)-systems of Deuber, Hindman and Lefmann and generalize a classical theorem of Rado regarding partition regularity of linear systems of equations over N to commutative semigroups.

On polynomial configurations in fractal sets

We show that subsets of $\mathbb{R}^n$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form \begin{align*} ( x ,\, x + A_1 y ,\, \dots,\, x + A_{k-1} y ,\, x + A_k y + Q(y) e_n ), \quad x \in \mathbb{R}^n,\ y \in \mathbb{R}^m, \end{align*} where $A_i$ are real $n \times m$ matrices, $Q$ is a real polynomial in $m$ variables and $e_n = (0,\dots,0,1)$.

Polynomial configurations in subsets of random and pseudo-random sets

We prove transference results for sparse random and pseudo-random subsets of ZN, which are analogous to the quantitative version of the well-known Furstenberg–Sárközy theorem due to Balog, Pintz, Steiger and Szemerédi.In the dense case, Balog et al. showed that there is a constant C>0 such that for all integer k≥2 any subset of the first N integers of density at least C(log⁡N)−14log⁡log⁡log⁡log⁡N contains a configuration of the form {x,x+dk} for some integer d>0.Let [ZN]p denote the random set obtained by choosing each element from ZN with probability p independently. Our first result shows that for p>N−1/k+o(1) asymptotically almost surely any subset A⊂[ZN]p (N prime) of density |A|/pN≥(log⁡N)−15log⁡log⁡log⁡log⁡N contains the polynomial configuration {x,x+dk}, 0<d≤N1/k. This improves on a result of Nguyen in the setting of ZN.Moreover, let k≥2 be an integer and let γ>β>0 be real numbers satisfyingγ+(γ−β)/(2k+1−3)>1. Let Γ⊆ZN (N prime) be a set of size at least Nγ and linear bias at most Nβ. Then our second result implies that every A⊆Γ with positive relative density contains the polynomial configuration {x,x+dk}, 0<d≤N1/k.For instance, for squares, i.e., k=2, and assuming the best possible pseudo-randomness β=γ/2 our result applies as soon as γ>10/11.

Optimisation of polynomial railway transition curves of even degrees

This paper represents new results obtained by its authors while searching for the proper shape of polynomial railway transition curves (TCs). The search for the proper shape means the evaluation of the curve properties based on chosen dynamical quantities and generation of such shape with use of mathematically understood optimisation methods. The studies presented now and in the past always had got a character of the numerical tests. For needs of this work advanced vehicle model, dynamical track-vehicle and vehicle-passenger interactions, and optimisation methods were exploited. In this software complete rail vehicle model of 2-axle freight car, the track discrete model, and non-linear description od wheel-rail contact are used. That part of the software, being vehicle simulation software, is combined with library optimisation procedures into the final computer programme. The main difference between this and previous papers by the authors are the degrees of examinated polynomials. Previously they tested polynomial curves of odd degrees, now they focus on TCs of 6th, 8th and 10th degrees with and without curvature and superelevation ramp tangence in the TC’s terminal points. Possibility to take account of fundamental demands (corresponding values of curvature in terminal points) concerning TC should be preserved. Results of optimisation are compared both among themselves and with 3rd degree parabola. The aim of present article is to find the polynomial TCs’ optimum shapes which are determined by the possible polynomial configurations. Only one dynamical quantities being the results of simulation of railway vehicle advanced model is exploited in the determination of quality function (QF1). This is: minimum of integral of vehicle body lateral acceleration.

Polynomial multiple recurrence over rings of integers

We generalize the polynomial Szemerédi theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new polynomial configurations in positive-density subsets of $\mathbb{Z}^{m}$ and strengthens and extends recent results of Bergelson, Leibman and Lesigne [Intersective polynomials and the polynomial Szemerédi theorem. Adv. Math.219(1) (2008), 369–388] on polynomials over the integers.

Polynomial Configurations in the Primes

The Bergelson-Leibman theorem states that if P_1, ..., P_k are polynomials with integer coefficients, then any subset of the integers of positive upper density contains a polynomial configuration x+P_1(m), ..., x+P_k(m), where x,m are integers. Various generalizations of this theorem are known. Wooley and Ziegler showed that the variable m can in fact be taken to be a prime minus 1, and Tao and Ziegler showed that the Bergelson-Leibman theorem holds for subsets of the primes of positive relative upper density. Here we prove a hybrid of the latter two results, namely that the step m in the Tao-Ziegler theorem can be restricted to the set of primes minus 1.

Polynomial configurations in difference sets

We prove a quantitative result on the existence of linearly independent polynomial configurations in the difference set of sparse subsets of the integers. This result is achieved by first establishing a higher dimensional analogue of a theorem of Sárközy and then applying a simple lifting argument.

A Polynomial-Based Linear Mapping Strategy for Feedforward Compensation of Hysteresis in Piezoelectric Actuators

A set of memory-based properties is employed in this paper for modeling multiple-path hysteresis response of piezoelectric actuators. These properties, namely, targeting turning points, curve alignment, and wiping-out effect, are applied in a linear mapping strategy to develop a mathematical framework for modeling the hysteresis phenomenon. More specifically, the locations of turning points are detected and recorded for the prediction of future hysteresis trajectory. An internal trajectory is assumed to follow a multiple-segmented path via a continuous connection of several curves passing through every two consequent turning points. These curves adopt their shapes via a linear mapping strategy from the reference hysteresis curves with polynomial configurations. Experimental implementation of the proposed method demonstrates slight improvement over the widely used Prandtl–Ishlinskii hysteresis operator. However, to maintain the level of precision during the operation, a sufficient number of memory units must be included to record the turning points. Otherwise, in the event of memory saturation, two memory-allocation modes, namely, “open” and “closed” strategies, can be implemented. It is shown that the closed memory-allocation strategy demonstrates better performance by keeping the most important target points. The proposed modeling framework is adopted in an inverse model-based control scheme for feedforward compensation of hysteresis nonlinearity. The controller is experimentally implemented on a three-dimensional nanopositioning stage for surface topography tracking, a problem typically encountered in scanning probe microscopy applications.

ERGODIC AVERAGES FOR INDEPENDENT POLYNOMIALS AND APPLICATIONS

Szemeredi's theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemeredi's theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in combinatorics for rationally independent polynomials, showing that their behavior differs sharply from the general situation.

Reliability analysis for a repairable parallel system with time-varying failure rates

To solve a real problem: how to calculate the reliability of a system with time-varying failure rates in industry systems, this paper studies a model for the load-sharing parallel system with time-varying failure rates, and obtains calculating formulas of reliability and availability of the system by solving differential equations. In this paper, the failure rates are expressed in polynomial configuration. The constant, linear and Weibull failure rate are in their special form. The polynomial failure rates provide flexibility in modeling the practical time-varying failure rates.

脳型情報処理を行う力学系の多項式設計法とそのヒューマノイドの全身運動生成への応用

In this paper, we develop the dynamics-based information processing system that is a brain-like information processing based on dynamical phenomenon. A nonlinear dynamics is designed with the polynomial configuration so that it has attractors on the closed carved lines in the N dimensional space, which means memorization, reproduction and association of the time sequence data of the whole body motions. The motion transition of the humanoid robot is realized by the transition of the dynamics from one attractor to another. In this paper, we propose a design strategy of the system that generates the humanoid whole body motions using entrainment and detrainment phenomenon of the nonlinear dynamics with a motion reduction method based on the principal component analysis.

Topological multiple recurrence for polynomial configurations in nilpotent groups

We establish a general multiple recurrence theorem for an action of a nilpotent group by homeomorphisms of a compact space. This theorem can be viewed as a nilpotent version of our recent polynomial Hales–Jewett theorem (Ann. Math. 150 (1999) 33) and contains nilpotent extensions of many known “abelian” results as special cases.