Quadratic Sequences Research Articles

Galois groups and prime divisors in random quadratic sequences

AbstractGiven a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $\gamma$ of elements of S, one can associate an arboreal representation to $\gamma$ , generalising the case of iterating a single polynomial. We study the probability that a random sequence $\gamma$ produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets S defined over $\mathbb{Z}[t]$ , and we conjecture a similar positive-probability result for suitable sets over $\mathbb{Q}$ . As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify all S possessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration.

Singularity avoidance method for dual-arm closed-chain system based on damped least squares

When two manipulators cooperate to clamp a rigid object, a closed chain with both ends fixed is formed. The singularity of a single manipulator causes the closed chain to be at a singularity, which results in uncertainty in the velocity mapping between the terminal and joint space. This study proposes a closed-chain singularity processing method for dual-arm collaboration, adopting the Jacobian pseudoinverse method based on damped least squares to obtain an approximate solution. During the solution process, the adaptive damping factor is improved by cubic polynomials, and the optimal damping factor is determined by quadratic sequence programming. Furthermore, this study analyzes the kinematic constraints of the closed-chain system and constructs a closed-chain Jacobian to judge whether the system is in a singular region. Finally, the effectiveness of the proposed method is verified through simulations.

High order accelerating method for quadratic sequences

High order accelerating method for quadratic sequences

The least common multiple of a bivariate quadratic sequence

The least common multiple of a bivariate quadratic sequence

Focusing ions at atmospheric pressure using nonlinear DC voltage sequences applied to a stacked ring ion guide

Many modern ion mobility (IM) and mass spectrometers (MS) operate under low pressure (≤10 Torr) and employ high voltage radiofrequencies (RF) to provide ion confinement. Unfortunately, RF effectiveness drastically decreases as pressure increases, and few techniques for focusing ions at elevated pressures exist. Here we demonstrate a new approach for focusing ions at atmospheric pressure (AP) by applying nonlinear DC voltage sequences following quadratic and power (exponential) functions to a stacked ring ion guide. We used ion trajectory simulations to rigorously explore how ions react to nonlinear electric fields and validate the simulations with a set of ion current measurements performed at AP. Ion trajectory simulations show that ions initially defocus near the entrance of the device but then become intensely focused as they travel through the device. Contour plots for both nonlinear voltage sequences show electric field lines that increasingly curve inwards as a function of distance, resulting in spatial ion focusing. Experimental ion current and spot size measurements were performed at AP using a 10-cm stacked ring ion guide and a segmented Faraday cup detector. Quadratic sequences produced ∼5% smaller spot sizes (∼22.8 mm) and ∼25% higher ion current compared to a linear voltage sequence (∼24.0 mm). Alternatively, power sequences produced ∼64% smaller spot sizes (∼8.7 mm), albeit with ∼10x lower ion current. However, both nonlinear voltage sequences produced similar ion currents at the center of the Faraday cup detector, indicating that higher ion densities are achieved when using nonlinear voltage gradients. These results demonstrate a new way to focus ions at AP, and the capabilities demonstrated here provide fundamental insights on how to keep ions inside analytical devices at elevated pressures without RF.

Numerical semigroups generated by quadratic sequences

We investigate numerical semigroups generated by any quadratic sequence with initial term zero and an infinite number of terms. We find an efficient algorithm for calculating the Apéry set, as well as bounds on the elements of the Apéry set. We also find bounds on the Frobenius number and genus, and the asymptotic behavior of the Frobenius number and genus. Finally, we find the embedding dimension of all such numerical semigroups.

Homological characterization of bounded [formula omitted]-regularity

Semi-regular sequences over F2 are sequences of homogeneous elements of the algebra B(n)=F2[X1,...,Xn]/(X12,...,Xn2), which have as few relations between them as possible. It is believed that most such systems are F2-semi-regular and this property has important consequences for understanding the complexity of Gröbner basis algorithms such as F4 and F5 for solving such systems. In fact even in one of the simplest and most important cases, that of quadratic sequences of length n in n variables, the question of the existence of semi-regular sequences for all n remains open. In this paper we present a new framework for the concept of F2-semi-regularity which we hope will allow the use of ideas and machinery from homological algebra to be applied to this interesting and important open question. First we introduce an analog of the Koszul complex and show that F2-semi-regularity can be characterized by the exactness of this complex. We show how the well known formula for the Hilbert series of a F2-semi-regular sequence can be deduced from the Koszul complex. Finally we show that the concept of first fall degree also has a natural description in terms of the Koszul complex.

Exploring Goldbach’s conjecture, the Pebonacci sequence and its five-dimensional vortex structure based on the logarithm of the circle


 
 
 A method based on circle logarithm to prove Goldbach’s conjecture and Pebonacci sequence is proposed. Its essence is to deal with the real infinite series, each of the finite three elements (prime numbers, number series) has asymmetry problems, forming a basic even function one-variable quadratic equation and odd function one-variable three-dimensional number sequence; it is converted to "The irrelevant mathematical model expands latently in a closed interval of 0 to 1," forming a five-dimensional vortex space structure.
 
 

Regularity of powers of quadratic sequences with applications to binomial ideals

In this article, we obtain an upper bound for the Castelnuovo-Mumford regularity of powers of an ideal generated by a homogeneous quadratic sequence in a polynomial ring in terms of the regularity of its related ideals and degrees of its generators. As a consequence, we compute upper bounds for the regularity of powers of several binomial ideals. We generalize a result of Matsuda and Murai to show that the regularity of JGs is bounded below by 2s+ℓ(G)−1 for all s≥1, where JG denotes the binomial edge ideal of a graph G and ℓ(G) is the length of a longest induced path in G. We compute the regularity of powers of binomial edge ideals of cycle graphs, star graphs, and balloon graphs explicitly. Also, we give sharp bounds for the regularity of powers of almost complete intersection binomial edge ideals and parity binomial edge ideals.

Finite Element Model Modification of an Arch Bridge Based on Response Surface Method

Usually the structure of the initial finite element model can not reflect the actual situation of structure, the finite element model updating is for the initial finite element model of the structure, according to the static and dynamic response of the structure of the measured optimize the model parameters are adjusted, the revised model response is more close to the actual structural response, so as to get a more accurate finite element model. In this paper, the finite element model of the bridge is updated by using the second order incomplete polynomial response surface model. Then determine the sample data, use the least squares method to fit the undetermined coefficients of the second-order incomplete polynomial, and perform the accuracy test. Then the objective function based on frequency is constructed, and the updated optimal design parameters are obtained by quadratic sequence programming. Finally, using the optimal design parameters for modal analysis, it is found that the frequency error has decreased. It is proved that the correction by the response surface method (RSM) can improve the accuracy of the finite element model.

Generalization Strategies in Finding the nth Term Rule for Simple Quadratic Sequences

In this study, we identify ways in which a sample of 18 graduates with mathematics-related first degrees found the nth term for quadratic sequences from the first values of a sequence of data, presented on a computer screen in various formats: tabular, scattered data pairs and sequential. Participants’ approaches to identifying the nth term were recorded with eye-tracking technology. Our aims are to identify their strategies and to explore whether and how format influences these strategies. Qualitative analysis of eye-tracking data offers several strategies: Sequence of Differences, Building a Relationship, Known Formula, Linear Recursive and Initial Conjecture. Sequence of Differences was the most common strategy, but Building a Relationship was more likely to lead to the right formula. Building from Square and Factor Search were the most successful methods of Building a Relationship. Findings about the influence of format on the range of strategies were inconclusive but analysis indicated sporadic evidence of possible influences.

Performance Analysis of Hybrid 2D Codes at Constant Weight Using (n, w, λa, λc) OOCs for OCDMA

Hybrid optical two-dimensional (2D) codes due to their large cardinality and improved system performance for constant weight have recently been designed and studied for optical CDMA system. In this manuscript, an attempt has been made to analyze the performance of four different 2D codes from a communication point of view; that describe their use to support a system under certain conditions. These hybrid 2D codes employ (n, w, λa, λc) optical orthogonal coding for one dimension that is for time spreading and synchronized prime sequences (SPS), synchronized quadratic congruence sequences (SQC), prime code sequences (PC), and synchronized prime procession (SPP) for other dimension that is for wavelength hopping respectively. The distinguishing property amongst the four codes lies in their values of maximum auto- and cross- correlation function. It has been observed that with the constant value of weight, the code with the largest code length parameter has shown the best performance while the code with the least code length gives the poorest performance. Investigations reveal that SPS/OOC code performs better in comparison to other codes with constant weight and for different number of wavelengths and SQC/OOC results in poor performance due to lesser code length.

Quadratic sequences of powers and Mohanty’s conjecture

We prove under the Bombieri–Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in arithmetic progression. This answers a question proposed in 2010 by Browkin and Brzezinski, and independently by Gonzalez-Jimenez and Xarles. We also show that under the Bombieri–Lang conjecture for surfaces, for every [Formula: see text] there is an absolute bound on the length of sequences formed by [Formula: see text]th powers with constant second differences. This gives a conditional result on one of Mohanty’s conjectures on arithmetic progressions in Mordell’s elliptic curves [Formula: see text]. Moreover, we obtain an unconditional result regarding infinite families of such arithmetic progressions. We also study the case of hyperelliptic curves of the form [Formula: see text]. These results are proved by unconditionally finding all curves of genus zero or one on certain surfaces of general type. Moreover, we prove the unconditional analogues of these arithmetic results for function fields by finding all the curves of low genus on these surfaces.

Using a Model to Describe Students’ Inductive Reasoning in Problem Solving

Introducción. Presentamos algunos aspectos de una investigación más amplia (Cañadas, 2007), cuyo principal objetivo es describir y caracterizar el razonamiento inductive utilizado por estudiantes españoles de 3º y 4º de Educación Secundaria Obligatoria cuando resuelven problemas que involucran sucesiones lineales y cuadráticas.Método. Propusimos un cuestionario de seis problemas de diferentes características relacionados con sucesiones a 359 estudiantes de Secundaria. Estos problemas podían ser resueltos mediante el razonamiento inductivo. Utilizamos un modelo de razonamiento inductivo compuesto por siete pasos (Cañadas and Castro, 2007) para analizar las respuestas de los estudiantes.Resultados. Mostramos algunos resultados relacionados son: (a) las frecuencias de los pasos que emplean los estudiantes, (b) las relaciones entre las frecuencias de los pasos según las características del problema y (c) el estudio de las relaciones de (in)dependencia entre los diferentes estados del modelo de razonamiento inductivo.Discusión. Podemos concluir que el modelo de razonamiento inductive fue útil para describir la actuación de los estudiantes. En este artículo ponemos de manifiesto que este modelo no es linear. Como ejemplo, los estudiantes lograron la generalización sin haber realizado algunos pasos previos en algunos problemas. Describir cómo los estudiantes llegan a pasos más avanzados sin haber realizado los considerados previos y analizar si la realización de los pasos intermedios puede ser útil para los estudiantes son tareas pendientes.

Nonuniform exponential dichotomies and Lyapunov functions

For the nonautonomous dynamics defined by a sequence of bounded linear operators acting on an arbitrary Hilbert space, we obtain a characterization of the notion of a nonuniform exponential dichotomy in terms of quadratic Lyapunov sequences. We emphasize that, in sharp contrast with previous results, we consider the general case of possibly noninvertible linear operators, thus requiring only the invertibility along the unstable direction. As an application, we give a simple proof of the robustness of a nonuniform exponential dichotomy under sufficiently small linear perturbations.

Proof Without Words: A Recursion for Triangular Numbers and More

SummaryWe visually demonstrate a recurrence satisfied by the triangular numbers and hence all quadratic sequences including all types of polygonal numbers.

NOTES ON QUADRATIC INTEGERS AND REAL QUADRATIC NUMBER FIELDS

It is shown that when a real quadratic integer $\xi$ of fixed norm $\mu$ is considered, the fundamental unit $\varepsilon_{d}$ of the field $\mathbb{Q}(\xi) = \mathbb{Q}(\sqrt{d})$ satisfies $\log \varepsilon_{d} \gg (\log d)^{2}$ almost always. An easy construction of a more general set containing all the radicands $d$ of such fields is given via quadratic sequences, and the efficiency of this substitution is estimated explicitly. When $\mu = -1$, the construction gives all $d$'s for which the negative Pell's equation $X^{2} - d Y^{2} = -1$ (or more generally $X^{2} - D Y^{2} = -4$) is soluble. When $\mu$ is a prime, it gives all of the real quadratic fields in which the prime ideals lying over $\mu$ are principal.

Comparative analysis of optical 2D codes using (n, w, λ a , λ c ) optical orthogonal codes for optical CDMA

Hybrid two dimensional (2D) wavelength-hopping time-spreading coding techniques have been focused now days for Optical Code Division Multiple Access (OCDMA) systems to increase the capacity of system. In this paper, design and comparative analysis of five different 2D coding techniques has been performed. These codes use Synchronized Quadratic Congruence sequences (SQC), Prime Code sequences (PC), Synchronized Prime Sequences (SPS) and One-Coincidence Frequency Hopping Code (OCFHC) for wavelength hopping and (n, w, źa, źc) one dimensional optical orthogonal codes for time spreading respectively along with Extended Reed Solomon (E-RS) codes. It has been observed that SQC/OOC code inspite of having larger value of cross correlation than other codes under consideration, out performs due to its ability to support a larger code weight. The comparison on the basis of BER shows that irrespective of the increase in number of hits due to higher code weight, SQC/OOC provides better code performance and hence supports large number of users in the system. The results are reported on the basis of maximum auto- and cross correlation function, cardinality and bit error rate (BER).

Neural mechanisms underlying the computation of hierarchical tree structures in mathematics.

Whether mathematical and linguistic processes share the same neural mechanisms has been a matter of controversy. By examining various sentence structures, we recently demonstrated that activations in the left inferior frontal gyrus (L. IFG) and left supramarginal gyrus (L. SMG) were modulated by the Degree of Merger (DoM), a measure for the complexity of tree structures. In the present study, we hypothesize that the DoM is also critical in mathematical calculations, and clarify whether the DoM in the hierarchical tree structures modulates activations in these regions. We tested an arithmetic task that involved linear and quadratic sequences with recursive computation. Using functional magnetic resonance imaging, we found significant activation in the L. IFG, L. SMG, bilateral intraparietal sulcus (IPS), and precuneus selectively among the tested conditions. We also confirmed that activations in the L. IFG and L. SMG were free from memory-related factors, and that activations in the bilateral IPS and precuneus were independent from other possible factors. Moreover, by fitting parametric models of eight factors, we found that the model of DoM in the hierarchical tree structures was the best to explain the modulation of activations in these five regions. Using dynamic causal modeling, we showed that the model with a modulatory effect for the connection from the L. IPS to the L. IFG, and with driving inputs into the L. IFG, was highly probable. The intrinsic, i.e., task-independent, connection from the L. IFG to the L. IPS, as well as that from the L. IPS to the R. IPS, would provide a feedforward signal, together with negative feedback connections. We indicate that mathematics and language share the network of the L. IFG and L. IPS/SMG for the computation of hierarchical tree structures, and that mathematics recruits the additional network of the L. IPS and R. IPS.

Teaching Functions Using a Realistic Mathematics Education Approach: A Theoretical Perspective

This paper discusses how the notion of a function can be developed for high school learners through the use of different representations and models; verbal, visual, graphical and symbolic. The realistic mathematics education approach provides the framework that guides the discussion. Using the matchstick problem as an example, multiple-representations of the function concept inherent in it are mathematised. The possible representations of the function are; geometric patterns, independent - dependent variables, ordered pairs, fish diagrams, number sequences (quadratic sequences, arithmetic sequences, geometric sequences), dual bar graphs, graphs on the Cartesian plane and the functional f(x) symbolism. It is argued that multiple representations that start with the informal and every day and then gradually progress to the formal and abstract, help learners to gain insight of the big idea functions in mathematics. The paper provides mathematics educators a platform which facilitates a realistic mathematics education approach to teaching functions which can be extended to other mathematical topics.