合理构造,巧解数列

数列是高中数学中的一个重要知识点,也是考试中经常考查的知识点.在解题过程中,合理构造数列可以帮助我们巧妙地解答问题.数列中的构造问题是历年高考的一个热点,常以主观题或客观题的形式出现,通常考查构造新的数列求数列的通项公式.本文针对数列构造问题进行分析,以期帮助学生掌握常见的构造方法,破解数列的构造难题.

利用韦达定理巧解数列通项

本文介绍了从一元二次方程的角度去看待连续两项的对称二次式,通过韦达定理实现条件的降次,化简,从而求出数列的通项。把方程的思想运用于数列中,实现了知识的交汇,有利于提高学生的解题能力。

构造新数列 巧解数列题

求数列的通项公式是各级各类考试中常见的题型,一般是选择题、填空题以及解答题中的一个小题.由于题设条件不同,题型本身也形式多样.其中有一类比较多见,即已知数列的递推公式(a_(n)与a_(n-1)的关系式)求其通项公式,解决这一类问题除验算-猜想-证明的方法外,就是利用所给递推公式变形构造出一个新数列,即它是一个新的等差或等比数列,然后运用其性质来求解.解题中需要抓住题设条件式的特点,运用代数手段进行恰当的变形.本文通过典例分析阐述此类问题的十种变形手段,供参考.

Pell方程x^2-(a^2-1)y^2=k的解集

应用本原解、解数列等概念,完整、清晰地表述了形如x2-(a2-1)y2=k(k∈Z,k≠0,a≥2)型Pell方程的整数解集.

Doubly Periodic Wave Solutions of Jaulent-Miodek Equations Using Variational Iteration Method Combined with Jacobian-function Method

One of the advantages of the variational iteration method is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some unknown parameters. The Jaulent-Miodek equations are used to illustrate effectiveness and convenience of this method, some new explicit exact travelling wave solutions have been obtained, which include bell-type soliton solution, kink-type soliton solutions, solitary wave solutions, and doubly periodic wave solutions.

On Smarandache Multiplicative Sequence and Its Generation

The main purpose of this paper is to introduce the general Smarandache mul- tiplicative sequence based on the Smarandache multiplicative sequence, and calculate the value of some infinite series involving these sequences.

Role of Metacognition in Mathematical Problem Solving Process of Permutations and Combinations--Basic Study for CAI Software Development

When solving a mathematical problem, we sometimes encounter a situation where we can not reach a correct answer in spite of acquiring knowledge and formula necessary for the solution. The reason can be attributed to the lack in metacognitive abilities. Metacognitive abilities consist of comparing the difficulty of problem with own ability, proper plan of solution process, and conscious monitoring and control of solution process. The role and importance of metacognitive ability in mathematical problem solving of permutations and combinations was explored. Participants were required to solve five practical problems related to permutations and combinations. For each problem, the solution process was divided into： （1） understanding （recognition） of mathematical problem; （2） plan of solution; （3） execution of solution. Participants were also required to rate the anticipation whether they could solve it or not, and to rate the confidence of their own answer. According to the total score of five problems, the participants were categorized into the group of the high test score and the group of the low test score. As a result, at the plan and the execution processes, statistically significant differences were detected between the high and the low score groups. As for the rating on the anticipation of result and the confidence of own answer, no significant differences were found between both groups. Moreover, the relationship between the score of plan process and the score of execution process was statistically correlated. In other words, the more proper the plan process was conducted, the more proper solution the participants reached. In such a way, the importance of metacognitive ability in the solving process, especially the plan ability, was suggested.

Solving the Sequence Problems by Constructing Matrices and Determinants

Higher mathematics is more extensive, profound and abstract than elementary mathematics; it is the development and sublimation of elementary mathematics. They have a deep connection, determinant and matrix theories originated in Elementary Mathematics, in tum, they also can be used as tools to solve related problems, and they have important roles in guiding the study of elementary mathematics. This paper will introduce the methods of solving some recursive sequence problems by constructing determinants and matrices.

Upper bound solutions of stability factor of shallow tunnels in saturated soil based on strength reduction technique

Based on the upper bound theorem of limit analysis,the factor of safety for shallow tunnel in saturated soil is calculated in conjunction with the strength reduction technique.To analyze the influence of the pore pressure on the factor of safety for shallow tunnel,the power of pore pressure is regarded as a power of external force in the energy calculation.Using the rigid multiple-block failure mechanism,the objective function for the factor of safety is constructed and the optimal solutions are derived by employing the sequential quadratic programming.According to the results of optimization calculation,the factor of safety of shallow tunnel for different pore pressure coefficients and variational groundwater tables are obtained.The parameter analysis shows that the pore pressure coefficient and the location of the groundwater table have significant influence on the factor of safety for shallow tunnel.

Analysis on Characteristics of Focal Mechanisms for the M_S7.3 Yutian Earthquake Sequence of February 12,2014

The paper inverts the focal mechanism solutions of the Yutian M_S7. 3 main shock,foreshocks and M_S≥3. 5 aftershocks by using the CAP method,based on the broadband waveforms recorded by the Xinjiang and Tibet Digital Seismic Networks. The results show that the M_S7. 3 strong earthquake is of strike-slip type with a normal faulting component,and combined with the analysis of focal structure and the aftershock distribution,the nodal plane I with strike 241°,dip 90° and rake- 22° is considered to be the seismogenic fault plane of the main shock. The direction of P-axis for the main shock is 194°,close to the near NS direction of the principal stress P-axis of historical strong earthquakes in this region. The focal mechanism solution of the M_S5. 4 foreshock has a good consistency with that of the main shock. Among the 18 aftershocks,10 are of strike-slip type,6 are of normal faulting type and 2 are of thrust type. 70% of the aftershocks in the sequence have a focal mechanism with P-axis in the near-NS direction. The focal depths of this M_S7. 3 earthquake sequences are distributed in the range of 5km- 28 km,with the majority in the depth range of 15km- 20 km,slightly deeper than the depth of 10 km of the main shock as calculated.

Activation of Prophenol Oxidase PHOX-S with Limited Proteolysis in Drosophila Melanogaster

Prophenol oxidase isoform A1 was isolated from Drosophila melanogaster （The sequence has been deposited in GenBank data base under accession number AB557586） PHOX-S strain, and its characteristics and activation mechanism were determined. The NH2-terminal region of PHOX-S A1 was determined to be comprised of 15 amino acids with the following sequence MTNMKMKMKAMMR. Comparison of an alignment in the known prophenol oxidase protein sequences from Drosophila melanogaster strains showed high homology in the copper-binding sequences at the Cu （A） site of the active center. Limited proteolysis takes place between Arg-50 and Val-51. Therefore, it is concluded that prophenol oxidase PHOX-S protein was evolved at the upstream, but no evolved at the central site in Drosophila melanogaster.

Quasi-binomial coefficients stemming from Nakayama algebras

It is known that there is a very closed connection between the set of non-isomorphic indecomposable basic Nakayama algebras and the set of admissible sequences.To determine the cardinal number of all nonisomorphic indecomposable basic Nakayama algebras,we describe the cardinal number of the set of all t-length admissible sequences using a new type of integers called quasi-binomial coefficients.Furthermore,we find some intrinsic relations among binomial coefficients and quasi-binomial coefficients.

Generalized dressing method for the extended two-dimensional Toda lattice hierarchy and its reductions

In this paper, we solve the extended two-dimensional Toda lattice hierarchy (ex2DTLH) by the generalized dressing method developed in Liu-Lin-Jin-Zeng (2009). General Casoratian determinant solutions for this hierarchy are obtained. In particular, explicit solutions of soliton-type are formulated by using the τ-function in the form of exponential functions. The periodic reduction and one-dimensional reduction of ex2DTLH are studied by finding the constraints. Many reduced systems are shown, including the periodic ex2DTLH, sinh-Gordon equation with self-consistent sources and one-dimensional Toda lattice hierarchy with self-consistent sources. The general solutions of reduced hierarchies are found from the Casoratian solutions of ex2DTLH, by considering additional constraints during the dressing procedure.

ADMISSIBLE ESTIMATES IN THE IMPORTANT CLASS OF ESTIMATES OF THE COVARIANCE MATRIX

Let X1,X2(where N>m)be independent Nm(μ,∑) random vectors, and putwhere T is upper- triangular with positive diagonal elements. The author considers the problemof estimating ∑,and restricts his attention to the class of estimates is any diagonal matrix and b* is any nonnegative constant} because it has the following attractive features:(a) Its elements are all quadratic forms of the sufficient and complete statistics (X, T).(b) It contains all estimates of the form(and), which constructa complete subclass of the class of nonnegative quadratic estimates (where X = (X1,…, XN)') for any strict convex loss function.(c)It contains all invariant estimates under the transformation group of upper-triangularmatrices.The author obtains the Characteristics for an estimate of the formof ∑ to be admissible in when the loss function is chosed as tr(∑-1∑-Ⅰ)2, and shows,by an example, that(and)is admissible in can not imply itsadmissibility in .

On the size of the intersection of two Lucas sequences of distinct type Ⅱ

Let a and b be positive integers, with a not perfect square and b > 1. Recently, He, Togband Walsh proved that the Diophantine equation x2-a((bk-1)/(b-1))2=1 has at most three solutions in positive integers. Moreover, they showed that if max{a,b} > 4.76·1051, then there are at most two positive integer solutions (x,k). In this paper, we sharpen their result by proving that this equation always has at most two solutions.