Powers and Exponents

<正>In mathematics,a power is represented with a base number and an exponent.The base number tells what number is being multiplied.The exponent,a small number written above and to the right of the base number,tells how many times the base number is being multiplied.Today we will talk about     

Effect of rotation on the onset of double diffusive convection in a Darcy porous medium saturated with a couple stress fluid

The effect of rotation on the onset of double diffusive convection in a horizontal couple stress fluid-saturated porous layer, which is heated and salted from below, is studied analytically using both linear and weak nonlinear stability analyses. The extended Darcy model, which includes the time derivative and Coriolis terms, has been employed in the momentum equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. The effect of Taylor number, couple stress parameter, solute Rayleigh number, Lewis number, Darcy–Prandtl number, and normalized porosity on the stationary, oscillatory, and finite amplitude convection is shown graphically. It is found that the rotation, couple stress parameter and solute Rayleigh number have stabilizing effect on the stationary, oscillatory, and finite amplitude convection. The Lewis number has a stabilizing effect in the case of stationary and finite amplitude modes, with a destabilizing effect in the case of oscillatory convection. The Darcy–Prandtl number and normalized porosity advances the onset of oscillatory convection. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer. The transient behavior of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge–Kutta method.     

ZFC的模型的上限序数的一个性质

S1 引言 Forcing方法假设存在ZFC的一个可数可传的模型M。记满足αM的最小序数α为,显然M中一切序数所成的集合即。由于M是ZFC的模型,故应具有某些性质。本文证明了它满足关系,故为ε数或1级关键数,进而证明了是H级关键数(H为任意自然数)。文中的记号等引用。     

树的四类控制参数的束缚数

图的束缚数是图的控制数研究中的一个重要方面，它在某种程度上反映了图的控制数对边数的敏感度．本文通过对图的结构特征的分析．研究了树的四类控制参数的束缚数，即控制数，强控制数，弱控制数．分数控制数的束缚数．分别给出了其紧的上界．     

Coloring of lattices

The concept of coloring is studied for graphs derived from lattices with 0. It is shown that, if such a graph is derived from an atomic or distributive lattice, then the chromatic number equals the clique number. If this number is finite, then in the case of a distributive lattice, it is determined by the number of minimal prime ideals in the lattice. An estimate for the number of edges in such a graph of a finite lattice is given.     

Bifurcation of multimode flows of a viscous fluid in a plane diverging channel

The pattern of steady multimode flow of a viscous incompressible fluid in a plane diverging channel is constructed and investigated. It is shown that odd-mode flows have velocity profiles that are symmetrical about the axis of the channel and from one to three different flows with a fixed number of modes exist. The even-mode flows are asymmetric and exist as pairs. The existence of a denumerable set of finite ranges adjoining one another, in which a single-type of complex bifurcation of the flow occurs, is established in the case of an unbounded range of values of the Reynolds number. As the Reynolds number increases, transitions to flows with an increasing number of modes, containing domains of forward and backward flows, occur successively. Flow patterns with a smaller number of modes do not occur. An increase in the number of an range corresponding to an increase in the Reynolds number leads to an unlimited increase in the length of the range and the number of modes of permissible flows.     

Richardson数对后台阶流动熵产的影响

后台阶流动是研究伴随有传热现象的分离流动的常用模型．虽然Richardson数的改变会明显影响分离流动的流动和传热特性,但是迄今为止关于Richardson数对后台阶流动熵产影响的研究依然很少．基于求解熵产方程,第一次系统研究Richardson数对后台阶流动熵产的影响．对于求解熵产方程所需的速度和温度等变量,通过格子Boltzmann方法来得到．通过上述工作可以发现,后台阶流动中熵产和Bejan数的分布随Richardson数变化显著．总熵产数是Richardson数的单调减函数而平均Bejan数是Richardson数的单调增函数．     

On Producing Multiple Solutions Using Repeated Trials

The number of trials that is required by an algorithm to produce a given fraction of the problem solutions with a specified level of confidence is analyzed. The analysis indicates that the number of trials required to find a large fraction of the solutions rapidly decreases as the number of solutions obtained on each trial by an algorithm increases. In applications where multiple solutions are sought, this decrease in the number of trials could potentially offset the additional computational cost of algorithms that produce multiple solutions on a single trial. The analysis framework presented is used to compare the efficiency of a homotopy algorithm to that of a Newton method by measuring both the number of trials and the number of calculations required to obtain a specified fraction of the solutions.     

AN UPPER BOUND ON AVERAGE TOUCHING NUMBER OF A VORONOI PARTITION

A Voronoi partition is decided bythe configurations of N centerepoints in n dimensional Euclidean space. The total number of nearest neighbor points for a given centerpoint in the partition is called its touching number. It is shown that the average touching number for all points in a Voronoi partition is not greater than the n dimensional kissing number, that is, the maximum uumber of unit spheres that can touch a given unit sphere without overlapping.     

Using Tabu search in CSPS

A heuristic method TABU-CSP using Tabu Search (TS) is described for solving Constraint Satisfaction Problems (CSPs). The method is started with a complete but inconsistent solution of a binary CSP and obtained in prespecified number of iterations either a consistent solution or a near optimal solution with an acceptable number of conflicts. The repair in the solution at each iterative step is done by using two heuristics alternatively. The first heuristic is a min-conflict heuristic that chooses a variable with the maximum number of conflicts and reassigns it the value which leads to the minimum number of conflicts. If the acceptable solution is not reached after the search continued for a certain number of iterations, the min-conflict heuristic is changed and the variable selected least number of times is chosen for repair. If an acceptable solution is not reached, the method switches back to the min-conflict heuristic and proceeds further. This allowed the method to explore a different region of search space for the solution as well as to prevent cycling. The demonstration of the method is shown on a toy problem [9] which has no solution. The method is then tested on various randomly generated CSPs with different starting solutions. The performance of the proposed method in terms of the average number of consistency is checked and the average number of conflicts is compared with that of the Branch and Bound (BB) method used to obtain the same solution. In almost all cases, the proposed method moves faster to the acceptable solution than BB.     

Development and Validation of a Computer‐Administered Number Sense Scale for Fifth‐Grade Children in Taiwan

To investigate the structure of number sense and then to assess its uses in fifth‐grade children's number sense development, a computerized number sense scale was developed and evaluated. The findings of the study indicate that the newly developed scale, with four dominant factors identified and reconfirmed, is internally consistent and substantially valid. It can be reliably used as a screening measure for a quick check of students' number sense development via online self‐assessment. Compared with our previous study, both qualitative and quantitative changes were detected in students' number sense development at different grades. The qualitative change in number sense development is manifested in different numbers of factor components produced at different grades. The quantitative change is manifested in different amounts of factor variance explained at different grades. Furthermore, among the four aspects of number sense, Taiwanese students perform best on recognizing the relative number size yet relatively worse on judging the reasonableness of computational results.     

Eliminating graphs by means of parallel knock-out schemes

In 1997 Lampert and Slater introduced parallel knock-out schemes, an iterative process on graphs that goes through several rounds. In each round of this process, every vertex eliminates exactly one of its neighbors. The parallel knock-out number of a graph is the minimum number of rounds after which all vertices have been eliminated (if possible). The parallel knock-out number is related to well-known concepts like perfect matchings, hamiltonian cycles, and 2-factors.We derive a number of combinatorial and algorithmic results on parallel knock-out numbers: for families of sparse graphs (like planar graphs or graphs of bounded tree-width), the parallel knock-out number grows at most logarithmically with the number n of vertices; this bound is basically tight for trees. Furthermore, there is a family of bipartite graphs for which the parallel knock-out number grows proportionally to the square root of n. We characterize trees with parallel knock-out number at most 2, and we show that the parallel knock-out number for trees can be computed in polynomial time via a dynamic programming approach (whereas in general graphs this problem is known to be NP-hard). Finally, we prove that the parallel knock-out number of a claw-free graph is either infinite or less than or equal to 2.     

Monotonicity of the search number in the Golovach problem

The Golovach problem, also known as the ɛ-search problem, is as follows. A team of pursuers pursues an evader on a topological graph. The objective of the pursuers is to catch the evader, that is, approach the evader to a distance not exceeding a given nonnegative number ɛ. It is assumed that the evader is invisible to the pursuers and is fully informed beforehand about the search program of the pursuers. The problem is to find the ɛ-search number, i.e., the least number of pursuers sufficient for capturing the evader. Graphs with monotone ɛ-search number are studied; the ɛ-search number of a graph G is said to be monotone if it is not exceeded by the ɛ-search numbers of all connected subgraphs H of G. It is known that the ɛ-search number of any tree is monotone for all nonnegative ɛ. The edgesearch number, which is equal to the 0-search number, is monotone for all connected subgraphs of an arbitrary graph. A sufficient monotonicity condition for the ɛ-search number of any graph is obtained. This result is improved in the case of complete subgraphs. The Golovach function is constructed for graphs obtained by removing one edge from complete graphs with unit edges.     

Factors affecting the process of proficient mental addition and subtraction: case studies of flexible and inflexible computers

The relationship between mental computation and number sense is complex: mental computation can facilitate number sense when students are encouraged to be flexible, but flexibility and number sense is neither sufficient nor necessary for accuracy in mental computation. It is possible for familiarity with a strategy to compensate for a lack of number sense and inefficient processes. This study reports on six case studies exploring Year 3 students’ procedures for and understanding of mental addition and subtraction, and understanding of number sense and other cognitive, metacognitive, and affective factors associated with mental computation. The case studies indicate that the mental computation process is composed of four stages in which cognitive, metacognitive and affective factors operate differently for flexible and inflexible computers. The authors propose a model in which the differences between computer types are seen in terms of the application of different knowledges in number facts, numeration, effect of operation on number, and beliefs and metacognition on strategy choice and strategy implementation.     

Poly-Cauchy numbers with a q parameter

The concept of poly-Cauchy numbers was recently introduced by the author. The poly-Cauchy number is a generalization of the Cauchy number just as the poly-Bernoulli number is a generalization of the classical Bernoulli number. In this paper we give some more generalizations of poly-Cauchy numbers and show some arithmetical properties.     

D-完全一致混合超图不可着色的一个充要条件

混合超图的上,下色数与C-超边和D-超边数有着必然联系.一般地,增加C边会使下色数x(H)增加,增加D-超边会使上色数(x)(H)减小.本论文对D-完全一致混合超图进行研究,利用组合数学中分划思想及方法得到的D-完全一致混合超图不可着色的一个充要条件,对D-完全一致混合超图能否着色找到了可行的依据,进一步揭示C-超边数...     

Surfactant Influence on Water Wave Packets

The influence of surfactant on water wave packets is investigated. An envelope equation for a slowly varying wave packet in the potential flow equations with variable Bond number is derived. The properties of this equation depend on the relative phases of the wave packet and the distribution of surface tension. We observe that small variations in the Bond number may change the focusing nature of the envelope equation from that of the constant Bond number problem. Variations in Bond number can thus suppress, or incite, the Benjamin‐Feir instability. The existence of envelope solitary waves depends in a similar way on the Bond number variation. The envelope equation is also derived in a larger class of models.     

临界图的星色数

1988年,Vince定义了图的色数的一个推广——图的星色数,本文研究了有围长限制或有最大度限制的临界图的星色数,得到了三个新结果。     

About one - an inquiry about the meanings and uses of the number one

The story of the number one is told. The long struggle to recognize one as a number is described. The role of the number one in philosophy, religion, mathematics and science is briefly discussed. Finally, the relationship between one and some other numbers is presented.     

The Optimal Preconditioning in the Domain Decomposition Method for Wilson Element

1.TheCollstructionofPreconditionerLetfil)eapolygolldolllaillillR',feL'(fl).Consi(lertheholllogeneousDiricllletboulldaryvalueProblenlofPoissonequation,Assllmethat,fordomainfi,thereareacoarsersubdivisionTHwitllIneshsizeHalldananotheroneThwithmeshsizeh,whichisobtainedbyrefiningTH'Thebotllsubdivisionssatisfythequasi-uniformityandtheillversehypothesis.FOragivenelemelltT,Pm(T)dellotesthespaceofallpolynomialswiththedegreenotgreaterthanm,Qm(T)denotesthespaceofallpolynomialswiththedegreecorres…