Read Carefully (1)——General Advice About Reading

To solve a problem, you must understand it. So, if the problem is written down, you must read it carefully. Here are three things to do to get the necessary information from reading.1. Know the meaning of all words and symbols in the problem.2. Sort the information into what is needed and is not needed.3. Determine if there is enough information to solve the problem.     

Proportions in Similar Figures

The image of afigure under a size change of magnitude k is similar to the original figure.,then how to determine the size change factor when given two similar figures.In this activity you will learn it.     

What are the anti-derivative and the indefinite integral?(1)

Since we know the derivative of the function,so it is the thinking way in math to find a function of F whose derivative is a known function f.If such a function Fexists,we can call it an anti-derivative of f.Let us think about it.For instance,let f(x)=x2.We can find an anti-derivative of f,if we use the Power Rule on it.What F(x)=1/3x1/3 is the one could be discovered,since it is satisfied with.Is there anyone else? Yes,you are right.More functions     

A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs

The metric dimension dim(G)of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices.The zero forcing number Z(G)of a graph G is the minimum cardinality of a set S of black vertices(whereas vertices in V(G)\S are colored white)such that V(G)is turned black after finitely many applications of"the color-change rule":a white vertex is converted black if it is the only white neighbor of a black vertex.We show that dim(T)≤Z(T)for a tree T,and that dim(G)≤Z(G)+1 if G is a unicyclic graph;along the way,we characterize trees T attaining dim(T)=Z(T).For a general graph G,we introduce the"cycle rank conjecture".We conclude with a proof of dim(T)-2≤dim(T+e)≤dim(T)+1 for e∈E(T).     

Putting-Together with Overlap Model(1)

If a quantity x is put together with a quantity y,and there is overlap z,the result is the quantity x y-z. When you know the value of x y-z and two of the variables in it,you can find the value of the third variable by solving an equation.     

What is the limit of a function in precise way?

The precise definition of a limit is represented in graphically,numerically,analytically and verbally by using algebraic way.The intuitive definition of a limit is given as "the limit of f(x),as x approaches a,equals L ",and we write lim x→af(x)=L.But the more questions will be asked if you think it broadly."Is the definition of the limit of function vague?" "How should be x close to a presented algebraically?" "How should be f(x) gets closer and closer to L presented algebraically?" "How close to a does x to     

Being a Good Problem Solver（1）

When solving problems, You need not feel as if you are lost in a maze. Strategies can help you find solutions. What Is an Algorithm?you may have learned to solve any equation of the form x a = b for x. The method was to add -a to each side and then simplify. This method is an example of an algorithm. An algorithm is a sequence of steps that leads to a desired result. Not all algorithms are short. The algorithm called long division can involve many steps.     

Monotone projected gradient methods for large-scale box-constrained quadratic programming

Inspired by the success of the projected Barzilai-Borwein (PBB) method for large-scale box-constrained quadratic programming, we propose and analyze the monotone projected gradient methods in this paper. We show by experiments and analyses that for the new methods, it is generally a bad option to compute steplengths based on the negative gradients. Thus in our algorithms, some continuous or discontinuous projected gradients are used instead to compute the steplengths. Numerical experiments on a wide variety of test problems are presented, indicating that the new methods usually outperform the PBB method.     

All good (bad) words consisting of 5 blocks

Generalized Fibonacci cube Q_d(f), introduced by Ilic, Klavzar and Rho, is the graph obtained from the hypercube Q_d by removing all vertices that contain f as factor. A word f is good if Q_d(f) is an isometric subgraph of Q_d for all d ≥ 1, and bad otherwise. A non-extendable sequence of contiguous equal digits in a word μ is called a block of μ. Ilic, Klavzar and Rho shown that all the words consisting of one block are good, and all the words consisting of three blocks are bad. So a natural problem is to study the words consisting of other odd number of blocks. In the present paper,a necessary condition for a word consisting of odd number of blocks being good is given, and all the good(bad) words consisting of 5 blocks is determined.     

ON QUINTIC K-3 SURFACES

If a non-normal quintic surface is birational to a K3 surface,then there are threepossibilities:either it is singular along a conic;or it is singular along two mutuallyintersecting lines;or it is singular along a line and has an isolated triple point outside theline.Conversely if a K3 surface contains a hyperelliptic curve of genus three with anode or simple cusp,then it is birational to a quintic surface of the first type mentionedabove.For the other two cases,the minimal models are also characterized.     

拟正规算子的Jord标准形

An operator on a Hilbert space is said to have (SI) decomposition if it is similar to the orthogonal direct sum of some (SI) operators. In this paper, we prove that every operator, which is similar to a quasinormal operator, has (SI)decomposition if and only if it is similar to D ( 0≤j相似文献   

旋转锥面$l_\infty$拟合的截断光滑化牛顿法

In this paper, the rotated cone fitting problem is considered. In case the measured data are generally accurate and it is needed to fit the surface within expected error bound, it is more appropriate to use l∞ norm than 12 norm. l∞ fitting rotated cones need to minimize, under some bound constraints, the maximum function of some nonsmooth functions involving both absolute value and square root functions. Although this is a low dimensional problem, in some practical application, it is needed to fitting large amount of cones repeatedly, moreover, when large amount of measured data are to be fitted to one rotated cone, the number of components in the maximum function is large. So it is necessary to develop efficient solution methods. To solve such optimization problems efficiently, a truncated smoothing Newton method is presented. At first, combining aggregate smoothing technique to the maximum function as well as the absolute value function and a smoothing function to the square root function, a monotonic and uniform smooth approximation to the objective function is constructed. Using the smooth approximation, a smoothing Newton method can be used to solve the problem. Then, to reduce the computation cost, a truncated aggregate smoothing technique is applied to give the truncated smoothing Newton method, such that only a small subset of component functions are aggregated in each iteration point and hence the computation cost is considerably reduced.     

Puzzle and Solution:Diagonal String

“Suppose there were one nail in each of thefour walls of this room and in addition one nail onthe ceiling and one in the floor,Suppose furtherthat we had to tie strings between these nails.Ihave tWO colors of strings which I use—red andblue.Each connection between any two nails iseither with a red string or with a blue string. “All these strings make up many triangles,that is,any three nails can be considered apexesof a triangle formed by the strings connecting these three nails.My problem is to see if I could distribute the colors so that no triangle has all three sides the same color.”Said young Nicho—las. “This is rather complicated,”mused the mathematician. “It must involve calculations of permutations and combinations,and the like.I didn't think you knew that much algebra,Nick.”“I don’t,”replied young Nick。“but I can still do the problem.”“Well,all right,tell us how。”said the eld—er. “It is really very simple.”said young Nicho—las,“You only have to know enough to start rea-soning, “The answer is that there will be at least onetriangle all sides of which are the same color.Iwill show that it is impossible to avoid this. “Consider any one nail. Out from it theremust stretch five strings,one to each of the otherfive nails。No matter how you distribute the col—ors in these five strings,at least three of themmust be the same,since you have only two colors.For the sake of argument let assume that three ofthe strings are red. “Consider now the triangle formed betweenthree nails at which the three red strings haveterminated. “If we are to try to avoid a triangle with all three sides the same color,it follows thatthese three nails cannot all be joined to each other with one color.Putting this more simply,thetriangle formed by the three terminal nailsshould not be all blue.At least one of the stringsbetween the three nails must be red.But if so,we have completed a red triangle from the origi—nal nail.”     

The Error in the Approximation

In applying any numerical method such as the bisection method to determine a root, it is important to realize that the best we can usually achieve is an approximation ofthe exact root. At each iteration of the method,we obtain a better estimate of the root. Thus it becomes desirable that we be able to estimate how accurate the approximation is at each stage so that we know when to stop the process.With the bisection method,suppose we know that there is a root in some interval [a,b], where a and b are successive integers, say 2 and 3. If we select the midpoint M1 of this interval, then it is obvious that the root R is     

三角形数表的性质及其应用

The calculation of the number of relations on a finite set is interesting,it is related to the world famous question “the number of topology on n”.In this paper,we discussed the relation between the numbers of a curve triangle number table,it is similar to Pascal triangle,gave the number of solutions of infinite equation(*)and obtained the new combination meaning of Catalan number;similar to the second class Striling number,it is the number of equivalent class of a relation.     

How to use a definite integral to find area(1)

Integration has a wide variety of applications.Today you’ll learn to use a definite integral to find the area of a region bounded by two curves.Area is nonnegative number.When we find the area of a region above theχ-axis,the area is positive.When we find the area of a region below theχ-axis,the area is negative.     

Being a Good Problem Solver（2）

George Polya was a mathematician at Stanford University who was famous for his writing about solving problems. He once wrote:Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solving problem, you have to observe and to imitate what other people do when solving problems and, finally, you learn to do problems by doing them.     

Engel Subalgebras of n-Lie Algebras

Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive 2-soluble n-Lie algebra is shown to split over its minimal ideal, and all the complements to its minimal ideal are conjugate. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements. Cartan subalgebras are shown to have a property analogous to intravariance.     

Generalization of the interaction between Haar approximation and polynomial operators to higher order methods

In applications it is useful to compute the local average empirical statistics on u. A very simple relation exists when of a function f（u） of an input u from the local averages are given by a Haar approximation. The question is to know if it holds for higher order approximation methods. To do so, it is necessary to use approximate product operators defined over linear approximation spaces. These products are characterized by a Strang and Fix like condition. An explicit construction of these product operators is exhibited for piecewise polynomial functions, using Hermite interpolation. The averaging relation which holds for the Haar approximation is then recovered when the product is defined by a two point Hermite interpolation.     

Read Carefully(2)——Finding the Meanings of Words and Symbols

If you do not know the meaning of a word, look it up in a dictionary. Some mathematics books have glossaries or indexes in the back that can help you locate words.Some terms have more than one meaning even in mathematics. The word divisor can mean the number divided by in a division problem. (In 12÷5=2.4, 5 is the divisor. ) But divisor also means a number that divides another with a zero remainder. For example, 7 is a divisor of 21. In this situation divisor has the same meaning as factor.