The Function Concept(二)

The domain of linear function f(x)=ax b(a≠0) is R, its range is R. The domain of inverse proportion function f(x)=(k/x)(k≠0) is {x|x≠0}, its range is B ={y|y≠0}. The domain of quadratic function, f(x)=ax~2 bx c(a≠0)is R, its range is B. B={y|y≥(4ac-b~2)/4a},when a>0;B={y|y≤(4ac-b~2/4a)},     

Inverse Functions of Number-Theoretic Functions (I)

If gf(x) =x for every x, then g is called a left inverse function of f and f is a right inverse function of g. If f is both left and right inverse function of g, then f and g are said to be mutually inverse to each other. We show that (§ 1) the following results hold. A function f has a left inverse if and only if f is univalent, a function g has a right inverse if and only if g is exhaustive, i. e., g takes every (natural) number as values. Hence f has both left and right inverse if and only if f is both univalent and exhaustive, i. e., f is a permutation on the domain of natural numbers. Let g_1 and g_2 be two left inverse functions of the function f. If for every left inverse g of f, we have $g_1(x) \leq g(x) \leq g_2(x)$, then g_1(x) is called the weak, and g_2(x) is the strong, left inverse function of f. Similarly we define the weak and the strong right inverse functions. We show that(§ 2) every strict increasing function f must possess weak and strong left inverse functions, and all of its left inverse functions must be exhaustive slow increasing (a function g(x) is slow increasing if and only if g(Sx) —Sg(x) =0, here s denotes the successor function). On the other hand, every exhaustive function g must possess weak and strong right inverse functions, and all of its right inverse functions must strict increasing. We show also that (§ 3): If f_1(x) and f_2(x) both take g(x) as their strong (weak) left inverse, then f_1(x)=f_2(x)(f_1(Sx)=f_2(Sx)). If g_1(x) and g_2(x) both take f(x) as their strong or weak right inverse, then g_1(x)=g_2(x). From these results we see that we may find a function from its strong (weak) left or right inverse function. Let there be f(c) \leq x 相似文献   

What is the differential equation？

An equation involving a derivative is called a differential equation.Such as,dy/dx=2x,and the function y=f（x）satisfies this equation.When we know the additional condition that y=2 when x=-1,the function y=f（x）will be find exactly.The additional condition is called the initial condition.It is used to evaluate constant of integration.     

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     

Optimal Delaunay Triangulations

The Delaunay triangulation, in both classic and more generalized sense, is studied in this paper for minimizing the linear interpolation error (measure in L^P-norm) for a given function. The classic Delaunay triangulation can then be characterized as an optimal triangulation that minimizes the interpolation error for the isotropic function ‖x‖^2 among all the triangulations with a given set of vertices. For a more general function, a functiondependent Delaunay triangulation is then defined to be an optimal triangulation that minimizes the interpolation error for this function and its construction can be obtained by a simple lifting and projection procedure. The optimal Delaunay triangulation is the one that minimizes the interpolation error among all triangulations with the same number of vertices, i.e. the distribution of vertices are optimized in order to minimize the interpolation error. Such a function-depend entoptimal Delaunay triangulation is proved to exist for any given convex continuous function.On an optimal Delaunay triangulation associated with f, it is proved that △↓f at the interior vertices can be exactly recovered by the function values on its neighboring vertices.Since the optimal Delaunay triangulation is difficult to obtain in practice, the concept of nearly optimal triangulation is introduced and two sufficient conditions are presented for a triangulation to be nearly optimal.     

Local generalized empirical estimation of regression

Let f(x) be the density of a design variable X and m(x) = E[Y\X = x] the regression function. Then m(x) - G(x)/f(x), where G(x) = m(x)f(x). The Dirac δ-function is used to define a generalized empirical function Gn (x) for G(x) whose expectation equals G(x). This generalized empirical function exists only in the space of Schwartz distributions, so we introduce a local polynomial of order p approximation to Gn(.) which provides estimators of the function G(x) and its derivatives. The density f(x) can be estimated in a similar manner. The resulting local generalized empirical estimator (LGE) of m(x) is exactly the Nadaraya-Watson estimator at interior points when p = 1, but on the boundary the estimator automatically corrects the boundary effect. Asymptotic normality of the estimator is established. Asymptotic expressions for the mean squared errors are obtained and used in bandwidth selection. Boundary behavior of the estimators is investigated in details. We use Monte Carlo simulations to show that the     

What is the integration by parts?

The anti-derivative or indefinite integral of a function f(x)is a function F(x)whose derivative is f(x).Then,each differentiation rule should have a corresponding integration rule.Last time we worked with the substitution rule for integration corresponds to the chain rule in differentiation.What is     

BOUNDARY VALUE PROBLEMS WITH INTEGRO CONDITION FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS

1 IntroductionIn recent years) there were many results of studying boundary value problemsfor functional differential equations [1-5]. But the boundary conditions are allDirichlet's type. In this paper, we will study a kind of boundary value problemswitb integro boundary condition fOr the functional differential equdtionswhere f(t, yt, x, y) is a continuous real function defined on the set fl = [0, T] xC. x R', and is continuously differentiable with respect to each variable x, y;p E C. = C…     

On Quasi-weakly Almost Periodic Points of Continuous Flows

Let X be a compact metric space, F : X ×R→ X be a continuous flow and x ∈ X a proper quasi-weakly almost periodic point, that is, x is quasi-weakly almost periodic but not weakly almost periodic. The aim of this paper is to investigate whether there exists an invariant measure generated by the orbit of x such that the support of this measure coincides with the minimal center of attraction of x? In order to solve the problem, two continuous flows are constructed. In one continuous flow,there exist a proper quasi-weakly almost periodic point and an invariant measure generated by its orbit such that the support of this measure coincides with its minimal center of attraction; and in the other,there is a proper quasi-weakly almost periodic point such that the support of any invariant measure generated by its orbit is properly contained in its minimal center of attraction. So the mentioned problem is sufficiently answered in the paper.     

A STONE-WEIERSTRASS THEOREM FOR CERTAIN DISCONTINUOUS FUNCTIONS

La Abe point separating uiatal subalgebra. of C(T) where T is a compact metric space. For each bounded function f:T which is continuous on the complement of a meagre subset of T there exists a sequence (wn) of elements of the algebra A such that the sequence (w) converges uniformly to the function f on each compact subset of the interior of the continuity points of the function f.     

On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

In this paper, some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type(T_λf)(x, y) = ∫_a~b ∫_a~b f(t, s)K_λ(t-x,s-y)dsdt, x,y ∈(a,b), λ∈Λ  [0,∞),(0.1)are given. Here f belongs to the function space L_1( a,b ~2), where a,b is an arbitrary interval in R. In this paper three theorems are proved, one for existence of the operator(T_λf)(x, y) and the others for its Fatou-type pointwise convergence to f(x_0, y_0), as(x,y,λ) tends to(x_0, y_0, λ_0). In contrast to previous works, the kernel functions K_λ(u,v)don't have to be 2π-periodic, positive, even and radial. Our results improve and extend some of the previous results of [1, 6, 8, 10, 11, 13] in three dimensional frame and especially the very recent paper [15].     

QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES

Let f : U(x0) (?) E→F be a. C1 map and f'(X0) be the Prechet derivative of /fat X0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties: f'(x0) is double splitting and R(f'(x))∩N(T0 ) = {0} near X0. However, in infinite dimensional Banach spaces, f'(x0) is not always double splitting (i.e., the generalized inverse of f'(xo) does not always exist), but its bounded outer inverse of f'(x0) always exists. Only using the C1 map f and the outer inverse T0# of f'(x0), the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if X0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.     

A Note on a Problem of M. S. Berger

In this paper, we discuss the problem concerning global and local structure of solutions of an operator equation posed by M. S. Berger. Let f : U (?)E→ F be a C1 map, where E and F are Banach spaces and U is open in E. We show that the solution set of the equation f(x)=y for a fixed generalized regular value y of f is represented as a union of disjoint connected C1 Banach submanifolds of U, each of which has a dimension and its tangent space is given. In particular, a characterization of the isolated solutions of the equation f(x) = y is obtained.     

有关M.S.Berger问题的注记

In this paper, we discuss the problem concerning global and local structure of solutions of an operator equation posed by M. S. Berger. Let f : U 真包含 E → F be a C1 map, where E and F are Banach spaces and U is open in E. We show that the solution set of the equation f(x) = y for a fixed generalized regular value y of f is represented as a union of disjoint connected C1 Banach submanifolds of U, each of which has a dimension and its tangent space is given. In particular, a characterization of the isolated solutions of the equation f(x) = y is obtained.     

Approximation integration by using rectangles

In calculus,two famous problems are well known.The one problem of finding the tangent line led us to the derivative.The other problem of finding area led us to find the definite integral.In this paper,we will learn how to approximation integration by using rectangles.1.What does the Riemann sum mean？A given function y=f（x）is continuous on the interval[a,b].     

DNA labelled graphs with DNA computing

Let k≥2, 1≤i≤k andα≥1 be three integers. For any multiset which consists of some k-long oligonucleotides, a DNA labelled graph is defined as follows: each oligonucleotide from the multiset becomes a point; two points are connected by an arc from the first point to the second one if the i rightmost uucleotides of the first point overlap with the i leftmost nucleotides of the second one. We say that a directed graph D can be(k, i;α)-labelled if it is possible to assign a label(l_1(x),..., l_k(x))to each point x of D such that l_j(x)∈{0,...,a-1}for any j∈{1,...,k}and(x,y)∈E(D)if and only if(l_k-i 1(x),..., l_k(x))=(l_1(y),..., l_i(y)). By the biological background, a directed graph is a DNA labelled graph if there exist two integers k, i such that it is(k, i; 4)-labelled. In this paper, a detailed discussion of DNA labelled graphs is given. Firstly, we study the relationship between DNA labelled graphs and some existing directed graph classes. Secondly, it is shown that for any DNA labelled graph, there exists a positive integer i such that it is(2i, i; 4)-labelled. Furthermore, the smallest i is determined, and a polynomial-time algorithm is introduced to give a(2i, i; 4)-labelling for a given DNA labelled graph. Finally, a DNA algorithm is given to find all paths from one given point to another in a(2i, i; 4)-labelled directed graph.     

Linearization Ill-Posedness for 2.5-D Ware Equation Inversion Model

Abstract For the weakly inhomogeneous acoustic medium in Ω={(x,y,z):z>0},we consider the inverse problemof determining the density function ρ(x,y).The inversion input for our inverse problem is the wave field givenon a line.We get an integral equation for the 2-D density perturbation from the linearization.By virtue of theintegral transform,we prove the uniqueness and the instability of the solution to the integral equation.Thedegree of ill-posedness for this problem is also given.     

Rough singular integral operators on Hardy-Sobolev spaces

The authors study the singular integral operator TΩ,αf(x)=p.v.∫R^nb(|y|Ω(y′)|y|^-n-αf(x-y)dy, defined on all test functions f, where b is a bounded function, α＞0, Ω(y′) is an integrable function on the unit sphere S^n-1 satisfying certain cancellation conditions. It is proved that, for n/(n α)＜p＜∞,TΩ,α is a bounded operator from the Hardy-Sobolev space H^pα to the Hardy space H^p. The results and its applications improve some theorems in a previous paper of the author and they are extensions of the main theorems in Wheeden‘s paper(1969). The proof is based on a new atomic decomposition of the space H^pα by Han, Paluszynski and Weiss(1995). By using the same proof,the singluar integral operators with variable kernels are also studied.     

Logarithmic Function

Recall that a one-to-one function y= f（x） has an inverse function that is defined （implicitly） by the equation x= f（y）, In particular, the exponential function：     

The Factor Decomposition Theorem of Bounded Generalized Inverse Modules and Their Topological Continuity

In this paper we obtain a Douglas type factor decomposition theorem about certain important bounded module maps. Thus, we come to the discussion of the topological continuity of bounded generalized inverse module maps. Let X be a topological space, x →Tx ： X→L（E） be a continuous map, and each R（Tx） be a closed submodule in E, for every fixed x C X. Then the map x→ Tx^＋： X→L（E） is continuous if and only if ｜｜Tx^＋｜｜ is locally bounded, where Tx^＋ is the bounded generalized inverse module map of Tx. Furthermore, this is equivalent to the following statement： For each x0 in X, there exists a neighborhood ∪0 at x0 and a positive number λ such that （0, λ^2）lohtatn in ∩x∈∪0C／σ（Tx^＋Tx）, where a（T） denotes the spectrum of operator T.