What is the Inverse of a Basis?

A formula is given for the inverse of the linear map, from coordinate space to a linear space, induced by a basis for that linear space, which is then connected to various basic Applied Linear Algebra constructs.This revised version was published online in October 2005 with corrections to the Cover Date.     

What is the Rees algebra of a module?

In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module may be computed in terms of a ``maximal' map from to a free module as the image of the map induced by on symmetric algebras. We show that the analytic spread and reductions of can be determined from any embedding of into a free module, and in characteristic 0--but not in positive characteristic!--the Rees algebra itself can be computed from any such embedding.

     

What is the differential equation?

<正>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=2when 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 is the integral test?

<正>In general,it is not easy to find the exact sum of a series.We could find geometric series with a simple formula for the n-th partial sum S_n.And we are also lucky can find the n-th partial sum S_n of the series ∞∑(n=1)1/(n(n+1)).     

What is the Substitution Rule?

When I learned the following basic formula of indefinite integrals in the calculus,at that time I was a teenager,the strange ideas had jump out my head.I would be a great girl in math if I can change all the ? in the following formula to the real functions     

What is the alternating series test?

<正>We learned all the convergence tests before,which all terms are positive.Please try to use your brain to imagine how to deal with series whose terms are not always positive.The most simplest series are alternating series,whose terms alternate in sign.     

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     

What is the basic integration rule?

<正>In this paper you will learn the basic integration rule from the basic differentiation rule.Last time we have learned anti-derivative,indefinite integral,and the power rule in integration.Let us recall what the power rule in differentiation is.If n is a real number,and y=xn,then y'=nxn-1.Proof:Let y=xn,we get ln|y|=ln|xn|,Since ln|xn|=n ln|x|,so we get ln|y|=n ln|x|.Take differentiation for both side above:y'y=n x.     

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     

What is the Limit of a Function in Intuitive Way?

In AP Calculus,if you want to learn the course,you will be able to work with the four waysof representation-graphically,numerically,analytically,and verbally.It is important to em-phasize and connect the relationship among these representations as you learn the course success     

What is the role and contribution of models to management and research in the health services? A view from Europe

Modelling in different forms has long been regarded as a cornerstone of health OR as in other fields of application. Models are now tending to become a standard tool in health services management and research. What are the lessons we as operational researchers have learnt during this development? How can health care managers and health service researchers benefit from modelling — with or without OR-analysts to guide them between the pitfalls? After an introductory overview concerning the nature and objectives of modelling, examples will be given of modelling applications from different health service areas in order to illustrate the versatility of the method. Further, the choice of methods and models is discussed with special attention to the problems of interpretation and implementation of results. From this overview some conclusions are drawn with regards to advantages and disadvantages of modelling as tools for policy planning and decision-making in the health area. Finally some observations are made concercing the conditions for the future development in the field.     

What is the theory without power set?

We show that the theory , consisting of the usual axioms of but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well‐ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of in which ω₁ is singular, in which every set of reals is countable, yet ω₁ exists, in which there are sets of reals of every size , but none of size , and therefore, in which the collection axiom fails; there are models of for which the ?o? theorem fails, even when the ultrapower is well‐founded and the measure exists inside the model; there are models of for which the Gaifman theorem fails, in that there is an embedding of models that is Σ₁‐elementary and cofinal, but not elementary; there are elementary embeddings of models whose cofinal restriction is not elementary. Moreover, the collection of formulas that are provably equivalent in to a Σ₁‐formula or a Π₁‐formula is not closed under bounded quantification. Nevertheless, these deficits of are completely repaired by strengthening it to the theory , obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach 18 .     

What is the integration by partial fractions?

<正>1.What a rational function?Do you remember what a polynomial function is?If you know a function of the form is a form as following f(x)=anxn+an-ixn-1+...+a1x+a0,where an,an-i,...a1,a0are real numbers and nis nonnegative integer.The domain of a polynomial function is the set of all real numbers.But here are some func-     

What is the alternating series estimation theorem?

<正>Now we'll learn what the alternating series estimation theorem is?A partial sumSnof any convergent series can be used as an approximation to the total sum.If we want estimate the accuracy of approximation,then the error and remainder should be considered.The error involved in using S≈Snis the remainder.In other words,error involved in using     

An inventory model: What is the influence of the shape of the lead time demand distribution?

In this paper the influence of the shape of the lead time demand distribution is studied for a specific inventory model which is described in a preceding paper by Heuts and van Lieshout [4]. This continuous review inventory model uses as lead time demand distribution a Schmeiser-Deutsch distribution (S-D distribution) [9]. In a previous paper [4] an algorithm was given to solve the decision problem.In the literature attention is given to the following problem: what information on the demand during the lead time is necessary and sufficient to obtain good decisions. Using a (s, S) policy; Naddor [8] concluded that thespecific form of the lead time demand distribution is negligible, and that only its first two moments are essential. For a simple (s, q) control system Fortuin [3] comes to the same conclusion. Both authors analysed the case with known lead times and with given demand distributions from the class of two parameter distributions. So in fact their results are obvious, as the lead time demand distributions resulting from their suppositions are all nearly symmetric. We shall demonstrate that the skewness of the lead time demand distribution in our inventory model is also an important measure, which should be taken into account, as the cost differences with regard to the case where this skewness measure is not used, can be considerable.     

What is continuity?

A continuous process is a person who draws the curve without a break,or,takes place gradually without interruption. Those are a daily language in continuity. However,in calculus,we notice that the limit of a function as x approaches a can often be found simply by calculating the value of the function at a. It means a continuous function     

Diagnostics of stochastic resonance using Poincaré recurrence time distribution

In the paper we calculate the distribution density of Poincaré recurrence times for a one-dimensional nonhyperbolic cubic map subjected to white noise and a harmonic signal. It is established that for small vicinities of recurrence the distribution density is not described by an exponential law and is periodically modulated with the external signal frequency. It is shown that the Fourier spectrum of the distribution density exhibits a well-distinguishable peak at the external signal frequency. The peak amplitude achieves its maximal value in the regime of stochastic resonance (SR), that can be used for detecting the SR effect.