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 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 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 so special about mathematical texts? Analyses of common claims in research literature and of properties of textbooks

This study surveys claims in research articles regarding linguistic properties of mathematical texts, focusing on claims supported by empirical or logical arguments. It also performs a linguistic analysis to determine whether some of these claims are valid for school textbooks in mathematics and history. The result of the survey shows many and varying claims that mainly describe mathematical texts as highly compact, precise, complex, and containing technical vocabulary. However, very few studies present empirical support for their claims, and the few empirical studies that do exist contradict the most common, and unsupported, claims, since no empirical study has shown mathematical texts to be more complex than texts from other subjects, and any significant differences rather indicate the opposite. The linguistic analysis in this study is in line with previous empirical studies and stands in contrast to the more common opinion in the unsupported claims. For example, the mathematics textbooks have significantly shorter sentences than the history textbooks.     

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 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 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 role of stochastic resonance?

The role of stochastic resonance (SR) in the detection of signals is discussed. It is shown that when a signal is not detectable using traditional methods, such as spectral analysis or averaging, it cannot be detected using SR. In other words by adding noise a far-from detectable signal cannot become detectable.     

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 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 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     

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 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     

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 optimal cutoff surface for ore bodies with more than one mineral?

In mine planning problems, cutoff grade optimization defines a threshold at every time period such that material above this value is processed, and the rest is considered waste. In orebodies with multiple minerals, which occur in practice, the natural extension is to consider a cutoff surface. We show that in two dimensions the optimal solution is a line, and in n dimensions it is a hyperplane.     

What is quasiconvex analysis?

We endeavour to answer the question of the title, or rather the question of how much one can extend convex analysis to a wider framework in which some convexity features remain. Subdifferential calculus and duality are the main directions we consider