The Pythagorean Theorem(一)

From the area of a square you can determine the length ofany of its sides.If the length of the side is √A.That is why√A is called the square root of A. Just as a plant rests on itsroots,a square can rest on its(positive)square root.side=√AThus if the area of a square is 400 square feet,its side has length 20cm.If the area is 13square feet,its side has length √13 feet,You can verify the latter of these with a calculator:     

The Pythagorean Theorem(三)

ProofThe original triangle withits translation and rota-tion images is below.Itslegs are a and b and thehypotenuse is c.The areaof the shaded square is     

Applying of The Pythagorean Theorem

~~Applying of The Pythagorean Theorem~~     

Pythagorean Converse Theorem

To make a right triangle,the ancient Egyptians took a rope with 12 equally spaced knots in it,and then bent it in two places to form a triangle with sides of lengths 3,4,and 5.     

Three-dimensional Pythagorean Theorem for Elliptic Space

Let M be an elliptic space with constant curvature K=1/R~2,and consider inM a right-angled tetrahedron,i.e.,a tetrahedron with three mutually perpendi-cular edges at a certain vertex.Denote the tetrahedron by VABC and assumethat each of the edges VA,VB and VC is perpendicular to others.The areas oftriangles VBC,VCA,VAB and ABC are denoted by S_1,S_2,S_3 and S,respective-ly.We obtain the following     

What is The Squeeze (or Sandwich) Theorem?

<正>Let's do an exercise as warming up.Ex:Find lim x→0sin1/x.lim x→0sin1/x=lim1/x→∞sin1/x=lim t→∞sint(Let t=1/x).We know lim t→∞sint does not exist,so lim x→0sin1/x DNE.We can use graphing calculator to show the result is DNE(from figure 1to figure 8).     

The Cameron–Martin Theorem for (p-)Slepian Processes

We show a Cameron–Martin theorem for Slepian processes \(W_t:=\frac{1}{\sqrt{p}}(B_t-B_{t-p}), t\in [p,1]\), where \(p\ge \frac{1}{2}\) and \(B_s\) is Brownian motion. More exactly, we determine the class of functions \(F\) for which a density of \(F(t)+W_t\) with respect to \(W_t\) exists. Moreover, we prove an explicit formula for this density. p-Slepian processes are closely related to Slepian processes. p-Slepian processes play a prominent role among others in scan statistics and in testing for parameter constancy when data are taken from a moving window.     

The Equivalence between Property (ω) and Weyl’s Theorem

We call T ∈ B(H) consistent in Fredholm and index (briefly a CFI operator) if for each B ∈ B(H),T B and BT are Fredholm together and the same index of B,or not Fredholm together.Using a new spectrum defined in view of the CFI operator,we give the equivalence of Weyl’s theorem and property (ω) for T and its conjugate operator T* .In addition,the property (ω) for operator matrices is considered.     

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)},     

The Osofsky–Smith Theorem for Modular Lattices and Applications (I)

The aim of this article is to present a latticial version of the renown module theoretical Osofsky–Smith Theorem.     

The Loomis–Sikorski Theorem revisited

We prove, constructively, that the Loomis–Sikorski Theorem for σ-complete Boolean algebras follows from a representation theorem for Archimedean vector lattices and a constructive representation of Boolean algebras as spaces of Carathéodory place functions. We also prove a constructive subdirect product representation theorem for arbitrary partially ordered vector spaces. Received August 10, 2006; accepted in final form May 30, 2007.     

The Central Limit Theorem on SO0(p, q)/SO(p)×SO(q)

We obtain a central limit theorem for the space SO ₀(p, q)/SO(p)×SO(q). To achieve this, we derive a Taylor expansion of the spherical function on the group SO ₀(p, q).     

The Radon-Nikodym Theorem of Bounded Set-Valued Measures

1. Notions and Lemmas Let (Ω,■) be a measurable space, R~m the m-dimensional Euclidean apace.     

The Jentzsch-Szegő Theorem and Balayage Measures

The numerous generalizations of the Jentzsch-Szeg? theorem on the location of zeros of Taylor polynomials have been based so far on the extremal properties satisfied by the corresponding approximants. We do away with those kinds of assumptions and prove the theorem for a general class of interpolating polynomials. This is possible thanks to the discovery presented here that the limit distribution of the zeros of the interpolants is governed by a balayage measure depending on the distribution of the interpolation points and the region of analyticity of the function being approximated.     

An Extension of The Handle Addition Theorem

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The Filippov-Ważewski Relaxation Theorem Revisited

The converse statement of the Filippov-Waewski relaxation theorem is proved. More precisely, two differential inclusions have the same closure of their solution sets if and only if the right-hand sides have the same convex hull. The idea of the proof is examining the contingent derivatives to the attainable sets.     

Property (aw) and Weyl’s Theorem

In this note we study the property (aw), a variant of Weyl’s theorem introduced by Berkani and Zariouh, by means of the localized single valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (aw) holds. We also relate this property with Weyl’s theorem, a-Weyl’s theorem and property (w). Finally, we show that if T is a-polaroid and either T or T* has SVEP then f(T) satisfies property (aw) for each ${f \in H_1(\sigma(T))}$ .     

The Berry-Esseen Theorem and Non-Uniform Estimate for U-Statistics

Let X_1,X_2,…,X_n be independent random variables. Define a U-statistic by U_n(?)~(-1)sum from 1≤i≤j≤n (h(X_i,X_j), where h(x,y) is a symmetric function of two variables x,y and that Eh(X_i,X_j)=0(i≠j, i,j=1,2,…,n). Write g_j(X_i)=E(h(x_i,x_j)|x_i),g(X_1)=1/n-1 sum from j=1 j≠i to n g_j(X_i) We give the following two theorem: Theorem 1 Suppore that