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.     

On Vincent''''s Theorem

Vincent's theorem is the theoretical foundation on which the computer algori-thms so far considered best to isolate polynomial real roots are based. In this pa-per, we present an improvement and an extension of Vincent's theorem. Several     

On Explicit Bounds in Picard's Theorem and Schottky's Theorem

Let λ(τ)be the elliptic function.We define the function t∶(0,∞);→(0,∞) byλ(1 it(α))=-α.The functiorl t(α)is a decreasing functiorl of a with t(1)= 1.In the following we need the function g(a)=ae~(πt).Let S denote the class of functions which are analytic in the unit disc D and omit 0 and 1.Schottky's theorem asserts the existence of a function K(α,γ)sueh that     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

A-Browder＇s Theorem and Generalized a-Weyl＇s Theorem

Two variants of the essential approximate point spectrum are discussed. We find for example that if one of them coincides with the left Drazin spectrum then the generalized a-Weyl＇s theorem holds, and conversely for a-isoloid operators. We also study the generalized a-Weyl＇s theorem for Class A operators.     

Improvements of LaSalle's Theorem

In this paper,we improve LaSalle's invariance theorem based on Li'swork(Li Yong,Asymptotic stability and ultimate boundedness,Northeast.Math.J.,6(1)(1990),53-59)by relaxing the restrictions,which make the theorem moreeasy to apply.In addition,we also improve LaSalle's theorem for stochastic differ-ential equation established by Mao(Mao Xuerong,Stochastic versions of the LaSalletheorem,J.Differential Equations,153(1999),175-195).     

A Unified Convergence Theorem

We prove a unified convergence theorem, which presents, in four equivalent forms, the famous Antosik-Mikusinski theorems. In particular, we show that Swartz‘ three uniform convergence principles are all equivalent to the Antosik-Mikusinski theorems.     

Remarks on a Noble''''s Theorem

Noble proved the following Theorem: If A (?) X with a nonisolated point and B (?) Y, then A × B is bounded in X×Y if and only if the projection map π : X × Y → X is a z-map with respect to A × B and A, A is bounded in X and B is bounded in Y. In this note, we give two examples showing the necessary and sufficient conditions of Noble's theorem are not right.     

A Note on Zorn's Theorem

In this note, we prove that if G is a finite group, then G is p-nilpotent if and only if there exists a positive integer n such that for any x,y∈G.     

Ergodic Retraction Theorem and Weak Convergence Theorem for Reversible Semigroups of Non-Lipschitzian Mappings

Let G be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banaeh space E with a Frechet differentiable norm, and T = {Tt ： t ∈ G} be a continuous representation of G as nearly asymptotically nonexpansive type mappings of C into itself such that the common fixed point set F（T） of T in C is nonempty. It is shown that if G is right reversible, then for each almost-orbit u（.） of T, ∩s∈G ^-CO{u（t） ： t ≥ s} ∩ F（T） consists of at most one point. Furthermore, ∩s∈G ^-CO{Ttx ： t ≥ s} ∩ F（T） is nonempty for each x ∈ C if and only if there exists a nonlinear ergodic retraction P of C onto F（T） such that PTs - TsP = P for all s ∈ G and Px ∈^-CO{Ttx ： s ∈ G} for each x ∈ C. This result is applied to study the problem of weak convergence of the net {u（t） ： t ∈ G} to a common fixed point of T.     

On a Theorem of Milin

Let S be class of functions f(z)=z a_2 z~2 … analytic and univalent in the unit disk D, and let 1.M.Milin proved that Theorem A. If f∈S and then where d_o(h)=1and In fact,the result is deduced from Milan′s Tauberian Theorem. Here     

Radon-Nikodym Theorem in Loeb Space

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On a Theorem of Drasin

D. Drasin in 1969 proved that a family of holomorphic functions is normal ifevery function in the family satisfies f' - af~3≠b, where a and b are two complexnumbers and a≠0. Now, we obtain two improvements of this criterion. Theorem 1 A family {f} of holomorphic functions is normal if every functionin {f} satisfies     

Applying of The Pythagorean Theorem

~~Applying of The Pythagorean Theorem~~     

A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem

Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \(\mathbf {RCA}_0\) to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman’s Theorem with apartness to the Increasing Polarized Ramsey’s Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman’s Theorem homogeneity is required only for finite sums of adjacent elements.     

‘Twice’ Equivariant C*-Index Theorem and the Index Theorem for Families

We prove the index theorem for elliptic operators acting on sections of bundles where the fiber is equal to a projective module over a C ^*-algebra in the situation of the action of a compact Lie group on this algebra as well as on the total space commuting with a symbol. For this purpose, we prove in particular the corresponding Thom isomorphism theorem. As an application, the equivariant family index theorem for a direct product of a base by the space of parameters is obtained.     

Cartan’s Second Main Theorem and Mason’s Theorem for Jackson Difference Operator

Let f : C → Pⁿ be a holomorphic curve of order zero. The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves. In addition, a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial. Furthermore, they extend the Mason’s theorem for m + 1 polynomials. Some examples are constructed to show that their results are accurate.     

Spectral Inclusion Theorem for Toeplitz Products

The spectral inclusion theorem for Toeplitz operator ([1], problem 196) plays a important role in studying problems of single Toeplitz operator. In this paper, we present the spectral inclusion theorem for Toeplitz products and give several corollaries. The notation and terminology in this paper are the same as in [1]. For example, denotes Laurent operator on L determined by (L~2 and L~∞ are     

Generalising Conduché’s Theorem

In a previous paper (Kasangian and Labella, J Pure Appl Algebra, 2009) we proved a form of Conduché’s theorem for LSymcat-categories, where L was a meet-semilattice monoid. The original theorem was proved in Conduché (CR Acad Sci Paris 275:A891–A894, 1972) for ordinary categories. We showed also that the “lifting factorisation condition” used to prove the theorem is strictly related to the notion of state for processes whose semantics is modeled by LSymcat-categories. In this note we resume the content of Kasangian and Labella (J Pure Appl Algebra, 2009) in order to generalise the theorem to other situations, mainly arising from computer science. We will consider PSymcat-categories, where P is slightly more general than a meet-semilattice monoid, in which the lifting factorisation condition for a PSymcat-functor still implies the existence of a right adjoint to its corresponding inverse image functor.     

The Pythagorean Theorem(二)

By translate the triangle we can get a figure like that right.The angle measures J. and y add to 90°, so the angles of the