Negative Theorems in Approximation Theory
نویسنده
چکیده
1. INTRODUCTION. Negative theorems have a rich tradition in mathematics. In fact mathematics seems to be unique among the sciences in that negative results are very much a part of the mathematical edifice. Most mathematical theories try to explain what is possible and also what is not. To understand the structure of a mathematical theory is also to understand its limitations. There are many different types of negative theorems. Some simply say that something is impossible. For example, one of the classical problems of Greek mathematics asks whether it is possible, using only a ruler and a compass, to square the circle, i.e., to construct a square with the same area as a given circle. This was answered in the negative in 1882 when Lindemann proved that π is transcendental. (Lambert had already proved that π is irrational, but this is insufficient to prove the impossibility of squaring the circle.) We recall that Abel established the insolvability of the general quintic equation, i.e., one cannot find a formula for the roots of fifth degree polynomi-als that involves only the coefficients of the polynomials and radicals. And, of course, there is Wiles's proof of Fermat's Last Theorem. Each of these problems and results is a fundamental part of our mathematical heritage. Each has had a profound influence. Nevertheless, the results themselves simply say that something is impossible. Other negative results delineate what cannot be done, as compared with what is possible. The best of these results also highlight the salient features that illustrate why they are not possible. For example, Gauss proved that a necessary and sufficient condition for the construction of a regular n-gon, using only ruler and compass, is that
منابع مشابه
Approximation theorems for fuzzy set multifunctions in Vietoris topology. Physical implications of regularity
n this paper, we consider continuity properties(especially, regularity, also viewed as an approximation property) for $%mathcal{P}_{0}(X)$-valued set multifunctions ($X$ being a linear,topological space), in order to obtain Egoroff and Lusin type theorems forset multifunctions in the Vietoris hypertopology. Some mathematicalapplications are established and several physical implications of thema...
متن کاملSOME RESULTS ON INTUITIONISTIC FUZZY SPACES
In this paper we define intuitionistic fuzzy metric and normedspaces. We first consider finite dimensional intuitionistic fuzzy normed spacesand prove several theorems about completeness, compactness and weak convergencein these spaces. In section 3 we define the intuitionistic fuzzy quotientnorm and study completeness and review some fundamental theorems. Finally,we consider some properties of...
متن کاملApproximation the fuzzy solution of the non linear fuzzy Volterra integro differential equation using fixed point theorems
متن کامل
Convergence theorems of iterative approximation for finding zeros of accretive operator and fixed points problems
In this paper we propose and studied a new composite iterative scheme with certain control con-ditions for viscosity approximation for a zero of accretive operator and xed points problems in areflexive Banach space with weakly continuous duality mapping. Strong convergence of the sequencefxng dened by the new introduced iterative sequence is proved. The main results improve andcomplement the co...
متن کاملS-APPROXIMATION SPACES: A FUZZY APPROACH
In this paper, we study the concept of S-approximation spaces in fuzzy set theory and investigate its properties. Along introducing three pairs of lower and upper approximation operators for fuzzy S-approximation spaces, their properties under different assumptions, e.g. monotonicity and weak complement compatibility are studied. By employing two thresholds for minimum acceptance accuracy and m...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- The American Mathematical Monthly
دوره 110 شماره
صفحات -
تاریخ انتشار 2003