scholarly journals When the Identity Theorem “Seems” to Fail

2014 ◽  
Vol 121 (1) ◽  
pp. 60 ◽  
Author(s):  
J. A. Conejero ◽  
P. Jiménez-Rodríguez ◽  
G. A. Muñoz-Fernández ◽  
J. B. Seoane-Sepúlveda
Keyword(s):  
Identity Theorem

scholarly journals Valiron’s Interpolation Formula and a Derivative Sampling Formula in the Mellin Setting Acquired via Polar-Analytic Functions

2020 ◽  
Vol 20 (3-4) ◽  
pp. 629-652
Author(s):  
Carlo Bardaro ◽  
Paul L. Butzer ◽  
Ilaria Mantellini ◽  
Gerhard Schmeisser
Keyword(s):  
Analytic Functions ◽  
Main Tool ◽  
Interpolation Formula ◽  
Residue Theorem ◽  
Differentiation Formula ◽  
Sampling Formula ◽  
Bernstein Spaces ◽  
Derivative Sampling ◽  
Classical Interpolation ◽  
Identity Theorem

AbstractIn this paper, we first recall some recent results on polar-analytic functions. Then we establish Mellin analogues of a classical interpolation of Valiron and of a derivative sampling formula. As consequences a new differentiation formula and an identity theorem in Mellin–Bernstein spaces are obtained. The main tool in the proofs is a residue theorem for polar-analytic functions.

COMPARING AND ZILBER’S EXPONENTIAL FIELDS: ZERO SETS OF EXPONENTIAL POLYNOMIALS

2014 ◽  
Vol 15 (1) ◽  
pp. 71-84 ◽  
Author(s):  
P. D’Aquino ◽  
A. Macintyre ◽  
G. Terzo
Keyword(s):  
Analytic Function ◽  
Function Theory ◽  
Research Programme ◽  
Exponential Functions ◽  
Exponential Polynomials ◽  
Zero Sets ◽  
Diophantine Geometry ◽  
Complex Exponential ◽  
Identity Theorem

We continue the research programme of comparing the complex exponential with Zilberś exponential. For the latter, we prove, using diophantine geometry, various properties about zero sets of exponential functions, proved for $\mathbb{C}$ using analytic function theory, for example, the Identity Theorem.

On convergence and quasiconformality of complex planar spline interpolants

1986 ◽  
Vol 99 (2) ◽  
pp. 347-356 ◽  
Author(s):  
H. P. Dikshit ◽  
A. Ojha
Keyword(s):  
Finite Elements ◽  
Integral Formula ◽  
Holomorphic Functions ◽  
The Other ◽  
Piecewise Function ◽  
Cauchy’S Integral Formula ◽  
Recent Origin ◽  
Image Position ◽  
Spline Interpolants ◽  
Identity Theorem

There appear to be two main approaches for developing complex splines. One of these, which has been in use for quite some time, consists in defining splines on the boundary of a given region which are then extended into the interior by Cauchy's integral formula (see e.g. [1]). The other approach, which is of a more recent origin, is motivated in spirit by the theory of finite elements (see e.g. [10], p. 320) and is contained in [8] and [9]. Observing that the foregoing extension into the interior is not easy to execute numerically, certain continuous piecewise non-holomorphic functions, called complex planar splines have been studied in [8] and [9]. The choice of non-holomorphic functions is justified, since if we take the pieces to be holomorphic functions like polynomials, then by the well known identity theorem ([5], p. 132, theorem 60) the continuity of such a piecewise function implies that all the pieces represent just one holomorphic function. Thus, we shall consider polynomials in z and its conjugate z¯ of the formwhich are generally non-holomorphic functions. The numberwill be called the degree of q. For simplicity we also write q(z) for q(z, z¯).

A topological proof in group theory

1963 ◽  
Vol 59 (2) ◽  
pp. 277-282
Author(s):  
D. E. Cohen
Keyword(s):  
Group Theory ◽  
Free Group ◽  
Related Result ◽  
Normal Closure ◽  
Single Element ◽  
Covering Spaces ◽  
Topological Theory ◽  
Topological Proof ◽  
Identity Theorem

The topological theory of covering spaces may be used to prove results in group theory, for instance, the Kuros-Reidemeister-Schreier theorem (1). It seems likely that such methods can be applied to prove the Freiheitsatz (4) and the identity theorem (3), and also perhaps Lyndon's conjecture, that the normal closure in a free group F of a, single element r is freely generated by conjugates of r. However, although these problems may easily be stated in topological terms, no such proof is at present known. In this paper we prove a related result.

scholarly journals On Roots of Polynomials and Algebraically Closed Fields

2017 ◽  
Vol 25 (3) ◽  
pp. 185-195 ◽  
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Algebraic Theory ◽  
Polynomial Rings ◽  
Multiple Roots ◽  
Real Numbers ◽  
The Real ◽  
Closed Fields ◽  
Roots Of Polynomials ◽  
Algebraically Closed Fields ◽  
Finite Domains ◽  
Identity Theorem

Summary In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].

scholarly journals Hybrid Proofs of the $q$-Binomial Theorem and Other Identities

10.37236/547 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Dennis Eichhorn ◽  
James McLaughlin ◽  
Andrew V. Sills
Keyword(s):  
Binomial Theorem ◽  
Restricted Version ◽  
Full Version ◽  
Analytic Methods ◽  
Summation Formulae ◽  
Identity Theorem

We give "hybrid" proofs of the $q$-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version. We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan. Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities.

An identity theorem for multi-relator groups

1991 ◽  
Vol 109 (2) ◽  
pp. 313-321 ◽  
Author(s):  
William A. Bogley
Keyword(s):  
Large Class ◽  
Finite Subgroups ◽  
Identity Theorem

In this paper, the Identity Theorem of R. C. Lyndon and the Freiheitssatz of W. Magnus are extended to a large class of multi-relator groups. Included are the two-relator groups introduced by I. L. Anshel in her thesis, where the Freiheitssatz was proved for those groups. The Identity Theorem provides cohomology computations and a classification of finite subgroups. The methods are geometric; technical tools include the original theorems of Magnus and Lyndon, as well as an amalgamation technique due to J. H. C. Whitehead.

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