Preliminaries

2021 ◽  
pp. 1-24
Author(s):  
Adel N. Boules
Keyword(s):  
Complex Number ◽  
Continuous Functions ◽  
Uniform Continuity ◽  
Weierstrass Theorem ◽  
Mathematical Concepts ◽  
Number Systems ◽  
Set Operations ◽  
Basic Set ◽  
Bounded Sets ◽  
Complex Number Field

In addition to giving a very brief reminder of set notation and basic set operations, this chapter provides a brief refresher on basic mathematical concepts. The natural, rational and real number systems are taken for granted. However, it does develop at length the Cauchy criterion and its equivalence to the completeness of the real line, and the Bolzano-Weierstrass theorem, as well as the complex number field, including its completeness. Embryonic manifestations of completeness and compactness can be seen in this chapter. Examples include the nested interval theorem and the uniform continuity of continuous functions on compact intervals, and the proof of the Heine-Borel theorem in chapter 4 is squarely based on the Bolzano-Weierstrass property of bounded sets.

scholarly journals Type II degenerations of K3 surfaces

1985 ◽  
Vol 99 ◽  
pp. 11-30 ◽  
Author(s):  
Shigeyuki Kondo
Keyword(s):  
Number Field ◽  
Complex Manifold ◽  
Complex Number ◽  
Three Dimensional ◽  
K3 Surfaces ◽  
Type Ii ◽  
Holomorphic Map ◽  
Canonical Class ◽  
Proper Holomorphic Map ◽  
Complex Number Field

A degeneration of K3 surfaces (over the complex number field) is a proper holomorphic map π: X→Δ from a three dimensional complex manifold to a disc, such that, for t ≠ 0, the fibres Xt = π-1(t) are smooth K3 surfaces (i.e. surfaces Xt with trivial canonical class KXt = 0 and dim H1(Xt, Oxt) = 0).

scholarly journals A Characterization of the Existence of a Fundamental Bounded Resolution for the Space Cc(X) in Terms of X

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Juan Carlos Ferrando
Keyword(s):  
Continuous Functions ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Bounded Sets

We characterize in terms of the topology of a Tychonoff space X the existence of a bounded resolution for CcX that swallows the bounded sets, where CcX is the space of real-valued continuous functions on X equipped with the compact-open topology.

On the Nonemptiness of the Adjoint Linear System of Polarized Manifolds

1998 ◽  
Vol 41 (3) ◽  
pp. 267-278 ◽  
Author(s):  
Yoshiaki Fukuma
Keyword(s):  
Linear System ◽  
Number Field ◽  
Complex Number ◽  
Complex Number Field

AbstractLet (X, L) be a polarized manifold over the complex number field with dim X = n. In this paper, we consider a conjecture of M. C. Beltrametti and A. J. Sommese and we obtain that this conjecture is true if n = 3 and h0(L) ≥ 2, or dim Bs |L| ≤ 0 for any n ≥ 3. Moreover we can generalize the result of Sommese.

Phased Bessel functions

2008 ◽  
Vol 86 (7) ◽  
pp. 863-870 ◽  
Author(s):  
X Hu ◽  
H Wang ◽  
D -S Guo
Keyword(s):  
Real Number ◽  
Number Field ◽  
Complex Number ◽  
Bessel Functions ◽  
Natural Extension ◽  
State Transitions ◽  
Photon State ◽  
Analytical Extension ◽  
Complex Number Field ◽  
Real Number Field

In the study of photon-state transitions, we found a natural extension of the first kind of Bessel functions that extends both the range and domain of the Bessel functions from the real number field to the complex number field. We term the extended Bessel functions as phased Bessel functions. This extension is completely different from the traditional “analytical extension”. The new complex Bessel functions satisfy addition, subtraction, and recurrence theorems in a complex range and a complex domain. These theorems provide short cuts in calculations. The single-phased Bessel functions are generalized to multiple-phased Bessel functions to describe various photon-state transitions.PACS Nos.: 02.30.Gp, 32.80.Rm, 42.50.Hz

scholarly journals Convolution operators on weighted spaces of continuous functions and supremal convolution

2019 ◽  
Vol 199 (4) ◽  
pp. 1547-1569
Author(s):  
T. Kleiner ◽  
R. Hilfer
Keyword(s):  
Continuous Functions ◽  
Weighted Spaces ◽  
Constructive Method ◽  
Convolution Operators ◽  
Linear Combinations ◽  
The Third ◽  
Bounded Set ◽  
Spaces Of Continuous Functions ◽  
Bounded Sets

AbstractThe convolution of two weighted balls of measures is proved to be contained in a third weighted ball if and only if the supremal convolution of the corresponding two weights is less than or equal to the third weight. Here supremal convolution is introduced as a type of convolution in which integration is replaced with supremum formation. Invoking duality the equivalence implies a characterization of equicontinuity of weight-bounded sets of convolution operators having weighted spaces of continuous functions as domain and range. The overall result is a constructive method to define weighted spaces on which a given set of convolution operators acts as an equicontinuous family of endomorphisms. The result is applied to linear combinations of fractional Weyl integrals and derivatives with orders and coefficients from a given bounded set.

A theorem of Stone-Weierstrass type

1966 ◽  
Vol 62 (4) ◽  
pp. 649-666 ◽  
Author(s):  
G. A. Reid
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Weierstrass Theorem ◽  
Necessary And Sufficient ◽  
Complex Valued

The Stone-Weierstrass theorem gives very simple necessary and sufficient conditions for a subset A of the algebra of all real-valued continuous functions on the compact Hausdorff space X to generate a subalgebra dense in namely, this is so if and only if the functions of A strongly separate the points of X, in other words given any two distinct points of X there exists a function in A taking different values at these points, and given any point of X there exists a function in A non-zero there. In the case of the algebra of all complex-valued continuous functions on X, the same result holds provided that we consider the subalgebra generated by A together with Ā, the set of complex conjugates of the functions in A.

scholarly journals On weighted spaces without a fundamental sequence of bounded sets

2002 ◽  
Vol 30 (8) ◽  
pp. 449-457 ◽  
Author(s):  
J. O. Olaleru
Keyword(s):  
Weighted Space ◽  
General Setting ◽  
Continuous Functions ◽  
Weighted Spaces ◽  
Fundamental Sequence ◽  
Spaces Of Continuous Functions ◽  
Bounded Sets

The problem of countably quasi-barrelledness of weighted spaces of continuous functions, of which there are no results in the general setting of weighted spaces, is tackled in this paper. This leads to the study of quasi-barrelledness of weighted spaces in which, unlike that of Ernst and Schnettler (1986), though with a similar approach, we drop the assumption that the weighted space has a fundamental sequence of bounded sets. The study of countably quasi-barrelledness of weighted spaces naturally leads to definite results on the weighted (DF)-spaces for those weighted spaces with a fundamental sequence of bounded sets.

Sign in / Sign up

Export Citation Format

Share Document