On the Cauchy Completeness of the Constructive Cauchy Reals

It is consistent with constructive set theory (without Countable Choice, clearly) that the Cauchy reals (equivalence classes of Cauchy sequences of rationals) are not Cauchy complete. Related results are also shown, such as that a Cauchy sequence of rationals may not have a modulus of convergence, and that a Cauchy sequence of Cauchy sequences may not converge to a Cauchy sequence, among others

Non-local image deconvolution by Cauchy sequence

We present the deconvolution between two smooth function vectors as a Cauchy sequence of weight functions. From this we develop a Taylor series expansion of the convolution problem that leads to a non-local approximation for the deconvolution in terms of continuous function spaces. Optimisation of this form against a given measure of error produces a theoretically more exact algorithm. The discretization of this formulation provides a deconvolution iteration that deconvolves images quicker than the Richardson-Lucy algorithm.Comment: 12 pages, 3 figure

Rough convergence in 2-normed spaces

In this work, we introduced the notions of rough convergence, rough Cauchy sequence and the set of rough limit points of a sequence and obtained rough convergence criteria associated with this set in 2-normed space. Later, we proved that this set is closed and convex. Finally, we examined the relations between rough convergence and rough Cauchy sequence in 2-normed space.In this work, we introduced the notions of rough convergence, rough Cauchy sequence and the set of rough limit points of a sequence and obtained rough convergence criteria associated with this set in 2-normed space. Later, we proved that this set is closed and convex. Finally, we examined the relations between rough convergence and rough Cauchy sequence in 2-normed space

Wijsman I-invariant convergence of sequences of sets

In this paper, we study the concepts of Wijsman I-invariant convergence (I^W_sihma ), Wijsman I*-invariant convergence (I*W_sigma ), Wijsman p-strongly invariant convergence ([WV_sigma]_p) of sequences of sets and investigate the relationships among Wijsman invariant convergence, [WV_sigma]_p, I^W_sigma and I*W_sigma . Also, we introduce the concepts of IW_sigma -Cauchy sequence and I*W_sigma -Cauchy sequence of sets

Wijsman I-invariant convergence of sequences of sets

In this paper, we study the concepts of Wijsman I-invariant convergence (I^W_σ ), Wijsman I*-invariant convergence (I*^Wσ ), Wijsman p-strongly invariant convergence ([W Vσ]p) of sequences of sets and investigate the relationships between Wijsman invariant convergence, [W Vσ]p, IWσ and I*WσAlso, we introduce the concepts of IWσ -Cauchy sequence and I*Wσ -Cauchy sequence of sets