We consider a theory of scalar QED on spatially compact $1+1$-dimensional spacetime. By considering constant electric field pointing down the dimension, we compute quantum effective action by integrating out degrees freedom in Euclidean sector. Working saddle-point approximation, uncover two novel branches/physical regimes upon analytically continuing back to real time and discover new result, ...

A classical question in the theory of functional equations is “when is it true that a mapping, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?” Such a problem, called a stability problem of the functional equation, was formulated by Ulam 1 in 1940. In the next year, Hyers 2 gave a partial solution of Ulam’s problem for the case of...

(i) CAC iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete. (ii) Every infinite subset X of R has a countably infinite subset iff every infinite sequentially closed subset of R includes an infinite closed subset. (iii) The statement “R is sequential” is equivalent to eac...

The aim of this paper is to define and emphasize a strong form D-compact sets in generalized metricspaces, namely D-precompact sets. Also with other sets, we shall study the relationships. Furthermore, wegive notions sequentially

Given a Hausdorff uniform space $$X$$ with the countable gage of pseudometrics uniformity , we introduce concept approximate variation function $$f$$ mapping subset $$T$$ reals into : this is infimum family Jordan-type variations all functions $$g:T\to X$$ which differ from in each pseudometric, generated by pseudometric gage, not greater than $$\varepsilon>0$$ . We prove following compactness ...

‎In this paper‎, ‎we characterize the class of orthogonality preserving operators on an infinite-dimensional Hilbert space $H$ as scalar multiples of unitary operators between $H$ and some closed subspaces of $H$‎. ‎We show that any circle (centered at the origin) is the spectrum of an orthogonality preserving operator‎. ‎Also‎, ‎we prove that every compact normal operator is a strongly orthogo...