In the present article, we study almost continuous operators on the set of left computably enumerable (left-c.e.) real numbers (called L -operators). In constructive analysis developed in Markov’s school, almost continuity is one of effective modifications of the classical continuity. We suggest a criterion for almost continuity of an L -operator F that increases on a segment [a,b] ; namely, for every positive real ε and every number α in the image of [a,b] under F , there exists a number β in the same image such that α <β <α +ε . We compare this criterion with the well-known criterion from classical calculus for an increasing function f on a segment [a,b] to be continuous; namely, every real in the segment [f(a),f(b)] belongs to the range of f . We show that there exists an increasing almost continuous L -operator whose range is the set of all left-c.e. reals that do not belong to a prescribed interval [c,d] with rational endpoints and c⩽ d . We construct an increasing L -operator that is not pseudocontinuous at 0 . We also construct an increasing almost continuous L -operator F such that F admits no inverse L -operator on [0,1] and is not continuous on this segment but each left-c.e. real in [F(0),F(1)] belongs to the range of F .
We consider the composition (or chain) operation on mappings of two-step Carnot groups and explicitly derive properties of mappings that guarantee that their composition is a Lipschitz continuous mapping. We present examples and consider an important case in which one of mappings is a projection.
This research proposes a statistical test that accounts for changes in binary outcomes over time—an exact repeated measures test for binary observations. The test is particularly well suited to settings in which the objective is to assess the temporal effect of a factor, such as a treatment, advertising, or a political campaign. The proposed test is not time-symmetric: interchanging the vectors of initial and final values may lead to different test results. The test follows the Barnard test approach in computing the p -values.
