A universal differentiability set in Banach spaces with separable dual

We show that any non-zero Banach space with a separable dual contains a totally disconnected, closed and bounded subset S of Hausdorff dimension 1 such that every Lipschitz function on the space is Fr\'echet differentiable somewhere in S.Comment: 41 pages, 1 figur

A dichotomy of sets via typical differentiability

We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function, namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has zero length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: In any such coverable set a typical Lipschitz function is everywhere severely non-differentiable.Comment: Accepted for publication in Forum of Mathematics Sigma. Some revisions made according to the referee's repor

Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces

Motivated by an attempt to find a general chain rule formula for differentiating the composition $f\circ g$ of Lipschitz functions $f$ and $g$ that would be as close as possible to the standard formula $(f\circ g)'(x) = f'(g(x))\circ g'(x)$, we show that this formula holds without any artificial assumptions provided derivatives are replaced by complete derivative assignments. The idea behind these assignments is that the derivative of $f$ at $y$ is understood as defined only in the direction of a suitable ``tangent space'' $U(f,y)$ (and so it exists at every point), but these tangent spaces are chosen in such a way that for any $g$ they contain the range of $g'(x)$ for almost every $x$. Showing the existence of such assignments leads us to a detailed study of derived sets and the ways in which they describe pointwise behavior of Lipschitz functions

Cognitive functions and patterns of brain activity in patients after simultaneous coronary and carotid artery revascularization

BackgroundOn-pump coronary artery bypass grafting (CABG) is associated with a high risk of neurological complications in patients with severe carotid stenosis. Moreover, early postoperative cognitive dysfunction (POCD) incidence remains high in patients undergoing simultaneous coronary and carotid surgery. Recent studies have shown that even moderate carotid stenosis (≥50%) is associated with postoperative cognitive decline after CABG. Data on brain health in the postoperative period of simultaneous coronary and carotid surgery are limited.ObjectivesThis study aimed to analyze early postoperative changes in the cognitive function and patterns of brain electrical activity in patients after simultaneous coronary and carotid artery revascularization.Materials and methodsBetween January 2017 and December 2020, consecutive patients were assigned to on-pump CABG with or without carotid endarterectomy (CEA) according to clinical indications. An extended neuropsychological and electroencephalographic (EEG) assessment was performed before surgery and at 7–10 days after CABG or CABG + CEA.ResultsA total of 100 patients were included [median age 59 (55; 65), 95% men, MMSE 27 (26; 28)], and among these, 46 underwent CEA. POCD was diagnosed in 29 (63.0%) patients with CABG + CEA and in 32 (59.0%) patients with isolated CABG. All patients presented with a postoperative theta power increase. However, patients with CABG + right-sided CEA demonstrated the most pronounced theta power increase compared to patients with isolated CABG.ConclusionThe findings of our study show that patients with CABG + CEA and isolated CABG have comparable POCD incidence; however, patients with CABG + right-sided CEA presented with lower brain activity

A compact null set containing a differentiability point of every Lipschitz function

We prove that in a Euclidean space of dimension at least two, there exists a compact set of Lebesgue measure zero such that any real-valued Lipschitz function defined on the space is differentiable at some point in the set. Such a set is constructed explicitly.Comment: 28 pages; minor modifications throughout; Lemma 4.2 is proved for general Banach space rather than for Hilbert spac