Recent zbMATH articles in MSC 26A15https://www.zbmath.org/atom/cc/26A152021-04-16T16:22:00+00:00WerkzeugPerturbing the mean value theorem: implicit functions, the Morse lemma, and beyond.https://www.zbmath.org/1456.260062021-04-16T16:22:00+00:00"Lowry-Duda, David"https://www.zbmath.org/authors/?q=ai:lowry-duda.david"Wheeler, Miles H."https://www.zbmath.org/authors/?q=ai:wheeler.miles-hOne can begin with the authors' abstract:
``The mean value theorem of calculus states that, given a differentiable function \(f\) on an interval \([a, b]\), there exists at least one mean value abscissa \(c\) such that the slope of the tangent line at \((c, f (c))\) is equal to the slope of the secant line through \((a, f (a))\) and \((b, f (b))\). In this article, we study how the choices of \(c\) relate to varying the right endpoint \(b\). In particular, we ask: When we can write \(c\) as a continuous function of \(b\) in some interval? As we explore this question, we touch on the implicit function theorem, a simplified version of the Morse lemma, and the theory of analytic functions.''
It is noted that the mean value theorem is one of the truly fundamental theorems of calculus. This theorem, auxiliary notions, and the main statements of the problem of the present research are briefly explained. Some examples are considered.
Several statements are proven with explanations. Some main results of this investigation are summarized in one theorem. Also, a special attention is given to further generalizations. It includes some explanations and proving certain statements.
Finally, some open questions related to the present research are given and discussed.
Reviewer: Symon Serbenyuk (Kyïv)On the weak computability of continuous real functions.https://www.zbmath.org/1456.030642021-04-16T16:22:00+00:00"Bauer, Matthew S."https://www.zbmath.org/authors/?q=ai:bauer.matthew-steven"Zheng, Xizhong"https://www.zbmath.org/authors/?q=ai:zheng.xizhongSummary: In computable analysis, sequences of rational numbers which effectively converge to a real number \(x\) are used as the (\(\rho\text{-})\) names of \(x\). A real number \(x\) is computable if it has a computable name, and a real function \(f\) is computable if there is a Turing machine \(M\) which computes \(f\) in the sense that, \(M\) accepts any \(\rho\)-name of \(x\) as input and outputs a \(\rho\)-name of \(f(x)\) for any \(x\) in the domain of \(f\). By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition.
For the entire collection see [Zbl 1391.03010].Separating sets by functions and by sets.https://www.zbmath.org/1456.260042021-04-16T16:22:00+00:00"Szyszkowska, Paulina"https://www.zbmath.org/authors/?q=ai:szyszkowska.paulinaThe topic of characterising sets that can be separated with a function from a given class \(\mathcal F\) (of generalised continuous functions), motivated by Urysohn's lemma, has originated with a paper by \textit{A. Maliszewski} [Fundam. Math. 175, No. 3, 271--283 (2002; Zbl 1017.26002)] and has been followed by a number of works related to Maliszewski's colleagues and students; for more references on this topic see the bibliography of the article under review.
This article is a follow-up to [the author, Topology Appl. 206, 46--57 (2016; Zbl 1345.26007)] and focuses, mainly, on the problem of characterising sets that can be separated with Darboux quasi-continuous and upper semicontinuous (\(\mathscr{D\!Q}\textup{usc}\)) functions.
Two sets \(A_0,A_1\subset\mathbb R\) are said to be (classically exactly) separated with a function \(f\colon\mathbb R\to\mathbb R\) if \(f^{-1}(0)=A_0\) and \(f^{-1}(1)=A_1\). A characterisation of pairs \((A_0,A_1)\) which can be separated with a \(\mathscr{D\!Q}\textup{usc}\) function has been provided by the author first in [loc. cit.] and sounds there (Theorem~3.7) as follows: there are semi-closed \(\mathscr F_\sigma\) sets \(A_0,A_1\) such that (i)~\(\operatorname{cl}A_1\cap A_0=\emptyset\) and (ii)~\(\mathbb R\setminus A_0\) and \(\mathbb R\setminus(A_1\cup G)\) are both bilaterally dense in themselves; here \(G\) is the union of all intervals contiguous to \(\operatorname{cl}A_1\) that meet \(A_0\). In the present work, the author points out an error in that characterisation (Example~3.2). Then, along the same lines as in [Szczuka, loc. cit.] she proves that the previously claimed characterisation becomes complete if one adds the condition that also \(A_1\cup G\) is semi-closed (the main result, Theorem 3.3). Some other modes of separation with \(\mathscr{D\!Q}\textup{usc}\) functions are also considered.
Reviewer: Piotr Sworowski (Bydgoszcz)A. Kharazishvili's some results of on the structure of pathological functions.https://www.zbmath.org/1456.260032021-04-16T16:22:00+00:00"Kirtadze, Aleks"https://www.zbmath.org/authors/?q=ai:kirtadze.aleks-p"Pantsulaia, Gogi"https://www.zbmath.org/authors/?q=ai:pantsulaia.gogi-rauliThe authors present a brief survey of A. Kharazishvili's works devoted to real-valued functions with strange, pathological and paradoxical structural properties, e.g. absolutely non-measurable functions, Sierpiński-Zygmund functions, sup-measurable and weakly sup-measurable functions of two real variables, and non-measurable functions of two real variables for which both iterated integrals exist.
Reviewer: George Stoica (Saint John)