Some remarks about homeomorphisms, "energy", and so on

Some remarks about Cauchy integrals

This article describes some aspects of Cauchy integrals and related geometry of sets and measures in Euclidean spaces, etc.

A few remarks about linear operators and disconnected open sets in the plane

Some aspects of analysis on disconnected open subsets of the plane with connected fractal boundary are discussed.

A few remarks about operator theory, topology, and analysis on metric spaces

Some basic facts about Fredholm indices are briefly reviewed, often used in connection with Toeplitz and pseudodifferential operators, and which may be relevant for operators associated to fractals.

Some remarks about Clifford analysis and fractal sets

Although Clifford analysis is like complex analysis in many ways, there are obvious differences related to noncommutativity, and a few aspects of this are considered here.

Remarks about the existence of conformable derivatives and some consequences

The aim of the present paper is to make some notes to the newly introduced conformable derivative as a type local fractional derivative and to present a surprising result about the relation between the conformable derivatives and the usual integer order derivatives.

关于可变指数的Calderón-Hardy空间的注记

A remark about Calderón-Hardy spaces with variable exponents

本文改进了可变指数Calderón-Hardy空间中参数q的值，提升了相关理论的精确性。

Some remarks about metric spaces, 2

This short survey has been prepared in connection with the workshop on discrete metric spaces and their applications at Princeton, August, 2003, and tries to convey some of the ways that one might look at functions on metric spaces in harmonic analysis, in terms of behavior at different scales and locations in the space.

Some remarks about metric spaces

The first section of this modest survey reviews some basic notions and describes some families of examples, and the second section briefly indicates some general aspects of analysis on metric spaces. The remaining three sections are concerned with some particular situations involving sub-Riemannian geometry, hyperbolic groups, and p-adic numbers.

Some remarks about Descartes' rule of signs

What can we deduce about the roots of a real polynomial in one variable by simply considering the signs of its coefficients? On one hand, we give a complete answer concerning the positive roots, by proposing a statement of Descartes' rule of signs which strengthens the available ones while remaining as elementary and concise as the original. On the other hand, we provide new kinds of restrictions on the combined numbers of positive and negative roots.

Some remarks about Cauchy integrals and totally real surfaces in C^m

There seems to be quite a bit of room for interesting things related to surfaces M in C^m with real dimension m which are totally real and aspects of several complex variables on C^m around M. A basic case occurs when m = 1, with Cauchy integral operators along curves, holomorphic functions on the complements of curves, and so on. In the present article we briefly indicate a few points concerning these matters.

A remark about critical sets in $R^3$

We give a necessary condition for a closed subset of $R^3$ to be the set of critical points of some smooth function. In particular we obtain that for example neither the Whitehead continuum nor the p-adic solenoid are such a critical sets.

Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results

The moments of the Lucas polynomials and of the Chebyshev polynomials of the first kind are (multiples of) central binomial coefficients and the moments of the Fibonacci polynomials and of the Chebyshev polynomials of the second kind are Catalan numbers. In this survey paper we present some generalizations of these results together with various q-analogues.

Some remarks about The Morse-Sard theorem and approximate differentiability

Let $n, m$ be positive integers, $n\geq m$. We make several remarks on the relationship between approximate differentiability of higher order and Morse-Sard properties. For instance, among other things we show that if a function $f:\mathbb{R}^n\to\mathbb{R}^m$ is locally Lipschitz and is approximately differentiable of order $i$ almost everywhere with respect to the Hausdorff measure $\mathcal{H}^{i+m-2}$, for every $i=2, \dots, n-m+1$, then $f$ has the Morse-Sard property (that is to say, the image of the critical set of $f$ is null with respect to the Lebesgue measure in $\mathbb{R}^m$).