We provide a finite basis for the class of Borel functions that are not in the first Baire class, as well as the class of Borel functions that are not $\sigma $ -continuous with closed witnesses.

Yuval Peres and Perla Sousi showed that the mixing times and average mixing times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on compact state spaces that satisfy the strong Feller property.

The article is devoted to Hardy type inequalities on closed manifolds. By means of various weighted Ricci curvatures, we establish several sharp Hardy type inequalities on closed weighted Riemannian manifolds. Our results complement in several aspects those obtained recently in the non-compact Riemannian setting.

In this paper, we prove contact Poincaré and Sobolev inequalities in Heisenberg groups $\mathbb{H}^{n}$, where the word ‘contact’ is meant to stress that de Rham’s exterior differential is replaced by the exterior differential of the so-called Rumin complex $(E_{0}^{\bullet },d_{c})$, which recovers the scale invariance under the group dilations associated with the stratification of the Lie algebra of $\mathbb{H}^{n}$. In addition, we construct smoothing operators for differential forms on sub-Riemannian contact manifolds with bounded geometry, which act trivially on cohomology. For instance, this allows us to replace a closed form, up to adding a controlled exact form, with a much more regular differential form.

We discuss an alternative approach to Fréchet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carathéodory. The approach allows many questions of differentiability to be reduced to questions of continuity. We demonstrate how that simplifies the theory of differentiation, including the rules of differentiation and the Schwarz lemma on the symmetry of second-order derivatives. We also provide a short proof of the differentiable dependence of fixed points in the Banach fixed point theorem.

Barnard and Steinerberger [‘Three convolution inequalities on the real line with connections to additive combinatorics’, Preprint, 2019, arXiv:1903.08731] established the autocorrelation inequality

$$\begin{eqnarray}\min _{0\leq t\leq 1}\int _{\mathbb{R}}f(x)f(x+t)\,dx\leq 0.411||f||_{L^{1}}^{2}\quad \text{for}~f\in L^{1}(\mathbf{R}),\end{eqnarray}$$

$0.411$

$0.37$

$f$

$$\begin{eqnarray}\max _{x_{1}\in \mathbb{R}}\min _{0\leq t\leq 1}\left[f(x_{1}-t)+f(x_{1}+t)\right]\leq \min _{x_{2}\in \mathbb{R}}\max _{0\leq t\leq 1}\left[f(x_{2}-t)+f(x_{2}+t)\right].\end{eqnarray}$$

$f$

$||f||_{L^{1}}$

$d,n\in \mathbb{Z}^{+}$

$f\in L^{1}$

$A$

$d\times n$

$a_{i}$

$1\leq i\leq n$

$C$

$A$

$f$

$$\begin{eqnarray}\min _{\mathbf{t}\in [0,1]^{d}}\int _{\mathbb{R}}\mathop{\prod }_{i=1}^{n}~f(x+\mathbf{t}\cdot a_{i})\,dx\leq C||f||_{L^{1}}^{n}.\end{eqnarray}$$

We completely characterize the validity of the inequality $\| u \|_{Y(\mathbb R)} \leq C \| \nabla^{m} u \|_{X(\mathbb R)}$, where X and Y are rearrangement-invariant spaces, by reducing it to a considerably simpler one-dimensional inequality. Furthermore, we fully describe the optimal rearrangement-invariant space on either side of the inequality when the space on the other side is fixed. We also solve the same problem within the environment in which the competing spaces are Orlicz spaces. A variety of examples involving customary function spaces suitable for applications is also provided.

We present some inequalities for the mappings defined by Dragomir [‘Two mappings in connection to Hadamard’s inequalities’, J. Math. Anal. Appl.167 (1992), 49–56]. We analyse known inequalities connected with these mappings using a recently developed method connected with stochastic orderings and Stieltjes integrals. We show that some of these results are optimal and others may be substantially improved.

We establish inequalities of Jensen’s and Slater’s type in the general setting of a Hermitian unital Banach $\ast$-algebra, analytic convex functions and positive normalised linear functionals.

The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with a certain symmetry.

The main result of this note implies that any function from the product of several vector spaces to a vector space can be uniquely decomposed into the sum of mutually orthogonal functions that are odd in some of the arguments and even in the other arguments. Probabilistic notions and facts are employed to simplify statements and proofs.

Zacharias [‘Proof of a conjecture of Merca on an average of square roots’, College Math. J.49 (2018), 342–345] proved Merca’s conjecture that the arithmetic means $(1/n)\sum _{k=1}^{n}\sqrt{k}$ of the square roots of the first $n$ integers have the same floor values as a simple approximating sequence. We prove a similar result for the arithmetic means $(1/n)\sum _{k=1}^{n}\sqrt[3]{k}$ of the cube roots of the first $n$ integers.

All non-negative, continuous, $\text{SL}(n)$, and translation invariant valuations on the space of super-coercive, convex functions on $\mathbb{R}^{n}$ are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, $\text{SL}(n)$, and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume, and polar volume are characterized.

The Friedgut–Kalai–Naor (FKN) theorem states that if ƒ is a Boolean function on the Boolean cube which is close to degree one, then ƒ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured version of the slice.

As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.

Let $p:\mathbb{C}\rightarrow \mathbb{C}$ be a polynomial. The Gauss–Lucas theorem states that its critical points, $p^{\prime }(z)=0$, are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that $p$ has $n+m$ roots, where $n$ are inside the unit disk,

$$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$

$p^{\prime }$

$n-1$

$m$

$(dn-m)/(n+m)>1$

$n$

$m$

${\sim}n^{-1}$

We obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process.

In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) related to the radial derivation (i.e., the derivation along the geodesic curves) on the Cartan–Hadamard manifolds. By Gauss lemma, our new Hardy inequalities are stronger than the classical ones. We also establish the improvements of these inequalities in terms of sectional curvature of the underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequalities on the hyperbolic space ℍn. Especially, we show that our new Rellich inequalities are indeed stronger than the classical ones on the hyperbolic space ℍn.

The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions N ⩾ 5. Then it was extended to lower dimensions N ∈ {3, 4} by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques.

In this note, we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy–Rellich inequality in any dimension N ⩾ 3. In addition, we provide minimizing sequences which were not explicitly mentioned in the quoted papers in lower dimensions N ∈ {3, 4}, emphasizing their symmetry breaking. We also show that the best constant is not attained in the proper functional space.

This paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.