Local behavior of homeomorphisms with a finite mean over spheres

It is well known that the modulus method is one of the most powerful tools for studying mappings. Distortion estimates of the modulus of paths families are established in many known classes, in particular, the modulus does not change under conformal mappings, is finitely distorted under qu\-a\-si\-con\-for\-mal mappings, at the same time, its behavior under mappings with finite distortion depends on the dilatation coefficient. One common case is the study of mappings for which this coefficient is integrable in the domain. In the context of our research, this case has been studied in detail in our previous publications and its consideration has mostly been completed. In particular, we obtained results on the local, boundary, and global behavior of homeomorphisms, the inverse of which satisfy the weight Poletsky inequality, provided that the corresponding majorant is integrable. In contrast, the focus in this paper is on mappings for which a similar inequality may contain non integrable weights. Study of the situation of non integrable majorants, in turn, is associated with the specific behavior of the weight modulus of the annulus, which is achieved on a certain function and up to constant is equal to $(n-1)$-degree of the Lehto integral. To the same extent, these results are also related to finding the extremal in the weight modulus of the ring. The basic theorem contains the result about equicontinuity of homeomorphisms with the inverse Poletsky inequality, when the corresponding weight has finite integrals on some set of spheres, and the set of corresponding radii of these spheres must have a positive Lebesgue measure. According to Fubini's theorem, the mentioned result summarizes the corresponding statement for any integrable majorants and is fundamental in the sense that it is easy to give examples of non integrable functions with finite integrals by spheres. In addition, since conformal and quasiconformal mappings satisfy the Poletsky inequality with a constant majorant in the forward and inverse directions, the basic theorem may be considered as a generalization of previously known statements in these classes. Note that the main result does not contain any geometric constraints on the definition and image domains of the mappings, in particular, the definition domain is assumed to be arbitrary, and the image domain is supposed to be only a bounded domain in Euclidean $n$-dimensional space. The proof of the main theorem is given by the contradiction, namely, we assume that the statement about equicontinuity of the corresponding family of mappings is incorrect, and we obtain a contradiction to this assumption due to upper and lower estimates of the modulus of families of paths.

Spread of COVID-19 in urban neighbourhoods and slums of the developing world

We study the spread of COVID-19 across neighbourhoods of cities in the developing world and find that small numbers of neighbourhoods account for a majority of cases ( k -index approx. 0.7). We also find that the countrywide distribution of cases across states/provinces in these nations also displays similar inequality, indicating self-similarity across scales. Neighbourhoods with slums are found to contain the highest density of cases across all cities under consideration, revealing that slums constitute the most at-risk urban locations in this epidemic. We present a stochastic network model to study the spread of a respiratory epidemic through physically proximate and accidental daily human contacts in a city, and simulate outcomes for a city with two kinds of neighbourhoods—slum and non-slum. The model reproduces observed empirical outcomes for a broad set of parameter values—reflecting the potential validity of these findings for epidemic spread in general, especially across cities of the developing world. We also find that distribution of cases becomes less unequal as the epidemic runs its course, and that both peak and cumulative caseloads are worse for slum neighbourhoods than non-slums at the end of an epidemic. Large slums in the developing world, therefore, contain the most vulnerable populations in an outbreak, and the continuing growth of metropolises in Asia and Africa presents significant challenges for future respiratory outbreaks from perspectives of public health and socioeconomic equity.

On Bennequin-type inequalities for links in tight contact 3-manifolds

We prove that a version of the Thurston–Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere [Formula: see text], whenever [Formula: see text] is tight. More specifically, we show that the self-linking number of a transverse link [Formula: see text] in [Formula: see text], such that the boundary of its tubular neighborhood consists of incompressible tori, is bounded by the Thurston norm [Formula: see text] of [Formula: see text]. A similar inequality is given for Legendrian links by using the notions of positive and negative transverse push-off. We apply this bound to compute the tau-invariant for every strongly quasi-positive link in [Formula: see text]. This is done by proving that our inequality is sharp for this family of smooth links. Moreover, we use a stronger Bennequin inequality, for links in the tight 3-sphere, to generalize this result to quasi-positive links and determine their maximal self-linking number.

On the Global Gaussian Lipschitz Space

It is well known that the standard Lipschitz space in Euclidean space, with exponent α ∈ (0, 1), can be characterized by means of the inequality , where is the Poisson integral of the function f. There are two cases: one can either assume that the functions in the space are bounded, or one can not make such an assumption. In the setting of the Ornstein–Uhlenbeck semigroup in ℝn, Gatto and Urbina defined a Lipschitz space by means of a similar inequality for the Ornstein–Uhlenbeck Poisson integral, considering bounded functions. In a preceding paper, the authors characterized that space by means of a Lipschitz-type continuity condition. The present paper defines a Lipschitz space in the same setting in a similar way, but now without the boundedness condition. Our main result says that this space can also be described by a continuity condition. The functions in this space turn out to have at most logarithmic growth at infinity.

Spectre et géométrie conforme des variétés compactes à bord

We prove that on any compact manifold $M^{n}$ with boundary, there exists a conformal class $C$ such that for any Riemannian metric $g\in C$ of unit volume, the first positive eigenvalue of the Neumann Laplacian satisfies ${\it\lambda}_{1}(M^{n},g)<n\,\text{Vol}(S^{n},g_{\text{can}})^{2/n}$. We also prove a similar inequality for the first positive Steklov eigenvalue. The proof relies on a handle decomposition of the manifold. We also prove that the conformal volume of $(M,C)$ is $\text{Vol}(S^{n},g_{\text{can}})$, and that the Friedlander–Nadirashvili invariant and the Möbius volume of $M$ are equal to those of the sphere. If $M$ is a domain in a space form, $C$ is the conformal class of the canonical metric.

UNCERTAINTY PRINCIPLE FOR WIGNER–YANASE–DYSON INFORMATION IN SEMIFINITE VON NEUMANN ALGEBRAS

In Ref. 9, Kosaki proved an uncertainty principle for matrices, related to Wigner–Yanase–Dyson information, and asked if a similar inequality could be proved in the von Neumann algebra setting. In this paper we prove such an uncertainty principle in the semifinite case.

On Extensions of an Inequality of Kolmogorov

The inequality of Kolmogorov (Sankhya, 1963) has been extended to a sequence of independent sub-Gaussian and other random variables. All the earlier results in the literature on this problem concerned only on the very special case of Bernoulli variables. We use martingale inequalities to establish a key result. A similar inequality is also proved for U-statistics based on exchangeable random variables.

Stečkin inequalities for summability methods

Stečkin proved an inequality on Fejér means of Fourier series He said, “Proving similar inequality for other summability method is an interesting problem.” We generalize Stečkin's inequality and give various applications to summability methods.

Effects of errors in the initial-time geometry on the solution of an equation from dynamo theory in an exterior domain

This paper studies the improperly posed backward-in-time problem, in addition to the forward-in-time problem, for a solution to a non-symmetric partial differential equation which arises in dynamo theory. Throughout, the spatial domain is unbounded and exterior to a compact region in three-space. Continuous dependence on changes in the initial-time geometry is established. For the forward-in-time problem, an explicit continuous dependence inequality depending solely on data is derived, while for the backward-in-time problem, a similar inequality is established but the bound depends also on a constraint set.

Resolvable (r, λ)-designs and the Fisher inequality

It is well known that in any (v, b, r, k, λ) resolvable balanced incomplete block design that b≧ ν + r − l with equality if and only if the design is affine resolvable. In this paper, we show that a similar inequality holds for resolvable regular pairwise balanced designs ((ρ, λ)-designs) and we characterize those designs for which equality holds. From this characterization, we deduce certain results about block intersections in (ρ, λ)-designs.