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- Carlos, Alegría (2)
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In this paper, we provide a new and more general filtration to the family of noncommutative rings known as skew PBW extensions. We introduce the notion of σ-filtered skew PBW extension and study some homological properties of these algebras. We show that the homogenization of a σ-filtered skew PBW extension A over a ring R is a graded skew PBW extension over the homogenization of R. Using this fact, we prove that if the homogenization of R is Auslander-regular, then the homogenization of A is a domain Noetherian, Artin–Schelter regular, and A is Noetherian, Zariski and (ungraded) skew Calabi–Yau. |
In this paper we obtain new estimates for bilinear pseudodifferential operators with symbol in the class BSm1,1
, when both arguments belong to Triebel-Lizorkin spaces of the type Fn/pp,q(Rn). The inequalities are obtained as a consequence of a refinement of the classical Sobolev embedding Fn/pp,q(Rn)↪bmo(Rn), where we replace bmo(Rn) by an appropriate subspace which contains L∞(Rn). As an application, we study the product of functions on Fn/pp,q(Rn) when 1 |
We investigate a nonlinear equation with quadratic nonlinearities, including a nonlinear model in Silva and Freire (J. Differ. Equ. 320:371–398, 2022). Using the classical energy estimate methods, we give a necessary and sufficient condition of blow-up of solutions to nonlinear equations. We answer a problem pointed out by Silva and Freire (J. Differ. Equ. 320:371–398, 2022). |
This paper is primarily concerned with the following random coverage problem. Given a specified compact region A in a d-dimensional Euclidean space, what is the probability that A is fully covered by a union of Euclidean balls of radius r centred on n points placed independently uniformly at random in A, in the large-n limit with r=r(n)
becoming small in an appropriate manner? |
The function a is assumed to be critical in the sense of Hebey and Vaugon. For low dimensions N∈(2s,4s)
, we prove that the Robin function ϕa satisfies infx∈Ωϕa(x)=0, which extends a result obtained by Druet for s=1
. In dimensions N∈(8s/3,4s), we then study the asymptotics of the fractional Brezis–Nirenberg energy S(a+εV)
for some V∈L∞(Ω) as ε→0+. We give a precise description of the blow-up profile of (almost) minimizing sequences and characterize the concentration speed and the location of concentration points. |
Cone-nets are conjugate nets on a surface such that along each individual curve of one family of parameter curves there is a cone in tangential contact with the surface. The corresponding conjugate curve network is projectively invariant and is characterized by the existence of particular transformations. We study properties of that transformation theory and illustrate how several known surface classes appear within our framework. We present cone-nets in the classical smooth setting of differential geometry as well as in the context of a consistent discretization with counterparts to all relevant statements and notions of the smooth setting. |
Let Ω⊂Rd be a C1 domain or, more generally, a Lipschitz domain with small Lipschitz constant and A(x) be a d×d
uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume u is harmonic in Ω, or with greater generality u solves div(A(x)∇u)=0 in Ω, and u vanishes on Σ=∂Ω∩B for some ball B. We study the dimension of the singular set of u in Σ, in particular we show that there is a countable family of open balls (Bi)i
such that u|Bi∩Ω does not change sign and K∖⋃iBi has Minkowski dimension smaller than d−1−ϵ for any compact K⊂Σ. We also find upper bounds for the (d−1)-dimensional Hausdorff measure of the zero set of u in balls intersecting Σ in terms of the frequency. As a consequence, we prove a new unique continuation principle at the boundary for this class of functi... |
Atherosclerosis is an inflammatory disease characterised by the formation of plaques, which are deposits of lipids and cholesterol-laden macrophages that form in the artery wall. The inflammation is often non-resolving, due in large part to changes in normal macrophage anti-inflammatory behaviour that are induced by the toxic plaque microenvironment. These changes include higher death rates, defective efferocytic uptake of dead cells, and reduced rates of emigration. We develop a free boundary multiphase model for early atherosclerotic plaques, and we use it to investigate the effects of impaired macrophage anti-inflammatory behaviour on plaque structure and growth. |
Motivated by several applications, we investigate the well-posedness of a switched system composed by a system of linear hyperbolic balance laws and by a system of linear algebraic differential equations. This setting includes networks and looped systems of hyperbolic balance laws. The obtained results are globally in time, provided that the inputs have finite (but not necessarily small) total variation. |
Continuous-time Markov chains are frequently used as stochastic models for chemical reaction networks, especially in the growing field of systems biology. A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs) is to understand the dependence of the stochastic behavior of these systems on the chemical reaction rate parameters. Towards solving this problem, in this paper we develop theoretical tools called comparison theorems that provide stochastic ordering results for SCRNs. These theorems give sufficient conditions for monotonic dependence on parameters in these network models, which allow us to obtain, under suitable conditions, information about transient and steady-state behavior. These theorems exploit structural properties of SCRNs, beyond those of gener... |
