More Like this

1. On the Convolution Inequality f ≥ f ⋆ f

   https://doi.org/10.1093/imrn/rnaa350  

   Carlen, Eric A; Jauslin, Ian; Lieb, Elliott H; Loss, Michael P (January 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We consider the inequality $$f \geqslant f\star f$$ for real functions in $$L^1({\mathbb{R}}^d)$$ where $$f\star f$$ denotes the convolution of $$f$$ with itself. We show that all such functions $$f$$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $$1 < p \leqslant 2$$, for which the convolution is defined. We also show that all solutions in $$L^1({\mathbb{R}}^d)$$ satisfy $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$$. Moreover, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$, then $$f$$ must decay fairly slowly: $$\int _{{\mathbb{R}}^{\textrm{d}}}|x| f(x)\ \textrm{d}x = \infty $$, and this is sharp since for all $r< 1$, there are solutions with $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$$ and $$\int _{{\mathbb{R}}^{\textrm{d}}}|x|^r f(x)\ \textrm{d}x <\infty $$. However, if $$\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$$, the decay at infinity can be much more rapid: we show that for all $$a<\tfrac 12$$, there are solutions such that for some $$\varepsilon>0$$, $$\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon |x|}f(x)\ \textrm{d}x < \infty $$. 

   more » « less
2. Quantitative Rigidity of Differential Inclusions in Two Dimensions

   https://doi.org/10.1093/imrn/rnad108  

   Lamy, Xavier; Lorent, Andrew; Peng, Guanying (May 2023, International Mathematics Research Notices)

   Abstract For any compact connected one-dimensional submanifold $$K\subset \mathbb R^{2\times 2}$$ without boundary that has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate $$\begin{align*} \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \qquad\forall u\in H^1(B_1;\mathbb R^2). \end{align*}$$This is an optimal generalization, for compact connected submanifolds of $$\mathbb R^{2\times 2}$$ without boundary, of the celebrated quantitative rigidity estimate of Friesecke, James, and Müller for the approximate differential inclusion into $SO(n)$. The proof relies on the special properties of elliptic subsets $$K\subset{{\mathbb{R}}}^{2\times 2}$$ with respect to conformal–anticonformal decomposition, which provide a quasilinear elliptic partial differential equation satisfied by solutions of the exact differential inclusion $$Du\in K$$. We also give an example showing that no analogous result can hold true in $$\mathbb R^{n\times n}$$ for $$n\geq 3$$. 

   more » « less
3. Notions of numerical Iitaka dimension do not coincide

   https://doi.org/10.1090/jag/763  

   Lesieutre, John (January 2021, Journal of Algebraic Geometry)

   null (Ed.)

   Let X be a smooth projective variety. The Iitaka dimension of a divisor D is an important invariant, but it does not only depend on the numerical class of D. However, there are several definitions of “numerical Iitaka dimension”, depending only on the numerical class. In this note, we show that there exists a pseuodoeffective R-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective R-divisor D_+ for which h^0(X,mD_+ + A)$ is bounded above and below by multiples of m^{3/2} for any sufficiently ample A. 

   more » « less
4. Zero Cycles on a Product of Elliptic Curves Over a p -adic Field

   https://doi.org/10.1093/imrn/rnab020  

   Gazaki, Evangelia; Leal, Isabel (March 2021, International Mathematics Research Notices)

   Abstract We consider a product $$X=E_1\times \cdots \times E_d$$ of elliptic curves over a finite extension $$K$$ of $${\mathbb{Q}}_p$$ with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has supersingular reduction. Under these assumptions, we prove that the Albanese kernel of $$X$$ is the direct sum of a finite group and a divisible group, extending work by Raskind and Spiess to cases that include supersingular phenomena. Our method involves studying the kernel of the cycle map $$CH_0(X)/p^n\rightarrow H^{2d}_{\acute{\textrm{e}}\textrm{t}}(X, \mu _{p^n}^{\otimes d})$$. We give specific criteria that guarantee this map is injective for every $$n\geq 1$$. When all curves have good ordinary reduction, we show that it suffices to extend to a specific finite extension $$L$$ of $$K$$ for these criteria to be satisfied. This extends previous work by Yamazaki and Hiranouchi. 

   more » « less
5. Investigating $D^0$ meson production in p-Pb collisions at 5.02 TeV with a multi-phase transport model

   https://doi.org/10.1140/epjc/s10052-024-13265-9  

   Zhang, Chao; Zheng, Liang; Shi, Shusu; Lin, Zi-Wei (September 2024, The European Physical Journal C)

   Abstract We study the production of$$D^0$$ $D^0$meson inp+pandp-Pb collisions using the improved AMPT model considering both coalescence and independent fragmentation of charm quarks after the Cronin broadening is included. After a detailed discussion of the improvements implemented in the AMPT model for heavy quark production, we show that the modified AMPT model can provide a good description of$$D^0$$ $D^0$meson spectra inp-Pb collisions, the$$Q_{\textrm{pPb}}$$ $Q_{\text{pPb}}$data at different centralities and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data in both mid- and forward (backward) rapidities. We also studied the effects of nuclear shadowing and parton cascade on the rapidity dependence of$$D^{0}$$ $D^0$meson production and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$. Our results indicate that using the same strength of the Cronin effect (i.e$$\delta $$ $\delta$value) as that obtained from the mid-rapidity data leads to a considerable overestimation of the$$D^0$$ $D^0$meson spectra and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data at high$$p_{\textrm{T}}$$ $p_{\text{T}}$in the backward rapidity. As a result, the$$\delta $$ $\delta$is determined via a$$\chi ^2$$ ${\chi }^2$fitting of the$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data across various rapidities. This work lays the foundation for a better understanding of cold-nuclear-matter (CNM) effects in relativistic heavy-ion collisions. 

   more » « less