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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)

   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 $, and this is sharp since for allmore »$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 $.« less
2. Bandgap and strain engineering in epitaxial rocksalt structure (Ti _0.5 Mg _0.5 ) _1−x Al _x N(001) semiconductors

   https://doi.org/10.1039/d0tc03598j  

   Wang, Baiwei ; Zhang, Minghua ; Adhikari, Vijaya ; Fang, Peijiao ; Khare, Sanjay V. ; Gall, Daniel ( September 2020 , Journal of Materials Chemistry C)

   Rocksalt structure nitrides emerge as a promising class of semiconductors for high-temperature thermoelectric and plasmonic applications. Controlling the bandgap and strain is essential for the development of a wide variety of electronic devices. Here we use (Ti 0.5 Mg 0.5 ) 1−x Al x N as a model system to explore and demonstrate the tunability of both the bandgap and the strain state in rocksalt structure nitrides, employing a combined experimental and computational approach. (Ti 0.5 Mg 0.5 ) 1−x Al x N layers with x ≤ 0.44 deposited on MgO(001) substrates by reactive co-sputtering at 700 °C are epitaxialmore »single crystals with a solid-solution B1 rocksalt structure. The lattice mismatch with the substrate decreases with increasing x , leading to a transition in the strain-state from partially relaxed (74% and 38% for x = 0 and 0.09) to fully strained for x ≥ 0.22. First-principles calculations employing 64-atom Special Quasirandom Structures (SQS) indicate that the lattice constant decreases linearly with x according to a = (4.308 − 0.234 x ) Å for 0 ≤ x ≤ 1. In contrast, the measured relaxed lattice parameter a o = (4.269 − 0.131 x ) Å is linear only for x ≤ 0.33, its composition dependence is less pronounced, and x > 0.44 leads to the nucleation of secondary phases. The fundamental (indirect) bandgap predicted using the same SQS supercells and the HSE06 functional increases from 1.0 to 2.6 eV for x = 0–0.75. In contrast, the onset of the measured optical absorption due to interband transitions increases only from 2.3 to 2.6 eV for x = 0–0.44, suggesting that the addition of Al in the solid solution relaxes the electron momentum conservation and causes a shift from direct to indirect gap transitions. The resistivity increases from 9.0 to 708 μΩ m at 77 K and from 6.8 to 89 μΩ m at 295 K with increasing x = 0–0.44, indicating an increasing carrier localization associated with a randomization of cation site occupation and the increasing bandgap which also causes a 33% reduction in the optical carrier concentration. The overall results demonstrate bandgap and strain engineering in rocksalt nitride semiconductors and show that, in contrast to conventional covalent semiconductors, the random cation site occupation strongly affects optical transitions.« less
3. Detailed-balance analysis of Yb ³⁺ :CsPb(Cl _1−x Br _x ) ₃ quantum-cutting layers for high-efficiency photovoltaics under real-world conditions

   https://doi.org/10.1039/c9ee01493d  

   Crane, Matthew J. ; Kroupa, Daniel M. ; Gamelin, Daniel R. ( January 2019 , Energy & Environmental Science)

   Yb 3+ -Doped lead-halide perovskites (Yb 3+ :CsPb(Cl 1−x Br x ) 3 ) have emerged as unique materials combining strong, tunable broadband absorption with near-infrared photoluminescence quantum yields (PLQYs) approaching 200% at ambient temperature. These remarkable properties make Yb 3+ :CsPb(Cl 1−x Br x ) 3 an extremely promising candidate for spectral shaping in high-efficiency photovoltaic devices. Previous theoretical assessments of such “downconversion” devices have predicted single-junction efficiencies up to 40%, but have been highly idealized. Real materials like Yb 3+ :CsPb(Cl 1−x Br x ) 3 have practical limitations such as constrained band-gap and PL energies, non-directional emission,more »and an excitation-power-dependent PLQY. Hence, it is unclear whether Yb 3+ :CsPb(Cl 1−x Br x ) 3 , or any other non-ideal quantum-cutting material, can indeed boost the efficiencies of real high-performance PV. Here, we examine the thermodynamic, detailed-balance efficiency limit of Yb 3+ :CsPb(Cl 1−x Br x ) 3 on different existing PV under real-world conditions. Among these, we identify silicon heterojunction technology as very promising for achieving significant performance gains when paired with Yb 3+ :CsPb(Cl 1−x Br x ) 3 , and we predict power-conversion efficiencies of up to 32% for this combination. Surprisingly, PL saturation does not negate the improved device performance. Calculations accounting for actual hourly incident solar photon fluxes show that Yb 3+ :CsPb(Cl 1−x Br x ) 3 boosts power-conversion efficiencies at all times of day and year in two representative geographic locations. Predicted annual energy yields are comparable to those of tandem perovskite-on-silicon technologies, but without the need for current matching, tracking, or additional electrodes and inverters. In addition, we show that band-gap optimization in real quantum cutters is inherently a function of their PLQY and the ability to capture that PL. These results provide key design rules needed for development of high-efficiency quantum-cutting photovoltaic devices based on Yb 3+ :CsPb(Cl 1−x Br x ) 3 .« less
4. Matching in $$ pp\to t\overline{t}W/Z/h+ $$ jet SMEFT studies

   https://doi.org/10.1007/JHEP06(2021)151  

   Goldouzian, Reza ; Kim, Jeong Han ; Lannon, Kevin ; Martin, Adam ; Mohrman, Kelci ; Wightman, Andrew ( June 2021 , Journal of High Energy Physics)

   A bstract In this paper, we explore the impact of extra radiation on predictions of $$ pp\to \mathrm{t}\overline{\mathrm{t}}\mathrm{X},\mathrm{X}=\mathrm{h}/{\mathrm{W}}^{\pm }/\mathrm{Z} $$ pp → t t ¯ X , X = h / W ± / Z processes within the dimension-6 SMEFT framework. While full next-to-leading order calculations are of course preferred, they are not always practical, and so it is useful to be able to capture the impacts of extra radiation using leading-order matrix elements matched to the parton shower and merged. While a matched/merged leading-order calculation for $$ \mathrm{t}\overline{\mathrm{t}}\mathrm{X} $$ t t ¯ X is not expected to reproduce themore »next-to-leading order inclusive cross section precisely, we show that it does capture the relative impact of the EFT effects by considering the ratio of matched SMEFT inclusive cross sections to Standard Model values, $$ {\sigma}_{\mathrm{SM}\mathrm{EFT}}\left(\mathrm{t}\overline{\mathrm{t}}\mathrm{X}+\mathrm{j}\right)/{\sigma}_{\mathrm{SM}}\left(\mathrm{t}\overline{\mathrm{t}}\mathrm{X}+\mathrm{j}\right)\equiv \mu $$ σ SMEFT t t ¯ X + j / σ SM t t ¯ X + j ≡ μ . Furthermore, we compare leading order calculations with and without extra radiation and find several cases, such as the effect of the operator $$ \left({\varphi}^{\dagger }i{\overleftrightarrow{D}}_{\mu}\varphi \right)\left(\overline{t}{\gamma}^{\mu }t\right) $$ φ † i D ↔ μ φ t ¯ γ μ t on $$ \mathrm{t}\overline{\mathrm{t}}\mathrm{h} $$ t t ¯ h and $$ \mathrm{t}\overline{\mathrm{t}}\mathrm{W} $$ t t ¯ W , for which the relative cross section prediction increases by more than 10% — significantly larger than the uncertainty derived by varying the input scales in the calculation, including the additional scales required for matching and merging. Being leading order at heart, matching and merging can be applied to all operators and processes relevant to $$ pp\to \mathrm{t}\overline{\mathrm{t}}\mathrm{X},\mathrm{X}=\mathrm{h}/{\mathrm{W}}^{\pm }/\mathrm{Z}+\mathrm{jet} $$ pp → t t ¯ X , X = h / W ± / Z + jet , is computationally fast and not susceptible to negative weights. Therefore, it is a useful approach in $$ \mathrm{t}\overline{\mathrm{t}}\mathrm{X} $$ t t ¯ X + jet studies where complete next-to-leading order results are currently unavailable or unwieldy.« less
5. Concatenating Bipartite Graphs

   https://doi.org/10.37236/8451  

   Chudnovsky, Maria ; Hompe, Patrick ; Scott, Alex ; Seymour, Paul ; Spirkl, Sophie ( April 2022 , The Electronic Journal of Combinatorics)

   Let $x,y\in (0,1]$, and let $A,B,C$ be disjoint nonempty stable subsets of a graph $G$, where every vertex in $A$ has at least $x|B|$ neighbours in $B$, and every vertex in $B$ has at least $y|C|$ neighbours in $C$, and there are no edges between $A,C$. We denote by $\phi(x,y)$ the maximum $z$ such that, in all such graphs $G$, there is a vertex $v\in C$ that is joined to at least $z|A|$ vertices in $A$ by two-edge paths. This function has some interesting properties: we show, for instance, that $\phi(x,y)=\phi(y,x)$ for all $x,y$, and there is a discontinuity inmore »$\phi(x,x)$ when $1/x$ is an integer. For $z=1/2, 2/3,1/3,3/4,2/5,3/5$, we try to find the (complicated) boundary between the set of pairs $(x,y)$ with $\phi(x,y)\ge z$ and the pairs with $\phi(x,y)1/3$.« less