More Like this

1. Hermitian–Yang–Mills Connections on Collapsing Elliptically Fibered K3 Surfaces

   https://doi.org/10.1007/s12220-021-00808-9  

   Datar, Ved ; Jacob, Adam ( January 2022 , The Journal of Geometric Analysis)

   Let$$X\rightarrow {{\mathbb {P}}}^1$$$X\to P^1$be an elliptically fiberedK3 surface, admitting a sequence$$\omega _{i}$$${\omega }_i$of Ricci-flat metrics collapsing the fibers. LetVbe a holomorphicSU(n) bundle overX, stable with respect to$$\omega _i$$${\omega }_i$. Given the corresponding sequence$$\Xi _i$$${\Xi }_i$of Hermitian–Yang–Mills connections onV, we prove that, ifEis a generic fiber, the restricted sequence$$\Xi _i|_{E}$$${\Xi }_i{|}_E$converges to a flat connection$$A_0$$$A_0$. Furthermore, if the restriction$$V|_E$$${V|}_E$is of the form$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$${\oplus }_{j=1}^n{O}_E(q_j-0)$forndistinct points$$q_j\in E$$$q_j\in E$, then these points uniquely determine$$A_0$$$A_0$.

    

   more » « less
2. Evidence of a liquid–liquid phase transition in H$$_2$$O and D$$_2$$O from path-integral molecular dynamics simulations

   https://doi.org/10.1038/s41598-022-09525-x  

   Eltareb, Ali ; Lopez, Gustavo E. ; Giovambattista, Nicolas ( April 2022 , Scientific Reports)

   We perform path-integral molecular dynamics (PIMD), ring-polymer MD (RPMD), and classical MD simulations of H$$_2$$${}_2$O and D$$_2$$${}_2$O using the q-TIP4P/F water model over a wide range of temperatures and pressures. The density$$\rho (T)$$$\rho \left(T\right)$, isothermal compressibility$$\kappa _T(T)$$${\kappa }_T\left(T\right)$, and self-diffusion coefficientsD(T) of H$$_2$$${}_2$O and D$$_2$$${}_2$O are in excellent agreement with available experimental data; the isobaric heat capacity$$C_P(T)$$$C_P\left(T\right)$obtained from PIMD and MD simulations agree qualitatively well with the experiments. Some of these thermodynamic properties exhibit anomalous maxima upon isobaric cooling, consistent with recent experiments and with the possibility that H$$_2$$${}_2$O and D$$_2$$${}_2$O exhibit a liquid-liquid critical point (LLCP) at low temperatures and positive pressures. The data from PIMD/MD for H$$_2$$${}_2$O and D$$_2$$${}_2$O can be fitted remarkably well using the Two-State-Equation-of-State (TSEOS). Using the TSEOS, we estimate that the LLCP for q-TIP4P/F H$$_2$$${}_2$O, from PIMD simulations, is located at$$P_c = 167 \pm 9$$$P_c=167\pm 9$ MPa,$$T_c = 159 \pm 6$$$T_c=159\pm 6$ K, and$$\rho _c = 1.02 \pm 0.01$$${\rho }_c=1.02\pm 0.01$ g/cm$$^3$$${}^3$. Isotope substitution effects are important; the LLCP location in q-TIP4P/F D$$_2$$${}_2$O is estimated to be$$P_c = 176 \pm 4$$$P_c=176\pm 4$ MPa,$$T_c = 177 \pm 2$$$T_c=177\pm 2$ K, and$$\rho _c = 1.13 \pm 0.01$$${\rho }_c=1.13\pm 0.01$ g/cm$$^3$$${}^3$. Interestingly, for the water model studied, differences in the LLCP location from PIMD and MD simulations suggest that nuclear quantum effects (i.e., atoms delocalization) play an important role in the thermodynamics of water around the LLCP (from the MD simulations of q-TIP4P/F water,$$P_c = 203 \pm 4$$$P_c=203\pm 4$ MPa,$$T_c = 175 \pm 2$$$T_c=175\pm 2$ K, and$$\rho _c = 1.03 \pm 0.01$$${\rho }_c=1.03\pm 0.01$ g/cm$$^3$$${}^3$). Overall, our results strongly support the LLPT scenario to explain water anomalous behavior, independently of the fundamental differences between classical MD and PIMD techniques. The reported values of$$T_c$$$T_c$for D$$_2$$${}_2$O and, particularly, H$$_2$$${}_2$O suggest that improved water models are needed for the study of supercooled water.

    

   more » « less
3. Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$

   https://doi.org/10.1007/s00208-022-02391-y  

   Sauermann, Lisa ( April 2022 , Mathematische Annalen)

   Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$$a_{j,1}{x}_1+\cdots +a_{j,k}{x}_k=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$$a_{j,i}\in F_p$. Suppose that$$k\ge 3m$$$k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$$a_{j,1}+\cdots +a_{j,k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$and that every$$m\times m$$$m\times m$minor of the$$m\times k$$$m\times k$matrix$$(a_{j,i})_{j,i}$$${\left(a_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$$A\subseteq F_p^n$of size$$|A|> C\cdot \Gamma ^n$$$\left|A\right|>C·{\Gamma }^n$contains a solution$$(x_1,\dots ,x_k)\in A^k$$$(x_1,\cdots ,x_k)\in A^k$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$$x_1,\cdots ,x_k\in A$are all distinct. Here,Cand$$\Gamma $$$\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma <p$. The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$$x_1,\cdots ,x_k$in the solution$$(x_1,\dots ,x_k)\in A^k$$$(x_1,\cdots ,x_k)\in A^k$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$$x_1,\cdots ,x_k$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.

    

   more » « less
4. Highly-excited Rydberg excitons in synthetic thin-film cuprous oxide

   https://doi.org/10.1038/s41598-023-41465-y  

   DeLange, Jacob ; Barua, Kinjol ; Paul, Anindya Sundar ; Ohadi, Hamid ; Zwiller, Val ; Steinhauer, Stephan ; Alaeian, Hadiseh ( December 2023 , Scientific Reports)

   Cuprous oxide ($$\hbox {Cu}{}_2\hbox {O}$$$\text{Cu}{}_2\text{O}$) has recently emerged as a promising material in solid-state quantum technology, specifically for its excitonic Rydberg states characterized by large principal quantum numbers (n). The significant wavefunction size of these highly-excited states (proportional to$$n^2$$$n^2$) enables strong long-range dipole-dipole (proportional to$$n^4$$$n^4$) and van der Waals interactions (proportional to$$n^{11}$$$n^{11}$). Currently, the highest-lying Rydberg states are found in naturally occurring$$\hbox {Cu}_2\hbox {O}$$${\text{Cu}}_2\text{O}$. However, for technological applications, the ability to grow high-quality synthetic samples is essential. The fabrication of thin-film$$\hbox {Cu}{}_2\hbox {O}$$$\text{Cu}{}_2\text{O}$samples is of particular interest as they hold potential for observing extreme single-photon nonlinearities through the Rydberg blockade. Nevertheless, due to the susceptibility of high-lying states to charged impurities, growing synthetic samples of sufficient quality poses a substantial challenge. This study successfully demonstrates the CMOS-compatible synthesis of a$$\hbox {Cu}{}_2\hbox {O}$$$\text{Cu}{}_2\text{O}$thin film on a transparent substrate that showcases Rydberg excitons up to$$n = 8$$$n=8$which is readily suitable for photonic device fabrications. These findings mark a significant advancement towards the realization of scalable and on-chip integrable Rydberg quantum technologies.

    

   more » « less
5. Multipliers on bi-parameter Haar system Hardy spaces

   https://doi.org/10.1007/s00208-024-02887-9  

   Lechner, R. ; Motakis, P. ; Müller, P_F_X ; Schlumprecht, Th ( May 2024 , Mathematische Annalen)

   Let$$(h_I)$$$\left(h_I\right)$denote the standard Haar system on [0, 1], indexed by$$I\in \mathcal {D}$$$I\in D$, the set of dyadic intervals and$$h_I\otimes h_J$$$h_I\otimes h_J$denote the tensor product$$(s,t)\mapsto h_I(s) h_J(t)$$$(s,t)\mapsto h_I\left(s\right)h_J\left(t\right)$,$$I,J\in \mathcal {D}$$$I,J\in D$. We consider a class of two-parameter function spaces which are completions of the linear span$$\mathcal {V}(\delta ^2)$$$V\left({\delta }^2\right)$of$$h_I\otimes h_J$$$h_I\otimes h_J$,$$I,J\in \mathcal {D}$$$I,J\in D$. This class contains all the spaces of the formX(Y), whereXandYare either the Lebesgue spaces$$L^p[0,1]$$$L^p[0,1]$or the Hardy spaces$$H^p[0,1]$$$H^p[0,1]$,$$1\le p < \infty $$$1\le p<\infty$. We say that$$D:X(Y)\rightarrow X(Y)$$$D:X\left(Y\right)\to X\left(Y\right)$is a Haar multiplier if$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$$D(h_I\otimes h_J)=d_{I,J}{h}_I\otimes h_J$, where$$d_{I,J}\in \mathbb {R}$$$d_{I,J}\in R$, and ask which more elementary operators factor throughD. A decisive role is played by theCapon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$$C:V\left({\delta }^2\right)\to V\left({\delta }^2\right)$given by$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$$C{h}_I\otimes h_J=h_I\otimes h_J$if$$|I|\le |J|$$$\left|I\right|\le \left|J\right|$, and$$\mathcal {C} h_I\otimes h_J = 0$$$C{h}_I\otimes h_J=0$if$$|I| > |J|$$$\left|I\right|>\left|J\right|$, as our main result highlights: Given any bounded Haar multiplier$$D:X(Y)\rightarrow X(Y)$$$D:X\left(Y\right)\to X\left(Y\right)$, there exist$$\lambda ,\mu \in \mathbb {R}$$$\lambda ,\mu \in R$such that$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$$\begin{array}{c}\lambda C+\mu (\text{Id}-C)\text{approximately 1-projectionally factors through}D,\end{array}$i.e., for all$$\eta > 0$$$\eta >0$, there exist bounded operatorsA, Bso thatABis the identity operator$${{\,\textrm{Id}\,}}$$$\text{Id}$,$$\Vert A\Vert \cdot \Vert B\Vert = 1$$$\Vert A\Vert ·\Vert B\Vert =1$and$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta $$$\Vert \lambda C+\mu (\text{Id}-C)-ADB\Vert <\eta$. Additionally, if$$\mathcal {C}$$$C$is unbounded onX(Y), then$$\lambda = \mu $$$\lambda =\mu$and then$${{\,\textrm{Id}\,}}$$$\text{Id}$either factors throughDor$${{\,\textrm{Id}\,}}-D$$$\text{Id}-D$.

    

   more » « less