Spherical Hall algebra of Spec (Z)

We study an arithmetic analog of the Hall algebra of a curve, when the curve is replaced by the spectrum of the integers compactified at infinity. The role of vector bundles is played by lattices with quadratic forms. This algebra H consists of automorphic forms with respect to GL_n(Z), n>0, with multiplication given by the parabolic pseudo-Eisenstein series map. We concentrate on the subalgebra SH in H generated by functions on the Arakelov Picard group of Spec(Z). We identify H with a Feigin-Odesskii type shuffle algebra, with the function defining the shuffle algebra expressed through the Riemann zeta function. As an application we study relations in H. Quadratic relations express the functional equation for the Eisenstein-Maass series. We show that the space of additional cubic relations (lying an an appropriate completion of H and considered modulo rescaling), is identified with the space spanned by nontrivial zeroes of the zeta function

Eigenfunctions of GL(N,\RR) Toda chain: The Mellin-Barnes representation

The recurrent relations between the eigenfunctions for GL(N,\RR) and GL(N-1,\RR) quantum Toda chains is derived. As a corollary, the Mellin-Barnes integral representation for the eigenfunctions of a quantum open Toda chain is constructed for the $N$-particle case.Comment: Latex+amssymb.sty, 7 pages; corrected some typos published in Pis'ma v ZhETF (2000), vol. 71, 338-34

Long-term follow-up of renal function in patients treated with migalastat for Fabry disease

The effect of migalastat on long-term renal outcomes in enzyme replacement therapy (ERT)–naive and ERT-experienced patients with Fabry disease is not well defined. An integrated posthoc analysis of the phase 3 clinical trials and open-label extension studies was conducted to evaluate long-term changes in renal function in patients with Fabry disease and amenable GLA variants who were treated with migalastat for ≥2 years during these studies. The analysis included ERT-naive (n = 36 [23 females]; mean age 45 years; mean baseline estimated glomerular filtration rate (eGFR), 91.4 mL/min/mL/1.73 m2) and ERT-experienced (n = 42 [24 females]; mean age, 50 years; mean baseline eGFR, 89.2 mL/min/1.73m2) patients with amenable variants who received migalastat 123 mg every other day for ≥2 years. The annualized rate of change from baseline to last observation in estimated glomerular filtration rate using the Chronic Kidney Disease Epidemiology Collaboration equation (eGFRCKD-EPI) was calculated by both simple linear regression and a random coefficient model. In ERT-naive patients, mean annualized rates of change from baseline in eGFRCKD-EPI were − 1.6 mL/min/1.73 m2 overall and − 1.8 mL/min/1.73 m2 and − 1.4 mL/min/1.73 m2 in male and female patients, respectively, as estimated by simple linear regression. In ERT-experienced patients, mean annualized rates of change from baseline in eGFRCKD-EPI were − 1.6 mL/min/1.73 m2 overall and − 2.6 mL/min/1.73 m2 and − 0.8 mL/min/1.73 m2 in male and female patients, respectively. Mean annualized rate of change in eGFRCKD-EPI in ERT-naive patients with the classic phenotype (defined by white blood cell alpha galactosidase A [α-Gal A] activity of <3% of normal and multiorgan system involvement) was −1.7 mL/min/1.73 m2. When calculated using the random coefficient model, which adjusted for sex, age, and baseline renal function, the annualized eGFRCKD-EPI change was minimal (mean: −0.1 and 0.1 mL/min/1.73 m2 in ERT-naive and ERT-experienced patients, respectively). In conclusion, patients with Fabry disease and amenable GLA variants receiving long-term migalastat treatment (≤8.6 years) maintained renal function irrespective of treatment status, sex, or phenotype

Phase diagram of two-lane driven diffusive systems

We consider a large class of two-lane driven diffusive systems in contact with reservoirs at their boundaries and develop a stability analysis as a method to derive the phase diagrams of such systems. We illustrate the method by deriving phase diagrams for the asymmetric exclusion process coupled to various second lanes: a diffusive lane; an asymmetric exclusion process with advection in the same direction as the first lane, and an asymmetric exclusion process with advection in the opposite direction. The competing currents on the two lanes naturally lead to a very rich phenomenology and we find a variety of phase diagrams. It is shown that the stability analysis is equivalent to an `extremal current principle' for the total current in the two lanes. We also point to classes of models where both the stability analysis and the extremal current principle fail

Machine Learning Assisted Design of Experiments for Solid State Electrolyte Lithium Aluminum Titanium Phosphate

Lithium-ion batteries with solid electrolytes offer safety, higher energy density and higher long-term performance, which are promising alternatives to conventional liquid electrolyte batteries. Lithium aluminum titanium phosphate (LATP) is one potential solid electrolyte candidate due to its high Li-ion conductivity. To evaluate its performance, influences of the experimental factors on the materials design need to be investigated systematically. In this work, a materials design strategy based on machine learning (ML) is employed to design experimental conditions for the synthesis of LATP. In the variation of parameters, we focus on the tolerance against the possible deviations in the concentration of the precursors, as well as the influence of sintering temperature and holding time. Specifically, models built with different design selection strategies are compared based on the training data assembled from previous laboratory experiments. The best one is then chosen to design new experiment parameters, followed by measuring the corresponding properties of the newly synthesized samples. A previously unknown sample with ionic conductivity of 1.09 × 10$^{-3}$ S cm$^{-1}$ is discovered within several iterations. In order to further understand the mechanisms governing the high ionic conductivity of these samples, the resulting phase compositions and crystal structures are studied with X-ray diffraction, while the microstructures of sintered pellets are investigated by scanning electron microscopy. Our studies demonstrate the advantages of applying machine learning in designing experimental conditions by the synthesis of desired materials, which can effectively help researchers to reduce the number of required experiments