Mathematical Sciences Research Institute

Fargues and Scholze proved the geometric Satake equivalence over the Fargues–Fontaine curve. On the other hand, Zhu proved the geometric Satake equivalence using a Witt vector affine Grassmannian. In this paper, we explain the relation between the two versions of the geometric Satake equivalence via nearby cycle.

Bayesian optimization (BO) can accelerate materials discovery by guiding expensive experiments toward the most promising processing conditions. We systematically compare five BO surrogate and framework combinations (Gaussian processes in Ax, Gaussian processes and Monte-Carlo neural networks in BayBE, random forests in Lolopy, and tree-structured Parzen (TPE) estimators in Hyperopt) on three benchmarks that mimic common materials design tasks (a discrete solid-electrolyte composition space, a hybrid discrete/continuous laminate-composite design problem solved with micromechanics modeling, and the continuous Ishigami analytic function which is a standard optimization benchmark). Each BO surrogate is paired with posterior mean, probability of improvement, and expected improvement acquisition functions and run for 100 trials from randomized initial samples with uniform random search providing a control. Across five random seeds per setting, BayBE’s Gaussian-process surrogate with expected improvement consistently reached ≥95 % of the known optimum in the fewest evaluations, while Lolopy’s random forest matched or exceeded GP performance on purely categorical or mixed spaces at a higher computational cost. Posterior mean alone often stagnated at local optima, underscoring the need for exploration, whereas probability and expected improvement balanced exploration and exploitation leading to better optimization in fewer trials. Execution times ranged from milliseconds for TPE to minutes for neural-network and random-forest surrogates. These results establish baseline expectations for BO in automated materials laboratories and highlight expected improvement with Gaussian processes as a reliable first choice, with random forests offering a strong alternative when categorical variables dominate. The benchmark suite and code are released to facilitate future surrogate, acquisition, and constraint-handling research in data-driven materials optimization.

Abstract The celebrated Haglund–Haiman–Loehr (HHL) formula provides an explicit monomial expansion of the Macdonald polynomials. In 1994, Butler introduced a refinement of the Macdonald polynomial and conjectured its Schur positivity. According to the Science Fiction conjecture by Bergeron and Garsia, this refinement represents the “intersection” of Macdonald polynomials. In this work, we introduce a novel combinatorial tool, the column exchange rule , which enables us to derive a positive monomial expansion for Butler's symmetric function , thereby refining the HHL formula. Additionally, we prove Butler's conjecture on the Schur positivity of in specific cases.

Purpose The objectives of this study are threefold: (1) to introduce the tempered Chebyshev wavelet (TCW), (2) to propose the TCW method for solving linear tempered Caputo fractional diffusion-type equations and (3) to develop a fast tempered Chebyshev wavelet (fTCW) method for solving one- and two-dimensional nonlinear tempered Caputo fractional diffusion-type equations. Design/methodology/approach The fTCW method integrates the TCW method with L2−1σ and sum-of-exponentials approximations of the tempered Caputo fractional time derivative. For this purpose, we introduce the TCW and derive its new operational matrices for tempered fractional integration by using the hypergeometric function. For nonlinear problems, we employ the interpolation technique in conjunction with operational matrices and fast approximations. The efficiency of the fTCW method is demonstrated through comparisons with the exact solution and the solution obtained using the TCW method. Findings We have derived the TCW operational matrix of tempered fractional integration and the TCW operational matrix of tempered fractional integration for boundary value problems. These matrices, in conjunction with the fast approximations of the tempered Caputo fractional time derivative and the interpolation technique, form the basis for the construction of the fTCW method. We present a detailed methodology for solving linear tempered Caputo fractional equations using the TCW method. Additionally, we provide a comprehensive methodology for solving both one- and two-dimensional nonlinear tempered Caputo fractional diffusion-type equations using the fTCW method. The convergence and error analysis of both methods are thoroughly discussed. Numerical simulations are presented to validate and illustrate the theoretical results. These simulations demonstrate the effectiveness and accuracy of the proposed methods by solving three test problems, including one linear and two nonlinear tempered Caputo fractional diffusion-type equations. The results are compared with analytical solutions or with each other to highlight the efficiency and precision of the TCW and fTCW methods. Originality/value Many engineers and scientists can utilize the presented methods for solving their linear and nonlinear tempered Caputo fractional diffusion-type models.

Cytoreductive surgery with hyperthermic intraperitoneal chemotherapy (CRS/HIPEC) is often associated with extensive surgical resection and has traditionally been performed via laparotomy. Data on minimally invasive robotic approaches remain limited. We evaluated surgical safety and survival outcomes of robotic CRS/HIPEC for peritoneal surface malignancies (PSM). A single-center propensity score-matched study was conducted using a prospectively collected database (2018-2025). PSM patients treated with robotic CRS/HIPEC were matched 1:1 to open CRS/HIPEC controls using propensity scores based on age, sex, BMI, histology, and PCI. Of 99 cases, 15 robotic and 71 open CRS/HIPECs were identified. After matching, 15 robotic and 15 open cases were balanced by age, sex, BMI, diagnosis, prior surgical score, and PCI. CC-0/1 rate was 100.0% in robotic and 93.3% in open CRS/HIPEC (p = 1.00). Median blood loss was lower in the robotic group (100mL vs 250mL, p = 0.04). Median hospital stay was shorter after robotic CRS/HIPEC (6 vs 9days, p = 0.05). No differences were observed in major complications (26.7% vs 33.3%, p = 1.00) and reoperation (0.0% vs 6.7%, p = 1.00). No 90-day mortality was observed in either group. Median follow-up was 58months. Median overall survival (not reached vs 52months, p = 0.20) and progression-free survival (not reached vs 42months, p = 0.19) did not differ between robotic and open CRS/HIPEC. In PSM patients with low tumor burden, robotic CRS/HIPEC is not associated with increased morbidity or worse survival. It may offer benefits such as reduced blood loss and shorter hospitalization. Multicenter studies are warranted to optimize patient selection.

Abstract Triple symbols are arithmetic analogues of the mod n triple linking number in topology, where $$n &gt; 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> is an integer. In this paper, we introduce a cohomological formulation of a mod n triple symbol for characters over a number field containing a primitive n -th root of unity. Our definition is motivated by the arithmetic Chern–Simons theory and in this respect it differs from earlier approaches to triple symbols. We show that our symbol agrees with that of Rédei when $$n=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and of Amano–Mizusawa–Morishita when $$n=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> .

The 2020 presidential election saw election officials experience physical and social media threats, harassment, and animosity. Although little research exists regarding animosity toward US election officials, observers noted a sharp increase in 2020 in animosity toward US election officials. The harassment of election officials hindered their work in administering a free and fair election and may have generated doubts about electoral integrity. Our study: (1) Proposes a unique measurement and modeling strategy applicable across many social media networks to study toxicity directed at officials, institutions, or groups; (2) Collects a novel dataset of social media conversations about election administration in the 2020 election; (3) Uses joint sentiment-topic modeling to identify toxicity from the reactions of the public and election officials, and uses dynamic vector autoregression models to determine the temporal structure of the toxic conversations directed at election officials; (4) Finds that the level of animosity toward election officials spikes immediately after the election, that hostile topics overall make up about a quarter of the discussion share during this period, increasing to about 60% following the election, and that hostile topics come from left- and right-wing partisans. Our article concludes by discussing how similar data collection and topic modeling approaches could be deployed in future elections to monitor trolling and harassment of election officials, and to mitigate similar threats to successful election administration globally.