We study a graded Lie algebra arising from the Galois action on the pro-$p$ fundamental group of a once-punctured elliptic curve with complex multiplication. Among other things, we provide a minimal generating set of the rationalized Lie algebra under suitable assumptions. The proof is based on a slight variant of the theory of weighted completion of profinite groups developed by Hain and Matsumoto.
Mathematical expressions play a central role in scientific discovery. Symbolic regression aims to automatically discover such expressions from given numerical data. Recently, Neural symbolic regression (NSR) methods that involve Transformers pre-trained on synthetic datasets have gained attention for their fast inference, but they often perform poorly, especially with many input variables. In this study, we analyze NSR from both theoretical and empirical perspectives and show that (1) ordinary token-by-token generation is ill-suited for NSR, as Transformers cannot compositionally generate tokens while validating numerical consistency, and (2) the search space of NSR methods is greatly restricted due to reproduction bias, where the majority of generated expressions are merely copied from the training data. We further examine whether tailored test-time strategies can reduce reproduction bias and show that providing additional information at test time effectively mitigates it. These findings contribute to a deeper understanding of the limitation of NSR approaches and provide guidance for designing more robust and generalizable methods. Code is available at https://github.com/Shun-0922
In this paper, we show that any biharmonic simple rotational surface in the four-dimensional Euclidean space is minimal. The proof is based on reducing the biharmonic equation to a system of ordinary differential equations for the profile curve and then excluding all possible non-minimal branches. This is a partial affirmative answer to Chen's conjecture.
Let $V$ be a vertex algebra and $g$ an automorphism of $V$ of order $T$. We construct a sequence of associative algebras $\tilde{A}_{g,n}(V )$ for any $n\in(1/T)\mathbb{N}$, which are not depend on the conformal structure of $V$. We show that for a vertex operator algebra, $g$-rationality, $g$-regularity, and twisted fusion rules are independent of the choice of the conformal vector.
In this paper, we study the Rasmussen-Tamagawa conjecture for abelian varieties with constrained prime power torsion. Previously, Rasmussen and Tamagawa have established the conjecture under the Generalized Riemann Hypothesis for abelian varieties of any dimension over any number field, and unconditionally for those over $\mathbb{Q}$ of dimension at most three. We prove several cases of the conjecture by giving partial refinements of their techniques. Among other things, we give an unconditional proof of the conjecture for abelian fivefolds over $\mathbb{Q}$.
In this paper, we classify complete, nontrivial shrinking and steady quasi-Yamabe gradient solitons whose scalar curvature is bounded below by the soliton constant. We also classify complete, nontrivial expanding and steady quasi-Yamabe gradient solitons whose scalar curvature is bounded above by the soliton constant.
We calculate the fundamental group of the regular part of certain compact Kahler symplectic orbifolds constructed by Fujiki, called Fujiki's examples. We determine which one is an irreducible symplectic orbifold among Fujiki's examples. This answers a question posed by A.Perego.
In this paper, we completely classify nontrivial non-flat two- and three-dimensional complete gradient Yamabe solitons.
In this paper we prove boundedness of propagators for Dirac equations with unbounded time-dependent potentials on Wiener amalgam spaces. In particular we deal with class of potentials such as including Stark and harmonic potentials.
For the fourth Painlevé transcendents we derive elliptic asymptotic representations, which were announced by late Professor Kapaev without proofs. Then we newly obtain related results including the correction function.
A metric measure space is a metric space with a Borel measure. In Gromov's theory of metric measure spaces, there are important invariants called the partial diameter and the observable diameter. We obtain the result that the partial diameter or the observable diameter equals zero if and only if there exists a point that has positive measure.
For the fifth Painlevé equation it is known that a general solution is represented asymptotically by an elliptic function in cheese-like strips near the point at infinity. We present an explicit asymptotic formula for the error term of this expression, which leads to an estimate for its magnitude as was conjectured. Analogous formula is obtained for the error term of the correction function associated with the Lagrangian.
For the first Painlevé transcendents Kitaev established an asymptotic representation in terms of the Weierstrass pe-function in cheese-like strips near the point at infinity. We present an explicit error bound of this asymptotic expression, which leads to the order estimate of exponent $-1$.
This article examines the impact of China's delayed retirement announcement on households' savings behavior using data from China Family Panel Studies (CFPS). The article finds that treated households, on average, experience an 8% increase in savings rates as a result of the policy announcement. This estimation is both significant and robust. Different types of households exhibit varying degrees of responsiveness to the policy announcement, with higher-income households showing a greater impact. The increase in household savings can be attributed to negative perceptions about future pension income.
In this paper, we show that any nontrivial complete shrinking gradient Yamabe soliton whose scalar curvature is bounded below by the soliton constant everywhere and is strictly greater than the constant at some point is rotationally symmetric. This assumption is optimal for higher dimensions. This result resolves the Yamabe-soliton analogue of Perelman's conjecture.
For a general solution of the degenerate third Painlevé equation we show the Boutroux ansatz near the point at infinity. It admits an asymptotic representation in terms of the Weierstrass pe-function in cheese-like strips along generic directions. The expression is obtained by using isomonodromy deformation of a linear system governed by the degenerate third Painlevé equation.
We propose embodied scene-aware human pose estimation where we estimate 3D poses based on a simulated agent's proprioception and scene awareness, along with external third-person observations. Unlike prior methods that often resort to multistage optimization, non-causal inference, and complex contact modeling to estimate human pose and human scene interactions, our method is one-stage, causal, and recovers global 3D human poses in a simulated environment. Since 2D third-person observations are coupled with the camera pose, we propose to disentangle the camera pose and use a multi-step projection gradient defined in the global coordinate frame as the movement cue for our embodied agent. Leveraging a physics simulation and prescanned scenes (e.g., 3D mesh), we simulate our agent in everyday environments (library, office, bedroom, etc.) and equip our agent with environmental sensors to intelligently navigate and interact with the geometries of the scene. Our method also relies only on 2D keypoints and can be trained on synthetic datasets derived from popular human motion databases. To evaluate, we use the popular H36M and PROX datasets and achieve high quality pose estimation on the c
The $A$-polynomial is conjectured to be obtained from the potential function of the colored Jones polynomial by elimination. The AJ conjecture also implies the relationship between the $A$-polynomial and the colored Jones polynomial. In this paper, we connect these conjectures from the perspective of the parametrized potential function.
We investigate the quantize and binning scheme, known as the Shimokawa-Han-Amari (SHA) scheme, for the distributed hypothesis testing. We develop tools to evaluate the critical rate attainable by the SHA scheme. For a product of binary symmetric double sources, we present a sequential scheme that improves upon the SHA scheme.