arXiv subjects

Wanlin Li

Publications and source records attributed to Wanlin Li.

At least 37 records · Page 2Linked to original sources

arXiv2024

Large-scale Deployment of Vision-based Tactile Sensors on Multi-fingered Grippers

Vision-based Tactile Sensors (VBTSs) show significant promise in that they can leverage image measurements to provide high-spatial-resolution human-like performance. However, current VBTS designs, typically confined to the fingertips of robotic grippers, prove somewhat inadequate, as many grasping and manipulation tasks require multiple contact points with the object. With an end goal of enabling large-scale, multi-surface tactile sensing via VBTSs, our research (i) develops a synchronized image acquisition system with minimal latency,(ii) proposes a modularized VBTS design for easy integration into finger phalanges, and (iii) devises a zero-shot calibration approach to improve data efficiency in the simultaneous calibration of multiple VBTSs. In validating the system within a miniature 3-fingered robotic gripper equipped with 7 VBTSs we demonstrate improved tactile perception performance by covering the contact surfaces of both gripper fingers and palm. Additionally, we show that our VBTS design can be seamlessly integrated into various end-effector morphologies significantly reducing the data requirements for calibration.

cs.RO↗

arXiv2024

Gel-OPTOFORT Sensor: Multi-axis Force/Torque Measurement and Geometry Observation Using GelSight and Optoelectronic Sensor Technology

Although conventional GelSight-based tactile and force/torque sensors excel in detecting objects' geometry and texture information while simultaneously sensing multi-axis forces, their performance is limited by the camera's lower frame rates and the inherent properties of the elastomer. These limitations restrict their ability to measure higher force ranges at high sampling frequencies. Besides, due to the coupling of the Gelsight sensor unit and multi-axis force/torque unit structurally, the force/torque measurement ranges of the Gelsight-based force/torque sensors are not adjustable. To address these weaknesses, this paper proposes the GEL-OPTOFORT sensor that combines a GelSight sensor and an optoelectronic sensor-based force/torque sensor.

cs.RO↗

arXiv2023

Surface bundles and the section conjecture

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of this conjecture. In so doing we produce many examples of curves satisfying the section conjecture over fields of geometric interest, and then over p-adic fields and number fields via a Chebotarev argument. We construct two Galois cohomology classes o_1 and o_2, which obstruct the existence of pi_1-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curves M_g studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to produce curves over p-adic fields and number fields where each class obstructs pi_1-sections and hence rational points. Among our geometric results are a new proof of the section conjecture for the generic curve of genus g>2, and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).

math.AG↗

arXiv2023

Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class

Let l be a prime and G a pro-l group with torsion-free abelianization. We produce group-theoretic analogues of the Johnson/Morita cocycle for G -- in the case of surface groups, these cocycles appear to refine existing constructions when l=2. We apply this to the pro-l etale fundamental groups of smooth curves to obtain Galois-cohomological analogues, and discuss their relationship to work of Hain and Matsumoto in the case the curve is proper. We analyze many of the fundamental properties of these classes and use them to give an example of a non-hyperelliptic curve whose Ceresa class has torsion image under the l-adic Abel-Jacobi map.

math.AG↗

arXiv2023

A Miniaturised Camera-based Multi-Modal Tactile Sensor

In conjunction with huge recent progress in camera and computer vision technology, camera-based sensors have increasingly shown considerable promise in relation to tactile sensing. In comparison to competing technologies (be they resistive, capacitive or magnetic based), they offer super-high-resolution, while suffering from fewer wiring problems. The human tactile system is composed of various types of mechanoreceptors, each able to perceive and process distinct information such as force, pressure, texture, etc. Camera-based tactile sensors such as GelSight mainly focus on high-resolution geometric sensing on a flat surface, and their force measurement capabilities are limited by the hysteresis and non-linearity of the silicone material. In this paper, we present a miniaturised dome-shaped camera-based tactile sensor that allows accurate force and tactile sensing in a single coherent system. The key novelty of the sensor design is as follows. First, we demonstrate how to build a smooth silicone hemispheric sensing medium with uniform markers on its curved surface. Second, we enhance the illumination of the rounded silicone with diffused LEDs. Third, we construct a force-sensitive mechanical structure in a compact form factor with usage of springs to accurately perceive forces. Our multi-modal sensor is able to acquire tactile information from multi-axis forces, local force distribution, and contact geometry, all in real-time. We apply an end-to-end deep learning method to process all the information.

eess.SP↗

arXiv2023

Exceptional biases in counting primes over functions fields

We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call "complete bias", "lower order bias" and "reversed bias", occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in $\mathbb{F}_q[x]$ as $q$, a power of a fixed prime, goes to infinity. The bounds given improve on a previous result of Kowalski, who studied a similar question along particular $1$-parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret-Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.

math.NT↗

arXiv2023

Doubly isogenous genus-2 curves with $D_4$-action

We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C and C' are curves over a finite field K, with K-rational base points P and P', and let D and D' be the pullbacks (via the Abel-Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C,P) and (C',P') are *doubly isogenous* if Jac(C) and Jac(C') are isogenous over K and Jac(D) and Jac(D') are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naive heuristics predict, and we provide an explanation for this phenomenon.

math.NT↗

arXiv2022

On the vanishing of twisted $L$-functions of elliptic curves over rational function fields

We investigate in this paper the vanishing at $s=1$ of the twisted $L$-functions of elliptic curves $E$ defined over the rational function field $\mathbb{F}_q(t)$ (where $\mathbb{F}_q$ is a finite field of $q$ elements and characteristic $\geq 5$) for twists by Dirichlet characters of prime order $\ell \geq 3$, from both a theoretical and numerical point of view. In the case of number fields, it is predicted that such vanishing is a very rare event, and our numerical data seems to indicate that this is also the case over function fields for non-constant curves. For constant curves, we adapt the techniques of Li and Donepudi--Li who proved vanishing at $s=1/2$ for infinitely many Dirichlet $L$-functions over $\mathbb{F}_q(t)$ based on the existence of one, and we can prove that if there is one $χ_0$ such that $L(E, χ_0, 1)=0$, then there are infinitely many. Finally, we provide some examples which show that twisted $L$-functions of constant elliptic curves over $\mathbb{F}_q(t)$ behave differently than the general ones.

math.NT↗

arXiv2022

The Ceresa class and tropical curves of hyperelliptic type

We define a new algebraic invariant of a graph $G$ called the Ceresa-Zharkov class and show that it is trivial if and only if $G$ is of hyperelliptic type, equivalently, $G$ does not have as a minor the complete graph on 4 vertices or the loop of 3 loops. After choosing edge-lengths, this class specializes to an algebraic invariant of a tropical curve with underlying graph $G$ that is closely related to the Ceresa cycle for an algebraic curve defined over $\mathbb{C}(\!(t)\!)$.

math.AG↗

arXiv2022

The Ceresa class: tropical, topological, and algebraic

The Ceresa cycle is an algebraic cycle attached to a smooth algebraic curve with a marked point, which is trivial when the curve is hyperelliptic with a marked Weierstrass point. The image of the Ceresa cycle under a certain cycle class map provides a class in étale cohomology called the Ceresa class. Describing the Ceresa class explicitly for non-hyperelliptic curves is in general not easy. We present a "combinatorialization" of this problem, explaining how to define a Ceresa class for a tropical algebraic curve, and also for a topological surface endowed with a multiset of commuting Dehn twists (where it is related to the Morita cocycle on the mapping class group). We explain how these are related to the Ceresa class of a smooth algebraic curve over $\mathbb{C}(\!(t)\!)$, and show that the Ceresa class in each of these settings is torsion.

math.AG↗

arXiv2022

On the Northcott property of zeta functions over function fields

Pazuki and Pengo defined a Northcott property for special values of zeta functions of number fields and certain motivic $L$-functions. We determine the values for which the Northcott property holds over function fields with constant field $\mathbb{F}_q$ outside the critical strip. We then use a case by case approach for some values inside the critical strip, notably $Re (s) < \frac{1}{2} - \frac{\log 2}{\log q}$ and for $s$ real such that $1/2 \leq s \leq 1$, and we obtain a partial result for complex $s$ in the case $1/2< Re(s)\leq 1$ using recent advances on the Shifted Moments Conjecture over function fields.

math.NT↗

arXiv2021

Nonvanishing of hyperelliptic zeta functions over finite fields

Fixing $t \in \mathbb{R}$ and a finite field $\mathbb{F}_q$ of odd characteristic, we give an explicit upper bound on the proportion of genus $g$ hyperelliptic curves over $\mathbb{F}_q$ whose zeta function vanishes at $\frac{1}{2} + it$. Our upper bound is independent of $g$ and tends to $0$ as $q$ grows.

math.NT↗

arXiv2021

Abelian varieties of prescribed order over finite fields

Given a prime power $q$ and $n \gg 1$, we prove that every integer in a large subinterval of the Hasse--Weil interval $[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$ is $#A(\mathbb{F}_q)$ for some geometrically simple ordinary principally polarized abelian variety $A$ of dimension $n$ over $\mathbb{F}_q$. As a consequence, we generalize a result of Howe and Kedlaya for $\mathbb{F}_2$ to show that for each prime power $q$, every sufficiently large positive integer is realizable, i.e., $#A(\mathbb{F}_q)$ for some abelian variety $A$ over $\mathbb{F}_q$. Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse--Weil interval. A separate argument determines, for fixed $n$, the largest subinterval of the Hasse--Weil interval consisting of realizable integers, asymptotically as $q \to \infty$; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if $q \le 5$, then every positive integer is realizable, and for arbitrary $q$, every positive integer $\ge q^{3 \sqrt{q} \log q}$ is realizable.

math.NT↗

arXiv2021

Data for Shimura varieties intersecting the Torelli locus

For infinitely many Hurwitz spaces parametrizing cyclic covers of the projective line, we provide a method to determine the integral PEL datum of the Shimura variety that contains the image of the Hurwitz space under the Torelli morphism.

math.NT↗

arXiv2020

Vanishing of Dirichlet L-functions at the central point over function fields

We give a geometric criterion for Dirichlet $L$-functions associated to cyclic characters over the rational function field $\mathbb{F}_q(t)$ to vanish at the central point $s=1/2$. The idea is based on the observation that vanishing at the central point can be interpreted as the existence of a map from the projective curve associated to the character to some abelian variety over $\mathbb{F}_q$. Using this geometric criterion, we obtain a lower bound on the number of cubic characters over $\mathbb{F}_q(t)$ whose $L$-functions vanish at the central point where $q=p^{4n}$ for any rational prime $p \equiv 2 \bmod 3$. We also use recent results about the existence of supersingular superelliptic curves to deduce consequences for the $L$-functions of Dirichlet characters of other orders.

math.NT↗

arXiv2019

Newton polygon stratification of the Torelli locus in PEL-type Shimura varieties

We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain PEL-type Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $p$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems which demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the twenty special PEL-type Shimura varieties found in Moonen's work, we prove that all Newton polygon strata intersect the open Torelli locus (if $p>>0$ in the supersingular cases).

math.NT↗

arXiv2019

Effective Bounds on the Dimensions of Jacobians Covering Abelian Varieties

We show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields, which allows us to show the existence of smooth curves arising as hypersurface sections of bounded degree and genus. Additionally, we show that for simple abelian varieties a better bound is possible. As an application of these results we show that if $E$ is an elliptic curve over a finite field then for any $n\in \mathbb{N}$ there exist smooth curves of bounded genus whose Jacobians have a factor isogenous to $E^n$.

math.AG↗

arXiv2018

Newton Polygons Arising for Special Families of Cyclic Covers of the Projective Line

By a result of Moonen, there are exactly 20 positive-dimensional families of cyclic covers of the projective line for which the Torelli image is open and dense in the associated Shimura variety. For each of these, we compute the Newton polygons, and the $μ$-ordinary Ekedahl--Oort type, occurring in the characteristic $p$ reduction of the Shimura variety. We prove that all but a few of the Newton polygons appear on the open Torelli locus. As an application, we produce multiple new examples of Newton polygons and Ekedahl--Oort types of Jacobians of smooth curves in characteristic $p$. Under certain congruence conditions on $p$, these include: the supersingular Newton polygon for genus $5,6,7$; fourteen new non-supersingular Newton polygons for genus $5-7$; eleven new Ekedahl--Oort types for genus $4-7$ and, for all $g \geq 6$, the Newton polygon with $p$-rank $g-6$ with slopes $1/6$ and $5/6$.

math.NT↗