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In a recent paper, the authors introduced the notion of an alternating snake and a corresponding family of finite dimensional modules for the quantum affine algebra associated to $A_n$. We prove that under some restrictions, an alternating snake defines a canonical monoidal category. We prove that this category has finitely many prime objects. As a consequence we prove that the Grothendieck ring is isomorphic to the Grothendieck ring of the category $\mathscr C_ξ$ for a suitable height function. In particular it follows that the special family of alternating snakes provides a monoidal categorification of a cluster algebra of type $A_N$ for a suitable value of $N$.
Recent demonstrations indicate that silicon-spin QPUs will be able to shuttle physical qubits rapidly and with high fidelity - a desirable feature for maximising logical connectivity, supporting new codes, and routing around damage. However it may seem that shuttling at the logical level is unwise: static defects in the device may 'scratch' a logical qubit as it passes, causing correlated errors to which the code is highly vulnerable. Here we explore an architecture where logical qubits are 1D strings ('snakes') which can be moved freely over a planar latticework. Possible scratch events are inferred via monitor qubits and the complimentary gap; if deemed a risk, remarkably the shuttle process can be undone in a way that negates any corruption. Interaction between logical snakes is facilitated by a semi-transversal method. We obtain encouraging estimates for the tolerable levels of shuttling-related imperfections.
We prove a scaling limit for globally centered discrete snakes on size-conditioned critical Bienaymé trees. More specifically, under a global finite variance condition, we prove convergence in the sense of random finite-dimensional distributions of the head of the discrete snake (suitably rescaled) to the head of the Brownian snake driven by a Brownian excursion. When the third moment of the offspring distribution is finite, we further prove uniform functional convergence under a necessary tail condition on the displacements. We also consider displacement distributions with heavier tails, for which we instead obtain convergence to a variant of the hairy snake introduced by Janson and Marckert. We further give two applications of our main result. Firstly, we obtain a scaling limit for the difference between the height process and the Łukasiewicz path of a size-conditioned critical Bienaymé tree. Secondly, we obtain a scaling limit for the difference between the height process of a size-conditioned critical Bienaymé tree and the height process of its associated looptree.
The magnetic metamaterials known as Artificial Spin Ice (ASI) are promising candidates for neuromorphic computing, composed of vast numbers of interacting nanomagnets arranged in the plane. Every computing device requires the ability to transform, transmit and store information. While ASI excel at data transformation, reliable transmission and storage has proven difficult to achieve. Here, we take inspiration from the Cellular Automaton (CA), an abstract computing model reminiscent of ASI. In CAs, information transmission and storage can be realised by the ``glider'', a simple structure capable of propagating while maintaining its form. Employing an evolutionary algorithm, we search for gliders in pinwheel ASI and present the simplest glider discovered: the ``snake''. Driven by a global field protocol, the snake moves strictly in one direction, determined by its orientation. We demonstrate the snake, both in simulation and experimentally, and analyse the mechanism behind its motion. The snake provides a means of manipulating a magnetic texture in an ASI with resolution on the order of 100 nm, which could in turn be utilised to precisely control other magnetic phenomena. The integra
We study the Steinberg algebra with coefficients in an arbitrary field K for the two- and three-headed snake groupoids, which are basic examples of non-Hausdorff groupoids. We are particularly interested in elements of this algebra that are no longer continuous, known as singular functions. We prove that the ideal of singular functions in the Steinberg algebra of the two-headed snake does not properly contain any non-zero ideal, regardless of the choice of the base field K. In the case of the three-headed snake, we prove that the ideal of singular functions of the Steinberg algebra properly contains non-zero ideals if, and only if, the base field K is a splitting field of x^2 + x + 1. In particular, there are always non-zero proper subset ideals when K has characteristic a prime not congruent to -1 mod 3.
This paper explores the Lipschitz geometric and combinatorial properties of germs of real semialgebraic surfaces (or, more generally, definable in a polynomially bounded o-minimal structure) with circular link (homeomorphic to the circle $\mathbb{S}^1$). We define and investigate the outer Lipschitz geometry of the so-called circular snakes, showing what results in the paper "Lipschitz geometry and combinatorics of abnormal surface germs" (by Andrei Gabrielov and Emanoel Souza) valid to snakes still holds for the circular case. We prove the existence of a canonical decomposition for the Valette link of a circular snake into finitely many segments and nodal zones and establish some necessary and sufficient criteria to determine when it is possible to obtain a snake from a circular snake by "removing" either one of its segments or a Hölder triangle whose Valette link is contained in one of its nodal zones. We construct a combinatorial object associated with a circular snake and prove a realization theorem for this combinatorial object. We also present a weakly outer Lipschitz classification for circular snakes. Finally, we show some results about the combinatorics of binary circular
We prove some technical results relating to the Brownian snake on a stable Lévy tree. This includes some estimates on the range of the snake, estimates on its occupation measure around its minimum and also a proof of the fact that the snake and the height function of the associated tree have no common increase points.
We construct a finitely generated group which is not virtually free, yet has decidable snake tiling problem. This shows that either a long-standing conjecture by Ballier and Stein (the characterization of groups with decidable domino problem as those virtually free ones) is false, or a question by Aubrun and Bitar has a positive answer (there exists a group for which the domino and snake problems are of different difficulty).
Accelerating polarized proton beams and storing them for many turns can lead to a loss of polarization when accelerating through energies where a spin rotation frequency is in resonance with orbit oscillation frequencies. First-order resonance effects can be avoided by installing Siberian Snakes in the ring, devices which rotate the spin by 180 degrees around the snake axis while not changing the beam's orbit significantly. For large rings, several Siberian Snakes are required. Here a criterion will be derived that allows to find an optimal choice of the snake axes. Rings with super-period four are analyzed in detail, and the HERA proton ring is used as an example for approximate four-fold symmetry. The proposed arrangement of Siberian Snakes matches their effects so that all spin-orbit coupling integrals vanish at all energies and therefore there is no first-order spin-orbit coupling at all for this choice, which I call snakes matching. It will be shown that in general at least eight Siberian Snakes are needed and that there are exactly four possibilities to arrange their axes. When the betatron phase advance between snakes is chosen suitably, four Siberian Snakes can be sufficien
The game of Gyan Chauper or snakes and ladders exists in many forms throughout history as a board game of varying size, structure, and game elements of snakes and ladders associated with various vices and virtues inscribed within the board. Here, three boards were analyzed via Monte Carlo simulation, including the 1998 Milton Bradley version, the 72-square Vaisnava board, and the 84-square Jaina board, with the goal of understanding the relationships between board design and associated behaviors and spiritual concepts. Game play on each board was simulated 100,000 times with variations that included individual removal of a snake or ladder, thereby quantifying the importance of that element towards achieving victory. Comparison of the weighted importance of each game element and associated vice and virtue then permitted analysis of their importance for the game designer and their associated culture.
We present a polynomial complexity, deterministic, heuristic for solving the Hamiltonian Cycle Problem (HCP) in an undirected graph of order $n$. Although finding a Hamiltonian cycle is not theoretically guaranteed, we have observed that the heuristic is successful even in cases where such cycles are extremely rare, and it also performs very well on all HCP instances of large graphs listed on the TSPLIB web page. The heuristic owes its name to a visualisation of its iterations. All vertices of the graph are placed on a given circle in some order. The graph's edges are classified as either snakes or ladders, with snakes forming arcs of the circle and ladders forming its chords. The heuristic strives to place exactly $n$ snakes on the circle, thereby forming a Hamiltonian cycle. The Snakes and Ladders Heuristic (SLH) uses transformations inspired by $k-$opt algorithms such as the, now classical, Lin-Kernighan heuristic to reorder the vertices on the circle in order to transform some ladders into snakes and vice versa. The use of a suitable stopping criterion ensures the heuristic terminates in polynomial time if no improvement is made in $n^3$ major iterations.
Snake moves across various terrains by bending its elongated body. Recent studies discovered that snakes can use vertical bending to traverse terrain of large height variation, such as horizontally oriented cylinders, a wedge (Jurestovsky, Usher, Astley, 2021, J. Exp. Biol.), and uneven terrain (Fu & Li, 2020, Roy. Soc. Open Sci.; Fu, Astley, Li, 2022 Bioinspiration & Biomimetics). Here, to understand how vertical bending generates propulsion, we developed a dynamic simulation of a snake traversing a wedge (height = 0.05 body length, slope = 27 degrees) and a half cylindrical obstacle (height = 0.1 body length). By propagating down the body an internal torque profile with a maximum around the obstacle, the simulated snake moved forward as observed in the animal. Remarkably, even when frictional drag is low (snake-terrain kinetic friction coefficient of 0.20), the body must push against the wedge with a pressure 5 times that from body weight to generate sufficient forward propulsion to move forward. This indicated that snakes are highly capable of bending vertically to push against the environment to generate propulsion. Testing different controllers revealed that contact fo
Springer numbers are an analog of Euler numbers for the group of signed permutations. Arnol'd showed that they count some objects called snakes, that generalize alternating permutations. Hoffman established a link between Springer numbers, snakes, and some polynomials related with the successive derivatives of trigonometric functions. The goal of this article is to give further combinatorial properties of derivative polynomials, in terms of snakes and other objects: cycle-alternating permutations, weighted Dyck or Motzkin paths, increasing trees and forests. We obtain the generating functions, in terms of trigonometric functions for exponential ones and in terms of J-fractions for ordinary ones. We also define natural q-analogs, make a link with normal ordering problems and combinatorial theory of differential equations.
Snakes can move through almost any terrain. Similarly, snake robots hold the promise as a versatile platform to traverse complex environments like earthquake rubble. Unlike snake locomotion on flat surfaces which is inherently stable, when snakes traverse complex terrain by deforming their body out of plane, it becomes challenging to maintain stability. Here, we review our recent progress in understanding how snakes and snake robots traverse large, smooth obstacles that lack anchor points for gripping or bracing. First, we discovered that the generalist variable kingsnake combines lateral oscillation and cantilevering. Regardless of step height and surface friction, the overall gait is preserved. Next, to quantify static stability of the snake, we developed a method to interpolate continuous body in three dimensions (both position and orientation) between discrete tracked markers. By analyzing the base of support using the interpolated continuous body 3-D kinematics, we discovered that the snake maintained perfect stability during traversal, even on the most challenging low friction, high step. Finally, we applied this gait to a snake robot and systematically tested its performance
A uniform distribution of the image force field around the object fasts the convergence speed of the segmentation process. However, to achieve this aim, it causes the force constructed from the heat diffusion model unable to indicate the object boundaries accurately. The image force based on electrostatic field model can perform an exact shape recovery. First, this study introduces a fusion scheme of these two image forces, which is capable of extracting the object boundary with high precision and fast speed. Until now, there is no satisfied analysis about the relationship between Snakes and Geometric Active Contours (GAC). The second contribution of this study addresses that the GAC model can be deduced directly from Snakes model. It proves that each term in GAC and Snakes is correspondent and has similar function. However, the two models are expressed using different mathematics. Further, since losing the ability of rotating the contour, adoption of level sets can limits the usage of GAC in some circumstances.
Terrestrial locomotion requires generating appropriate ground reaction forces which depend on substrate geometry and physical properties. The richness of positions and orientations of terrain features in the 3-D world gives limbless animals like snakes that can bend their body versatility to generate forces from different contact areas for propulsion. Despite many previous studies of how snakes use lateral body bending for propulsion on relatively flat surfaces with lateral contact points, little is known about whether and how much snakes use vertical body bending in combination with lateral bending in 3-D terrain. This lack had contributed to snake robots being inferior to animals in stability, efficiency, and versatility when traversing complex 3-D environments. Here, to begin to elucidate this, we studied how the generalist corn snake traversed an uneven arena of blocks of random height variation 5 times its body height. The animal traversed the uneven terrain with perfect stability by propagating 3-D bending down its body with little transverse motion (11° slip angle). Although the animal preferred moving through valleys with higher neighboring blocks, it did not prefer lateral
Maximal snake polyominoes are difficult to study numerically in large rectangles, as computing them requires the complete enumeration of all snakes for a specific rectangle size, which corresponds to a brute force algorithm. This hinders the study of maximal snakes in larger rectangles. Moreover, most enumerable snakes lie in small rectangles, obscuring large-scale patterns. In this paper, we investigate the contribution of a deep neural network to the generation of maximal snake polyominoes from a data-driven training, where the maximality and adjacency constraints are not encoded explicitly, but learned. To this extent, we experiment with a denoising diffusion model, which we referred as Structured Pixel Space Diffusion (SPS Diffusion). We find that SPS Diffusion generalizes from small rectangles to larger ones, generating valid snakes up to 28x28 squares and producing maximal snake candidates on squares close to the current computational limit. The model is, however, prone to errors such as branching, cycles, or multiple snake components. Overall, the diffusion model is promising and suggests that complex combinatorial objects can be understood by deep neural networks, which is
Snake robots are inspired by the ability of biological snakes to move over rock, grass, leaves, soil, up trees, along pavement and more. Their ability to move in multiple distinct environments is due to their legless locomotion strategy, which combines distinct gaits with a skin that exhibits frictional anisotropy. Designing soft robotic snakes with similar capabilities requires an understanding of how this underlying frictional anisotropy should be created in engineered systems, and how variances in the frictional anisotropy ratio affect locomotion speed and direction on different surfaces. While forward and backward frictional ratios have been characterized for previous scale designs, lateral friction and the associated ratios are often overlooked. In this paper, our contributions include: (i) the development of a novel articulated pseudo-skin design that is modular, easy to construct and has removable or replaceable scales; (ii) experimental measurement of the frictional characteristics of otherwise-identical scales at varying angles of attack (15°, 25°, 35°, 45°) on different surfaces of interest (grass, bark, smooth surface, carpet);(iii) separate measurements of locomotion sp
The snake-in-the-box problem, introduced by Kautz in 1958, asks for the longest induced (chordless) path, called a snake, in the hypercube graph $Q_n$. The maximum length $a(n)$ is known in each dimension $n \leq 8$. We give snakes that are longer than the previous best-known in every dimension from $9$ to $13$, improving the lower bound on $a(n)$. All record-length paths are provided in a computer-verifiable dataset.
This research introduces a novel heuristic algorithm known as the Snake Locomotion Learning Search algorithm (SLLS) designed to address optimization problems. The SLLS draws inspiration from the locomotion patterns observed in snakes, particularly serpentine and caterpillar locomotion. We leverage these two modes of snake locomotion to devise two distinct search mechanisms within the SLLS. In our quest to mimic a snake's natural adaptation to its surroundings, we incorporate a learning efficiency component generated from the Sigmoid function. This helps strike a balance between exploration and exploitation capabilities throughout the SLLS computation process. The efficacy and effectiveness of this innovative algorithm are demonstrated through its application to 60 standard benchmark optimization problems and seven well-known engineering optimization problems. The performance analysis reveals that in most cases, the SLLS outperforms other algorithms, and even in the remaining scenarios, it exhibits robust performance. This conforms to the No Free Lunch Theorem, affirming that the SLLS stands as a valuable heuristic algorithm with significant potential for effectively addressing spec