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We introduce \emph{Quantum Frog}, a two-player cooperative game built on a novel \emph{quantized-time} mechanic in which the environment advances only when a player acts. Inspired by the classic arcade game Frogger, Quantum Frog requires two frogs to cross an 8$\times$8 grid of traffic and reach the far side together. We use reinforcement learning (RL) as an analytical lens to answer four design questions: (1) how does game difficulty scale with traffic density, (2) what is the optimal single-agent policy and why, (3) how large is the cooperation gap between independent and cooperative two-agent play, and (4) what joint strategy emerges when agents are incentivised to cooperate? We train agents through five escalating stages, Tabular Q-Learning, Deep Q-Network (\DQN), Independent \DQN~(\IDQN), and Multi-Agent Proximal Policy Optimisation (\MAPPO\ with a centralised critic), evaluating each against traffic densities of one to six cars. Our key findings are: (i) the quantized-time mechanic makes a \emph{rush strategy} (moving directly upward at every step) universally optimal, as time exposure to traffic is minimised; (ii) adding an uncoordinated second player is harder than sextupli

We present a method for quantifying pulse-shape instability in a train of pulses using multi-shot Second-Harmonic-Generation Frequency-Resolved Optical Gating (SHG FROG). All versions of multi-shot FROG have previously shown the ability to distinguish stable from unstable pulse trains, as systematic differences appear between measured and retrieved traces when instability is present. This has proved possible because the recently introduced Retrieved-Amplitude N-grid Algorithmic (RANA) approach provides highly reliable pulse retrieval, even for unstable pulse trains and in the presence of noise, thus eliminating the possibility that algorithm stagnation, which mimics the effects of pulse-shape instability, could be confused for it. In other words, RANAs excellent performance ensures that any non-random discrepancies between measured and retrieved FROG traces reflect physical pulse-shape instability, rather than algorithmic stagnation. To begin to quantify such instability, we now introduce an instability parameter, R. It involves the use of the well-known statistical Runs test, which tests for systematic error in fits to one-dimensional (1D) data. A runs test counts the runs consecu

The frog model with a Bernoulli initial configuration is an interacting particle system on the $d$-dimensional lattice ($d \geq 2$) with two types of particles: active and sleeping. Active particles perform independent simple random walks. In contrast, although the sleeping particles do not move at first, they become active and start moving once touched by the active particles. Initially, only the origin has a single active particle, and the other sites have sleeping particles according to a Bernoulli distribution. After the original active particle starts moving, further active particles are gradually generated under the above rule and propagate across the lattice. The time required for the propagation of active frogs is expected to increase as the parameter of the Bernoulli distribution decreases, since fewer frogs are available. The aim of this paper is to investigate this increase in the vanishing density limit. In particular, we observe that it diverges and the rate of divergence differs significantly between $d=2$ and $d \geq 3$.

The fungal disease Chytridiomycosis poses a threat to frog populations worldwide. It has driven over 90 amphibian species to extinction and severely affected hundreds more. Difficulties in disease management have shown a need for novel conservation approaches. We present a novel mathematical model for chytridiomycosis transmission in frogs that includes the natural history of infection, to test the hypothesis that sunlight-heated refugia reduce transmission. The model is fit using approximate Bayesian computation to experimental data where a cohort of frogs, a fixed subset of which had cleared a prior infection, were provided access to either sunlight-heated or shaded refugia. Using our model, we can estimate the extent to which prior chytridiomycosis infection protects against subsequent infection, and quantify the effect of sunlight-heating of refugia. Results estimate a 40% reduction in chytridiomycosis transmission when frogs have access to sunlight-heated refugia, compared to shaded refugia. This strongly supports the hypothesis that the sunlight-heated refugia reduce disease transmission. Frogs that were infected and recovered were estimated to have a reduction in susceptibil

Ultrashort laser pulses enable attosecond-scale measurements and drive breakthroughs across science and technology, but their routine use hinges on reliable pulse characterization. Frequency-Resolved Optical Gating (FROG) is a leading solution, forming a spectrogram by scanning the delay between two pulse replicas and recording the nonlinear signal spectrum. In online settings, however, dense delay-frequency scans are costly or impractical-especially for long pulses, wavelength regimes with limited spectrometer coverage (e.g., UV), or hardware with coarse resolution, yielding severely undersampled FROG traces. Existing reconstruction methods struggle in this regime-iterative algorithms are computationally heavy, convolutional networks blur fine structure, and sequence models are unstable when inputs are discontinuous or sparse. We present a generative diffusion framework tailored to recover ultrafast pulse intensity and phase from incomplete FROG measurements. Our model infers missing spectro-temporal content with high fidelity, enabling accurate retrieval from aggressively downsampled inputs. On a simulated benchmark of FROG-pulse pairs, the diffusion approach surpasses strong CNN

Monitoring species distribution is vital for conservation efforts, enabling the assessment of environmental impacts and the development of effective preservation strategies. Traditional data collection methods, including citizen science, offer valuable insights but remain limited in coverage and completeness. Species Distribution Modelling (SDM) helps address these gaps by using occurrence data and environmental variables to predict species presence across large regions. In this study, we enhance SDM accuracy for frogs (Anura) by applying deep learning and data imputation techniques using data from the "EY - 2022 Biodiversity Challenge." Our experiments show that data balancing significantly improved model performance, reducing the Mean Absolute Error (MAE) from 189 to 29 in frog counting tasks. Feature selection identified key environmental factors influencing occurrence, optimizing inputs while maintaining predictive accuracy. The multimodal ensemble model, integrating land cover, NDVI, and other environmental inputs, outperformed individual models and showed robust generalization across unseen regions. The fusion of image and tabular data improved both frog counting and habitat

Write $W^{(n)}$ to mean the $n$-letter word obtained by repeating a fixed word $W$ and let $R_n$ denote a uniformly random $n$-letter word sampled from the same alphabet as $W$. We are interested in the average length of the longest common subsequence between $W^{(n)}$ and $R_n$, which is known to be $γ(W)\cdot n+o(n)$ for some constant $γ(W)$. Bukh and Cox recently developed an interacting particle system, dubbed the frog dynamics, which can be used to compute the constant $γ(W)$ for any fixed word $W$. They successfully analyzed the simplest case of the frog dynamics to find an explicit formula for the constants $γ(12\cdots k)$. We continue this study by using the frog dynamics to find an explicit formula for the constants $γ(12\cdots kk\cdots 21)$. The frog dynamics in this case is a variation of the PushTASEP on the ring where some clocks are identical. Interestingly, exclusion processes with correlated clocks of this type appear to have not been analyzed before. Our analysis leads to a seemingly new combinatorial object which could be of independent interest: frogs with hats!

We consider a slight modification of the frog model. For a given graph, each vertex has $\mathrm{Poisson}(λ)$ particles (or frogs). At time zero, only the particles at the origin are active, and all the other particles are sleeping. Each active particle performs an independent, continuous-time simple random walk, becoming inactive after time $t$. Once an active frog jumps to a vertex, it activates all of its particles. The survival of active particles can be studied as a dependent percolation model with two parameters $λ$ and $t$. In the present work, we establish the existence of a phase transition with respect to each parameter for non-amenable graphs of bounded degrees and quasi-transitive graphs of superlinear polynomial growth, as well as prove the sharpness of the phase transition for transitive graphs.

The continuous Frechet distance between two polygonal curves is classically computed by exploring their free space diagram. Recently, Har-Peled, Raichel, and Robson [SoCG'25] proposed a radically different approach: instead of directly traversing the continuous free space, they approximate the distance by computing paths in a discrete graph derived from the discrete free space, recursively bisecting edges until the discrete distance converges to the continuous Frechet distance. They implement this so-called frog-based technique and report substantial practical speedups over the state of the art. We revisit the frog-based approach and address three of its limitations. First, the method does not compute the Frechet distance exactly. Second, the recursive bisection procedure only introduces the monotonicity events required to realise the Frechet distance asymptotically, that is, only in the limit. Third, the applied simplification technique is heuristic. Motivated by theoretical considerations, we develop new techniques that guarantee exactness, polynomial-time convergence, and near-optimal lossless simplifications. We provide an open-source C++ implementation of our variant. Our prim

Consider a Poisson process on $\mathbb{R}$ with intensity $f$ where $0 \leq f(x)<\infty$ for ${x}\geq 0$ and ${f(x)}=0$ for $x<0$. The "points" of the process represent sleeping frogs. In addition, there is one active frog initially located at the origin. At time ${t}=0$ this frog begins performing Brownian motion with leftward drift $λ$ (i.e. its motion is a random process of the form ${B}_{t}-λ{t}$). Any time an active frog arrives at a point where a sleeping frog is residing, the sleeping frog becomes active and begins performing Brownian motion with leftward drift $λ$, independently of the motion of all of the other active frogs. This paper establishes sharp conditions on the intensity function $f$ that determine whether the model is transient (meaning the probability that infinitely many frogs return to the origin is 0), or non-transient (meaning this probability is greater than 0). A discrete model with $\text{Poiss}(f(n))$ sleeping frogs at positive integer points (and where activated frogs perform biased random walks on $\mathbb{Z}$) is also examined. In this case as well, we obtain a similar sharp condition on $f$ corresponding to transience of the model.

In this paper, we study the upper tail large deviation for the one-dimensional frog model. In this model, sleeping and active frogs are assigned to vertices on $\mathbb Z$. While sleeping frogs do not move, the active ones move as independent simple random walks and activate any sleeping frogs. The main object of interest in this model is the asymptotic behavior of the first passage time ${\rm T}(0,n)$, which is the time needed to activate the frog at the vertex $n$, assuming there is only one active frog at $0$ at the beginning. While the law of large numbers and central limit theorems have been well established, the intricacies of large deviations remain elusive. Using renewal theory, Bérard and Ramírez have pointed out a slowdown phenomenon where the probability that the first passage time ${\rm T}(0,n)$ is significantly larger than its expectation decays sub-exponentially and lies between $\exp(-n^{1/2+o(1)})$ and $\exp(-n^{1/3+o(1)})$. In this article, using a novel covering process approach, we confirm that $1/2$ is the correct exponent, i.e., the rate of upper large deviations is given by $n^{1/2}$. Moreover, we obtain an explicit rate function that is characterized by prope

Consider a population of infinitesimally small frogs on the real line. Initially the frogs on the positive half-line are dormant while those on the negative half-line are awake and move according to the heat flow. At the interface, the incoming wake frogs try to wake up the dormant frogs and succeed with a probability proportional to their amount among the total amount of involved frogs at the specific site. Otherwise, the incoming frogs also fall asleep. This frog model is a special case of the infinite rate symbiotic branching process on the real line with different motion speeds for the two types. We construct this frog model as the limit of approximating processes and compute the structure of jumps. We show that our frog model can be described by a stochastic partial differential equation on the real line with a jump type noise.

We prove that the probability the frog model with death and drift on the $d$-ary tree is recurrent can be made positive and thus is not monotone in the drift parameter.

Fuzzy reasoning is vital due to the frequent use of imprecise information in daily contexts. However, the ability of current large language models (LLMs) to handle such reasoning remains largely uncharted. In this paper, we introduce a new benchmark, FRoG, for fuzzy reasoning, featuring real-world mathematical word problems that incorporate generalized quantifiers. Our experimental findings reveal that fuzzy reasoning continues to pose significant challenges for LLMs. Moreover, we find that existing methods designed to enhance reasoning do not consistently improve performance in tasks involving fuzzy logic. Additionally, our results show an inverse scaling effect in the performance of LLMs on FRoG. Interestingly, we also demonstrate that strong mathematical reasoning skills are not necessarily indicative of success on our benchmark.

Spiral waves are a well-known phenomenon in excitable media, playing critical roles in biological systems such as cardiac tissues, where they are involved in arrhythmias, and in slime molds, where they guide collective cell migration. However, their presence in the cytoplasm of cells has not been reported to date. In this study, we present the observation of spiral waves in a Xenopus laevis frog egg extract reconstituting periodic cell cycle transitions. We find that the emergence of these spiral waves accelerates the cell division cycle nearly twofold. Using two distinct computational models, we demonstrate that this behavior arises from generic principles and is driven primarily by time-scale separation in the cell cycle oscillator. Additionally, we investigate the interplay between these spiral waves and the more commonly observed target pattern waves in the frog cytoplasm, providing new insights into their dynamic interactions.

While frequency-resolved optical gating (FROG) is widely used in characterizing the ultrafast pulse in optics, analytic signals are often considered in time-frequency analysis and signal processing, especially when extracting instantaneous features of events. In this paper we examine the phase retrieval (PR) problem of analytic signals in $\Bbb{C}^N$ by their FROG measurements. After establishing the ambiguity of the FROG-PR of analytic signals, we found that the FROG-PR of analytic signals of even lengths is different from that of analytic signals of odd lengths, and it is also different from the case of $B$-bandlimited signals with $B \leq N/2$. The existing approach to bandlimited signals can be applied to analytic signals of odd lengths, but it does not apply to the even length case. With the help of two relaxed FROG-PR problems and a translation technique, we develop an approach to FROG-PR for the analytic signals of even lengths, and prove that in this case the generic analytic signals can be uniquely (up to the ambiguity) determined by their $(3N/2+1)$ FROG measurements.

We present the design and operation of tadpole-mimetic robots prepared for a study of the parenting behaviors of poison frogs, which pair bond and raise their offspring. The mission of these robots is to convince poison frog parents that they are tadpoles, which need to be fed. Tadpoles indicate this need, at least in part, by wriggling with a characteristic frequency and amplitude. While the study is in progress, preliminary indications are that the TadBots have passed their test, at least for father frogs. We discuss the design and operational requirements for producing convincing TadBots and provide some details of the study design and plans for future work.

The frog model starts with one active particle at the root of a graph and some number of dormant particles at all nonroot vertices. Active particles follow independent random paths, waking all inactive particles they encounter. We prove that certain frog model statistics are monotone in the initial configuration for two nonstandard stochastic dominance relations: the increasing concave and the probability generating function orders. This extends many canonical theorems. We connect recurrence for random initial configurations to recurrence for deterministic configurations. Also, the limiting shape of activated sites on the integer lattice respects both of these orders. Other implications include monotonicity results on transience of the frog model where the number of frogs per vertex decays away from the origin, on survival of the frog model with death, and on the time to visit a given vertex in any frog model.

Interaction mechanism in the acoustic communication of actual animals is investigated by combining mathematical modeling and empirical data. Here we use a deterministic mathematical model (a phase oscillator model) to describe the interaction mechanism underlying the choruses of male Japanese tree frogs (Hyla japonica) in which the male frogs attempt to avoid call overlaps with each other due to acoustic communication. The mathematical model with a general interaction term is identified by a Bayesian approach from multiple audio recordings on the choruses of three male frogs. The identified model qualitatively reproduces the stationary and dynamical features of the empirical data, supporting the validity of the model identification. In addition, we quantify the magnitude of attention paid among the male frogs from the identified model, and then analyze the relationship between the attention and behavioral parameters by using a statistical model. The analysis demonstrates the biologically valid relationship about the negative correlation between the attention and inter-frog distance, and also indicates the existence of a behavioral strategy that the male frogs selectively pay attent