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The main purpose of this paper is to attract the attention of researchers working in the field of physiological processes, towards crucial events. Crucial events are often confused with extreme events thereby generating the misleading impression that their treatment should be based on quantum mechanical formalism. We show that crucial events are invisible and should not be confused with catastrophes. Crucial events are generated by self-organization processes yielding a form of swarm intelligence, and signal their action with fluctuations characterized by anomalous scaling and 1/f spectrum. The existence or the lack of crucial events can be revealed with an entropic method of analysis called the Diffusion Entropy Analysis (DEA). However, anomalous scaling and 1/f spectrum are not a compelling signature of efficient self-organization, and physiological processes with anomalous scaling and 1/f noise spectrum without crucial events are a signature of collapsing physiological organizations. In the case of physiological processes like cancer dynamics, the existence of crucial events is a signal of intelligence that must be destroyed rather than reinforced.

The scarcity of data in various scenarios, such as medical, industry and autonomous driving, leads to model overfitting and dataset imbalance, thus hindering effective detection and segmentation performance. Existing studies employ the generative models to synthesize more training samples to mitigate data scarcity. However, these synthetic samples are repetitive or simplistic and fail to provide "crucial information" that targets the downstream model's weaknesses. Additionally, these methods typically require separate training for different objects, leading to computational inefficiencies. To address these issues, we propose Crucial-Diff, a domain-agnostic framework designed to synthesize crucial samples. Our method integrates two key modules. The Scene Agnostic Feature Extractor (SAFE) utilizes a unified feature extractor to capture target information. The Weakness Aware Sample Miner (WASM) generates hard-to-detect samples using feedback from the detection results of downstream model, which is then fused with the output of SAFE module. Together, our Crucial-Diff framework generates diverse, high-quality training data, achieving a pixel-level AP of 83.63% and an F1-MAX of 78.12% on

Reinforcement Learning (RL) offers promising solutions for control tasks in industrial cyber-physical systems (ICPSs), yet its real-world adoption remains limited. This paper demonstrates how seemingly small but well-designed modifications to the RL problem formulation can substantially improve performance, stability, and sample efficiency. We identify and investigate key elements of RL problem formulation and show that these enhance both learning speed and final policy quality. Our experiments use a one-degree-of-freedom (1-DoF) helicopter testbed, the Quanser Aero~2, which features non-linear dynamics representative of many industrial settings. In simulation, the proposed problem design principles yield more reliable and efficient training, and we further validate these results by training the agent directly on physical hardware. The encouraging real-world outcomes highlight the potential of RL for ICPS, especially when careful attention is paid to the design principles of problem formulation. Overall, our study underscores the crucial role of thoughtful problem formulation in bridging the gap between RL research and the demands of real-world industrial systems.

Millisecond pulsars, like pulsars, have led to major advances in many areas of astronomy and physics. The discovery in 2023 of a cosmological gravitational wave (GW) stochastic background using millisecond pulsar timing arrays has focused attention on the importance of millisecond pulsars, to both multi-messenger astronomy and cosmology, and for identifying the origin of the GW stochastic background, which is hypothesized to be due to supermassive black hole binaries (SMBHBs). Unlike pulsars, however, for which the details of the discovery are well-known, those of millisecond pulsars are not well known. In particular, the details of the first crucial step in the discovery of millisecond pulsars, namely the discovery of interplanetary scintillation (IPS) in the radio source 4C 21.53, are known only to the author. This article presents a first-hand account of this crucial first step, which resulted ultimately in the discovery of millisecond pulses from this object. A brief description of interplanetary and interstellar scintillation and scattering is given in the Appendix.

Noise-induced phase transitions are common in various complex systems, from physics to biology. In this article, we investigate the emergence of crucial events in noise-induced phase transition processes and their potential significance for understanding complexity in such systems. We utilize the first-passage time technique and coordinate transformations to study the dynamics of the system and identify crucial events. Furthermore, we employ Diffusion Entropy Analysis, a powerful statistical tool, to characterize the complexity of the system and quantify the information content of the identified events. Our results show that the emergence of crucial events is closely related to the complexity of the system and can provide insight into its behavior. This approach may have applications in diverse fields, such as climate modeling, financial markets, and biological systems, where understanding the emergence of crucial events is of great importance.

A permutation is square-free if it does not contain two consecutive factors of length two or more that are order-isomorphic. A permutation is bicrucial with respect to squares if it is square-free but any extension of it to the right or to the left by any element gives a permutation that is not square-free. Bicrucial permutations with respect to squares were studied by Avgustinovich et al., who proved that there exist bicrucial permutations of lengths $8k+1, 8k+5, 8k+7$ for $k\ge 1$. It was left as open questions whether bicrucial permutations of even length, or such permutations of length $8k+3$ exist. In this paper, we provide an encoding of orderings which allows us, using the constraint solver Minion, to show that bicrucial permutations of even length exist, and the smallest such permutations are of length 32. To show that 32 is the minimum length in question, we establish a result on left-crucial (that is, not extendable to the left) square-free permutations which begin with three elements in monotone order. Also, we show that bicrucial permutations of length $8k+3$ exist for $k=2,3$ and they do not exist for $k=1$. Further, we generalise the notions of right-crucial, left-cru

A crucial permutation is a permutation that avoids a given set of prohibitions, but any of its extensions, in an allowable way, results in a prohibition being introduced. In this paper, we introduce five natural types of crucial permutations with respect to monotone patterns, notably quadrocrucial permutations that are linked most closely to Erdős-Szekeres extremal permutations. The way we define right-crucial and bicrucial permutations is consistent with the definition of respective permutations studied in the literature in the contexts of other prohibitions. For each of the five types, we provide its characterization in terms of Young tableaux via the RSK correspondence. Moreover, we use the characterizations to prove that the number of such permutations of length $n$ is growing when $n\to\infty$, and to enumerate minimal crucial permutations in all but one case. We also provide other enumerative results.

Unsupervised Domain Adaptation (UDA), which aims to explore the transferrable features from a well-labeled source domain to a related unlabeled target domain, has been widely progressed. Nevertheless, as one of the mainstream, existing adversarial-based methods neglect to filter the irrelevant semantic knowledge, hindering adaptation performance improvement. Besides, they require an additional domain discriminator that strives extractor to generate confused representations, but discrete designing may cause model collapse. To tackle the above issues, we propose Crucial Semantic Classifier-based Adversarial Learning (CSCAL), which pays more attention to crucial semantic knowledge transferring and leverages the classifier to implicitly play the role of domain discriminator without extra network designing. Specifically, in intra-class-wise alignment, a Paired-Level Discrepancy (PLD) is designed to transfer crucial semantic knowledge. Additionally, based on classifier predictions, a Nuclear Norm-based Discrepancy (NND) is formed that considers inter-class-wise information and improves the adaptation performance. Moreover, CSCAL can be effortlessly merged into different UDA methods as a

Image restoration is a challenging ill-posed problem which estimates latent sharp image from its degraded counterpart. Although the existing methods have achieved promising performance by designing novelty architecture of module, they ignore the fact that different regions in a corrupted image undergo varying degrees of degradation. In this paper, we propose an efficient and effective framework to adapt to varying degrees of degradation across different regions for image restoration. Specifically, we design a spatial and frequency attention mechanism (SFAM) to emphasize crucial features for restoration. SFAM consists of two modules: the spatial domain attention module (SDAM) and the frequency domain attention module (FDAM). The SFAM discerns the degradation location through spatial selective attention and channel selective attention in the spatial domain, while the FDAM enhances high-frequency signals to amplify the disparities between sharp and degraded image pairs in the spectral domain. Additionally, to capture global range information, we introduce a multi-scale block (MSBlock) that consists of three scale branches, each containing multiple simplified channel attention blocks (

The analysis of glioblastoma (GB) cell locomotion and its modeling inspired by Levy random walks is presented herein. We study such walks occurring on a two-dimensional plane where the walk is similar to the motion of a bird flying with a constant velocity, but with random changes of direction in time. The intelligence of the bird is signaled by the instantaneous changes of flying direction, which become invisible in the time series obtained by projecting the 2D walk either on the x axis or the y axis. We establish that the projected 1D time series share the statistical complexity of time series frequently used to monitor physiological processes, shedding light on the role of crucial events (CE-s) in pathophysiology. Such CE-s are signified by abrupt changes of flying direction which are invisible in the 1D physiological time series. We establish a connection between the complex scaling index δgenerated by the CE-s through μ_{R} = 2 - δ, where μ_{R} is the inverse power law index of the probability density function of the time interval between consecutive failures of the process of interest. We argue that the identification of empirical indices along with their theoretical relation

This paper introduces a dataset for improving real-time object recognition systems to aid blind and low-vision (BLV) individuals in navigation tasks. The dataset comprises 21 videos of BLV individuals navigating outdoor spaces, and a taxonomy of 90 objects crucial for BLV navigation, refined through a focus group study. We also provide object labeling for the 90 objects across 31 video segments created from the 21 videos. A deeper analysis reveals that most contemporary datasets used in training computer vision models contain only a small subset of the taxonomy in our dataset. Preliminary evaluation of state-of-the-art computer vision models on our dataset highlights shortcomings in accurately detecting key objects relevant to BLV navigation, emphasizing the need for specialized datasets. We make our dataset publicly available, offering valuable resources for developing more inclusive navigation systems for BLV individuals.

This paper is devoted to the study of the interaction between two distinct forms of non-stationary processes, which we will refer to as non-stationarity of first and second kind. The non-stationarity of first kind is caused by criticality-generated events that we call crucial events. Crucial events signal ergodicity breaking emerging from the interaction between the units of the complex system under study, indicating that the non stationarity of first kind has internal origin. The non-stationarity of second kind is due to the influence on the system of interest of an environment changing in time, thereby implying an external origin. In this paper we show that the non-stationarity of first kind, measured by an inverse power law index μ is characterized by singularities at μ = 2 and μ = 3. We realize the interaction between the non-stationarity of first kind and the non-stationarity of second kind with a model frequently adopted to study earthquakes, namely, a system of mainshocks, assumed to be crucial events, generating a cascade of after-shocks simulating the changing in time environment. We prove that the after-shocks significantly affects the detection of anomalous scaling, with

A pattern $τ$ is a permutation, and an arithmetic occurrence of $τ$ in (another) permutation $π=π_1π_2...π_n$ is a subsequence $π_{i_1}π_{i_2}...π_{i_m}$ of $π$ that is order isomorphic to $τ$ where the numbers $i_1<i_2<...<i_m$ form an arithmetic progression. A permutation is $(k,\ell)$-crucial if it avoids arithmetically the patterns $12... k$ and $\ell(\ell-1)... 1$ but its extension to the right by any element does not avoid arithmetically these patterns. A $(k,\ell)$-crucial permutation that cannot be extended to the left without creating an arithmetic occurrence of $12... k$ or $\ell(\ell-1)... 1$ is called $(k,\ell)$-bicrucial. In this paper we prove that arbitrary long $(k,\ell)$-crucial and $(k,\ell)$-bicrucial permutations exist for any $k,\ell\geq 3$. Moreover, we show that the minimal length of a $(k,\ell)$-crucial permutation is $\max(k,\ell)(\min(k,\ell)-1)$, while the minimal length of a $(k,\ell)$-bicrucial permutation is at most $2\max(k,\ell)(\min(k,\ell)-1)$, again for $k,\ell\geq3$.

Some extremely low-dimensional yet crucial geometric eigen-lengths often determine the success of some geometric tasks. For example, the height of an object is important to measure to check if it can fit between the shelves of a cabinet, while the width of a couch is crucial when trying to move it through a doorway. Humans have materialized such crucial geometric eigen-lengths in common sense since they are very useful in serving as succinct yet effective, highly interpretable, and universal object representations. However, it remains obscure and underexplored if learning systems can be equipped with similar capabilities of automatically discovering such key geometric quantities from doing tasks. In this work, we therefore for the first time formulate and propose a novel learning problem on this question and set up a benchmark suite including tasks, data, and evaluation metrics for studying the problem. We focus on a family of common fitting tasks as the testbed for the proposed learning problem. We explore potential solutions and demonstrate the feasibility of learning eigen-lengths from simply observing successful and failed fitting trials. We also attempt geometric grounding for

A word is "crucial" with respect to a given set of "prohibited words" (or simply "prohibitions") if it avoids the prohibitions but it cannot be extended to the right by any letter of its alphabet without creating a prohibition. A "minimal crucial word" is a crucial word of the shortest length. A word W contains an "abelian k-th power" if W has a factor of the form X_1X_2...X_k where X_i is a permutation of X_1 for 2<= i <= k. When k=2 or 3, one deals with "abelian squares" and "abelian cubes", respectively. In 2004 (arXiv:math/0205217), Evdokimov and Kitaev showed that a minimal crucial word over an n-letter alphabet A_n = {1,2,..., n} avoiding abelian squares has length 4n-7 for n >= 3. In this paper we show that a minimal crucial word over A_n avoiding abelian cubes has length 9n-13 for n >= 5, and it has length 2, 5, 11, and 20 for n=1, 2, 3, and 4, respectively. Moreover, for n >= 4 and k >= 2, we give a construction of length k^2(n-1)-k-1 of a crucial word over A_n avoiding abelian k-th powers. This construction gives the minimal length for k=2 and k=3. For k >= 4 and n >= 5, we provide a lower bound for the length of crucial words over A_n avoiding abe

$f \propto r^{-α} \cdot (r+γ)^{-β}$ has been empirically shown more precise than a naïve power law $f\propto r^{-α}$ to model the rank-frequency ($r$-$f$) relation of words in natural languages. This work shows that the only crucial parameter in the formulation is $γ$, which depicts the resistance to vocabulary growth on a corpus. A method of parameter estimation by searching an optimal $γ$ is proposed, where a ``zeroth word'' is introduced technically for the calculation. The formulation and parameters are further discussed with several case studies.

Facial expression recognition (FER) is still one challenging research due to the small inter-class discrepancy in the facial expression data. In view of the significance of facial crucial regions for FER, many existing researches utilize the prior information from some annotated crucial points to improve the performance of FER. However, it is complicated and time-consuming to manually annotate facial crucial points, especially for vast wild expression images. Based on this, a local non-local joint network is proposed to adaptively light up the facial crucial regions in feature learning of FER in this paper. In the proposed method, two parts are constructed based on facial local and non-local information respectively, where an ensemble of multiple local networks are proposed to extract local features corresponding to multiple facial local regions and a non-local attention network is addressed to explore the significance of each local region. Especially, the attention weights obtained by the non-local network is fed into the local part to achieve the interactive feedback between the facial global and local information. Interestingly, the non-local weights corresponding to local regio

Recently, Avgustinovich, Kitaev, and Taranenko defined five types of $(k, \ell)-$crucial permutations, which are maximal permutations that do not contain an increasing subsequence of length $k$ or a decreasing subsequence of length $\ell$. Further, Avgustinovich, Kitaev, and Taranenko began the enumeration of the $(k, \ell)-$crucial permutations of the minimal length and the next minimal length and the $(k, 3)-$crucial permutations of all lengths for each of the five types of $(k,\ell)-$crucial permutations. In this paper, we complete the enumeration that Avgustinovich, Kitaev, and Taranenko began.

Although deep learning-based methods have achieved excellent performance on SAR ATR, the fact that it is difficult to acquire and label a lot of SAR images makes these methods, which originally performed well, perform weakly. This may be because most of them consider the whole target images as input, but the researches find that, under limited training data, the deep learning model can't capture discriminative image regions in the whole images, rather focus on more useless even harmful image regions for recognition. Therefore, the results are not satisfactory. In this paper, we design a SAR ATR framework under limited training samples, which mainly consists of two branches and two modules, global assisted branch and local enhanced branch, feature capture module and feature discrimination module. In every training process, the global assisted branch first finishes the initial recognition based on the whole image. Based on the initial recognition results, the feature capture module automatically searches and locks the crucial image regions for correct recognition, which we named as the golden key of image. Then the local extract the local features from the captured crucial image regi