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Post-processing synchrosqueezing transform and synchroextracting transform methods can improve TFR resolution for fault diagnosis. The normal and fault signal can be described by infinite variance process, and 1 < α ≤ 2, even the background noise belongs to the process under complex conditions. The effect of traditional SST and SET methods is greatly reduced and even lost in infinite variance process environment. Several robust post-processing methods are proposed including FSET, FSSET, FSOSET and FMSST technology employing infinite variance process statistical model and FLOS, and their mathematical derivation are completed in this paper. The proposed methods are compared with the conventional methods, and the results show that the proposed methods achieve better results than the existing methods. In addition, the new methods are applied to diagnose the bearing outer race DE signals polluted by infinite variance process, the result demonstrates that they have performance advantages. Finally, the characteristics, shortcomings and application scenarios of the improved algorithms are summarized. [ABSTRACT FROM AUTHOR]

The authors investigate the complete moment convergence for arrays of rowwise independent random elements in Rademacher p Banach spaces. The results obtained in this paper improve the corresponding theorems of Hu et al. (Hu, T.-C., Rosalsky, A., Volodin, A., Zhang, S., 2021. A complete convergence theorem for row sums from arrays of rowwise independent random elements in Rademacher type p Banach spaces. II, Stochastic Anal. Appl., 39 (1), 177-193). Some corollaries and examples are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we present a more general approach based on a Picard integral scheme for nonlinear partial differential equations with a variable time-fractional derivative of order α (x , t) ∈ (1 , 2) and space-fractional order s ∈ (0 , 1) , where v = u ′ (t) is introduced as the new unknown function and u is recovered using the quadrature. In order to get rid of the constraints of traditional plans considering the half-time situation, integration by parts and the regularity process are introduced on the variable v. The convergence order can reach O (τ 2 + h 2) , different from the common L 1 , 2 − α schemes with convergence rate O (τ 2 , 3 − α (x , t)) under the infinite norm. In each integer time step, the stability, solvability and convergence of this scheme are proved. Several error results and convergence rates are calculated using numerical simulations to evidence the theoretical values of the proposed method. [ABSTRACT FROM AUTHOR]

The multi-task allocation has become the key to be solved urgently in applying a coal gangue sorting robot system (CGSRS) in raw coal with high throughput. This paper constructs a multi-task allocation framework of CGSRS for time-varying raw coal flow (TVRCF), including task decomposition, task allocation, and collaborative strategy. Establish the TVRCF decomposition model using the tumbling time window method to obtain the gangue model. A multi-task allocation model based on M/M/s queuing model with impatient customers is proposed because arrival time, granularity, and spatial position of the gangue model are uncertain. By introducing trajectory function and designing benefit function, to solve environment states of sorting area. Creating an iterative mechanism for environment states based on the infinite Markov process, using service time and arrival time interval. An adaptive weights rule based on the greedy algorithm (GAW) is proposed to improve the benefit of multi-task allocation. Simulating 18 groups of TVRCF with 9 productivity. Experimental results show that the proposed framework suits the multi-task allocation of CGSRS to TVRCF. The GAW rule always ensures the optimal sorting rate, the near-optimal task completion success rate, and the manipulator utilization rate. In addition, the stability of the GAW rule is best. [ABSTRACT FROM AUTHOR]

There are two distinct but interrelated questions concerning Aristotle's account of infinity that have been the subject of recurring debate. The first of these, what I call here the interpretative question, asks for a charitable and internally coherent interpretation of the limited pieces of text where Aristotle outlines his view of the 'potential' (and not 'actual') infinite. The second, what I call here the philosophical question, asks whether there is a way to make Aristotle's notion of the potential infinite coherent and rigorous with modern tools that can stand as a rival to the widely-accepted view of the infinite as characterized in a mathematical theory of sets. In this paper, I argue that the theoretical roles that Aristotle intends his account of the potential infinite to fulfill lead to a deep and irresoluble tension that can help explain the persistence of debates on both of these questions. I do so by turning to the places where Aristotle attempts to argue for or against the existence of particular infinite processes to show that he slides between different underlying notions of when changes are possible. Making these underlying notions clear can help us better understand the role of Aristotle's account in the history of philosophy, the possible pitfalls for a contemporary theory of the potential infinite, and what each of these debates might learn from each other. [ABSTRACT FROM AUTHOR]

Mechanical fault vibration signal is a typical non‐Gaussian process, they can be characterized by the infinite variance process, and the noise within these signals may also be the process in complex environments. The performance of the traditional cross‐term reduction algorithm is compromised, sometimes yielding incorrect results under the infinite variance process environment. Several robust fractional lower order time–frequency representation methods are proposed including fractional low‐order smoothed pseudo Wigner (FLOSPW), fractional low‐order multi‐windowed short‐time Fourier transform (FLOMWSTFT), and improved fractional low‐order multi‐windowed short‐time Fourier transform (IFLOMWSTFT) utilizing fractional low‐order statistics and short‐time Fourier transform (STFT) to mitigate cross‐terms, enhance time–frequency resolution, and accommodate the infinite variance process environment. When compared to traditional methods, simulation results indicate that they effectively suppress the pulse noise and function effectively in lower mixed signal noise ratio (MSNR) in an infinite variance process environment. The efficacy of the proposed time–frequency algorithm is validated through its application to mechanical bearing outer ring fault vibration signals contaminated with Gaussian noise and subjected to an α infinite variance process. [ABSTRACT FROM AUTHOR]

Fractional order filters offer greater freedom of design and a precise control over stopband attenuation in electronic circuits and systems. This paper presents the design of a fractional order low pass Chebyshev filter (FOLCF) that achieves near-ideal response characteristics. The methodology introduced utilizes metaheuristic optimization methods, including particle swarm optimization, firefly algorithm, and grey wolf optimization. These techniques are employed to precisely adjust the filter coefficients for the orders (1+α), (2+α), and (3+α). The adjustment is carried out by comparing the desired behaviour of the FOLCF with generalized fractional order low pass transfer functions. Throughout these instances, the parameter α is varied within the range of (0, 1). The designed filters are then tested and compared on the basis of various factors. Simulation results demonstrate that the designed filters closely follow the behaviour of an ideal Chebyshev filter with maximum passband and stopband magnitude errors being −31.93 dB and −52.74 dB respectively for (1+α) order filters. These values for (2+α) and (3+α) order FOLCF have been observed to be −30.04 dB and −55.91 dB; −17.72 dB and −49.52 dB respectively. Furthermore, it has been observed that the proposed work outperforms existing state-of-the-art approaches in various aspects, including magnitude error, stopband attenuation, and cut-off frequency. The stability of the designed filters has been verified through stability analysis. Additionally, practical feasibility of the proposed FOLCF is demonstrated through SPICE simulations for α = [0.2,0.5,0.8] using second generation current conveyor (CCII) and operational transconductance amplifier (OTA) based topologies while approximating the constant phase element using fifth order continued fraction expansion. The SPICE implementations closely follow the behaviour of ideal filter with −48.67 dB and −62.8 dB as mean square errors for CCII and OTA circuits respectively, showcasing the proposed filters' superiority and practical applicability in advanced electronic design. • A generalized design of Fractional order low Pass Chebyshev filters having near idle response has been proposed. • Nature inspired Algorithms: particle swarm optimization (PSO), firefly algorithm (FA) & grey wolf optimization (GWO) are used. • Comparisons with existing literature on basis of magnitude errors, stopband attenuation and cut off frequencies has been made. • Stability analysis for all the designed filters has been done by converting s-plane to W-plane. • Realization of proposed filters has been done by using CCII and OTA based circuits to test feasibility. [ABSTRACT FROM AUTHOR]

This paper is concerned with the eigenvalue decay of solution operators to operator Lyapunov equations, a relevant topic in the context of model reduction for parabolic control problems. We mainly focus on the Gramian operator, which arises in the context of control and observation of heat processes in infinite time, which is normally the first step towards observations in a finite time horizon. By improving existing energy and observability estimates for parabolic equations, we obtain both upper and lower bounds on the convergence rate of the eigenvalues of the Gramian operator towards zero. Both bounds follow the same polynomial decay rate, up to a multiplicative constant, which ensures their optimality. This confirms the slow decay of the eigenvalues and limits the efficiency of model reduction. The theoretical findings are supported by numerical results. [ABSTRACT FROM AUTHOR]

The article studies a single-type critical Markov branching process with infinite variance of the offspring distribution. The process admits also an immigration component at the time points of a non-homogeneous Poisson process. If additionally the mean number of immigrants is infinite, then proper limit distributions are obtained, under suitable normalization of the sample paths, depending on the rate of change of the intensity of the Poisson process. [ABSTRACT FROM AUTHOR]

This paper focuses on a class of regime-switching functional diffusion processes with infinite delay and develops a central limit theorem (CLT) for additive functionals under uniform mixing conditions. In addition, a law of iterated logarithm (LIL) for the additive functionals is also established by using the square integrable martingale difference sequences. Finally, two examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]