Search Results (11)

The problem of recovering the relaxation spectrum of the process described by the stretched exponential Kohlrausch–Williams–Watts (KWW) model is considered using the Post–Widder Laplace transform inversion rule. Based on the specific properties of the stretched exponential function, an analytical formula was derived that describes an arbitrarily high-order Post–Widder approximation of the spectrum directly in terms of the relaxation modulus, without any—neither analytical nor numerical—differentiation of the modulus as a product of finite power series of the relaxation times and the relaxation modulus. An alternative recurrence formula defining a sequence of the Post–Widder approximate models of the relaxation spectrum was developed. The positive definiteness, multiple differentiability, and zero asymptotic properties of the spectrum model are demonstrated; its extreme properties are discussed, and a sensitivity analysis with respect to the model parameters is conducted. The developed algorithm ensures fast convergence of the generated model sequence by applying a simple adaptive rule to select subsequent model orders, which relates them to the discrepancy between successive models and results in high-order models in at most a dozen iterations. Detailed numerical studies performed for nine values of the stretching exponent for KWW spectra covering relaxation times from 3 to 41 decades show that the proposed approach allows for generating an excellent approximation of the KWW spectrum by using only the relaxation modulus values in simple algebraic calculations. Full article

This article investigates the lower bounds of analytic functions defined by absolutely convergent Dirichlet series in the left half-plane and establishes conditions under which such functions exhibit the pits property. The study extends classical results for entire Dirichlet series and lacunary power series by refining assumptions on the sequence of exponents and resolving gaps in earlier proofs found in the literature. Central to the analysis is the introduction of a modified function $k\left(\sigma \right)$, whose behavior determines the size of exceptional sets surrounding the zeros of the series. Under suitable growth conditions on the exponents, the authors prove that outside small disks centered at the zeros, the modulus of the Dirichlet series admits explicit lower bounds involving its maximum term. Several auxiliary lemmas provide sharp estimates for the maximum modulus, the distribution of zeros, and the behavior of truncated Dirichlet polynomials. The main theorem demonstrates that the pits property holds uniformly in vertical strips approaching the imaginary axis. The paper concludes with a discussion of how the imposed step-size condition on exponents might be weakened, formulating a conjecture regarding a more general condensation index. Full article

We develop a convergence framework for Grünwald–Letnikov (GL) fractional and classical integer difference operators acting on sequences in fuzzy-paranormed (fp) spaces, motivated by data that are imprecise and contain sporadic outliers. Fuzzy paranorms provide a resolution-dependent notion of proximity, while statistical and lacunary statistical convergence downweight sparse deviations by natural density; together, they yield robust criteria for difference-filtered signals. Within this setting, we establish uniqueness of fp–${\mathsf{\Delta }}^m$ statistical limits; an equivalence between fp-statistical convergence of ${\mathsf{\Delta }}^m$ (and its GL extension ${\mathsf{\Delta }}^{\alpha }$) and fp-strong p-Cesàro summability; an equivalence between lacunary fp-${\mathsf{\Delta }}^m$ statistical convergence and blockwise strong p-Cesàro summability; and a density-based decomposition into a classically convergent part plus an fp-null remainder. We also show that GL binomial weights act as an ${\ell }^1$ convolution, ensuring continuity of ${\mathsf{\Delta }}^{\alpha }$ in the fp topology, and that nabla/delta forms are transferred by the discrete Q–operator. The usefulness of the criteria is illustrated on simple engineering-style examples (e.g., relaxation with memory, damped oscillations with bursts), where the fp-Cesàro decay of difference residuals serves as a practical diagnostic for Cesàro compliance. Beyond illustrative mathematics, we report engineering-style diagnostics where the fuzzy Cesàro residual index correlates with measurable quantities (e.g., vibration amplitude and energy surrogates) under impulsive disturbances and missing data. We also calibrate a global decision threshold ${\tau }_{\mathrm{glob}}$ via sensitivity analysis across $(\alpha ,p,m)$, where $m\in \mathbb{N}$ is the integer difference order, $\alpha >0$ is the fractional order, and $p\ge 1$ is the Cesàro exponent, and provide quantitative baselines (median/M-estimators, ${\ell }_1$ trend filtering, Gaussian Kalman filtering, and an $\alpha$-stable filtering structure) to show complementary gains under bursty regimes. The results are stated for integer m and lifted to fractional orders $\alpha >0$ through the same binomial structure and duality. Full article

The aim of this article is to investigate the relations between the exponent of the convergence of sequences and other characteristics defined for monotone sequences of positive numbers. Another main goal is to characterize such monotone sequences $\left(a_n\right)$ of positive numbers that, for each $n\ge 2$, satisfy the equality $a_n=K(a_{n-1},a_{n+1})$, where the function $K:{\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is the mean, i.e., each value of $K(x,y)$ lies between $min\{x,y\}$ and $max\{x,y\}$. Well-known examples of such sequences are, for example, arithmetic (geometric) progression, because starting from the second term, each of its terms is equal to the arithmetic (geometric) mean of its neighboring terms. Furthermore, this accomplishment generalized and extended previous results, where the properties of the logarithmic sequence $\left(a_n\right)$ are referred to, i.e., in such a sequence that every $n\ge 2$ satisfies $a_n=L(a_{n-1},a_{n+1})$, where $L(x,y)$ is the logarithmic mean of positive numbers $x,y$ defined as follows: $L(x,y):=\left\{\begin{array}{ccc}{\displaystyle \frac{y-x}{lny-lnx}}\hfill & \mathrm{if}\hfill & x\ne y,\hfill \\ x\hfill & \mathrm{if}\hfill & x=y.\hfill \end{array}\right.$ Full article

The Lyapunov exponent serves as a measure of the average divergence or convergence between chaotic trajectories from the perspective of Lyapunov exponents (LEs). Chaotic systems with more and larger positive LEs have more complex dynamical behavior and can weaken the degeneration of digital chaos. Some existing control algorithms for chaos need more and larger preset parameters, which are not favorable for practical application; others require the original system to satisfy specific conditions, which lack generality. To address the deficiencies of these algorithms, this paper proposes a construction algorithm of N-dimensional discrete non-degenerate chaos based on two-parameter gain (ND-NCTG), which can realize the non-degenerate or non-chaotic control of chaotic systems by only two control parameters. We take a 3D chaotic system as an example and analyze the relationship between control parameters and LEs, as well as the characteristics of chaotic sequences, to verify the effectiveness and reliability of the algorithm. In addition, since the initial value sensitivity of the chaotic system coincides with the sensitivity in input information for the hash function, this paper takes the proposed chaotic construction algorithm as the basis to design a bidirectional diffusion chaotic hash function. The effectiveness and security of this hash algorithm are verified by sensitivity, statistical distribution and collision analysis. Compared with similar algorithms, both the non-degenerate chaotic construction algorithm and the hash function algorithm proposed in this paper have better performance and can meet the application requirements of secure communication. Full article

To explore the mixed traffic characteristics of vehicles and aircraft on the airport surface and solve the problem of real-time conflict detection at key intersections. According to the actual taxiing procedures and airport control rules in China, this paper focus on abstracting the mixed motion process of aircraft and vehicles in the maneuvering area, defining the convergent cross-safety operation scenario. To quantify the driver’s attention to safety separation and the degree of conservatism in adjusting speed, the vehicle deceleration rate and acceleration rate are defined with α as the exponent. Under the same spacing, the vehicle deceleration rate is directly proportional to α, and α is named the vehicle safety sensitivity. At the same time, the rules of speed change of vehicles and aircraft will be designed, and a convergent operation model of aircraft and vehicles will be proposed. Based on real-time speed and separation dynamic assessment of safety and efficiency under different traffic strategies, the computer simulation results show that vehicle safety sensitivity and deceleration rules determine the sequence of vehicles and aircraft when they are passing through in the short term and can affect the proportion of mixed traffic flow in the long run. The safety probability of vehicles passing the intersection first is negatively correlated with vehicle safety sensitivity, while the safety probability of aircraft passing the intersection first is correlated positively with vehicle safety sensitivity. The efficiency of passing without conflict with mixed traffic has an inverse relationship with vehicle safety sensitivity. When the vehicle safety sensitivity takes the value in the interval [0.4, 0.6], a mixed traffic flow with higher safety and efficiency, better stability, and a balanced ratio of locomotives can be obtained. Full article

Considering the security of a communication system, designing a high-dimensional complex chaotic system suitable for chaotic synchronization has become a key problem in chaotic secure communication. In this paper, a new 5-D hyperchaotic system with high order nonlinear terms was constructed and proved to be hyperchaotic by dynamical characterization characteristics, the maximum Lyapunov exponent was close to 2, and there was a better permutation entropy index, while a valid chaotic sequence could be generated in three cycles in the FPGA (Field Programmable Gate Array)-based implementation. A multivariable nonlinear feedback synchronous controller based on FPGA was proposed to design and implement synchronization of high order complex hyperchaotic systems. The results show that the error signal converged to 0 rapidly under the effect of the nonlinear feedback synchronous controller. This lays the foundation for the synchronization of high order complex chaotic systems. Full article

Our first aim is to clarify the results obtained by Lidskii devoted to the decomposition on the root vector system of the non-selfadjoint operator. We use a technique of the entire function theory and introduce a so-called Schatten–von Neumann class of the convergence exponent. Considering strictly accretive operators satisfying special conditions formulated in terms of the norm, we construct a sequence of contours of the power type that contrasts the results by Lidskii, where a sequence of contours of the exponential type was used. Full article

A key factor limiting our understanding of rock slope behavior and associated geohazards is the interaction between internal and external system controls on the nature, rates, and timing of rockfall activity. We use high-resolution, monthly terrestrial light detection and ranging (LiDAR) surveys over a 2 year monitoring period to quantify rockfall patterns across a 0.6 km-long (15.3 × 10³ m²) section of a limestone rock cliff on the northeast coast of England, where uncertainty in rates of change threaten the effective planning and operational management of a key coastal cliff top road. Internal system controls, such as cliff material characteristics and foreshore geometry, dictate rockfall characteristics and background patterns of activity and demonstrate that layer-specific analyses of rockfall inventories and sequencing patterns are essential to better understand the timing and nature of rockfall risks. The influence of external environmental controls, notably storm activity, is also evaluated, and increased storminess corresponds to detectable rises in both total and mean rockfall volume and the volumetric contribution of large (>10 m³) rockfalls at the cliff top during these periods. Transient convergence of the cumulative magnitude–frequency power law scaling exponent (ɑ) during high magnitude events signals a uniform erosion response across the wider cliff system that applies to all lithologies. The tracking of rockfall distribution metrics from repeat terrestrial LiDAR in this way demonstrably improves the ability to identify, monitor, and forecast short-term variations in rockfall hazards, and, as such, provides a powerful new approach for mitigating the threats and impacts of coastal erosion. Full article

In view of the fact the accuracy of the third-degree Cubature Kalman Filter (CKF) used for initial alignment under large misalignment angle conditions is insufficient, an improved fifth-degree CKF algorithm is proposed in this paper. In order to make full use of the innovation on filtering, the innovation covariance matrix is calculated recursively by an innovative sequence with an exponent fading factor. Then a new adaptive error covariance matrix scaling algorithm is proposed. The Singular Value Decomposition (SVD) method is used for improving the numerical stability of the fifth-degree CKF in this paper. In order to avoid the overshoot caused by excessive scaling of error covariance matrix during the convergence stage, the scaling scheme is terminated when the gradient of azimuth reaches the maximum. The experimental results show that the improved algorithm has better alignment accuracy with large misalignment angles than the traditional algorithm. Full article