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Dengue fever poses a significant global health threat, with over 50 million annual infections spanning more than 100 countries. Given the absence of a specific treatment, medical intervention primarily targets symptom alleviation. The present study utilizes a Caputo-type fractional-order derivative operator to investigate and analyze the dynamics of dengue virus spread within a host with adaptive immune responses. The developed model describes and analyzes the dynamics of immune cells, free dengue particles, infected monocytes, and susceptible monocytes in the presence of cytotoxic T-Lymphocytes. A range of analytical methods is employed to probe the fractional-order within-host model. The application of the generalized mean value theorem aids in investigating the model's solutions, employing positivity and boundedness theory. Furthermore, the Banach fixed-point approach is utilized to establish the existence and uniqueness of solutions. Employing the normalized forward sensitivity approach, the fractional-order system's response to various model parameters is scrutinized. The study reveals that the dynamics of the viral model are significantly influenced by the transmission rate and parameters representing adaptive immune responses. Numerical simulations underscore the critical role of transmission rates and adaptive immune responses in the model. Additionally, the study examines the impact of memory on the density of susceptible monocytes, infected monocytes, free dengue particles, and immune cells to optimize immune responses. Through simulations, the study illustrates the influence of memory on immune dynamics. [ABSTRACT FROM AUTHOR]

Our goal is to prove the Euler–Maclaurin summation formula using only the Taylor formula. Furthermore, the Euler–Maclaurin summation formula will be considered for the case of functions of the class C 2 k ([ a , b ]) . A stronger version of the estimation of the rest for functions of class C n ([ a , b ]) is given. We will also present the application of the obtained Euler–Maclaurin summation formulas to determine the asymptotics of Riemann sums for uniform partitions of intervals of integration. This paper also introduces the concepts of the asymptotic smoothing of the graph of a given function, as well as the asymptotic uniform distribution of the positive and negative values of the given function (generalizing the concept of the symmetric distribution of these values with respect to the x-axis, as in the case of the Peano–Jordan measure). [ABSTRACT FROM AUTHOR]

In this paper, we show that there are infinitely many primes p such that p + 2 has at most two prime factors and p + 6 has at most 11 prime factors. This result constitutes an improvement on that of Cai. [ABSTRACT FROM AUTHOR]

In the era of big data, the exponential increase in complex-valued data generation requires overwhelming resources such as storage capacity, device-to-device communications, data processing costs, and power consumption. Hence, in this study, we raise the question of whether all training data is beneficial to the training of fully complex-valued neural networks (FCVNNs). To this end, exploiting the online censoring (OC) strategy, we introduce novel cost-effective learning algorithms for training the FCVNNs, called the OC-based fully complex nonlinear gradient descent (OC-FCNGD) for the FCVNN and the OC-based augmented FCNGD (OC-AFCNGD) for the augmented FCVNN (AFCVNN). These algorithms are derived using the Wirtinger calculus to simplify their derivations, define them in more compact forms, and thus remove Schwarz symmetry restrictions on their complex activation functions. The proposed OC-FCNGD and OC-AFCNGD algorithms considerably reduce the training times of the FCVNN and AFCVNN by censoring less informative training data, that is, by avoiding unnecessary weight updating. In this way, the OC-FCNGD and OC-AFCNGD also achieve superior testing results than their standard counterparts. Importantly, the proposed algorithms are not limited to the training of shallow architectures such as the FCVNN and AFCVNN, they are also extended and applied to the fully complex-valued deep neural network (FCVDNN) and augmented FCVDNN (AFCVDNN) in this paper, maintaining the efficiency and performance advantages arising from the OC strategy even in deep neural network models. Furthermore, we rigorously analyze the deterministic convergence of the OC-FCNGD and OC-AFCNGD, including both weak and strong convergence, by considering the unified mean value theorem. Finally, comprehensive experiments validate the theoretical and practical advantages of the proposed algorithms. [ABSTRACT FROM AUTHOR]