Search

An explicit transformation for the series $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ , or equivalently, $\sum \limits _{n=1}^{\infty }d(n)\log (n)e^{-ny}$ for Re $(y)>0$ , which takes y to $1/y$ , is obtained for the first time. This series transforms into a series containing the derivative of $R(z)$ , a function studied by Christopher Deninger while obtaining an analog of the famous Chowla–Selberg formula for real quadratic fields. In the course of obtaining the transformation, new important properties of $\psi _1(z)$ (the derivative of $R(z)$) are needed as is a new representation for the second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$ evaluated at $b=1$ , all of which may seem quite unexpected at first glance. Our transformation readily gives the complete asymptotic expansion of $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ as $y\to 0$ which was also not known before. An application of the latter is that it gives the asymptotic expansion of $ \displaystyle \int _{0}^{\infty }\zeta \left (\frac {1}{2}-it\right)\zeta '\left (\frac {1}{2}+it\right)e^{-\delta t}\, dt$ as $\delta \to 0$. [ABSTRACT FROM AUTHOR]

Shared electric scooters have become a popular and flexible transportation mode in recent years. However, managing these systems, especially the rebalancing of scooters, poses significant challenges due to the unpredictable nature of user demand. To tackle this issue, we developed a stochastic optimization model (M0) aimed at minimizing transportation costs and penalties associated with unmet demand. To solve this model, we initially introduced a mean-value optimization model (M1), which uses average historical values for user demand. Subsequently, to capture the variability and uncertainty more accurately, we proposed a data-driven optimization model (M2) that uses the empirical distribution of historical data. Through computational experiments, we assessed both models' performance. The results consistently showed that M2 outperformed M1, effectively managing stochastic demand across various scenarios. Additionally, sensitivity analyses confirmed the adaptability of M2. Our findings offer practical insights for improving the efficiency of shared electric scooter systems under uncertain demand conditions. [ABSTRACT FROM AUTHOR]

Traditionally, the reliability of a rock slope is evaluated based on the linear Mohr–Coulomb strength criterion, neglecting the nonlinear shear strength of the majority of geotechnical materials. The paper proposes a probability method to assess the reliability of rock slopes based on the power-type nonlinear strength criterion and the mean-value first-order second-moment method. In the proposed method, a novel analytical method to determine the safety factors of slopes with plane failure is derived based on the power-type nonlinear strength criterion. The local mean value of a random variable on the failure surface is determined based on the two-dimensional random field theory, avoiding the random field simulation. To quickly search the minimum reliability index βmin, a dichotomic search method is proposed. The validity of the proposed method is demonstrated in two cases. The study results from the two cases illustrate that the proposed method has high computational efficiency and accuracy. The proposed method offers a fresh way to examine the reliability of rock slopes when the nonlinear strength criterion is adopted. [ABSTRACT FROM AUTHOR]

Passive samplers have long been used to measure atmospheric pollutants in both indoor and outdoor environments. They are simple to operate, and can now monitor several chemical species. However, their use is limited because they usually require a long exposition time and provide a mean value that cannot control or evidence expected or non-expected events of environmental significance. A new apparatus specifically developed for exposing Analyst© passive samplers has been used to monitor ozone and nitrogen dioxide by automatically selecting a sampling duration of 8 h, as most legislation requires. The instrument was designed to accumulate ozone or NO2 in one passive sampler for 8 h over each day, and in another passive sampler for the remaining hours. This allows for a long-time accumulation of the 8 h ozone or nitrogen dioxide in a dedicated sampler. Measurements were carried out NE of Rome at a rural site. A description of the experiments is given, with special emphasis on the quality controls. Very low uncertainties and good comparability of the data with the reference methods were obtained for both pollutants. [ABSTRACT FROM AUTHOR]

In this paper, we study mean values for exponential sums of divisor functions. We improve previous results of [M. Pandey, Moment estimates for the exponential sum with higher divisor functions, C. R. Math. Acad. Sci. Paris  360 (2022) 419–424]. [ABSTRACT FROM AUTHOR]

We consider the Laplacian on a metric graph, equipped with Robin (δ -type) vertex condition at some of the graph vertices and Neumann–Kirchhoff condition at all others. The corresponding eigenvalues are called Robin eigenvalues, whereas they are called Neumann eigenvalues if the Neumann–Kirchhoff condition is imposed at all vertices. The sequence of differences between these pairs of eigenvalues is called the Robin–Neumann gap. We prove that the limiting mean value of this sequence exists and equals a geometric quantity, analogous to the one obtained for planar domains by Rudnick et al. (Commun Math Phys, 2021. arXiv:2008.07400). Moreover, we show that the sequence is uniformly bounded and provide explicit upper and lower bounds. We also study the possible accumulation points of the sequence and relate those to the associated probability distribution of the gaps. To prove our main results, we prove a local Weyl law, as well as explicit expressions for the second moments of the eigenfunction scattering amplitudes. [ABSTRACT FROM AUTHOR]

Assuming the Generalized Riemann Hypothesis, we calculate ∑ 0 < γ χ ≤ T L (μ) (ρ χ + i δ , χ) L (ν) (1 − ρ χ − i δ , χ ‾) , which extends the result proposed by Gonek in 1984. [ABSTRACT FROM AUTHOR]

Given a monic polynomial f(X)∈Zm[X] over a residue ring Zm modulo an integer m≥2 and a discrete interval I={1,...,H} of H≤m consecutive integers, considered as elements of Zm, we obtain a new upper bound for the additive energy of the set f(I), where f(I) denotes the image set f(I)={f(u): u∈I}. We give an application of our bounds to multiplicative character sums, improving some previous result of Shkredov and Shparlinski~(2018).. [ABSTRACT FROM AUTHOR]

This paper presents an analytical method to solve the optimal control problem for affine nonlinear systems with unknown drift dynamics. A new non‐quadratic cost function over an infinite horizon is presented that considers input constraints and includes the cost of the feed‐forward component of the control law. The mean value theorem for vector‐valued functions has been used to derive an integral form of this theorem. Based on this theorem, a rigorous proof is provided demonstrating that the cost function can be converted into another form. In the presence of input constraints, this converted form enables extracting the optimal control solution without solving the HJB equation. Additionally, unknown nonlinearity effects in drift dynamics are compensated in the control input. This is accomplished by estimating the unknown drift dynamics via an adaptive neural network (NN) approach. It is proven that the states and weights of NN are uniformly ultimately bounded based on a Lyapunov technique. The necessary and sufficient conditions are provided that ensure the optimality of the infinite horizon optimal control problem with a discount factor. As a result, it is demonstrated that the proposed approach satisfies the optimality criteria. To evaluate the effectiveness of the proposed approach, simulation examples are provided. [ABSTRACT FROM AUTHOR]

This paper copes with a matter of event-triggered fast finite-time prescribed tracking control for non-affine stochastic nonlinear systems, in which deception attacks are taken into consideration. The main objective of the proposed controller is that all the closed-loop variables are bounded in probability, and the tracking error is constrained to the predefined funnel functions by choosing suitable parameters. Primarily, the mean value theorem is utilised to transform the considered systems into the equivalent systems with affine structures. Next, based on the backstepping algorithm, a modified event-triggered scheme is developed to save the network resources. Meanwhile, the unknown parts in the designed systems are approximated by neural networks. Finally, an actual simulation is conducted to verify the validity of the theorem. [ABSTRACT FROM AUTHOR]

This paper aims to address the problem of event‐based adaptive robust control for a class of nonaffine nonlinear systems against actuator faults and unknown deception attacks. Firstly, different from the widely used relative threshold triggering mechanism, an improved event‐triggered mechanism with the time‐varying function threshold is devised for effectively saving more communication costs between controller and actuator. Then, based on a class of modified filter‐based coordinate transformations and the improved event‐triggered mechanism, the intermediate virtual control laws are devised to construct actual controllers, which not only guarantees robust stabilization of the system but also relaxes the universal assumptions on control coefficients and attack weights. Besides, the design challenges resulting from the presence of coupling terms in the nonaffine nonlinear system are successfully overcome by virtue of mean value theorem and the property of fuzzy basis function. Theoretical analysis proves that all signals of the closed‐loop system are semiglobally uniformly ultimately bounded (SUUB) and the Zeno behavior can be averted. Finally, the effectiveness of the proposed strategy is verified via comprehensive simulation results. [ABSTRACT FROM AUTHOR]

We give an improved lower bound for the average of the Erdős–Hooley function Δ(n)$\Delta (n)$, namely ∑n⩽xΔ(n)≫εx(loglogx)1+η−ε$\sum _{n\leqslant x} \Delta (n) \gg _\varepsilon x(\log \log x)^{1+\eta -\varepsilon }$ for all x⩾100$x\geqslant 100$ and any fixed ε$\varepsilon$, where η=0.3533227⋯$\eta = 0.3533227\dots$ is an exponent previously appearing in work of Green and the first two authors. This improves on a previous lower bound of ≫xloglogx$\gg x \log \log x$ of Hall and Tenenbaum, and can be compared to the recent upper bound of x(loglogx)11/4$x (\log \log x)^{11/4}$ of the second and third authors. [ABSTRACT FROM AUTHOR]

We extend and simplify proofs of some recent results on stability in the sense of Hyers and Ulams for mean value points that come from differential expressions, like Rolle theorem and co. Then, classical mean value theorems due to Lagrange and Cauchy are generalized in the setting of divided differences with multiplicities and we prove Hyers-Ulam stability for the corresponding mean value points. [ABSTRACT FROM AUTHOR]

In this paper, a LuGre model–based robust adaptive control (RAC) approach is presented for a pump-controlled hydraulic actuator. We first decompose the LuGre friction model into its steady-state model and a lumped dynamic part applying the mean value theorem, which are compensated by a feedforward term and a robust adaptive term, respectively. The robust adaptive term also plays a part in mismatched disturbance attenuation. In addition, parametric uncertainties and matched disturbances are handled by σ -modified adaptation laws and a robust control law, respectively. The stability of the closed-loop system is proved via the Lyapunov analysis. The efficacy and robustness of the proposed approach are validated by comparative experiments. Compared with common adaptive friction compensation methods, the proposed method has a simpler structure, less computational burden, better control performance, and stronger robustness. Moreover, since the available information is separated from the LuGre model and acts as a model-based compensation term, the design conservativeness of RAC is effectively reduced. [ABSTRACT FROM AUTHOR]

This paper deals with the quasi-instantaneous determination (in a single-step response time) of apparent, active, and reactive (i.e., blind and distortion) power mean values including the total power factor, total harmonic distortion, and phase shift of fundamentals of a power electronic and electrical system (PEES) using the ip-iq method, which is the main contribution of the paper. The power components' mean values are investigated during the transient and steady states. The power components' mean values can be determined directly from phase current and voltage quantities, using an integral calculus over one period within the next calculation step and using moving average and moving rms techniques (or digital filtering). Consequently, the power factor can be evaluated with known values of a phase shift of fundamentals (using a Fourier analysis). The results of this study show how a distortion power component during transients is generated even under a harmonic supply and linear resistive–inductive load. The paper contains a theoretical base, modeling, and simulation for the three and single phases of the transients in power electronic systems. The worked-out results can be used to determine and size any PES. The presented approach brings a detailed time waveform verified by simulations in Matlab/Simulink 2022a and the Real-time HW Simulator Plecs RT Box 1. [ABSTRACT FROM AUTHOR]

In this work, we proposed a method to extract the distribution of the built-in field (Eb) from the switching dynamics of the TiN/HfZrO/TiN capacitor. The relationship between reversal polarization and the distribution of Eb is established based on the classic inhomogeneous field mechanism model. Both positive and negative write pulses with different amplitudes and durations are applied to obtain the distribution parameters of Eb. The distribution of Eb is fitted by a Gaussian-type distribution, and the mean value and standard deviation are about −0.02 MV/cm and 0.28 MV/cm, respectively. This work provides an effective approach to analyze Eb directly from the electrical measurement and helps optimize the device design from the polarization switching point of view. [ABSTRACT FROM AUTHOR]

This paper investigates a tracking control issue for a class of time-varying output constrained nonlinear systems subject to non-affine nonlinear faults and virtual control coefficients. In the controller design process, the nonlinear function is approximated by fuzzy logic systems. Utilizing a feasible function to convert the system, a new transformation strategy is proposed to handle the system with time-varying output constraints or without output constraints. The mean value theorem is applied to split non-affine faults, and the Nussbaum-type function is used to eliminate the effect of the unknown directional affine variable gain of input resulted from non-affine faults. Combined with adaptive fuzzy backstepping technology and the error transformation function, a new control strategy is presented, which not only deals with the time-varying output constraint but also handles non-affine faults, effectively. Finally, all closed-loop system signals are bounded, and the tracking error can converge to a small preset set. Two simulations are performed to confirm the validity of the presented strategy. • Nonlinear transformation functions are adapted to systems with or without output constraints. • Tracking error can converge to a small adjustable set and maintain within a prescribed range. • Updating estimates of virtual control coefficient online reduces system limitations. [ABSTRACT FROM AUTHOR]

We show that almost all sectors of the disc (z 2 C W 2 ) of area .log X/15:1 contain products of exactly two Gaussian primes, and that almost all sectors of area .log X/1C" contain products of exactly three Gaussian primes. The argument is based on mean value theorems, large value estimates and pointwise bounds for Hecke character sums. [ABSTRACT FROM AUTHOR]

In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical differential calculus are introduced in terms of this new derivative. Thus, we establish and demonstrate the basic operations with derivatives, chain rule, mean value theorems with their immediate applications and inverse function's derivative. We complete the theory of generalized FF calculus by proposing a notion of integration and presenting two important results of integral calculus: the fundamental theorem and Barrow's rule. Finally, we analytically solve interesting FF ordinary differential equations by applying our proposed definition. [ABSTRACT FROM AUTHOR]

In this paper, a fractional model in the Caputo sense is used to characterize the dynamics of HPV with cervical cancer. Generalized mean value theorem has been used to examine whether the infection model has a unique positive solution. The model has two equilibrium points: the disease-free point and the endemic point. The examination of the system's local and global stability is provided in terms of the basic reproductive number R p ° . The global stability analysis has been carried out using an appropriate Lyapunov function and the LaSalle invariant principle. The results demonstrate that in the infection model, if R p ° < 1 , then the solution converges to the disease-free equilibrium, which is both locally and globally asymptotically stable. Whilst R p ° > 1 , the endemic equilibrium is considered to exist. Simulations are implemented via a finite difference method with Grünwald-Letnikov discretization approach for Caputo derivative operator to define how changes in parameters impact the dynamic behavior of the system using Matlab. [ABSTRACT FROM AUTHOR]

In the present paper, the expected-value (EV) reformulation of a class of stochastic vector variational inequalities (SVVI) is investigated. By using the regularized gap function, the EV reformulation of SVVI is transformed into a constrained optimization problem. Then a sample average approximation (SAA) method is proposed for solving the constrained optimization problem. Under suitable assumptions, the limiting behaviors of the optimal values and optimal solutions of the approximation problem are investigated. Finally, the rates of convergence in the different senses of optimal solutions for sample average approximation problem are discussed under the error bound condition. [ABSTRACT FROM AUTHOR]

In this study, a mathematical model for the transmission dynamics of malaria among different socioeconomic groups in the human population interacting with a susceptible-infectious vector population is presented and analysed using a fractional-order derivative of the Caputo type. The total human population is stratified into two distinguished classes of lower and higher income individuals, with each class further subdivided into susceptible, infectious, and recovered populations. The socio hierachy-structured fractional-order malaria model is analyzed through the application of different dynamical system tools. The theory of positivity and boundedness based on the generalized mean value theorem is employed to investigate the basic properties of solutions of the model, while the Banach fixed point theory approach is used to prove the existence and uniqueness of the solution. Furthermore, unlike the existing related studies, comprehensive global asymptotic dynamics of the fractional-order malaria model around both disease-free and endemic equilibria are explored by generalizing the usual classical methods for establishing global asymptotic stability of the steady states. The asymptotic behavior of the trajectories of the system are graphically illustrated at different values of the fractional (noninteger) order. [ABSTRACT FROM AUTHOR]

We consider an elliptic inclusion that is driven by $ (-\Delta)^s_{\mathrm {a}(\cdot)} $ (− Δ) a (⋅) s called fractional $ \mathrm {a}(\cdot) $ a (⋅) -Laplacian operator, with Dirichlet-type boundary conditions. We take two different assumptions on the Carathéodory function $ \mathrm {f} $ f . One of them is where $ \mathrm {f} $ f satisfies the localy Lipchitz condition and the other one is when $ \mathrm {f} $ f does not satisfy an assumption of growth. With a regular Clarke subdifferential $ \partial _C $ ∂ C and Lebourg's mean value theorem, by applying a variational approach together with the critical point theory, we obtain a weak solution to the inclusion problem in appropriate fractional Orlicz–Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study an initial value problem with a weakly singular nonlinear fractional differential equation of higher order. First, we establish the existence of global solutions to the problem within the appropriate function space. We then introduce a generalized Riemann-Liouville mean value theorem. Using this theorem, we prove the Nagumo-type uniqueness theorem for the stated problem. Moreover, we give two examples to illustrate the applicability of the existence and uniqueness theorems. [ABSTRACT FROM AUTHOR]

Given a matrix A\in \mathrm {GL}_d(\mathbb {Z}). We study the pseudorandomness of vectors \mathbf {u}_n generated by a linear recurrence relation of the form \begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots, \end{equation*} modulo p^t with a fixed prime p and sufficiently large integer t \geqslant 1. We study such sequences over very short segments of length which has not been accessible via previously used methods. Our technique is based on the method of N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp. 654–670, 672] of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford [Proc. London Math. Soc. (3) 85 (2002), pp. 565–633]. This is combined with some ideas from the work of I. E. Shparlinski [Proc. Voronezh State Pedagogical Inst., 197 (1978), 74–85 (in Russian)] which allows us to construct polynomial representations of the coordinates of \mathbf {u}_n and control the p-adic orders of their coefficients in polynomial representations. [ABSTRACT FROM AUTHOR]

The problem of asymptotic behavior of the expectation of a pairwise interaction type functional on branching random walk is considered. [ABSTRACT FROM AUTHOR]

Summary: This paper presents neural networks (NNs) adaptive controller for an uncertain fractional‐order nonlinear system in strict‐feedback form, subject to input saturation, unavailable states for measurement, and external disturbances. The fractional‐order adaptive laws are derived based only on the output tracking error thanks to the implementation of the strictly positive real (SPR) property, differently from the existing results in the literature of fractional‐order strict‐feedback systems where all system states are used in the adaptive laws. The proposed design approach addresses the nonaffine nature of the control input due to saturation nonlinearity by using the mean value theorem and follows a nonrecursive design by using a state transformation where the advantage is twofold. First, it eliminates the explosion complexity found in back‐stepping control‐based approaches design, and second, it reduces the approximators units and parameters in controller implementation. Furthermore, an observer is introduced to estimate the unavailable newly defined states, and then an output adaptive feedback control design is ensured. An NN is used to approximate the unknown ideal control law, and an auxiliary control term is appended to deal with saturation effect, unknown disturbance, and approximation errors. The tracking error is proved to converge asymptotically to a bounded set using the Lyapunov theory. Simulation results on three examples show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups. [ABSTRACT FROM AUTHOR]

Let h, b, c, and q ≥ 3 be integers, and let a ¯ satisfy the equation a a ¯ ≡ 1 mod q. The general mth Cochrane sum is defined as C m h , c , q = ∑ a = 1 ′ q c a ¯ / q ha m / q. The purpose of this paper is to study the hybrid mean value involving hyper-Kloosterman sums and the mth Cochrane sum ∑ x = 1 ′ q I m + 1 , h ; q C m h , c , q by applying the properties of Gauss sums, primitive characters, and a mean value theorem of Dirichlet L-functions. Similarly, we also have the hybrid mean value involving mth Cochrane sum and the general Kloosterman sum defined by Ye [Y.B. Ye, Hyper-Kloosterman sums and estimation of exponential sums of polynomials of higher degrees, Acta Arith., 86(3):255–267, 1998] as K m b ; c ; q : = ∑ x = 1 ′ q e bx m + c x ¯ / q. For m = 1, we obtain a better asymptotic formula than that of Zhang in [W.P. Zhang, On a Cochrane sum and its hybrid mean value formula, J. Math. Anal. Appl., 267(1):89–96, 2002]. [ABSTRACT FROM AUTHOR]

Yıldırım obtained an asymptotic formula of the discrete moment of | ζ (1 2 + i t) | over the zero of the higher derivatives of Hardy's Z -function. We give a generalization of his result on Hardy's Z -function. [ABSTRACT FROM AUTHOR]

The adaptive fuzzy backstepping control problem is studied for Itô-type nonlinear switched systems subject to unknown hysteresis input. Compared with existing works, the unknown hysteresis and stochastic disturbances are considered in the pure-feedback switched systems. The mean value theorem tackles the non-affine functions. The backstepping technique introduces an auxiliary virtual controller. In addition, the Nussbaum function is employed to solve the difficulty caused by the unknown hysteresis under arbitrary switching. Based on a fuzzy logic system and backstepping technique, a new adaptive control proposal is obtained, which ensures that the system states satisfy semiglobally uniformly ultimately bounded (SGUUB) in probability and that the tracking error converges to a region of the origin. Finally, we provide two examples to show the validity of the presented scheme. [ABSTRACT FROM AUTHOR]

The distributed adaptive fuzzy coordination control (DAFCC) method for the leader-following for non-affine nonlinear (NAN) multi-agent systems (MASs) with actuator failures (including partial failures and actuator bias failures) under the unreliable switching communication environments is proposed in this article. First, based on the neighborhood state information, a group of distributed reference observer (DRO) is constructed so that all followers can easily estimate the leader's information. Second, the implicit function theorem and the mean value theorem are utilized to convert the non-affine term into the corresponding affine term in order to solve the non-affine term in the NAN system. Third, to achieve global synchronization, we design an adaptive fuzzy coordination fault-tolerant controller with local estimation. Besides, the DAFCC method presented in this paper solves the cooperative control problem when switching undirected networks with intermittent communications. Finally, simulation is used to validate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the standard semilinear heat equation$ \begin{eqnarray*} \partial_t u = \Delta u + |u|^{p-1}u, \quad p >1. \end{eqnarray*} $The determination of the (believed to be) generic blowup profile is well established in the literature, with the solution blowing up only at one point. Though the blow-up of the gradient of the solution is a direct consequence of the single-point blow-up property and the mean value theorem, there is no determination of the final blowup profile for the gradient in the literature, up to our knowledge. In this paper, we refine the construction technique of Bricmont-Kupiainen [4] and Merle-Zaag [20], and derive the following profile for the gradient:$ \nabla u(x,T) \sim - \frac{\sqrt{2b}}{p-1} \frac{x}{|x| \sqrt{ |\ln|x||}} \left[\frac{b|x|^2}{2|\ln|x||} \right]^{-\frac{p+1}{2(p-1)}} {\text{ as }} x \to 0, $where $ b = \frac{(p-1)^2}{4p} $, which is as expected the gradient of the well-known blowup profile of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some of important qualitative properties of solutions of secondorder hyperbolic equations, whose coefficients of the terms involving the second-order derivatives are independent of the desired function and its derivatives. Solutions of these equations have a special property called curvilinear parallelogram identity (or mean-value property), which can be used to solve some initial-boundary value problems. [ABSTRACT FROM AUTHOR]

This paper deals with the equation - Δ u + μ u = f on high-dimensional spaces R m , where the right-hand side f (x) = F (T x) is composed of a separable function F with an integrable Fourier transform on a space of a dimension n > m and a linear mapping given by a matrix T of full rank and μ ≥ 0 is a constant. For example, the right-hand side can explicitly depend on differences x i - x j of components of x. Following our publication (Yserentant in Numer Math 146:219–238, 2020), we show that the solution of this equation can be expanded into sums of functions of the same structure and develop in this framework an equally simple and fast iterative method for its computation. The method is based on the observation that in almost all cases and for large problem classes the expression ‖ T t y ‖ 2 deviates on the unit sphere ‖ y ‖ = 1 the less from its mean value the higher the dimension m is, a concentration of measure effect. The higher the dimension m, the faster the iteration converges. [ABSTRACT FROM AUTHOR]

In this paper, we study Schur convexity and concavity, its properties like Schur, Schur harmonic convexity(concave)on the ratio of difference of means obtained by arithmetic mean, geometric mean, harmonic mean, contraharmonic mean, heron mean, root square mean, subsidiary means. [ABSTRACT FROM AUTHOR]

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group, in this case, the irreducible representation spaces of homogeneous harmonic polynomials. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with L p boundary data, 1 ≤ p ≤ ∞ . We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy's estimates, the mean-value property, Liouville's Theorem, etc. [ABSTRACT FROM AUTHOR]

A simple phase error criterion (PEC-)-based adaptive algorithm for estimating the frequency of a complex sinusoidal signal in additive white Gaussian and impulsive noises is proposed. The proposed technique makes use of the instantaneous phase response of a first-order complex linear predictor (CLP) as a driving function to update the frequency parameter of the CLP. The proposed PEC is attractive due to its simplicity and high impulsive noise robustness. Theoretical analysis for the mean value of the estimated frequency and the steady-state mean square error (MSE) of the frequency estimate are derived in closed forms. Computer simulations are drawn to show the performance of the proposed frequency estimator. [ABSTRACT FROM AUTHOR]

We present a new classifier that uses a generalized Heronian mean (GHM) operator, and fuzzy robust principal component analysis (FRPCA) algorithms. The similarity classifier was earlier studied with other aggregation operators, including: the ordered weighted averaging (OWA), generalized mean, arithmetic mean among others. Parameters in the GHM operator makes the new classifier suitable for handling a variety of modeling problems involving parameter settings. Motivated by the nature of the GHM operator, we examine which FRPCA algorithm is suitable for use to achieve optimal performance of the new classifier. The effects of dimensionality reduction and fuzziness variable on classification accuracy are examined. The performance of the new classifier is tested on three real-world datasets: fertility, horse-colic, and Haberman's survival. Compared with previously studied similarity classifiers, the new method achieved improved classification accuracy for the tested datasets. In fertility dataset, the new classifier achieved improvements in accuracy of 1 4. 6 0 % , 1 9. 7 3 % , and 2 3. 0 0 % compared with the OWA, generalized mean, and arithmetic mean based classifiers respectively. The new classifier is simpler to implement since it does not require any weight generation criteria as the case is for the OWA based classifier. [ABSTRACT FROM AUTHOR]

In this short note, we present a generalization of Wayment's Mean Value Theorem for Integrals. [ABSTRACT FROM AUTHOR]

The dynamic performance of high-speed maglev trains, a next-generation rapid transit system, has received continuous attention in recent years. In this study, a dynamic reliability analysis method for a high-speed maglev train–guideway coupled system was proposed. First, a refined model of the maglev vehicle–bridge interaction system was established, where the vehicle subsystem was simulated as a rigid body-spring-damper model with 101 degrees of freedom. The guideway subsystem was simulated as a finite element model, and these two subsystems were coupled as an entire system through a magnet–rail interaction model with a proportional-derivative (PD) controller. Second, a dimension-reduction method for the simulation of representative samples of track irregularities was developed, and thus the number of random variables in the system was reduced to four. Finally, an efficient method for the calculation of the dynamic reliability of a maglev train–guideway coupled system was proposed using the probability density evolution method-based equivalent extreme value principle. With numerical examples, the accuracy of the maglev train–guideway interaction model was verified by comparing it with field measurement data from the Shanghai high-speed maglev line. The accuracy of the proposed dynamic reliability analysis method was confirmed by comparing three types of results, that is, the mean value time-history curve, the probability density function, and the cumulative distribution function of extreme values, obtained by the Monte Carlo method. Finally, the dynamic reliability of the running safety and stability of the maglev vehicle at a speed of 430 km/h and the variation laws of the dynamic reliabilities with train speed were examined in detail. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of detecting cancer using such biomarkers as blood protein markers. The purpose of this research is to propose an approach for making decisions in the diagnosis of cancer through the creation of cost-sensitive SVM classifiers on the basis of datasets with a variety of features of different nature. Such datasets may include compositions of known features corresponding to blood protein markers and new features constructed using methods for calculating entropy and fractal dimensions, as well as using the UMAP algorithm. Based on these datasets, multiclass SVM classifiers were developed. They use cost-sensitive learning principles to overcome the class imbalance problem, which is typical for medical datasets. When implementing the UMAP algorithm, various variants of the loss function were considered. This was performed in order to select those that provide the formation of such new features that ultimately allow us to develop the best cost-sensitive SVM classifiers in terms of maximizing the mean value of the metric M a c r o F 1 − s c o r e . The experimental results proved the possibility of applying the UMAP algorithm, approximate entropy and, in addition, Higuchi and Katz fractal dimensions to construct new features using blood protein markers. It turned out that when working with the UMAP algorithm, the most promising is the application of a loss function on the basis of fuzzy cross-entropy, and the least promising is the application of a loss function on the basis of intuitionistic fuzzy cross-entropy. Augmentation of the original dataset with either features on the basis of the UMAP algorithm, features on the basis of the UMAP algorithm and approximate entropy, or features on the basis of approximate entropy provided the creation of the three best cost-sensitive SVM classifiers with mean values of the metric M a c r o F 1 − s c o r e increased by 5.359%, 5.245% and 4.675%, respectively, compared to the mean values of this metric in the case when only the original dataset was utilized for creating the base SVM classifier (without performing any manipulations to overcome the class imbalance problem, and also without introducing new features). [ABSTRACT FROM AUTHOR]

In this paper, we establish a mean value result of arithmetic functions over shorter intervals with the Selberg–Delange method using the Hooley–Huxley contour. [ABSTRACT FROM AUTHOR]

We give characterizations of the Dunkl polyharmonic functions, i.e., solutions to the iteration of the Dunkl-Laplace operator Δ κ which is a differential-reflection operator associated with a Coxeter–Weil group W generated by a finite set of reflections and an invariant multiplicity function κ , in terms of integral means over Euclidean balls and spheres. [ABSTRACT FROM AUTHOR]

This paper mainly studied the distribution of values of Hardy sums involving Chebyshev polynomials. By using the method of analysis and the arithmetic properties of Hardy sums and Chebyshev polynomials of the first kind, we obtained a sharp asymptotic formula for the hybrid mean value of Hardy sums S5(h, q) involving Chebyshev polynomials of the first kind. In addition, we also gave the value of Hardy sums S (h, q) and S 3(h, q) involving Chebyshev polynomials. Finally, we found the reciprocal formulas of S3(h, q) and S 4(h, q) involving Chebyshev polynomials of the first kind. [ABSTRACT FROM AUTHOR]

Quality‐related fault detection and diagnosis are crucial in the data‐driven process monitoring field. Most existing methods are based on principal component analysis (PCA) or partial least squares (PLS), which will miss high‐order statistical information when the industrial process does not satisfy a Gaussian distribution. Meanwhile, the traditional contribution plot is difficult to directly apply to nonlinear processes in some cases due to its limitation of convergence. As such, a modified kernel independent component regression (MKICR) model, which considers high‐order statistical information, is proposed for quality‐related fault detection and faulty variable identification. First, the relationship between the independent components and quality variables is established by kernel independent component regression, and the correlation matrix is obtained. Then, the kernel independent components can be suitably divided into quality‐related and quality‐unrelated parts. Finally, an analysis of the contribution of each variable to the statistics based on Lagrange's mean value theorem is presented. In addition, a numerical case and the Tennessee Eastman process (TEP) demonstrate the efficacy and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, a new observer design technique is proposed for the general affine class of fractional-order nonlinear multi-input/multi-output (MIMO) systems. This study is presented based on the differential mean value theorem. One significant characteristic of the suggested approach is that the nonlinear dynamic of observer error is converted into a linear parameter-varying system. Stability and convergence of observation error are shown using the Lyapunov direct method leading to feasibility and existence of a solution for some linear matrix inequalities. The performance and efficacy of the proposed method are evaluated through some illustrated simulations. [ABSTRACT FROM AUTHOR]

Let |$k_i\ (i=1,2,\ldots ,t)$| be natural numbers with |$k_1>k_2>\cdots >k_t>0$|⁠ , |$k_1\geq 2$| and |$t