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In this paper, we derived that a sufficiently large integer $ N $ can always be represented as the sum of three generalized D. H. Lehmer numbers. As a consequence, we deduced Lu and Yi's original result (Monatsh. Math., 159 (2010), 45–58).

Summary: Primary Sjögren’s syndrome (pSS) is a complex autoimmune disease characterized by lymphocytic infiltration and exocrine dysfunction, particularly affecting the salivary gland (SG). We employed single-cell RNA sequencing to investigate cellular heterogeneity in 11 patients with pSS and 5 non-SS controls. Notably, patients with pSS exhibited downregulated SOX9 in myoepithelial cells, potentially associated with impaired epithelial regeneration. An expanded ACKR1+ endothelial subpopulation in patients with pSS suggested a role in facilitating lymphocyte transendothelial migration. Our analysis of immune cells revealed expanded IGHD+ naive B cells in peripheral blood from patients with pSS. Pseudotime trajectory analysis outlined a bifurcated differentiation pathway for peripheral B cells, enriching three subtypes (VPREB3+ B, BANK1+ B, CD83+ B cells) within SGs in patients with pSS. Fibroblasts emerged as pivotal components in a stromal-immune interaction network, potentially driving extracellular matrix disruption, epithelial regeneration impairment, and inflammation. Our study illuminates immune and stromal cell heterogeneity in patients with pSS, offering insights into therapeutic strategies.

Denote by $ \chi $ a Dirichlet character modulo $ q\geq 3 $, and $ \overline{a} $ means $ a\cdot\overline{a} \equiv 1 \bmod q $. In this paper, we study Dirichlet characters of the rational polynomials in the form $ \sum\limits^{q}_{a = 1}'\chi(ma+\overline{a}), $ where $ \sum\limits_{a = 1}^{q}' $ denotes the summation over all $ 1\le a\le q $ with $ (a, q) = 1 $. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity [6] under a more relaxed situation. We also derive some new identities for the fourth power mean of it by adding some new ingredients.

The main purpose of this paper is to obtain some exact computational formulas or upper bounds for hybrid mean value involving Hardy sums $S_{3}(h, p)$ and general Kloosterman sums $K(r, l, \lambda; p)$. By applying the properties of Gauss sums and the mean value theorems of Dirichlet $L$-function, we derive some new identities. As the special cases, we also deduce some exact computational formulas for hybrid mean value involving $S_{3}(h, p)$ and classical Kloosterman sums $K(n, p)$.

In this paper, the issue of adaptive neural event-triggered control (ETC) is studied for uncertain block-structure multi-input multi-output (MIMO) constrained non-affine nonlinear systems with unmodeled dynamics. A dynamic signal produced by the auxiliary system based on the property of unmodeled dynamics is employed to solve the dynamical disturbances. The unknown continuous function obtained at each step of recursion is estimated by using radial basis function neural networks (RBFNNs). Utilizing logarithmic function as an invertible mapping, the uncertain constrained MIMO non-affine system is changed into a novel unconstrained block-structure MIMO nonaffine system. Using improved dynamic surface control (DSC) strategy, adaptive event-triggered control scheme is developed for the transformed non-affine system based on relative threshold mechanism. According to the Lyapunov method, all the signals in the closed-loop system are shown to be semi-globally uniformly ultimately bounded (SGUUB). Output constraint requirements are not triggered, and Zeno behavior is avoided. A constrained pure-feedback system and a kind of 2-DOF flexible manipulator system are used to illustrate the theoretical findings.

In this paper, finite time adaptive neural network (NN) tracking control is investigated for a class of uncertain nonstrict feedback systems with unknown dead-zone and unmodeled dynamics as well as full state restrictions. Using the linearized representation of dead-zone, finite-time adaptive control is presented by the aid of dynamic surface control (DSC) technique and the property of Gaussian function. By introducing logarithmic function as invertible nonlinear mapping, the full state restriction problem is solved. The dynamic signal generated by the low-pass filter is used to handle unmodeled dynamics. Through finite-time stability theory and introducing the bounded closed set in the proof, all the signals in the closed-loop system are proved to be semi-globally practical finite time stable (SGPFS). The restriction conditions are not triggered for all the states. Finally, two numerical examples are used to illustrate the feasiability of the finite time adaptive control strategy.

An approach to quantify the lifetime and reliability of dual-mode ion thruster is presented based on coupling analyses of thruster failure modes, throttle levels, life models, input uncertainties, and throttle profiles. This approach involves lifetime characterization based on each failure mode, a conservative life margin assessment followed by a failure mode reduction and a quantification of life margins and uncertainties which gives a further reduction of the failure modes, Weibull fitting of the sampled lifetimes based on each of the remained failure modes to obtain their reliability functions, and derivation of a combined reliability function according to the throttle profile. The results for LIPS-200E completing a 13 kh mission show that the lifetime distribution and mission reliability strongly depend on the throttle profile.

An approach to quantify the lifetime and reliability of dual-mode ion thruster is presented based on coupling analyses of thruster failure modes, throttle levels, life models, input uncertainties, and throttle profiles. This approach involves lifetime characterization based on each failure mode, a conservative life margin assessment followed by a failure mode reduction and a quantification of life margins and uncertainties which gives a further reduction of the failure modes, Weibull fitting of the sampled lifetimes based on each of the remained failure modes to obtain their reliability functions, and derivation of a combined reliability function according to the throttle profile. The results for LIPS-200E completing a 13 kh mission show that the lifetime distribution and mission reliability strongly depend on the throttle profile.

We performed molecular dynamics simulations of the high voltage pulse explosion of single aluminum wires with the energy ratio of 0.6 in vacuum and studied the role of wire radial dimension. Simulation results show that large-diameter wires having a large material depth and a small specific surface can maintain a higher deposition energy density and effectively reduce the influence of the radial difference in thermodynamic parameters, leading to higher explosion velocity and a lower vaporization rate in the large-diameter wire. The most significant effect is that the larger diameter wire has a longer explosion development time. In addition, the propagation and reflection of the rarefaction waves in the wire result in two explosion regimes: the spinodal decomposition propagating inward from the surface and the cavitation boiling from the center to the surface. Increasing the diameter will increase the domination range of the spinodal decomposition mechanism.

Bombardment of the keeper electrode by high-energy ions limits the lifetime of the ion thruster. To investigate its sputtering mechanism, this paper established a two-dimensional simulation model and applied the Particle-In-Cell method to simulating the generation process of the high-energy ions and their sputtering process on the keeper electrode surface. The Monte Carlo Collision method was used to address the collisions between particles. The results showed that the plasma potential downstream of the keeper electrode oscillated; therefore, ions in this area were accelerated and their energy was increased from 0.085 to 35 eV, which was enough to erode the keeper electrode’s surface. double charged xenon (Xe++) ions and charge exchange xenon (CEX) ions were the two kinds of high-energy ions. The most severe position was the center of the keeper electrode surface, and its erosion rate was about 2.0 × 10−4 m/1000 h, while the average erosion rate was about 10−5 m/1000 h when the frequency and the amplitude of the plasma potential were 50 Hz and 35 eV, respectively. Plasma potential oscillation greatly affected the sputtering mechanism of the keeper electrode.

This study aimed at discussing the laws of the design parameters of a radio frequency (RF) ion thruster that influence the thruster performance, guide the performance optimization-oriented design of the thruster, and realize the high-accuracy continuous adjustment of the thrust performance. The key influencing parameters of the thrust performance were analyzed by the numerical simulation method. The influencing laws of RF parameters on key plasma parameters, as well as the thermal characteristics of the thruster under the rated parameters, were explored. Moreover, a LRIT-30 RF ion thruster was developed and subjected to the performance adjustment test. The research results demonstrated that the simulation model can describe the plasma parameter distribution of the discharge chamber and the thermal distribution of key components. The 3 cm RF ion thruster performs well under 2 MHz operational frequency. The RF power and gas flow rate are the key influencing factors of beam extraction and are appropriate for the accurate adjustment of parameters. The wide-range adjustment of thrust (0.5–2.3 mN) and specific impulse (869–2564 s) can be realized when the screen-grid voltage, decelerating screen voltage, RF power, and gas flow rate are 1500 V, −200 V, 40–65 W, and 0.4–0.8 SCCM, respectively. The performance indices of the proposed RF ion thruster are close to the international advanced level.

Abstract Let q ≥ 2 $q\geq2$ be a fixed integer, A = A ( q ) ≤ q $A=A(q)\leq q$ , B = B ( q ) ≤ q $B=B(q)\leq q$ , and H = H ( q ) ≤ q $H=H(q)\leq q$ . Define ħ ( A , B , H ) = { a ∈ Z ∣ ( a , q ) = 1 , a b ≡ 1 ( mod q ) , 1 ≤ a ≤ A , 1 ≤ b ≤ B , | a − b | ≤ H } . $$\hbar(A,B,H)= \bigl\{ a\in\mathbb{Z}\mid (a,q)=1, ab\equiv1\pmod{q}, 1\leq a\leq A, 1\leq b\leq B, \vert a-b\vert \leq H \bigr\} . $$ With the aid of the estimates for the general Kloosterman sums and the properties of trigonometric sums, we obtain an upper bound of the general partial Gaussian sums over the number set ħ ( A , B , H ) $\hbar(A,B,H)$ .

Abstract Let q > 2 $q>2$ be an integer, n ⩾ 2 $n\geqslant2$ be a fixed integer with ( n , q ) = 1 $(n,q)=1$ , ψ be a non-principal Dirichlet character modq. An upper bound estimate for character sums of the form ∑ a ∈ C ( 1 , q ) ψ ( a ) $$\sum_{a\in\textit{C}(1,q)}\psi(a) $$ is given, where C ( 1 , q ) = { a ∣ 1 ⩽ a ⩽ q − 1 , a a ‾ ≡ 1 ( mod q ) , n ∤ ( a + a ‾ ) } $\mathcal{C}(1,q)=\{a \mid 1\leqslant a\leqslant q-1, a\overline{a}\equiv1 (\bmod q), n\nmid(a+\overline{a})\}$ .

Using the properties of Gauss sums, the orthogonality relation of character sum and the mean value of Dirichlet L-function, we obtain some exact computational formulas for the hybrid mean value involving general Kloosterman sums K ( r , l , λ ; p ) and certain Hardy sums S 1 ( h , q ) ∑ m = 1 p − 1 ∑ s = 1 p − 1 K ( m , n , λ ; p ) K ( s , t , λ ; p ) S 1 ( 2 m s ¯ , p ) , ∑ m = 1 p − 1 ∑ s = 1 p − 1 | K ( m , n , λ ; p ) | 2 | K ( s , t , λ ; p ) | 2 S 1 ( 2 m s ¯ , p ) . Our results not only cover the previous results, but also contain something quite new. Actually the previous authors just consider the case of the principal character λ modulo p, while we consider all the cases.

In this article, the disturbance estimation and rejection problem are discussed for the yaw channel dynamics of small-scale unmanned helicopter subject to mismatched disturbances. Following Takagi–Sugeno fuzzy description for unknown nonlinear disturbances, a composite control input is designed to attenuate the influence of mismatched disturbances in system output channel by combining state feedback control law, disturbance observer, and well-designed disturbance compensator. The satisfactory control performances can be verified using linear inequality optimization algorithm. Simulation results on a yaw channel dynamical system can reflect the feasibility and effectiveness of the proposed disturbance rejection approach.

The high-dimensional D. H. Lehmer problem over quarter intervals is studied. By using the properties of character sum and the estimates of Dirichlet L-function, the previous result is improved to be the best possible in the case of q = p, an odd prime with p≡1(mod 4), which is shown by studying the mean square value of the error term.