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Let [Formula: see text] be any integers with [Formula: see text], and [Formula: see text] be a Dirichlet character modulo [Formula: see text]. The general Kloosterman sums [Formula: see text] are defined as follows: [Formula: see text] where [Formula: see text], and [Formula: see text] denotes the multiplicative inverse of [Formula: see text] modulo [Formula: see text]. By combining elementary methods and analytic methods, along with Montgomery and Vaughan’s clever trick on primitive characters, we derive some new identities for [Formula: see text] under the condition [Formula: see text], where [Formula: see text] denotes the summation over all [Formula: see text] with [Formula: see text]. Previously only the case of [Formula: see text] was studied.

Summary In this article, the issue of adaptive finite‐time dynamic surface control (DSC) is discussed for a class of parameterized nonlinear systems with full state constraints. Using the property ...

In this article, a mathematical derivation is made to develop a nonlinear static model for the thermal post-buckling characteristics of an annular system reinforced with graphene nanoplatelets (GPL...

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 $K(n, p)$.

An approach to quantify the lifetime and mission reliability of multi-mode ion thruster is presented. The reliability of LIPS-300S thruster completing the Asteroid Exploration Mission is assessed. The procedure of this approach is (i) assess the life margin factor (LMF) against each failure mode according to the wear test results, then identify the engineering failure modes whose LMFs are less than 1.5; (ii) quantify the uncertainty in the lifetime based on each engineering failure mode and identify the key failure mode with the minimum LMF; (iii) sample the lifetime based on the key failure mode and obtain its 3-parameter Weibull distribution; (iv) simplify the throttle profile as operations at several representative throttle levels, then obtain the conditional lifetime distribution for each operation based on previous operations; (v) considering all possible failure modes, integrate the probability of completing each operation as the total reliability index. The results show that LIPS-300S can complete the Asteroid Exploration Mission with a reliability of 100%.