The trade-in program is a common marketing strategy adopted by companies to encourage customers to return their used products in exchange for a discount or rebate on the purchase of new items. A trade-in program can significantly boost sales by lowering customers’ purchasing costs while enabling the collection of returned products for recycling or remanufacturing. However, it also introduces additional operational and financial costs. Therefore, the company must carefully balance profit gains with associated costs to ensure that the trade-in program remains economically viable. This paper explores the company’s optimal non-exclusive trade-in program, under which both existing customers and competitors’ customers can trade their on-hand products for new ones. The innovation level of new generation products, the previous pricing strategy of the company and its competitor, the durability of the products, and the relative perceived value of the old generation products by customers are considered in the paper. The results show that the optimal trade-in strategy design depends on the previous pricing strategy, and setting high trade-in rebates is not always optimal to maximize profit. Moreover, whether the company should provide a higher trade-in rebate to its existing customers depends on the customers’ perceived value of the company’s past products compared to competitors’ offerings, along with both parties’ past pricing strategies, and the trade-in program does not always benefit the environment.

We propose a general approach to construct cryptographic significant Boolean functions of (r + 1)m variables based on the additive decomposition F2rm× F2mof the finite field F2(r+1)m, where r ≥ 1 is odd and m ≥ 3. A class of unbalanced functions is constructed first via this approach, which coincides with a variant of the unbalanced class of generalized Tu-Deng functions in the case r = 1. Functions belonging to this class have high algebraic degree, but their algebraic immunity does not exceed m, which is impossible to be optimal when r > 1. By modifying these unbalanced functions, we obtain a class of balanced functions which have optimal algebraic degree and high nonlinearity (shown by a lower bound we prove). These functions have optimal algebraic immunity provided a combinatorial conjecture on binary strings which generalizes the Tu-Deng conjecture is true. Computer investigations show that, at least for small values of number of variables, functions from this class also behave well against fast algebraic attacks.

In this paper, the authors study binomial skew polynomial ring A. We can find a Gröbner basis of a two-sided ideal (as left ideal). Thus, we solve the question 2 in [2]: Is it true that A has a decidable membership problem for finitely generated two-sided ideals?

Let K[x1,…,xn]be a polynomial ring over a field K in variables x1,…,xn, and K[y1,…,ym] be a polynomial ring over a field K in variables y1,…,ym. m n n. Let T = (t1,…,tn) be an ordered n-tuple of non-constant polynomials in K[y1,…,ym]. For any finite set F of K[x1,…,xn], let F o T be the set obtained from F by replacing xi; by ti, thus for any FeK[x1,…,xn], FoT eK[y1,…,ym]. With the above notations, Hong's main theorem [9] and the main theorem of [6] are generalized to general cases with some new proofs.

In this paper, we discuss Wu's well ordering principle and theorem proving over finite fields, try to prove some theorems in the geometry over finite fields.