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The field containing exactly q elements is referred to as q , where q is a power of a prime. The class of permutations polynomials (PP) with the formula f(x) = b(x q +ax+δ) i +c(x q +ax+δ) j +ux q +vx and its compositional inverse over F q is examined in this work, where b, c ∈ q , a, δ, u and v ∈ q 2 with a 1+q = 1, (av q + u)(au − v) ≠ 0 and u q + v = a(u + av q ).

The field containing exactly q elements is referred to as q, where q is a power of a prime. The class of permutations polynomials (PP) with the formula f(x) = b(xq+ax+δ)i+c(xq+ax+δ)j+uxq+vx and its compositional inverse over Fq is examined in this work, where b, c ∈ q, a, δ, u and v ∈ q2 with a1+q = 1, (avq + u)(au − v) ≠ 0 and uq + v = a(u + avq).

Recently, P. Yuan presented a local method to find permutation polynomials and their compositional inverses over finite fields. The work of P. Yuan inspires us to construct some classes of permutation polynomials and their compositional inverses by the local method.

Recently, P. Yuan presented a local method to find permutation polynomials and their compositional inverses over finite fields. The work of P. Yuan inspires us to compute the compositional inverses of three classes of the permutation polynomials: (a) the permutation polynomials of the form \\(ax^q+bx+(x^q-x)^k\\) over \\(\\mathbb{F}_{q^2},\\) where \\(a+b \\in \\mathbb{F}_q^*\\) or \\(a^q=b;\\) (b) the permutation polynomials of the forms \\(f(x)=-x+x^{(q^2+1)/2}+x^{(q^3+q)/2} \\) and \\(f(x)+x\\) over \\(\\mathbb{F}_{q^3};\\) (c) the permutation polynomial of the form \\(A^{m}(x)+L(x)\\) over \\(\\mathbb{F}_{q^n},\\) where \\({\\rm Im}(A(x))\\) is a vector space with dimension \\(1\\) over \\(\\mathbb{F}_{q}\\) and \\(L(x)\\) is not a linearized permutation polynomial.

In this paper, we present the compositional inverses of several classes permutation polynomials of the form \\(\\sum_{i=1}^kb_i\\left({\\rm Tr}_m^{mn}(x)^{t_i}+\\delta\\right)^{s_i}+f_1(x)\\), where \\(1\\leq i \\leq k,\\) \\(s_i\\) are positive integers, \\(b_i \\in \\mathbb{F}_{p^m},\\) \\(p\\) is a prime and \\(f_1(x)\\) is a polynomial over \\(\\mathbb{F}_{p^{mn}}\\) satisfying the following conditions: (i) \\({\\rm Tr}_m^{mn}(x) \\circ f_1(x)=\\varphi(x) \\circ {\\rm Tr}_m^{mn}(x),\\) where \\(\\varphi(x)\\) is a polynomial over \\(\\mathbb{F}_{p^m};\\) (ii) For any \\(a \\in \\mathbb{F}_{p^m},\\) \\(f_1(x)\\) is injective on \\({\\rm Tr}_m^{mn}(a)^{-1}.\\)

In this paper, we further investigate the local criterion and present a class of permutation polynomials and their compositional inverses over \\( \\mathbb{F}_{q^2}\\). Additionally, we demonstrate that linearized polynomial over \\(\\mathbb{F}_{q^n}\\) is a local permutation polynomial with respect to all linear transformations from \\(\\mathbb{F}_{q^n}\\) to \\(\\mathbb{F}_q ,\\) and that every permutation polynomial is a local permutation polynomial with respect to certain mappings.

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over \\(\\mathbb{F}_{2^m}\\) in Zieve's paper. We prove a conjecture proposed by Gupta and Sharma and obtain some new permutation trinomials over \\(\\mathbb{F}_{2^m}\\). Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials.

In this paper, we present the compositional inverses of several classes permutation polynomials of the form \\(\\sum_{i=1}^kb_i(x^{p^m}+x+\\delta)^{s_i}-x\\) over \\(\\mathbb{F}_{p^{2m}}\\), where for \\(1\\leq i \\leq k,\\) \\(s_i, m\\) are positive integers, \\(b_i, \\delta \\in \\mathbb{F}_{p^{2m}},\\) and \\(p\\) is prime.

R. Gupta, P. Gahlyan and R.K. Sharma presented three classes of permutation trinomials over \\(\\mathbb{F}_{q^3}\\) in Finite Fields and Their Applications. In this paper, we employ the local method to prove that those polynomials are indeed permutation polynomials and provide their compositional inverses.

A MIL-53(Fe)/g-C3N4 heterogeneous composite was synthesized and applied in photocatalytic oxidation of 5-hydroxymethylfurfural (5-HMF) to 2,5-diformylfuran (DFF). The systematic investigation indicated that the introduction of MIL-53(Fe) into g-C3N4 increased the specific surface area, broadened the visible-light response region, and promoted the separation efficiency of the photo-generated electron-hole pairs. The 10% MIL-53(Fe)/g-C3N4 heterogeneous composite achieved the best photocatalytic oxidation activity with 74.5% of 5-HMF conversion under simulated sunlight, which was much higher than that of pristine g-C3N4 and MIL-53(Fe). The MIL-53(Fe)/g-C3N4 composite displayed good photocatalytic reusability and stability. Based on the characterization results and photocatalytic performance, a Z-scheme photocatalytic mechanism of the MIL-53(Fe)/g-C3N4 composite was suggested, and a possible reaction route was deduced.