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In this paper, we provide a local method to find compositional inverses of all PPs, some new PPs and their compositional inverses are given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we give a new proof using Newton polygons for an irreducibility criterion established by Singh and Kumar. Then we prove a conjecture of Koley and Reddy. In addition, we give a new proof for the statement that x n ± c x n − 1 ± c with c ⩾ 2 is irreducible unless (n , c) = (2 , 4) , which is a conjecture of Harrington. [ABSTRACT FROM AUTHOR]

An element a of a ring R is called a quasi-idempotent if a 2 = k a for some central unit k of R , or equivalently, a = k e , where k is a central unit and e is an idempotent of R. A ring R is called a quasi-Boolean ring if every element of R is quasi-idempotent. A ring R is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or R has no image isomorphic to ℤ 2 ; For an indecomposable commutative semilocal ring R with at least two maximal ideals, n (R) (n ≥ 2) is strongly quasi-clean if and only if n (R) is quasi-clean if and only if min { | R / m | , m is a maximal ideal of R } > n + 1. For a prime p and a positive integer n ≥ 2 , n (ℤ (p)) is strongly quasi-clean if and only if p > n. Some open questions are also posed. [ABSTRACT FROM AUTHOR]

For a prime p and positive integers m, n, let F q be a finite field with q = p m elements and F q n be an extension of F q. Let h(x) be a polynomial over F q n satisfying the following conditions: (i) Tr m nm (x) ∘ h (x) = τ (x) ∘ Tr m nm (x) ; (ii) For any s ∈ F q , h(x) is injective on Tr m nm (x) - 1 (s) , where τ (x) is a polynomial over F q. For b , c ∈ F q , δ ∈ F q n , and positive integers i, j, d with q ≡ ± 1 (mod d) , we propose a class of permutation polynomials of the form b (Tr m nm (x) + δ) 1 + i (q n - 1) d + c (Tr m nm (x) + δ) 1 + j (q n - 1) d + h (x) over F q n by employing the Akbary–Ghioca–Wang (AGW) criterion in this paper. Accordingly, we also present the permutation polynomials of the form b (Tr m nm (x) + δ) 1 + i (q n - 1) d + h (x) by letting c = 0 and choosing some special i, which covered some known results of this form. [ABSTRACT FROM AUTHOR]

Let q be a prime power and F q be a finite field with q elements. In this paper, we employ the AGW criterion to investigate the permutation behavior of some polynomials of the form b (x q + a x + δ) 1 + i (q 2 - 1) d + c (x q + a x + δ) 1 + j (q 2 - 1) d + L (x) over F q 2 with a 1 + q = 1 , q ≡ ± 1 (mod d) and L (x) = - a x or x q - a x. Accordingly, we also present the permutation polynomials of the form b (x q + a x + δ) s - a x by letting c = 0 and choosing some special exponent s, which generalize some known results on permutation polynomials of this form. [ABSTRACT FROM AUTHOR]

We show that an order m dimension 2 tensor is primitive if and only if its majorization matrix is primitive, and then we obtain the characterization of order m dimension 2 strongly primitive tensors and the bound of the strongly primitive degree. Furthermore, we study the properties of strongly primitive tensors with n≥3 and propose some problems for further research. [ABSTRACT FROM AUTHOR]

Let m≥5 be an odd integer. For d=2m+2(m+1)/2+1 or d=2m+1+3, Blondeau et al. conjectured that the power function Fd=xd over GF(22m) is differentially 8-uniform in which all values 0,2,4,6,8 appear. In this paper, we confirm this conjecture and compute the differential spectrum of Fd for both values of d. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known results about various spectral radii, including the adjacency spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph. [ABSTRACT FROM AUTHOR]

Permutation polynomials (PPs) of the form $$(x^{q} -x + c)^{\frac{q^2 -1}{3}+1} +x$$ over $$\mathbb {F}_{q^2}$$ were presented by Li et al. (Finite Fields Appl 22:16-23, 2013). More recently, we have constructed PPs of the form $$(x^{q} + bx + c)^{\frac{q^2 -1}{d}+1} -bx$$ over $$\mathbb {F}_{q^2}$$ , where $$d=2, 3, 4, 6$$ (Yuan and Zheng in Finite Fields Appl 35:215-230, 2015). In this paper we concentrate our efforts on the PPs of more general form where $$a,b,c,u,v \in \mathbb {F}_{q^2}$$ , $$r \in \mathbb {Z}^{+}$$ , $$\phi (x)\in \mathbb {F}_{q^2}[x]$$ and d is an arbitrary positive divisor of $$q^2-1$$ . The key step is the construction of a commutative diagram with specific properties, which is the basis of the Akbary-Ghioca-Wang (AGW) criterion. By employing the AGW criterion two times, we reduce the problem of determining whether f( x) permutes $$\mathbb {F}_{q^2}$$ to that of verifying whether two more polynomials permute two subsets of $$\mathbb {F}_{q^2}$$ . As a consequence, we find a series of simple conditions for f( x) to be a PP of $$\mathbb {F}_{q^2}$$ . These results unify and generalize some known classes of PPs. [ABSTRACT FROM AUTHOR]

Let q be a prime power. [ABSTRACT FROM AUTHOR]

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection (*-invariant idempotent). Recently, Gao, Chen, and Li obtained necessary and sufficient conditions forRGto be *-clean, whereRis a commutative local ring andGis one ofC3,C4,S3, andQ8. Most recently, Li, Yuan, and Parmenter gave a complete characterization of when the group algebraFCpis *-clean, whereFis a field andCpis a cyclic group of prime orderp. In this article, we extend the above mentioned result fromFCptoFqCpk, whereFqis a finite field andCpkis a cyclic group of an odd prime power orderpk. For the general case whenG = Cnis cyclic group of ordern, we also provide a necessary condition and a few sufficient conditions forFqCnto be *-clean. [ABSTRACT FROM PUBLISHER]

Forn ∈ N, we denote by π(n) the set of prime divisors ofn. LetGbe a finite group. Denote by IBrp(G) and cl(Gp′) the set of irreduciblep-Brauer characters and the set ofp-regular conjugacy classes ofG, respectively. Set ρp(G) = {q prime: q|ϕ(1), ϕ ∈IBrp(G)},, σp(G) = max{|π(ϕ(1))|: ϕ ∈IBrp(G)}, and. For a solvable groupG, we prove that |ρp(G)| andare bounded by σp(G) and, respectively, if a well-known Diophantine equation has finite integer solutions. [ABSTRACT FROM PUBLISHER]

Let be an additively written finite cyclic group of order and let be a minimal zero-sum sequence with elements of , i.e. the sum of elements of is zero, but no proper nontrivial subsequence of has sum zero. is called unsplittable if there do not exist an element in and two elements in such that and the new sequence is still a minimal zero-sum sequence. In this paper, we investigate long unsplittable minimal zero-sum sequences over . Our main result characterizes the structures of all such sequences and shows that the index of is at most 2, provided that the length of is greater than or equal to where is a positive integer with least prime divisor greater than . [ABSTRACT FROM AUTHOR]

In an additively written abelian group, a sequence is called zero-sum free if each of its nonempty subsequences has sum different from the zero element of the group. In this paper, we consider the structure of long zero-sum free sequences and n-zero-sum free sequences over finite cyclic groups $${\mathbb{Z}_n}$$ . Among which, we determine the structure of the long zero-sum free sequences of length between $${n/3+1 }$$ and $${n/2}$$ , where $${n\ge 50}$$ is an odd integer, and we provide a general description on the structure of n-zero-sum free sequences of length n + l, where $${\ell\geq n/p+p-2}$$ and p is the smallest prime dividing n. [ABSTRACT FROM AUTHOR]

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 픽Cp is *-clean, where 픽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 픽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring. [ABSTRACT FROM AUTHOR]

We obtain all positive integer solutions ( m, m, a, b) with a > b, gcd( a, b) = 1 to the system of Diophantine equations $$km_1^2 - la^{t_1 } b^{t_2 } a^{2r} = C_1$$, $$km_2^2 - la^{t_1 } b^{t_2 } b^{2r} = C_2$$ with C, C ∈ {−1, 1,−2, 2, −4, 4}, and k, l, t, t, r ∈ ℤ such that k > 0, l > 0, r > 0, t > 0, t ⩾ 0, gcd( k, l) = 1, and k is square-free. [ABSTRACT FROM AUTHOR]

A natural number $n$ is called $k$-perfect if $\sigma (n)= kn$. In this paper, we show that for any integers $r\geq 2$ and $k\geq 2$, the number of odd $k$-perfect numbers $n$ with $\omega (n)\leq r$ is bounded by $\left({\lfloor {4}^{r} { \mathop{ \log } \nolimits }_{3} 2\rfloor + r\atop r} \right){ \mathop{ \sum } \nolimits }_{i= 1}^{r} \left({\lfloor kr/ 2\rfloor \atop i} \right)$, which is less than ${4}^{{r}^{2} } $ when $r$ is large enough. [ABSTRACT FROM AUTHOR]

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. The monotone catenary degree c ( H) of H is the smallest integer m with the following property: for each $${a \in H}$$ and each two factorizations z, z′ of a with length | z| ≤ | z′|, there exist factorizations z = z, ... , z = z′ of a with increasing lengths-that is, | z| ≤ ... ≤ | z|-such that, for each $${i \in [1,k]}$$ , z arises from z by replacing at most m atoms from z by at most m new atoms. Up to now there was only an abstract finiteness result for c ( H), but the present paper offers the first explicit upper and lower bounds for c ( H) in terms of the group invariants of G. [ABSTRACT FROM AUTHOR]

Let H be a Krull monoid with class group G and suppose that every class contains a prime divisor. The set of distances Δ (H) of H is the set of all d∈ℕ with the following property: there are irreducible elements u1, …, uk, v1, …, vk+d such that u1·…·uk=v1·…·vk+d, but u1·…·uk cannot be written as a product of l irreducible elements for any l with k

Let a, b, c be integers. In this paper, we prove the integer solutions of the equation axy + byz + czx = 0 satisfy max{|x|, |y|, |z|} ≤ 2 max{a, b, c} when a, b, c are odd positive integers, and when a = b = 1, c = -1, the positive integer solutions of the equation satisfy max{x, y, z} < exp(exp(exp(5))). [ABSTRACT FROM AUTHOR]

We obtain all solutions of the equation $\frac{ax^{n+2l}+c}{abt^{2}x^{n}+c} = by^{2}$ with c∈{±1,±2,±4}. [ABSTRACT FROM AUTHOR]

In 1992, Ma made a conjecture related to Pell equations. In this paper, we use Störmer's theorem and related results on Pell equations to prove some particular cases of Ma's conjecture. [ABSTRACT FROM AUTHOR]

Let p be a prime. We assign to each positive number k a digraph G whose set of vertices is {1, 2, ..., p − 1} and there exists a directed edge from a vertex a to a vertex b if a ≡ b (mod p). In this paper we obtain a necessary and sufficient condition for $$G_p^{{k_1}} \simeq G_p^{{k_2}}$$. [ABSTRACT FROM AUTHOR]

Let a ⩾ 1 be an integer. In this paper, we will prove the equation in the title has at most three positive integer solutions. [ABSTRACT FROM AUTHOR]

Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over $$ \mathbb{F}_{3^{3m} } $$ are presented. These polynomials have simple shapes, and they are related to planar functions. [ABSTRACT FROM AUTHOR]

Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel’s method. As an application of this result, it is proved that: if D>0 is not a square, and ε = x 0 denotes the fundamental solution of x 2− Dy 2=−1, then x 2+1= Dy 4 is solvable if and only if y 0= A 2, where A is an integer. Moreover, if ε>64, then x 2+1= Dy 4 has at most one positive integral solution ( x, y). [ABSTRACT FROM AUTHOR]

The main result in the earlier paper (by the first author) is improved as follows. The number of odd multiperfect numbers with at most $r$ distinct prime factors is bounded by ${4}^{{r}^{2} } / {2}^{r+ 2} (r- 1)!$. [ABSTRACT FROM AUTHOR]