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Register a new userA characterization of arithmetic functions satisfying $f(u^{2}+kv^{2})=f^{2}(u)+kf^{2}(v)$
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In this paper, we mainly discuss the characterization of a class of arithmetic functions $f: N rightarrow C$ such that $f(u^{2}+kv^2)=f^{2}(u)+kf^{2}(v)$ $(k, u, v in N)$. We obtain a characterization with given condition, propose a conjecture and show the result holds for $k in {2, 3, 4, 5 }$.
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We compute the one-loop divergences in a theory of gravity with Lagrangian of the general form $f(R,R_{mu u}R^{mu u})$, on an Einstein background. We also establish that the one-loop effective action is invariant under a duality that consists of changing certain parameters in the relation between the metric and the quantum fluctuation field. Finally, we discuss the unimodular version of such a theory and establish its equivalence at one-loop order with the general case.
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=F_{q}+vF_{q}+v^{2}F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over $R$. Further, we give bounds on the minimum distance of LCD codes over $F_q$ and extend these to codes over $R$.
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, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16--23]. 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 $d=2, 3, 4, 6$ [Finite Fields Appl. 35 (2015) 215--230]. In this paper we concentrate our efforts on the PPs of more general form [ f(x)=(ax^{q} +bx +c)^r phi((ax^{q} +bx +c)^{(q^2 -1)/d}) +ux^{q} +vx~~text{over $mathbb{F}_{q^2}$}, ] 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.
The aim of this paper is to classify the finite nonsolvable groups in which every irreducible character of even degree vanishes on at most two conjugacy classes. As a corollary, it is shown that $L_2(2^f)$ are the only nonsolvable groups in which every irreducible character of even degree vanishes on just one conjugacy class.
We re-analyze the production of seed magnetic fields during Inflation in (R/m^2)^n F_{mu u}F^{mu u} and I F_{mu u}F^{mu u} models, where n is a positive integer, R the Ricci scalar, m a mass parameter, and I propto eta^alpha a power-law function of the conformal time eta, with alpha a positive real number. If m is the electron mass, the produced fields are uninterestingly small for all n. Taking m as a free parameter we find that, for n geq 2, the produced magnetic fields can be sufficiently strong in order to seed dynamo mechanism and then to explain galactic magnetism. For alpha gtrsim 2, there is always a window in the parameters defining Inflation such that the generated magnetic fields are astrophysically interesting. Moreover, if Inflation is (almost) de Sitter and the produced fields almost scale-invariant (alpha simeq 4), their intensity can be strong enough to directly explain the presence of microgauss galactic magnetic fields.
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