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In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.

In the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions e 0 {e_{0}} and e 1 {e_{1}} . It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.

We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s = 0 s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.

It is well known that SOR iterative methods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices). However, the same can not be true in case of those iterative methods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices). This paper proposes some necessary and sufficient conditions such that SOR iterative methods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.

We consider iterated integrals of $\log\zeta(s)$ on certain vertical and horizontal lines. Here, the function $\zeta(s)$ is the Riemann zeta-function. It is a well known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of $\int_{0}^{t} \log \zeta(1/2 + it')dt'$ under the Riemann Hypothesis. Moreover, we show that, for any $m\geq 2$, the denseness of the values of an $m$-times iterated integral on the critical line is equivalent to the Riemann Hypothesis., Comment: 15 pages

In 1977, Colville, Davis, and Keimel [Positive derivations on f-rings, J. Aust. Math. Soc. Ser. A 23 1977, 3, 371–375] proved that a positive derivation on an Archimedean f-algebra A has its range in the set of nilpotent elements of A. The main objective of this paper is to obtain a generalization of the above Colville, Davis and Keimel result to general derivations. Moreover, we give a new version of the Singer–Wermer conjecture for the class of second-order derivations acting on uniformly complete almost f-algebras.

In this paper we discuss coefficient problems for functions in the class 𝒞 0 ⁢ ( k ) {{\mathcal{C}}_{0}(k)} . This family is a subset of 𝒞 {{\mathcal{C}}} , the class of close-to-convex functions, consisting of functions which are convex in the positive direction of the real axis. Our main aim is to find some bounds of the difference of successive coefficients depending on the fixed second coefficient. Under this assumption we also estimate | a n + 1 | - | a n | {|a_{n+1}|-|a_{n}|} and | a n | {|a_{n}|} . Moreover, it is proved that Re ⁡ { a n } ≥ 0 {\operatorname{Re}\{a_{n}\}\geq 0} for all f ∈ 𝒞 0 ⁢ ( k ) {f\in{\mathcal{C}}_{0}(k)} .

In this paper, we consider the long time behavior of solutions for 3D incompressible MHD equations with fractional Laplacian. Firstly, in a periodic bounded domain, we prove the existence of a global attractor. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and magnetic fields. Finally, in the whole space ℝ 3 {\mathbb{R}^{3}} , we establish the sharp algebraic decay rate of solutions to the generalized MHD system provided that the parameters satisfy α , β ∈ ( 0 , 2 ] {\alpha,\beta\in(0,2]} .

In this paper we define and consider some familiar subsets of analytic functions associated with sine functions in the region of unit disk on the complex plane. For these classes our aim is to find the Hankel determinant of order three.

In this paper, we concern with a 2nth-order discrete system. Using the critical point theory, we establish various sets of sufficient conditions for the existence of periodic solutions with prescribed minimal period. To the best of our knowledge, this is the first time to discuss the periodic solutions with prescribed minimal period for a 2nth-order discrete system.

This paper is devoted to study the global well-posedness of solutions for the Cauchy problem of the fractional Cahn–Hilliard equation in ℝ N {\mathbb{R}^{N}} ( N ∈ ℕ + {N\in\mathbb{N}^{+}} ), provided that the initial datum is sufficiently small. In addition, the L p {L^{p}} -norm ( 1 ≤ p ≤ ∞ {1\leq p\leq\infty} ) temporal decay rate for weak solutions and the higher-order derivative of solutions are also studied.

Let H be a Hilbert space and B(H) the set of all bounded linear operators on H. In the paper we consider the generalized concept of order relations on B(H) which was proposed by Šemrl and which covers the star partial order, the left-star partial order, the right-star partial order, and the minus partial order. We also connect this concept with the sharp partial order.

A homogeneous polynomial of degree $d$ in $n+1$ variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of $d$ and $n$ for which a general polynomial of degree $d$ in $n+1$ variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces., 25 pages. Proof of Proposition 17 and Lemma 18 fixed, some typos corrected

The theory of multivariate neural network operators in a Kantorovich type version is here introduced and studied. The main results concerns the approximation of multivariate data, with respect to the uniform and Lp norms, for continuous and Lp functions, respectively. The above family of operators, are based upon kernels generated by sigmoidal functions. Multivariate approximation by constructive neural network algorithms are useful for applications to neurocomputing processes involving high dimensional data. At the end of the paper, several examples of sigmoidal functions for which the above theory holds have been presented.

In the present paper we prove a separation theorem for t-Wright convex and t-Wright concave functionals. We also consider the problem of supporting the t-Wright concave functions by t-Wright affine ones. As a consequence of these results we prove some decomposition theorems for t-Wright convex functions.

The purpose of this paper is to establish the rate of convergence in terms of the weighted modulus of continuity and Lipschitz type maximal function for the q-Szász-beta operators. We also study the rate of A-statistical convergence. Lastly, we modify these operators using King type approach to obtain better approximation.

In this paper, we first establish some refinements of the weighted generalized trapezoid inequality for functions of bounded variation in terms of a cumulative variation function, and then obtain some perturbed versions of the weighted generalized trapezoid inequality. Several applications for selfadjoint operators on complex Hilbert spaces are also given.

In this paper we prove a stability result for the functional equation 1 L ⁢ ∑ λ ∈ K F ⁢ ( x + λ ⁢ y ) = α ⁢ ( y ) ⁢ G ⁢ ( x ) + H ⁢ ( y ) , x , y ∈ S , \frac{1}{L}\sum_{\lambda\in K}F(x+\lambda y)=\alpha(y)G(x)+H(y),\quad x,y\in S, for the functions F , G , H : S → X {F,G,H\colon S\to X} , α : S → 𝕂 {\alpha\colon S\to\mathbb{K}} , where ( S , + ) {(S,+)} is an abelian semigroup, K is a finite subgroup of the automorphism group of S, L := | K | {L:=|K|} and ( X , ∥ ⋅ ∥ ) {(X,\|\cdot\|)} is the Banach space over a field 𝕂 ∈ { ℝ , ℂ } {\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}} .

Let π be a unitary cuspidal automorphic representation for GL m ⁢ ( 𝔸 ℚ ) {\mathrm{GL}_{m}(\mathbb{A}_{\mathbb{Q}})} , and let L ⁢ ( s , π ) {L(s,\pi)} be the automorphic L-function attached to π, which has a Dirichlet series expression in the half-plane ℜ ⁡ s > 1 {\Re s>1} , i.e. L ⁢ ( s , π ) = ∑ n = 1 ∞ λ π ⁢ ( n ) n s . L(s,\pi)=\sum_{n=1}^{\infty}\frac{\lambda_{\pi}(n)}{n^{s}}. In this paper we are interested in the upper bound of the fourth power moment of λ π ⁢ ( n ) {\lambda_{\pi}(n)} , i.e. ∑ n ≤ x λ π ⁢ ( n ) 4 {\sum_{n\leq x}\lambda_{\pi}(n)^{4}} . If m = 2 {m=2} , we are able to consider the sixteenth power moment of λ π ⁢ ( n ) {\lambda_{\pi}(n)} . As an application, we consider the lower bound of ∑ n ≤ x | λ π ⁢ ( n ) | {\sum_{n\leq x}\lvert\lambda_{\pi}(n)\rvert} , which improves previous results.

In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.

In this paper, we discuss mean values of some Witten L-functions with respect to the “character aspect”. In particular, we focus on the SU ⁢ ( 2 ) ${\mathrm{SU}(2)}$ case. We prove that such mean values turn out to be 0, 1 or to diverge.

In this paper, we give estimates of the L p ⁢ ( ℝ n ) ${L^{p}(\mathbb{R}^{n})}$ ( 1 < p < ∞ ${1 ) and weak type ( 1 , 1 ) ${(1,1)}$ norm of singular integral operator T Ω ${T_{\Omega}}$ with homogeneous kernel defined by T Ω ⁢ f ⁢ ( x ) = p . v . ∫ ℝ n Ω ⁢ ( x - y ) | x - y | n ⁢ f ⁢ ( y ) ⁢ 𝑑 y , $T_{\Omega}f(x)=\mathrm{p.v.}\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n}}% f(y)\,dy,$ where Ω satisfies L 1 ${L^{1}}$ -Dini condition with mean value zero on 𝕊 n - 1 ${\mathbb{S}^{n-1}}$ . More precisely, we show that the following two estimates hold: ∥ T Ω ⁢ f ∥ p ≲ ( ( log ⁡ n ) ⁢ ∥ Ω ∥ 1 + ∫ 0 1 n ω ~ 1 ⁢ ( δ ) δ ⁢ 𝑑 δ ) ⁢ ∥ f ∥ p , $\|T_{\Omega}f\|_{p}\lesssim\Biggl{(}(\log n)\|\Omega\|_{1}+\int_{0}^{\frac{1}{% n}}\frac{\tilde{\omega}_{1}(\delta)}{\delta}\,d\delta\Biggr{)}\|f\|_{p},$ and for any λ > 0 ${\lambda>0}$ , λ ⁢ m ⁢ ( { x ∈ ℝ n : | T Ω ⁢ f ⁢ ( x ) | > λ } ) ≲ ( ( log ⁡ n ) ⁢ ∥ Ω ∥ 1 + ∫ 0 1 n ω ~ 1 ⁢ ( δ ) δ ⁢ 𝑑 δ ) ⁢ ∥ f ∥ 1 , $\lambda m(\{x\in\mathbb{R}^{n}:|T_{\Omega}f(x)|>\lambda\})\lesssim{\Bigg{(}(% \log n)\|\Omega\|_{1}+\int_{0}^{\frac{1}{n}}\frac{\tilde{\omega}_{1}(\delta)}{% \delta}\,d\delta\Biggr{)}}\|f\|_{1},$ where ω ~ 1 ${\tilde{\omega}_{1}}$ denotes the L 1 ${L^{1}}$ integral continuous modulus of Ω defined by translation in ℝ n ${\mathbb{R}^{n}}$ . Moreover, some similar estimates are also established for the Marcinkiewicz integral μ Ω ${\mu_{\Omega}}$ , which was introduced by Stein in [13].

In this paper we obtain the classification of p-nilpotent restricted Lie algebras of dimension at most 4 over a perfect field of positive characteristic p.

The goal of this paper is to show that the equivariant Tamagawa number conjecture implies the extended abelian Stark conjecture contained in [12] and [11]. In particular, this gives the first proof of the extended abelian Stark conjecture for the base field ℚ {\mathbb{Q}} , since the equivariant Tamagawa number conjecture away from 2 was proved in this context by Burns and Greither in [8] and Flach completed their results at 2 in [13] and [14].

Let E $E$ be an elliptic curve over ${\mathbb {Q}}$ without complex multiplication. For each prime p $p$ of good reduction, let | E ( p ) | $|E({\mathbb {F}}_p)|$ be the order of the group of points of the reduced curve over p ${\mathbb {F}}_p$ . According to a conjecture of Koblitz, there should be infinitely many such primes p $p$ such that | E ( p ) | $|E({\mathbb {F}}_p)|$ is prime, unless there are some local obstructions predicted by the conjecture. Suppose that E $E$ is a curve without local obstructions (which is the case for most elliptic curves over ${\mathbb {Q}}$ ). We prove in this paper that, under the GRH, there are at least 2 . 778 C E twin x / ( log x ) 2 $2.778 C_E^{\rm twin} x / (\log x)^2$ primes p $p$ such that | E ( p ) | $|E({\mathbb {F}}_p)|$ has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [20, 21] and Miri & Murty [15]. This is also the first result where the dependence on the conjectural constant C E twin $C_E^{\rm twin}$ appearing in Koblitz's conjecture (also called the twin prime conjecture for elliptic curves) is made explicit. This is achieved by sieving a slightly different sequence than the one of [20] and [15]. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant of Zywina [24] appearing in the upper bound for the number of primes p $p$ such that | E ( p ) | $|E({\mathbb {F}}_p)|$ is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH.