On the Sum of a Prime and the k-th Power of a Prime

In this paper, we investigate HUA’s Theorem for short intervals under GRH. Let E k(x)=#{{n≤x;2|n,k is odd, n≠p 1+p k 2}∪{n≤x;2|n,2|k,(p-1)|k, n1(modp),n≠p 1+p k 2}}. Assume GRH. For any k≥2, any A】0 and any 0【ε【14,E k(x+H)-E k(x)≤H(log x) -Aholds for x 12-14k+ε≤H≤x, here the implies constant depends at most on A and ε.

One Additive Diophantine Inequality with Mixed Powers 2 and 4

In this paper, it is shown that： if λ1 ,……λs axe nonzero real numbers, not all of the same sign, such that A1/A2 is irrational, then for any real number η and ε 〉 0 the inequality ｜λ1x1^2 ＋ λ2x2^2＋ λ3x3^4＋ λsx3^4＋……λsx8^4 ＋η〈 ε has infinitely many solutions in positive integers x1,... ,xs.

Energy-Efficient Deployment of Water Quality Sensor Networks

Water quality sensor networks are promising tools for the exploration of oceans.Some key areas need to be monitored effectively.Water quality sensors are deployed randomly or uniformly,however,and understanding how to deploy sensor nodes reasonably and realize effective monitoring of key areas on the basis of monitoring the whole area is an urgent problem to be solved.Additionally,energy is limited in water quality sensor networks.When moving sensor nodes,we should extend the life cycle of the sensor networks as much as possible.In this study,sensor nodes in non-key monitored areas are moved to key areas.First,we used the concentric circle method to determine the mobile sensor nodes and the target locations.Then,we determined the relationship between the mobile sensor nodes and the target locations according to the energy matrix.Finally,we calculated the shortest moving path according to the Floyd algorithm,which realizes the redeployment of the key monitored area.The simulation results showed that,compared with the method of direct movement,the proposed method can effectively reduce the energy consumption and save the network adjustment time based on the effective coverage of key areas.

On the Sum of Two Piatetski-Shapiro Primes

In this paper, we study the binary Goldbach problem in the set of the Piatetski-Shapiro primes. We obtain that for all most all large even integer n, the equationn = p1+p2, pi∈Pγi, i = 1, 2has solutions, where 0 < γ1,γ2 ≤ 1 are fixed real numbers, such that 73(1 - γ2) < 9, 73(1 - γ1) + 43(1 - γ2) < 9.

Screening of Cellulase Producing Fungus ZDTJ097-1-(7) from Bohai Bay and Its Yielding Conditions

Abstract [ Objective ] This study aimed to screen a marine-derived cellulase producing filamentous fungus and to investigate the yielding conditions and celluase activity. [ Method] A total of 25 marine-derived filamentous fungus were screened out of from Bohai Bay, and one producing cellulase was obtained through Congo red transparent circle method and cellulase activity assay. After that, optimum conditions were examined for cellulase producing and catalyzing. [ Result] CMC-Na could induce ZDTJ097-1-（7 ） strain to produce filter paper cellulase, CMC cellulase and 13-glucosidase. The optimum conditions for ZDTJ097-1-（7） to produce en- zymes were 0.5% CMC-Na, 0. 2% NH4NO3 , pH 6.0 at 25 ℃. [ Conclusion] This study provided reference for furfller development and utilization of cellulase produ-cing fungus ZDTJ097-1-（7）.

On Sums of a Prime, and a Square of Prime, and a κ-power of Prime

In 1923, Hardy and Littlewood[1] conjectured that each integer n can be written asp+m12+ m22 = n,and Linnik[2,3] proved that this conjecture is true. But if these mi with i = 1,2 are restricted to primes Pi, the corresponding result is out of reach at present. We consider the following Diophantine equation

Analytical solution for a circular roadway considering the transient effect of excavation unloading

The rocks surrounding a roadway exhibit some special and complex phenomena with increasing depth of excavation in underground engineering.Quasi-static analysis cannot adequately explain these engineering problems.The computational model of a circular roadway considering the transient effect of excavation unloading is established for these problems.The time factor makes the solution of the problem difficult.Thus,the computational model is divided into a dynamic model and a static model.The Laplace integral transform and inverse transform are performed to solve the dynamic model and elasticity theory is used to analyze the static model.The results from an example show that circumferential stress increases and radial stress decreases with time.The stress difference becomes large gradually in this progress.The displacement increases with unloading time and decreases with the radial depth of surrounding rocks.It can be seen that the development trend of unloading and displacement is similar by comparing their rates.Finally,the results of ANSYS are used to verify the analytical solution.The contrast indicates that the laws of the two methods are basically in agreement.Thus,the analysis can provide a reference for further study.

On infinite arithmetic progressions in sumsets

Let k be a positive integer.Denote by D_(1/k)the least integer d such that for every set A of nonnegative integers with the lower density 1/k,the set(k+1)A contains an infinite arithmetic progression with difference at most d,where(k+1)A is the set of all sums of k+1 elements(not necessarily distinct)of A.Chen and Li(2019)conjectured that D_(1/k)=k~2+o(k~2).The purpose of this paper is to confirm the above conjecture.We also prove that D_(1/k)is a prime for all sufficiently large integers k.

On the saturation number for singular cubic surfaces

We investigate the distribution of rational points on singular cubic surfaces,whose coordinates have few prime factors.The key tools used are universal torors,the circle method and results on linear equations in primes.

HUA’S THEOREM WITH FIVE ALMOSTE QUAL PRIME VARIABLES

It is proved that each su?ciently large integer N ≡ 5 (mod24) can be written as N = p21+p22+p23+p24+p25 with |pj? N/5| ≤ U = N 12? 135+ε, where pj are primes. This result, which is obtained by an iterative method and a hybrid estimate for Dirichlet polynomial, improves the previous results in this direction.

The Exceptional Set in Hua's Theorem for Three Squares of Primes

It is proved that with at most O(N^(11/12+ε)) exceptions, all positiveintegers n ≤ N satisfying some necessary congruence conditions are the sum of three squares ofprimes. This improves substantially the previous results in this direction.

Exponential sums over primes in short intervals

In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12+p22+p32+p42+p52, with |pj-(N/5)^(1/2)|≤U = N1/2-1/20+ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.

The Ternary Goldbach Problem with Primes in Arithmetic Progressions

In this paper, we give an explicit numerical upper bound for the moduli of arithmetic progressions, in which the ternary Goldbach problem is solvable. Our result implies a quantitative upper bound for the Linnik constant.

Hua's Theorem on Prime Squares in Short Intervals

It is proved that every large integer N≡5（mod24）can be written as N=p12+…+p52 with each prime pj satisfying |pJ-（N/5|）1/2≤N11/23.This gives a short interval version of Hua’s theorem on the quadratic Waring-Goldbach problem

Values of binary linear forms at prime arguments

We investigate the exceptional set of real numbers not close to some value of a given binary linear form at prime arguments. Let λ1 and λ2 be positive real numbers such that λ1/λ2 is irrational and algebraic. For any （C, c） well-spaced sequence v and δ 〉 0, let E（v, X,δ ） denote the number of ν∈v with v ≤ X for which the inequality ｜λ1P1 ＋ λ2P2 - v｜ 〈 v-δ has no solution in primes P1,P2. It is shown that for any ε 〉 0, we have E（V, X, δ） 〈〈 max（X3/5＋2δ＋ε, X1/3＋4/3δ＋ε）.

Prime in Quadratic Progressions on Average

In this paper we study the distribution of primes in quadratic progressions on average. Our result improves a previous result of Baier and Zhao in some aspects.

Small prime solutions to cubic equations

Let a_1,..., a_9 be nonzero integers not of the same sign, and let b be an integer. Suppose that a_1,..., a_9 are pairwise coprime and a_1 + + a_9 ≡ b(mod 2). We apply the p-adic method of Davenport to find an explicit P = P(a_1,..., a_9, n) such that the cubic equation a_1p_1~3+ + a9p_9~3= b is solvable with p_j 《 P for all 1 ≤ j ≤ 9. It is proved that one can take P = max{|a_1|,..., |a_9|}~c+ |b|^(1/3) with c = 2. This improves upon the earlier result with c = 14 due to Liu(2013).

Quadratic forms connected with Fourier coefficients of Maass cusp forms

For the normalized Fourier coefficients of Maass cusp forms λ（n） and the normalized Fourier coefficients of holomorphic cusp forms a（n）, we give the bound of

A Hybrid of Theorems of Goldbach and Piatetski-Shapiro

It is proved that for almost all sufficiently large even integers n, the prime variable equation n = p1 ＋ p2, p1 ∈ Pγ is solvable, with 13/15 〈γ≤ 1, where Pγ = {p ｜p = [m^1/γ], for integer m and prime p} is the set of the Piatetski-Shapiro primes.