Search

The pavements are subjected to the moving vehicle, pore pressure and temperature loadings, the analytical solutions of thermo-hydro-mechanical response for the pavements are paid more attentions by the road engineers while the pavements are maintained. The analytical solutions of the multi-layered pavement model are derived by the novel DVP (duel variable and position) method in this paper. Starting with the basic equations of the pavement system, obeying Biot's dynamic poroelastic theory and generalized thermoelastic theory, the original problem is divided into two sub-systems and solved twice to avoid tangling two different loading frequencies. With the aid of moving coordinate and vector functions, the decomposition expressions of stress, displacement, temperature and pore pressure variations are deduced and propagated by using DVP method, and the transformed solutions are numerically inverted. In the numerical examples, the obtained solutions are firstly verified. Then the difference among the results of several pavement models are compared and analyzed. Finally, the effects of the surface temperature, moving load speed and interface conditions on the displacement, stress, and strain distributions are discussed by the solutions of thermo-hydro-mechanical pavement models in the numerical results.

Let $$1< c < \frac{59}{44},\, c\ne \frac{4}{3}$$ . In this paper it is proved that for any sufficiently large real N, for almost all real $$T\in (N,2N]$$ (in the sense of Lebesgue measure), the inequality $$|p_1^{c} + p_2^{c} - T|< T^{-\frac{9}{10c}(\frac{59}{44}-c)}$$ is solvable in primes $$p_1, p_2$$ . This result constitutes an improvement upon that of Kumchev and Laporta.

Let 1 0 such that for each real numberN>N(c) the inequality\(|p_1^c + p_2^c + p_3^c - N|< N^{ - \tfrac{1}{c}(\tfrac{{11}}{{10}} - c)} \log ^{c_1 } N\) is solvable in prime numbersp 1,p 2,p 3, wherec 1 is some absolute positive constant.

In this paper it is proved that every sufficiently large even integer N satisfying one of the congruence conditions N ≡ 10, 58, 130, or 178 (mod 240) may be represented as the sum of one square and nine fourth powers of prime numbers.

Let [Formula: see text]. In this paper, it is proved that for every sufficiently large real number [Formula: see text], the Diophantine inequality [Formula: see text] is solvable in primes [Formula: see text]. This result constitutes an improvement upon that of Baker and Weingartner.

Let $$[\theta ]$$ denote the integral part of the real number $$\theta $$ . In this paper it is proved that for $$1

Assume that $\psi$ is a function of positive variable $t$, monotonically increasing to infinity and $0 \lt \psi(t) \ll \log t/(\log \log t)$. Let $\mathcal{R}_{3}(n)$ denote the number of representations of the integer $n$ as sums of two squares and three biquadrates of primes and we write $\mathcal{E}_{3}(N)$ for the number of integers $n$ satisfying $n \leq N$, $n \equiv 5, 53, 101 \pmod{120}$ and \[ \left| \mathcal{R}_{3}(n) - \frac{\Gamma^{2}(1/2) \Gamma^{3}(1/4)}{\Gamma(7/4)} \frac{\mathfrak{S}_{3}(n) n^{3/4}}{\log^{5}n} \right| \geq \frac{n^{3/4}}{\psi(n) \log^{5}n}, \] where $0 \lt \mathfrak{S}_{3}(n) \ll 1$ is the singular series. In this paper, we prove \[ \mathcal{E}_{3}(N) \ll N^{23/48+\varepsilon} \psi^{2}(N) \] for any $\varepsilon \gt 0$. This result constitutes a refinement upon that of Friedlander and Wooley [2].

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that for every large odd integer N , and 5 ≤ a ≤ 8 , a ≤ b , 13 60 1 a + 1 b ≤ 1 3 , the equation N = x 2 + p 2 + p 1 3 + p 2 4 + p 3 5 + p 4 a + p 5 b is solvable with x being an almost-prime P r ( a , b ) and the other variables primes, where r ( a , b ) is defined in the Theorem, in particular, r ( 6 , 7 ) = 5 . This result constitutes an refinement upon that of J. Brűdern.

Let $P_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved, among other results, that, for every sufficiently large, even integer $N$ satisfying the congruence condition $N \not \equiv 2\pmod 3$, the equation \[ N=x^2+p^2+p_1^3+p_2^4+p_3^4+p_4^4 \] is solvable with $x$ being a $P_{5}$ and the other variable primes. This result constitutes an enhancement upon that of Vaughan \cite {s12} and Mu \cite {s9}.

Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that for every sufficiently large odd integer N, the equation $$N = x^2+ p_{2}^{2}+ p_{3}^{3}+ p_{4}^{4}+ p_{5}^{5}+ p_{6}^{6}+ p_{7}^{7}$$ is solvable with x being an almost-prime P6 and the other terms powers of primes. This result constitutes a refinement upon that of C. Hooley [7].

Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper we show that the inequality \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {\sqrt p } \right\} < p^{ - \tfrac{1} {{15.5}}}$$ \end{document} has infinitely many solutions in primes p such that p + 2 = P4.

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer N, the equation $$N=x^2+p^3_1+p^3_2+p_3^3+p_4^3+p_5^3 $$ is solvable with x being an almost-prime P 36 and the p j ’s primes. This result constitutes a refinement upon that of G.L. Watson.

Let λ1, λ2 be positive real numbers such that \({\frac{{\lambda_1}}{{\lambda_2}}}\) is irrational and algebraic. For any (C, c) well-spaced sequence \({\mathcal {V} = \{{v_i}\}_{i = 1}^\infty}\) and δ > 0 let \({E( {\mathcal {V},X,\delta})}\) denote the number of elements \({v \in \mathcal {V}, v \le X}\) for which the inequality $$| {\lambda_1 p_1 + \lambda_2 p_2 - v} | < X^{- \delta}$$ is not solvable in primes p 1, p 2. In this paper it is proved that $$E( {\mathcal {V},X,\delta}) \ll X^{\frac{4}{5} + \delta + \varepsilon}$$ for any \({\varepsilon > 0}\). This result constitutes an improvement upon that of Brudern, Cook, and Perelli for the range \({\frac{2}{{15}} < \delta < \frac{1}{5}}\).

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity, and let E 3(N) denote the number of natural numbers not exceeding N that are congruent to 4 modulo 24 yet cannot be represented as the sum of three squares of primes and the square of one P 5. Then we have E 3(N)≪log1053 N. This result constitutes an improvement upon that of D. I. Tolev, who obtained the same bound, but with P 11 in place of P 5.

Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that there exist infinitely many primes of the form p = [nc] such that p + 2 = Pr, where r is the least positive integer satisfying certain inequalities. In particular for [Formula: see text] we have r = 5. This result constitutes an improvement upon that of T. P. Peneva.

Let N be a sufficiently large even integer. Let p denote a prime and P2 denote an almost prime with at most two prime factors. In this paper, it is proved that the equation N = p + P2 (p ≤ N0.945) is solvable.

Let N denote a sufficiently large integer satisfying N ≡ 4 (mod 24), and Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we proved that the equation [Formula: see text] is solvable in one prime and three P42, or in four P13. These results constitute improvements upon that of Heath-Brown and Tolev.

Let p denote a prime and P 2 denote an almost prime with at most two prime factors. The author proves that for sufficiently large x, where the constant 1.13 constitutes an improvement of the previous result 1.104 due to J. Wu.

Let $N$ denote a sufficiently large even integer. In this paper itis proved that for $0.941 \leq \theta \leq 1$, the equation$$N=p+P_2,\hspace{10mm} p\leq N^{\theta}$$is solvable, where $p$ is a prime and $P_2$ is an almost prime withat most two prime factors. The range $0.941 \leq \theta \leq 1$ extended the previous one $0.945 \leq \theta \leq 1$.

Soit Γ = PSL(2, Z). On demontre que π Γ (x) = lix + O(x +e ), e > 0, ou l'exposant 71/102 ameliore l'exposant 7/10 precedemment obtenu par W. Z. Luo et P. Sarnak.

Let $N$ denote a sufficiently large odd integer. In this paper it is proved that $N$ can be represented as the sum of three primes, one of which is $\leq N^{\frac{11}{400}+\varepsilon}$ for any $\varepsilon>0$. This result constitutes an improvement upon that of K.C. Wong, who obtained the exponent $\frac{7}{216}$.

Let N be a sufficiently large even integer and S ( N ) denote the number of solutions of the equation N = p + P 2 , where p denotes a prime and P 2 denotes an almost-prime with at most two prime factors. In this paper we obtain S ( N ) > 0.867 C ( N ) N log 2 N , where C ( N ) = ∏ p > 2 ( 1 − 1 ( p − 1 ) 2 ) ∏ p | N p > 2 p − 1 p − 2 , and thus improved the previous result S ( N ) > 0.836 C ( N ) N log 2 N due to J. Wu.

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that any sufficiently large integer N satisfying the congruence condition N ≡ 13 ( mod 240 ) can be represented as the sum of twelve fourth powers of primes and the fourth power of one P 5 . This result constitutes an improvement upon that of Ren and Tsang.