A note on the simultaneous Pell equations $$x^2-(a^2-1)y^2=1$$ x 2 - ( a 2 - 1 ) y 2 = 1 and $$y^2-bz^2=1$$ y 2 - b z 2 = 1

Periodica Mathematica Hungarica - Let $$a>1,b$$ be two positive integers where the square-free part of b is 2pq with p, q two distinct odd primes. Recently, Cipu (Proc Am Math Soc...     

The quaternary Piatetski-Shapiro inequality with one prime of the form p = x2 + y2 + 1

Lithuanian Mathematical Journal - In this paper, we show that for any fixed 1 < c < 967/805, every sufficiently large positive number N, and a small constant ?? >...     

Automorphism subgroups for designs with $$\lambda =1$$ λ = 1

Designs, Codes and Cryptography - Given an integer $$k\ge 3$$ and a group G of odd order,  if there exists a 2-(v, k, 1)-design and if v is sufficiently large then there...     

A note on the Lebesgue–Ljunggren–Nagell equation $$ax^2+b^{2m}=4y^n$$ a x 2 + b 2 m = 4 y n

Periodica Mathematica Hungarica - Let a, b be fixed positive integers such that $$2\not \mid {ab}$$ , $$\gcd (a,b)=1$$ and a is square-free, and let $$h(-a)$$ denote the class number of...     

A ternary Diophantine inequality by primes with one of the form $$p=x^2+y^2+1$$ p = x 2 + y 2 + 1

The Ramanujan Journal - In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed $$1<\frac{427}{400}$$...     

Il sistema diofanteo N+1=x2, 3N+1=y2, 8N+1=z2

Sunto Nel 1969 A. Baker e H. Davenport hanno dimostrato che il sistema3x ² −2=y ² ,8x ² −7=z ² ha soltanto le soluzioni |x|=|y|=|z|=1; |x|=11, |y|=19, |z|=31. Una seconda dimostrazione di P. Kanagasabapathy — Tharmambikai Ponmudurai è apparsa nell'ottobre 1975. L'A. dà una terza dimostrazione che consente, con l'uso dei metodi classici dell'aritmetica, di controllare i ragionamenti in ogni punto. A Dario Graffi nel suo 70° compleanno Entrata in Redazione il 26 novembre 1975.     

Quadratic optimization with switching variables: the convex hull for $$n=2$$

Mathematical Programming - We consider quadratic optimization in variables (x, y) where $$0\le x\le y$$ , and $$y\in \{ 0,1 \}^n$$ . Such binary variables are commonly referred to as...     

Horizontal Joinability in Canonical 3-Step Carnot Groups with Corank 2 Horizontal Distributions

Siberian Mathematical Journal - We prove that on each 2-step Carnot group with a corank 1 horizontal distribution two arbitrary points can be joined with a horizontal...     

椭圆曲线y~2=x~3+27x-62的整数点

根据四次Diophantine方程的已知结果,运用初等数论方法证明了:椭圆曲线y~2=x~3+27x-62仅有整数点(x,y)=(2,0)和(28844402,±154914585540).     

On the determination of solutions of simultaneous Pell equations $$x^{2}-(a^{2}-1)y^{2}=y^{2}-pz^{2}=1$$ x 2 - ( a 2 - 1 ) y 2 = y 2 - p z 2 = 1

Periodica Mathematica Hungarica - In this paper, we consider the simultaneous Pell equations 0.1 $$\begin{aligned} x^{2}-(a^{2}-1)y^{2}= & {} 1, \nonumber \\ y^{2}-pz^{2}= & {} 1,...     

On the Diophantine equation x 2+5 m =y n

In this paper we consider the Diophantine equation x ²+5 ^m =y ⁿ , n>2, m>0. We prove that the equation has no positive integer solutions when 2∤ m, nor when 2∣m under the additional condition (x,y)=1, with the help of Bilu, Hanrot, and Voutier’s deep result in (J. Reine Angew. Math. 539:75–122, 2001). Supported by the 973 Grant of P.R.C and SRFDP 20040284018.     

On the Diophantine equation $$Cx^{2}+D=2y^{q}$$ C x 2 + D = 2 y q

The Ramanujan Journal - Let C and D denote positive integers such that $$CD>1$$ . In this paper we investigate the solvability of the Diophantine equation $$Cx^{2}+D=2y^{q}$$ , in positive...     

On the integrability of the differential systems in dimension two and of the polynomial differential systems in arbitrary dimension

This is a survey on recent results providing sufficient conditions for the existence of a first integral, first for vector fields defined on real surfaces, and second for polynomial vector fields in \(R^n\) or \(C^n\) with \(n\geq 2\). We also provide an open question and some applications based on the existence of such first integrals.               

Matrix solutions of the equation\mathfrak{B}U_t = - U_{xx} + 2U^3 + \mathfrak{B}[U_x,U] + 4cU: extension of the method of the inverse scattering problem

Complex solution matrices of the nonlinear Schrödinger equation \(\mathfrak{B}U_t = - U_{xx} + 2U^3 + \mathfrak{B}[U_x ,U] + 4cU:\) are found and the method of the inverse scattering problem is subjected to a natural extension. That is, for the nonself-conjugateL-A Lax doublet that arises for this equation, the presence of chains of adjoint vectors for the operatorL is taken into account by means the corresponding normed chains. A uniqueness theorem for the Cauchy problem for the above Schrödinger equation is obtained. Here \(\mathfrak{B} = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right),[M,N] = MN - NM\) , and c is a parameter.     

Cauchy problem for the Zakharov System Arising from Ion-Acoustic Modes with low regularity data

We prove local well-posedness results for the Zakharov System Arising from Ion-Acoustic Modes in two spacial dimension with large initial data in low regularity Sobolev space \(   (\dot{H}^1 \bigcup H^{\frac{1}{2}})\times L^2 \times H^{-1}  \).  Using ”derivative sharing”, the local well-posedness results in \( (\dot{H}^1 \bigcup H^{\frac{1}{2}-\delta})\times H^{\delta} \times H^{-1+\delta}\)  are also obtained, for any  0 \(\leq \delta \leq 1/2 \).     

Effective primality tests for integers of the formsN=k3 n +1 andN=k2 m 3 n +1

Using third roots of unity Proth's theorem for primality testing is generalized to integers of the formN=k3 ⁿ +1, avoiding the use of Lucas sequences which are more suitable ifN+1 is factored instead ofN–1. This approach has the advantage of being easily combined with Proth's test and gives polynomial time algorithms for testing integers of the formN=k2 ^m 3 ⁿ +1.     

Double Hopf bifurcation and chaos in Liu system with delayed feedback

In this paper, we consider the stability of equilibria, Hopf and double Hopf bifurcation in Liu system with delay feedback. Firstly, we identify the critical values for stability switches and Hopf bifurcationusing the method of bifurcation analysis. When we choose appropriate feedback strength and delay, two symmetrical nontrivial equilibria of Liusystem can be controlled to be stable at the same time, and the stable bifurcating periodic solutions occur in the neighborhood of the two equilibria at the same time. Secondly, by applying the normal form method and center manifold theory,the normal form near the double Hopf bifurcation, as well as classifications of local dynamics are analyzed. Furthermore, we give the bifurcation diagram to illustrate numerically that a family of stable periodic solutions bifurcated from Hopf bifurcation occur in a large region of delay and the Liu system with delay can appear the phenomenon of ``chaos switchover''.     

A Class of Invertible Subelliptic Operators in S(m,g)-Classes

Results in Mathematics - Given a self-adjoint second order differential operator L with positive characteristic and subellipticity of order 1 ≤ τ < 2....     

Exact tail asymptotics for fluid models driven by an M/M/c queue

Queueing Systems - In this paper, we investigate exact tail asymptotics for the stationary distribution of a fluid model driven by the M / M / c queue, which is...