Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

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 normalized Ricci flow and smooth structures on four-manifolds with b + = 1

We find an obstruction to the existence of non-singular solutions to the normalized Ricci flow on four-manifolds with b⁺ = 1. By using this obstruction, we study the relationship between the existence or non-existence of non-singular solutions of the normalized Ricci flow and exotic smooth structures on the topological 4-manifolds , where . Received: 13 November 2008     

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...     

The equation $$x^4+2^ny^4=z^4$$ x 4 + 2 n y 4 = z 4 in algebraic number fields

Acta Mathematica Hungarica - Let n be a positive integer. We show that if the equation $$(1) \qquad \qquad \qquad x^4+2^ny^4=z^4$$ has a solution (x,y,z) in a cubic number field K with $$xyz \neq...     

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,...     

Counter-examples to the Dunford–Schwartz pointwise ergodic theorem on $$\varvec{L^1+L^\infty }$$ L 1 + L ∞

Extending a result by Chilin and Litvinov, we show by construction that given any $$\sigma $$ -finite infinite measure space $$(\Omega ,\mathcal {A}, \mu )$$ and a function $$f\in L^1(\Omega )+L^\infty (\Omega )$$ with $$\mu (\{|f|>\varepsilon \})=\infty $$ for some $$\varepsilon >0$$ , there exists a Dunford–Schwartz operator T over $$(\Omega ,\mathcal {A}, \mu )$$ such that $$\frac{1}{N}\sum _{n=1}^N (T^nf)(x)$$ fails to converge for almost every $$x\in \Omega $$ . In addition, for each operator we construct, the set of functions for which pointwise convergence fails almost everywhere is residual in $$L^1(\Omega )+L^\infty (\Omega )$$ .     

On values of n 2 + 1 free of large prime factors

Let P ⁺(m) denote the greatest prime factor of the positive integer m. Improving and simplifying work of Dartyge [3] we show that

On asymptotic behaviour of the difference equationX_{N + 1} = \alpha + \frac{{X_{N - 1} ^P }}{{X_N ^P }}

In this paper, we investigate local stability, oscillation and boundeness character of positive solutions of the difference equation $$x_{n + 1} = \alpha + \frac{{x_{n - 1} ^p }}{{x_n ^p }},n = 0,1,...$$ under specified conditions.     

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.     

On the Hilbert scheme of linearly normal curves in $$\mathbb {P}^4$$ P 4 of degree $$d = g+1$$ d = g + 1 and genus g

Archiv der Mathematik - We denote by $$\mathcal {H}_{d,g,r}$$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and...     

On a family of diophantine triples {k, A 2 k + 2A, (A + 1)2 k + 2 (A + 1)} with two parameters

Let A and k be positive integers. We study the Diophantine quadruples $$ \{ k,A^2 k + 2A,(A + 1)^2 k + 2(A + 1),d\} $$ . We prove that if d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then $$ \begin{gathered} d = (4A^4 + 8A^3 + 4A^2 )k^3 + (16A^3 + 24A^2 + 8A)k^2 \hfill \\ + (20A^2 + 20A + 4)k + (8A + 4) \hfill \\ \end{gathered} $$ when 3 ≦ A ≦ 10. This extends a theorem obtained by Dujella [7] for A = 1, and also, a classical theorem of Baker and Davenport [2] for A = k = 1.     

On the recursive sequencex_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation $$x_{n + 1} = \frac{{ax_{n - 2m + 1}^p }}{{b + cx_{n - 2k}^{p - 1} }}, n = 0, 1,...$$ wherem, k ∈ N, 2k > 2m?1,a, b, c are nonnegative real numbers andp < 1, are investigated.     

On the difference equationx_{n + 1} = \frac{{a + bx_{n - k} - cx_{n - m} }}{{1 + g(x_{n - 1} )}}

In this paper we consider the difference equation $$x_{n + 1} = \frac{{a + bx_{n - k} - cx_{n - m} }}{{1 + g(x_{n - 1} )}},$$ wherea, b, c are nonegative real numbers,k, l, m are nonnegative integers andg(x) is a nonegative real function. The oscillatory and periodic character, the boundedness and the stability of positive solutions of the equation is investigated. The existence and nonexistence of two-period positive solutions are investigated in details. In the last section of the paper we consider a generalization of the equation.     

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,&nbsp; if there exists a 2-(v,&nbsp;k,&nbsp;1)-design and if v is sufficiently large then there...     

On the diophantine equationD 1 x 2+D 2 m =4y n

For anyD ₁,D ₂, leth(-D ₁ D ₂) denote the class number of the imaginary quadratic field . In this paper we prove that the equationD ₁ x ²+D ₂ ^m =4y ⁿ.D ₁,D ₂,x, y, m, n, gcd (D ₁x,D ₂y=1,2m,n an odd prime,nh(-D ₁ D ₂, has only a finite number of solutions (D ₁,D ₂,x,y,m,n) withn>5. Moreover, the solutions satisfy 4y ⁿ相似文献   

Some notes on the multiplicative order of $$\alpha + \alpha ^{-1}$$α+α-1 in finite fields of characteristic two

Periodica Mathematica Hungarica - In this paper we prove some results on the possible multiplicative order of $$\alpha + \alpha ^{-1}$$ when $$\alpha $$ is a non-zero element of a finite field of...