Periodica Mathematica Hungarica最新文献

Let T be a tree; a vertex of degree 1 is a leaf of T and a vertex of degree at least 3 is a branch vertex of T. The reducible stem of T is the smallest subtree that contains all branch vertices of T. In this paper, we give some sharp sufficient conditions for (K_{1,4})-free graphs to have a spanning tree whose reducible stem has few leaves.

In this paper, we study q-statistical convergence for double sequences. The definitions of q-analog of statistical Cauchy and statistical pre-Cauchy for double sequences are given. The necessary and sufficient condition for a double sequence to have different statistical limits is also obtained. We show that a q-statistical convergent sequence is q-statistical Cauchy and vice-versa.

In this paper, we show some results on the non-vanishing of the generalized local cohomology modules (H^i_I(M,N)). In a Cohen–Macaulay local ring ((R,mathop {mathfrak {m}})), we prove, by using induction on (dim N), that if M, N are two finitely generated R-modules with ({text {id}},M<infty ) and ({text {Gid}},N<infty ), then (H^{dim R-grade _R({text {Ann}}_RN,M)}_{mathop {mathfrak {m}}}(M,N)ne 0). We also study the I-cofiniteness of the generalized local cohomology module (H^i_{mathop {mathfrak {m}}}(M,N)).

From S. Banach’s results it follows that even for the function (f(x)=1) ((xin [0,1])) the general partial sums of its general Fourier series are not bounded a.e. on [0, 1]. In the present paper, we find conditions for the functions (varphi _n) of an orthonormal system ((varphi _n)) under which the partial sums of functions from some differentiable class are bounded. We prove that the obtained results are best possible. We also investigate the properties of subsequences of general orthonormal systems.

In this paper we investigate the 1:1 resonance and the hybrid control in a discrete predator–prey model with Holling-IV functional response, which is derived from a 2-dimensional continuous one of Gause type. When the parameters satisfy some conditions, this discrete model has a positive fixed point, which has a double eigenvalue 1 with geometric multiplicity 1. With the Picard iteration and the time-one map, this discrete one is converted into an ordinary differential system. It is shown that a Bogdanov–Takens bifurcation for this ordinary differential system happens by the bifurcation theory. This implies that this discrete model undergoes a Neimark–Sacker bifurcation and a homoclinic bifurcation. The stability of its fixed point is obtained. Then, the hybrid control strategy is applied to control the stability of this fixed point. Finally, the local phase portraits of these systems are also simulated by the Matlab software.

For each (sin {mathbb {R}}) and (nin {mathbb {N}}), let (sigma _s(n) = sum _{dmid n}d^s). In this article, we study the number of sign changes in the difference (sigma _s(an+b)-sigma _s(cn+d)) where a, b, c, d, s are fixed, the vectors (a, b) and (c, d) are linearly independent over ({mathbb {Q}}), and n runs over all positive integers. We also give several examples and propose some problems.

We deeply investigate the Diophantine equation (cx^2+d^{2m+1}=2y^n) in integers (x, yge 1, mge 0) and (nge 3), where c and d are coprime positive integers satisfying (cdnot equiv 3 pmod 4). We first solve this equation for prime n under the condition (gcd (n, h(-cd))=1), where (h(-cd)) denotes the class number of the imaginary quadratic field ({mathbb {Q}}(sqrt{-cd})). We then completely solve this equation for both c and d primes under the assumption (gcd (n, h(-cd))=1). We also completely solve this equation for (c=1) and (dequiv 1 pmod 4) under the condition (gcd (n, h(-d))=1). For some fixed values of c and d, we derive some results concerning the solvability of this equation.

Let ({a_1,b,c}) and ({a_2,b,c}) be Diophantine triples with (a_1<b<a_2<c) and (a_2ne b+c-2sqrt{bc+1}). Put (d_2=a_2+b+c+2a_2bc-2r_2st), where (r_2=sqrt{a_2b+1}), (s=sqrt{ac+1}) and (t=sqrt{bc+1}). In this paper, we prove that if (c le 16mu ^2 b^3), where (mu =min {a_1,d_2}), then ({a_1,a_2,b,c}) is a Diophantine quadruple. Combining this result with one of our previous results implies that if ({a_i,b,c,d}) ((iin {1,2,3})) are Diophantine quadruples with (a_1<a_2<b<a_3<c<d), then (a_3=b+c-2sqrt{bc+1}). It immediately follows that there does not exist a septuple ({a_1,a_2,a_3,a_4,b,c,d}) with (a_1<a_2<b<a_3<a_4<c<d) such that ({a_i,b,c,d}) ((i in {1,2,3,4})) are Diophantine quadruples. Moreover, it is shown that there are only finitely many sextuples ({a_1,a_2,a_3,b,c,d}) with (a_1<b<a_2<a_3<c<d) such that ({a_i,b,c,d}) ((i in {1,2,3})) are Diophantine quadruples.

We call a semigroup ({mathcal {R}})-noetherian if it satisfies the ascending chain condition on principal right ideals, or, equivalently, the ascending chain condition on ({mathcal {R}})-classes. We investigate the behaviour of the property of being ({mathcal {R}}text {-noetherian}) under the following standard semigroup-theoretic constructions: semidirect products, Schützenberger products, free products, Rees matrix semigroups, Brandt extensions, Bruck–Reilly extensions and semilattices of semigroups.