Annali di Matematica Pura ed Applicata最新文献

In this paper, we define a class of braces that we call module braces or R-braces, which are braces for which the additive group has also a module structure over a ring R, and for which the values of the gamma functions are automorphisms of R-modules. This class of braces has already been considered in the literature in the case where the ring R is a field; we generalise the definition to any ring R, reinterpreting it in terms of the so-called gamma function associated with the brace, and prove that this class of braces enjoys all the natural properties one can require. We exhibit explicit example of R-braces, and we study the splitting of a module braces in relation to the splitting of the ring R, generalising thereby Byott’s result on the splitting of a brace with nilpotent multiplicative group as a sum of its Sylow subgroups. The core of the paper is in the last two sections, in which, using methods from commutative algebra and number theory, we study the relations between the additive and the multiplicative groups of an R-brace showing that if a certain decomposition of the additive group is small (in some sense which depends on R), then the additive and the multiplicative groups have the same number of elements of each order. In some cases, this result considerably broadens the range of applications of the results already known on this issue.

In this paper, we investigate the precise behavior of orbits inside attracting basins. Let f be a holomorphic polynomial of degree (mge 2) in (mathbb {C}), (mathcal {A}(p)) be the basin of attraction of an attracting fixed point p of f, and (Omega _i (i=1, 2, cdots )) be the connected components of (mathcal {A}(p)). Assume (Omega _1) contains p and ({f^{-1}(p)}cap Omega _1ne {p}). Then there is a constant C so that for every point (z_0) inside any (Omega _i), there exists a point (qin cup _k f^{-k}(p)) inside (Omega _i) such that (d_{Omega _i}(z_0, q)le C), where (d_{Omega _i}) is the Kobayashi distance on (Omega _i.) In paper Hu (Dynamics inside parabolic basins, 2022), we proved that this result is not valid for parabolic basins.

We establish optimal (p, q) ranges for the weighted convolution operator associated with a complex polynomial curve. Establishing this estimate comes down to establishing a lower bound for the Jacobian of a mapping associated with the complex curve in question.

We classify, up to isometric congruence, the homogeneous hypersurfaces in the Riemannian symmetric spaces (textrm{SL}(3,mathbb {H})/textrm{Sp}(3), textrm{SO}(5,mathbb {C})/textrm{SO}(5),) and (textrm{Gr}^*(2,mathbb {C}^{n+4}) = textrm{SU}(n+2,2)/textrm{S}(textrm{U}(n+2)textrm{U}(2)), , n geqslant 1).

We prove the weak type (1,1) estimate for maximal function of the truncated rough Hilbert transform considered in Paluszy’nski (ASNSCS 910:679-704, 2019), Paluszy’nski (CM 164(2):305-325, 2021).

For the curve shortening flow in ({mathbb {R}}^3) several new monotonicity formulas are derived. All of them share one main feature: the dependence of the “energy” term on the angle between the position vector and the plane orthogonal to the tangent vector. The first formula deals with the projection of the curve on the unit sphere, and computes the derivative of its length. The second formula is the generalization of the classical formula of G. Huisken, while the third one is the generalization of the monotonicity formula with logarithmic terms previously derived by the author for planar curves.

We show that contact reductions can be described in terms of symplectic reductions in the traditional Marsden–Weinstein–Meyer as well as the constant rank picture. The point is that we view contact structures as particular (homogeneous) symplectic structures. A group action by contactomorphisms is lifted to a Hamiltonian action on the corresponding symplectic manifold, called the symplectic cover of the contact manifold. In contrast to the majority of the literature in the subject, our approach includes general contact structures (not only co-oriented) and changes the traditional view point: contact Hamiltonians and contact moment maps for contactomorphism groups are no longer defined on the contact manifold itself, but on its symplectic cover. Actually, the developed framework for reductions is slightly more general than purely contact, and includes a precontact and presymplectic setting which is based on the observation that there is a one-to-one correspondence between isomorphism classes of precontact manifolds and certain homogeneous presymplectic manifolds.

In this paper we study the properties of an algorithm, introduced in Browkin (Math Comput 70:1281–1292, 2000), for generating continued fractions in the field of p-adic numbers (mathbb Q_p). First of all, we obtain an analogue of the Galois’ Theorem for classical continued fractions. Then, we investigate the length of the preperiod for periodic expansions of square roots. Finally, we prove that there exist infinitely many square roots of integers in (mathbb Q_p) that have a periodic expansion with period of length 4, solving an open problem left by Browkin in (Math Comput 70:1281–1292, 2000).