Monatshefte für Mathematik最新文献

Abstract

A classical result of Fathi and Herman from 1977 states that a smooth compact connected manifold without boundary admitting a locally free action of a 1-torus, respectively, an almost free action of a 2-torus, admits a minimal diffeomorphism, respectively, a minimal flow. In the first part of our paper we study the existence of locally free and almost free actions of tori on homogeneous spaces of compact connected Lie groups, thus providing new examples of spaces admitting minimal diffeomorphisms or flows. In the second part we combine the ideas of Fathi and Herman with our recent ideas to study the existence of minimal skew products over certain minimal flows with general connected Lie groups as acting groups. Our results apply to so called flows with free cycles. In the last part of our work we study the existence of free cycles in homogeneous flows.

introduce the spectral decomposition property for measures and prove that a homeomorphism has the spectral decomposition property if and only if every Borel probability measure has the property too. Furthermore, we show that all shadowable measures for expansive homeomorphisms have the spectral decomposition property. Additionally, we provide illustrative examples relevant to these results.

Diaconis and Gamburd computed moments of secular coefficients in the CUE ensemble. We use the characteristic map to give a new combinatorial proof of their result. We also extend their computation to moments of traces of symmetric powers, where the same result holds but in a wider range. Our combinatorial proof is inspired by gcd matrices, as used by Vaughan and Wooley and by Granville and Soundararajan. We use these CUE computations to suggest a conjecture about moments of characters sums twisted by the Liouville (or by the Möbius) function, and establish a version of it in function fields. The moral of our conjecture (and its verification in function fields) is that the Steinhaus random multiplicative function is a good model for the Liouville (or for the Möbius) function twisted by a random Dirichlet character. We also evaluate moments of secular coefficients and traces of symmetric powers, without any condition on the size of the matrix. As an application we give a new formula for a matrix integral that was considered by Keating, Rodgers, Roditty-Gershon and Rudnick in their study of the k-fold divisor function.

In this paper, we establish the monotonicity of the period map in terms of the energy levels for certain periodic solutions of the equation (-varphi ''+varphi -varphi ^{k}=0), where (k>1) is a real number. We present a new approach to demonstrate this property, utilizing spectral information of the corresponding linearized operator around the periodic solution and tools related to Floquet theory.

Abstract

In this essay, we investigate the blow-up scenario, global solution and propagation speed for a modified Camassa–Holm (MCH) equation both dissipation and dispersion in Sobolev space (H^{s,p} (mathbb {R})) , (sge 1) , (pin (1,infty )) . First of all, by the mathematical induction of index s, we establish the precise blow-up criteria, which extends the result obtained by Gui et al. in article (Comm Math Phys 319: 731–759, 2013). Secondly, we derive the global existence of the strong solution of MCH equation both dissipation and dispersion. Eventually, the propagation speed of the equation is studied when the initial data are compactly supported.

In this paper, we first construct some explicit solutions to the b-family of equations, which will become unbounded in a finite time. Then, we investigate the asymptotic stability of the aforementioned singular solutions of the b-family of equations in the Sobolev space (H^s) with (s>frac{7}{2}). It is also interesting to point out that this stability highly depends on the values of parameter b, that is, (bin (-1,2]). The proof is based on the detailed analysis on the estimates of the perturbed solutions and the properties of the corresponding linear operators.

Abstract

We describe the inverse image of the Riemannian exponential map at a basepoint of a compact symmetric space as the disjoint union of so called focal orbits through a maximal torus. These are orbits of a subgroup of the isotropy group acting in the tangent space at the basepoint. We show how their dimensions (infinitesimal data) and connected components (topological data) are encoded in the diagram, multiplicities, Weyl group and lattice of the symmetric space. Obtaining this data is precisely what we mean by counting geodesics. This extends previous results on compact Lie groups. We apply our results to give short independent proofs of known results on the cut and conjugate loci of compact symmetric spaces.

In this work, we propose new contributions to a complex system of quadratic heat equations with a generalized kernel of the form: (partial _t z=mathfrak {L},z+ widetilde{z}^{2},;partial _t widetilde{z}=mathfrak {L},widetilde{z}+ z^2,;t>0,) with initial conditions (z_{0}=u_0+v_0,;widetilde{z}_{0}=widetilde{u}_0+widetilde{v}_0), and (mathfrak {L}) is a linear operator with (e^{tmathcal {L}}) its semigroup having a generalized heat kernel G satisfying in particular (G(t,x)= t^{-frac{N}{d}} G(1,xt^{-1/d}),,d>0,, t>0) and (xin mathbb {R}^N.) Under conditions on the parameters (sigma _{1},,widetilde{sigma }_{1},,rho _{1},,) and (widetilde{rho _{1}}) we show results on global-in time solution for small data (u_{0}(x)sim c|x|^{-dsigma _{1}},,v_{0}(x)sim c|x|^{-drho _{1}},,widetilde{u}_{0}(x)sim c|x|^{-dwidetilde{sigma }_{1}}) and (widetilde{v}_{0}(x)sim c|x|^{-dwidetilde{rho }_{1}}) as (|x|rightarrow infty ), ( |c| is sufficiently small ). We investigate the global existence of solutions to the given system.