Resonance Identities for Closed Characteristics on Compact Star-shaped Hypersurfaces in R2n{\bf R}^{2n}

Resonance relations among periodic orbits on given energy hypersurfaces are very important for getting deeper understanding of the dynamics of the corresponding Hamiltonian systems. In this paper, we establish two new resonance identities for closed characteristics on every compact star-shaped hypersurface $\Sigma$ in ${\bf R}^{2n}$ when the number of geometrically distinct closed characteristics on $\Sigma$ is finite, which extend those identities established by C. Viterbo in 1989 for star-shaped hypersurfaces assuming in addition that all the closed characteristics and their iterates are non-degenerate, and that by W. Wang, X. Hu and Y. Long in 2007 for strictly convex hypersurfaces in ${\bf R}^{2n}$.Comment: arXiv admin note: substantial text overlap with arXiv:math/0701608. To appear in Journal of Functional Analysis, this is the final versio

The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form

Let $M=S^n/ \Gamma$ and $h$ be a nontrivial element of finite order $p$ in $\pi_1(M)$, where the integer $n\geq2$, $\Gamma$ is a finite group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractible homologically visible minimal closed geodesics of the class $[h]$ on every Finsler compact space form $(M, F)$ when there exist only finitely many distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$ with a bumpy Finsler metric, which improves a result of Taimanov in [Taimanov 2016] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on $\mathcal{R}P^n$ in [arXiv:1607.02746] to general compact space forms.Comment: 33 pages, All comments are welcome. arXiv admin note: substantial text overlap with arXiv:1607.0274

Multiplicity of closed characteristics on symmetric convex hypersurfaces in R2n\R^{2n}

Let $\Sigma$ be a compact $C^2$ hypersurface in $\R^{2n}$ bounding a convex set with non-empty interior. In this paper it is proved that there always exist at least $n$ geometrically distinct closed characteristics on $\Sigma$ if $\Sigma$ is symmetric with respect to the origin.Comment: 16 page

Enterprise Comprehensive Budget Informatization Management Based on Cloud Accounting and Blockchain Technology

With the continuous expansion of enterprise scale and economic development, the traditional comprehensive budget management lacks strategic guidance and information communication is becoming increasingly prominent, making it difficult to adapt to the needs of enterprise development in the new era. In response to this, research is conducted on modeling time series algorithms based on autoregressive moving average models, and a prediction model based on backpropagation neural networks is also established. And the error variance weighted average method is used to organically integrate two single prediction models to obtain a combined prediction model. A combination forecasting model is utilized to predict and analyze the budget of G Company. The analysis results ia applied to budget management. Finally, based on cloud accounting and blockchain technology, the enterprise budget management is optimized, and a comprehensive budget information management model for enterprises is constructed based on cloud accounting and blockchain technology. The experimental findings denote that after the model is put into use, a total of three projects complete 101.1%, 102.3%, and 106.0% of the budget, all of which are able to achieve the budget goals. The model can effectively improve the accuracy of enterprise budgeting and the effectiveness of budget execution