Periodic, Irrational and Transcendental Numbers in the Factorial Series

The main aim of the article is to investigate the irrational and transcendental properties of certain real numbers by means of the factorial series and the factorial number system. The difference between the factorial series and the factorial system is that the factorial series does not set an upper bound at a given place after the radix point, while in the factorial system (i – 1) is the maximal possible value for ri after the radix point. I give an extended definition of periodic numbers, and show the relationship between periodic and irrational numbers. I prove the transcendence of e by means of the factorial series and the factorial number system.

On two actuarial quantities for the compound Poisson risk model with tax and a threshold dividend strategy

In this paper, we consider a compound Poisson risk model with taxes paid according to a loss-carry-forward system and dividends paid under a threshold strategy. First, the closed-form expression of the probability function for the total number of taxation periods over the lifetime of the surplus process is derived. Second, analytical expression of the expected accumulated discounted dividends paid between two consecutive taxation periods is provided. In addition, explicit expressions are also given for the exponential individual claims.

Least Number of Periodic Points of Self-maps of Lie Groups

There are two algebraic lower bounds of the number of n-periodic points of a self-map f ： M → M of a compact smooth manifold of dimension at least 3： NFn（f） = min{#Fix（gn）;g - f; g is continuous} and NJDn（f） = min{#Fix（gn）; g - f; g is smooth}. In general, NJDn（f） may be much greater than NFn（f）. If M is a torus, then the invariants are equal. We show that for a self-map of a nonabelian compact Lie group, with free fundamental group, the equality holds 〈=〉 all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.

Banana Xanthomonas wilt dynamics with mixed cultivars in a periodic environment Juliet

In this paper,we study a deterministic model with non-autonomous system for mixed cultivars to assess the effect of cultivar susceptibility and seasonal variation on banana Xanthomonas wilt(BXW)disease dynamics.A special case of two cultivars classified as highly susceptible for inflorescence infection(ABB)and less susceptible(AAA)cultivar is considered.The basic reproduction number corresponding to the non-autonomous sys­tem is derived and numerically computed to determine disease dynamics.Results showed that the disease dies out whenever the periodic basic reproduction number is less than unity and a periodic solution is obtained when it is greater than one.Results further showed that for both cultivars,the basic reproduction number increases with increas­ing values of the transmission rates and declines exponentially with increasing values of roguing rates.The critical roguing rate of ABB-genome cultivar was higher than that of AAA-genome cultivars.The peaks in disease prevalence indicate the importance of effective implementation of controls during the rainy season.We conclude that highly susceptible cultivars play an important role in the spread of BXW and control measures should be effectively implemented during the rainy season if BXW is to be eradicated.

Self-maps of S2 Homotopic to a Smooth Map with a Single n-periodic Point

We show for which （d, n） ∈ Z × N there exists a smooth self-map f ： S2 → S2 so that deg（f） = d and Fix（fn） is a point.