Some properties of τ If we put ∆(z) = D(e), Im (z) > 0, (3) then it is known that the function ∆ is, up to a constant factor, the unique cusp form of weight 12 for the group SL (2,Z). In particular, the function ∆ is, for each prime number p, an eigenfunction of the Hecke operator Tp, with corresponding eigenvalue τ(p) (cf. e.g. Hecke [6], p. 644–671). This implies the following properties, whi...

For a distribution F * τ of a random sum S τ = ξ 1 +. .. + ξ τ of i.i.d. random variables with a common distribution F on the half-line [0, ∞), we study the limits of the ratios of tails F * τ (x)/F (x) as x → ∞ (here τ is an independent counting random variable). We also consider applications of obtained results to random walks, compound Poisson distributions, infinitely divisible laws, and su...

In this paper we prove that the spherical subalgebra eH1,τ e of the double affine Hecke algebra H1,τ is an integral Cohen-Macaulay algebra isomorphic to the center Z of H1,τ , and H1,τ e is a Cohen-Macaulay eH1,τ emodule with the property H1,τ = EndeH1,τ e(H1,τ e) when τ is not a root of unity. In the case of the root system An−1 the variety Spec(Z) is smooth and coincides with the completion o...

Isospin violating signals in the τ − → (3π) − ν τ decay mode are discussed. For the τ − → π − π − π + ν τ decay mode, isospin violation arises from the vector current contribution in the τ − → ωπ − ν τ decay with the subsequent isospin violating ω decay into π + π −. We demonstrate that such effects may be observed in presently available data through the measurement of the interference effects ...

We study lower limits for the ratio F * τ (x) F (x) of tail distributions where F * τ is a distribution of a sum of a random size τ of i.i.d. random variables having a common distribution F , and a random variable τ does not depend on summands. Let τ be a counting random variable which does not depend on {ξ n } n≥1. Denote by F * τ the distribution of a random sum S τ = ξ 1 +. .. + ξ τ. In this...

Lovász and Stein (independently) proved that any hypergraph satisfies τ ≤ (1+ln ∆)τ * , where τ is the transversal number, τ * is its fractional version, and ∆ denotes the maximum degree. We prove τ f ≤ 3.17τ * max{ln ∆, f } for the f-fold transversal number τ f. Similarly to Lovász and Stein, we also show that this bound can be achieved non-probabilistically, using a greedy algorithm. As a com...

In this paper, we take physical and health integration as the entry point. Firstly, based on transformation mechanism of adolescent idiopathic scoliosis construct a time delay differential model. Moreover, using theory characteristic equation discuss stability positive equilibrium under delays τ=0 τ≠0. Furthermore, through numerical simulation, it has been verified delay, τ, exceeds critical va...

ionAx ∪ {x : τ ′} e : τA λx.e : τ ′ → τ LetA e : σ Ax ∪ {x : σ} e′ : τA (let x = e in e′) : τ The meaning of the Taut-rule is simply that a free variable has the type assignedto it in the boundary condition. If there is no assignment the expression x is nottypable, and is barred from the universe.The meaning of rule App is also clear. It is useful, however, to know h...

In this paper, the synchronization problem of uncertain neutral-type neural networks (NTNNs) with sampled-data control is investigated. First, a mixed-delay-dependent augmented Lyapunov–Krasovskii functional (LKF) proposed, which not only considers interaction between transmission delay and communication delay, but also takes interconnected relationship neutral into consideration. Then, two-sid...

We study the statistical properties of recurrence times in the self-excited Hawkes conditional Poisson process, the simplest extension of the Poisson process that takes into account how the past events influence the occurrence of future events. Specifically, we analyze the impact of the power law distribution of fertilities with exponent α, where the fertility of an event is the number of trigg...