Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GL_m(Z)

Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(n1,..., nm-1). In this paper,an asymptotic expansion of Voronoi's summation formula for Af(n1,..., nm-1) is established. As applications of this formula, a smoothly weighted average of Af(n, 1,..., 1) against e(α|n|β) is proved to be rapidly decayed when 0 < β < 1/m. When β = 1/m and α equals or approaches ±mq1/mfor a positive integer q, this smooth average has a main term of the size of |Af(1,..., 1, q) + Af(1,..., 1,-q)|X1/(2m)+1/2, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients Af(n, 1,..., 1). Similar estimate is also proved for a sharp-cut sum.

The Insphere of a Tetrahedron

A contiguous derivation of radius and center of the insphere of a general tetrahedron is given. Therefore a linear system is derived. After a transformation of it the calculation of radius and center can be separated from each other. The remaining linear system for the center of the insphere can be solved after discovering the inverse of the corresponding coefficient matrix. This procedure can also be applied in the planar case to determine radius and center of the incircle of a triangle.

Quadratic forms connected with Fourier coefficients of Maass cusp forms

For the normalized Fourier coefficients of Maass cusp forms λ（n） and the normalized Fourier coefficients of holomorphic cusp forms a（n）, we give the bound of

Exponential Sums over Primes Formed with Coefficients of Primitive Cusp Forms

Let Af（n） be the coefficient of the logarithmic derivative for the Hecke L-function. In this paper we study the cancellation of the function Ay（n） twisted with an additive character e（α√n）, α 〉0, i.e. Ef（x） = Σx〈n〈2x Af（n）e（α√n）.

Exponential sums involving Maass forms

We study the exponential sums involving l：burmr coeffcients ot Maass forms and exponential functions of the form e（anZ）, where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponential sum ∑x〈n≤2x λg（n）e（αnβ）, when β = 1/2 and ｜α｜ is close to 2√ q C Z＋, where Ag（n） is the normalized n-th Fourier coefficient of a Maass cusp form for SL2 （Z）. The similar natures of the divisor function 7（n） and the representation function r（n） in the circle problem in nonlinear exponential sums of the above type are also studied.

Spectral square moments of a resonance sum for Maass forms

Let f be a Maass cusp form for Г0（N） with Fourier coefficients 1 k2. λf（n） and Laplace eigenvalue 1/4 ＋k2 For real α≠0 and β 〉 0, consider the sum Sx（f; α,β） = ∑n λf（n）e（αnβ）φ（n/X）, where φ is a smooth function of compact support. We prove bounds for the second spectral moment of Sx （f;α, β）, with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond X1/2＋ε, the standard resonance main term for Sx（f; ±2√q 1/2）, q ∈Z＋, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL（2） × GL（2）. It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of Kε≤ L≤ K1-ε. The same bounds can be proved in a similar way for holomorphie cusp forms.

Effects of B addition on glass forming ability and thermal behavior of FePC-based bulk metallic glasses

The FePC-based bulk metallic glasses（BMGs）have been demonstrated to possess high plasticity and good soft magnetic properties.However,the relatively poor glass forming ability（GFA）and thermal stabilities limited their application in industries.The effects of microalloying with B in FePC-based BMGs on the GFA and thermal behaviors were systematically investigated.It was found that a small amount of B addition can dramatically enhance the GFA of FePC-based BMGs,which in turn leads to the critical maximum diameter up to 2 mm for full glass formation even using low cost raw materials.The underlying mechanism of the enhancement of GFA from the competing crystalline phase with amorphous phase,the average thermal expansion coefficient and dynamic viscosity were discussed in detail.

Oscillations of coefficients of symmetric square L-functions over primes

Let L（s, sym2f） be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL（2, Z）, with tf（n, 1） denoting the nthcoefficient of the Dirichlet series for it. It is proved that, forN≥2and anyα∈ there exists an effective positive constant c such that ∑n≤N∧（n）tf（n,1）e（nα）〈〈N exp （-c√logN,where ∧（n） is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov＇s three primes theorem associated to the coefficients of Rankin-Selberg L-functions.

On the Fourth Moment of Coefficients of Symmetric Square L-Function

t Let f（z） be a holomorphic Hecke eigencuspform of weight k for the full mod- ular group. Let Af（n） be the nth normalized Fourier coefficient of f（z）. Suppose that L（sym2f, s） is the symmetric square L-function associated with f（z）, and Asym2f（n） de- notes the nth coefficient L（sym2f, s）. In this paper, it is proved that where P2 （t） is a polynomial in t of degree 2. Similarly, it is obtained that where P2（t） is a polynomial in t of degree 2.

Bifurcations of a discrete-time SIR epidemic model with logistic growth of the susceptible individuals

This study focuses on the stability and local bifurcations of a discrete-time SIR epidemic model with logistic growth of the susceptible individuals analytically,and numerically.The analytical results are obtained using thenormal form technique and numerical results are obtained using the numerical continuation method.For this model,a number of bifurcations are studied,including the transcritical(pitchfork)and fip bifurcations,the Neimark-Sacker(NS)bifurcations,and the strong resonance bifurcations.We especially determine the dynamical behaviors of the model for higher iterations up to fourth-order.Numerical simulation is employed to present a closed invariant curve emerging about an NS point,and its breaking down to several closed invariant curves and eventuality giving rise to a chaotic strange attractor by increasing the bifurcation parameter.