We consider the differential operator Λk defined by Λk(y) = Ψk(y) + ak−1Ψk−1(y) + . . .+ a1Ψ1(y) + a0 , where a0, . . . , ak−1 are analytic functions of restricted growth and Ψk(y) is a differential operator defined by Ψ1(y) = y and Ψk+1(y) = yΨk(y) + (Ψk(y)) for k ∈ N. We suppose that k ≥ 3, that F is a meromorphic function on an annulus A(r0), and that Λk(F ) has all its zeros on a set E such...

We derive the large n asymptotics of zeros of sections of a generic exponential sum. We divide all the zeros of the nth section of the exponential sum into “genuine zeros,” which approach, as n → ∞, the zeros of the exponential sum, and “spurious zeros,” which go to infinity as n → ∞. We show that the spurious zeros, after scaling down by the factor of n, approach a “rosette,” a finite collecti...

The notions of decoupling zeros of positive discrete-time linear systems are introduced. The relationships between the decoupling zeros of standard and positive discrete-time linear systems are analyzed. It is shown that: 1) if the positive system has decoupling zeros then the corresponding standard system has also decoupling zeros, 2) the positive system may not have decoupling zeros when the ...

The Katz-Sarnak Density Conjecture states that zeros of families L-functions are well-modeled by eigenvalues random matrix ensembles. For suitably restricted test functions, this correspondence yields upper bounds for the families' order vanishing at central point. We generalize results on nth centered moment distribution to arbitrary functions. On computational side, we use our improved formul...

We introduce a new integrable hierarchy of nonlinear differential-difference equations which we call constrained Toda (C-Toda). It can be regarded as certain subhierarchy the 2D lattice obtained by imposing constraint $\bar {\cal L}={\cal L}^{\dag}$ on two Lax operators (in symmetric gauge). prove existence tau-function C-Toda and show that it is square root tau-function. In this some other res...

Random matrix theory has successfully modeled many systems in physics and mathematics, and often analysis in one area guides development in the other. Hughes and Rudnick computed 1-level density statistics for low-lying zeros of the family of primitive Dirichlet L-functions of fixed prime conductor Q, as Q→∞, and verified the unitary symmetry predicted by random matrix theory. We compute 1and 2...

the aim of this note is to characterize the finite groups in which all non-linear irreducible characters have distinct zero entries number.

A commonly accepted fact is that the diagonal structure of the controller poses fundamental limitations on the achievable performance, but few quantitative results are available for measuring these limitations. This paper provides a lower bound on the achievable quality of disturbance rejection using a decentralized controller for stable discrete time linear systems with time delays, which do n...

The partition function of the Potts model can be expressed in terms of famous combinatorial graph polynomials for restricted Hamiltonians. We investigate the Hamiltonian with variable edge interaction and external magnetic field, for which the partition function is the V-polynomial. Because the partition function is a polynomial, we can use motivic methods from algebraic geometry to characteriz...

The notions of poles and zeros at infinity and their relations are extended to the case of linear continuous time-varying systems. This study is based on the notion of a “newborn system” which is, in a mathematical point of view, a graded module extension over the noncommutative ring of differential operators. It is proved to be a relevant generalization to the time-varying case of the equivale...